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Modeling of Asphalt Concrete
About the Editor Y. Richard Kim, Ph.D., P.E., is a professor in the Department of Civil, Construction, and Environmental Engineering at North Carolina State University in Raleigh, North Carolina.
Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.
Modeling of Asphalt Concrete Y. Richard Kim, Ph.D., P.E.
Editor
Professor, North Carolina State University
New York Chicago San Francisco Lisbon London Madrid Mexico City Milan New Delhi San Juan Seoul Singapore Sydney Toronto
Copyright © 2009 by the American Society of Civil Engineers. All rights reserved. Manufactured in the United States of America. Except as permitted under the United States Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written permission of the publisher. 0071596518 The material in this eBook also appears in the print version of this title: 007146462X. All trademarks are trademarks of their respective owners. Rather than put a trademark symbol after every occurrence of a trademarked name, we use names in an editorial fashion only, and to the benefit of the trademark owner, with no intention of infringement of the trademark. Where such designations appear in this book, they have been printed with initial caps. McGrawHill eBooks are available at special quantity discounts to use as premiums and sales promotions, or for use in corporate training programs. For more information, please contact George Hoare, Special Sales, at george_hoare@ mcgrawhill.com or (212) 9044069. TERMS OF USE This is a copyrighted work and The McGrawHill Companies, Inc. (“McGrawHill”) and its licensors reserve all rights in and to the work. Use of this work is subject to these terms. Except as permitted under the Copyright Act of 1976 and the right to store and retrieve one copy of the work, you may not decompile, disassemble, reverse engineer, reproduce, modify, create derivative works based upon, transmit, distribute, disseminate, sell, publish or sublicense the work or any part of it without McGrawHill’s prior consent. You may use the work for your own noncommercial and personal use; any other use of the work is strictly prohibited. Your right to use the work may be terminated if you fail to comply with these terms. THE WORK IS PROVIDED “AS IS.” McGRAWHILL AND ITS LICENSORS MAKE NO GUARANTEES OR WARRANTIES AS TO THE ACCURACY, ADEQUACY OR COMPLETENESS OF OR RESULTS TO BE OBTAINED FROM USING THE WORK, INCLUDING ANY INFORMATION THAT CAN BE ACCESSED THROUGH THE WORK VIA HYPERLINK OR OTHERWISE, AND EXPRESSLY DISCLAIM ANY WARRANTY, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO IMPLIED WARRANTIES OF MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. McGrawHill and its licensors do not warrant or guarantee that the functions contained in the work will meet your requirements or that its operation will be uninterrupted or error free. Neither McGrawHill nor its licensors shall be liable to you or anyone else for any inaccuracy, error or omission, regardless of cause, in the work or for any damages resulting therefrom. McGrawHill has no responsibility for the content of any information accessed through the work. Under no circumstances shall McGrawHill and/or its licensors be liable for any indirect, incidental, special, punitive, consequential or similar damages that result from the use of or inability to use the work, even if any of them has been advised of the possibility of such damages. This limitation of liability shall apply to any claim or cause whatsoever whether such claim or cause arises in contract, tort or otherwise. DOI: 10.1036/007146462X
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To the glory of God For my better half, Jee Hye, and my children, Frances, Daniel, and Christie
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Contents Contributors 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
Modeling of Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Performance Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future of Asphalt Concrete Modeling .......................... Pavement Response Model versus Performance Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiscale Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Virtual Testing of Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . Organization Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 1—Asphalt Binder Rheology . . . . . . . . . . . . . . . . . . . . . . . . . Part 2—Stiffness Characterization . . . . . . . . . . . . . . . . . . . . . . . . . Part 3—Constitutive Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 4—Models for Rutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 5—Models for Fatigue Cracking and Moisture Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part 6—Models for LowTemperature Cracking . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 1 2 3 3 4 5 5 5 6 6 6 7 7 7 7
Part 1 Asphalt Rheology 2
Modeling of Asphalt Binder Rheology and Its Application to Modiﬁed Binders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modeling Critical Properties of Asphalt Binders . . . . . . . . . . . . . . . . . . Traditional Rheological Properties . . . . . . . . . . . . . . . . . . . . . . . . . Asphalt Susceptibility Parameters . . . . . . . . . . . . . . . . . . . . . . . . . Linear Viscoelastic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Viscoelastic Nature of Asphalt Binders . . . . . . . . . . . . . . . . . . . . . . Asphalt Viscoelastic Properties and Pavement Performance . . . . . . . Modeling of the Viscoelastic Properties of Asphalts . . . . . . . . . . . . . . . The Asphalt Viscoelastic and Failure Properties Selected in the SHRP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Need for Asphalt Modiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . Asphalt Modiﬁcation Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . Asphalt Modiﬁers Currently Used . . . . . . . . . . . . . . . . . . . . . . . . .
11 11 12 12 16 19 20 22 23 25 29 29 30
vii
viii
Contents Critical Properties of Modiﬁed Asphalts . . . . . . . . . . . . . . . . . . . . . . . . Effects of Modiﬁcation on Viscoelastic Properties . . . . . . . . . . . . Effects of Modiﬁcation on Failure Properties . . . . . . . . . . . . . . . . Effects of Modiﬁcation on the Superpave Grading Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complexity of Modiﬁed Binders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Assumptions in the Superpave Binder System . . . . . . . . . . . New Classiﬁcation of Asphalt Binders . . . . . . . . . . . . . . . . . . . . . Damage Resistance Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of New Tests for Binder Damage Behaviors . . . . . . . . . Rutting Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binder Fatigue Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection of New Damage Behavior Parameters . . . . . . . . . . . . . . . . . . Binder Rutting Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Binder Fatigue Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of Findings for the Damage Resistance Parameters . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31 31 34 36 38 38 41 42 45 45 48 50 50 52 55 55 56 56
Part 2 Stiffness Characterization 3
Comprehensive Overview of the Stiffness Characterization of Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asphalt Concrete Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods of Measuring Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Location of Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry of the Test Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . Loading Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate, Temperature, and Age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Importance of Asphalt Concrete Stiffness . . . . . . . . . . . . . . . . . . . . . . . . Use of Stiffness in Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tests to Determine Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factors on Which Asphalt Concrete Stiffness Depends . . . . . . . . . . . . Temperature and Rate of Loading . . . . . . . . . . . . . . . . . . . . . . . . . Stress State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aggregate Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asphalt Binder Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air Voids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65 65 65 66 66 66 66 66 67 67 67 68 68 68 69 70 70 70 74 74 77 77 77
Contents Characterization of Asphalt Concrete Stiffness . . . . . . . . . . . . . . . . . . . Undamaged Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effects of Microcracks on Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78 78 80 87 87
4
Complex Modulus Characterization of Asphalt Concrete . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test Protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressive Axial versus SSTShear Modulus . . . . . . . . . . . . . . Stiffness as the Asphalt Mix Performance Indicator . . . . . . . . . . Analysis of Cyclic Sinusoidal Test Data . . . . . . . . . . . . . . . . . . . . Mastercurve Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Endnote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89 89 90 90 92 96 98 101 111 116 116 117 117
5
Complex Modulus from the Indirect Tension Test . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Viscoelastic Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic Modulus Testing of HMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specimen Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testing and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Dynamic Moduli Values . . . . . . . . . . . . . . . . . . . . . . . . . Graphical Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Statistical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of Phase Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
121 121 121 122 122 126 126 128 128 131 131 131 134 135 135 137 137
6
Interrelationships among Asphalt Concrete Stiffnesses . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of LVE Response Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creep Compliance and Relaxation Modulus . . . . . . . . . . . . . . . . Complex Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of LVE Response Functions . . . . . . . . . . . . . . . . . . . . . . Analytical Representation of LVE Response Functions . . . . . . . . . . . . PowerLaw Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pure Power Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139 139 139 140 140 141 142 142 143 143
ix
x
Contents Generalized Power Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modiﬁed Power Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prony Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting of Experimental Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Presmoothing Experimental Data Prior to Prony Series Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interconversion between LVE Response Functions . . . . . . . . . . . . . . . . Approximate Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . Approximate Analytical Methods . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
143 144 144 145 145 146 148 149 155 157
Part 3 Constitutive Models 7
VEPCD Modeling of Asphalt Concrete with Growing Damage . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analytical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TTS with Growing Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The VECD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . StrainHardening Viscoplastic Model . . . . . . . . . . . . . . . . . . . . . . VEPCD Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Calibration of the VEPCD Model in Tension . . . . . . . . . . . . . . . . Validation of the VEPCD Model in Tension . . . . . . . . . . . . . . . . . . . . . . TSRST Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VEPCD Model in Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Veriﬁcation of the VEPCD Model in Compression . . . . . . . . . . . . . . . . Finite Element Implementation of the VEPCD Model . . . . . . . . . . . . . Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
163 163 163 164 165 167 179 180 180 186 187 191 194 198 200 200 200
8
Uniﬁed Disturbed State Constitutive Modeling of Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approaches for Pavement Analysis and Design . . . . . . . . . . . . . Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Review of RM Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniﬁed Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factors in Mechanistic Uniﬁed Model . . . . . . . . . . . . . . . . . . . . . . Disturbed State Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disturbed State Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
205 205 206 206 206 209 209 210 212 213 213 213 214
Contents Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Capabilities and Hierarchical Options . . . . . . . . . . . . . . . . . . . . . . . . . . Plasticity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disturbance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Creep Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rate Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fracture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Healing or Stiffening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interfaces and Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation for Laboratory Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation for Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . Computer Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validations and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reﬂection Cracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniﬁed Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
DBN Law for the ThermoViscoElastoPlastic Behavior of Asphalt Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solicitations in Bituminous Roadways . . . . . . . . . . . . . . . . . . . . . . . . . . General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Trafﬁc Inﬂuence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Inﬂuence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Behavior of Bituminous Materials in Pavement Structures . . . . Presentation of the DBN Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Description of the EPi Body Behavior . . . . . . . . . . . . . . . . . . . . . . Description of the Vi Body Behavior . . . . . . . . . . . . . . . . . . . . . . . Brittle Behavior of the Spring of Rigidity, E0 . . . . . . . . . . . . . . . . . How the Same Formalism Can Be Adapted to Describe the Different Typical Kinds of Mix Behavior . . . . . . . . . . . . . . . . . . . . Current Developments for the DBN Law . . . . . . . . . . . . . . . . . . . . . . . . Small Strains and Small Number of Cycles: Linear Viscoelastic (LVE) Behavior . . . . . . . . . . . . . . . . . . . . . . . Large Strains and Small Number of Cycles: Nonlinearity and Viscoplastic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large Strains at Low Temperatures: Brittle Failure . . . . . . . . . . .
215 215 218 220 220 221 222 222 222 223 223 224 225 225 226 227 229 229 231 236 238 241 241 241 245 245 245 246 246 246 247 248 249 250 251 251 252 252 252 254 257
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Contents Developments in Progress for the DBN Law . . . . . . . . . . . . . . . . . . . . . Small Strains and Great Number of Cycles: Fatigue Damage Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclic Loading from Stress Control Tests: Accumulation of Permanent Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thixotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Healing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ThreeDimensional Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
258 258 259 260 261 261 262 263
Part 4 Models for Rutting 10
11
Rutting Characterization of Asphalt Concrete Using Simple Shear Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanics of Permanent Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . Volume Change versus Shape Distortion . . . . . . . . . . . . . . . . . . . Representative Volume Element and Laboratory Test Specimen Size Considerations . . . . . . . . . . . . . . . . . . . . . . . Nonlinear Response Characteristics of Asphalt Aggregate Mixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Laboratory Simple Shear Test to Characterize Permanent Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test Equipment and Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . Test Variability and Reliability Considerations . . . . . . . . . . . . . . Test Specimen Size and Preparation Considerations . . . . . . . . . . Mix Design and Analysis, Performance Evaluation . . . . . . . . . . . . . . . Mix Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Warrantied Pavement, I5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Long Life Pavement Rehabilitation, I710 . . . . . . . . . . . . . . . . . . . Mix Evaluations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of Mix Design Criteria for Taxiway Subjected to Heavy Aircraft Loading . . . . . . . . . . . . . . . . . . . . . Creep versus Repeated Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Recursive Rut Depth Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . Constitutive Relationship Formulation for HMA Behavior at Elevated Temperatures . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permanent Deformation Assessment for Asphalt Concrete Pavement and Mixture Design . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
269 269 269 270 271 275 279 280 280 285 286 288 288 292 293 295 296 299 301 308 311 313 317 317 317
Contents MechanisticEmpirical Rutting Models . . . . . . . . . . . . . . . . . . . . . . . . . . Subgrade Rutting Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permanent Strain Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Permanent to Resilient Strain Ratio Models . . . . . . . . . . . . . . . . . Regression Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advanced Constitutive Models for Rutting . . . . . . . . . . . . . . . . . . . . . . Linear Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Viscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuum Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Performance Test for Mixture Rutting . . . . . . . . . . . . . . . . . . . . Static Creep Permanent Deformation Test . . . . . . . . . . . . . . . . . . Repeated Load Permanent Deformation Test . . . . . . . . . . . . . . . . Testing Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mix Design Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
318 318 319 322 325 326 326 327 328 332 336 338 339 340 342 342 344 346 347 347
Part 5 Models for Fatigue Cracking and Moisture Damage 12
Micromechanics Modeling of Performance of Asphalt Concrete Based on Surface Energy . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schapery’s Fundamental Law of Viscoelastic Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adhesion and Cohesion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Micromechanics Modeling of Fatigue and Healing of Asphalt Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture Damage Prediction Model . . . . . . . . . . . . . . . . . . . . . . . Measurement of Surface Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface Energy of Asphalt Binder . . . . . . . . . . . . . . . . . . . . . . . . . Surface Energy of Aggregate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validation of the Fatigue and Healing Models . . . . . . . . . . . . . . . . . . . Cohesive and Adhesive Bond Energies . . . . . . . . . . . . . . . . . . . . . Cohesive Fatigue and Healing . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adhesive Fatigue and Healing . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moisture Damage Modeling and Its Mechanistic Validation . . . . . . . Stripping Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accelerated Moisture Damage Testing on AsphaltAggregate Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . .
355 355 355 357 357 357 358 359 363 366 366 372 377 377 378 380 382 384 385
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Contents
13
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
387 388 389
Field Evaluation of Moisture Damage in Asphalt Concrete . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanisms of Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mechanical Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chemical Reaction and Molecular Orientation Theory . . . . . . . Surface Energy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Typical Failures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evaluation of Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Field Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Visual Inspection of Split Cores . . . . . . . . . . . . . . . . . . . . . . . . . . . Image Analysis of Split Cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strength Measurements of Cores . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
391 391 391 392 392 392 393 393 394 394 395 395 395 396 401
Part 6 Models for LowTemperature Cracking 14
Prediction of Thermal Cracking with TCMODEL . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Cracking Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . Previous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pavement Response Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Governing Constitutive Equation for Stress Prediction and Relaxation Modulus . . . . . . . . . . . . . . . . . . . . . . Viscoelastic Interconversion: Creep Compliance to Relaxation Modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Methods for Applying Hereditary Integral: Hourly Stress Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pavement Distress Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stress Intensity Factor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crack Growth Prediction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . Probabilistic Crack Amount Model . . . . . . . . . . . . . . . . . . . . . . . . Model Calibration and Sample Output . . . . . . . . . . . . . . . . . . . . . . . . . . Calibration Methods and Results . . . . . . . . . . . . . . . . . . . . . . . . . . Model Changes Implemented under NCHRP 137A . . . . . . . . . . . . . . Use of Three Input Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modiﬁed Calibration Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . Sample Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
405 405 405 406 407 408 408 411 411 413 413 414 414 415 416 419 419 419 419 421 422
Contents Summary and Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
424 425
LowTemperature Fracture in Asphalt Binders, Mastics, and Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thermal Stress Buildup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Polymers and the Thermal Stress Restrained Specimen Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interface Studies in Asphalt Systems . . . . . . . . . . . . . . . . . . . . . . . LowTemperature Failure in Asphalt Binders and Mastics . . . . . . . . . Toughness, Brittleness, and Ductility . . . . . . . . . . . . . . . . . . . . . . . BrittletoDuctile Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toughening Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enhanced Yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toughness in the Brittle State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LowTemperature Fracture in Asphalt Mixtures . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
430 431 431 432 433 435 436 443 446 448 449 449
Index
453
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429 429 429 430
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Contributors Professor, Department of Civil and Environmental Engineering, University of Wisconsin, Madison, Wisconsin (Chap. 2)
Hussain U. Bahia
Amit Bhasin Associate Research Scientist, Texas Transportation Institute, Texas A&M University, College Station, Texas (Chap. 12)
William G. Buttlar Associate Professor, Department of Civil and Environmental Engineering, University of Illinois at UrbanaChampaign, Urbana, Illinois (Chap. 14) Assistant Professor, Department of Civil and Environmental Engineering, Pennsylvania State University, University Park, Pennsylvania (Chaps. 6, 7)
Ghassan R. Chehab
Assistant Professor, Department of Civil Engineering, University of New Hampshire, Durham, New Hampshire (Chap. 7)
Jo S. Daniel
Regents’ Professor, Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, Arizona (Chap. 8)
Chandrakant S. Desai
Hervé Di Benedetto Professor, Département Génie Civil et Bâtiment, URA CNRS 1652, Ecole Nationale des TPE, VaulxenVelin, France (Chap. 9) John T. Harvey Professor, Department of Civil and Environmental Engineering, University of California, Davis, California (Chap. 10) Associate Professor, Department of Chemistry, Queens University, Kingston, Ontario, Canada (Chap. 15)
Simon A. M. Hesp
Associate Professor, Department of Civil and Coastal Engineering, University of Florida, Gainesville, Florida (Chap. 14)
Dennis R. Hiltunen
Kamil E. Kaloush Assistant Professor, Department of Civil and Environmental Engineering, Arizona State University, Tempe, Arizona (Chap. 11) Y. Richard Kim Professor, Transportation Systems and Materials, Department of Civil, Construction, and Environmental Engineering, North Carolina State University, Raleigh, North Carolina (Chaps. 1, 5, 6, 7) Department of Civil & Environmental Engineering, Sejong University, Kwangjin Ku, Seoul, Republic of Korea (Chap. 7)
H. J. Lee
Dallas N. Little E. B. Snead Chair Professor, Department of Civil Engineering, Texas A&M University, College Station, Texas (Chap. 12)
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Contributors Robert L. Lytton Benson Chair Professor, Department of Civil Engineering, Texas A&M University, College Station, Texas (Chaps. 3, 12)
G. W. Maupin, Jr. Principal Research Scientist, Virginia Transportation Research Council, Charlottesville, Virginia (Chap. 13)
Mostafa Momen North Carolina Department of Transportation, North Carolina State University, Raleigh, North Carolina (Chap. 5)
Carl L. Monismith Robert Horonjeff Professor of Civil Engineering, Institute for Transportation Studies, University of California, Berkeley, California (Chap. 10)
François Olard Research and Development Direction, EIFFAGE Travaux Publics, Corbas, France (Chap. 9) Terhi K. Pellinen Professor, Highway Engineering, Department of Civil and Environmental Engineering, Helsinki University of Technology, Espoo, Finland (Chap. 4)
Reynaldo Roque Professor, Department of Civil and Coastal Engineering, University of Florida, Gainesville, Florida (Chap. 14)
Charles W. Schwartz Associate Professor, Department of Civil and Environmental Engineering, University of Maryland, College Park, Maryland (Chap. 11) Senior Researcher, Highway & Transportation Technology Institute, Hwaseong, Gyeonggi, South Korea (Chap. 5)
Youngguk Seo
Shane Underwood Research Scientist, Department of Civil, Construction, and Environmental Engineering , North Carolina State University, Raleigh, North Carolina (Chap. 7) Shmuel L. Weissman
President and CEO, Symplectic Engineering Corporation, Berkeley,
California (Chap. 10)
T. Y. Yun Department of Civil, Construction, and Environmental Engineering, North Carolina State University, Raleigh, North Carolina (Chap. 7)
Modeling of Asphalt Concrete
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CHAPTER
1
Modeling of Asphalt Concrete Y. Richard Kim
Introduction Asphalt concrete pavement, one of the largest infrastructure components in the United States, is a complex system that involves multiple layers of different materials, various combinations of irregular traffic loading, and varying environmental conditions. Therefore, a realistic prediction of the longterm service life of asphalt pavements is one of the most challenging tasks for pavement engineers. The performance of asphalt concrete pavements is closely related to the performance of asphalt concrete. It is performance models of asphalt concrete that provide the links among various processes involved in asphalt mixture design, pavement design, construction, and rehabilitation. Various factors affect the deformation behavior and performance of asphalt concrete, including time (i.e., rate of loading, loading time, rest period), temperature, stress state, mode of loading, aging, and moisture. Models have been developed to capture the effects of these factors on asphalt concrete performance. Most of these models, developed prior to the Strategic Highway Research Program (SHRP), are empirical in nature. The primary reason for the empirical nature of these models is the lack of computing power necessary to calculate the longterm performance of asphalt concrete and, therefore, asphalt pavements. The SHRP recognized the importance of mechanistic models for material specifications, mixture design, and pavement design and developed a range of research products based on the principles of mechanics. The paradigm shift from empiricism to mechanics during the SHRP made a significant impact on the role of models in asphalt pavement engineering. Development of a fundamentally sound performance model serves two important purposes. For pavement engineers, such a model can provide accurate information about the performance of asphalt concrete under realistic loading conditions, thus leading to a better assessment of the service life of a new pavement or the remaining life of an existing pavement. For materials engineers, the performance model founded on
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Chapter One basic principles of mechanics provides relationships between material properties (chemical or mechanical) and model parameters, which can be used for the selection or design of better performing binders or mixtures.
Performance Characteristics Performance of asphalt concrete can be categorized into two major types of distress: cracking and permanent deformation. Cracking of asphalt concrete can be caused by mechanical loading from repetitive traffic and/or thermal loading from changes in temperature. When the asphalt concrete is subjected to repeated loading, whether it is mechanical or thermal, distributed microstructural damage occurs primarily in the form of microcracks. A microscopic video image of the cracking area in asphalt concrete is presented in Fig. 11 to display the formation of micro and macrocracks under tensile stress. As shown in this figure, microcracks exist ahead of the macrocrack tip, forming a socalled damage zone. Propagation, coalescence, and rebonding of these microcracks in the damage zone affect the macrocrack growth and healing and, thus, the fatigue behavior of asphalt concrete. That is to say, the modeling of the fatigue behavior of asphalt concrete requires an evaluation of the effects of both micro and macrocracks and their interaction on the global behavior of the mixture. At high temperatures and/or slow loading rates, the asphalt binder becomes too soft to carry the load and, thus, the principal type of damage is permanent deformation due to volume change (i.e., densification) and rearrangement of aggregate particles caused by shear flow. The degree of aggregate interlocking and anisotropy in asphalt concrete caused by aggregate orientation under compaction become important factors in the accurate prediction of the permanent deformation behavior of asphalt concrete.
FIGURE 11 Microscopic surface image of cracking area in asphalt concrete. (Kim et al. 1997, with permission from International Society for Asphalt Pavements.)
Modeling of Asphalt Concrete
Future of Asphalt Concrete Modeling The modeling of asphalt concrete is an evolving subject. Continuing developments and improvements in computational power and test techniques will allow asphalt materials and pavement engineers to use more realistic, powerful models to predict the performance of asphalt materials and pavements. The following subsections attempt to shed some light on the possibilities for future models of asphalt concrete.
Pavement Response Model versus Performance Model A traditional approach to asphalt pavement performance prediction is divided into two steps: pavement response prediction and pavement performance prediction. In this approach, responses of an undamaged pavement (e.g., tensile strain at the bottom of the asphalt layer) are estimated from a structural model (e.g., the multilayered elastic theory) using initial, undamaged properties of the layer materials. Asphalt concrete performance models are developed using laboratory test results and relate the initial response of the asphalt concrete specimens to the life of those specimens. The responses estimated from the structural model are then input to the performance model to determine the life of the pavement. This approach is the stateofthepractice method that is adopted in most recent mechanisticempirical pavement design methods, including the MechanisticEmpirical Pavement Design Guide (MEPDG) developed under the NCHRP project 137A (2004). These models are simple to use because the only measured response of the mixture is at the initial stage of fatigue testing. Such models deserve credit for the basic foundation of current mechanisticempirical pavement designs. However, there are several weaknesses in this traditional approach. First, damage evolution in complex structures and modified materials may not be captured accurately. For example, complex combinations of layer material types and thicknesses in perpetual pavements make it more difficult to accurately predict the failure mechanisms using conventional hot mix asphalt (HMA) performance prediction models and pavement response models. Secondly, most performance models used in the twostep approach are modeofloading dependent. These models are developed using results obtained from laboratory tests, which are conducted either in controlled stress mode or in controlled strain mode. Currently available twostep approaches do not have the ability to discern the mode of loading in a mechanistic manner and, therefore, could result in an unreliable performance prediction. Thirdly, the laboratory test methods used in the traditional twostep approach are designed to simulate the boundary conditions of pavement structures rather than to define the material’s constitutive behavior in the representative volume element (RVE). Often these laboratory test methods predict the performance under only some selected pavement conditions. So, because the test methods simulate the pavement boundary conditions rather than capture the behavior of the RVE, the number of tests needed to cover the wide range of pavement conditions that are expected in the field is undesirably large. The weaknesses of the twostep approach can be overcome using a mechanistic approach that combines HMA material models and the pavement response model. In this approach, the material model describes the stressstrain behavior of the material in the RVE. The material model is then implemented into the pavement response model where boundary conditions of the pavement structure in question are applied. This
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Chapter One approach allows the accurate evaluation of the effects of changes in layer stiffnesses due to damage growth on pavement performance. Prediction of multiple performance characteristics and their interactions is possible in a realistic manner, although the material models in both tension and compression are needed. The lack of computing power needed to calculate damage evolution for the entire life of the pavement forced earlier researchers to develop the twostep approach to pavement performance prediction, as opposed to the more realistic onestep integrated approach. However, improvements in computing power and numerical techniques now allow modelers to implement more powerful material models into the pavement response model and to predict the pavement performance directly from the integrated model.
Multiscale Model Two general approaches in mechanics can be used for modeling the changes in the stressstrain behavior of asphalt concrete: a micromechanical approach and a continuum approach. In the micromechanical approach, defects that constitute the damage are described by microscopic geometrical parameters, such as microcrack size, orientation, and density. These parameters are evaluated through an appropriate microstructural evolution law, such as the microcrack growth law. Mechanics is then applied typically on an idealized RVE to determine the effects of the distribution of microdefects on the macroscopic constitutive parameters, such as the effective stiffness of the damaged body. Such analyses are, in general, difficult to perform because of the intrinsic complexity of the microstructure and the micromechanisms and also due to the interactions among the defects. Therefore, without proper simplifications and assumptions both in modeling and analysis, the micromechanical approach may fail to provide realistic information about the macroscopic constitutive framework for modeling the progressive degradation of the mechanical properties of solids (Park et al. 1996). On the other hand, in the continuum approach, or socalled continuum damage mechanics, the damaged body is represented as a homogeneous continuum on a scale that is much larger than the flaw sizes. The state of damage is quantified by internal state variables (ISVs) within the context of the thermodynamics of irreversible processes. That is, the growth of damage is governed by an appropriate damage evolution law. The choice and interpretation of the ISVs are somewhat arbitrary, and the functional form of the thermodynamic potential (typically Helmholtz or Gibbs free energy) and the resulting stressstrain relations are postulated usually on a phenomenological basis. The stiffness of the material, which varies with the extent of damage, is determined as a function of the ISVs by fitting the theoretical model to the available experimental data. The phenomenological continuum damage models thus provide a viable constitutive framework for the efficient modeling of macroscopic mechanical behavior of materials with distributed damage without requiring explicit descriptions of microstructural evolution kinetics (Park et al. 1996). Recently, significant advancements in the modeling of asphalt concrete have been made in both micromechanics and continuum damage mechanics. In future models of asphalt concrete, micromechanical and continuum damage models will be coupled to describe the behavior and performance of asphalt pavements using the properties of their component materials (i.e., binder and aggregate). This multiscale model will take advantage of the strengths of both micromechanics and continuum damage mechanics, that is, the ability of the micromechanical model to describe mixture behavior using component material properties and that of the continuum damage model to describe
Modeling of Asphalt Concrete the global stressstrain behavior of asphalt concrete in predicting the pavement performance. The challenge in this combined approach is to determine the material properties at the proper scales, as it is expected that some material properties are scale dependent.
Virtual Testing of Asphalt Concrete One of the fastest growing techniques that can aid asphalt concrete modeling is the imaging technique, including digital imaging, laser, and xray tomography, to name a few. These techniques allow engineers to view and construct two and threedimensional microstructures of the mixture. The imaging techniques can be combined with advanced models of asphalt concrete and provide the tools to perform virtual testing of asphalt concrete. In this approach, virtual microstructures of asphalt concrete are generated from the imaging technique, and virtual testing is conducted on the virtual microstructure using the advanced numerical models. These virtual testing techniques will help asphalt material and pavement engineers to evaluate the effects of any change in the component material properties on the mixture behavior and performance without any laboratory testing. The virtual testing will also be an efficient tool in undergraduate and graduate asphalt materials courses to demonstrate the effects of changing testing conditions and mixture design parameters on the behavior and performance of asphalt concrete.
Organization Summary Part 1 (Chap. 2) is dedicated to asphalt binder modeling. Various aspects of the stiffness characterization of asphalt concrete are described in Part 2 (Chaps. 3 through 6). Part 3 (Chaps. 7 to 9) presents different constitutive modeling approaches for asphalt concrete. Part 4 (Chaps. 10 and 11) examines models for rutting. Part 5 (Chaps. 12 and 13) addresses models for fatigue cracking and moisture damage. Last, Part 6 (Chaps. 14 and 15) addresses models for lowtemperature cracking. Note that this book does not necessarily describe all the models that are currently available within each aspect of asphalt concrete modeling (as outlined in the six parts, accordingly). However, it should provide sufficient information about a wide range of models that are available. The following descriptions provide a summary of the contents of each part.
Part 1—Asphalt Binder Rheology Part 1 of this book deals with issues pertaining to asphalt binder rheology. The historical use of rheological indices in the asphalt industry is discussed to provide perspectives on the development and rationale for the findings from the SHRP project. The influence of binder properties on mixture performance is discussed. Background on polymer modification of asphalt binders is presented and, subsequently, an argument for the enhancement of asphalt binder performance by adding polymermodifying agents is given. A rheological modeling approach capable of capturing the beneficial aspects of polymermodified binders is presented along with results from an accompanying experimental study. Since the focus of this book is the modeling of asphalt concrete, only one chapter is allotted to a discussion of asphalt binders.
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Chapter One
Part 2—Stiffness Characterization Part 2 of this book focuses on asphalt concrete stiffness. Stiffness is critically important for mechanistic modeling of both the pavement response and the pavement performance. Chapter 3 discusses explicitly the importance of this factor for such analysis and also details the major factors affecting the material stiffness. Particular attention is paid in Chaps. 4 and 5 to the stiffness characterization of asphalt concrete via the complex modulus. Two different test methods are demonstrated. The first is part of the proposed simple performance test protocol and involves testing cylindrical asphalt concrete specimens in the axial direction. The second method strives to overcome shortcomings associated with using the geometry of the first method to evaluate the stiffness of field cores using the indirect tension test. There are numerous advantages to assessing material stiffness via the dynamic modulus in the frequency domain; however, many of the mechanistic models presented in this book require stiffnesses in the time domain. Linear viscoelastic theory and mathematical manipulation are used in Chap. 6 to demonstrate different methods of converting the dynamic modulus into time domain functions such as creep compliance and the relaxation modulus.
Part 3—Constitutive Models Part 3 of this book focuses on the constitutive modeling of asphalt concrete. Three approaches are presented in detail in this part. These approaches utilize different principles to describe the deformation behavior and performance of asphalt concrete, but are similar in that they attempt to form a unified model encompassing different performance characteristics by accounting for various constitutive factors. Chapter 7 in this part incorporates the theory of viscoelasticity, continuum damage mechanics, and the theory of viscoplasticity to arrive at a socalled viscoelastoplastic continuum damage (VEPCD) model as a constitutive relationship for the behavior of asphalt concrete. Implementation of the VEPCD model into the finite element program is discussed. Chapter 8 presents a constitutive model based on the hierarchical disturbed state concept (DSC). The chapter describes the capabilities of the DSC for various pavement distresses such as permanent deformation and different types of cracking. Analysis of both two and threedimensional pavement problems is given using the DSC model, and a unified methodology with DSC for design, maintenance, and rehabilitation of pavement structures is proposed. Chapter 9 uses the DBN (Di Benedetto and Neifar) law to describe the behavior of asphalt concrete under a broad range of conditions. It explains how the different types of behavior can be modeled using the same formulation.
Part 4—Models for Rutting In this part, the mechanisms of permanent deformation are described and modeled in two chapters. Information documented in Chap. 10 is the result of the SHRP A003 study and illustrates that shear deformation contributes a significantly greater portion of total permanent deformation (rutting) in asphalt concrete than volume change. Based on these findings, the shear test was proposed to measure the propensity of a mix for rutting. The issue of sample size is discussed in the light of RVEs. The data presented illustrate the efficacy of the simple shear test, performed in the repeated load, constant height mode, for mix design and performance evaluation. Chapter 11 summarizes the findings from the more recent NCHRP 919 project. This chapter is composed of three main sections: (a) a review of mechanisticempirical modeling approaches, and in particular the permanenttoresilient strain ratio model adopted for the NCHRP 137A MEPDG; (b) the VEPCD
Modeling of Asphalt Concrete model for the compression behavior of asphalt concrete; and (c) a simple performance test to identify the rutting potential of mixtures during the design process, based on the measurement of fundamental engineering responses and properties. It is noted that the VEPCD model adopted in this chapter employs the same principles as found in Chap. 7 in Part 3, except that the HiSSPerzyna model is used to describe the viscoplastic strain of asphalt concrete instead of the strainhardening model used in Chap. 7.
Part 5—Models for Fatigue Cracking and Moisture Damage The detrimental effects of moisture and fatigue damage are discussed in Part 5 of this book. Chapter 12 focuses primarily on the fatigue damage mechanisms with a particular interest in the acceleration of such damage growth with additional moisture damage. Surface energy principles, fracture mechanics, and continuum damage mechanics are utilized for this argument. In Chap. 13 more attention is given to the moisture damage phenomenon. A review of current procedures for moisture damage assessment is given as a precursor to more advanced, objective techniques for the assessment of moisture damage.
Part 6—Models for LowTemperature Cracking Part 6 of this book discusses thermal cracking of asphalt concrete pavements. Mechanisms and events leading to thermal cracking are discussed in detail in Chaps. 14 and 15. Chapter 14 presents the TCMODEL, which has been implemented into the NCHRP 137A MEPDG to predict thermal cracking performance. The second chapter, Chap. 15, casts the phenomenon in the light of fracture mechanics and presents experimental results of multiscale modeling efforts encompassing binder, mastic, and mixture modeling.
Concluding Remarks This book attempts to document models of asphalt concrete and should be regarded as an evolving document, as some of the models are still being refined and improved. It also may be noted that this book focuses mostly on continuum models in order to maintain a reasonable length. Significant advancements in micromechanical modeling of asphalt concrete have also been made. Nonetheless, this book should provide a fair presentation and sufficient review of mechanistic models that are currently available at the time of publication.
References Kim, Y. R., H. J. Lee, Y. Kim, and D. N. Little, Mechanistic Evaluation of Fatigue Damage Growth and Healing of Asphalt Concrete: Laboratory and Field Experiments, Proceedings of the Eighth International Conference on Asphalt Pavements, International Society for Asphalt Pavements, University of Washington, Seattle, Washington, 1997, pp. 1089–1107. NCHRP 137A Research Team, “Guide for MechanisticEmpirical Design of New and Rehabilitated Pavement Structures,” Final Report, NCHRP 137A, ARA, Inc. and ERES Consultants Division, 2004. Park, S. W., Y. R. Kim, and R. A. Schapery, “A Viscoelastic Continuum Damage Model and Its Application to Uniaxial Behavior of Asphalt Concrete,” Mechanics and Materials, Vol. 24, No. 4, December 1996, pp. 241–255.
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PART
Asphalt Rheology CHAPTER 2 Modeling of Asphalt Binder Rheology and Its Application to Modiﬁed Binders
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CHAPTER
2
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders Hussain U. Bahia
Introduction Rheology is the science of flow and deformation. It is a science for the study of materials whose deformation characteristics vary not only with load but also with time rate of load application. Asphalt binders are rheological materials; that is, their behavior depends on temperature and rate (or time) of loading. At any combination of time and temperature, the timedependent (i.e., viscoelastic) behavior of asphalt binders, within the linear range, is best characterized by two properties: the total resistance to deformation under load and the relative distribution of that deformation between elastic and viscous parts (Bahia and Anderson 1995). Although there are many methods of characterizing viscoelastic properties, cyclic (oscillatory) testing and creep testing are two of the best techniques to represent the uniqueness of the behavior of this class of materials. What is unique about asphalt binders is their high sensitivity to temperatures within the range of applications. The stiffness of asphalt can vary by as much as eight orders of magnitude, and their phase angle (relative distribution of response between elastic and viscous) by as much as 85°, between peak summer and peak winter conditions. It can also vary by similar amounts in response to standing traffic and highspeed traffic (Anderson et al. 1994). Similar to viscoelastic properties, failure properties and damage resistance properties of asphalt binders are also very sensitive to temperature and loading rate. Stress and strain at failure can change by an order of magnitude by a change of only 10°C (Dongre et al. 1995). Fatigue life of binders can change by orders of magnitude in response to a change of a few degrees of temperature or by changing loading frequency within the application range (fast versus slow traffic) (Pell and Cooper 1975; Bahia et al. 1999; Bonnetti et al. 2002). Conventional methods of asphalt refining have their limitations which prompted the introduction of modified binders. In 2005 it is estimated that 20 to 25 percent of the
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Chapter Two asphalts used in the United States are modified asphalts. Modification of asphalts is, therefore, an integral part of this field and understanding its variables is necessary. Modification of asphalt binders is usually performed to improve one or more of the basic asphalt properties that are related to one or more of the pavement distress modes (Bahia 1995; Terrel and Epps 1989). The basic properties that have been targeted include Rigidity: Total resistance to deformation which can be measured by complex moduli like G∗ under dynamic loading or by creep stiffness, S(t), under quasistatic loading. Higher rigidity is favorable at high temperatures or low loading rates to resist rutting while lower rigidity is favorable at intermediate and low temperatures to resist fatigue and thermal cracking, respectively. Elasticity: Recovery of deformation using stored energy applied. It can be measured either by the phase angle (d) or by the logarithmic creep rate (m). To resist rutting and fatigue damage more elasticity is favorable. To resist thermal cracking, less elasticity and more ability of relaxing stress by flow is favorable. Brittleness: Failure at low strains is the best definition of brittleness. To improve resistance for fatigue and thermal cracking, brittleness should be reduced by enhancing strain tolerance or ductility. Storage stability and durability: Oxidative aging, physical hardening, and volatilization are key durability properties. Resistance to all of these changes is favorable. Resistance to accumulated damage: Rutting and fatigue damage are recognized as the two most important load induced types of distress. They represent progressive failure mode which is not necessarily measured using small stress or small strain testing.
Modeling Critical Properties of Asphalt Binders Critical properties of asphalt binders can be classified into two main groups: mechanical properties and durability properties. Testing of mechanical properties can be further sorted, based on progress in testing technologies into three groups: traditional rheological, linear viscoelastic, and damage resistance characterization. Durability properties include aging during production and construction and for modified binders, compatibility or storage stability are important properties. The following sections describe these different properties.
Traditional Rheological Properties Traditional or index rheological properties include many standardized tests that were used mostly before the completion of the Strategic Highway Research Program (SHRP) in the early 1990s. Although in North America and some European countries a new set of rheological test methods are rapidly being introduced, the traditional tests continue to be used widely in other parts of the world. Table 21 includes a list of the main traditional tests (Isacsson and Lu 1995). The most common independent traditional measures used for asphalts include penetration, ductility, softening point, and viscosity. The problem with the first three measures, in addition to being one point measurements, is that the empirical nature complicates the derivation of any meaningful engineering property. The problem with
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders
Standard Protocol
Test
Purpose of Test and Applicability for Modified Asphalt Binders (MABs)
Index (Empirical) Tests Softening Point (R&B)
ASTM D36
An index of consistency at very high temperatures. Used widely around the world for compliance but not considered reliable for assessing the high temperature performance in paving applications. Test can be used to measure instability of modified asphalts.
Penetration @ 25°C (dmm)
ASTM D5 NF T 66004
Considered an index for consistency of binder at intermediate temperature. Although test is questionable and of little value, it is used in many MAB specifications in Europe, Australia, and Japan. The effect of modifiers can be an increase or a decrease in penetration.
Frass Breaking Point
IP 80
An index of resistance to cracking at low temperatures under repeated flexing and reducing temperature. Test is mostly used in Europe as an indicator of brittleness. Modifiers are known to result in decreasing Frass point, particularly for elastomers.
Viscosity (Resistance to Flow) Tests Rotational Viscosity @ 135– 165°C
ASTM D402
A measure of resistance to flow at production and construction temperatures of HMA. Used widely to measure workability of MABs.
Absolute Viscosity @ 60°C
ASTM D2170
Measured using the capillary tube under vacuum. Because many modified asphalts are nonNewtonian, it has been used with caution and is not used in specification widely. Modifiers are known to increase the value of this viscosity and also increase shear rate dependency.
Cone and Plate Apparent Viscosity @ 25–60°C
ASTM D3205
Developed to measure creep response under increasing creep loads to estimate apparent viscosity at intermediate to high temperature. Not used widely and standard was discontinued in 2000. Most polymer modifiers are expected to increase viscosity.
Tensile (Extensional) Properties Ductility @ 4°C and @ 25°C
TABLE 21
ASTM D113
An index of flexibility at low temperatures when used at 4°C. Also considered an index of compatibility particularly when used at 25°C. Effect of modifiers on ductility varies significantly depending on nature of modifier. Elastomeric modifier tends to increase ductility at these temperatures, plastomers have minimal or negative effect, and chemical or oxidation tends to decrease ductility particularly at low temperatures. Used in Europe at various temperatures. Also used in North America in some states.
List of Asphalt Traditional Tests
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Chapter Two
Standard Protocol
Test
Purpose of Test and Applicability for Modified Asphalt Binders (MABs)
Tensile (Extensional) Properties Forced Ductility @ 4°C
Not a standard
An index of tensile strength and energy required for complete failure. Specifically developed for polymer modified asphalts and used widely in North America. Response usually includes an initial and a secondary peak of the stress which is used to calculate a ratio showing effect of the modifier. Test taken from joint sealant testing field.
Elastic Recovery @ 25°C
D8084
An index of the capability of modified asphalt for elastic recovery. Measured using the conventional ductility set up but sample is stretched and then cut to measure recovery of cut ends. One of the most widely used to determine if modified binder includes elastomers. Used in North America, Australia and Europe. Method has been modified several times and is run using sliding plate rheometer, ARRB Elastometer, Consistometer, and torsional loading setup.
Toughness and Tenacity @ 25°C
Not a standard test
An index of energy to failure used to detect modifiers and assess their contribution to toughness. A hemispherical head is inserted in an asphalt container and then pulled out. The area under the load deformation curve is divided into an initial peak area and a terminal tenacity area. The sum is the toughness. Elastomeric modifiers could have a significant effect on tenacity and on toughness particularly if they are crosslinked.
TABLE 21
List of Asphalt Traditional Tests (Continued)
these measures has been recognized by several researchers, and many attempts have been made to correlate them with more fundamental rheological properties. For example, formulas were proposed to calculate the coefficient of viscosity from penetration (Saal and Labout 1958; Van der Poel 1954; Heukelom 1973; Davies 1981). Ductility was considered by some researchers as an indicator of internal structure (Halstead and Zenewitz 1961; Barth 1962), while others showed that it may be correlated with certain shear susceptibility parameters (Traxler 1961; Traxler et al. 1944; Kandhal and Wenger 1975). Other researchers tried to combine the two measures (penetration and ductility) and relate them directly to pavement performance (Halstead 1961; Serafin et al. 1967). The softening point was also correlated with more fundamental measures; Van der Poel (1954) indicated that it can be considered as equipenetration temperature, while Jongepier and Kuilman (1969) indicated that softening point is an equimodulus temperature rather than an equiviscous temperature. These correlation studies have relied on some general correlations that suffer in many instances from exceptions, low statistical significance, and limited sampling of asphalts. Viscosity, also a single point measure, is a fundamental material property expressed in absolute units. However, the coefficient of viscosity is only a fundamental absolute measure for Newtonian fluids. Newtonian fluids are those whose properties are
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders independent of rate of loading or stress level. Asphalts exhibit Newtonian behavior only at very high temperatures, (above softening point), or at very low shear rates, which are seldom seen by asphalts in their applications in road pavements. At low temperatures, or short loading times asphalts are not Newtonian and cannot be described by an absolute value of coefficient of viscosity. To solve this problem and make it more relevant to the domain of application conditions, the “apparent viscosity,” a measure dependent on shear rate, was introduced. Questions then arise regarding where, in the time, stress and temperature senses, that the measurement should be taken. The selection of appropriate ranges varied amongst researchers and became a matter of experimental convenience. Traxler and his coworkers (Traxler and Schweyer 1936; Romberg and Traxler 1947; Traxler 1947) selected a temperature level of 77°F and a constant power input (constant value of the product of stress times strain rate) was proposed. For these experimental studies a power input of 1000 ergs was used because aged and unaged asphalts could be measured at that value of power input, with the available viscometer without any extrapolation (Romberg and Traxler 1947). Several researchers used this approach in asphalt aging studies, good examples are given in Moavenzadeh and Stander (1967), Majidzadeh (1969), and Page and coworkers (1985). With the introduction of the Shell sliding plate viscometer (Griffin et al. 1955), apparent viscosity at 77°F and a constant shear rate of 0.05 s−1 was introduced for experimental convenience and suitability of the device for this measurement. A great number of research groups followed this approach; apparent viscosity at 77°F and 0.05s −1 became the most common measure for evaluating asphalt rheology of aged and unaged asphalts. Early as well as recent studies have used this measure (Heithus and Johnson 1958; Gallaway 1957; Kemp and Predoehl 1981; Button and Epps 1985). Despite the wide acceptance of the constant strain rate viscosity measure Mack (1965) indicated that stress level is equally important as the shear rate, and that apparent viscosity should not only be compared at a constant temperature and strain rate but also at a constant stress level. Chipperfield and Welch (1967) suggested that using a constant stress level is more accurate than constant strain rate. Based on an extensive field study, the authors indicated that a constant stress apparent viscosity, although not the ultimate choice, is a much better indicator of asphalt hardening due to aging than the constant strain rate apparent viscosity. This other approach also had its followers (Schmidt 1972). The apparent viscosity was one of the measures adopted into the American Society for Testing and Materials (ASTM ) standards. It is, however, subject to many questions. The common method of determining apparent viscosity is by incremental creep tests, where a series of loads are added in sequence and the strain is measured with time. At each load level the strain is monitored until it shows a constant rate which is selected to calculate viscosity from the corresponding stress application. Then, the next load is added and the procedure is repeated to calculate the viscosity at the new shear rate. With several measured viscosities at different strain rates, the viscosity is calculated by interpolation (ASTM D 3205). For a nonNewtonian material, such as asphalts at temperatures below 140°F, the strain rate is a strong function of loading time and becomes more so as the temperature is reduced. It may take several hours or even days of loading time so that the strain rate reaches a constant and an asphalt starts behaving like a truly viscous material. Moreover, the proximity to the steadystate viscosity is highly asphalt specific. The other fundamental problem with apparent viscosity is the possibility of reaching the nonlinear region because of the geometry of the specimen or the stress level being
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Chapter Two used. Keeping the load for long times so as to reach an approximately constant strain rate, may easily put the material in the material nonlinearity state (high stains), or in the geometrically nonlinear region which complicates the interpretation of the measurement. Increasing the load level to very high values, in order to reach a constant strain rate, may also result in stress nonlinearity. These fundamental problems with apparent viscosity are not new subjects; the same researchers who promoted the measure have warned continuously of these problems. Wood and Miller (1960) indicated that testing with the sliding plate microviscometer may require several hours before a constant strain rate may be reached. Labout and van Ort (1956), Griffin and coworkers (1955), Fink and coworkers (1961), and Evans and Griffin (1963), all suggested using certain geometries for the sliding plates and used different film thicknesses to limit the strain rates and to show that the measure may be significantly affected by the geometry or the total amount of deflections are exceeded. Others have also reported on the effect of stress level and more specifically the stress history on the apparent viscosity; Puzinauskas (1967, 1979), and Majidzadeh and Schweyer (1965) are among those who thoroughly evaluated the effect of stress history on apparent viscosity and reported sufficient evidence of the stress dependence. Shortfalls of the traditional tests have been recognized for some time, and various attempts were made to use these tests to estimate fundamental rheological properties. These attempts started more than 50 years ago and have demonstrated that it is best that asphalt cements be characterized as linear viscoelastic materials that are thermorheologically simple. However, the difficulty of conducting fundamental rheological testing has led many to simplify the rheology of asphalts, and depend on what is available in the traditional labs, despite the empiricism involved and the insufficient characterization that followed. Van der Poel introduced his widely used nomograph in 1954 and indicated that asphalts can be successfully characterized using simple rheological indices, derived from empirical measures (Van der Poel 1954). The concepts of using the socalled “temperature susceptibility parameters, and shear susceptibility parameters” was introduced to model behavior of asphalts.
Asphalt Susceptibility Parameters To advance modeling of asphalt, rheological derivatives of the traditional tests were introduced. The motivation was to derive a better relationship with performance that the traditional tests fail to predict. These estimated, pseudorheological properties, can be divided into two sets: the temperature susceptibility parameters and the shear susceptibility parameters.
The Temperature Susceptibility Parameters The change in consistency with temperature is the general definition of temperature susceptibility. Several types of such parameters have been proposed and used. They vary basically in two aspects: the type of consistency measurement used and the range of temperature covered. The early works used penetration as the consistency measure. The penetration at different temperatures was measured and the ratio of penetration, the difference in penetration, the temperature required to increase the penetration by a certain number, or the slope of log penetration versus temperature have been used to characterize temperature susceptibility (Pfieffer and van Doormaal 1936; Van der Poel 1954; Neppe 1952; Barth 1962). More fundamental approaches used viscosity as the consistency
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders measure and to define a temperature susceptibility parameter. Again several types of relations have been used: slope of log viscosity versus temperature (Traxler and Schweyer 1936), log viscosity versus reciprocal of temperature raised to a variable power (Traxler and Schweyer 1936; Cornelissen and Waterman 1956), and loglog viscosity versus log temperature (Fair and Volkmann 1943). The latter measure is probably the most widely accepted and is referred to as viscositytemperature susceptibility or VTS. A third type of approach used a combination of consistency measures. The penetration index (PI) for example combines penetration and softening point (Van der Poel 1954), and the penetration viscosity number (PVN) combines penetration and viscosity (McLeod 1972). The best indication of how well these different susceptibility measures represent the rheological properties of asphalts is probably found in durability studies that used these measures to define asphalt performance with age hardening. The conclusions about the effect of aging on temperature susceptibility are controversial: Pfieffer (1950) and Blokker and Hoorn (1959), using the penetration index, reported a decrease in temperature susceptibility upon oxidative aging. Halstead and Zenewitz (1961) using plots of loglog viscosity versus temperature, in the range of 60 to 95°F, indicated that some asphalts show a decrease in temperature susceptibility while others show an increase upon aging. McLeod (1972) used the penetration index (PI) and his penetration viscosity number (PVN) and observed that after 9 years of service asphalts showed either no change or a significant increase in temperature susceptibility. However, the PI did not show the same change in temperature susceptibility. Puzinauskas (1979) used the VTS and concluded that the majority of asphalts show an increase in temperature susceptibility. Anderson and his coworkers (1983) used three different parameters (PI, PVN, and VTS) and concluded that PI and VTS show a general increase in temperature susceptibility, while PVN does not show significant changes with aging. The authors, experimenting on a very large number of asphalts, indicated that these measures do not measure the same property, and no simple explanation can justify the difference in the effect of oxidative aging on the values of these parameters. Button and his coworkers (1983) confirmed Anderson’s findings. The authors found that, using the PVN (estimated from pen77 and vis140) as the temperature susceptibility measure, asphalts that were originally highly temperature susceptible became more susceptible with oven aging, while those that were less temperature susceptible became even less susceptible with aging. Others have also used apparent viscosity as the measure of consistency. Moavenzadeh and Stander (1967) measured apparent viscosity at a constant power input (1000 ergs) over a range of temperatures between 10 and 160°C. The authors indicated that aged asphalts are less temperature susceptible in the lower temperature range while more susceptible in the higher temperature range. In other words, the temperature susceptibility shows different changes at different ranges of temperature. Kandhal and his coworkers (Kandhal et al. 1973; Kandhal and Wenger 1975) confirmed Moavenzadeh and Stander findings. Here researchers measured apparent viscosity at 39, 77, 140, and 275°F and observed that temperature susceptibility measured in terms of loglog viscosity versus log temperature, decreased in the lower temperature range with age hardening while it increased in the higher temperature range. No doubt the above review is confusing. Two fundamental reasons can be given to explain the problems with temperature susceptibility parameters: 1. The temperature susceptibility, as shown in linear viscoelastic characterization studies, is both temperature range and time of loading dependent. The relations are not linear nor can be approximated by a linear relation such that a few
17
18
Chapter Two measurements, even if done properly, can be used to represent them. Therefore, any two measures should consider both time of loading and temperature range. In addition oxidative hardening may change the shape of the relation differently at different times of loading. This observation dictates then that unless the parameters of susceptibility are referring to the same narrow temperature range and times of loading they will naturally give different indications of the effect of aging. Except for the VTS, none of these requirements are satisfied in the used parameters. 2. The penetration, softening point, and apparent viscosity have been shown in the previous section to involve serious problems. Using them for temperature susceptibility measurements may only lead to more difficulties. The simple reasoning is that as the temperature decreases, or as the asphalt is aged, the time required to reach the specified shear rate will be different which means we are using measures at the same temperature but two different times of loading are being used. In summary, it is rather difficult to consider the temperature susceptibility parameters discussed here as reliable measures to characterize asphalts or to even measure the effect of oxidative hardening.
Shear Susceptibility Parameters Two types of parameters have been used to represent the shear susceptibility of asphalt cements: The degree of complex flow “C” and the shear index. Traxler and coworkers (1944) were the first to introduce the use of this parameter for paving asphalts. They indicated that measurements on different asphalts show that the slope of log shear stress versus log strain rate may be considered constant and, therefore, concluded that asphalt rheological properties may be approximated by a complex flow equation that is usually used for power low fluids:
Degree of Complex Flow
M=
T SC
(21)
where M = constant T = shear stress S = shear rate C = degree of complex flow For C = 1, the asphalt is a Newtonian fluid and M is the steadystate coefficient of viscosity. Therefore, C was considered a good measure of nonNewtonian behavior. A large number of researchers accepted the approximation and the change in the C value was used as an indicator of the asphalt rheology and an indicator of effect of oxidative aging on the properties of asphalts (Gallaway 1957; Moavenzadeh and Stander 1967; Jimenez and Gallaway 1961). The degree of complex flow C, in fact, looks very attractive for the study of asphalt rheology. However, the measure also has its own problems that have been shown in several studies: 1. It assumes the relation between shear stress and shear rate to be linear when C = 1. This is true only over a small range of stress or strain rate. At very low shear rates or at very small stress levels almost all asphalts will show Newtonian behavior (C = 1.0). This behavior, depending on the type of asphalt, will start
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders changing to nonNewtonian gradually as the shear rate or stress level increases (Puzinauskas 1967). Therefore, the relation is not linear and the value of C will depend on which range is referred to. 2. The parameter, being a measure of nonNewtonian behavior, will naturally depend on the stress history. Majidzadeh and Schweyer (1965) conducted an experiment where the C value was measured for the same asphalt under the same conditions, but the sequence of loading applications was changed. They compared Ci measured when the load increments where added in a decreasing sequence of load values with “Cd “ measured with an increasing sequence of load step values. 3. In addition to the above problems with the C measure another point may be added here. Constructing the stressshear rate relation is subjected to the same controversy associated with apparent viscosity. From a standard creep curve measured for a nonNewtonian material, the shear rate may never come to an equilibrium within reasonable time or strain level, therefore, the value of the shear rate used to construct the stressshear rate curve will depend on the time of loading at which this rate is determined. In other words the C value is time of loading dependent. Apparent viscosity is plotted versus shear rate on a loglog scale. The slope of the relation between two different shear rates is determined and considered as the shear susceptibility parameter. Although early works tried to approximate the relation as linear (Jimenez and Gallaway 1961; Gallaway 1959), a large number of more recent studies showed clearly that this is not true. Apparent viscosity is constant up to the Newtonian behavior limit after which the viscosity starts decreasing continuously with increasing shear rate. For asphalt aging studies, the shear index has been used by many researchers. Zube and Skog (1969), Culley (1969), and Kandhal et al. (1973) all compared viscosityshear rate plots of unaged with lab or field aged asphalts. However, since the plots are nonlinear and temperature dependent researchers selected different ranges of shear rates and different temperatures. All agreed that an increase in the shear susceptibility is observed upon aging. As with other susceptibility parameters, the shear index may be considered as an arbitrary value. The exact value depends, to a great extent, on the range of shear rate considered and the temperature at which it is measured. A single shear index may only indicate the type of behavior expected for an asphalt at the temperature and shear rate range it is measured at; no simple extrapolation or even interpolation can be made. Moreover, it still uses the apparent viscosity which is time of loading dependent and if this is not constant then the plot is very difficult to interpret. In summary, shear susceptibility parameters are not much better than temperature susceptibility parameters, or even the single measurements. They are also arbitrary in nature and do not show high promise for use in asphalt rheological characterization.
The Shear Index
Linear Viscoelastic Properties Although using rheological concepts to characterize asphalts dates back more than 50 years, the cost and availability of equipment hampered the spread of using viscoelasticity to qualify asphalts and study effect of modifiers. This shortcoming was overcome
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20
Chapter Two through the SHRP project where major emphasis was placed on the development of testing equipment and/or test protocols to standardize the use of asphalt tests that were relatively blind to the binder composition. The test protocols, although developed with the intention of being suitable to unmodified and modified binders, focused on linear viscoelasticity as a compromise between the difficulty of rheological testing and the need to produce practical and realistic test protocols for an industry that has historically used mainly simplistic index testing to qualify asphalts (Anderson et al. 1991). In 1992–1993 a new set of testing techniques and a new grading system was introduced. The testing and grading systems are based on fundamental properties that are related in a more rational way to pavement performance. The viscoelastic properties of asphalt binders are measured at application temperatures and grading criteria are based on sound understanding of pavement failure mechanisms.
The Viscoelastic Nature of Asphalt Binders At any combination of time and temperature, viscoelastic behavior, within the linear range, must be characterized by at least two properties: the total resistance to deformation and the relative distribution of that resistance between an elastic part and a viscous part. Although there are many methods of characterizing viscoelastic properties, dynamic (oscillatory) testing is one of the best techniques to represent the behavior of this class of materials. In the shear mode, the dynamic modulus (G∗, for simplicity, denoted as G∗ hereinafter) and phase angle (d) are measured. G∗ represents the total resistance to deformation under load, while d represents the relative distribution of this total response between an inphase component and an outofphase component. The inphase component is an elastic component and can be related to energy stored in a sample for every loading cycle, while the outofphase component represents the viscous component and can be related to energy lost per cycle in permanent flow. The relative distribution of these components is a function of the composition of the material, loading time, and temperature. Rheological properties can be represented either by the variation of G∗ as a function of frequency at a reference temperature (commonly referred to as a mastercurve) or by the variation of G∗ and d with temperature at a selected frequency or loading time, commonly called isochronal curve. Although time and temperature dependency can be related using a temperaturefrequency shift function (Ferry 1980), for practical purposes it is much easier to present data with respect to one of the variables. Figure 21 depicts typical rheological properties of an AC40 and an AC5 asphalt binder at a wide range of temperatures and frequencies. Figure 21a is a mastercurve at 25°C and Fig. 21b is an isochronal curve at 10 rad/s. Some common unique characteristics of the rheological behavior of asphalt can be seen in the typical plots of Fig. 21: • At low temperatures or high frequencies, both asphalts tend to approach a limiting G∗ value of approximately 1.0 GPa and a limiting d value of 0°. The 1.0 GPa reflects the rigidity of the carbon hydrogen bonds as the asphalts reach their minimum thermodynamic equilibrium volume. The 0° value d represents the completely elastic nature of the asphalts at these temperatures. • As the temperature increases or as the frequency decreases, G∗ decreases continuously while d increases continuously. The first reflects a decrease in resistance to deformation (softening) while the second reflects a decrease in elasticity or ability to store energy. The rate of change is, however, dependent
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders 11
100 AC40 (G∗) AC5 (G∗)
AC5 (Delta) 80
9 8 7
60
6 40
5 4 3
Phase Angle at 25°C
Complex Modulus at 25°C, log Pa
10
AC40 (Delta)
20
2 1 −5 −3
−1 1 3 5 7 9 11 13 Loading frequency, log rod/s
0 15
(a) Frequency master curves 11 AC40 (Delta) AC5 (Delta)
9
100
80
8 7
60
6 40
5
Phase Angle, °
Complex Modulus at 10 rad/s, log Pa
10
AC40 (G∗) AC5 (G∗)
4 3
20
2 1 −50
−30
−10
10 30 50 Temperature, °C
70
0 90
(b) Isochronal curves
FIGURE 21
Typical rheological spectra for two asphalt binders.
on the composition of the asphalt. Some will show a rapid decline with temperature and frequency; others will show a gradual change. Asphalts within this range may show significantly different combinations of G∗ and d. • At high temperatures, the d values approach 90° for all asphalts, which reflects the approach to complete viscous behavior or complete dissipation of energy in viscous flow. The G∗ values, however, vary significantly, reflecting the different consistency properties (viscosity) of the asphalts.
21
Chapter Two
Asphalt Viscoelastic Properties and Pavement Performance Figure 22 is an isochronal plot that depicts rheological properties of asphalt in its unaged condition and after aging in the field under a moderate climate for approximately 16 years. To relate asphalt properties to pavement performance, reference can be made to three temperature zones. At temperatures in the range of 45 to 85°C, typical of highest pavement inservice temperatures, the main distress mechanism is rutting (Chaps. 10 and 11), and, therefore, the G∗ and d need to be measured. A measure of viscosity alone cannot be sufficient, since viscosity measurements are done on the assumption that asphalt response has only a viscous component. For rutting resistance, a high G∗ value is favorable because it represents a higher total resistance to deformation. A lower d is favorable because it reflects a more elastic (recoverable) component of the total deformation. Within the intermediate temperature zone, asphalts are generally harder and more elastic than at higher temperatures. The prevailing failure mode at these temperatures is fatigue damage (Chaps. 12 and 13). For viscoelastic materials, like asphalt binders, both G∗ and d play a role in damage caused by fatigue. They are both important because during every cycle of loading the damage is dependent on how much strain or stress is developed by the cyclic load and how much of that deformation can be recovered or dissipated. Under straincontrolled conditions, a softer material and a more elastic material will be more favorable to resist fatigue damage because the stress developed for a given deformation is lower and the asphalt will be more capable of recovering to its preloading condition. Similar to rutting, a single measure of hardness or viscosity cannot be sufficient to select betterperforming asphalts with respect to fatigue resistance. Rutting and fatigue damage are both functions of frequency of loading, and, therefore, the rate Unaged (G∗) Field aged (G∗)
Unaged (delta) Field aged (delta)
10
100
9
90
8
80
7
70
6
60
5
50
4
40
3
30
2
20
1
10
0 −1 −50
Thermal cracking
−30
−10
Fatigue cracking
30 50 10 Temperature, °C
Rutting
70
Delta at 10 rad/s, °
G∗ at 10 rad/s, log Pa
22
0 −10 90
FIGURE 22 Typical rheological behavior of asphalt binders before and after aging in the ﬁeld in relation to pavement main distress modes.
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders of loading of the pavement under traffic needs to be simulated in measurement to obtain a reliable estimate of binder contribution to pavement performance. The third temperature zone is the lowtemperature zone at which thermal cracking (Chaps. 14 and 15) is the prevailing failure mode. During thermal cooling, asphalt stiffness increases continuously and thus results in higher stresses for a given shrinkage strain. Simultaneously, thermal stresses relax due to viscoelastic flow of the binder. To reliably predict binder contribution to cracking, both the stiffness of a binder and its rate of relaxation need to be evaluated. The stiffness of the binder is directly proportional to G∗ and the rate of relaxation is directly related to d. A lower stiffness and higher rate of relaxation are favorable for resistance to thermal cracking. As with other temperature zones, a single measure of the stiffness or viscosity of the binder is not sufficient to select better binders that will resist cracking at the lowest pavement temperatures. The above discussion of the relation between asphalt binder properties and pavement performance is further complicated by the aging phenomenon. Asphalts are hydrocarbon materials that oxidize in the presence of oxygen from the environment. This oxidation process changes the rheological and failure properties of the asphalt. As shown in Fig. 22, the rheological mastercurve slope decreases with aging, which indicates higher G∗ and smaller d values for the unaged binder at all temperatures. These changes translate into less sensitivity of G∗ and d to temperatures or loading frequency and into more elastic component (lower d). Significant oxidation effects usually appear after considerable service life. Increased G∗ values and lower d values are favorable changes with respect to rutting performance, but they are unfavorable for thermal cracking performance. For fatigue cracking, the increase in G∗ is not favorable while the decreased d is generally favorable, depending on the type of pavement and mode of fatigue damage.
Modeling of the Viscoelastic Properties of Asphalts Many attempts have been made to use simple mechanical analog, such as the generalized Burgers model and the Prony series, and phenomenological models, defined by curve fitting of experimental data, to describe the viscoelastic properties. The latter approach has seen more acceptance particularly with the advancement of computers and the flexibility of these models. Some of the most notable models that followed the pioneering work by Van der Poel in the early 1950s include work by Jongepier and Kuilman (1969) who proposed that asphalts can be considered as simple liquids whose rheological behavior can be approximated by log Gaussian distribution of relaxation times. These authors used a width parameter to represent the dependency of rheological behavior on loading time and an equiviscous temperature to represent the temperature dependence. Dickinson and Witt (1974) reported a study related to the work done by Dobson (1969). The authors proposed a new mathematical function for representing the loadingtime dependency of the rheological parameters and adopted the same mathematical functions developed by Dobson for the temperature dependency (Dobson 1969). These early models were evaluated in several following works, and the accuracy of the models have been tested using many types of unaged as well as aged asphalts (Pink et al. 1980; de Bats and Gooswilligen 1995; Maccarrone 1987). These researchers, although sometimes not consistent, in their observation all agree that asphalts can indeed be represented as linear viscoelastic materials that are thermorheologically simple. They also agree that to characterize such materials two behaviors need to be defined: the dependency of rheology on loading time, and the dependency of rheology on temperature.
23
24
Chapter Two More recently, during the SHRP, Christensen and Anderson (1992) proposed using a function derived from the Weibel distribution to represent asphalt rheology. Following that work, Marasteanu, working with Anderson (Marasteanu and Anderson 1999), offered a modification of the original Christensen–Anderson model and introduced what is called the CAM model. The CAM model was used in many studies and is considered an effective phenomenological model for unmodified asphalt binders whose properties are within the linear viscoelastic range. As a result of studying the rheology of modified binders Zeng et al. 2001 introduced a generalized model to represent the complex behavior of modified binders and mixtures, which allows for a shift for the nonlinearity (strain dependency) and the plateau region at high temperatures or very long loading times (Zeng et al. 2001). The model reduces dynamic test data, measured at multiple temperatures and strains, by constructing single complex modulus and phase angle curves. The model was considered universal because it is used to reduce the test data for the binders and mixtures with four formulations for complex modulus mastercurve, phase angle mastercurve, temperature shift factor, and strain shift factor. Details of the development of the model are found elsewhere (Zeng et al. 2001). The following is a brief review: The complex modulus mastercurve of both asphalt binder and mixture may be expressed by the following equation:
G* = Ge* +
where
Gg* − Ge*
(22)
[1 + ( fc f ′)k ]m /k e
G∗e = G∗(f → 0), equilibrium complex modulus G∗g = G∗(f → ∞), glass complex modulus fc = location parameter with dimensions of frequency f´ = reduced frequency k, me = dimensionless shape parameters
The phase angle mastercurve is represented by the following equation: ⎧⎪ ⎡ log( fd f ′) ⎤ 2 ⎫⎪ δ = 90I − (90I − δ m ) ⎨ 1 + ⎢ ⎥⎦ ⎪⎬ Rd ⎪⎩ ⎣ ⎭ where
−md
2
(23)
dm = phase angle constant f´ = reduced frequency fd = location parameter with dimensions of frequency Rd and md = dimensionless shape parameters I = 0 if f > fd I = 1 if f < fd for binders I = always 0 for mixtures
Equation (23) satisfies the requirement that the phase angle varies from 90 to 0° when the frequency is increased from zero to infinity for asphalt binders. For asphalt mixtures, this equation satisfies the requirement that the phase angle increases from 0° to a peak value and returns to 0° when the frequency is increased from zero to infinity.
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders Formulatifon of temperature and strain dependencies followed the WLF (WilliamsLandelFerry) equation (Williams et al. 1955) to express the temperature shift factor log
c (T − T0 ) aT (T ) =− 1 aT (T0 ) c2 + (T − T0 )
(24)
where T0 = reference temperature c1 = constant c2 = temperature constant In view of the similarity of strain and temperature dependencies for constructing single curve purpose, the WLF equation is utilized to express the strain shift factor: log
aγ (γ ) d (γ − γ 0 ) =− 1 aγ (γ 0 ) d2 + (γ − γ 0 )
(25)
where g0 = reference strain d1 = constant d2 = strain constant The reduced frequency in Eqs. (22) and (23) is defined as follows: log
f = log a = log aT + log aγ f′
(26)
where a = overall shift factor aT = strain shift factor ag = strain shift factor The reference temperature, T0, and reference strain, g0, in Eqs. (24) and (25) can be arbitrarily chosen at convenience. At the reference values, the respective shift factors are unity, or their logarithms are zero. The logarithms of both shift factors are the amounts of shift to form a single curve; one unit represents a shift of one logarithmic decade, positive to the direction of high frequency or low loading time. Based on testing a relatively large number of binders and mixtures Zeng and Bahia characterized the models in Eqs. (22) through (25). The reference temperature and reference strain were chosen to be 52°C and 0 percent, respectively. The reference strain is chosen assuming that material properties at zero strain represent linear properties, which cannot be directly measured but can be projected from measurements at strains greater than zero. For binders, constants G∗g = 1.0 GPa, G∗e = 0, and me = 1 were assigned as it has been shown that Gg ≈ Eg/3 ≈ 3.0/3, GPa = 1.0 GPa and that asphalt binders are nearly a linear viscoelastic fluid; parameters k and fc were estimated using a minimum square of error fitting procedure. For mixtures, all five parameters were estimated to give the best fit by minimizing the square error routine. An example of the fitting results for binder and mixture is presented in Figs. 23 and 24. The binder in the example is the PG 7622 modified by ethylene terpoly; the aggregate in the mixture is crushed limestone with fine gradation.
The Asphalt Viscoelastic and Failure Properties Selected in the SHRP Although there are many methods of characterizing viscoelastic properties, SHRP researchers selected the dynamic (oscillatory) testing as the best technique to represent
25
26
Chapter Two
FIGURE 23 Example of binder and mixture complex modulus and phase angle master curves (PG 7622 ethylene terpoly modiﬁed binder and ﬁne limestone aggregate).
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders
FIGURE 24
Temperature and strain shift factors associated with Fig. 23.
the behavior of this class of materials. As previously discussed, in the shear loading mode, the dynamic modulus (G∗) and the phase angle (d) are measured. G∗ represents the total resistance to deformation under load, while d represents the relative distribution of this total response between an inphase component and an outofphase component. In addition to the prefailure properties of asphalts as measured by rheology, their failure properties need to be characterized. Asphalt’s failure behavior is also highly dependent on temperature and time of loading (Dongre 1994). At low temperatures,
27
28
Chapter Two failure occurs in a brittle manner with a plateau zone showing a strain at failure that is relatively small (limiting value of approximately 1.0 percent strain). As temperature increases, a transition from brittle to ductile failure can be observed which, at high temperatures, converts into a flow zone. The most critical part of this behavior for pavement applications is the temperature and loading rate at which the transition from the brittle to the ductile behavior occurs. For many unmodified asphalts, there is some correlation between stuffiness measured at small strains (rheological prefailure properties) and this transition. The correlation, however, does not hold for modified asphalts or specially produced asphalts (Bahia 1995). Failure properties can be measured using the direct tension test (DTT) to measure the strength and strain tolerance of asphalts. In the original version of the SHRP binder specification it was determined that most unmodified asphalts show very similar strength value and thus strength is not needed as a specification parameter. Strain tolerance, however, was found to vary significantly depending on asphalt source (chemistry) and aging. It was also observed that asphalts vary significantly in the temperature zone at which their failure behavior transitions from brittle to ductile behavior. To ensure that asphalts at minimum pavement temperatures are within the ductile region, a minimum value for the strain at failure was included in the specifications. A minimum value of 1 percent strain at failure measured at a specified strain rate was selected as a suitable criterion to ensure ductility at minimum pavement temperature. Figure 25 shows a schematic explaining the concept of brittle to ductile transition of asphalt binders. From the earlier discussion of asphalt properties, it is expected that without measuring the rheological and failure properties at the temperature and loading frequency ranges that correspond to pavement climatic and loading conditions, selection of asphalt binders for betterperforming pavements and selection of modifiers that can improve the properties of these binders is very difficult. In the NCHRP 910 project the use of rheological and failure properties to differentiate between modified asphalts played a different role (Bahia et al. 2001). It was observed that although binders can show similar linear viscoelastic behavior, their nonlinear behavior and resistance to damage can vary significantly. It was clearly observed that the assumption that energy dissipated during viscoelastic testing (particularly within the linear range) could be dissipated in more
FIGURE 25
Schematic showing the concept of brittle—ductile transition for asphalt binders.
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders than one form. A significant part could be dissipated in damping as opposed to damage and thus testing for accumulated damage is necessary to differentiate between the two mechanisms. This issue was found to be particularly important for modified asphalts since most are manufactured to enhance elasticity thus causing an increase in delayed elasticity (cause for damping). New testing protocols were, therefore, introduced to capture the rate of damage accumulation in the linear and nonlinear viscoelastic range. Depending on G∗sind and G∗/sind parameters has proven to be insufficient because these parameters are derived based on the concept that all energy dissipated is lost in damage. The subject of damage resistance is covered in a subsequent section.
The Need for Asphalt Modification Modification of asphalts is prompted mainly by the limitation of the conventional refining practices used today in producing asphalts from crude petroleum that can resist distresses. The chemical composition of asphalt and, in consequence its properties, are largely dependent on the crude source and the refining process. Asphalt production in most refineries is a secondary process that cannot compete with fuel and other products in revenue generation. Therefore, production of betterperforming asphalts is not one of the common strategies in petroleum refining. When the produced asphalt does not meet climate, traffic, and pavement structure requirements, modification has been used as one of the attractive alternatives to improve asphalt properties. In effect, what is called conventional asphalts or straight run asphalts have a range of rheological and durability properties that are not sufficient for resistance on distresses caused by the increase in traffic and total loading on current highways. Modification by specialized refining practices, chemical reaction, and/or additives has been found to improve contribution of asphalt binders to resistance of asphalt mixtures to various modes or pavement distress. This improvement is recognized to result in lifecycle cost savings and thus use of modified asphalts has been steadily increasing for the last 20 years or so. In a recent survey of the state highway agencies in the United States, 35 out of 47 agencies that responded indicated that they plan on increasing the use of modified binders in road construction, 12 were expecting to use the same amount of modified asphalt, and none indicated that they plan on reducing amount of modified asphalts. The majority of the agencies have cited premature distress such as rutting and fatigue cracking as the main reasons for justifying the use of modified binders, which on average increase the initial cost of construction (Bahia et al. 1998). Asphalt modification using additives dates back to the last century (King et al. 1999). Patents for using polymers to modify asphalts date back to 1823 (Isacsson and Lu 1995). Test projects were placed in Europe in the 1930s and in North America in the 1950s. In the early and mid1980s the introduction of newer polymers and European technologies resulted in proliferation of asphalt modification in the United States. By 1982 more than 1000 technical articles had been published on polymer modified asphalts or mixtures, and more recently there is a continuing emphasis on this subject (Bahia et al. 2001).
Asphalt Modification Strategies A modifier can be selected to improve one or more of the main performance related properties of asphalts. Also, different modifiers that affect different properties can be combined to improve several properties. There have been numerous tests used to quantify each of the properties and measure the effectiveness of certain additives to
29
30
Chapter Two
FIGURE 26 Schematics shown the target change in rheological and failure properties expected from modiﬁcation.
improve asphalt properties. This chapter covers some of the key techniques to measure properties of modified asphalts and gives the background of these techniques. Ideally a modifier will change rheological properties to match requirements as defined by resistance to pavement distresses as shown in the first part of Fig. 26. It would also change failure properties such that binders would tolerate higher stresses and strains before failure due to static or repeated loading as shown in the second part of Fig. 26.
Asphalt Modifiers Currently Used There are currently a large number of modifiers used for paving grade asphalts. Table 22 lists several published surveys of asphalt modifiers and identifies the general types of modifiers that each study had identified. Asphalt modifiers can be classified based on the mechanism by which the modifier alters the asphalt properties, based on composition and physical nature of the modifier, or based on the target asphalt property that needs to be modified. In the NCHRP 910 project (Bahia et al. 2001) a list of modifiers classified based on the nature of the modifier and the mechanism by which it alters asphalt properties was developed. A total of 55 modifiers classified in 17 generic classes were identified. The information gathered about these modifiers indicates that they vary in many respects. Some modifiers are particulate matters while others disperse completely or
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders
Terrel and Epps 1989
Peterson 1993
Romine et al. 1991
Moratzai and Moulthrop 1993
McGennis 1995
Isacsson and Lu 1995
Banasiak and Geistlinger 1996
Reference
Thermoplastic Polymers
X
X
X
X
X
X
X
Thermoset Polymers
X
X
X
X
X
X
X
Fillers/Reinf. Agents/ Extenders
X
X
X
X
X
X
X
Adhesion Promoters
X
X
X
X
X
X
Catalysts or Chemical Reaction Modifiers
X
X
X
X
X
X
X
Aging Inhibitors
X
X
X
X
X
X
X
Others
X
X
X
X
Total number of existing brands or types


82 (27 ASA)
48
31
62
Modifier
46
ASA: Antistripping additives
TABLE 22
Recent Surveys of Asphalt Modifiers
dissolve in the asphalt. The modifiers range from organic to inorganic materials, some of which react with the asphalt while others are added as inert fillers. The modifiers generically vary in their specific gravity as well as other physical characteristics. They are expected to react differently to environmental conditions, such as oxidation and moisture effects. Not all these types of modifiers are being currently used. The frequency of use varies significantly depending on marketing of the modifiers, experience of contractors and agencies, and cost. In a survey conducted in 1997, state highway agencies indicated that polymer modifiers are among the most widely known and used in practice. Table 23 depicts the results of the survey (Bahia 1998b). Table 23 shows that the elastomeric polymer styrenebutadienestyrene (SBS) is the most frequently used to target a variety of distresses among which permanent deformation is the most common. As indicated in Table 23, many agencies did not indicate the specific target distress for which they are using the modifier.
Critical Properties of Modified Asphalts Effects of Modification on Viscoelastic Properties Measuring properties of modified binders has in general followed the testing technology used for unmodified asphalts but in many specifications expanded to capture some
31
32
Chapter Two
Target Distress/Property (No. of Agencies) Type
No. of Agencies
PD∗
FC†
LTC‡
MD§
AR¶
Styrene Butadiene Styrene (SBS)
28
18
8
10
3
6
Styrene Butadiene (SB)
16
13
5
5
0
2
Styrene Butadiene Rubber Latex (SBR)
17
10
4
4
1
2
Tire Rubber
3
1
Ethyl Vinyl Acetate (EVA)
6
3
Fatty Amidoamines
8
4
Polyamines
6
4
Hydrated Lime
4
3
Others
7
3
Natural Asphalts
6
5
12
3
1
7
4
1
Class
PolymerElastomer
PolymerPlastomer
AntiStripping Agents
Hydrocarbons
Cellulose Polypropylene
Fibers
1 1
1
Polyester
6
4
Mineral
3
1
ProcessedBased
Air Blowing
4
2
Mineral Fillers
Lime
4
1
AntiOxidants
Hydrated Lime
7
4
Extenders
Sulfur
4
∗
Permanent Deformation. Aging Existence.
†
Fatigue Cracking.
‡
1 1
Low Temperature Cracking.
1
1
§
Moisture Damage.
¶
TABLE 23
Modifiers Most Commonly Used by State Highway Agencies
unique properties that modifiers were used to induce or enhance in the base asphalt (Bahia 1995; King et al. 1992; Brule and Maze 1995). As discussed at the beginning of this chapter modification targets included rigidity, elasticity, strength, brittleness, and damage resistance. To show the general trends in effect of modifiers on asphalt viscoelastic properties, typical examples are shown in Fig. 27. The figure includes two plots, plot (a) depicts
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders 10
100 c%
2c % 80
8
60 6 40 4
Delta at 10 r/s, °
G* at 10 r/s, log Pa
0%
20
2
−30
0 30 Temperature, °C
60
0
(a) Modified with an SBpolymer
9
Phase angle, °
80 G* CRM 60 40
Delta CRM G* AC
7 5
Delta AC 3
20 0 −5
0 5 10 Reduced frequency, log(s)
G* at 25°C, log Pa
100
1 15
(b) Modified with crumb rubber
FIGURE 27 Isochronal rheological curves for an asphalt before and after modiﬁcation with two types of additives. (a) Modiﬁed with an SB polymer; (b) modiﬁed with crumb rubber.
rheological mastercurves measured using a dynamic shear rheometer for an asphalt before and after modification with an styrenebutadiene (SB) polymer at two different concentrations (c = 3 and 2c = 6 percent). Changes in both G∗ and d as a function of temperature are shown. The effects of this modifier show favorable trends of change: at high temperatures, G∗ is higher while d is lower. This indicates an increase in rigidity and in elasticity, which results in better resistance to permanent deformation. At intermediate temperatures (0 to 30°C) lower values of G∗ can be observed while d values remain indifferent. The reduction in G∗ values is favorable for fatigue cracking under straincontrolled conditions, typical of conditions for thin pavements. At low temperatures (−20 to 0°C), a similar, or more pronounced, reduction is observed for G∗ and a minor increase in d is seen. Both these effects are favorable since they make the binder less rigid and less elastic or more prone to stress relaxation under load. The changes shown appear to improve the properties with respect to pavement performance at all temperatures. Considering the relative changes in the G∗ and d, it is evident that the main effect is the change in the rigidity of the binder as measured by G∗. The data shown in Fig. 27(a) indicate that while the G∗ value is increased at 60°C by 100 to 200 percent, the d value is reduced by approximately 16 to 30 percent. At low temperatures the same trend can be
33
34
Chapter Two observed; the G∗ value is reduced by 40 to 50 percent while the d value is increased by only few degrees. Similar trends in changes were observed for the other types of polymers that were used in the study. Considering the fact that energy dissipation and rate of relaxation of binders are functions of sind or tand, it appears that effects of these commonly used polymeric additives on binders at small strains or stresses are mainly caused by change in rigidity while only secondary effects are caused by changes in elasticity. It is, however, recognized that type of polymer and concentration can have different effects (Bouldin et al. 1991; Brule et al. 1986 and 1988; King et al. 1992; Anderson and Lewandowski 1993; Collins and Bouldin 1991; Lesueur et al. 1997; Masson and Lauzier 1993). Figure 27(b) depicts master rheological curves for an asphalt before and after modification with a crumb rubber modifier (CRM), at 15 percent weight concentration (Bahia and Davies 1994). The figure is in terms of loading frequency rather than temperature. As discussed earlier, frequency and temperature are interchangeable; the effect of high temperature corresponds to that of low frequencies, and vice versa. Changes in mastercurves are similar to the changes observed for polymer modification shown in Fig. 27(a). G∗ values increase at low frequencies (high temperatures) while they decrease at intermediate and high frequencies (intermediate and low temperatures). The d values are lower at low frequencies but higher at high frequencies. The relative changes of either parameter are of the same order of magnitude as for the polymer modification. The effects of CRM can, therefore, be also described as mainly changes in rigidity of the asphalt. The mechanism by which CRM changes properties is, however, different; while for most polymer modifiers the polymer is completely dispersed in the asphalt and causes changes in the molecular structure of the asphalt, the CRM is observed to keep its physical identity and behave as a flexible particulate filler in the asphalt. The overall effect of CRM on rheological mastercurve is reduction of dependency of G∗ and d on frequency. This effect is similar in nature to the effect of polymer modification despite the difference in the nature of material. Polymer modification usually results in a more homogeneous binder which is more favorable than the nonhomogeneous CRM modification. The tradeoff, however, is the relatively higher cost of the polymer modifiers compared to the CRM. Various researchers have conducted extensive evaluation of CRM modifications (Oliver 1982; Chehoveits et al. 1982; Bahia and Davies 1994).
Effects of Modification on Failure Properties Using the direct tension test developed by SHRP, the binders modified with the different additives were tested at temperatures ranging between −30 and 0°C. The tests were conducted at a deformation rate of 1.0 mm/min in three replicates and the stress and strain at failure were calculated. To evaluate the effect of the modifiers, the stress and strain values at failure of the base and the modified binders are compared. Figure 28 depicts the results for the same additives discussed in the pervious section. Figure 28(a) shows strain at failure and stress at failure plots as a function of temperature for an asphalt before and after modification with 3 and 6 percent of the SBbased polymer. The strain curves show that the polymer increases the strain at failure within the brittle and the brittleductile zones but converge to the same values as the flow zone is approached. The effect can be considered as shifting the strain at failure curve horizontally to lower temperatures without significant changes in shape of curve. The effect of polymer addition is favorable as it tends to increase the strain at failure within the critical region. The results shown also indicate that the effect is more favorable with higher concentration of the polymer. The stresses at failure are similar for all binders which indicate that the polymer does not result in significant changes in strength of binders.
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders 100 Unmodified 3 % SB 6 % SB
10
10
Stress
1
1
Stress at failure, MPa
Starin at failure, %
100
Strain
Test Temperature, °C −30
−25 −20 −15 −10 −5 (a) Modified with SBPolymer
100
100
Starin at failure, %
.1 0
Unodified CRM1 CRM2
10
10
Stress
1
1
Stress at failure, MPa
.1 −35
Strain
Test Temperature, °C .1 −35
−30 −25 −20 −15 −10 −5 (b) Modified with crumb rubber
.1 0
FIGURE 28 Failure strain isochronal curves for an asphalt before and after modiﬁcation. (a) Modiﬁed with SBpolymer; (b) modiﬁed with crumb rubber.
The results shown may not apply to all types of polymer modifiers. The effect of different polymers on failure properties is expected to depend largely on the type of interaction between the asphalt and polymer, the molecular nature of the polymer additives, and the way it is dispersed in the asphalt. The effect of polymers on failure properties can be hypothesized in different ways. One hypothesis is that polymers form some kind of molecular network inside the asphalts resulting in more straintolerant
35
36
Chapter Two material. Another hypothesis is that the dispersed polymer particulates may serve as reinforcements, arresting microcrack propagation and increasing toughness of binders. The typical trend that can be observed from review of polymer modification works, however, is that not many polymers used currently improve lowtemperature failure properties. This may be attributed to the fact that until recently there has been no simple technique to measure the brittle failure of asphalt, and also to the fact that none of the used binder specifications address the brittleness of asphalt in a rational and fundamental form. These issues did not encourage many polymer modifier producers to concentrate on designing a modifier to mainly enhance lowtemperature failure properties. Figure 28(b) depicts failure plots for an asphalt before and after modification with crumb rubber at 10 percent (CRM1) and 20 percent (CRM2) concentration by weight of total binder. The effect of the CRM is similar to the polymer modification with respect to the strain at failure values; higher strains are observed at low temperatures but similar strains are observed as the flow region is reached by the binders. The effect also represents a shift of the failure curve along the temperature scale toward lower temperatures. The shift is larger for the higher CRM content. The stresses at failure, however, show a trend different from the polymer modification. The CRM results in stress values that are significantly higher than the unmodified asphalt at all temperatures. This behavior can be attributed to the reinforcing effect of the rubber particles. The crumb rubber particles do not dissolve in asphalt; the particles maintain their integrity and tend to swell in asphalts resulting in effective volumes that are larger than their initial volume (Bahia and Davies 1994; Chehoveits 1982). It is speculated that the swelling results in selective absorption and/or adsorption of certain components of the asphalt. Such interactions are expected to reinforce the matrix of the binder and result in higher strength, as observed in the figure. The increase in strain and stress at failure is favorable for paving grade asphalts, particularly when it is not accompanied by an increase in stiffness.
Effects of Modification on the Superpave Grading Properties The Superpave performance grading (PG) system was developed to evaluate binder properties at specific temperatures with specific testing systems (Anderson and Kennedy 1993; Anderson et al. 1994; McGennis 1995). The AASHTO MP1 procedure includes details of testing and grading of binders. Figure 29 is a schematic describing
FIGURE 29
The Superpave grading system.
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders the different test systems and pavement distresses covered in the PG system. Workability is measured at 135°C using a rotational viscometer; resistance to permanent deformation and fatigue cracking is measured using a dynamic shear rheometer (DSR) at maximum pavement temperature and intermediate pavement temperatures. For evaluating the resistance to lowtemperature cracking, the bending beam rheometer (BBR) and the direct tension testing (DTT) device are used. Numerous studies have been conducted to evaluate how different modifiers affect the grades of conventional asphalts. Figure 210 depicts an example of effect of three different types of crumb rubber based additives on performance measures. The effect is calculated in terms of ratios of the value of the parameter after modification to the parameter before modification. It is apparent that for the rutting parameter (G∗/sind) the ratios are higher than one while for the fatigue parameter, the ratios of G∗sind is much lower than one. S(60) ratios are also lower than one while the ratios of m(60) and strain at failure are very close to the value of 1.0 (no change). The data in Fig. 210 show that this type of modification has its main effects at high temperatures. This is expected when the nature of the CRM modifier is considered. Crumb rubber acts mainly as a flexible filler; at high temperatures it is stiffer than the asphalt and thus contributes significantly to the increased moduli. With decreasing temperatures, the asphalt becomes stiffer while the crumb rubber properties do not change significantly. At a certain temperature, the asphalt may become stiffer than the crumb rubber and thus a reduction in stiffness can be observed for the modified binder. Crumb rubber, at moderate concentrations that are used in practice (10 to 20 percent), however, cannot reduce stiffness by large margins because of its own relatively high stiffness at low temperatures. It is, therefore, expected that the main effects of crumb rubber remain to be seen at higher temperatures and to affect mainly the rutting parameter. The use of the Superpave technology to evaluate modification is rather widespread today and it became the standard for estimating the relative value of additives and to justify some of the added initial cost to buy and use selected modifiers. Concerns, however, were raised about using this technology without verification that it applies to modified binders. The concerns were the results of a few field problems and because the original SHRP did not include a variety of modified asphalts used in practice after the program was completed. The following sections address these concerns (Bahia et al. 1998b; Bahia et al. 2001).
Relative change (CRM/neat)
10 RB1
RB2
RB3
8 6 4 2 0 G*/sind
FIGURE 210
G*/sind
S(60)
m(60)
F.strain
Typical effects of CRMmodiﬁer on the Superpave grading parameters.
37
38
Chapter Two
Complexity of Modified Binders As discussed previously, one of the main objectives of the SHRP was to develop test methods for characterization of asphalts that are equally applicable to unmodified or modified asphalt cements, collectively called asphalt binders (Anderson et al. 1994). There were, however, two problems that raised concerns about applicability of PG specification to all asphalt binders. The first was that the majority of the testing during SHRP was done on unmodified asphalts of certain PG grades that did not cover the extreme grades required by the new specifications. A review of the asphalts included in the SHRP Materials Reference Library (MRL) indicates that they range between a PG 6428 and a PG 4634 with one PG 7022. This range of grades does not cover extreme grades that are being specified for highvolume traffic in warm regions and grades being considered in many cold regions. The second problem was the fact that these extreme grades, such as 7622, 8222, 6434, and 5840 did not exist at the time SHRP research was active. The concerns about the Superpave binder specification applicability to all asphalt binders resulted in the initiation of the NCHRP 910 project, “Superpave Protocols for Modified Asphalt Binders” (Bahia et al. 2001). The first phase of the project included a survey of users and producers of modified binders to identify the types of asphalt additives most commonly used in practice, to summarize concerns about the use of Superpave protocols for modified asphalts, and to define the current and future needs for modified asphalts. It also included a comprehensive literature review to evaluate the research done to evaluate modified binders using the Superpave protocols. The first phase resulted in recommendation for classifying asphalt binders into simple and complex binders. Based on this classification, it is recommended that the Superpave binder specification be used for asphalts that exhibit simple rheological behavior. The first phase has also resulted in the recommendation for the addition of new or revised testing procedures to characterize specific properties that are important for asphalts modified with additives. These procedures include modification of the rolling thin film oven test (RTFOT) procedure, development of the particulate additive test (PAT) and the laboratory asphalt stability test (LAST) (Bahia et al. 1998). The following sections cover the details of the deficiencies in the existing SHRP PG grading protocols and the recommended modifications.
The Assumptions in the Superpave Binder System The Superpave binder specification contains criteria based on assumptions that were made to simplify the testing required and evaluate characteristics that are most critical to pavement performance. These assumptions although were validated for neat asphalts, may not be valid for asphalts modified with different additives. Based on detailed review of the SHRP Project A002A report (Anderson and Kennedy 1993; Anderson et al. 1994) and other recent published literature (Bahia et al. 1998; Bahia et al. 1999), the following assumptions are found to be the most important that are related to the behavior of modified binders. 1. No strain/stress dependency of rheological response (wide linear viscoelastic range). 2. No shear rate dependency of viscosity (wide Newtonian range). 3. Testing at one loading rate is sufficient (similar loading rate dependency).
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders 4. Binders are homogeneous and isotropic (no sample geometry or particulate additives effects). 5. Similar timetemperature equivalency for all binders (one shift is used). 6. Binders are not thixotropic (no effect of mechanical working). 7. Stability of asphalts is affected mainly by oxidation. The essence of the above assumptions is that asphalt binders are simple systems that can be characterized using linear viscoelasticity and simple geometry within which stress and strain fields are simple to calculate. Except for number 7, the assumptions in the specification are related to rheological and thermorheological behavior. Assumption number 7 may not be very important for neat asphalt, but it is critical for asphalts modified with additives. In a recent study conducted as part of the NCHRP 910 project, each one of the listed assumption was challenged and data were collected to prove that many modified binders used in practice do not satisfy these assumptions. There were, however, important reasons why researchers in SHRP had considered these assumptions to be reasonable among which the two main are: (1) Most asphalts tested during the SHRP showed a relatively wide linear range. (2) Asphalt pavement structural design should be selected such that materials will be exposed to small stresses and strains. Therefore, the specification testing was designed to measure behavior within the linear range expecting that asphalts in the field will be mainly performing within this region. Some of the important behaviors that will need to be characterized were, however, not addressed due to need for simplification. Of particular importance are the possible thixotropic effect and the dissimilarity of the effect of repeated loading. Figure 211 is an example of possible effects of repeated loading at two different strain levels. It is clear that some modified asphalts can show significant reductions in G∗
FIGURE 211 Thixotropic behavior of modiﬁed binder at 10 percent strain.
39
40
Chapter Two due to mechanical working (repeated loading). In the field of asphalt emulsions, technology is available to produce such thixotropic materials, such as highfloat emulsions. It is reasonable to assume that this same technology will find its way in the production of paving grade asphalts. The dependence of rheological response on loading rates and temperature is known in many fields to be material specific. In the asphalt field, behavior of modified and unmodified asphalts varies significantly in responding to change of traffic speed in the field, or testing rate in the lab. Because of this variation, using common timetemperature equivalency factors could result in significant errors in estimating the effect of traffic speed on response of various asphalts. Figure 212 depicts that the relationship between effect of changing the loading rate and effect of changing temperature on G∗ for typical paving grade asphalts. The R2 values are 0.17 for the shift from 5 to 10 rad/s and 0.30 for the shift from 1 to 10 rad/s. This clearly indicates that there is hardly a correlation between the effects of loading time and temperature changes on changes in G∗ values. In other words, the practice of changing the temperature grade to compensate for traffic speed cannot be justified. The polymermodified asphalts appear to be significantly less affected by a change in traffic speed (frequency) than the unmodified asphalts. Also, the effect on G∗ of changing temperature by 6°C is highly variable, even for the unmodified asphalts. For the asphalts examined, which include 32 sources tested during the SHRP, the change ranged from 1.7 to 2.6 folds. This result is expected, since temperature susceptibility is highly asphalt source specific. The simplification used in the current specification is, therefore, not acceptable. Regarding the assumption of homogeneity, the Superpave binder testing is done using selected geometries for different temperatures and response types. These geometries were selected to give stress fields that can be easily estimated and thus used in calculating the material response. The concept is based on the assumption that binders are homogeneous materials that exhibit isotropic behavior. Therefore, testing in one mode of loading, using a single geometry, can give a comprehensive evaluation of the material behavior under different geometric and loading conditions. For some modified asphalts, however, the additive used results in an anisotropic binder and thus
FIGURE 212
Relationship between 6°C temperature change and loading rate.
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders the response under loading can be highly dependent on sample geometry. Additives such as fibrous materials with high aspect ratio, and other particulate additives of relatively large random shapes are examples that can result in anisotropic behavior of the modified binders. It is known that as the ratio of particle size to sample size increases, and/or as the volume concentration of the additive increases, the geometry of the sample can interfere with the measurement. The size and the volume concentration of particulate additives are not fully controlled in the current specification. The only existing limitation is particulates should not exceed 250 μm, which was selected arbitrarily (AASHTO TP5). Size and volume concentrations are both required in limiting the amount of additive that can result in interference with measurements. It is, therefore, necessary that the presence of solid additives be detected and the nature of the additive be checked. A new test, called the PAT is described in this article to address this need. Stability of binders in the current Superpave binder specification has also been oversimplified by considering oxidation as the only mechanism by which asphalt binder properties can change in service. Modified asphalt binders, however, can undergo changes due to factors other than oxidation. It is recognized that modified binders are multiphase systems in which the modifiers are dispersed into the asphalt cement phase. This dispersion is generally accompanied by a degree of incompatibility that is affected by various physical, thermal, and chemical factors. Excessive incompatibilities can negatively affect the performance of these binders as gross separation occurs during storage and handling of the binder, production and transportation of asphalt mixtures, and during the construction of pavement layers. There are four separation mechanisms (physical, thermal, chemical, and oxidation) that need to be considered. In the current Superpave system, there are protocols for characterizing the oxidative stability [Pressure Aging Vessel PAV and RTFO]. There are also proposed protocols for measuring physical separation (cigar tube test). There are, however, no provisions to separate these effects and to take into account the different physical, thermal, and chemical treatments expected in the field.
New Classification of Asphalt Binders Two main conclusions can be drawn from the review of the Superpave binder test protocols and specification and the existing knowledge of modified binders: 1. The existing protocols cannot be used to fully characterize all asphalt binders modified with different additives. The main reason is that they are based on simplifying assumptions that cannot be reliably extended to modified binders. 2. Some additives can result in binders that are too complex to be evaluated by any binderonly protocols. Such additives will result in anisotropy or interference with testing geometry such that only actual replication of films that will exist in mixtures will allow reliable estimation of their role in pavement performance. To apply the current Superpave binder protocols for modified binders these two conditions have to be satisfied. In other words, the modified binders have to be “simple” rheological systems. Based on this concept, asphalt binders should be classified into simple binders and complex binders, as follows. Simple Binders: Asphalt binders with simple behavior that do not violate the assumptions, which the PGgrading system is based upon; these assumptions include
41
42
Chapter Two 1. Wide linear range (independence of strain) 2. Nonthixotropy (independence of mechanical working effects) 3. Isotropy and independence of sample geometry (no additives that result in geometric effects) Complex Binders: Asphalt binders that cannot be classified as simple binders because their behavior violates one or more of the PGgrading system assumptions. This new classification is based on the hypothesis that the role of simple binders in mixture and pavement performance can be estimated using the existing (or slightly revised) Superpave binder protocols, regardless of their constituents or the method of production. The role of complex binders in mixture and pavement performance, on the other hand, cannot be estimated using binder testing. Mixture testing will have to be used. An asphalt binder can be classified as a complex binder because of the physical characteristics of the modifier or because of the nature of the effect of the modifier. Binders modified with particulate matter can be complex because of their dependency on sample geometry. Other binders can be complex because they are thixotropic or strain dependent. To qualify asphalt as a simple binder, two initial tests were recommended: (a) strain sweep, and (b) time sweep. Each should be done at high and intermediate temperatures. Both these tests are easily performed using the current version of the Dynamic Shear Rheometer (DSR). A change by more than 10 percent for either test will indicate a complex behavior. If the binder passes both tests, an evaluation of the nature of additive was recommended. A new test called the PAT is proposed to separate the additive and evaluate its nature. If the binder does not contain more than 2 percent by volume of a particulate additive, and passes the strain and time sweep tests, it can be graded as a simple binder and can be graded according to the PG system. The current Superpavetesting protocols do not cover stability under nonoxidative conditions. The stability under nonoxidative conditions requires a new test method. The LAST was proposed to measure potential for phase separation and thermal degradation of asphalts. The LAST can be used to simulate the conditions of hot storage with and without agitation under minimal exposure to oxidation. It is proposed that potential rate factors for degradation (Kd) and for separation (Ks) be calculated from the results of this test used to evaluate the stability of asphalts. In addition to these new tests, the RTFO test needs modification to make it more effective in handling the highly viscous polymer modified asphalts. The following sections give a brief overview of the PAT and the LAST. A detailed description of the background and the initial data collected is available in another publication by the author (Bahia 1999). The finding of the study on modified asphalts in the NCHRP 910 project (Bahia et al. 2001) resulted in considering a different approach to testing of asphalts in general. The approach is to evaluate damage resistance as apposed to linear viscoelastic properties. It is becoming clear that the critical role of modification is not reflected in the linear viscoelastic range but is more profoundly shown in the damage resistance range. The following section describes this approach.
Damage Resistance Characterization In the previous section the limitations of applying the Superpave binder specification to modified binders was described. It was shown that modified asphalts are generally nonlinear viscoelastic materials that exhibit complex rheology. They also exhibit unique
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders stability and composition characteristics that are not typical of unmodified asphalts and thus require special testing. In this section a new approach to test performance related properties in the linear and nonlinear domains is described. The approach is based on the concept of characterizing resistance to damage as a better alternative to link lab testing results to pavement performance. To understand the causes of pavement distresses and study their effects on the pavement performance has been the goal of asphalt researchers attentions for several decades (Kim et al. 1997; Majidzadeh et al. 1972; Dijk 1975; Dijk and Visser 1977; Moavenzadeh et al. 1974; Monismith and Deacon 1969). Among these distresses, rutting and fatigue, which are recognized as associated with the increasing traffic volume, have led to an apparent reduction in the longterm performance of flexible pavement. To predict these damages efficiently, many research projects have been designed to evaluate the effects of different factors on the performance of the asphalt pavement. Although it is recognized that these distresses are mainly caused by the deformation and/or damage within the asphalt binders, very few studies have used binder testing to evaluate damage behaviors of binders under simulated testing conditions (Bahia et al. 2001). While it is recognized that mixture factors and pavement structure factors can have important effects, efforts to understand damage behaviors are very limited. In the existing Superpave specifications, the parameter G∗sind is used to rate the binder contribution to fatigue damage resistance, while G∗/sind is used to evaluate the rutting damage resistance (Bahia and Anderson 1995). Both parameters were selected based on the dissipated energy concept as applied to linear viscoelastic range. However, there is a significant lack of information about the role of binder composition or rheological properties of binders in damage progression under cyclic loading. As part of the NCHRP 910 project (Bahia et al. 2001) two sources (gravel and crushed limestone) and two gradations (12.5 mm coarse and 12.5 mm fine) of aggregates were used to study the relationship between mixture damage behavior and the Superpave binder parameters. One asphalt binder content was used throughout the testing. Nine modified asphalts were included in the study. All of these asphalts were modified from one base asphalt. Five asphalts were modified with elastomers: ethylenepropylene diene terpolymer (Ethylene terpoly), styrenebutadienestyrene (SBS) radial, styrenebutadiene diblock (SB), styrenebutadienestyrene (SBS) linear, and styrenebutadiene rubber (SBR). One asphalt was modified using stabilized polyethylene (PE) and three oxidized asphalts were also included in this study. The oxidized asphalts were produced by steam distilled, oxidized by back blending (BB) and oxidized by straight run (SR). All of the asphalts were aged using an RTFO prior to conducting the binder tests. To study the effect of modified asphalts on the rutting behavior of asphalt mixtures the Repeated Shear Constant Height (RSSTCH) test (Chap. 10) was used. In order to evaluate the effect of binder modification on the results of the RSCH test results, certain mixture behavior indicators are needed. These indicators are commonly derived from the typical power law model recommended for representing rutting by the SHRP. The model, defined in Eq. (27), includes an initial strain factor (ep(1)) and a slope (S) factor. log ε p = log ε p(1) + S log N
(27)
where ep(1) is the total accumulated permanent strain and N is the number of cycles.
43
Chapter Two
FIGURE 213 Correlation between average rate of accumulation of strain (S) and G ∗/sind.
It is believed that the initial permanent strain can be affected by many mixture factors that are not related to the binders and this effect is carried to the total strain that is commonly used in comparing mixtures. Based on this, the value of the logarithmic slope (S) is selected as the more representative parameter that needs to be considered for studying the role of the binders in permanent deformation of the mixtures. Figure 213 shows the correlation between the average mixture rate of accumulated strain (S) and the parameter G∗/sind measured at 10 rad/s. As can be seen, there is hardly any reasonable correlation. Correlations with mixture rutting of individual aggregate blends showed similar lack of correlations. These results made it necessary to explain the reason of this lack of correlations and search for a better indicator of binders’ contribution to mixtures’ rutting performance. To study effect on the fatigue behavior of mixtures, the flexural beam fatigue test was used in the straincontrolled mode and the test was terminated at 50 percent reduction in modulus of mixture (N50). To look at the overall relations, the fatigue lives of mixtures (N50) were averaged across the aggregate types and correlated with the binder G∗sind values as shown in Fig. 214. Average for all Aggregates 1.1E+07 1.0E+07 Binder G∗ sin delta
44
y = −27.925x + 8E + 06 R2 = 0.1874
9.0E+06 8.0E+06 7.0E+06 6.0E+06 5.0E+06 4.0E+06 0
40000 60000 80000 20000 Average mixture fatigue life (N50)
100000
FIGURE 214 Correlation between G ∗sind at 10 rad/sec and mixture fatigue life measured at 10 Hz.
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders The same lack of correlation is observed, as the Rsquared value is lower than 20 percent. Similar values of correlation were observed for the individual aggregates. These results show clearly that the G∗sind of the binders is a poor indictor of the fatigue life of the mixtures (N50), as defined by the number of cycles at which the initial mixture modulus reduces by 50 percent. These findings started the process of looking for a better indicator of the role of binder in fatigue damage and resulted in putting more emphasis on the development of a better binder test for the evaluation of binder fatigue behavior, as will be discussed next.
Development of New Tests for Binder Damage Behaviors Rutting Test Several protocols were investigated to select a test procedure and a rheological parameter that could be used as a more effective indicator of the role of binders in mixture rutting than the parameter G∗/sind. The selection process was based on two main hypotheses: 1. The strains in the binder domains within the mix are significantly larger than those at which the binders are subjected to in the DSR. 2. The cyclic loading with complete reversal in strain or stress is not appropriate for rating the binder’s contribution to rutting resistance caused by cyclic irreversible loading (also called nonsteadystate cyclic deformation or more simply repeated creep). The first hypothesis is based on the data collected in a previous study by the author with coworkers (Bahia et al. 1999), which indicated that modified binders vary significantly in their strain dependency. It is also based on the finding that mixture rheological behavior was found highly sensitive to strain level (Monismith 1994). The second hypothesis was based on the concept of the RSCH and the review of literature related to the concept of energy dissipation. To test these hypotheses different testing protocols were used to find a relationship with mixture rutting performance. The protocols included strain sweeps, stress sweeps, time sweeps at constant strain and time sweeps at constant stress. In addition, a repeated creep test was developed and conducted to measure permanent strain behavior of binders. The analysis of the results led to believe that these strain sweeps and time sweeps are not promising. None of the cyclicreversible tests showed a clear potential to differentiate between binders and to relate strongly to mixture performance. As a result of this finding a detailed review of the dissipated energy concept and the derivation of the binder parameters were initiated. This review indicated that although the cyclic reversible loading could be used to estimate the total energy dissipated during a loading cycle, for viscoelastic materials that combines permanent deformation and delayed elasticity, this type of test does not allow the separation between these two different types of dissipated energy. As shown in Fig. 215, during the cyclic reversible loading only the total energy dissipated is possible to estimate. The rutting mechanism, as described by many research efforts and measured in the field, does not include reversible loading required to bring the pavement material to zero deformation. As shown in Fig. 216, rutting is in fact a repeated creep mechanism with sinusoidal loading pulses. In this case the pavement layer is not forced back to zero deflection but would
45
46
Chapter Two
FIGURE 215
Current concept used in deriving the G ∗/sind for the binder speciﬁcation (MP1).
FIGURE 216 Improved application of the dissipated energy concept to derive a fundamental rutting parameter for binders.
recover some deformation due to elastic stored energy in the material of the layers. Under this type of loading, the energy is dissipated in damping (also called viscoelasticity or hysteresis) and in permanent flow. The damping energy is mostly recoverable but requires time to be effectively utilized. The energy related to the permanent flow, however, is lost and thus called permanent. The permanent part of the dissipated energy is believed to be the main contributor to the rutting behavior of asphalt mixtures and pavements. The main problem with the reversible cyclic loading used today is the inability to distinguish between these two mechanisms that result in energy dissipation. Based on this analysis the focus of this study shifted toward developing a creep and recovery testing procedure to better simulate the contribution of binders to rutting resistance based on more fundamental understanding of rutting behavior. Realizing that the cyclic reversible tests that have been tried in the project do not offer a good indicator, and recognizing that there is a fundamental problem with estimating energy dissipation during repeated creep from the cyclic reversible loading, it was decided that a new approach that utilizes repeated creep testing is required. Evaluation of the DSR software and capabilities indicated that a repeated creep test could be conducted on binders using same geometry and temperature range. The testing was, therefore, started to develop a repeated creep test using the DSR.
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders Creep Tests at 70°C, 300 Pa Shear Stress (Loading is Recovery 9s) 100 cycles 14.0
Accumulated strain
12.0 10.0
PG 82Oxidized
8.0
PG 82PEs
6.0 4.0 PG82SBSr
2.0 0.0
0
200
400
600 Time (s)
800
1000
FIGURE 217 Results of the accumulated strain under repeated creep testing for 3 PG 82 binders at selected conditions of 1second loading and 9 seconds recovery.
Figure 217 depicts the results of the repeated creep testing of 3 binders of the PG 82 grade at selected loading conditions of 1 second loading and 9 seconds unloading. The results show clear distinction between the accumulated permanent strains of the binders which could not be detected using the G∗/sind. It can be observed that the elastomeric binder (SBS) is offering significantly higher recovery during the creep testing and thus results in less accumulated deformation compared to the other binders. The results of the creep and recovery and the G∗ and phase angle are listed in Table 24. As shown, the accumulated strain after 100 cycles at 70°C is only 20 percent SBS Modified PG 8222 (R1B02)
PEs Modified PG 8222 (R1B09A)
Oxidized PG 8222 (R1B15)
2.389
6.948
11.599
G at 300 Pa
10989
11379
15272
d at 300 Pa
56.2
60.3
73.9
sind
0.831
0.869
0.961
G∗/sind
13224
13100
15895
Ratio of strain at end of loading cycle (eL) to permanent strain at end of recover (ep) at 1 cycle
2.787
1.656
1.169
eL/ep at 100 cycle
5.571
1.764
1.179
Total strain at 100 cycles, etotal ∗
TABLE 24 Comparison of the Creep and Recovery Indicators with the Values of G ∗ and sind Measured with the DSR at Small Strains
47
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Chapter Two
FIGURE 218 Correlation between binder rutting parameter and the average mixture rutting parameter. Slope measured in strain per stress (1/kPa).
of the accumulated strain of the oxidized binder of the same grade. The G∗/sind of the oxidized binder at the testing temperature is 15,900 Pa which is higher than the elastomeric binder that has a G∗/sind value of 13,000 Pa. This inversion of ranking is very critical and can be well explained by the ability of the elastomeric binder to recover under the testing conditions. The recovery, however, is not being captured by the G∗/sind due to the fact that the parameter cannot distinguish between total energy dissipated and the energy dissipated in permanent flow. The results from the creep and recovery test can also be explained by what is known about the molecular nature for these materials. To evaluate the effectiveness of using the creep and recovery binder test, 9 binders of various grades were aged in the RTFO and tested at condition that matches the temperatures and loadingtime conditions at which the RSCH testing was conducted. The rates of accumulation of permanent strain of the mixtures were plotted versus the rate from the new binder test, as shown in Fig. 218. The correlation between the mixture and binder properties has improved significantly from 23 percent to approximately 68 percent. It appears that the new approach based on the creep and recovery testing is very promising. The correlations with the individual aggregate blends varied between higher than the average value for the crushed limestone to very low correlations for the gravel aggregates. The poor correlations for certain binders were expected since aggregates play a main role in the rutting performance. An analytical approach was used to isolate the effect of binders from the effect of aggregates based on modeling the relationship between mixture and binder accumulation of permanent strain. This type of analysis enabled the separation of the main effect of binders which showed correlations of the mixture and binder rutting behavior in the range of 80 to 90 percent (Bahia et al. 2001).
Binder Fatigue Test As discussed earlier, it is found that the parameter G∗sind is not well related to the accumulation of fatigue damage of mixtures as measured in a beam fatigue test, under straincontrolled conditions. It is believed that the main reason is that the parameter G∗sind is measured in the linear viscoelastic range using small strains. There is a
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders fundamental problem with this approach because it is unlikely to be useful in representing the effect of repeated cyclic loading and the changes in binder properties with accumulation of damage. The fatigue damage behavior was the subject of a previous publication that showed the effect of modification on the nonlinear behavior and specifically the damage behavior (Bonnetti et al. 2002). The effort to develop a new test was focused on simulating the fatigue phenomenon in a binderonly fatigue test. The DSR was used to conduct what is called a time sweep test. The test provides a simple method of applying repeated cycling of stress or strain loading at selected temperatures and loading frequency. The initial data collected were very promising and showed that the time sweeps are effective in measuring binder damage behavior under repeated loading in shear (Bahia et al. 1999). To understand the test results and to establish the best testing conditions that could lead to effective characterization of binder fatigue behavior, all nine binders used in the production of mixtures were tested at conditions that match the mixture beam fatigue conditions. The binders were aged in the RTFO to simulate the effect of mixing and compaction and the testing was conducted at 10 Hz at temperatures as close as possible to the mixture beam fatigue temperatures. The testing of binders was conducted in straincontrolled mode, and to match the mixture strain level, an estimated strain of 3 percent was used for all binders. Figure 219 shows the results of the binder testing and indicates that although the initial G∗ values are similar; these binders show significantly different fatigue behavior. Some of these binders did not reach 50 percent of the initial G∗ even after applying close to 1,000,000 cycles while others reached this level of G∗ after only 10,000 cycles. In all these tests the maximum strain at edge of plate was kept constant at 3 percent. To see if the binder fatigue life measured in the straincontrolled binder test has any relationship to the mixture straincontrolled fatigue life, Fig. 220 was prepared to show
FIGURE 219 Binder fatigue results at 10 Hz, 3 percent strain, and the temperatures selected for the mixture beam fatigue testing (binders were RTFO aged).
49
50
Chapter Two
FIGURE 220 Correlation between binder fatigue life and average mixtures fatigue life measured at the same temperature and frequency.
the relationship between the average mixture performance and the binder fatigue life as determined by the number of cycles to 50 percent of the initial G∗ value. As shown in the figure, there is a high correlation (Rsquared = 84 percent) for the nine binders that showed binder fatigue failure. This result is very encouraging and indicates that the newly developed binder fatigue test is promising and could be a better indicator of fatigue damage of mixtures.
Selection of New Damage Behavior Parameters Binder Rutting Parameter Based on the above analyses of the repeated creep tests, it was determined that a creep and recovery test would significantly improve the estimation of resistance to accumulation of permanent strain of binders and their contribution to resistance of mixture rutting. To derive a new parameter for rutting resistance, a basic approach based on viscoelasticity was followed. The approach is based on the wellknown power law model that represents the relationship between secondary creep rate as a function of number of cycles of loading [Eq. (27)]. Although this concept has been used extensively for asphalt mixtures, it has not been used for binders before. The data collected in this study indicate that modified asphalt binders can be evaluated using the same concept. The binder data, however, indicated that the rate of secondary creep of binders is a simple direct function of the number of cycles and thus the logarithmic transformation is not needed. The following model proved to be reliable:
εa = I + S × N
(28)
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders where ea = accumulated permanent strain I = intercept with permanent strain axis (arithmetic strain value, not log value) N = number of load applications S = slope of the linear portion of the logarithmic relation It is rather difficult to use the parameter of creep rate S as a specification parameter because it is an experimental parameter that is affected by a few testing attributes such as stress, loading time, and number of cycles. A better choice for a specification parameter is to use rheological models that combine fundamental behaviors to understand the performance of the material. Although several models have been used to describe the behavior of asphalt binders, the “fourparameter” (Burgers) model, shown in Fig. 221, was shown to offer a good representation of the binder behavior (Bahia et al. 2001). This model is a combination of a Maxwell model and a Voigt model. The total shear strain versus time is expressed as follows:
γ (t) = γ 1 + γ 2 + γ 3 =
τ0 τ0 τ + (1 − e − t/τ ) + 0 t G0 G1 η0
(29)
By normalizing the strain to the stress applied, the following equation representing the creep compliance, J(t), in terms of its elastic component (Je), the delayedelastic (Jde), and the viscous component (Jv) could be defined: J(t) = J e + J de + J v
(210)
The viscous component is inversely proportional to the viscosity (h) and directly proportional to stress and time of loading. Based on this separation of the creep response, the compliance could be used as an indicator of the contribution of binders to rutting resistance. Instead of using the compliance (Jv), which has a strange unit of 1/Pa, and to be compatible with the concept of stiffness introduced during SHRP, the inverse of the compliance (Gv) could be used. This term is defined as the viscous component of the
FIGURE 221
Fourelement (burgers) model and its response.
51
52
Chapter Two creep stiffness. The creep and recovery response measured with the DSR could be used to estimate the Gv value and the accumulated permanent strain for any selected combination of loading and unloading times. This finding implies that the accumulated permanent deformation is a function of viscosity, load, and loading time.
γ 1 = f (η , τ , t)
(211)
S = f (η , τ , t)
(212)
By selecting the appropriate testing stress (t) and the appropriate time of loading (t) the viscous component of the stiffness Gv could be directly related to the rate of accumulation of permanent deformation S and thus used as a fundamental indicator of rutting resistance of asphalt binders.
Binder Fatigue Parameter Although there are different loading modes that could be used in fatigue testing, a reliable indicator of fatigue failure should be independent of the loading mode. It should provide a consistent indication of the level of damage and progression of damage in the material in terms of changes in mechanical behavior under any loading conditions. The most commonly used definition of fatigue failure in asphalt mixtures is a decrease in the initial stiffness by 50 percent, as was indicated in the previous sections. This arbitrary definition, however, does not allow evaluation of the distinctly different mechanism by which a material would respond to the energy input during a loading history for the different loading modes. Researchers have, therefore, focused on using the concept of dissipated energy to explain fatigue behavior of asphalt mixtures. For many decades researchers have used the loss modulus as an indicator of fatigue resistance because of the relationship between this modulus (G∗sind) and the energy dissipated per cycle. The success of this approach has been questioned, however, in many studies because this parameter tends to give different results at different loading conditions. Recent advancements in fatigue research have indicated that a better indicator of fatigue is the rate of change of dissipated (distortion) energy per load cycle. There are several approaches to present the criterion of fatigue based on the rate of change in the dissipated energy. The most promising approaches are presented by Carpenter and coworkers (Carpenter and Jansen 1996; Ghuzlan and Carpenter 2000), and by Pronk and coworkers (Pronk 1995; Pronk and Hopman 1990).
Rate of Change of Dissipated Energy Ghuzlan and Carpenter (2000) defined the ratio of dissipated energy as ΔDE Wi − Wi+1 = DE Wi
(213)
where Wi is the total dissipated energy at cycle i calculated by area within hysteresis loop and Wi+1 is the total dissipated energy at cycle i+1. Plotting the values of this ratio versus loading cycles gives a curve that can be used to determine the fatigue life (Np) by identifying the sudden change in the rate. The problem with this approach is that the data points, especially for the constant stress tests, are scattered widely, which makes it difficult to determine an accurate Np value.
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders
FIGURE 222 Application of rate of change of energy dissipation concept to binder straincontrolled testing data.
Applying this concept to binder data, it is observed that the approach is useful for the stresscontrolled testing, but is not useful for the straincontrolled testing. As shown in Fig. 222, the straincontrolled measurements result in scattered points, which makes it almost impossible to clearly define the Np value. Conceptually, it is difficult to apply this approach for most binders because of the nature of the straincontrolled test. Since the test is straincontrolled, the rate of energy dissipation will stabilize when the damage starts accumulating because the material will soften due to damage resulting in reduction in stress required to cause same strain. The test can, therefore, take a long time to result in transition of material from the crack initiation stage to the propagation stage.
Cumulative Dissipated Energy Ratio Pronk (1995) defined the dissipated energy ratio as follows: n
Dissipated energy ratio =
∑W i=1
Wn
i
(214)
where Σ nn=1Wi is the total sum of the dissipated energy up to cycle n and Wn is the dissipated energy at cycle n. Figure 223 shows examples of applying this concept to binder data. The results shown indicate that binders can be evaluated effectively using this method. The curves of the binder data are similar to the mixture data published earlier by other researchers, which is very promising. This also implies that the main factor in the fatigue behavior of mixtures could be well related to the fatigue damage in the binder. It is also observed that the slope of the relationship between the energy ratio and the number of cycles to failure is equal to 1.0 when the material is not undergoing
53
54
Chapter Two
FIGURE 223
Using the dissipated energy ratio to analyze fatigue data.
fatigue damage as indicted by the constancy of the dissipated energy per cycle. This actually can be derived from Eq. (215) by assuming that Wi is constant and is equal to Wn: n
∑W = N ⋅ W i=1
i
n
(215)
Based on this derivation, an understanding of the fatigue curve could be introduced. It is assumed that the first portion represents the stage during which the energy per cycle is dissipated in viscoelastic damping with negligible damage. In the next stage, cracking initiation consumes an additional amount of energy beyond the viscoelastic damping. In the third stage, crack propagation starts and a noticeable increase in dissipated energy per cycle is observed. This is assumed to be the most critical stage during which damage per cycle is so high that healing and recovery from damage are difficult to occur.
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders
Summary of Findings for the Damage Resistance Parameters Based on the results and analysis of the binder rutting and fatigue studies, the following points provide a summary of the findings: 1. There are critical questions about the validity of the binder parameter G∗/sind. The correlation between the mixture rutting indicators and G∗/sind is very poor. The parameter is derived from testing that does not provide a good representation of traffic loading in the field. The parameter could not be found useful in describing the accumulation of permanent flow, which is important in rutting evaluation. 2. Repeated creep testing of binders is introduced as a better method for estimating binder resistance to permanent strain accumulation. The viscous component of the creep stiffness (Gv) is found to be a good indicator of the rate of permanent strain accumulation for binders. It is proposed as a better specification parameter. 3. Compared to the current binder protocol, the repeated creep test protocol for measuring binder accumulated permanent strain represents improvements in the theoretical and practical concepts for better rating of binder properties as related to rutting of pavements. 4. The lack of correlation between the mixture fatigue life and current binder fatigue resistance indicator, G∗sind, at the intermediate temperatures, indicates that a new test for the binder fatigue resistance is needed to determine the relation between mixture fatigue life and binder rheological properties. 5. Time sweep test is introduced as a promising binderonly fatigue test to evaluate the fatigue resistance of the binder. The test can be conducted using the current DSR within a relatively short testing time. 6. The dissipated energy ratio approach is employed as the method to determine the fatigue life of binders, because of the independent nature of this approach to the loading modes. Although the geometry has certain effects on the results, by selecting the proper test conditions, this approach can give reliable results that are found to correlate well with mixture performance.
Acknowledgments A major part of the work reported in this chapter in regard to modified asphalts is based on the SHRP and the NCHRP 910 project, which is sponsored by the American Association of State Highway and Transportation officials, in cooperation with the Federal Highway Administration. It was conducted as part of the National Cooperative Highway Research Program, which is administrated by the Transportation Research Board of the National Research Council. The author gratefully acknowledges this support and also acknowledges the support of the NCHRP Project Officer Dr. E. Harrigan, and the members of the project panel for their continuous encouragement and feedback. Also the author gratefully acknowledges the technical support of the engineers and scientist that contributed directly and indirectly to the results presented.
55
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Chapter Two
Disclaimer The opinions and conclusions expressed or implied in the report are those of the author. They are not necessarily those of the Transportation Research Board, the National Research Council, the Federal Highway Administration, the American Association of State Highway and Transportation Officials, or of the individual states participating in the National Cooperative Highway Research Program.
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Chapter Two Dijk, W., and W. Visser, “The Energy Approach to Fatigue for Pavement Design,” Journal of the Association of Asphalt Paving Technologists, Vol. 46, 1977, pp. 1–37. Dobson, G. R., “The Dynamic Mechanical Properties of Bitumen,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 38, 1969, p. 123. Dongre, R., “Development of Direct Tension Test Method to Characterize Failure properties of Asphalt Cements,” Ph.D. Thesis, the Pennsylvania State University, 1994. Dongre, R., M. G. Sharma, and A. A. Anderson, “Chracterization of Failure Properties of Asphalt Binders,” Physical Properties of Asphalt Cement Binders, ASTM STP 1241, 1995, pp. 117–136. Evans, C. G., and R. L. Griffin, “Modified Sample Plates for Tests of High Consistency Materials with the Microviscometer,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 32, 1963, pp. 64–81. Fair, W. F., Jr., and Volkmann, W., Ind. Eng. Chem., Anal. Ed., Vol. 15, 1943, pp. 240–242. Ferry, J. D., Viscoelastic Properties of Polymers, chapter 11, New York: John Wiley and Sons, 1980. Gallaway, B. M., “Durability of Asphalt Cements Used in Surface Treatments,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 26, 1957, p. 151. Gallaway, B. M., “Factors Relating Chemical Composition and Rheological Properties of Paving Asphalts with Durability,” Proceedings, The Association of Asphalt Pavino Technologists, Vol. 28, 1959, pp. 280–293. Ghuzlan, K. A., and S. H. Carpenter, “An EnergyDerived/DamageBased Failure Criteria for Fatigue Testing,” Preprints of the 79th TRB Annual Report, 2000. Griffin, R. L., T. K. Miles, and C. J. Penther, “Microfilm Durability Test for Asphalt,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 24, 1955, pp. 31–62. Halstead, W. J., and J. A. Zenewitz, “Changes in Asphalt Viscosities During the Thin Film Oven and Microfilm Durability Tests,” American Society for Testing and Materials, ASTM Special Technical Publication, No. 309, 1961, p. 133. Heithaus, J. J., and R. W. Johnson, “A Microviscometer Study of Road Asphalt Hardening in Field and Laboratory, “ Proceedings of the Association of Asphalt Paving Technologists, Vol. 27, 1958, pp. 17–34. Heukelom, W., ”An Improved Method of Characterizing Asphaltic Bitumens with the Aid of their Mechanical Responses,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 42, 1973, pp. 67–98. Isacsson, U., and X. Lu, “Testing and Appraisal of Polymer Modified Road BitumensState of the Art,” Material and Structures, Vol. 28, 1995, pp. 139–159. Jimenez, R. A., and B. M. Gallaway, “Laboratory Measurements of Service Connected Changes in Asphaltic Cement,” Proceedingsof the Association of Asphalt Paving Technologists, Vol. 30, 1961, p. 328. Jongepier, R., and B. Kuilman,”Characterization of the Rheology of Bitumens,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 38, 1969, pp. 98–122. Kandhal, P. S., L. D. Sandvig, and M. E. Wenger, “Shear Susceptibility of Asphalts in Relation to Pavement Performance,” Proceedings, Association of Asphalt Paving Technologists, Vol. 42, 1973, pp. 99–111. Kandhal, P. S., and M. E. Wenger, “Asphalt Properties in Relation to Pavement Performance,” Transportation Research Record_544, 1975, pp. 1–13. Kemp, G. R., and N. H. Predoehl, “A Comparison of Field and Laboratory Environments on Asphalt Durability,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 50, 1981, pp. 492–533.
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders Kim, Y. R., H. J. Lee, and D. N. Little, “Fatigue Characterization of Asphalt Concrete Using Viscoelasticity and Continuum Damage Theory,” Journal of the Association of Asphalt Paving Technologists, Vol. 66, 1997, pp. 633–685. King, G. N., H. W. King, O. Harders, P. Chavenot, J. P. Planche, “Influence of Asphalt Grade and Polymer Concentration on the High Temperature Performance of Polymer Modified Asphalt,” Asphalt Paving Technology, Vol. 60, 1992. King, G. N., H. W. King, O. Harders, A. Wolfgang, J. P. Planche, and P. Pascal., “Influence of Asphalt Grade and Polymer Concentration on the Low Temperature Performance of Polymer Modified Asphalt,” Journal of the Association of Asphalt Paving Technologists, Vol. 62, 1993, pp. 1–22. King, G. N., H. King, R. D. Pavlovich, A. L. Epps, and P. Kandhal, “Additives in Asphalt,” Journal of the Association of Asphalt Paving Technologists, Vol. 68A, 1999, pp. 32–69. Labout, J. W. A., and W.P. van Ort, “Micromethod for Determining Viscosity of High Molecular Weight Materials,” Analytical Chemistry, Vol. 28, 1956, p. 1147. Lesueur, D., J. F. Gerard, P. Claudy, et al., “Relationships between the Structure and the Mechanical Properties of Paving Grade Asphalt Cements,” Preprint of the AAPT Annual Meeting, Salt Lake City, Utah, 1997. Mack, C., “An Appraisal of Failure in Bituminous Pavement, “Proceedings of the Association of Asphalt Paving Technologists. Vol. 34, 1965, pp. 234–247. Maccarrone, S., “Rheological Properties of Weathered Asphalts Extracted from Sprayed Seals Nearing Distress Conditions,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 56, 1987, pp. 654–687. Majidzadeh, K., and H. E. Schweyer, “NonNewtonian Behavior of Asphalt Cements,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 34, 1965, p. 20. Majidzadeh, K., “Rheological Aspects of Aging: Part II,” Highway Research Record, 273, 1969, pp. 28–41. Majidzadeh, K., E. M. Kaufmann, and C.L. Saraf, “Analysis of Fatigue of Paving Mixtures from the Fracture Mechanics Viewpoint,” Fatigue of Compacted Bituminous Aggregate Mixtures, ASTM, STP 508, 1972, pp. 67–83. Marasteanu, M. O., and Anderson, D. A., “Improved Model for Bitumen Rheological Characterization,” Eurobitume Workshop on Performance Related Properties for Bituminous Binders, Luxembourg, May 1999. Masson, JF, and C. Lauzier, Methods for the Analysis of Polymers in Polymer Modified Asphalts, National Research Council Canada, Institute for research in Construction, Ottawa, A2053.2, 1993. McGennis, R. B., “Asphalt Modifiers are Here to Stay,” Asphalt Contractor Magazine, April 1995, pp. 38–41. McLeod, N. W., “A 4Year Survey of Low Temperature Transverse Pavement Cracking on the Three Ontario Roads,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 41, 1972, p. 424. Moavenzadeh, F., and R. R. Slander, “Durability Characteristics of Asphaltic Materials.” Research Report EES259, 1966, p. 236. Moavenzadeh, J., and R. R. Stander, Jr., “Effect of Aging on Flow Properties of Asphalts,” Hwy. Res. Brd. Rec., Vol. 178, 1967, pp. 1–29. Moavenzadeh, F., J. E. Soussou, and H. K. Findakly, “Synthesis for Rational Design,” Final Report for FHWA, Contract 7776, Vol. 2, 1974. Monismith, C. L., and J. A. Deacon, “Fatigue of Asphalt Paving Mixtures,” Transportation Engineering Journal, Proceedings of the ASCE, Vol. 95, TE2, May, 1969, pp. 317–346.
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Chapter Two Monismith, C. L., “Fatigue Response of AsphaltAggregate Mixes,” Strategic Highway Research Program, SHRPA404, National Research Council, 1994. Mortazavi, M., and J. S. Moulthrop, The SHRP Materials Reference Library, SHRPA646 Report, The Strategic Highway Research Program, National Research Council, Washington, D.C., 1993. Neppe, S. L., “Durability of Asphaltic Bitumen as Related to Rheological Characteristics,” Transaction, South African Institute of Civil Engineers, Vol. 2, 1952, p. 103. Oliver, J. W. H., “Optimizing the Improvements Obtained by the Digestion of Comminuted Scrap Rubbers in Paving Asphalts,” Proceedings of the Association of Asphalt Paving Technologies, Vol. 51, 1982, p. 169. Page, G. C., K. H. Murrphy, B. E. Ruth, and R. Roque, “Asphalt Binder Hardening— Causes and Effects,”Proceedings of the Association of Asphalt Paving Technologists, Vol. 54, 1985, pp. 140–167. Pell, P. S., and K. E. Cooper, “The Effect of Testing and Mix Variables on the Fatigue Performance of Bituminous Materials,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 44 (1975), pp. 1–37. Peterson, K., “Specific’s Guide to Asphalt Modifiers,” Roads and Bridges Magazine, May 1993, pp. 42–46. Pfeiffer, J. Ph., and P. M. van Doormaal, “The Rheological Properties of Asphaltic Bitumen,” Journal of Institute of Petroleum Technologists, Vol. 22, 1936, p. 414. Pfeiffer, J. Ph., The Properties of Asphaltic Bitumen, Section II, Elsevier Publishing Co., Amsterdam, 1950. Pink, H. S., R. E. Merz, and D. S. Bosniak, “Asphalt Rheology: Experimental Determination of Dynamic Moduli at Low Temperature,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 49, 1980, p. 64. Pronk, A. C., and P. C. Hopman, “Energy Dissipation: The Leading Factor of Fatigue,” Proceedings of Strategic Highway Research Program: Sharing the Benefits, London, 1990. Pronk, A. C., “Evaluation of the Dissipated Energy concept for the Interpretation of Fatigue Measurements in the Crack Initiation Phase,” The Road and Hydraulic Engineering Division (DWW), Netherlands, PDWW95001, 1995. Puzinauskas, V. P., ”Evaluation of Properties of Asphalt Cements with Emphasis on Consistencies at Low Temperature,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 36, 1967, p. 489. Puzinauskas, V. P., “Properties of Asphalt Cements,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 48, 1979, pp. 646–710. Romberg, J. W., and R. N. Traxler, “Rheology of Asphalt,” Journal of Colloid Science, Vol. 2, 1947, pp. 33–47. Romine, R. A., M. Tahmoressi, R. D. Rowlett, and D. F. Martinez, “Survey of State Highway Authorities and Asphalt Modifier Manufacturers on Performance of Asphalt Modifiers,” Transportation Research Record no. 1323, 1991, p. 61. Saal, R. N. J., and J. W. A. Labout, “Rheological Properties of Asphalts,” in Rheology: Theory and Applications, Vol. 2, 1958, F.R. Eirich, Ed., New York, Academic Press. Schmidt, R. J., “Laboratory Measurement of the Durability of Paving Asphalts,” ASTM SPT 532, 1973, pp. 79–99. Serafin, P. J., L. L. Kole, and A.P. Chirtz, “Michigan Bituminous Experimental Road— Final Report,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 36, 1967, pp. 582–614. Terrel, R. L., and I. A. Epps, Using Additives and Modifiers in Hot Mix Asphalt, QI Series 114, National Asphalt Pavement Association (NAPA), Riverdale, Md., 1989.
Modeling of Asphalt Binder Rheology and Its Application to Modified Binders Traxler, R. N., Asphalt: Its Composition, Properties and Uses, Reinhold, New York, 1961. Traxler, R. N., H. E. Schweyer, and H. W. Romberg, “Rheological Properties of Asphalt,” Industrial and Engineering Chemistry, Vol. 36, No. 9, 1944, p. 823. Traxler, R. N., and H. E. Schweyer, “Increase in Viscosity of Asphalts with Time,” Proceedings of the American Society for Testing and Materials, Vol. 36, Part II, 1936, pp. 544–551. Traxler, R. N., “Review of the Rheology of Bituminous Materials,” Journal of Colloidal Science, Vol. 2, 1947, p. 49. Van der Poel, C., “A General System Describing the ViscoElastic Properties of Bitumens and Its Relation to Routine Test Data,” Journal of Applied Chemistry, Vol. 4, 1954, p. 221. Williams, M. L., R. F. Landel, and J. D. Ferry, “The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other GlassForming Liquids,” The Journal of the American Chemical Society, Vol. 77, 1955, pp. 3701–3707. Wood, P. R., and H. C. Miller, “Rheology of Bitumens and the Parallel Plate Microviscometer,” Highway Research Board Bulletin, National Research Council, D.C., Vol. 270, 1960, pp. 38–46. Zeng, M., H. U. Bahia, H. Zhai, M. Anderson, and P.Turner, “Rheological Modeling of Modified Asphalt Binders and Mixtures,” Journal of the Association of Asphalt Paving Technologists, Vol. 70, 2001, pp. 403–441. Zube, E., and J. Skog, “Final Report on ZacaWigmore Asphalt Test Road,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 38, 1969, pp. 1–39.
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PART
2
Stiffness Characterization CHAPTER 3 Comprehensive Overview of the Stiffness Characterization of Asphalt Concrete
CHAPTER 5 Complex Modulus from the Indirect Tension Test
CHAPTER 4 Complex Modulus Characterization of Asphalt Concrete
CHAPTER 6 Interrelationships among Asphalt Concrete Stiffnesses
Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.
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CHAPTER
3
Comprehensive Overview of the Stiffness Characterization of Asphalt Concrete Robert L. Lytton
Abstract Stiffness of asphalt concrete is a material property that is central to the performance of asphalt pavements. This chapter reviews various factors affecting the stiffness of asphalt concrete, methods of measuring stiffness, and importance of stiffness in pavement response and performance prediction. Special attentions are given to describe how asphalt concrete stiffness is affected by different factors in a mechanics sense.
Introduction The first part of this chapter is conceptual and only the latter part becomes analytical. The chapter starts with five basic questions concerning asphalt stiffness and some conceptual answers to those questions. The questions are What is asphalt concrete? What is stiffness? How is stiffness measured? Why is stiffness important? How is stiffness used in computations? Although the first of these questions has been answered in the previous chapters, it is answered again here as a preparation for the conceptual answers concerning stiffness. The remaining parts of the chapter present the tests that can determine asphalt concrete
65 Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.
66
Chapter Three stiffness in the laboratory and in the field, the composition and condition factors that stiffness depends upon, and finally the methods of mathematical characterization of asphalt concrete stiffness in both the damaged and undamaged states.
Asphalt Concrete Asphalt concrete is a composite material made up of aggregate particles, bitumen, air, and other components such as additives, modifiers, fines, and water in either liquid or vapor forms. The emphasis in this chapter is that the material is a composite and its composition is important in determining its useful properties for engineering and construction applications.
Asphalt Concrete Stiffness Asphalt concrete stiffness is a material property. Precisely, it is a slope of a stressstrain curve of the asphalt concrete. What is unique about a material property is that it is independent of the test apparatus, or the sample size or geometry that is used to measure it. The converse of this is that if a measured result of a test does depend upon the test apparatus or the sample size or geometry, then that measured result is not and cannot be a material property. There are several types of stressstrain curves from which asphalt concrete stiffness can be measured as a material property.
Methods of Measuring Stiffness Asphalt concrete stiffness can be measured either in the laboratory, as will be discussed in Chaps. 4 and 5, or in the field. The geometry of the test specimen is important because it determines whether the material property can be measured directly with instruments mounted on the sample or must be inferred by analyzing the response of the test sample to the imposed load. The loading pattern that is applied to the test sample also determines what the measured stiffness will be. The rate of loading and the temperature and moisture at which the loading is applied will also determine the measured stiffness. Finally, the age of the test sample will also affect the measured stiffness.
Location of Test In the field, asphalt concrete stiffness may be measured either destructively or nondestructively. In the laboratory, tests can be conducted on cores taken from the field or from samples that were compacted in the laboratory. Destructive tests in the field include “small aperture” testing instruments such as cone penetration or pressure meter devices or “large aperture” methods such as test pits, or accelerated loading tests using fullscale or scalemodel vehicle simulation devices. Nondestructive tests include static, cyclic, impulse, and surface wave instruments. In general, the test data from all of these devices must be analyzed by some form of inverse analysis in order to produce a material property of the asphalt concrete, and specifically, its stiffness.
Geometry of the Test Specimen The geometry of a test specimen, either in the field or in the laboratory, is an important consideration because if some geometries are selected, material properties can be inferred directly from the test measurements without the need for the inverse analysis
Overview of the Stiffness Characterization of Asphalt Concrete of the test data. Test specimen geometry in situ is usually dictated by the pavement structure. In the laboratory, there are many more options such as uniaxial tension and compression, triaxial tests in compression and extension (Chaps. 4, 7, and 11), various forms of shear tests (Chap. 10), bending and torsion tests, and indirect tension tests (Chap. 5) of a test sample. Some test specimens are constructed and tested in the laboratory with load, layer, and test conditions that are intended to be severe simulations of field conditions such as moisture or temperature exposure. These are commonly referred to as torture tests. It is uncommon to be able to extract material properties, and asphalt concrete stiffness specifically, from torture tests. Torture tests are commonly used as screening tests for the suitability of the composition of the asphalt concrete. Other test specimens are constructed with such simple geometries that the direct measurement of both stresses and strains make the direct determination of asphalt stiffness possible. Examples of these are the uniaxial and triaxial tension and compression tests and torsional tests on cylindrical samples. These tests are characterized by a high degree of accuracy, precision, and repeatability, as will be shown in subsequent chapters.
Loading Pattern A variety of loading patterns are used to measure asphalt concrete stiffness under a variety of loading conditions. These include monotonic loading of both stresscontrolled or straincontrolled tests, frequency sweep tests, impulse and wave propagation tests, repeated load tests, creep, relaxation, and creep and recovery tests. A creep test is one in which the applied stress is held constant and the strain is measured as it grows with time. The ratio of the strain divided by the constant stress is termed the creep compliance. A relaxation test is one in which the applied strain is held constant and the diminishing stress is measured with time. The ratio of the stress divided by the constant strain is the relaxation modulus.
Rate, Temperature, and Age The stiffness of asphalt concrete rises as the rate of loading increases and it decreases as the temperature increases. The age of the asphalt concrete is determined less by its chronological age than by its exposure to air, heat, and solar radiation conditions which will increase its rate of reaction with oxygen and make its stiffness increase along with its susceptibility to brittle fracture.
Moisture The stiffness of asphalt concrete is affected by the amount of moisture that is held within the asphalt binder; the solubility of various components of the asphalt; the amount of asphalt that becomes emulsified; the strength of the adhesive bond between the asphalt and aggregates, with and without water present on the interface; and the strength of the cohesive bond within the asphalt, with and without water present on the surface of microcracks within the asphalt. The fact that asphalt can absorb water within the thin films of mastic that are present within a mixture, together with the fact that different asphalts can absorb widely differing amounts of water at a given level of water vapor pressure, makes the effect of water on the stiffness of asphalt concrete very highly dependent on the composition of the bitumen. It is also a fact that the rate at which water can diffuse through asphalt films differs greatly with the composition of the bitumen.
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Chapter Three
Summary With the stiffness of asphalt concrete varying with so many different factors and conditions, there is the possibility of an endless round of confusion about how each relates to the other and how to combine the effects of separate investigations of these different effects on stiffness into a practical, usable predictable result. The secret to combining all of these effects in a realistic synthesis is by the use of mechanics, as will be discussed later in this chapter.
Importance of Asphalt Concrete Stiffness Subsequent chapters present constitutive, mechanistically based models for describing the behavior of asphalt concrete. An important theme in each of these models is the need for accurate characterization of the fundamental material stiffness. Just as Young’s modulus is paramount for predicting the deflection of steel beams in a structure, the stiffness of asphalt concrete is critical for predicting the behavior of the material in pavement structures. The materials that pavements are built with are very complex despite the fact that they are so commonplace. Such complexity necessitates the use of numerical procedures, such as finite elements, which require significant computation effort. Technology advances in recent years have resulted in great improvements in computation speed, and in principle it is now possible to predict the time of appearance and the rate of deterioration of asphalt concrete distresses. That is not to say that all distresses can be predicted reliably at the present time with existing models but it is now possible to construct a model that will provide reliable predictions in a rapid manner. The only impediment to developing such a model is the willingness to understand the mechanics of the selected type of distress, identify the relevant materials properties, devise test methods that will provide these materials properties, select the most appropriate numerical method to use, and assemble the computer model. Having such a model available makes it possible to interpret correctly and quantitatively the results of inservice pavement, test track, and accelerated pavement tests and to extrapolate these results to other pavements. It also makes possible coordinated laboratory tests that are simple, accurate, precise, and inexpensive with which to obtain materials properties. It also makes possible the coordinated use of nondestructive testing in the field that will provide on site measurements of the same materials properties that are the central focus of all of these mechanicsrelated activities. Computerpredicted distresses make possible and practical the use of performancebased specifications, warranties, and construction quality assurance and quality control. They also make possible the prediction of remaining life to allow the planning of pavement maintenance and rehabilitation activities as well as the overall asset management of the pavement network, including the effects of safety and cost profiles with time. All of this is possible if stiffness, as a material property, is measured and used properly in numerical prediction methods employing the computers that are now capable of handling the computational tasks that pavements require.
Use of Stiffness in Computations Asphalt concrete stiffness is used in numerical computations to calculate both the primary responses and the mechanisms of distress. The primary responses are the deflections, stresses, and strains in a pavement structure under the variety of
Overview of the Stiffness Characterization of Asphalt Concrete loading, thermal, and moisture conditions that are applied to pavements. The principal mechanisms of distresses are fracture and flow. Fracture is predicted by the use of fracture mechanics while flow is predicted by the use of the various forms of plasticity theory. Both processes use energy in causing the asphalt to deteriorate and this fact is used in damage theory, which describes in mathematical form how the various components of energy are used in predicting the rate and magnitude of the damage that is done by the loads and environmental stress that are applied to the asphalt concrete. Because asphalt concrete is a viscoelastic material, the components of energy that affect its response to loading are that which is stored and can be recovered and that which is dissipated in a loading and unloading process. Some of that energy is used in overcoming the viscous resistance of the material while the rest of the dissipated energy is available to damage the material. The correct representation, measurement, and use of asphalt concrete stiffness as a material property are essential to being able to predict correctly this partitioning of the energy.
Tests to Determine Stiffness In order to be useful in mechanics and computerbased numerical predictions of both primary responses and distress of pavements, asphalt concrete stiffness must be a material property and not an index property. In the laboratory, the material property must be measured accurately, precisely, and repeatably by testing machines and transducers that are operating within the response range of which they are capable including response time or frequency, stroke, and magnitude. The test measurements must be made in that part of the test specimen where there is a uniform stress field, a uniform strain field, and the sample is held in a condition of uniform temperature and moisture. Examples of this are measurements in the middle third of triaxial compression and tensile tests and measurements in the center of the indirect tension test. It is only in this way that the measured material property can be assured to be independent of test apparatus, sample size, and geometry. Failure to do this produces large variances of the test results, low repeatability, large variances in the predicted results, and higher levels of risk to those who must rely upon them. Material properties such as asphalt concrete stiffness can be backcalculated from tests where there is no uniformity of stress, strain, temperature, or moisture but this requires the use of a computer program that is based upon the mechanics of the sample loading and geometry and may need to include the buildup of damage in all three dimensions. The material properties that are determined by this indirect method cannot be determined directly from the test measurements and this is a major source of their higher variability. The best tests to use for determining the asphalt concrete stiffness and, in fact, all of the material properties of asphalt concrete are those tests in which both the load and the displacement of the test sample are both known precisely by actual measurement on the sample and by the use of feedback control of the test with such rapid response that it is much faster than the loading or frequency rate being used in the test. These are general principles. It is understood that nondestructive tests in the field will always require the use of back calculation in order to arrive at material properties. This is inherently the reason for the larger variances of the results.
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Chapter Three
Factors on Which Asphalt Concrete Stiffness Depends Asphalt concrete depends upon the strain rate of loading, the temperature, the moisture content of the bitumen, the stress state, the aggregate particles, the bitumen itself, the fines in the mastic, the water in liquid and vapor form and its location with the mixture, the air in the mixture, its age, and its reactivity with oxygen, and any additives or modifiers that have been added to the mixture. Each of these is discussed subsequently.
Temperature and Rate of Loading The stiffness of asphalt concrete depends on the temperature and strain rate of loading. At any given temperature, asphalt concrete will deform slowly and permanently if it is loaded slowly, while if it is loaded at a higher rate, it will be much stiffer and will be subject to fracture. At any given strain rate of loading, there is a temperature above which the material will relax quickly enough that no stress will accumulate in the test sample. These two known facts about materials in general can be illustrated in a qualitative graph of strain rate versus temperature that shows the stressfree temperature and strain rate above which the material will experience microcracking and healing and below which the material will undergo plastic flow and the properties of the aggregates in the mixture will be important in limiting the size and shape of the flow patterns. This conceptual graph is shown in Fig. 31. Moisture has a similar effect on the stiffness and type of damage that occurs in asphalt concrete. Although the mechanisms are distinctively different, high moisture and high temperature both result in plastic flow. At any given strain rate, there is a moisture content in the asphalt, above which the material relaxes faster than the stress can build up in the material. On a graph of strain rate versus moisture content there will be a similar stressfree boundary between the brittle, fractureprone behavior and the soft, plastic flowprone behavior.
Stress State The state of stress of asphalt concrete modifies its stiffness. In an isotropic mixture, the stiffness depends upon the level of the first and the second deviatoric stress invariants I1 and J2′. In an anisotropic mixture, which is the usual case, the stiffness is directional and depends upon both the two stress state invariants just mentioned and the Asphalt Concrete Damage
Microcracking healing
te m p ra era te tur e
Fracture (Binder in tension)
in fre stra s s d re n St a
e
Strain rate
70
Plasticity flow (Aggregate properties important)
Temperature (Moisture)
FIGURE 31
Asphalt concrete damage dependence on temperature and loading rate.
Overview of the Stiffness Characterization of Asphalt Concrete
B C A
I1
D
J ′2
FIGURE 32
Loading and unloading stress path for an isotropic stress sensitive material.
components of the stress tensor. Because of the shape of the particles and the way that an asphalt concrete mixture is compacted, there is a vertical modulus, a horizontal modulus, and a shear modulus. There are also two Poisson’s ratios, one in the vertical plane and the other in the horizontal plane. Testing and analysis of triaxial test data have been able to determine all five of the moduli of this crossanisotropic case. The fact of crossanisotropy has several important implications for the way that pavements resist both fracture and plastic deformation. In a stresssensitive material like asphalt concrete, an elastic material is one in which loading and unloading follow a different stress path but one which closes upon itself as is illustrated schematically in Fig. 32. The figure is a graph of the first stress invariant plotted against the second deviatoric stress invariant and illustrates how the invariants during loading and unloading follow the path ABCDA. The elastic work potential for this kind of material is given by (Lade and Nelson 1989) W=
∫
ABCDA
dJ 2′ ⎞ ⎛ I l dI1 ⎜⎝ 9K + 2G ⎟⎠
(31)
where W = elastic work potential I1 = first invariant of the stress tensor J 2′ = second invariant of the deviatoric stress tensor K,G = bulk and shear modulus of the material If the elastic modulus of the material depends upon the same two stress invariants according to the equation, E = K1 ( I1 )K ( J 2′ )K 2
3
(32)
The requirement that the elastic work potential results in no net work produces a partial differential equation that Poisson’s ratio must satisfy (Lytton et al. 1993). −
k ⎞ ⎛ 1 k3 k ⎞ 2 ∂ν 1 ∂ν ⎛ 2 k3 + 22 ⎟ + =ν ⎜ + 22 ⎟ + ⎜ − I1 ∂I1 3 ∂J 2′ I1 ⎠ ⎝ 3 J 2′ I1 ⎠ ⎝ 3 J 2′
(33)
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Chapter Three
Computed Poisson’s ratio
1.2 1.0 0.8 0.6 0.4 0.4
FIGURE 33
Where
0.8 0.6 Measured Poisson’s ratio
1.0
1.2
Measured versus predicted stress dependent Poisson’s ratio.
n = Poisson’s ratio k2, k3 = coefficients which satisfy the boundary conditions of the partial differential equation
One of the implications of the solution of this differential equation is that if the modulus is stress dependent, then the Poisson’s ratio must be also. A graph of the Poisson’s ratio measured by Allen (1973) and predicted by the solution of Eq. (33) is shown in Fig. 33. The figure shows Poisson’s ratios that rise well above 0.5 which is the maximum that it can be in a material that has a constant elastic modulus. The measurement of Poisson’s ratios that are above 0.5 is a common observation in stressdependent materials such as asphalt concrete and unbound aggregate base course materials. It also varies with the frequency and direction of loading. A typical pattern is shown in Fig. 34. The tensile Poisson’s ratios remain below 0.5 while the compressive Poisson’s ratios rise above 0.5 once the loading frequency rises above about 1 Hz. Highway traffic loading is typically above 8 Hz and this means that an asphalt concrete layer, when loaded by traffic traveling at normal highway speeds, tries to expand laterally. When prevented from doing so, the asphalt concrete layer builds up a confining pressure that
1.0
Poisso’s ratio
72
0.1 0.5
0.2
2
res
si o
n
10
Tensio n
Intersection 0.0
FIGURE 34
1
p Com
Street
Frequency of loading, Hz
Poisson’s ratios of asphalt concrete.
Highway
20
Overview of the Stiffness Characterization of Asphalt Concrete Tire
Confinement pressure
Expansion
FIGURE 35
Effect of large compressive Poisson’s ratios.
stiffens the asphalt concrete, resists lateral plastic deformation, and presses closed any microcracks that may be growing in the asphalt. This is illustrated in Fig. 35. More recent work on crossanisotropic pavement materials has shown that the same phenomenon of large radial strains can be predicted using the elastic work potential shown in Eq. (34) without requiring either of the Poisson’s ratios to rise above 0.5 (Lytton 2000).
W=
where
∫
ABCDA
α
I1 dI + βτ zx dτ zx dJ 2′ 9 1 + Exx 2Gxy
(34)
1 − 4n − 2r m s β = 2 + 2r − m Eyy m= Exx n = nxy r = nxz E s = xx Gxy
α =2+
Exx = horizontal modulus Eyy = vertical modulus W = crossanisotropic elastic work potential It is still useful to use an “effective” Poisson’s ratio to describe the formation of large radial strains in a crossanisotropic elastic asphalt concrete. The determination of all five of the material properties of a crossanisotropic material cannot be determined from the measurements of the axial and radial stresses and strains of a triaxial test using the stressstrain relations of the material alone. Instead, it requires the use of an extra relation which solves for the shear modulus using the deviatoric strain energy that is measured during a test in which only the second invariant of the deviatoric stress tensor is applied (AduOsei 2000). Other methods using a constrained optimization approach to get a realistic estimate of the shear modulus have been suggested (Tutumluer and Seyhan 2002).
73
Chapter Three 100 e
Percent passing
74
lin
n
um xim
a
M
0
FIGURE 36
y sit
de
Large departure Moderate departure
Aggregate size
Effect of particle size distribution on “effective” poisson’s ratio.
Aggregate Particles The principal reason for the emergence of the crossanisotropic formulation of the stiffness of asphalt concrete is because of the shape of the aggregate particles. When they are compacted, the oblong particles tend to lie flat resulting in a modulus that is greater in the vertical direction than in the horizontal direction. In addition to the shape, the size and size distribution of the particles and the texture of the particles also have an effect upon the directional stiffness and the effective Poisson’s ratio of asphalt concrete. The graph in Fig. 36 shows qualitatively the effect of particle size distribution on the “effective” Poisson’s ratio of the mixture. The closer that the particle size distribution approaches the maximum density distribution line, the higher will be the “effective” Poisson’s ratio.
Asphalt Binder Properties The material properties of the asphalt binder which affect the stiffness of asphalt concrete are the compliance, the mastic film thickness, the aging, and the wetting and dewetting components of the surface energy. The compliance is a material response to a constant applied stress. When a stress is applied and held constant in a uniaxial test, the strain increases with the time after the load was applied. This timedependent strain divided by the constant stress is the creep compliance. A commonly used relation of the creep compliance to the loading time is given in the modified power law, which is given in Eq. (35) (Daniel and Kim 1998): D(t) = D0 +
D∞ − D0
τ0 ⎞ ⎛ ⎝1+ t ⎠
n
(35)
where D0 , D∞ are minimum and maximum compliance and t0, n are modified power law coefficient and exponent. A loglog graph of the modified power law is shown in Fig. 37. The curve begins with a nearly horizontal slope at short loading times and rises to approach a straight line in its midrange and finally can return to horizontal at long loading times as it
Overview of the Stiffness Characterization of Asphalt Concrete
Creep compliance Iog D(t)
D∞
n 1 Do
Do, D∞, to, n
FIGURE 37
Log (time)
Creep compliance.
approaches the asymptote of D∞. The slope n can never be greater than 1.0. The creep compliance is related to the relaxation modulus. This is determined in a uniaxial test in which a strain is imposed on the material and held constant. The stress in the material then relaxes as the time after the strain was first applied increases. The relaxation modulus is the timedependent stress divided by the constant strain. Chapter 6 deals exclusively with the relationship between this property and the creep compliance. At this point it simply needs to be known that the theory of linear viscoelasticity may be used to convert between any of the linear viscoelastic response functions as long as the measurements actually represent linear viscoelastic behavior.
Surface Energies The surface energies of the asphalt can be measured in a number of different test devices. The instrument which was used in measuring the surface energies that are tabulated in Table 31 were measured with the Wilhelmy plate apparatus. The apparatus is pictured in Fig. 123, and a detailed explanation of the testing procedure is given there. There are three components of surface energy: the nonpolar or Lifshitzvan der Waals, and the Lewis acid and Lewis base polar components. There is a hysteresis effect between the wetting and the dewetting surface energies. The wetting surface energies are associated with the healing of the microfractures in the asphalt while the dewetting surface energies are associated with the fracture of the asphalt. The total surface energy is made Surface Energy Components, ergs/cm2
Water
Wetting
Dewetting
Wetting
ΓLW
21.6
8.80
13.62
14.91
6.76
Γ–
25.5
1.50
18.87
1.74
15.28
Γ
25.5
2.81
10.52
1.07
15.03
Γ
51.0
4.13
28.18
2.64
30.30
Γ
72.6
12.93
41.80
17.55
37.06
+ AB TOTAL
TABLE 31
New Asphalt
Aged Asphalt
Wetting and Dewetting Surface Energies of Asphalt and Water
Dewetting
75
76
Chapter Three up of a sum of the nonpolar and the combined effect of the polar components. The relation is shown in Eq. (36) (Good and van Oss 1991): Γ = Γ LW + 2 Γ + Γ − where
(36)
Γ = total surface energy ΓLW = nonpolar Lifshitzvan der Waals surface energy Γ+ = acid component of the polar surface energy Γ− = basic component of the polar surface energy
In Table 31, the wetting and dewetting components and the total surface energies of new and aged asphalts are given along with the same components of water. Table 31 shows that as an asphalt ages, the surface energies change so as to reduce healing and to make fracture easier. Thus, the nonpolar portion of the wetting surface energies grows larger and the polar portions grow smaller as the asphalt ages. At the same time, both of the portions of the dewetting surface energies decrease so as to decrease the work of fracture with age. More detailed tables and figures illustrating these effects on the asphalt binder are presented in Chap. 12. Once a microcrack, or a crack, forms in the asphalt binder, the surface energies on each face of the crack interact to provide the cohesive bond strength against fracture and to provide the surface energy to promote healing. The computation of the cohesive bond strength on this interface, both dry and in the presence of water, is presented in detail in Chap. 12. The aggregate particles also have surface energy components which interact with the surface energies of the asphalt binder to produce the adhesive bonding strength at the interface between the two. The method used to measure the surface energies of the aggregates is the universal sorption device (USD) which is illustrated in Figs. 125 and 126. The USD is used to deposit vapor molecules on the surface of the aggregate particles in a vacuum. The accumulation of the mass of the vapor molecules on the particle surfaces at different levels of vapor pressure is used to calculate the specific surface area of the aggregate particles and to determine the wetting and dewetting components of the aggregate surface energies. The method of calculating the adhesive bond strength of an asphalt binder with an aggregate particle, both when the two surfaces are dry and when there is water present on the interface, is presented in Chap. 12. It is demonstrated there, and noted here for emphasis, that when water is on the interface between asphalt and an aggregate surface, the water acts to destroy the adhesive bond. The intensity of the action of the water varies greatly between various combinations of asphalt and aggregates as is demonstrated in Table 1216. This is the scientific basis for determining which combinations of asphalt and aggregate will strip and which will not. It explains why some aggregates will strip with some asphalt but not with others. There are two components of moisture damage to asphalt concrete stiffness: one due to soaking and the other due to repeated loading progressively opening adhesive debonding interface zones along the surface of the aggregates in the asphalt concrete mixture. The soaking damage depends upon the rate of moisture diffusion and the amount of water that the asphalt film can hold. The rate of moisture diffusion depends upon the relative vapor pressure in the immediate vicinity of each aggregate particle and the thickness of the mastic film surrounding the aggregate. Each asphalt has a unique water versus relative vapor pressure characteristic curve. Some asphalts hold more water at the same level of relative vapor pressure than others. Based upon measurements made to the present, this vapor pressure characteristic curve is the crucial
Overview of the Stiffness Characterization of Asphalt Concrete element in determining how much damage will be done to the asphalt concrete by moisture diffusion. Those asphalts which hold more moisture will experience more damage due to soaking. Also, based upon the results of recent measurements, those asphaltaggregate mixtures in which the work of the water in assisting the fracture of the interface is the greatest will be the ones that are most damaged by repeated loading in the presence of moisture.
Fines The fines in an asphalt concrete mix are all of the particles that are smaller than 0.075 mm. They make up around half of the volume of the mastic. The stiffness of the mix will be affected significantly by how well the fines bond with the asphalt binder, the size and size distribution of the fines, how well they are dispersed in the mastic, and their surface energy compatibility with the asphalt both with and without water. The size, size distribution, and dispersion all work together to arrest microcracks when they are small and easier to stop. Microcracks start out as a cloud of many small, dispersed flaws in the mastic. They grow as strain energy from repeated loading is made available to the mixture to extend the cracks. If any of these microcracks encounters a fine particle blocking its path, the particle will act as a crack arrester, and that microcrack will stop growing. If there are many welldispersed fine particles in the mastic, many of these microcracks will be arrested. One of the major effects of microcracks growing in an asphaltaggregate mixture is a progressive reduction of stiffness and one of the major effects of a welldispersed distribution of fines is to retain the stiffness, precisely because of its actions in arresting cracks. This also means that the fine particles must bond well with the asphalt, especially with water present. As discussed before, the adhesive bond strength between the two is determined by the surface energy characteristics of both the asphalt and the fine particles. The same can be said for additives that are expected to improve the stiffness, strength, and ductility of asphaltaggregate mix. Regardless of their composition, the particles of the additive must be small and well dispersed and must bond well with the asphalt in order to improve the mechanical properties of the mix.
Air Voids Air voids may be viewed with some accuracy as small particles with zero stiffness. Small, welldispersed air voids in the asphalt concrete mixture will provide several benefits to the mix, including acting as microcrack arresters and providing welldispersed volumes for the asphalt to expand into at high temperatures. Too much air will accelerate the growth of microcracks and too little air will cause bleeding (or flushing) and promote large plastic deformations. Too much air will also provide ready access of both air and water into the interior of an asphalt concrete layer and will accelerate aging and moisture damage. As with the fine particles, the air voids must be small and well dispersed in order to have its desired effect on the stiffness of the mixture.
Summary The stiffness of asphalt concrete depends upon numerous factors such as stress state, temperature, rate of loading, composition, and the mechanical and surface energy properties of the components of the mixture. It would be impossible to anticipate empirically how all of these factors will interact to provide an instantaneous value of
77
78
Chapter Three the asphalt concrete stiffness. It is fortunate that the disciplines of micromechanics and viscoelastic fracture mechanics have provided principles that may be used to determine how these factors will affect the mixture stiffness. An overview of these principles as applied to an asphalt concrete mixture is presented in the next section.
Characterization of Asphalt Concrete Stiffness Characterization of asphalt concrete stiffness refers to the mathematical relations between stress, strain, temperature, loading rates, moisture, and composition that are found in the disciplines of mechanics, including micromechanics and fracture mechanics. The stiffness of asphalt in an undamaged state is different from that in a damaged state and this difference is directly a result of the surface energy, adhesive and cohesive fracture and healing, plasticity and viscoplasticity, and moisture damage characteristics of the material. There is a large body of technical literature relating to the micromechanics of representing the properties of a composite material when the properties of the component materials are known. The discussion that follows will provide some of the high points that are relevant to the characterization of both damaged and undamaged asphalt concrete stiffness.
Undamaged Stiffness There are several approaches in micromechanics, all of which require that the strain energy that is put in to a composite material when it is being loaded is completely accounted for in being stored in the component materials of the composite. The same is true of energy that is released when the composite is unloaded. The objective is to arrive at a single stiffness that treats the composite as a homogeneous material but one that absorbs and releases strain energy in the same way that the composite does. Several of the principal micromechanics approaches give an upper and lower bound to this stiffness. One result of this mathematical process is the ratio of the shear modulus of the composite G* to the shear modulus of the matrix in which a solid inclusion is imbedded Gm. The relation is in Eq. (37) (Christensen 1991; Aboudi 1991):
G = 1− Gm
⎡ ⎛ G 15(1 − ν m ) ⎢ 1 − ⎜ i ⎝ Gm ⎣
⎞⎤ ⎟⎠ ⎥ ci ⎦ ⎛ Gi ⎞ 7 − 5ν m + 2(4 − 5ν m ) ⎜ ⎝ Gm ⎟⎠
(37)
where Gm , Gi = shear moduli of the matrix and the inclusion, respectively , nm = Poisson s ratio of the matrix ci = fraction of the total volume occupied by the inclusion Two wellknown results are when the inclusion is rigid and when it is a void, corresponding roughly to aggregate particles in an asphalt matrix and air voids in an asphalt medium. The rigid particle relation is in Eq. (38): G 5 = 1 + ci Gm 2
(38)
This relation was first published by Einstein (Einstein 1956). It has been used in a somewhat modified form in the Shell nomograph to estimate the stiffness of a mixture
Overview of the Stiffness Characterization of Asphalt Concrete from the stiffness of the bitumen (Heukelom and Klomp 1964, Van der Poel 1954). The air void relation is in Eq. (39): G 5 = 1 − ci Gm 2
(39)
It is possible to use these equations to get a rough estimate of how the shear stiffness of the asphalt concrete is related to the shear stiffness of the asphalt [Eq. (38)] or how air bubbles in the asphalt will alter the stiffness of the asphalt [Eq. (39)]. A corresponding relation for the bulk modulus of the composite is in Eq. (310) (Christensen 1991): K − Km (4Gm + 3K m ) = ci Ki − K m [4Gm + 3Ki + 3(K m − Ki )ci ] where
(310)
K = bulk modulus of the composite Km, Ki = bulk modulus of the matrix and inclusion, respectively Gm = shear modulus of the matrix material ci = volume fraction occupied by the inclusion
There are other, different relations predicting the bulk modulus of the composite that have been developed by using slightly different mathematical approaches, all of which have their limitations. Using the bulk and shear moduli of the composite as estimated from one of these formulas, it is possible to make further estimates of the Young’s modulus and Poisson’s ratio. Recognizing these limitations, the Method of Cells was developed to provide a numerical method for estimating these and other material properties, taking into account the one, two, or threedimensional geometry of the inclusion (Aboudi 1991). The numerical results are able to match measured results very closely. Other properties that are estimated by use of these micromechanics methods include the coefficient of thermal expansion, thermal conductivity, creep compliance and relaxation modulus, the timetemperature shift function, electrical conductivity and dielectric constant, and anisotropic yield strength of the composite among others (Christensen 1991). Because asphalt concrete stiffness is a viscoelastic, rather than an elastic property, it is frequently necessary to convert the viscoelastic properties of the components of the composite into the effective viscoelastic property of the composite. This can be done using the same micromechanics formulas and the correspondence principle which is illustrated in Eqs. (311), (312), and (313). The effective elastic bulk modulus formula in Eq. (310) is rewritten with bars over the modulus terms as in Eq. (311) (Christensen 1991): K − Km 4G m + 3K m = ci Ki − Km [4G m +3K i +3(K m − K i )ci ]
(311)
The meaning of the bar over either K or G is the Laplace transform of the relaxation modulus multiplied by s (Carson transform), the Laplace transform parameter in Eq. (312). ∞
K (s) = s ∫ K (t) e − st dt o
(312)
Equation (311) is solved for the Laplace transform of the composite K (s), and the entire expression is inverted to produce the effective bulk relaxation function of the composite K(t). A similar exercise will produce the effective shear relaxation function of the composite G(t). The inversion of this expression is usually done numerically although closed forms are possible using Schapery’s approximate inverse Laplace transform
79
80
Chapter Three method (Schapery 1962, 1965). Because there are different methods of estimating the bulk and shear moduli of elastic composite, it is usually necessary to verify that the conversion produces reliable results by comparison with actual measurements made on the composite. A similar transformation can be used to convert the complex moduli of the components into the effective complex moduli of the composite. In this way, the formula for the elastic bulk modulus of the composite is converted into the formula for the complex bulk modulus of the composite as in Eq. (313) (Christensen 1991). 4Gm* + 3K m* K * − K m* * * = ci * Ki − K m [4Gm + 3Ki* + 3(K m* − Ki* )ci ]
(313)
The K and G terms with the asterisks in this equation are the complex bulk and shear modulus of the matrix and the inclusion, all of which have a real and an imaginary component as in Eqs. (314a), (314b), and (314c) (Christensen 1991): K m* (ω ) = K m′ (ω ) + i K m′′ (ω )
(314a)
Ki* (ω ) = Ki′(ω ) + i Ki′′(ω )
(314b)
Gm* (ω ) = Gm′ (ω ) + i Gm′′ (ω )
(314c)
If the material is nonlinear viscoelastic, the equations given above must be treated as approximations, but they provide correct forms of equations that take into account the strainenergy storage of each of the components of the composite material.
Effects of Microcracks on Stiffness Because of the ability to use the correspondence principle to convert elastic solutions into viscoelastic equations for asphalt concrete stiffness, it is possible to derive relations using elastic theory with the confidence that they can be converted into the appropriate viscoelastic form, either the creep compliance, the relaxation modulus, the complex compliance, or the complex modulus. When a repeated load test is made on an asphalt concrete, its stiffness appears to decrease with increasing numbers of load applications. However, what is really happening is that small microcracks are forming in the material, producing an apparently smaller modulus. This is illustrated in Fig. 38 with two straps being subjected to the same tensile stress.
Microcrack size Actual E
Apparent E′
2c
Same stretch E′ < E
FIGURE 38
Effect of microcrack size on apparent modulus.
Overview of the Stiffness Characterization of Asphalt Concrete On the left is the actual strap with a crack in it of length 2c. On the right is an intact strap with an apparent elastic modulus E′. If it is required to find the modulus E′ that will store as much strain energy as the real strap on the left where the actual modulus E remains unchanged, the ratio between the apparent and the real modulus of the material is given by Eq. (315): E′ = E where
1 c2 ⎡ 8ΓE ⎤ 1 + 2π 1− 2 ⎥ bh ⎢⎣ σ c⎦
(315)
c = b, h = Γ= E, E′ =
half crack length width and length of the strap, respectively fracture surface energy of the material undamaged and apparently damaged mod dulus of the material, respectively σ = tensile stress applied to the strap
If the same strap is loaded repeatedly and the crack grows with each load repetition, the relation between the real and the apparent modulus is in Eq. (316): E′ = E
where
1 ⎡ c2 1 + 2π ⎢ 1 − 2Γ bh ⎢ ⎢ ⎣
1 ⎤ ⎛ 4At ⎞ 1+ n ⎥ ⎜ dW ⎟ ⎥ ⎜⎝ ⎟ dN ⎠ ⎥⎦
(316)
t = thickness of the strap A, n = ParisErdogan fracture law parameters dW = rate of change of dissipated strain eneergy per load cycle dN N N = number of load repetitions
If, instead of there being a single large crack of length 2c, there is a distribution of microcracks of various sizes with a crack density of (m/bh), the ratio between the apparent and the real modulus is given in Eq. (317): E′ = E
where
1 ⎡ 2 ⎛ m ⎞ 1 + 2π c ⎜ ⎟ ⎢ 1 − 2Γ ⎝ bh ⎠ ⎢ ⎢ ⎣
1 ⎤ ⎛ 4At ⎞ 1+ n ⎥ ⎜ dW ⎟ ⎥ ⎟ ⎥ ⎜⎝ dN ⎠ ⎦
(317)
m = number of microcracks in the strap (m/bh) = microcrack density c = mean crack size
Equations (316) and (317) both show that the apparent loss of stiffness depends upon the fracture properties of the material, A and n, the microcrack density (m/bh), the instantaneous rate of change of the dissipated strain energy per load cycle (dW/dN), and the average microcrack size c.
81
Chapter Three
FIGURE 39
Dissipated pseudostrain energy in a beam fatigue test.
If the material is viscoelastic instead of being elastic as in the previous examples, a certain amount of the energy that is expended with each load cycle is used up in overcoming the viscous resistance of the material and not in contributing to the damage of the material. In order to have the correct relation between the apparent and real relaxation moduli of the material, it is necessary to correct for the amount of dissipated energy that is not used directly in damaging the material. In order to make this correction, the concept of pseudostrain energy has been introduced. Although the pseudostrain concept will be presented in detail in Chap. 7, it is repeated here to compliment slight differences in its application to the work presented in this chapter. Pseudostrain energy is the amount of dissipated energy that is available to damage the material. A typical fatigue test on a beam fatigue sample is illustrated in Fig. 39. As the load is applied and then reduced it is necessary to apply compressive force in the opposite direction in order to return the beam to its original unstrained location. The graph of applied load versus deflection must be corrected to subtract the amount of energy that has been used to overcome the viscous resistance of the beam to upward and downward movement. This can be done by first finding out what the relaxation modulus of the beam material is by running a relaxation test on the material at a low stress level. Then the beam strain rate history is combined with the convolution integral to predict the linear viscoelastic stress history of the beam sLVE(t). If this calculated linear viscoelastic stress is plotted against the measured stress and a straight line such as shown in Fig. 310
Measured stress s
82
Linear x x x
Nonlinear
x x Calculated viscoelastic stress s LVE
FIGURE 310
Measured viscoelastic stress versus calculated linear viscoelastic stress.
Overview of the Stiffness Characterization of Asphalt Concrete
Measured stress s
Dissipated pseudostrain energy x x
x
x x
x x
x x
x x x
e R = s LVE/ER = “Pseudo’’ or reference strain
FIGURE 311
Illustration of dissipated pseudostrain energy.
is the result, it means that the material is behaving as a linear viscoelastic material. On the other hand, if a closed, curved figure such as is also shown in Fig. 310 is the result, then the material is behaving as a nonlinear viscoelastic material. If the calculated linear viscoelastic stress is divided by a reference modulus, the result has the dimensions of strain and is called pseudostrain. By this means, the graph of measured against calculated linear viscoelastic stress has been converted into a stress versus pseudostrain graph and the area within the closed, curved loop has been converted into dissipated pseudostrain energy, as illustrated in Fig. 311. This dissipated pseudostrain energy is the energy that has been lost in loading and unloading the material minus the energy that has been lost in overcoming the viscous resistance of the material, and therefore it represents the energy that is available to do damage to the material. However, if the material is loaded and unloaded repeatedly and the closed loop does not change its size or area, then that is an indication that the material is not changing and is not being damaged. Damage to the material is indicated by a change in the shape and area of the dissipated pseudostrain energy loop. Figure 312 illustrates a slight change in the dissipated pseudostrain energy loop.
Actual stress
Dissipated Pseudostrain Energy Change
Pseudostrain =
Dissipated pseudostrain energy
Linear viscoelastic stress (from strain history) Reference modulus
FIGURE 312
Change in dissipated pseudostrain energy indicating damage.
83
84
Chapter Three Changes this slight with one loading and unloading cycle can accumulate with repeated load cycles and the damage can grow accordingly. In order for the pseudostrain energy to match actual measured energy expenditures, the reference modulus ER must be a real modulus of the material which is found by dividing the maximum measured stress by the maximum measured strain. The rate of change of this dissipated pseudostrain area with respect to the change of microcrack surface area is defined as the pseudoJIntegral and is used in the fundamental law of fracture mechanics from which all fracture predictions are derived. This fundamental law was stated by Schapery (Schapery 1984) to be 2Γ = ER D(tα ) J R where
(318)
Γ = cohesive fracture surface energy of the material JR = pseudoJintegral D(ta) = compliance of the material ta = time required for the crack to grow through the distance of the fracture process zone which is of length, a ER = reference modulus of the material
Several examples of the application of this fundamental law of fracture are given in Chap. 12 and will not be repeated here. The law given in Eq. (318) is for cohesive fracture. There is a slightly different formulation for adhesive fracture which takes into account the interaction of the materials that are bonded at an interface, and even include the effect of a third material, such as water, which may be present on the interface. The calculation of the adhesive and cohesive bond strength from the individual surface energies of the component materials is also illustrated in Chap. 12. Thus the expression of the ratio between the damaged and undamaged modulus of a nonlinear viscoelastic material, taking into account the rate of change of dissipated pseudostrain energy in repeated loading is given in Eq. (319): ⎧ ⎡ E ⎛ m ⎞ ⎪ 2 ⎢ 1 − 2 Γ ⎛ 4 At = 1 + 2π ⎜ ⎟ ⎨ ∫ c ⎢ ⎜ dW ⎝ bh ⎠ ⎪ E′ R ⎜⎝ ⎢ dN ⎣ ⎩
1 ⎤ ⎫ ⎞ 1+ n ⎥ ⎪ ⎟ ⎥ p(c)dc ⎬ ⎪ ⎟⎠ ⎥ ⎦ ⎭
(319)
If a material that has been damaged by cracking is allowed time between repeated loads for the cracks to close and to reform the broken bonds, it will heal. Healing is a complementary process to fracture except that the polar and nonpolar components of wetting or healing surface energy play differing roles than they do in fracture. In fracture, they both resist the fracture. In healing, the stronger the polar wetting surface energies, the more they assist in forming healing bonds whereas the stronger the nonpolar surface energies of wetting or healing, the more they resist the reforming of the broken bonds. The nonpolar surface energies, which are primarily Lifshitzvan der Waals forces, affect the rate of short term healing, in the range of seconds. The polar surface energies, which are primarily hydrogen bonds, affect the longterm rate of healing in the range of minutes and hours. Thus, it is possible for a pavement to have microcracks and even large propagating shear cracks develop under repeated traffic loading, as illustrated in Fig. 313. But if the pavement is built of a good asphalt concrete which both resists fracture and heals well, it is possible to have the microcracks and even some of the propagating cracks to heal substantially and to recover much of the original strength and stiffness of the material during low traffic periods.
Overview of the Stiffness Characterization of Asphalt Concrete
Propagating
Microcracks
Shear cracks
FIGURE 313
Microcracks and propagating cracks under trafﬁc loading. Microcracking “Tertiary Creep”
Plastic pseudostrain e R
e 0R
N No. of load cycles
FIGURE 314 Plastic pseudostrain accumulation with repeated loading.
Another concern in the performance of asphalt concrete pavements is its plastic stiffness. A material which ruts substantially or at a high rate is undesirable. A typical graph of the plastic strain that an asphalt concrete undergoes under repeated loading is shown in Fig. 314. The rate of rutting is rapid initially and slows down as the asphalt work hardens and acquires lower plastic compliance, eventually approaching a horizontal asymptote at a plastic strain level of ε 0P . Occasionally, the asphalt concrete becomes too stiff and brittle and becomes susceptible to microcracking. At that point, the plastic strain curve begins to climb steeply as the asphalt concrete softens in accordance with the microcracking process described previously. The equation that describes this process of work—hardening and slowing the rate of increase of the plastic compliance of the material is in Eq. (320): ρ ⎛ ε P ⎞ −⎛⎜ ⎞⎟ DP (N ) = ⎜ 0 ⎟ e ⎝ N ⎠ ⎝ Δσ ⎠
(320)
where DP(N) = plastic compliance of the asphalt concrete which increases at a decreasing rate with repeated loading ε 0P = maximum or asymptote value of the plastic strain r, b = scale and shape factor of the plastic compliance curve Δs = repeated stress
85
86
Chapter Three The coefficient of this equation r is a scale factor and the exponent b is a shape factor which has a considerable physical significance. It measures the logarithmic rate of decrease of the plastic compliance of the asphalt concrete. The coefficient ε 0p is the asymptote that is approached by the accumulating plastic strain. If this asymptote plastic strain is divided by the repeated stress that causes it, it is the maximum plastic compliance of that material. The maximum plastic compliance is given in Eq. (321): D∞P =
ε 0P Δσ
(321)
The logarithmic rate of change of the plastic compliance D of the material is a constant b, as shown in Fig. 315. The figure also shows that as the plastic compliance gets smaller, it becomes stiff enough to promote the formation of microcracks. The curve departs from the straight line curving upward as microcracks grow and multiply, softening the material and allowing plastic deformation to accelerate. It can be shown mathematically that the exponent b is equal to the creep compliance exponent n if the rate of change of the plastic compliance DP(N) with respect to log(N) is a constant. Even if this rate of change is not constant, the slope b is closely approximated by the creep compliance exponent n. This exponent plays an important role in the fracture and healing of asphalt concrete, and it is also a close approximation of the logarithmic rate of accumulation of plastic strain in the same material. It also governs the rate at which the microcracks soften the asphalt concrete and accelerate the permanent deformation of the material in a process that has been called tertiary creep. This means that these two exponents are very important measures of the rate at which a pavement will deteriorate. It will be wise to devise ways of measuring these properties in the field in inservice pavements. Microcracking and plasticity are closely related phenomena. The first reduces the stiffness of asphalt concrete while the second increases it. Other types of damage, which reduce this stiffness, include moisture damage and aging, the latter of which makes the asphalt concrete more brittle and more susceptible to microcracking. The causes, measurement, and prediction of moisture damage are discussed in detail in Chap. 12.
Plastic deformation 1 log
∂ (log D p)
b
Microcracking
∂ (log N )
Iog N
FIGURE 315 Microcracking and plastic deformation.
Overview of the Stiffness Characterization of Asphalt Concrete
Conclusions The stiffness of asphalt concrete is a material property that is central to the performance of asphalt pavements. It depends upon many factors including stress state, temperature, moisture, strain rate, and damage condition. Being able to measure it precisely and accurately in the laboratory and the field is essential to making the design, construction, and management of pavements possible in the present and in the future. Subsequent chapters present methodologies and considerations for measuring the stiffness of asphalt concrete. A large part of the need for making these measurements is the greatly increased role that numerical predictions using mechanics models on computers will have on all engineering and construction operations related to pavements. Mechanics models require material properties of which asphalt concrete stiffness is one of the more important. The composition of the asphalt concrete determines its stiffness in both its damaged and undamaged conditions and also determines its response to the various stresses that are imposed upon it under traffic. Tiny variations of critical components of the mixture can have large scale effects on how the mix behaves under load and this is the reason that construction quality control will become an increasingly critical factor in the future. Defining in a mechanics sense how asphalt concrete stiffness is affected by all of these factors is what makes it possible to identify the critical components and to incorporate them in specifying the performance that those pavements will need to provide in order to meet taxpayer, safety, and public policy expectations.
References Aboudi, J. (1991), Mechanics of Composite Materials: A Unified Micromechanical Approach, Elsevier, New York. AduOsei, A. (2000), “Characterization of Unbound Granular Layers in Flexible Pavements,” Ph.D. dissertation, Texas A&M University, College Station, Tex., December. Allen, J. J. (1973), “The Effects of NonConstant Lateral Pressures on Resilient Response of Granular Materials,” Ph.D. dissertation, University of Illinois at UrbanaChampaign. Christensen, R. M. (1991), Mechanics of Composite Materials, Krieger Publishing Company, Malabar, Fla. Daniel, J. S., and Kim, Y. R. (1998), “Relationships among RateDependent Stiffnesses of Asphalt Concrete Using Laboratory and Field Test Methods,” Transportation Research Record No. 1630, Transportation Research Board, National Research Council, Washington, D.C., pp. 3–9. Einstein, A. (1956), Investigations of the Theory of Brownian Movement, Dover, New York. Findley, W. N., Lai, J. S., and Onaran, K. (1989), Creep and Relaxation of Nonlinear Viscoelastic Materials, Dover, New York. Good, R. J., and van Oss, C. J. (1991), “The Modern Theory of Contact Angles and the Hydrogen Bond Components of Surface Energies,” in Modern Approaches to Wettability (M. E. Schrader and G. Loeb, eds.), Plenum Press, New York. Heukelom, W., and Klomp, A. J. G. (1964), “Road Design and Dynamic Loading,” Proceedings, Association of Asphalt Paving Technologists, Ann Arbor, Mich. Lade, P. V., and Nelson, R. D. (1987), “Modeling the Elastic Behavior of Granular Materials,” International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 11, No. 5, pp. 521–542.
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Chapter Three Lytton, R. L., Uzan, J., Fernando, E. M., Roque, R., Hiltunen, D., and Stoffels, S. M. (1993), “Development and Validation of Performance Prediction Models and Specifications for Asphalt Binders and Paving Mixes,” SHRP A357 Report, National Research Council, Washington, D.C. Lytton, R. L. (2000), “Characterizing Asphalt Pavements for Performance,” Transportation Research Record No. 1723, Transportation Research Board, National Research Council, Washington, D.C., pp. 5–16. Schapery, R. A. (1962), “Approximate Methods of Transform Inversion for Viscoelastic Stress Analysis,” Proceedings, 4th U.S. National Congress of Applied Mechanics, p. 1075. Schapery, R. A. (1965), “A Method of Viscoelastic Stress Analysis Using Elastic Solutions,” Journal of the Franklin Institute, Vol. 279, No. 4, pp. 268–289. Schapery, R. A. (1984), “Correspondence Principles and a Generalized JIntegral for Large Deformation and Fracture Analysis of Viscoelastic Media,” International Journal of Fracture, Vol. 25, pp. 195–223. Tseng, K.H., and Lytton, R. L. (1989), “Prediction of Permanent Deformation in Flexible Pavement Materials,” in Implications of Aggregates in the Design, Construction, and Performance of Flexible Pavements, STP 106, ASTM, Philadelphia, Pa., pp. 154–172. Tutumluer, E., and Seyhan, U. (2002),“Characterization of CrossAnisotropic Aggregate Base Behavior from Stress Path Tests,” Proceedings, 15th ASCE Engineering Mechanics Division Conference, Columbia University, New York. Van der Poel, C. (1954), “A General System Describing the ViscoElastic Properties of Bitumens and Its Relation to Routine Test Data,” Shell Bitumen Reprint No. 9, Shell LaboratoriumKoninklijke, Amsterdam, Netherlands.
CHAPTER
4
Complex Modulus Characterization of Asphalt Concrete Terhi K. Pellinen
Abstract There are two complex modulus tests that have been used for characterization of asphalt mixtures in the United States: The dynamic modulus E∗ test and the dynamic shear modulus G∗ test, better known as the simple shear tester (SST) shear frequency sweep test. The dynamic modulus test has been selected to be used in the new AASHTO pavement design guide and it will replace the resilient modulus test currently used in pavement design. A new test protocol development for the dynamic modulus test has involved improving specimen instrumentation techniques and analysis methods for the test data. Research has shown that the modulus is less sensitive for test data imperfections than the phase angle when obtained using different signal analysis methods. However, a limit for feedback waveform for deviations of a perfect sine wave would improve data quality for both the modulus and phase angle. Comparison between uniaxial E∗ and SST shear modulus G∗ has shown that measured shear modulus was 2 to 30 times lower than theoretically would be expected, and deviations increased as the test temperature increased. Also, the phase angle values obtained from the SST testing were substantially higher than that of the uniaxial testing. It is assumed that instrumentation and specimen size problems in the SST testing are contributing to the differences. Also, differences in the specimen loading modes may have affected obtained parameter values. A new method of constructing mastercurves for cyclic modulus data has been developed that uses a sigmoidal fitting function and experimental shifting. The sigmoidal function approaches asymptotically to the limiting mix stiffness at cold and warm temperatures. At cold temperatures, the limiting stiffness is dependent of the glassy modulus of the binder and at warm temperatures it is dependent of the modulus of aggregate skeleton. Research has also shown that the stiffness correlates well with rutting and fatigue cracking and could be used as a simple performance test to complement the Superpave volumetric mix design method.
89 Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.
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Introduction Pavement design using the elastic layer theory needs two elastic parameters for each material layer used: Young’s modulus (stiffness) and Poisson’s ratio. One of the more widely used stiffness parameters for asphalt mixtures employed in mechanisticempirical structural pavement design procedures has been the dynamic modulus E∗. The dynamic modulus has also been selected to characterize the asphalt mixtures in the new AASHTO 2002 Guide for the Design of Pavement Structures, which has been in development in the NCHRP 137A project at Arizona State University (ASU). Additionally, the importance of dynamic modulus for mechanistic modeling will be discussed in other chapters in this book. Dynamic modulus will replace the resilient modulus test currently used for pavement design. This chapter discusses a new test protocol development for the dynamic modulus test and presents some considerations of the analysis of imperfect sinusoidal cyclic test data. Also, stiffness as a performance indicator for hot mix asphalt (HMA) will be discussed. A key feature in the material characterization is to construct a mastercurve of the mix. Through the mastercurve it is possible to integrate traffic speed, climatic effects, and aging for the pavement response and distress models. A new method to construct an asphalt mix mastercurve by using a sigmoidal fitting function and experimental shifting is discussed and a stressdependent mastercurve construction method is introduced. There are two complex modulus tests that have been used for characterization of asphalt mixtures in the United States: The dynamic modulus E∗ test and the shear modulus G∗ test. Differences of these two tests are discussed related to the mix design and pavement design applications.
Complex Modulus Complex mathematics gives a convenient tool to solve the viscoelastic behavior of the asphalt mixtures and binders in cyclic loading. The sinusoidal onedimensional loading can be represented by a complex form:
σ ∗ = σ 0 e iω t
(41)
ε ∗ = ε 0 e i(ω t−ϕ )
(42)
and the resulting strain
The axial complex modulus E∗(iw) is defined as the complex quantity
σ∗ ⎛σ ⎞ = E∗ (iω ) = ⎜ 0 ⎟ e iϕ = E1 + iE2 ε∗ ⎝ ε0 ⎠
(43)
in which σ0 is the stress amplitude, e0 is strain amplitude, and w is angular velocity, which is related to the frequency by
ω = 2π f
(44)
In the complex plane, the real part of the complex modulus E∗(iw) is called the storage or elastic modulus E1 while the imaginary part is the loss or viscous modulus E2, shown
Complex Modulus Characterization of Asphalt Concrete
FIGURE 41
Complex plane.
in Fig. 41. For elastic materials j = 0, and for viscous materials j = 90°. The alternative nomenclature is to call the storage modulus as E′ and loss modulus as E″. If a linearly viscoelastic material is subjected to a uniaxial compressive, tensile, or shear loading, the resulting steadystate strain ε = ε 0 sin(ω t − ϕ ) will be out of phase with the stress by the lag angle j, as shown in Fig. 42. The ratios of stress and strain amplitudes s0/e0 define the dynamic (or cyclic1) modulus E∗(w), shown in Eq. (45): E* (ω ) = E12 + E2 2 =
σ0 ε0
(45)
where E1 and E2 can be expressed as a function of phase lag or lag angle E1 =
σ 0 cos ϕ ε0
E2 =
and
σ 0 sin ϕ ε0
(46)
The loss tangent defines the ratio of lost and stored energy in a cyclic deformation: tan ϕ =
E2 E1
(47)
Figure 41 shows that the quantity dynamic modulus presents a magnitude, that is, length of the complex modulus vector E∗ in a complex plane. It should be noted that since the test can be done using either normal or shear stress, the norm of the complex modulus can be defined either E∗ or G∗. In applied viscoelasticity for the asphaltic
FIGURE 42
Sinusoidal stress and strain in cyclic loading.
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FIGURE 43 E∗ and j shown (a) in complex plane and (b) in the Black space. (Pellinen et al. 2002, ASCE.)
materials, the phase angle is usually denoted by j or f for the mix and d for the binder. The complex plane also called Cole and Cole plane (Di Benedetto and de la Roche 1998), or the Black space, can be used to check the quality of test data. In complex plane, the storage modulus E1 is plotted to the real axis (xaxis), and the loss modulus E2 is plotted to the imaginary axis (yaxis). Figure 43(a) presents an example of the dynamic modulus test results in the complex plane that allows assessment of data at intermediate and low temperatures. The plotted complex modulus E∗ points on a complex plane should form one unique curve, which is independent of frequency or temperature. In Black space, the modulus values are plotted in log space and phase angle values in arithmetic space, which gives better assessment of the data at high temperatures, as shown in Fig. 43(b). Similar to complex plane, the Black space shows the frequency and temperature independent relation of the complex modulus and phase angle. It also allows one to estimate the pure elastic component E(j = 0) of the complex modulus at very low temperatures. The Black space diagram has been shown in the literature with either parameter E∗ or j in the xaxis.
Test Protocols There are two complex modulus tests that have been used for material characterization of asphalt mixtures in the United States; the dynamic modulus E∗ test, and the shear modulus G∗ test, better known as the simple shear tester (SST) shear frequency sweep test. There are other test configurations and specimen geometry that can be used to obtain the complex modulus of the mix, such as bending tests using beams and various shearing tests (Di Benedetto et al. 2001). Some of the tests are homogeneous, that is, tests have direct access to stress and strain and therefore to the constitutive law. Some of the tests are nonhomogeneous, that is, they call for postulating the constitutive law first (such as linear elasticity), and specimen geometry needs to be taken into account to get parameters to the constitutive law. This chapter concentrates on the axial dynamic modulus and SST shear modulus tests although some of the data analysis and instrumentation issues can be applied to any cyclic testing. Both of these tests are homogeneous tests.
Complex Modulus Characterization of Asphalt Concrete
FIGURE 44
Schematic of shear frequency sweep test.
SSTShear Frequency Sweep Test The shear frequency sweep test conducted with the SST was developed in the SHRP research program. The test protocol was first introduced as SHRP Designation M003: “Standard Method of Test for Determining the Shear Stiffness Behavior of Modified and Unmodified Hot Mix Asphalt with Superpave Shear Test Device” (Harrigan et al. 1994). Later the test protocol was adopted by the American Association of State Highway and Transportation Officials (AASHTO) as a Provisional Standard: AASHTO Designation: TP794 (AASHTO 1994). The shear frequency sweep at constant height is a straincontrolled test; the maximum shear strain is limited to 100 microstrains. During the test, a horizontal shear strain is applied at a frequency of 10 to 0.01 Hz using a sinusoidal straining pattern. At the same time the specimen height is kept constant by compressing or pulling the specimen axially based on the closed loop feedback given by the vertical LVDTs (linear variable differential transformer) attached to the sides of the specimen. The specimen is sheared from the bottom as Fig. 44 schematically presents. The cylindrical test specimen diameter is 150 mm, height 50 mm and it is glued between two aluminum platens. Based on the TP7 protocol, testing is conducted at 4, 20, and 40°C, but higher temperatures have been used. The strain control mode in both actuators makes the test difficult to run and very soft mixtures can cause severe control problems at high temperatures. Also, test temperatures are limited above 4°C because at colder temperatures mix stiffness may exceed the stiffness of the glue and specimens may shear off the platens.
Compressive Dynamic Modulus Test Background The complex dynamic modulus test was originally adopted in 1979 by the American Society of Testing and Materials (ASTM) as a standard method: “Test Method for Dynamic Modulus of Asphalt Concrete Mixtures” ASTM D 349779 (ASTM 1979). The test was conducted applying a haversine load between 0 and 241 kPa using temperatures of 5, 25, and 40°C and frequencies of 1, 4, and 16 Hz. Additional temperatures and frequencies have been added to the protocol by Witczak et al. at University of Maryland (UMd) (Witczak et al. 1996; Witczak and Kaloush 1998; Pellinen and Witczak 1998) to enable construction of a full mastercurve
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FIGURE 45
Loading pattern for compressive dynamic modulus testing.
of the asphalt mix. They used test temperatures ranging from 15 to 55°C and frequencies from 25 to 0.1 Hz. In the original ASTM protocol, capped cylindrical specimens, having a height to diameter ratio of ≥2, are loaded with a uniaxial compressive haversine stress pattern shown in Fig. 45. Testing is performed in load control keeping the load constant throughout the frequency sweep. However, load levels are decreased as a function of test temperature to keep measured strain levels small and to achieve steadystate straining pattern after the initial creep, that is, recoverable and nonrecoverable (permanent) strain accumulation, of the specimen. This same loading approach was used by Witczak et al. in their testing program at UMd, where they created a large database that was used to develop the predictive dynamic modulus equation (latest form by Andrei et al. 1999), which will be used in the new pavement design guide as an initial approximation of the stiffness of asphalt mixtures. In recent years, the simple performance test and the new pavement design guide work has led to a new research and protocol development for the dynamic modulus test. The next sections discuss protocol issues that should be addressed when conducting cyclic sinusoidal testing.
Specimen Fabrication and Instrumentation Different instrumentation techniques have evolved in the testing over the years. In the early development of the test, strain gauges were used to measure displacements, whereas today, the deformations are measured using springloaded LVDTs. In the early approach used by Witczak et al. at UMd, these LVDTs were clamped vertically on diametrically opposite specimen sides. A new method of specimen instrumentation has been developed by the Superpave Models Management research team (Witczak et al. 2000). Axial deformations are measured with LVDTs mounted between gauge points glued to the specimen, as shown schematically in Fig. 46. LVDTs are secured in place using brackets screwed to the studs. In addition, guiding rods, which prevent the LVDTs from bulging out at high temperatures, were added to the instrumentation. This instrumentation allows the use of membrane around the specimen for confined testing. It also prevents any constraining that the clamp around the specimen may have at higher temperatures. The deformations are measured at a minimum of two locations 180° apart; however, three locations located 120° apart are recommended to minimize the number of replicate specimens required for testing. This new approach has been found to be mandatory for asphalt specimens that exhibit large nonrecoverable deformation during testing. Compaction methods for fabricating test specimens in the laboratory have varied over the years. The early compaction methods included kneading compactor and rolling
Complex Modulus Characterization of Asphalt Concrete
FIGURE 46 General schematic of gauge points (not to scale). (Witczak et al. 2000, with permission from Association of Asphalt Paving Technologists.)
wheel compaction method. In the Superpave volumetric mix design method, the specimens are compacted using Superpave Gyratory Compactor (SGC), which is also used for compacting test specimens for the simple performance testing. In addition, the SGC will be used to compact test specimens for the new pavement design guide material characterization testing. The standard protocol for laboratory determination of asphalt concrete stiffness dictates that the specimen size and boundary conditions produce a homogeneous stress distribution in the test specimen. A study by Witczak et al. (2000) determined the minimum test specimen dimensions that provide measured responses and material properties that are independent of the test specimen size (end effects) and aggregate size using gyratory compacted laboratory test specimens. The experiment included four heighttodiameter ratios being 1, 1.5, 2, and 3, and three specimen diameters being 70, 100, and 150 mm. The specimen diameter was also used as gauge length for the LVDTs. Testing was conducted with three mixtures with nominal aggregate sizes of 12.5, 19.0, and 37.5 mm. For the dynamic modulus test, the 70mmdiameter specimens with heighttodiameter ratio of 1.5 or greater gave acceptable test results when tested at 4 and 40°C. The specimen size of 100 mm in diameter and 150 mm in height has been selected for the new dynamic modulus test protocol development at ASU. The gauge length used is 100 mm. After compaction, the 100mmdiameter cylindrical test specimens are cored from the center of the 150mmdiameter gyratory specimens. If the desired specimen height is 150 mm, the gyratory specimen has to be compacted to a height of at least 170 mm in the gyratory mold. This gives 10 mm of each specimen end that can be sawed to obtain smooth ends perpendicular to the axis of the specimen. A recommended tolerance for perpendicular ends is 0.05 mm across any diameter, which can be checked using a straight edge and feeler gauges. The specimen end should not depart from perpendicular to the axis of the specimen by more than 0.5°. The ASTM protocol required capping specimens with sulfur mortar to ensure parallel specimen ends to prevent any misalignment and rocking during testing. Witczak et al. (2000) have recommended that specimens should not be capped and frictionreducing treatments should be placed between the specimen ends and the platens (hardened steel disks). This treatment is needed to have homogenous stress distribution in the specimens by avoiding any restraints capping may introduce. The end treatment consists of two 0.5mmthick latex sheets separated with silicone or vacuum grease.
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Chapter Four Control Mode In the original ASTM protocol, the dynamic modulus test was conducted in load control by applying constant compressive cyclic loading throughout the frequency sweep. This same approach was planned to be adapted to the new test protocol. However, the applied constant stress causes the subsequent resilient strain to increase as frequencies decrease in the frequency sweep due to the viscoelastic nature of the mix. The proposed new test protocol considers 150 microstrains as the limit for the linear viscoelastic region. However, at warm temperatures the lowfrequency strain can be up to three times larger than strain at high frequencies, when constant stress is used. Then, at high temperatures the magnitude of the strain may exceed 150 microstrains at low frequencies although the strain at high frequencies stays under 50 microstrains. This may cause some damage accumulation in the specimen, in addition to a creep imposed to the cyclic straining pattern. This needs to be accounted for in the data analysis when obtaining modulus and phase angle values. Therefore, some European researchers (Di Benedetto and de la Roche 1998; Doubbaneh 1995) prefer to do the cyclic testing in strain control mode by applying tensioncompression loading. However, specimens need to be glued to the platens to apply tension. This loading model gives zero average stress, which enables the steadystate straining pattern and, thus, eliminates the creep. The strain amplitude can also be controlled to stay below desired limit during testing. In pure tension testing, strain keeps creeping as a function of time without achieving steadystate pattern. A rest period between frequencies in the frequency sweep is not discussed in the proposed new test protocol. However, some of the controllers used in the past and perhaps some current ones cannot produce continuous frequency sweeps; each frequency must be programmed separately in such a manner that there will be some lag time or rest period between each frequency. The rest period helps to prevent specimen from heating during cyclic testing, although for modulus tests the number of cycles is usually limited to fewer than 200 and heat increase may not be a problem. However, the rest period allows some of the transient strains to recover during testing which may have some effect on the measured modulus values and selection of suitable data analysis methods.
Rest Period in the Frequency Sweep
Compressive Axial versus SSTShear Modulus Correlation of Modulus and Phase Angle Values A study conducted by Witczak et al. (2000) compared SST frequency sweep and uniaxial dynamic modulus tests and concluded that the shear modulus value G∗ obtained from the SST test was not a true fundamental property such as E∗ due to the instrumentation and specimen size problems. Thus, according to Witczak et al., test results from the SST device can be categorized as representing an index value of shear modulus of the asphalt mixture, not a fundamental material property. This means that the G∗ cannot be used directly to replace the E∗ in the pavement design applications where mixture stiffness is needed. Equation (48) gives a conversion model developed by Christensen, Pellinen, and Bonaquist (2003), which can be used to convert E∗ to G∗. Both moduli values have units of psi in the equation: G∗ = 0.0603E∗ 1.0887
R 2 = 0.93
(48)
Complex Modulus Characterization of Asphalt Concrete A study by Pellinen and Witczak (2002a) suggests similar results. The overall linear correlation between measured dynamic modulus E∗ and SSTshear modulus G∗ values within the linear viscoelastic region for the various dense graded mixtures was R2 = 0.87. Figure 47(a) shows that the deviations in test results are greater at low and hightemperature extremes, greatly exceeding the stiffness ratio of E∗ ≈ 2.5 to 3G∗, which is based on theoretical equivalency using Poisson’s ratio values from 0.2 to 0.5. The overall linear correlation of the phase angle values from dynamic modulus and SSTshear modulus measurements was R2 = 0.61 and, at best, is indicative of a fair correlation. More importantly, it can be observed from Fig. 47(b) that there is no direct
FIGURE 47 Correlation of axial and shear (a) modulus and (b) phase angle. (Pellinen and Witczak 2002a, with permission from National Academies Press.)
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Chapter Four agreement between the phase angles measured in axial and shear conditions. The SST device measured phase angle values ranging from 12 to 70°, while the axial testing gave values ranging between 12 and 42°. It can be hypothesized that two possibilities contribute to these findings: the SST device may have control problems at very high and low test temperatures, or the difference in the specimen loading mode may contribute to the different material responses. In compressive axial loading, the average stress amplitude is always larger than zero, but in the SST device, the average shear strain and stress amplitude are zero due to the shear straining through zero, as Fig. 44 shows. This may cause compressive testing to be more influenced by the aggregate skeleton effect (which is elastic) in the higher test temperatures, causing lower phase angle values and higher modulus values. Also material inhomogeneity and anisotropy may cause some differences.
Stiffness as the Asphalt Mix Performance Indicator Introduction A new mix design procedure, the Superpave volumetric mix design, was developed in the Strategic Highway Research Program (SHRP) in the mid1990s. However, unlike the Marshall mix design method, the new Superpave volumetric mix design procedure did not include any mechanical test to check the mixture performance after the volumetric part of the design procedure had been completed. Experience from the implementation process over recent years has shown that the volumetric mix design procedure without a performance test is inadequate for ensuring acceptable mix performance. Work leading to the development of a simple performance test has been in progress through the NCHRP 919 project: “Superpave Support and Performance Models Management,” TaskC at ASU (Pellinen and Witczak, 2002a). The focus of the proposed simple performance test (SPT) has been to measure a fundamental engineering material property that can be linked back to the advanced material characterization measurements that are needed for a detailed distress analysis. The three main asphalt mixture distresses considered in the design process are permanent deformation, fatigue cracking, and thermal cracking. The main objective of the overall NCHRP 919 TaskC research effort was to recommend from several candidate tests the most promising fundamental SPT for use with the Superpave volumetric mix design procedure. The potential simple performance tests that have been studied can be categorized as stiffnessrelated tests, deformability tests, and cracking tests.
Stiffnessrelated Tests: Recommendations for SPT The laboratory test program for stiffnessrelated test by Pellinen (2001) and Pellinen and Witczak (2002a) included duplicating mixtures from three different experimental test sites in the United States. These sites were the MnRoad, FHWAALF, and WesTrack sites. All mixtures from these sites were dense graded mixtures. However, two stone mastic asphalt (SMA) mixtures and two dense graded mixtures from the Finnish Asphalt Pavements Research Project (ASTO) were also studied. ASTO project was conducted from 1987 to 1992 including comprehensive laboratory and field research effort on Finnish asphalt mixtures (Saarela, 1993). The ASTO test specimens were fabricated by the Technical Research Center of Finland and tested at ASU. Testing was conducted using two replicate specimens. Specimen instrumentation was as shown in Fig. 46. Each specimen was tested in an increasing order of temperature
Complex Modulus Characterization of Asphalt Concrete using dynamic stress levels of 138 to 965 kPa for colder temperatures of −9, 4.4, and 21.1°C. For warmer temperatures of 37.8 and 54.4°C, stress levels of 46–68 kPa and about 21 kPa were used, respectively. For each temperature level, specimens were tested in a decreasing order of frequency; frequencies used were 25, 10, 5, 1, 0.5, and 0.1 Hz. A 60second rest period was used between each frequency to allow some specimen recovery before applying new loading at lower frequency. This was done to reduce the possible damage and heat accumulation during testing. This testing was conducted trying to stay under 150 microstrains at all temperature and frequency levels. For the confined testing program, the testing frame was equipped with a triaxial cell capable of applying cell pressure up to 690 kPa. Hightemperature stress levels were determined based upon the stress to strength ratios determined using cohesion and friction parameters from the triaxial strength test at 54.4°C. At colder temperatures, the deviatoric stress was changed depending upon the stiffness of the mix. A 60second rest period was used between each frequency similar to the unconfined testing. The instrumentation and testing setup is illustrated in two photos for confined dynamic modulus testing. Figure 48(a) demonstrates the loading of a specimen to the pressure cell, and Fig. 48(b) shows the pressure cell sealed and ready for testing. Based on the research, a provisional recommendation for the stiffness parameter for rutting was the dynamic modulus E∗ of the mix. For dense graded mixtures, the E∗ parameter could be obtained at an unconfined stress state using small stress levels to stay in the linear viscoelastic region. The provisional recommended test temperature was 54.4°C and the frequency was 5 Hz. These were the conditions under which all mixtures exhibited the best correlation to rutting. Timetemperature superposition principle can be used to translate test results to the desired climatic conditions and traffic speed for the performance criteria. The analysis of the Finnish SMA mixtures indicated that confinement is needed to capture performance of SMA mixtures compared to dense graded mixtures. Therefore, confinement was recommended for testing of SMA and opengraded mixtures. However, the level of confinement needs to be determined for mixtures constructed in the United States. In addition to the axial stiffness, the SST shear modulus G∗ was also included in the research program. Testing was conducted by Advanced Asphalt Technologies. The overall correlation to rutting was the same for both E∗ and G∗ values, the correlation coefficient being 0.79. However, the order or ranking of the mixtures was different. This indicates that these two modulus values are not interchangeable, as discussed above.
(a)
FIGURE 48
Conﬁned E∗ test set up.
(b)
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100
Chapter Four Most likely mixtures that have binder as a dominating stiffness contributor will be ranked similarly, while mixtures that have different gradations, such as dense graded versus SMA, may be ranked differently. The recommended provisional stiffness parameter for fatigue cracking was also found to be the unconfined compressive dynamic modulus of the mix. However, this conclusion is based on limited data. The dynamic modulus of asphalt mixtures was not an adequate performance indicator for thermal cracking.
E∗ versus E∗/sinj The Superpave binder specification defines and places requirements on a rutting factor, G∗/sind, which presents a measure of the hightemperature stiffness or rutting resistance of asphalt binder. G∗ is a shear modulus of binder and d is the phase lag between stress and strain. According to Bahia and Anderson (1995), the work W dissipated per loading cycle is inversely proportional to the parameter G∗/sind: 1 ⎡ ⎤ W = πσ2 ⎢ * ⎥⎦  /sin G δ ⎣
(49)
To minimize permanent deformation, the work dissipated during each load cycle should be minimized. In a similar manner, a rutting factor, E∗/sinj, can be defined for asphalt mixtures, where j is the phase angle of the mix. For fatigue cracking, a performance factor in the Superpave binder specification is G∗sind. Thus, the equivalent performance factor for the mix is E∗sinj. The Superpave binder specification has a minimum value for the G∗/sind parameter against rutting, and a maximum value for the G∗sind for fatigue cracking. The correlation between rutting and rutting factor E∗/sinj was R2 = 0.91, and between rutting and G∗/sinj only R2 = 0.74, although the modulus itself gave similar correlations. As discussed earlier, Fig. 47(b), the correlation between phase angles of the axial and shear modulus was poor to fair, which may explain the poorer correlation of G∗/sinj and rutting. However, it should be noted that these results apply only to the dense graded mixtures used in the testing program. The reason that the E∗/sinj was not recommended as the SPT, regardless that it gave better correlation to rutting than modulus itself, is that the phase angle of the asphalt mix is dependent of frequency and temperature differently than the phase angle of the conventional binder. For conventional binders, the phase angle is an increasing function of temperature, while the asphalt mix phase angle first increases as temperature increases and then starts to decrease. This is illustrated in Fig. 49 that shows binder and mix data in the Black space. The binder phase angle approaches 90° at high temperatures when the mix phase angle approaches some limiting value initiated by the damping of aggregate skeleton. This occurs because the elastic effect of aggregate skeleton pushes through the viscous effect of binder at high temperatures. Therefore, if the phase angle decreases it may be due to more elastic binder or more viscous binder that allows the elastic effect of aggregate skeleton to influence the phase angle value. Therefore E∗/sinj is not a stable performance parameter for asphalt mixtures. This same phenomenon can also be seen for some of the modified binders for which the phase angle is not an increasing function of temperature.
Effect of Confinement and High Stress Levels One potential limitation of using stiffness measurements as a performance indicator at low stress/strain test applications is that the effect of aggregate shape and hence internal
Complex Modulus Characterization of Asphalt Concrete
FIGURE 49
Binder and mix behavior in the Black space.
friction may not be accounted for adequately enough. To assess this possibility, axial testing was conducted employing both low and high levels of deviatoric stress with different levels of confinement. It was hypothesized that high stress levels with confinement would mobilize the internal friction in the mixtures. However, confinement (138 to 206 kPa) or high stress levels (up to 552 kPa) did not improve the correlation to rutting.
Analysis of Cyclic Sinusoidal Test Data It is often difficult to obtain a perfectly sinusoidal feedback signal from the highfrequency testing due to the test equipment limitations and operator errors. If the feedback signal is not a perfect sine wave, it is noisy, or if there is transient recoverable and permanent deformation imposed over the sinusoidal signal, the computed modulus and phase angle values may differ depending on the method used for filtering and phase referencing the signal. Fast Fourier Transform is one of the filtering methods that can be used to process the stress and strain signals. Also, different regression techniques have been used to smooth the data.
Imperfections in the Cyclic Test Data Figure 410 shows some examples of various dynamic modulus test data imperfections in cyclic testing by Pellinen and Crockford (2003). Figure 410(a) shows data where the applied load signal (stress) is slightly skewed to the left of the strain signals denoted as Axial1 and Axial2 in the legend. This imperfection causes a large deviation (linear regression standard error of 8.9% from a perfect sine wave) as Table 41 later on shows. Figures 410(b) and 410(d) show fairly good data, which is creeping due to the transient recoverable and nonrecoverable deformation. Linear regression standard error for load from perfect sine wave is between 1.5% and 2.1%, and between 2.8% and 6.5% for displacement. Figure 410(c) shows noisy displacement and load data in which standard error for load is 11.6% and for displacement 14.5%. Figure 410(e) shows load data that is skewed left and displacement transducers that are deviating from each other in large amounts. Standard error is 8.3% for load and 9% for displacement. Figure 410(f) shows fairly good load data signal and somewhat noisy displacement signals, standard error for load is 3.7% and for average displacement 6.2%.
101
102
Chapter Four
FIGURE 410 Examples of different imperfections of the sinusoidal test data. (Pellinen and Crockford 2003, with permission from RILEM Publications.)
Fast Fourier Transform The following short introduction of signal processing and Fast Fourier Transform (FFT) has been summarized from the Fundamentals of Signal Analysis by Hewlett Packard (1989) and from a paper by Ramsey (1975). The FFT is an algorithm for transforming data from time domain to frequency domain. A finite number of data points need to be sampled using discrete intervals of time separated by Δt. Then, time record is N equally spaced samples of input signal and the requirement for the Fast Fourier analysis is that the dataset have exactly N = 2m data
Complex Modulus Characterization of Asphalt Concrete
Standard Error for Load, %
Avg. Standard Error for Displ., %
Hz Temp.
25
Hz 10
5
1
0.5
0.1
25
10
5
1
0.5
0.1
9.6
14.4
5.8
3.8
10.4
3.3
−9°C
8.9
6.0
3.4
1.2
0.8
0.6
4.4°C
8.6
3.7
2.3
1.2
1.0
1.5
8.7
4.9
4.0
3.1
3.0
2.8
21.1°C
9.5
5.6
3.6
2.4
2.0
2.2
12.0
9.5
3.9
3.1
6.9
6.5
37.8°C
11.6
7.3
5.5
3.2
3.1
2.1
14.5
7.6
6.4
6.8
7.4
8.4
54.4°C
8.3
9.1
6.8
5.3
4.4
3.7
10.0
9.0
7.2
6.8
6.1
6.2
Average
9.4
6.3
4.3
2.7
2.3
2.0
11.0
9.1
5.5
4.7
6.8
5.4
St.Dev.
1.3
2.0
1.8
1.7
1.5
1.1
2.3
3.5
1.5
1.9
2.7
2.3
Source: Pellinen and Crockford 2003, with permission from RILEM Publications.
TABLE 41
Standard Error of Estimate for Deviation of Sine Wave
points (m > 2). To completely describe a given frequency, two values are required: the magnitude and phase, or real part and imaginary part. Consequently, N points in timedomain can yield N/2 complex quantities in the frequency domain. The linear Fourier spectrum is a complex valued function that results from the Fourier transform of a time waveform. A System Transfer Function H(f ) is used to transfer the input data x(t) and output data y(t) to frequency domain Sx(f ) and Sy(f ), where Sx and Sy are linear Fourier spectrums of x(t) and y(t). Thus, Sx and Sy have real (in phase or coincident) and imaginary (quadrature) parts, respectively. In general, the result of any continuous linear system on any time domain input signal x(t) may be determined from the convolution of the system impulse response h(t), with the input signal x(t), to give the output y(t). y(t) =
∫
∞
−∞
h(τ )(t − τ )dτ
(410)
Applying Fourier transform to the convolution integral Sy ( f ) = Sx ( f )H ( f )
(411)
the transfer function H can be defined H=
OUTPUT Sy = INPUT Sx
(412)
Power spectrum of the input x(t) is defined as Gxx = SxSx∗, where Sx∗ is complex conjugate of Sx. Power spectrum of output y(t) is defined as Gyy = SySy∗, where Sy∗ is a complex conjugate of Sy. Gross power spectrum is Gyx = SySx∗ and it contains the phase information. Then, the transfer function H, which can be applied to any waveform, can be defined as: H=
Sy Sx* Gyx ⋅ = Sx Sx* Gxx
(413)
103
104
Chapter Four The quality of transfer function determines if the system output is totally caused by the system input. Noise and/or nonlinear effects can cause large errors at various frequencies, thus including errors when estimating the transfer function. Coherence function g 2 can be used to estimate the quality of system, where
γ2 =
Response power caused by applied input Measuredd response power
γ = 2
Gyx
(414)
2
GxxGyy
where 0 ≤ g 2 ≤ 1
(415)
If g 2 = 1 at any specific frequency, the system is said to have perfect causality at that frequency. If g 2 < 1, then extraneous noise is also contributing to the output power.
Differences in Employed Analysis Methods: Time Domain Methods A study by Pellinen and Crockford (2003) compared three different filtering methods and two different phase referencing methods of computing modulus and phase angle from compressive dynamic modulus test data. The methods discussed were limited to time domain techniques (other than FFT method) applied to cyclic loading in compression. Study showed that the computed modulus values were less sensitive to different analysis techniques than the phase angles. Dynamic modulus test data of a dense graded asphalt mix obtained at five different temperatures and six different frequencies were analyzed in this study. This type of a data set is needed to construct a mastercurve of the mix required in the new pavement design guide. Testing was conducted as a stress controlled test applying compressive stress with a frequency sweep of 25, 10, 5, 1, 0.5, and 0.1 Hz. The testing system capable of maximum sampling rates of 1 kHz and sampling rates adequate to eliminate aliasing at the frequency of loading were used. Additional firmware and sampling techniques were employed to minimize skewing of sequential samples and noise in the incoming signal. In the presence of the noise remaining after firmware filtering, the peaks of the waveforms generally exhibit the largest noise amplitude in a single cycle and are therefore the worst locations to perform analyses that determine phase angles. Greater noise at the peaks is due in part to the test machine actuator and the transducers reversing their direction of movement at those times. It is assumed that it is much better to determine phase angle from the “middle” of the waveform where the machine and transducers are all moving in a relatively “steady state.” Using unfiltered peaks can easily create phase shifts that are due to noise instead of fundamental signal peaks. The modulus must be computed from the peaks, so additional software filtering is usually performed to improve the peak measurement. This filtering must be carefully done so that time skewing and alteration of the fundamental signal magnitude are minimized. When any type of cyclic forcing function is applied to a material such as asphalt concrete under load control, a strain response that mirrors the forcing function but with different amplitude and a phase shift is expected. This is an oversimplification even in the case of strictly compressive loading: • Even if the forcing function’s wave shape is perfect, there is a nonzero average stress level during the cyclic loading which causes the cyclic strain response curve to be superimposed on a creep curve. For asphalt and polymers, this is more apparent at high temperatures. Reducing the load may minimize the creep, but typical load amplitude requirements do not eliminate the creep.
Complex Modulus Characterization of Asphalt Concrete • Engineering behavioral characteristics of the material can cause the response curve to deviate from one which mirrors the forcing function. Characteristics that are important in causing these deviations include anisotropy (transversely isotropic and orthotropic degrees of anisotropy are particularly relevant to asphalt which has been compacted in the field or in a gyratory compactor), and what are sometimes referred to as bimodular properties. An additional response occurs if there are phenomena such as damage or strain softening embedded in the creep response. If, in each cycle of loading, the stress changes from compression to either extension or tension, an additional response called the Bauschinger effect (Chen and Han 1988) arises from plasticity theory. Since it is relatively unimportant to differentiate between strain softening and the Bauschinger effect, a single plasticity/damage response is the final component of a generalized strain response curve in which plasticity/damage effects may cause changes in response amplitude over time.
Studied Analysis Techniques The three studied data filtering methods were (1) no filtering, (2) Spencer’s 15 point, and (3) regression; the twophase referencing methods were (1) peak picking and (2) central waveform bracketing. These produced seven combinations of analyzed methods. Methods A and B deviate only in how the phase angle is obtained, as do methods E and F: • Method A: Spencer’s 15point data filtering and central waveform bracketing • Method B: Spencer’s 15point data filtering and peak picking • Method C: Secondorder polynomial over 25% of data and peak picking • Method D: Secondorder polynomial over 10% of data and peak picking • Method E: No filtering and central waveform bracketing • Method F: No filtering and peak picking • Method G: Sinusoidal over 100% of data and regression coefficients Figure 411(a) shows an example of the filtering and phase referencing for methods C and D. These methods were used in the UMd testing program (Mirza and Witczak
FIGURE 411 Example of (a) method C and (b) method A.
105
106
Chapter Four 1994). Signals were filtered using second order polynomial over 25% of data, and + marks in the signal peaks show the data points used in the modulus and phase angle computations. The number of points included in the regression analysis was based on the total number of points available in a cycle. The second order polynomial seems to be an adequate representation of the waveform in the local region of the peak even though a perfect waveform would be sinusoidal. The peak picking method has embedded the filtering method within it. That is, the peak with no filtering (method F) is the peak of the noise and the peak with regression filtering over less than 100% of the data coincident with a zero in the first derivative of the regression equation (methods C and D). Figure 411(b) shows an example of the filtering and phase referencing for method A. Signals were filtered using Spencer’s 15point filter and + marks in the stress signal peaks show the data points used in the modulus computations. Phase angle computations were done using data points from the central part of the wave signals averaging the lag time between stress and strain. Spencer’s 15point filter (Kendall 1951) is similar to a running cubic spline, and it seems to be quite effective at reducing the tendency to skew that is normally seen with running mean techniques. Cubic spline is an interpolation function that passes a curve through a set of points in such a way that the first and second derivatives of the curve are continuous across each point. Central waveform bracketing combines regression across peaks with numerical search methods to find a line (or curve) that defines the waveform period and location of central reference points to compute the phase (Crockford 2001). An assumption is made that the underlying creep in the signal is linear. This is known to be an approximation, but the number of cycles used for the analyses is relatively small justifying the assumption. The sinusoidal regression over 100% of the data (method G) compares the data to the perfect sine wave (AAT 2001). The phase reference comes from the regression parameter estimates, and again the underlying creep is assumed to be linear.
Variability of Modulus and Phase Angle Values An analysis of variability of modulus and phase angle values was performed for methods A, B, C, D, E, and F. At each loading frequency, analysis was done using the last five cycles. The modulus and phase angle values were determined for each five cycles separately and then values were averaged from the test data obtained at temperatures ranging from –9 to 54.4°C and frequencies between 25 to 0.1 Hz. Standard error of estimate Se% was computed as percent of average value. This analysis could not be performed for the method G because the analysis calculation scheme used produced only a single estimate of amplitude and phase over a group of cycles. Table 42 shows results of the analysis for all of the studied methods for the modulus of the mix using 25, 5, and 0.1 Hz test data at all test temperatures. For the modulus computations, the method A and B were the same and produced the same modulus values, as did the methods E and F. The Se% varied between 0.3 and 1.2 with the lower frequency values and higher temperature data giving slightly higher variation. Table 43 shows analysis results for the phase angle values. Methods B and F turned out to be very unstable, Se% were 12.8 and 19.7, respectively. Both methods used the peak picking technique to obtain data points, but signals for method B were filtered while method F had unfiltered data. For the other methods Se% was between 1.5 and 4.1.
Complex Modulus Characterization of Asphalt Concrete
−9°C ∗
4.4°C
21.1°C
E  (MPa)
Se (%)
E  (MPa)
Se (%)
E  (MPa)
Se (%)
E  (MPa)
Se (%)
E∗ (MPa)
Se (%)
25
A,B
16825
0.3
9762
0.2
4229
0.8
814
0.7
280
0.5
TABLE 42
∗
54.4°C
Case
0.1
∗
37.8°C
Hz
5
∗
C
16759
0.3
9686
0.3
4234
0.7
810
0.6
276
0.4
D
16876
0.4
9748
0.4
4243
0.7
815
0.9
282
0.4
E,F
16576
0.5
9563
0.3
4074
1.5
777
0.6
282
0.5
G
16734
–
9673
–
4233
–
810
–
272
–
A,B
14129
0.3
7553
0.4
2544
0.6
451
1.1
199
0.8
C
14401
0.2
7618
0.1
2606
0.2
452
0.5
195
0.6
D
14186
0.2
7581
0.1
2558
0.4
444
0.8
194
1.0
E,F
13682
0.4
7464
0.5
2533
0.7
452
1.2
200
0.8
G
14324
2567
–
7562
–
–
450
–
194
–
A,B
8928
0.3
3185
0.2
831
0.4
173
0.6
128
0.4
C
8881
0.2
3181
0.1
828
0.2
173
0.4
124
0.3
D
8923
0.3
3185
0.2
827
0.1
173
0.5
128
0.4
E,F
8795
0.3
3160
0.2
817
1.0
174
0.9
131
0.9
G
8720
–
2973
–
750
–
166
–
118
–
Variation of Modulus Values for Different Analysis Methods
Based on this analysis, it is clear that methods B and F did not produce stable phase angle parameter values. An analysis of variance and a Tukey test were conducted for methods A, C, D, and G to assess statistical differences of the average modulus and phase angle values. The Tukey test showed that, generally, there were only slight statistical differences in the modulus values between the methods A, C, D, and G at 25 or 5Hz test data. However, method G was computing systematically lower modulus values up to 11% for the 0.1Hz data (a = 5%). For the phase angle, the most deviations occurred at 54.4°C and 0.1Hz frequency range being up to 22% between method C and G.
Deviations from Perfect Sine Wave Test data was analyzed using method G provided by AAT (2001) to estimate the deviations from a perfect sine wave. Table 43 summarizes analysis results. The load feedback data, obtained at 25 and 10 Hz frequency, systematically exceeded a 5% standard error value from a perfect sine wave, which has been considered to be a cutoff value for rejecting the data in the proposed new test protocol (AAT 2001). The closest match for the perfect sine wave was with 0.1Hz load data at all test temperatures. This trend may be explained by incorrect PID parameters that adjust the waveshape in the feedback loop, failure to include an adaptive level control, or incapacity of the hardware (servovalve, actuator, and associated hydraulic flow controls) to deliver the desired waveshape.
107
108
Chapter Four
−9°C
4.4°C Se (%)
Se (%)
Se (%)
Se (%)
j (°)
Se (%)
j (°)
25
A
10.2
1.7
17.4
2.4
31.0
1.4
34.1
4.2
32.7
1.1
B
7.2
25.0
19.8
9.1
41.4
5.3
41.4
8.7
43.2
7.8
C
9.0
5.0
18.2
1.6
32.0
1.6
34.5
1.2
33.2
0.6
D
9.9
6.8
20.0
4.5
42.8
7.8
39.0
4.3
40.7
3.7
E
9.9
3.2
17.3
1.8
31.8
2.4
34.4
7.0
32.9
1.8
F
7.2
61.2
14.4
15.3
43.2
12.1
48.6
12.6
50.4
4.4
G
10.8
–
19.5
–
33.1
–
36.8
–
34.7
–
A
12.1
1.7
21.1
0.4
33.3
1.1
31.9
1.3
28.7
2.9
B
12.2
37.9
22.3
18.0
38.2
14.2
40.3
18.4
31.0
7.9
C
12.0
1.8
22.7
1.5
34.0
0.9
31.3
1.5
29.3
1.5
D
12.6
6.2
23.9
2.8
34.0
2.7
34.7
8.2
29.3
4.0
E
11.9
3.4
20.9
0.7
34.0
1.2
32.4
1.3
28.7
3.7
F
7.9
73.9
25.2
20.7
41.0
17.7
40.3
24.0
38.2
13.5
G
12.1
–
21.8
–
32.3
–
30.7
–
27.3
–
A
16.1
1.0
29.9
0.5
31.5
1.3
23.6
1.7
21.7
1.2
B
17.3
7.8
32.4
3.5
33.1
7.2
27.4
6.7
21.6
13.9
C
17.2
1.1
33.3
0.2
36.5
0.7
27.4
1.2
25.1
1.4
D
17.8
3.2
33.4
0.5
36.2
2.1
26.9
1.6
22.3
3.6
E
15.6
1.1
30.0
0.5
31.4
1.5
23.5
1.7
21.2
1.8
F
14.4
11.2
31.7
6.6
36.0
7.7
35.3
8.2
27.4
6.7
G
16.3
–
29.8
–
31.3
–
23.0
–
20.6
–
TABLE 43
j (°)
54.4°C
Case
0.1
j (°)
37.8°C
Hz
5
j (°)
21.1°C
Variation of Phase Angle Values for Different Analysis Methods
In Fig. 412, the same test data is plotted in the Black space, where the data should form a single temperature and frequency independent curve. Figure 412(a) shows method F and the combination of methods C and D, which was obtained by using either method based on the accuracy of the peaks at a given temperature and frequency. Method F produced very noisy data and should not be used for data analysis. Figure 412(a) also shows methods A and G. Method C and D gives higher phase angles compared to methods A and G, which seem to give very similar results, although method A has a slightly better correlation. Figure 412(b) presents a subset of the original data set that excludes the 25 and 10Hz data in which the load waveform standard error exceeded 5%. The selection of this subset noticeably cleaned the A and G data sets. This analysis suggests that the most robust method to obtain the parameters from the dynamic modulus test data is the combination of methods C and D, because data reduction increased the R2 value only 0.5%. However, this method seems to overpredict
Complex Modulus Characterization of Asphalt Concrete
FIGURE 412 Test data plotted in the Black space. (Pellinen and Crockford 2003, with permission from RILEM Publications.)
the phase angle values at intermediate temperatures compared to the methods A and G. This analysis also suggests that method A is slightly more robust than method G because the data reduction increased the R2 value only 2.4% compared to 5.7% increase for the method G. Overall, based on this analysis it seems reasonable to have some limit for deviations of the controlling load waveform from a perfect sine wave to obtain good quality data, but further study would be necessary to determine whether that limit should be 5% or some other value.
Differences in Employed Analysis Methods: FFT versus Time Domain Methods The same data set described above was also analyzed using Fast Fourier Transform in the Mathcad software. The original dataset did not have the required amount of data points, and interpolation functions cspline(x,y) and interp(S,X,Y,x) were needed to create the required amount of data points. After manipulation, the stress and strain dataset were analyzed using FFT function fft(v) in Mathcad. If the dataset would have exactly N = 2m data points, Excel spreadsheet and a Fourier analysis toolset could also be used to analyze the data. Before applying FFT analysis, the measured strain signals were modified to remove the drift caused by the creep to obtain steadystate sine wave. The drift was removed by assuming linear creep upon the cyclic complex modulus signal. Also, stress and strain datasets were normalized through zero to transfer compressive haversine waveform to be sinusoidal waveform. This is illustrated in Fig. 413. For more complex data manipulation, the following model presented by Neifar, Di Benedetto, and Dogmo (2003) can be used for modeling axial cyclic strain:
ε ax (N , t) = α ax (N )t + ε ax av (N ) + ε 0 (N )* sin[ω t + ϕ ε (N )] ax
where 0 ≤ t < 2T and ω = 2π/T.
ax
(416)
109
110
Chapter Four
FIGURE 413 (a) Raw cyclic data and (b) manipulated data for FFT analysis.
T is the period of the cyclic loading and j is the phase angle of the mix. The formula ε ax av (N ) represents the axial permanent deformation and ε 0 (N ) is the amplitude of the axial sinusoidal strain component at cycle N, which can be considered as linear viscoelastic response (complex modulus) when creep is eliminated. α ax (N ) is the slope of the average deformation at cycles N (and N+1). A similar approach for modeling the cyclic dynamic modulus signals is used by AAT (2001) in the method G discussed earlier. Figure 414 compares FFT and time domain techniques discussed above. Two different ways of obtaining modulus and phase angle values using FFT analysis are shown in the figure. A method designated as FFT was conducted by manipulating stress and strain signals as discussed above, and a method designated as FFThaversine was conducted by applying fft(v) function to the stress and strain data, which was not normalized through zero (rectified sinusoidal data). As an example, Fig. 413 shows rectified data on the left and normalized data on the right. A quadratic polynomial function was fitted through each data set to investigate relative variation of data points among them. ax
FIGURE 414 FFT versus time domain techniques.
Complex Modulus Characterization of Asphalt Concrete The FFT analysis seems to be very close to the method G. However, if the data is analyzed without normalizing the data through zero (FFThaversine), the analysis results were closer to the methods C and D. This indicates that the increased phase angle values are related to the phase referencing method. The peak picking produces larger phase angle values than central waveform bracketing due to the skewness of the stress and strain signals, as was hypothesized in the beginning of the research. However, the moduli values are not affected in the same extent as Fig. 414 shows. This also explains why the moduli values are not so sensitive for the analysis method compared to the phase angle values. Therefore, if other than FFT or method G is used for the data analysis, the phase referencing should be done from the center of the signal to obtain the phase angle. Based on this analysis, methods A, G, and FFT should be used for analyzing less perfect sinusoidal cyclic test data. Although this research was done using dynamic modulus test data, these results can be applied also for shear modulus data or any sinusoidal cyclic test data.
Mastercurve Development A full characterization of asphalt mixtures requires one to construct a mastercurve, which defines the viscoelastic material behavior as a function of both temperature and loading time.
Time Temperature Superposition Principle Test data collected at different temperatures can be “shifted” relative to the time of loading or frequency, so that the various curves can be aligned to form a single mastercurve. The shift factor a(T) defines the required shift at a given temperature, that is, a constant by which the time must be divided to get a reduced time for the mastercurve. In frequency domain, the frequency must be multiplied by shift factor a(T) to obtain reduced frequency x:
ξ = f ⋅ a(T )
or
log(ξ ) = log( f ) + log[a(T )]
(417)
Mastercurve can be constructed using an arbitrarily selected reference temperature T0 to which all data are shifted. At the reference temperature, the shift factor a(T0) = 1 or log a(T0) = 0. The advantage of this procedure is that once the mastercurve is established, it is possible to derive interpolated values of asphalt mixture stiffness for any combination of temperature T or time of loading (frequency) inside the range covered by the measurements. In addition, this gives the possibility of comparing the results obtained by two laboratories with different sets of test conditions such as frequencies and temperatures.
Shifting Techniques The three different functions that commonly have been used to model the timetemperature superposition relationship in the bituminous viscoelastic materials are loglinear, Arrhenius, and Williams, Landel, and Ferry (WLF) equations. The Arrhenius and WLF equations have been used to characterize the relationship with a(T) and temperature for asphalt binders, and the Arrhenius and loglinear equations have been used for asphalt mixtures as well (Partl and Francken 1998; Huang 1993). There are two different functional forms that mainly have been used by different researchers to mathematically model the response of bituminous mixtures to construct
111
112
Chapter Four the mastercurve. For time or frequency dependency, the generalized power law has been used at low to intermediate temperatures when shifting creep or relaxation test data for asphalt mixtures (Rogue and Buttlar 1992; Christensen 1998). As the higher temperature data is included, polynomial fitting functions have been used to capture the form of mastercurve initiated by the material behavior (Francken and Verstraeten 1998). Gordon and Shaw (1994) have used piecewise fitting of polynomial functions through the test data to construct a mastercurve. Rowe and Sharrock (2000) have modified this approach by adding the cubic spline method to shift the data to the reference temperature.
Experimental Shifting and Sigmoidal Fitting Function A study by Pellinen, Witczak, and Bonaquist (2002) and Pellinen (2001) developed a method of constructing the full mastercurve using an “experimental” shifting technique using a sigmoidal fitting function. The experimental shift solves shift factors simultaneously with the coefficients of the fitting function. In this way the form of the shift function is not forced to the mastercurve. However, shift factors may absorb some of the experimental error from the test data. As mentioned earlier, polynomial fitting functions have been used to shift the asphalt mix test data using piecewise fitting approach. However, a single polynomial model cannot be used for fitting the whole mastercurve because the polynomial swing at low and high temperatures causes irrational modulus value predictions when extrapolating outside the range of data. To avoid this problem a new functional form, sigmoidal function Eq. (418), was selected to fit the dynamic modulus test data obtained from temperatures ranging from −18°C to 55°C. log(E∗ ) = δ +
α 1 + e β −γ log(ξ )
(418)
where E∗ = dynamic modulus x = reduced frequency d = minimum modulus value a = span of modulus values b, g = shape parameters Parameter g influences the steepness of the function (rate of change between minimum and maximum) and b the horizontal position of the turning point, shown in Fig. 415.
FIGURE 415 Sigmoidal function (Pellinen et al. 2002, ASCE).
Complex Modulus Characterization of Asphalt Concrete The justification of using a sigmoidal function for fitting the compressive dynamic modulus data is based on the physical observations of the mix behavior. The upper part of the sigmoidal function approaches asymptotically to the maximum stiffness of the mix, which is dependent on the limiting binder stiffness (glassy modulus) at cold temperatures. At high temperatures, the compressive loading causes aggregate influence to be more dominant than the viscous binder influence causing mix stiffness to approach a limiting equilibrium value, which is dependent of the aggregate gradation. Thus, the sigmoidal function captures the physical behavior of the asphalt mixtures observed in the mechanical testing using compressive cyclic loading through the entire temperature range. The advantage of using the sigmoidal fitting function is that a mastercurve can be constructed using Excel spreadsheets and Solver Function. The Solver Function is a tool for performing nonlinear least squares regression in the Excel spreadsheet. However, it should be noted that if the dataset does not include modulus values for full temperature range, caution should be used if the sigmoidal function is employed in the mastercurve construction. One way is to confine the asymptotic high and low modulus values to some assumed default values. Then, the asymptotic parameter values d and d +a need to be constrained to proper modulus values to obtain adequate mastercurve. Witczak et al. (Fonseca and Witczak 1996; Andrei et al. 1999) introduced the sigmoidal function to model the behavior of asphalt mixtures in conjunction with the dynamic modulus predictive equation, which predicts mixture stiffness from volumetric and raw material information.
StressDependent Mastercurve for HMA The stiffness of the hot mix asphalt (HMA) varies as a function of test temperature and loading frequency as discussed above, however, the applied stress levels also affect the measured modulus values. Figure 416 presents three separate mastercurves constructed using test data obtained by applying four different combinations of dynamic deviatoric stress sd and confinement sc. The mastercurves were constructed using a sigmoidal fitting function and experimental shifting. All mastercurves approached the same
FIGURE 416 Master curves for varying conﬁnement levels.
113
114
Chapter Four asymptotic mix stiffness values at cold temperatures. At warm temperatures mastercurves deviated indicating that the mixture stiffness was affected by the applied stress state, that is, stress to strength ratio and bulk stress q. At cold and intermediate temperatures the measured strain levels stayed under 100 microsstrains for all applied stress levels, but at high temperatures the high stress levels produced resilient strains up to 1000 microstrains.
Model Development Equation (419) shows the universal material model, also called the k1k3 model, proposed by Witczak and Uzan (1988) for unbound materials k2
⎛ θ ⎞ ⎛τ ⎞ E = (k1 pa ) ⎜ ⎟ ⎜ oct ⎟ ⎝ pa ⎠ ⎝ pa ⎠
k3
(419)
where E = resilient elastic modulus pa = atmospheric pressure q = bulk stress = I1 (first stress invariant) (= s1+ s2 + s3 with si = principal stress) toct = octahedral shear stress = √(2/3)√J2 (second stress invariant) {= [(√2)/3]sd with sd = deviatoric stress} ki = model coefficients In a study conducted by Pellinen (2001) and Pellinen and Witczak (2002b) it was hypothesized that the three separate mastercurves shown in Fig. 416 can be combined to form a stressdependent mastercurve of the mix using the approach suggested by Witczak and Uzan for unbound materials. A proposed model form to fit a stressdependent mastercurve of a mix is described in Eqs. (420) and (421): 6 logE∗ = δ +
α −δ 1 + exp( β −γ log(ξ ) )
k k ⎡ θ ⎛τ ⎞ ⎤ δ = ⎢ (k1 pa ) ⎛⎜ ⎞⎟ ⎜ oct ⎟ ⎥ ⎝ pa ⎠ ⎝ pa ⎠ ⎦ ⎣ 2
(420)
3
(421)
where logE∗ = log of stressdependent dynamic modulus, pa = atmospheric pressure q = bulk stress toct = octahedral shear stress ki = regression coefficients a, b, g = regression coefficients for sigmoidal function X = reduced frequency The stress dependency has been incorporated into the equilibrium modulus value, that is, parameter d in the mastercurve, instead of incorporating it into the shift factors. This approach differs from the approach that Schapery (1969) has proposed for constructing a mastercurve for nonlinear viscoelastic materials. He incorporates the strain dependency to the reduced time by combining vertical straindependent ae (e) and horizontal timedependent aT(T) shift to a combined shift factor aeT(eT).
Complex Modulus Characterization of Asphalt Concrete The studied mixtures were from the FHWAALF, MnRoad and WesTrack experimental test sites. The calibrated model coefficients for the studied mixtures were k1 = −0.0124, k2 = −0.59063, k3 = 0.54011, and a = 1.395045, b = 0.464119, and g = −0.04893. Both the bulk stress and octahedral shear stress had an effect on the modulus values as the coefficients k2 and k3 indicate. A separate verification of the model was conducted by testing a single specimen of asphalt mixture with randomly varying confinement and deviatoric stress levels. The difference between measured and predicted modulus values is less than a factor of 1.5 indicating relatively good prediction accuracy. It can be speculated that deviations between predicted and measured values were caused by the damage accumulation because applied stress states were in the dilative side of the phase change line most of the time.
StressDependent Stiffness Predictive Equation Stiffness of asphalt mixture can be predicted using models such as the dynamic modulus predictive equation (Andrei et al. 1999) and the Hirsch model (Christensen et al. 2003). The advantage of these models is that they provide an approximate but useful way of estimating mixture stiffness (modulus) for various design purposes. These models, however, only model the modulus obtained in the linear viscoelastic region. Furthermore, there are no simple models available to estimate the effect of nonlinearity and confinement on HMA modulus values. Pellinen and Witczak (2002b) also developed the stressdependent stiffness predictive equation which is based on Eqs. (420) and (421). Equations (422), (423), and (424) show the model form, and Eq. (425) describes mix gradation (Ga) as an average of the percent passing four sieve sizes of 0.074 mm, 4 mm, 9.5 mm, and 19 mm (No. 200, No. 4, 3/8 in, and 3/4 in). Table 44 gives the model coefficients. The model expects a minimum bulk stress value of 21 kPa and octahedral shear stress value of 9.9 kPa for unconfined linear viscoelastic stress case predictions. log(E∗ ) = δ + A +
α − (δ + A) + a4Ga + a5 VFA 1 + exp β +γ log( f )− c log(η )
A = a0 + a1Ga + a2 VFA + a3 log(η)
(422) (423)
Material Coefficients
k1–k3 Model Coefficients
Sigmoidal Model Coefficients
a0 = −10.429150 a1 = 0.004106 a2 = −0.015376 a3 = 0.013351 a4 = −0.000808 a5 = 0.001594 c = 0.625379
k1 = 0.099088 k2 = 0.0217941 k3 = −0.011816
a = 1.324132 b = 0.615775 g = −0.584201
Source: Pellinen and Witczak 2002b, with permission from Association of Asphalt Paving Technologists.
TABLE 44
Predictive Model Coefficients
115
116
Chapter Four where
logE∗ = log of stressdependent dynamic modulus (106 kPa) d = equilibrium modulus Ga = average gradation (passing %) VFA = Voids filled with asphalt (volume %) h = binder viscosity (106 P) F = frequency (Hz) a0, a1, a2, a3, a4, a5 = regression coefficients k k ⎡ θ ⎛τ ⎞ ⎤ δ = ⎢ (k1 pa ) ⎛⎜ ⎞⎟ ⎜ oct ⎟ ⎥ ⎝ pa ⎠ ⎝ pa ⎠ ⎦ ⎣ 2
where
3
(424)
d = equilibrium modulus q = bulk stress (kPa) toct = octahedral shear stress (kPa) pa = atmospheric pressure, 103.3 kPa k1, k2, k3 = regression coefficients Ga =
p200 + p4 + p3/8 + p3/4 4
(425)
where Ga = average gradation (%) p200 = passing 0.074 mm (%) p4 = passing 4.36 mm (%) p3/8 = passing 9.5 mm (%) p3/4 = passing 19 mm (%)
Summary This chapter discusses the use of the axial dynamic modulus E∗ test and the SST shear modulus G∗ from the shear frequency sweep test to obtain complex modulus of asphalt mixture. More specifically, specimen fabrication, instrumentation, test control modes, and analysis of test data to obtain modulus and phase angle are discussed. In addition, a new method of constructing a mixture mastercurve is presented. Research has shown that these two test methods are not interchangeable for pavement design applications because the theoretical linear elastic law of E∗ being approximately three times larger than the G∗ when the Poisson’s ratio is 0.5, is not fulfilled. A conversion equation needs to be used to estimate the E∗ from the SST G∗ test results. Research has also shown that both of these test methods can be used as performance indicators for asphalt mixtures. However, they do not rank the mixtures necessarily in the same order for rutting performance. Most likely mixtures that have binder as a dominating stiffness contributor will be ranked similarly, while mixtures that have different gradations, such as dense graded versus SMA, may be ranked differently.
Acknowledgments The author wishes to acknowledge Dr. Donald W. Christensen from Advanced Asphalt Technologies, LLC, for his help in analyzing data using the method G he developed for National Cooperative Highway Research Project contract, NCHRP 929, for testing
Complex Modulus Characterization of Asphalt Concrete equipment development. The author also wishes to acknowledge Mr. Bill Crockford from ShedWorks for his help in writing the section of Analysis of Cyclic Sinusoidal Test Data. In addition, the author wishes to acknowledge professor Matthew W. Witczak from Arizona State University for his help and advice in the course of dynamic modulus testing development.
Endnote 1. According to Di Benedetto and de la Roche (1998), the word dynamic should be used in the tests with nonnegligible inertia effects inside the sample (i.e., when wave propagation is observed). Then, a complex modulus test is not a dynamic test but a cyclic test with repeated loading and is therefore interpreted as a static test. However, the author is using the word dynamic as it is traditionally used in literature in the United States.
References Advanced Asphalt Technologies, LLC (AAT). (2001). First Article Equipment Specifications for the Simple Performance Test System. NCHRP Project 929: Simple Performance Tester for Superpave Mix Design, Sterling, Va.: Advanced Asphalt Technologies, LLC, November 2001, p. 89. American Association of State Highway and Transportation Officials (AASHTO). (1994). “Shear Device, AASHTO Designation: TP794.” AASHTO Provisional Standards. American Society for Testing and Materials (ASTM). (1979). “Test Method for Dynamic Modulus of Asphalt Concrete Mixture.” Annual Book of ASTM Standards, ASTM D Vol. 04.03, 349779. Andrei, D., Witczak, M. W., and Mirza, M. W. (1999). Development of Revised Predictive Model for the Dynamic (Complex) Modulus of Asphalt Mixtures. Development of the 2002 Guide for the Design of New and Rehabilitated Pavement Structures, NCHRP 137A. Interim Team Technical Report. Department of Civil Engineering, University of Maryland of College Park, Md. Bahia, H., and Anderson, D. (1995). “The SHRP Binder Rheological Parameters: Why they are Required and How They Compare to Conventional Properties.” Transportation Research Board, 75th Annual Meeting, Washington, D.C. Chen, W. F., and Han D. J. (1988). Plasticity for Structural Engineers. SpringerVerlag, New York, p. 10. Christensen, D. W., Pellinen, T., and Bonaquist, R. F. (2003). “Hirsch Model for Estimating the Modulus of Asphalt Concrete.” Proceedings of the Association of Asphalt Paving Technologists. March 10–12, 2003, Lexington, Ky. Christensen, D. W. (1998). “Analysis of Creep Data from Indirect Tension Test on Asphalt Concrete.” Journal of the Association of Asphalt Paving Technologists, Vol. 67, 458–492. Crockford, W. W. (2001). Data Analysis—Load Controlled Dynamic Tests. ShedWorks Inc. Technical Note, 2nd ed. Di Benedetto, H., and de la Roche, C. (1998). “State of the Art of Stiffness Modulus and Fatigue of Bituminous Mixtures.” RILEM Report 17, Bituminous Binders and Mixes, Edited by L. Francken, 137–180, London. Doubbaneh, E. (1995). “Comportement Mécanique Des Enrobes Bitumineux Des “Petites” Aux ‘Grandes‘ Déformations.” Grade de Docteur. L’institut Nations Des Sciences
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Chapter Four Appliquées De Lyon, Laboratoire Géomatériaux du Département de Génie Civil, Lyon. Di Benedetto, H., Partl, M., and de la Roche, C. (2001). “Stiffness Testing for Bituminous Mixtures.” RILEM TC 182PEB Performance Testing and Evaluation of Bituminous Materials. Materials and Structure, Vol. 34 (236), March, 2001, pp. 65–70. Francken, L., and Verstraeten, J. (1998). “Interlaboratory Test Program on Complex Modulus and Fatigue.” RILEM Report 17, Bituminous Binders and Mixes, Edited by L. Francken, 182–215 London. Fonseca, O. A., and Witczak, M. W. (1996). “A Prediction Methodology for the Dynamic Modulus of InPlaced Aged Asphalt Mixtures.” Journal of the Association of Asphalt Paving Technologists, Vol. 65, 532–572. Gordon, G. V., and Shaw, M. T. (1994). Computer Programs for Rheologists. Hanser/Gardner Publishers, New York. Harrigan, E. T., Leahy, R. B., and Youtcheff, J. S. (1994). The Superpave Mix Design Manual Specifications, Test Methods, and Practices, SHRPA379. Strategic Highway Research Program. National Research Council. Washington, D.C.: National Academy of Science. HewlettPackard Co. (1989). The Fundamentals of Signal Analysis. Application Note 243. HewlettPackard Co. Printed in USA, 59528898. Huang, Y. (1993). Pavement Analysis and Design. Appendix A, Theory of Viscoelasticity. Prentice Hall, Englewood Cliffs, N.J. Kendall, M. G. (1951). The Advanced Theory of Statistics. Vol. II, Hafner, New York, p. 377. NCHRP 137A Draft Document. (2002). 2002 Guide for the Design of New and Rehabilitated Pavement Structures. ERES Division of ARA Inc., Champaign, Ill. Mirza, M. W., and Witczak, M. W. (1994). Bituminous Mix Dynamic Material Characterization Data Acquisition and Analysis Programs Using 458.20 MTS Controller. University of Maryland, Department of Civil and Environmental Engineering, College Park, Md. Neifar, M., Di Benedetto, H., and Dogmo, B. (2003). Permanent Deformation and Complex Modulus: Two Different Characteristics from a Unique Test. Proceedings in the 6th International RILEM Symposium, April 14–17, 2003, Zurich, Switzerland, 316–323. Partl, M. N., and Francken, L. (1998). “Introduction.” RILEM Report 17, Bituminous Binders and Mixes. edited by L. Francken, 2–10 London. Pellinen, T. (2001). Investigation of the Use of Dynamic Modulus as an Indicator of HotMix Asphalt Performance. A Dissertation Presented in Partial Fulfillment of the Requirements of the Degree of Doctor of Philosophy. Submitted to the Faculty of Graduate School of the Arizona Sate University, Tempe, Ariz. Pellinen, T., and Crockford, B. (2003). “Comparison of Analysis Techniques to Obtain Modulus and Phase Angle from Sinusoidal Test Data.” Proceedings in the 6th International RILEM Symposium, April 14–17, Zurich, 2003, Switzerland, 307–309. Pellinen, T., and Witczak, M. W. (1998). Assessment of the Relationship Between Advanced Material Characterization Parameters and the Volumetric Properties of A MDOT Superpave Mix. Submitted to Maryland Department of Transportation State Highway Administration. Department of Civil Engineering, University of Maryland of College Park, Md. Pellinen, T, K., and Witczak, M. W. (2002a). “Use of Stiffness of HotMix Asphalt as a Simple Performance Test.” Journal of Transportation Research Records, No 1789, 80–90.
Complex Modulus Characterization of Asphalt Concrete Pellinen, T. K., and Witczak, M. W. (2002b). “Stress Dependent Mastercurve Construction for Dynamic (Complex) Modulus.” Journal of the Association of Asphalt Paving Technologists, Vol. 71, 281–309. Pellinen, T., Witczak M. W., and Bonaqusit, R. (2002). “Master Curve Construction Using Sigmoidal Fitting Function with Nonlinear Least Squares Optimization Technique.” Proceedings of the 15th ASCE Engineering Mechanics Division Conference, Columbia University, June 2–5, 2002, New York. Ramsey, K. A. (1975). “Effective Measurements for Structural Dynamics Testing.” Sound and Vibrations, November 1975, 24–35. Rogue, R., and Buttlar, W. G. (1992). “The Measurement and Analysis System to Accurately Determine Asphalt Concrete Properties Using the Indirect Tensile Mode.” Journal of the Association of Asphalt Paving Technologists, Vol. 61, 304–328. Rowe G. M., and Sharrock, M. J. (2000). “Development of Standard Techniques for the Calculation of Master Curves for LinearVisco Elastic Materials.” The 1st International Symposium on Binder Rheology and Pavement Performance. The University of Calgary, August 14–15, 2000, Alberta, Canada. Saarela, A., (1993). Asfalttipäällysteiden tutkimusohjelma ASTO 19871992 (Asphalt Pavements Research Program ASTO 19871992), Loppuraportti. Technical Research Centre of Finland. Road, Traffic and Geotechnical Laboratory. Espoo. Schapery, R. (1969). “On the Characterization of Nonlinear Viscoelastic Materials.” Polymer Engineering and Science, Vol. 9, 295–310. Witczak, M. W., and Kaloush, K. (1998). Performance evaluation of Asphalt Modified Mixtures Using Superpave and P401 Mix Grading. Submitted to: Maryland Department of Transportation, Maryland Port Administration. Department of Civil Engineering, University of Maryland of College Park, Md. Witczak, M. W., and Uzan, J. (1988). The Universal Airport Pavement Design System— Report I of IV: Granular Material Characterization. Department of Civil Engineering, University of Maryland of College Park, Md. Witczak, M. W., Bonaquist, R., Von Quintus, H., and Kaloush, K. (2000). “Specimen Geometry and Aggregate Size Effects in Uniaxial Compression and Constant Height Shear Test.” Journal of the Association of Asphalt Paving Technologists, Vol. 69, 733–793. Witczak, M. W., Hafez, I., Ayres, H., and Kaloush, K. (1996). Comparative Study of MSHA Asphalt Mixtures Using Advanced Dynamic Material Characterization Tests. Submitted to: Maryland Department of Transportation, Maryland Port Administration, Department of Civil Engineering, University of Maryland of College Park, Md.
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CHAPTER
5
Complex Modulus from the Indirect Tension Test Y. Richard Kim, Youngguk Seo, and Mostafa Momen
Abstract This chapter presents the results from an analytical/experimental study on the dynamic modulus testing of hot mix asphalt (HMA) using the indirect tension (IDT) mode. An analytical solution for the dynamic modulus in the IDT mode is developed using the theory of linear viscoelasticity. To verify the analytical solution, temperature and frequency sweep tests were conducted on 24 asphalt mixtures commonly used in North Carolina, using both axial compression and IDT test methods. Graphical and statistical comparisons of results from the axial compression and IDT test methods show that the dynamic modulus mastercurves, phase angles, and shift factors derived from the two methods are in good agreement.
Introduction One of the issues related to the role of dynamic modulus in pavement management is its use in forensic studies and pavement rehabilitation design. The current dynamic modulus protocol, AASHTO TP62, calls for the uniaxial compression testing of 100mmdiameter and 150mmtall asphalt concrete specimens. It is often impossible to obtain this size specimen from actual pavements. Given that a typical asphalt layer thickness is less than a few inches and that coring is the most effective method of obtaining specimens from actual pavements, the indirect tension (IDT) test of cores seems to be more appropriate for the evaluation of existing pavements. In this chapter, linear viscoelastic solutions are presented for the determination of dynamic modulus (E∗) of asphalt concrete from the IDT test data. The IDT E∗ values are then compared with the values determined from the axial compression complex modulus testing for 24 different asphalt mixtures typically used in North Carolina.
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122
Chapter Five
Theoretical Background Historically, the pavement community utilized the elastic solutions for IDT testing that Hondros (1959) derived using the plane stress assumption until Roque and Buttlar (1992) introduced correction factors that accounted for the bulging effect of the specimen. Later, Kim et al. (2000) introduced viscoelastic solutions for the IDT creep test using the theory of linear viscoelasticity. Unlike those of the uniaxial test specimens, the stress and strain distributions in IDT specimens are biaxial. This biaxial state of stress and strain can cause errors in determining the material properties obtained from the IDT test unless the derivation of the properties is carefully handled. To illustrate this point more clearly, Hooke’s law, the governing equation for elastic materials, is presented below for both uniaxial and biaxial cases: Uniaxial case:
σ y = E × εy
or
εy =
σy E
(51)
Biaxial case:
εx =
1 (σ − νσ y ) E x
(52)
where x and y denote the loading direction (i.e., the vertical direction) and the direction perpendicular to the loading direction (i.e., the horizontal direction), respectively. In the uniaxial case (i.e., the axial compression dynamic modulus test) in Eq. (51), one can divide the axial stress (sy ) by the axial strain (ey ) to obtain the modulus. However, in the biaxial case (i.e., the IDT dynamic modulus test) in Eq. (52), one cannot obtain the modulus by dividing the horizontal stress (sx) by the horizontal strain (ex). Rather, the correct way to determine the modulus of the material is to divide the biaxial stress (i.e., sx–nsy) by the horizontal strain (ex). If the incorrect solution (i.e., sx /ex) is used to represent the modulus of the material, then that modulus should not be considered the same as the modulus determined from the axial test.
Linear Viscoelastic Solution The linear viscoelastic solution for the complex modulus of HMA under the IDT mode has been developed by Kim et al. (2004), and is presented in this section. Assuming the plane stress state, Hondros (1959) developed the following expressions for stresses and strains along the horizontal diameter of the IDT specimen subjected to a strip load, shown in Fig. 51:
εx =
1 (σ − νσ y ) E x
(53)
with
σ x ( x) = =
2 2 2P ⎡ (1 − x 2 / R 2 )sin 2α ⎫⎤ −1 ⎧ 1 − x / R ⎨ ⎢ 4 − tan 2 2 tan α ⎬ ⎥ 2 2 4 π ad ⎣ 1 + 2 x / R cos 2α + x / R 1 + x R / ⎩ ⎭⎦
2P [ f (x) − g(x)] π ad
(54)
C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t
FIGURE 51 Schematic of the IDT specimen subjected to a strip load. (Kim et al. 2004, with permission from Transportation Research Board.)
σ y ( x) = − =−
1 − x2 / R2 2P ⎡ (1 − x 2 / R 2 ) sin 2α ⎫⎤ −1 ⎧ + tan tan α ⎬ ⎥ ⎨ ⎢ 2 2 π ad ⎣ 1 + 2 x 2 / R 2 cos 2α + x 4 / R 4 + 1 x R / ⎩ ⎭⎦ 2P [ f (x) + g(x)] π ad
(55)
where x = horizontal distance from the center of the specimen face P = applied load a = loading strip width, m d = thickness of specimen, m R = radius of specimen, m a = radial angle E = Young’s modulus n = Poisson’s ratio For the viscoelastic materials subjected to the sinusoidal load in a steady state, Eq. (53) can be rewritten as
εx =
1 (σ − νσ y ) E∗ x
(56)
where E∗ is the complex modulus. It is often helpful to have E∗ in polar form, E ∗ = E∗ ⋅ e iφ
(57)
123
124
Chapter Five where E∗ is the dynamic modulus and f is the phase angle calculated from the time lag between the load and the displacement. The response to the sinusoidal load applied in the complex modulus test is the imaginary component of the response due to the complex load P shown below: P = P0 e iwt = P0 (cos wt + i sin wt)
(58)
where P0 and w are the amplitude and the angular frequency of the sinusoidal load used in the complex modulus test, respectively. Substituting Eqs. (54), (55), (57), and (58) into Eq. (56) results in 2 P0 e i( wt−φ ) [(1 + ν ) f (x) + (ν − 1) g(x)] E∗ π ad
ε x ( x , t) =
(59)
Integrate Eq. (59) over the gauge length to determine the horizontal displacement U(t) and obtain l
U (t) = ∫ ε x (x , t) dx = −l
l l ⎡ ⎤ 2 P0 i ( wt −φ ) e ( + ν ) f ( x ) dx + ( ν − 1 ) g(x) dx ⎥ 1 ⎢ ∫ ∫ E∗ π ad ⎢⎣ ⎥⎦ −l −l
(510)
where l is half of the gauge length. One may extract the response that only occurs due to the sinusoidal input by taking an imaginary part of the total response. Therefore, the dynamic modulus derived can be expressed using the horizontal displacement U(t) as follows: E∗ =
2 P0 sin(wt − φ ) A π ad ⋅ U (t)
(511)
where l l ⎡ ⎤ A = ⎢ (1 + ν ) ∫ f (x)dx + (ν − 1) ∫ g(x)dx ⎥ ⎢⎣ ⎥⎦ −l −l
(512)
with f ( x) =
and
(1 − x 2 / R 2 )sin 2α 1 + 2 x 2 / R 2 cos 2α + x 4 / R 4
⎫ ⎧ 1 − x2 / R2 g(x) = tan −1 ⎨ 2 2 tan α ⎬ ⎭ ⎩1+ x /R
(513)
(514)
Similarly, the analogous expression for the dynamic modulus using the vertical displacement V(t) is E∗ =
2 P0 sin(wt − φ ) B π ad ⋅ V (t)
(515)
C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t where l l ⎡ ⎤ B = ⎢ (ν − 1) ∫ n( y )dy − (1 + ν ) ∫ m ( y ) dy ⎥ ⎢⎣ ⎥⎦ −l −l
(516)
with m( y ) =
(1 − y
/ R 2 ) sin 2α 1 − 2 y 2 / R 2 cos 2α + y 4 / R 4 2
⎫ ⎧ 1 + y 2 / R2 n( y ) = tan −1 ⎨ 2 2 tan α ⎬ − / y R 1 ⎩ ⎭
and
(517)
(518)
By equating Eqs. (511) and (515), one can obtain A ⋅ V (t) = B ⋅ U (t)
(519)
Then, one may derive the expression for Poisson’s ratio as follows:
β 1U (t) − γ 1V (t) − β 2U (t) + γ 2V (t)
ν=
(520)
where l
l
β1 = − ∫ n( y )dy − ∫ m( y )dy −l
−l
l
l
−l
−l
l
l
−l
−l
β2 = ∫ n( y)dy − ∫ m( y )ddy γ1 =
∫ f (x)dx − ∫ g(x)dx l
γ2 =
∫
−l
−l
f (x) dx + ∫ g(x) dx
(521)
−l
Combining Eqs. (511) and (515) yields a single form of the dynamic modulus, as shown below: E∗ =
P0 sin(wt − φ ) AV (t) + P0 sin(wt − φ ) BU (t) π ad ⋅ V (t) ⋅U (t)
(522)
After substituting Eqs. (512) and (516) into Eq. (522), one can obtain E∗ = 2
P0 sin(wt − φ ) β1γ 2 − β 2γ 1 π ad γ 2V (t) − β 2U (t)
(523)
125
126
Chapter Five
Specimen Diameter (mm)
Gauge Length (mm)
b1
b2
g1
g2
101.6
25.4
−0.0098
−0.0031
0.0029
0.0091
101.6
38.1
−0.0153
−0.0047
0.0040
0.0128
101.6
50.8
−0.0215
−0.0062
0.0047
0.0157
152.4
25.4
−0.0065
−0.0021
0.0020
0.0062
152.4
38.1
−0.0099
−0.0032
0.0029
0.0091
152.4
50.8
−0.0134
−0.0042
0.0037
0.0116
Source: Kim et al. 2004, with permission from Transportation Research Board.
TABLE 51
Coefficients for Poisson’s Ratio and Dynamic Modulus
The vertical and horizontal displacements can be expressed in sine functions as follows: V (t) = V0 sin(wt − φ )
(524)
U (t) = U 0 sin(wt − φ )
(525)
where V0 and U0 are the constant amplitudes of vertical and horizontal displacements, respectively. Therefore, the final form of the dynamic modulus is E∗ = 2
P0 β1γ 2 − β 2γ 1 π ad γ 2V0 − β 2U 0
(526)
Likewise, the expression for Poisson’s ratio can be simplified as
ν=
β1U 0 − γ 1V0 − β 2U 0 + γ 2V0
(527)
The coefficients, b1, b2, g1, and g2, in Eqs. (526) and (527), are calculated for different specimen diameters and gauge lengths, and are presented in Table 51. Equations (526) and (527) are based on the plane stress assumption. Kim et al. (2000) used the threedimensional finite element analysis to calculate the center strain in the IDT specimen and concluded that the error due to the plane stress assumption is negligible.
Dynamic Modulus Testing of HMA Included in this section are the HMAs selected for testing, the procedure for specimen fabrication, and axial compression and IDT dynamic modulus test methods.
Materials A total of 24 asphalt mixtures with varying aggregate and binder characteristics were tested in the axial compression and IDT modes. These mixtures are typical HMAs used in paving construction in North Carolina. Table 52 summarizes the mixture variables for all the mixtures. Granite aggregates from six different sources from the mountains
C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t
Agg. Source
Binder Source
Binder Grade
Binder Content
Gmm
S 9.5 A C
Morganton
Inman, SC
PG 6422
5.8%
2.615
S9.5AF
CharlottePineville
CitgoWilmington
PG 6422
6.4%
2.668
S9.5BC
Haw River
CitgoWilmington
PG 6422
5.9%
2.579
S9.5BF
Morganton
Inman, SC
PG 6422
6.3%
2.579
S9.5BF
CharlottePineville
CitgoWilmington
PG 6422
5.8%
2.689
S9.5CC
Holly Springs
CitgoWilmington
PG 7022
5.3%
2.486
S9.5CF
Garner
CitgoWilmington
PG 7022
5.0%
2.456
S12.5BC
Haw River
CitgoWilmington
PG 6422
5.5%
2.595
S12.5BF
Holly Springs
CitgoWilmington
PG 6422
5.3%
2.480
S12.5CC
Morganton
Inman, SC
PG 7022
4.6%
2.663
S12.5CF
ConcordCabarrus
CitgoWilmington
PG 7022
5.0%
2.570
S12.5DC
ConcordCabarrus
CitgoWilmington
PG 7022
5.0%
2.582
S12.5DF
ConcordCabarrus
AASalisbury
PG 7622
4.7%
2.571
I19.0BC
Haw River
AlpasoApex
PG 6422
5.0%
2.633
I19.0BF
Garner
CitgoWilmington
PG 6422
5.4%
2.441
I19.0BF
CharlottePineville
CitgoWilmington
PG 6422
4.3%
2.773
I19.0CC
Garner
CitgoWilmington
PG 6422
4.7%
2.472
I19.0CF
ConcordCabarrus
AlpasoCharlotte
PG 6422
4.8%
2.582
I19.0DC
CharlottePineville
CitgoWilmington
PG 7022
4.3%
2.770
I19.0DF
ConcordCabarrus
AASalisbury
PG 7022
4.1%
2.597
B25.0BC
Holly Springs
CitgoWilmington
PG 6422
4.5%
2.503
B25.0BF
Garner
CitgoWilmington
PG 6422
4.2%
2.485
B25.0CC
Haw River
CitgoWilmington
PG 6422
4.0%
2.678
B25.0CF
ConcordCabarrus
AlpasoCharlotte
PG 6422
4.4%
2.599
Mixture ID *
† ‡
§
∗
S for surface mix, I for intermediate mix, and B for base mix. Nominal maximum aggregate size (NMAS) in mm. ‡ Traffic volume indicator. § Aggregate gradation type (C for coarse and F for fine). Source: Kim et al. 2004, with permission from Transportation Research Board. †
TABLE 52
Summary of Mixture Characteristics
to the coast of North Carolina were utilized with four different nominal maximum aggregate sizes (NMASs) (9.5, 12.5, 19.0, and 25.0 mm). Also, binders from six different sources with performance grades (PG) of 6422, 7022, and 7622 were used in making these mixtures. Superpave volumetric mix designs were performed on these mixtures and the resulting optimum asphalt content and the maximum specific gravity (Gmm) for each of the mixtures are shown in Table 52.
127
128
Chapter Five
Specimen Fabrication Asphalt mixtures were mixed and compacted at temperatures that were in accordance with the requirements for each binder. All mixtures were aged at 135°C for 4 hours (i.e., shortterm oven aging) before compaction. For mixtures tested in the axial compression mode, samples were compacted into gyratory plugs of 150 mm in diameter by 178 mm in height. Later, they were cut and cored to cylindrical specimens with dimensions of 100 mm in diameter and 150 mm in height. For mixtures tested in the IDT mode, samples were compacted into gyratory plugs of 150 mm in diameter by 60 mm in height, and they were cut to 38 mm height. Both ends were cut to ensure a more consistent air void distribution along the height of the test specimens. The target air voids for the final cut and cored specimen were 4% ± 0.5%. In order to achieve this density, the target air voids for the gyratory plug had to be higher than that of the cut and cored specimen. The difference between the target air voids for the gyratory plug and for the cut and cored specimen typically increased as the NMAS increased with an average being about 1.5% to 2%. For IDT specimens, a small study was conducted using four mixtures with different NMASs to estimate the reduction in air voids when the specimen is cut from a 60 mm height to 38 mm. The study showed that, in general, the reduction in air voids is around 1% for 9.5mm NMAS and the reduction increases by approximately 0.5% increments as the NMAS increases to 12.5, 19, and 25 mm. Air voids were measured using the Corelok vacuum sealing device. Specifications provided in ASTM D675203 were followed in taking these measurements and making the calculations. Appropriate adjustments were made to account for the density of water when measurements were taken at temperatures other than 25°C. For each of the 24 mixtures there were 3 replicates of each tested. During fabrication or testing, if errors were made or densities were not met, then the specimen was discarded and an additional specimen was manufactured and tested.
Testing and Analysis Test Setup Testing was performed using a closedloop servohydraulic machine, manufactured by Material Testing System (MTS). A temperature chamber, cooled by liquid nitrogen, was used to control the test temperature. Dummy specimens with thermocouples embedded in the middle of the specimen were used to monitor the temperatures to which the specimens were subjected. For axial compression testing, 100mmdiameter metal end plates attached to the top and bottom rams were used to apply the compressive load to the specimen. Frictionreducing end treatments were used to lessen the confinement that occurs as a result of the friction against the specimen and the metal end plates. The end treatments were made of two latex membranes, each 0.0125 in. thick. A very small amount of silicon grease was smeared between the latex membranes to allow the specimen to move (dilate) with respect to the metal end plates. Caution was used to apply only the minimum amount of grease necessary because excess grease creates a slick interface that causes the specimen to slip out of the test setup under high load and high frequency combinations. Also, an approximate 25mmdiameter hole was introduced in the center of the membrane to allow a small amount of contact between the metal plate and the specimen, thereby increasing the friction and reducing the movement of the specimen during testing.
C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t For the IDT test, the Load Guide Device (LGD), developed from the Strategic Highway Research Program (SHRP), was used as the loading apparatus. This device is shown in Fig. 52(a). From the NCHRP 128 study, it was found that when compared against other loading devices with no column or four columns, the SHRP LGD with
(a)
(b)
FIGURE 52 (a) IDT test setup with SHRP LGD and (b) surfacemounted LVDTs. (Kim et al. 2004, with permission from Transportation Research Board.)
129
130
Chapter Five
FIGURE 53 LVDT mounting and spacing in axial compression testing.
two guide columns resulted in the least amount of “rocking” of the IDT specimen without causing significant friction between the upper loading plate and guide columns under repetitive loading (Barksdale et al., 1997). For axial compression testing, vertical deformations were measured using four loosecore, CD type LVDTs (linear variable differential transducers) at 90° radial intervals. Targets were glued to the specimen face in the middle twothirds of the specimen (100 mm), and the LVDTs were mounted to the targets. A gluing device was used to maintain consistent spacing between the LVDT targets. The LVDT setup for axial compression testing is shown in Fig. 53. For IDT specimens, the vertical and horizontal deformations were measured using loosecore type miniature XSB LVDTs. These were mounted on each of the specimen faces using a 50.8mm gauge length, as shown in Fig. 52(b).
Data Acquisition System The data acquisition system used in this project is composed of LabView software and a 16bit data acquisition board by National Instruments. One channel was dedicated to the load cell on the machine, one to the actuator LVDT, and four to the onspecimen LVDTs. The data acquisition rate was 100 points per cycle.
Test and Analysis Methods In principle, the AASHTO TP62 protocol was followed. To ensure that the testing captures the linear viscoelastic behavior of the material, 75 microstrain was used as the maximum allowable axial and horizontal strains for the axial compression and IDT testing, respectively. Testing was performed by applying sinusoidal loadings at different frequencies and temperatures. Prior to applying the first frequency at each temperature, the preconditioning cycles were applied at 25 Hz and onehalf the load used in actual
C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t testing for 25 Hz. Following each loading frequency, a 5minute rest period was allowed before the next frequency was applied. Averaged deformations were used to calculate the dynamic modulus and phase angle. In the uniaxial case, readings from four LVDTs spaced at 90° intervals were averaged. In IDT, the vertical deformations and horizontal deformations from two surfaces were averaged to determine the deformation in each axis.
Comparison of Dynamic Moduli Values Graphical Comparison The IDT test data of the 24 mixtures were analyzed using the viscoelastic solutions in Eq. (526). The resulting dynamic modulus mastercurves from these analyses are plotted in Figs. 54 to 56 for the 12 representative mixtures. The reference temperature of 10°C was used as the basis of shifting the data. The data presented in these figures are the averages of the three replicates. The rest of the data can be found in Kim et al. (2005). It can be observed from these figures that the dynamic modulus mastercurves developed from the IDT test using the biaxial linear viscoelastic solution are generally in good agreement with those determined from the axial compression test. It was also found that the timetemperature shift factors obtained during the construction of the mastercurves are essentially identical for the axial compression and IDT tests.
Statistical Analysis Using PValue Recognizing that a sampletosample variation exists, a statistical analysis was conducted using the unequal variance ttest for each mixture at two frequencies for each testing temperature. In this analysis, all the individual replicates (three from the axial compression and three from the IDT tests) were used. The null hypothesis is that the dynamic modulus from the IDT test is the same as that from the axial compression test. The Pvalue was calculated and compared with the critical value of 0.05 to reject or accept the null hypothesis. The Pvalue indicates the extent to which a computed test statistic is unusual in comparison with what would be expected under the null hypothesis. Therefore, in this study a Pvalue greater than 0.05 indicates that the dynamic modulus from the IDT test is statistically the same as that from the axial compression test. A summary of the Pvalues for 144 tests (2 frequencies × 3 temperatures × 24 mixtures) is given in Table 53. About 19% of the tests indicate that the dynamic modulus from the IDT test is statistically different from the dynamic modulus from the axial compression test.
Using Percent Difference In addition to the statistical analysis, the percent difference was calculated for the dynamic moduli determined from the axial compression and IDT tests for 288 combinations of temperature and frequency (8 frequencies × 3 temperatures × 12 mixes). These values are also summarized in Table 53. A comparison of the data in this table and further investigation of individual test data resulted in several important observations.
131
132
Chapter Five
FIGURE 54 Dynamic modulus mastercurves for (a) S9.5AFine; (b) S9.5BCoarse; (c) S9.5CFine; and (d) S9.5CCoarse. (Kim et al. 2004, with permission from Transportation Research Board.)
FIGURE 55 Dynamic modulus mastercurves for (a) S12.5CFine; (b) S12.5DCoarse; (c) I19.0BFine; and (d) I19.0CCoarse. (Kim et al. 2004, with permission from Transportation Research Board.)
C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t
FIGURE 56 Dynamic modulus mastercurves for (a) I19.0DFine, (b) I19.0DCoarse, (c) B25.0BFine, and (d) B25.0BCoarse. (Kim et al. 2004, with permission from Transportation Research Board.)
NMAS
PValue
% Difference
9.5 mm
12.5 mm
19.0 mm
25.0 mm
All
0.05
85%
86%
81%
67%
81%
20%
10%
0%
4%
38%
8%
TABLE 53 Summary of PValues and Percent Difference in Dynamic Moduli from IDT and Axial Compression Tests
First, about 70% of the tests had a percent difference below 10%. Secondly, although it is not shown in Table 53, further investigation of the testing conditions that have high percent differences reveals that most of the high percent difference come from 35°C data due to very small dynamic modulus values in the denominator of the percent difference calculation. Finally, it can be observed in Table 53 that the percent difference becomes greater as the NMAS increases. An investigation of the individual test data reveals the same trend; that is, the variability among the replicates increases as the NMAS increases. These observations may be related to the ratio of gauge length to NMAS. Typically, a factor of 3 is recommended to maintain a representative volume element (RVE). Meeting
133
134
Chapter Five this requirement is not a problem in the axial compression test geometry. However, in the IDT tests with a 50.8mm gauge length, this requirement is satisfied for the 9.5 and 12.5mm mixes, but not for the 19.0 and 25.0mm mixes, resulting in a higher variability among replicates and a higher percent difference in the 19.0 and 25.0mm mixes. Another observation made from a detailed data analysis is that, in some replicates of the 25.0mm mix, a significant difference was found between displacements from the front and back surfaces of the IDT specimens. These observations suggest that the positions of large aggregate particles within the gauge length affect the data, and that a larger gauge length is required for 25.0mm mixes. The visual observation of the average mastercurves in Figs. 54 to 56 and further statistical analysis suggest that the dynamic modulus determined from the IDT test using the linear viscoelastic solution in Eq. (526) is statistically the same as the one measured from the axial compression test. A question may arise regarding why the effect of different relationships between the compaction direction and the direction in which the stressstrain analysis is performed in the axial compression and the IDT tests seems to be insignificant. This difference and possibly anisotropy may exist when the axial compression cylinders and the IDT specimens are compared. However, due to very small strain levels used in these tests (50 to 80 microstrains), the dynamic modulus test more or less “tickles” the mastic and does not fully capture the effect of these differences that are mostly related to the large aggregate orientation.
Comparison of Phase Angles It was observed that the phase angle obtained from axial compression testing is normally between the phase angles calculated from the horizontal and vertical strains in IDT testing. Based on this observation, the phase angles calculated from the horizontal and vertical strains were averaged and plotted in Figs. 57 and 58. The
FIGURE 57 Phase angle mastercurves for S9.5AFine mixture. (Kim et al. 2004, with permission from Transportation Research Board.)
C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t
FIGURE 58 Phase angle mastercurves for I19.0DFine mixture. (Kim et al. 2004, with permission from Transportation Research Board.)
averaged phase angles are close to the values from the axial compression testing. This finding needs to be refined further using a more rigorous approach. Kim et al. (2005) contains more phase angle comparisons between IDT and axial compression tests.
Poisson’s Ratio In order to investigate the cause(s) for the difference in the phase angles determined from the horizontal and vertical strains, Poisson’s ratios of the four representative mixtures were calculated from the IDT test results and plotted in Fig. 59. First to note from this figure is that the average values of Poisson’s ratio at different temperatures seem to be reasonable, that is, about 0.18 at −10°C, about 0.25 at 10°C, and around 0.45 at 35°C. Some of Poisson’s ratio values at lower frequencies and 35°C exceeded the linear elastic limit of 0.5, indicating that at these conditions specimens were damaged during the dynamic modulus test. Another important observation from Fig. 59 is that there is a slight dependence of Poisson’s ratio on the loading frequency. This frequency dependence of Poisson’s ratio results in the phase lag between the vertical and horizontal strains, which in turn produces different phase angles calculated from the vertical and horizontal strains in Figs. 57 and 58.
Conclusions In this chapter, an analytical solution for the dynamic modulus of asphalt mixtures tested in the IDT mode is developed using the theory of linear viscoelasticity. The accuracy of this solution was successfully validated using the experimental data
135
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Chapter Five
FIGURE 59 Poisson’s ratios at (a) −10°C; (b) 10°C; and (c) 35°C. (Kim et al. 2004, with permission from Transportation Research Board.)
obtained from the axial compression and IDT testing of 24 commonly used North Carolina mixtures. It was also found that Poisson’s ratio is a weak function of loading frequency. This frequency dependence of Poisson’s ratio results in different phase angles, depending on whether the horizontal strain or the vertical strain is used. An average of these two phase angles yields similar values to the axial compression phase angle. The findings presented in this chapter provide the analytical solutions that may be used in a standard test and analysis protocol for the determination of dynamic modulus and phase angle mastercurves of asphalt mixtures using the IDT test. This protocol will be an essential means of using the dynamic modulus in forensic analysis that utilizes the mechanistic principles in the NCHRP 137A Guide for Design of New and Rehabilitated Pavement Structures developed under the NCHRP project 137A (2004).
C o m p l e x M o d u l u s f r o m t h e I n d i r e c t Te n s i o n Te s t
Acknowledgment This material is based on work supported by the North Carolina Department of Transportation under Project No. HWY200309. The authors gratefully acknowledge this support.
References ASTM D675203, 2003, Standard Test Method for Bulk Specific Gravity and Density of Compacted Bituminous Mixtures Using Automatic Vacuum Sealing Method. American Society for Testing and Materials. Barksdale, R. D., J. Alba, N. P. Khosla, Y. R. Kim, P. C. Lambe, and M. S. Rahman, 1997, Laboratory Determination of Resilient Modulus for Flexible Pavement Design. Final Report, National Cooperative Highway Research Program 128 Project, National Research Council, Washington, D.C. Hondros, G, 1959, Evaluation of Poisson’s Ratio and the Modulus of Materials of a Low Tensile Resistance by the Brazilian (Indirect Tensile) Test with Particular Reference to Concrete. Australian Journal of Applied Science, Vol. 10, No. 3, pp. 243–268. Kim, Y. R., J. Daniel, and H. Wen, 2000, Fatigue Performance Evaluation of WesTrack and Arizona SPS9 Asphalt Mixtures Using Viscoelastic Continuum Damage Approach. Final report to Federal Highway Administration/North Carolina Department of Transportation. Kim, Y. R., Y. Seo, M. King, and M. Momen, 2004, Dynamic Modulus Testing of Asphalt Concrete in Indirect Tension Mode. Journal of Transportation Research Board, No. 1891, National Research Council, Washington, D.C., pp. 163–173. Kim, Y. R., M. Momen, and M. King, 2005, Typical Dynamic Moduli for North Carolina Asphalt Concrete Mixtures. Final report to the North Carolina Department of Transportation, Report No. FHWA/NC/200503. NCHRP 137A Research Team, 2004, Guide for MechanisticEmpirical Design of New and Rehabilitated Pavement Structures, Final Report. NCHRP 137A, ARA, Inc. and ERES Consultants Division. Roque, R., and W. G. Buttlar, 1992, The Development of a Measurement and Analysis System to Accurately Determine Asphalt Concrete Properties Using the Indirect Tensile Mode. Proceedings, The Association of Asphalt Paving Technologist, pp. 304–333.
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CHAPTER
6
Interrelationships among Asphalt Concrete Stiffnesses Ghassan R. Chehab and Y. Richard Kim
Abstract The chapter discusses the three major response functions used for characterizing the linear viscoelastic behavior of asphalt concrete mixtures. Definitions and analytical representations of those functions: creep compliance, relaxation modulus, and complex modulus, are presented. Methods for the determination of the functions’ analytical parameters through experimental tests are introduced. Additionally, numerical and analytical interconversion techniques to determine one LVE response function from the other are also presented and compared. Numerical examples and plots are included to supplement the methodologies presented.
Introduction Asphalt concrete exhibits time/rate dependence, where the material response is not only a function of the current input, but also of the current and past input history. When the loading conditions do not cause damage to the asphalt mixture, the response could be defined as linear viscoelastic and expressed through the convolution (hereditary) integral. While viscoelasticity is typically associated with a system’s timedependent response, linearity is associated with systems where the conditions of homogeneity and superposition are satisfied: Homogeneity: R {AI} = A R {I} Superposition: R {I1 + I2} = R {I1} + R {I2}
(61) (62)
where I, I1, I2 = input histories R = response A = arbitrary constant
139 Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.
140
Chapter Six The brackets { } indicate that the response is a function of the input history. The homogeneity condition, also referred to as proportionality, essentially states that the output is directionally proportional to the input, for example, if the input is doubled, the response doubles as well. The superposition condition states that the response to the sum of two inputs is equivalent to the sum of the responses from the individual inputs. For linear viscoelastic (LVE) materials, the inputresponse relationship is expressed through the hereditary integral which allows the response to any input history to be calculated as follows: t
R=
∫R
−∞
H
(t , τ )
dI dτ dτ
(63)
where RH = unit response function I = input t = time of interest t = integration variable With a known unit response function, the lower limit of the integration can be reduced to 0− (just before time zero) if the input starts at time t 0 and both the input and response are equal to zero at t < 0. The value of 0− is used instead of 0 to allow for the possibility of a discontinuous change in the input at t 0. For notational simplicity, 0 is used as the lower limit in all successive equations and should be interpreted as 0− unless specified otherwise. Equation (63) is applicable to an aging system in which the time zero is the time of production of the material rather than the time of load application. In a nonaging system, time zero corresponds to the onset of load application, irrespective of when the material was produced. In this text, asphalt concrete will be treated as a nonaging system; thus, Eq. (63) reduces to t
R = ∫ RH (t − τ ) 0
dI dτ dτ
(64)
Types of LVE Response Functions Several viscoelastic response functions can be used to characterize the LVE behavior of asphalt concrete (AC), the most fundamental ones being the relaxation modulus E(t), creep compliance D(t), and complex modulus E*. The need for different types of response functions is attributed to a number of factors including type of loading application, conditions under which the material is characterized, and the experimental testing difficulties that constrain the determination of those functions. While those response functions, or linear viscoelastic properties, are fundamental for characterizing the AC in the linear viscoelastic range, they also serve as key components in constitutive models that describe the nonlinear behavior of AC under damage. Additionally, when AC specimens are used in experimental testing, response functions can be used as “viscoelastic fingerprints” to evaluate the specimentospecimen variation and/or determine if the material has been damaged or not.
Creep Compliance and Relaxation Modulus The creep compliance D(t) is the ratio of strain response to a constant stress input; while the relaxation modulus E(t) is the ratio of stress response to a constant strain input. If
Interrelationships among Asphalt Concrete Stiffnesses asphalt concrete were purely elastic, then D(t) and E(t) would be the reciprocal of each other. However, due to the viscoelastic nature of asphalt concrete this is only true in the Laplace transform domain. In equation form D(t) =
ε (t) , σ0
E(t) =
σ (t) ε0
and
(65)
(66)
where D(t) and E(t) = creep compliance and relaxation modulus, respectively s0 and e0 = constant input stress and strain, respectively s (t) and e(t) = stress and strain response, respectively For uniaxial loading, and nonaging, isothermal conditions, the linear viscoelastic stressstrain relationships are represented by the Boltzmann convolution integral as follows: t
σ = ∫ E(t − τ ) 0
t
ε = ∫ D(t − τ ) 0
dε dτ dτ
(67)
dσ dτ dτ
(68)
where t is an integration variable. Substituting e in the righthand side of Eq. (67) by its equivalent from Eq. (68) results in the constitutive equation relating D(t) and E(t) as follows: t
1 = ∫ E(t − τ ) 0
dD(t) dτ dτ
(69)
Complex Modulus The complex modulus E∗ is a response function that relates stresses and strains for a linear viscoelastic material subjected to sinusoidal loading. It is composed of two components: dynamic modulus E∗ and phase angle f, defined as follows:
E* =
σ amp ε amp
,
and
φ = 2π fΔt where samp = stress amplitude eamp = strain amplitude f = loading frequency Δt = time lag between stress and strain response
(610)
(611)
141
142
Chapter Six Being a complex function E* is composed of real and imaginary components, referred to as the storage and loss moduli, respectively, and mathematically expressed as follows: E * = E′ + iE′′ where
(612)
E′ = E * cos φ storage modulus E″ = E * sin φ loss modulus E * = (E′)2 + (E″ )2 i = −1
Determination of LVE Response Functions Viscoelastic response functions can be determined either through experimental testing conducted in the LVE range or through interconversion from other known response functions. From the theory of viscoelasticity, it can be shown that all LVE response functions are interrelated; thus, any function can be obtained if another is known. Both the creep compliance and complex modulus tests are simple mechanical tests that allow for the accurate characterization of AC in the LVE range. From the creep test D(t) is determined as a function of time; whereas, from the complex modulus test E* and f are determined as a function of frequency. The simplicity of obtaining D(t) and E* from mechanical tests is countered with a difficulty in obtaining E(t) from the relaxation test which is more difficult to conduct and requires a high capacity and robust testing machine. Therefore, it is often the case where E(t) is obtained through interconversion from D(t) or E*. Interconversion can also be necessary where one material function cannot be determined from a single test type over the entire range of the domain needed. For example, E(t) and D(t) cannot be determined at very short times; in this case, E* is determined by conducting a complex modulus test for the corresponding range of interest in frequency domain and then converted to E(t) and D(t). Prior to elaborating on the different interconversion techniques, it is necessary to present and discuss the analytical representation of the response functions since that will impact the choice and accuracy of the interconversion method used.
Analytical Representation of LVE Response Functions For accurate material characterization to be achieved, it is essential that representative analytical expressions of LVE response functions be established regardless of how those functions are obtained. For example, if an analytical expression is to be established for E*, complex modulus tests are first conducted at several temperatures and frequencies. The timetemperature superposition principle is then applied to obtain a single mastercurve for E* and f as a function of reduced frequency at a reference temperature of choice. This has been covered in more depth in Chap. 4. A mathematical function is then fit to the mastercurve to arrive to a representative analytical expression of that response for a broad frequency (time) range.
Interrelationships among Asphalt Concrete Stiffnesses
PowerLaw Representations Various forms of powerlaw expressions have been used to represent the viscoelastic response. Usually, those representations, although rough and simplistic, yield fits that are globally smooth and stable (Park et al. 1996). Presented below are some of the common powerlaw expressions commonly used in fitting viscoelastic response functions. Although the expressions are presented for D(t), they can similarly apply to E(t).
Pure Power Law The pure power law (PPL) is the simplest among all powerlaw representations; it is given by the following form: D(t) = D1 t n
(613)
where the constant D1 is the value of the creep compliance at t = 1 and the exponent n is obtained by identifying the representative slope of the experimental data over the transient region plotted on a loglog scale. The limitation of the PPL is its inability of representing the data in regions other than the transient zone (Park and Kim 1996).
Generalized Power Law The generalized power law (GPL) can also be used to represent the response functions. It is expressed as D(t) = Dg + D1 t n
(614)
where Dg is the glassy compliance: Dg = lim D(t) . A sample graphical representation is t→ 0 given in Fig. 61 with Dg = 7.0E − 5 MPa−1, D2 = 2.3E − 3 MPa−1, and n = 0.45. The GPL fits the response data more accurately than the PPL in that it simulates the shorttime
FIGURE 61 Different analytical representations of the experimental creep data. (Park et al. 1996.)
143
144
Chapter Six behavior more realistically due to the presence of the factor Dg ; however, it fails to simulate the data at long times.
Modified Power Law The representation of the modified power law (MPL) (Williams 1964) is of the following form: D(t) = Dg +
De − Dg
τ⎞ ⎛ ⎜⎝ 1 + t ⎟⎠
n
(615)
and is plotted in Fig. 61 with Dg = 7.0E − 5 MPa−1, De = 6.6E − 2 MPa−1, t = 1.5E + 3 seconds, and n = 0.45. The constant De is the longtime equilibrium or rubbery compliance which is defined by De = lim D(t). Constants Dg and De can be determined through inspection of the t→∞ experimental data. As for t and n, their determination typically requires nonlinear analysis; however, both can be reasonably estimated by an appropriate simplified procedure. The MPL fits the data much better than the two aforementioned power laws by generating a characteristic, broadband, Sshaped curve. In particular, the MPL is capable of describing the glassy and rubbery behavior at short and long times, respectively. The exponent n gives the slope of the creep curve through the transition region between the glassy and rubbery behavior, and t fixes a characteristic retardation time (Park et al. 1996). As observed, the MPL representation ceases to fit the data satisfactorily at the top and bottom asymptotes (where the curvatures are maximum), a shortcoming attributed to the limited degrees of freedom associated with the expression. Enhancing the fit requires expanding the degrees of freedom such as in the case of a powerlaw series representation (Park et al. 1996). To enhance the fit of the experimental data, Park et al. (1996) investigated the powerlaw series representation in the following form: M
D(t) = Dg + ∑ i =1
ˆ D i
τˆi ⎞ ⎛ ⎜⎝ 1 + t ⎟⎠
n
(616)
ˆ and τˆ (i = 1, …, M), n, M are all constants. A fixed value for n is chosen a priori where D i i in order to allow only two degrees of freedom per term in series [Eq. (616)] just as in the PPL. Although a single n yields a satisfactory fitting, one may consider a discrete set of exponents, ni ( i = 1,…, M) instead of using a single, fixed n. Four cases with different Ms and ti ’ s were considered in fitting creep mastercurves. In general, the fit improves as the number of terms increases; however, the fiveterm (M = 5) representation was found to be sufficient in accurately fitting the experimental data as shown in Fig. 61.
Prony Series A Prony (Dirichlet) series consisting of a sequence of decaying exponentials has been widely used to represent viscoelastic response (Schapery 1961; Tschoegl 1989). The popularity of this series representation is attributed mainly to its ability of describing a wide range of viscoelastic response and to the relatively simple and rugged computational efficiency associated with its exponential basis functions. In addition, the Prony series representation of an LVE response function has a physical basis in the theory of mechanical models dealing with linear springs and dashpots (Park et al. 1996).
Interrelationships among Asphalt Concrete Stiffnesses
Analytical Representation For creep compliance, the Prony series representation is of the following form: M
D(t) = D0 + ∑ Dm [1 − e − t/τm ]
(617)
m
where tm = retardation time Dm = regression coefficient D0 = glassy compliance D(t) . The glassy compliance denotes the shorttime creep behavior, that is, D0 = lim t→ 0 The righthand side of Eq. (617) represents a mechanical model often referred to as the generalized Voigt model. It implies that the viscoelastic retardation process of the material can be regarded as a superposition of elementary processes in which the strain retards exponentially. For relaxation modulus, the Prony series expression takes a similar form: M
E(t) = E∞ + ∑ Em e − t/ρ
m
(618)
m= 1
where E∞, rm, and Em are longtime equilibrium modulus, relaxation time, and Prony regression coefficients, respectively. Physically, this representation is related to the Wiechert (or generalized Maxwell) model. Figure 62 provides a graphical illustration of the assembly of springs and dashpots associated with the Voigt and Wiechert models, respectively.
Fitting of Experimental Data There are a number of ways of fitting the Prony series expression to the given experimental data. Cost and Becker (1970) developed the socalled multidata method based on the least squares scheme and applied it to fit the Laplacetransformed Prony series to the Laplacetransformed data. Schapery (1961) used the collocation method to fit a Prony series model to experimental data obtained from tensile relaxation of polymethyl methacrylate and dynamic shear creep tests of polyisobutylene, while Park and Kim (1996) used it on data from relaxation and creep tests of sandasphalt and asphalt concrete mixtures. An illustration of the collocation method follows. In fitting a creep mastercurve using a Prony series expression, there are 2N unknowns including Di and ti (i = 1,…, N) and a corresponding system of 2N nonlinear equations. However, in order to avoid the complexity that often arises in solving for the unknowns, the relaxation times ti are usually specified a priori from experience, thus leaving Di the only unknowns to be determined by solving the resulting linear system of equations. Usually, onedecade intervals of ti are adequate (Schapery 1974), and the glassy compliance D0 is obtained by extending the experimental curve asymptotically to t = 0. Although the aforementioned methods are simple and straightforward; however, they often yield negative values for the Prony coefficients, a case that is physically unrealistic and often results in oscillations in the reconstructed curve. In an effort to overcome the problem of negative coefficients in a Prony series representation, a number of improved fitting methods have been proposed. Emri and Tschoegl (1993, 1994, 1995) and Tschoegl and Emri (1992, 1993) designed a recursive algorithm to avoid negative coefficients by using only welldefined subsets of the complete set of experimental data. Kashhta and Schwarzl (1994a, 1994b) developed a method that ensures positive coefficients through an interactive adjustment of relaxation or retardation times. Others
145
146
Chapter Six
FIGURE 62 (a) Generalized Voigt model: where h m is the coefﬁcient of viscosity and Dm is the compliance for the mth term and (b) Wiechert model: where h m is the coefﬁcient of viscosity and Em is the stiffness for the mth term.
applied the socalled Tikhonov regularization techniques (Honerkamp and Weese 1989, Elster et al. 1991) and the maximum entropy method (Elster and Honerkamp 1991) to overcome the difficulties in finding the coefficients due to the illposed nature (or nonuniqueness) of the problem. Baumgaertel and Winter (1989) found a series representation through a nonlinear regression in which the spectra, time constants, and the number of terms in the series were variable; they started with a large number of relaxation modes (usually 0.5 to 1 decade apart) and merged or eliminated unnecessary ones whenever negative coefficients occurred. Mead (1994) presented a numerical method of determining discrete line spectra based on a constrained linear regression.
Presmoothing Experimental Data Prior to Prony Series Fit It is very desirable to use the Prony series representation of linear viscoelastic response due to its amenability to mathematical operations as long as the local waviness of the fit and negative values of some coefficients that might result are resolved. Since those problems are mainly caused by the wide scatter of unsmoothed data, it would be advantageous to presmooth the experimental data to eliminate the scatter prior to fitting it using the Prony series.
Interrelationships among Asphalt Concrete Stiffnesses As concluded previously, the powerlaw series provides a globally smooth and broadband representation of the LVE response data. Its main drawback is the associated analytical difficulty when conducting mathematical operations. However, the latter fact does not eliminate the importance of powerlaw series as an ideal candidate to presmooth the data to which a Prony series is subsequently fit. The resulting Prony series fit to the presmoothed experimental data yields a smoother reconstructed curve that is free of local waviness and that is analytically simple for mathematical operations. In an exercise conducted by (Park and Schapery (1999). on AC relaxation data, a fiveterm powerlaw series representation (M = 5) was used to presmooth the experimental data that was later fit to a Prony series. The reconstructed curve from the resulting fit was graphically indistinguishable from the fiveterm powerlaw series representation. Figure 63 illustrates the effect of presmoothing on the quality of the Prony series fit to relaxation data. The above mentioned fitting techniques also hold true for other response functions such as relaxation modulus. After fitting the relaxation data using an MPL series, a Prony series is fit with the coefficients determined using the collocation method (Chehab 2002), as follows: Formulating Eq. (618) in column vectors ({A} and {C}) and matrix [B], the regression coefficients are determined using the following equation (Mun et al. 2007): M
E(tn ) − E∞ = ∑ exp(−tn / ρm ) E m , n 1,…, N m= 1 {C} {A}
(619)
[B]
The nonnegative coefficients {C} are solved for using the embedded linear programming function provided by MATLAB. The following rearranged form forces the coefficients to be positive while still satisfying Eq. (619): minimize [B]{C}−{A}
such that {C} ≥ 0
FIGURE 63 Prony series ﬁts to unsmoothed and smoothed E(t).
(620)
147
148
Chapter Six
FIGURE 64
MPL and Prony series ﬁts to E(t).
Figure 64 shows the MPL series fit of the experimental data and the Prony series fit. As observed, both fits match closely, with the Prony fit being more advantageous due to the analytical simplicity in mathematical applications.
Interconversion between LVE Response Functions Being mathematically equivalent, LVE response functions can be obtained from one another using several mathematical interconversion techniques (Schapery et al. 1999). This holds true for both shear and uniaxial modes of loading. As presented earlier, interconversion may be required when conditions do not allow a response function to be determined through direct experimental testing. A common example is the difficulty in obtaining E(t) from the relaxation test which requires a robust testing machine which may or may not be available. It is therefore more common to obtain E(t) through interconversion of D(t) or E*, both of which are generally easier to obtain from experimental testing. In other instances, a response function cannot be determined over the complete range of its domain from a single test type; in these cases, the response can be obtained for the desired broad range through the superposition of responses obtained through different types of tests. Interconversion is also used when an accurate shorttime response which is difficult to obtain from a test with a transient excitation is alternatively obtained from a test with steadystate sinusoidal excitation. This normally requires an interconversion between responses in time and frequency domains (Park and Schapery 1999). Hopkins and Hamming (1957) were among the early researchers who dealt with the subject of interconversion between linear viscoelastic functions by developing a numerical technique for relating E(t) and D(t). The approach was later improved by Knoff and Hopkins (1972) and Baumgaertel and Winter (1989) who established analytical conversion techniques using interrelationships in the Laplace transform domain and
Interrelationships among Asphalt Concrete Stiffnesses the Prony series representation of both the source and target response functions. Mead (1994) presented a numerical interconversion method based on constrained linear regression with regularization, while Ramkumar et al. (1997) proposed a regularization technique that used a quadratic programming technique. The aforementioned methods as well as others not mentioned have been covered extensively by Schwarzl and Struik (1967), Ferry (1980), and Tschoegl (1989). This chapter sheds light on some of the mostly used techniques in converting between LVE response functions for asphalt materials. Both the E(t) and D(t) are essential elements for characterizing viscoelastic responses. They have been incorporated in several constitutive models that have been developed and used over the years including some of those presented in this book (Schapery 1961 and 1974; Ferry 1980; Christensen 1982; Tschoegl 1989; Kim et al. 1990 and 1997; Lee and Kim 1998; Uzan 1996; Bahia et al. 2000; Daniel and Kim 2002; Roque et al. 2002; Chehab et al. 2003). As for E*, its acceptance as an indicator of the AC mix behavior has been on the rise especially after it was chosen as a Superpave simple performance test for mix design, a QC/QA tool, in addition to an indicator of stiffness in the ME design guide. It is also used for depicting the material’s response under sinusoidal loading. Thus, with the simplicity and wide use of the complex modulus test, using interconversion methods to obtain E(t) and D(t) from E* becomes more favorable than conducting additional experimental tests to obtain them. Interconversion is not always simple and straightforward. Using exact solutions may require integration over an infinite range, a task that is often complicated whether done analytically or numerically. In addition, experimental data required for the interconversion may only be available for a limited range of the required time or frequency domain. To overcome such difficulties, adopting approximate analytical and numerical techniques becomes necessary. In what follows, both approximate and analytical methods of interconversion are presented.
Approximate Numerical Methods Several numerical methods have been used in converting between LVE response functions. According to Taylor et al. (1970) and Park and Schapery (1999), those methods are especially useful when the response functions are expressed in the Prony series form. In what follows, t and w are used as symbols for time and radial frequency, respectively, or as their corresponding temperaturereduced quantities.
Interconversion between E(t) and D(t) The exact relationship between relaxation modulus E(t) and creep compliance D(t) in time domain is given by the following integral: t
∫ E(t − τ ) 0
dD(τ ) dτ = 1 dτ
(621)
Taking the Laplace transform of Eq. (621) yields E(s) D(s) = ∞
1 (for t > 0) s2
(622)
where f (s) ≡ ∫ f (t)e − st dt denotes the Laplace transform of f(t) and s is the transform 0 parameter. In solving the integral, a typical numerical approach requires that the integral be divided into a large number of time segments. This can be easily achieved when the functions are presented in their Prony series form. This being the case, when
149
150
Chapter Six {ri, Ei(i = 1,…, m) and Ee} or {tj, Dj (j = 1,…, n), Dj and h0} are known and the target time constants specified, the unknown set of constants can be determined through a system of linear algebraic equations. For example, creep compliance in its Prony series form, Dj (j 1,…, n), can be determined from relaxation modulus E(t) as follows: [ A]{D} = {B}
(623)
or AkjDj = Bk (summed on j; j = 1,…, n; k = 1,…, p) where
ρi Ei −(t /ρ ) (e − e − ( t /ρ ) ) ρi − τ j
m
{
Akj = Ee (1 − e −(t /τ ) ) + ∑ k
j
k
i=1
i
k
i
ρi ≠ τ j
(624)
or Ee (1 − e −(t
k
/τ j )
m
tk Ei −(t (e τj i=1
)+∑
k
/ ρi )
)
ρi = τ j
and m ⎛ ⎞ Bk = 1 − ⎜ Ee + ∑ Ei e −(t /ρ ) ⎟ ⎝ ⎠ i =1 k
i
m ⎛ ⎞ + E Ei ⎟ ∑ ⎜⎝ e ⎠
(625)
i =1
The symbol tk (k 1,…, p) denotes a discrete time corresponding to the upper limit of integration in Eq. (621). Once the model constants Dg, Dj, and tj are determined, then function D(t) can be obtained in its Prony series form. Similarly, E(t) can be determined from D(t) by determining the unknown constants of the Prony series representation of E(t): Ei (i 1,…, m). As for Dg and Ee, they are related as follows: Dg ≡
1 m
Ee + ∑ Ei i =1
Ee ≡
1
(626)
n
Dg + ∑ D j j =1
Equations (623) through (626) were used in converting E(t) data of an AC mixture to D(t). The source and target Prony series coefficients are presented in Table 61, with corresponding fitted data plotted in Fig. 65. For additional details and derivations of the above equations, the reader is referred to Park and Schapery (1999) where other approximate interconversion techniques available in literature are documented and compared. Only a limited number of common techniques are presented in the next section.
QuasiElastic Interrelationship The most intuitive and crude interrelationship between the E(t) and D(t) is the one based on quasielastic approximation: E(t)D(t) ≅ 1 (for t > 0)
(627)
Interrelationships among Asphalt Concrete Stiffnesses
Relaxation Modulus (MPa) 34.5
E•
Creep Compliance (1/MPa) D0
4.00E–05
ti (sec)
Di
Prony Coefficients ri (s)
Ei
1.0E−06
9.34E+02
1.0E−06
1.26E−06
1.0E−05
4.09E+03
1.0E−05
6.53E−06
1.0E−04
5.39E+03
1.0E−04
1.54E−05
1.0E−03
4.84E+03
1.0E−03
2.48E−05
1.0E−02
5.19E+03
1.0E−02
5.27E−05
1.0E−01
2.96E+03
1.0E−01
1.64E−04
1.0E+00
1.31E+03
1.0E+00
3.33E−04
1.0E+01
1.33E+02
1.0E+01
2.93E−03
1.0E+02
1.14E+02
1.0E+02
3.21E−03
1.0E+03
3.57E+01
1.0E+03
1.63E−02
1.0E+04
1.44E+00
1.0E+04
1.0E+05
−3.00E–01
1.0E+05
TABLE 61
Prony Series Coefficients for E(t) and D(t) for an AC Mix
FIGURE 65 E(t) and D(t) from interconversion for an AC mixture.
7.48E–03 –1.52E–0
151
152
Chapter Six The interconversion in Eq. (627) provides a good relation between the response functions when the material exhibits mostly elastic behavior with minimal viscoelasticity.
PowerLawBased Interrelationship An LVE material can be approximately represented by simple power law for small ranges in their transition zone. Representing E(t) and D(t) in pure powerlaw form [Eqs. (628) and (629), respectively], interrelationship Eq. (630) is obtained. E(t) = E1t − n
(628)
D(t) = D1t n
(629)
E(t)D(t) =
sin nπ nπ
(630)
where E1, D1, and n are all positive constants. Equation (630) was first given by Leaderman (1958). It is accurate in regions in which E(t) and D(t) are represented approximately by straight lines on loglog scales, with the exponent n being the absolute value of the slope of these lines. When n approaches zero, that is, for an elastic material, the righthand side of Eq. (630) becomes unity and Eq. (627) becomes exact. In equation form, n is represented as n=
d log RH (t) d log t
(631)
where RH(t) is the unit response function of interest.
Interrelationship by Christensen Christensen (1982) developed an approximate interconversion method between E(t) and D(t) using approximate relationships between the real and imaginary parts of a complex material function, and between the transient function and the real part of the complex material function: If E(t) is known then D(t) can be determined according to Eq. (632): E(t)
D(t) ≅ E 2 (t) +
π 2 t 2 ⎧ dE(t) ⎫ ⎨ ⎬ 4 ⎩ dt ⎭
2
(632)
Equation (632) is also applicable when D(t) and E(t) are interchanged. In terms of n, Eq. (632) can be expressed as follows: E(t)D(t) ≅
1 n2 π 2 1+ 4
(633)
Interrelationship by Denby An approximation similar to that presented above was proposed by Denby (1975), yielding the following interrelationship: E(t)D(t) ≅
1 n2 π 2 1+ 6
(634)
Interrelationships among Asphalt Concrete Stiffnesses Park et al. (1996) found that Eqs. (630), (633), and (634) yield comparable and accurate relationships for interconversion. This holds true as long as both functions E(t) and D(t) exhibit broadband and smooth variations on loglog scales. On that same premise, the following interconversion scheme has also been proposed: D(t) =
1 , E(α t)
E(t) =
1 ⎛ t⎞ D⎜ ⎟ ⎝α⎠
and
(635)
(636)
where ⎛ sin nπ ⎞ α =⎜ ⎝ nπ ⎟⎠
1
n
(637)
Interconversion between E* and D* The significance of using response functions in their complex domain in characterizing the LVE behavior of asphalt materials has been highlighted here as well as in other chapters. On that premise, it thus becomes important to establish a relationship between the material functions in the complex domain, specifically between E* and D*. Deriving such a relationship is more convenient when E(t) and D(t) are represented in the Laplace domain as follows (Tschoegl 1989): ∞
E (s) ≡ s ∫ E(t)e − st dt 0
(638)
∞
(s) ≡ s D(t)e − st dt D ∫ 0
(639)
where the integrals in Eqs. (638) and (639) are the Laplace transforms of E(t) and D(t), respectively. Incorporating the above equations with Eq. (621) yields the following relationship between the relaxation and creep compliance in the Laplace domain: (s) = 1 E (s)D
(640)
Complex material functions arise from the response to steadystate sinusoidal loading with angular frequency w, and are related to the Laplacetransformed functions according to the following relationship proposed by Tschoegl (1989): E * (w) = E (s) (s) D * (w ) = D
s→iw
s→iw
(641) (642)
Hence, the relationship between E* and D* can be deduced from Eqs. (640) to (642): E * (w)D* (w) = 1
(643)
153
154
Chapter Six The real and imaginary parts, denoted with primes and double primes, respectively, are expressed as follows: E* (w) ≡ E ′(w) + iE ′′(w)
(644)
D* (w) ≡ D′(w) − iD′′(w)
(645)
where D″ is positive. The real component of the response functions is also referred to as the storage component, while the imaginary is referred to as the loss component. The Prony series expression of the storage and loss components of modulus and creep are presented below (Park and Schapery 1999): m
E′(w) = Ee + ∑ i=1
m
E′′(w) = ∑ i=1
w 2 ρi2Ei w 2 ρi2 + 1
(646)
wρiEi w 2 ρi2 + 1
(647)
Dj w τ 2j + 1 j=1 n
D′(w) = Dg + ∑ D′′(w) =
(648)
2
n wτ j Dj 1 +∑ 2 2 η0 w j=1 w τ j + 1
(649)
It is observed from Eqs. (646) to (649) that if the Prony series of either the real or imaginary components of a complex function are known, then the series representation of the other component can be determined. The components of E* and D* can be interrelated using Eqs. (643), (644), and (645): D′ =
E′ (E ′)2 + (E ′′)2
(650)
D″ can then be obtained in terms of the same set of constants. E* and D* can be related by expressing E′ E″, and D′ in Eq. (650) by their Prony series representations in Eqs. (646) to (648), respectively. The interconversion can be achieved using the same form of Eq. (623) with the following Akj and Bk: Akj =
1 , wk2τ 2j + 1
and m
Bkj =
Ee + ∑ i=1
(651)
w 2 ρi2Ei w 2 ρi2 + 1 2
⎛ w ρ Ei ⎞ ⎛ wρiEi ⎞ ⎜⎝ Ee + ∑ w 2 ρ + 1 ⎟⎠ + ⎜⎝ ∑ w 2 ρ 2 + 1 ⎟⎠ i i=1 i=1 m
2
2 i 2 i
m
2
−
1 m
Ee + ∑ Ei
(652)
i=1
where wk (k = 1,…, p) is the angular frequency. It is selected using the same procedure used in selecting tk in Eqs. (624) and (625). Similarly, Dg can be computed from Ee and Ei according to Eq. (626).
Interrelationships among Asphalt Concrete Stiffnesses
Approximate Analytical Methods Numerous approximate analytical interconversion methods with different bases, simplifications, assumptions, and accuracies have been proposed by Tschoegl (1989), Schapery (1962), and Park et al. (1996), and others. Methods of primary interest are those that have been widely used by asphalt researchers to relate between the time, frequency, and Laplace domain representations of modulus and compliance. Schapery (1962) presented two approximate analytical methods for interconversion between the uniaxial relaxation modulus E(t) and the operational modulus E (s), defined as the Carson transform or the smultiplied Laplace transform of E(t); refer to Eq. (653). ∞
∞
0
−∞
E (s) ≡ s ∫ E(t)e − st d(ln t) = s ∫ E(t)te − st d(ln t)
(653)
The first relationship is given by E(t) ≅ E (s)
or
s =α t
E (s) ≅ E(t) t=α
s
(654)
where E (s) ≡ sE(s) , E(s) is the Laplace transform of E(t); and a = e−C, where C is Euler’s constant, thus yielding a ≅ 0.56. The second relationship is given as follows: E (s) ≅ E(t) t= β
(655)
s
−1
where β = {Γ(1 − n)} n , Γ(.) is the gamma function, and n is the local loglog slope of the E ( s ) E( t ) source function given either as n ≡ d log or n ≡ d log . If the moduli in Eqs. (654) d log s d log t ∼ and (655) are replaced by compliances, analogous relationships between D(t) and D (s) are obtained. Christensen (1982) proposed an approximate interconversion between E(t) and the storage modulus E′(w) as follows: E(t) ≅ E ′(w) w =2 π t
or
E ′(w) ≅ E(t) t= 2 π w
(656)
Analogous relationships hold for compliance functions when E′s in Eq. (656) are replaced by D′s. Staverman and Schwarzl (1955) gave the following approximate conversion from storage modulus E′(w) to the loss modulus E″(w): E′′(w) ≅
π dE ′(w) 2 d ln(w)
(657)
Booij and Thoone (1982) proposed the following conversion from E″(w) to E′(w): E ′(w) ≅ Ee −
π w d[E ′′(w)/w] , d ln w 2
E ′(w) ≅ Ee +
d ln E ′′ ⎞ π⎛ E ′′(w) 1− d ln w ⎟⎠ 2 ⎜⎝
or
(658) (659)
where Ee is the equilibrium modulus. Equations (657) and (659) also apply to compliance components when E′(w) and E″(w) are replaced by D′(w) and −D″(w), respectively.
155
156
Chapter Six A more recent interconversion technique has been developed by Schapery and Park (1999). The method employs variable adjustment factors dictated by the slope of the source function on a loglog scale. Presented below is a set of relationships for interconversion of relaxation modulus and its components. Similar relationships apply for creep compliance when the appropriate changes in parameters are made (Schapery ∼ ∼ and Park 1999), as follows: E( ) → D( ), n → −n, E → D, E′ → D′, and E″ → −D″. The sign change in E″ → −D″ necessitates a change in the sign of n so that the arguments of the trigonometric functions appearing in the set of l parameters used in the interconversion remain the same when compliance and modulus are interchanged. 1 E(t) ≅ E (s) s=(1/t ) λ E(s) ≅ λ E (t)
λ = Γ(1 − n)
(660)
t = (1 s)
E(t) ≅
1 E ′ (w ) w = ( 1 t ) λ′
E ′(w) ≅ λ ′E (t) t = (1 w )
E(t) ≅
1 E ′′ (w) w =(1 t ) λ ′′
E ′′(w) ≅ λ ′′E (t) t=(1/w ) 1 E (s) ≅ E ′ (w) w = s λ E ′(w) ≅ λ E (s) s= w 1 E (s) ≅ E ′′ (w) w = s λ E ′′(w) ≅ λ E (s) s= w 1 E ′(w) ≅ E ′′ (w ) w = w λ E ′′(w) ≅ λ E ′ (w ) w = w
λ ′ = Γ(1 − n)cos(nπ 2)
(661)
λ ′′ = Γ(1 − n)sin(nπ/2)
(662)
λ = cos(nπ /2)
(663)
λ = sin(nπ 2)
(664)
λ = tan(nπ 2)
(665)
In obtaining a Prony series representation for E(t) of an AC mix, the approximate interconversion in Eq. (661) was applied to experimental data obtained from E* testing. The E(t) obtained from interconversion were then fit to a Prony series as shown in Fig. 66.
Interrelationships among Asphalt Concrete Stiffnesses
FIGURE 66
Conversion from E′ to E(t) for an AC mix.
References Bahia, H., Zeng, M., and Nam, K. (2000), “Consideration of Strain at Failure and Strength in Prediction of Pavement Thermal Cracking,” Journal of Asphalt Paving Technology, AAPT, Vol. 69, pp. 497–539. Baumgaertel, M., and Winter, H. H. (1989), “Determination of Discrete Relaxation and Retardation time Spectra from Dynamic Mechanical Data,” Rheologica Acta, Vol. 28, pp. 511–519. Booij, H. C., and Thoone, G. P. (1982), “Generalization of Kramers–Kronig Transforms and Some Approximations of Relations between Viscoelastic Quantities,” Rheologica Acta, Vol. 21, pp. 15–24. Chehab, G. R. (2002), “Characterization of Asphalt Concrete in Tension Using a Viscoelastoplastic Model,” Ph.D. dissertation, North Carolina State University, Raleigh, N.C. Chehab, G. R., Kim, Y. R., Schapery, Y. R., Witczack, M., and Bonaquist R. (2002), “TimeTemperature Superposition Principle for Asphalt Concrete Mixtures with Growing Damage in Tension State,” Journal of Asphalt Paving Technology, AAPT, Vol. 71, pp. 559–593. Chehab, G. R., Kim, Y. R., Schapery, R. A., Witczack, M., and Bonaquist, R. (2003), “Characterization of Asphalt Concrete in Uniaxial Tension Using a Viscoelastoplastic Model,” Journal of Asphalt Paving Technology, AAPT, Vol. 72, pp. 315–355. Christensen, R. M. (1982), Theory of Viscoelasticity, 2d ed., Academic Press, New York, 1982, Section 4.6. Cost, T. L., and Becker, E. B. (1970), “A MultiData Method of Approximate Laplace Transform Inversion,” International Journal for Numerical Methods in Engineering, Vol. 2, pp. 207–219.
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Chapter Six Daniel, J. S., and Kim, Y. R. (2002), “Development of a Simplified Fatigue Test and Analysis Procedure Using a Viscoelastic Continuum Damage Model,” Journal of Asphalt Paving Technology, Vol. 71, pp. 619–650. Elster, C., and Honerkamp, J. (1991), “Modified Maximum Entropy Method and Its Application to Creep Data,” Macromolecules, Vol. 24, pp. 310–314. Elster, C., Honerkamp, J., and Weese, J. (1991), “Using Regularization Methods for the Determination of Relaxation and Retardation Spectra of Polymeric Liquids,” Rheologica Acta, Vol. 30, pp. 161–174. Emri, I., and Tschoegl, N. W. (1993), “Generating Line Spectra from Experimental Responses, Part I. Relaxation Modulus and Creep Compliance,” Rheoogica Acta, Vol. 32, pp. 311–312. Emri, I., and Tschoegl, N. W. (1994), “Generating Line Spectra from Experimental Responses, Part IV. Application to Experimental Data,” Rheologica Acta, Vol. 33, p. 6070. Emri, I., and Tschoegl, N. W. (1995), “Determination of Mechanical Spectra from Experimental Responses,” International Journal of Solid Structures, Vol. 32, pp. 817–826. Ferry, J. D. (1980), Viscoelastic Properties of Polymers, 3d ed., John Wiley & Sons, Inc., New York. Honerkamp, J., and Weese, J. (1989), “Determination of the Relaxation Spectrum by a Regularization Method,” Macromolecules, Vol. 22, pp. 4372–4377. Hopkins, I. L., and Hamming, R. W. (1957), “On Creep and Relaxation,” Journal of Applied Physics, Vol. 28, pp. 906–909. Kashhta, J., and Schwarzl, F. R. (1994a), “Calculation of Discrete Retardation Spectra from Creep Data: I. Method,” Rheologica Acta, Vol. 33, pp. 517–529. Kashhta, J., and Schwarzl, F. R. (1994b), “Calculation of Discrete Retardation Spectra from Creep Data: II. Analysis of Measured Creep Curves,” Rheologica Acta, Vol. 33, pp. 530–541. Kim, Y. R., and Little, D. L. (1990), “OneDimensional Constitutive Modeling of Asphalt Concrete,” ASCE Journal of Engineering Mechanics, Vol. 116, No. 4, pp. 751–772. Kim, Y. R., Lee, H. J., and Little, D. N. (1997), “Fatigue Characterization of Asphalt Concrete Using Viscoelasticity and Continuum Damage Theory,” Journal of Asphalt Paving Technology, Vol. 66, pp. 520–569. Knoff, W. L., and Hopkins, I. L. (1972), “An Improved Numerical Interconversion for Creep Compliance and Relaxation Modulus,” Journal of Applied Polymer Science, Vol. 16, pp. 2963–2972. Leaderman, H. (1958), “Viscoelasticity Phenomena in Amorphous High Polymeric Systems,” Rheology, Vol. II, F. R. Eirich, Academic, New York. Lee, H. J., and Kim, Y. R. (1998), “A Uniaxial Viscoelastic Constitutive Model for Asphalt Concrete under Cyclic Loading,” ASCE Journal of Engineering Mechanics, Vol. 124, No. 11, pp. 1224–1232. Mead, D. W. (1994), “Numerical Interconversion of Linear Viscoelastic Material Functions,” Journal of Rheology, Vol. 38, pp. 1769–1795. Mun, S., Chehab, G. R., and Kim, Y. R. (2007), “Determination of TimeDomain Viscoelastic Functions Using Optimized Interconversion Techniques,” Road Materials and Pavement Design, Lavoisier, Vol. 8, No. 2, pp. 351–365. Park, S. W., and Schapery, R. A. (1999), “Methods of Interconversion between Linear Viscoelastic Material Functions, Part I—A Numerical Method Based on Prony Series,” International Journal of Solids and Structures, Vol. 36, pp. 1653–1675.
Interrelationships among Asphalt Concrete Stiffnesses Park, S. W., Kim, Y. R., and Schapery, R. A. (1996), “A Viscoelastic Continuum Damage Model and Its Application to Uniaxial Behavior of Asphalt Concrete,” Mechanics of Materials, Vol. 24(4), pp. 241–255. Ramkumar, D. H. S., Caruthers, J. M., Mavridis, H., and Shroff, R. (1997), “Computation of the Linear Viscoelastic Relaxation Spectrum from Experimental Data,” Journal of Applied Polymer Science, Vol. 64, pp. 2177–2189. Roque, R., Birgisson, B., Sangpetngam, B., and Zhang, Z. (2002), “Hot Mix Asphalt Fracture Mechanics: A Fundamental Crack Growth Law for Asphalt Mixtures,” Journal of Asphalt Paving Technology, AAPT, Vol. 71. Schapery, R. A. (1961), “A Simple Collocation Method for Fitting Viscoelastic Models to Experimental Data,” Report GALCIT SM 6123A, California Institute of Technology, Pasadena, California. Schapery, R. A. (1962), “Approximate Methods of Transform Inversion for Viscoelastic Stress Analysis,” Proc. 4th U.S. Nat. Cong. Appl. Mech., pp. 1075–1085. Schapery, R. A. (1974), “Viscoelastic Behavior and Analysis of Composite Materials,” Composite Materials, Chap. 4, Vol. 2, G. P. Sendeckyj Ed., Academic Press, pp. 85–168. Schapery, R. A., and Park, S. W. (1999), “Methods of Interconversion between Linear Viscoelastic Material Functions. Part II—An Approximate Analytical Method,” International Journal of Solids and Structures, Vol. 36, pp. 1677–1699. Schwarzl, F. R., and Struik, L. C. E. (1967), “Analysis of Relaxation Measurements,” Advances in Molecular Relaxation Processes; Vol. 1, pp. 201–255. Taylor, R. L., Pister, K. S., and Goudreau, G. L. (1970), “Thermomechanical Analysis of Viscoelastic Solids,” International Journal for Numerical Methods in Engineering, Vol. 2, pp. 45–49. Tschoegl, N. W. (1989), The Phenomenological Theory of Linear Viscoelastic Behavior, SpringerVerlag, Berlin. Tschoegl, N. W., and Emri, I. (1992), “Generating Line Spectra from Experimental Responses. Part III: Interconversion between Relaxation and Retardation Behavior,” International Journal of Polymeric Materials, Vol. 18, pp. 117–127. Tschoegl, N. W., and Emri, I. (1993), “Generating Line Spectra from Experimental Responses. Part II: Storage and Loss Functions,” Rheologica Acta, Vol. 32, pp. 322–327. Uzan, J. (1996), “Asphalt Concrete Characterization for Pavement Performance Prediction,” Journal of Asphalt Paving Technology, AAPT, Vol. 65, pp. 573–607. Williams, M. L. (1964), “Structural Analysis of Viscoelastic Materials,” AIAA Journal, Vol. 2(5), pp. 785–808.
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PART
Constitutive Models CHAPTER 7 VEPCD Modeling of Asphalt Concrete with Growing Damage
CHAPTER 9 DBN Law for the ThermoViscoElastoPlastic Behavior of Asphalt Concrete
CHAPTER 8 Uniﬁed Disturbed State Constitutive Modeling of Asphalt Concrete
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CHAPTER
7
VEPCD Modeling of Asphalt Concrete with Growing Damage Y. Richard Kim, Shane Underwood, Ghassan R. Chehab, Jo S. Daniel, H. J. Lee, and T. Y. Yun
Abstract This chapter presents the development of a viscoelastoplastic continuum damage (VEPCD) model for the behavior of asphalt concrete in tension and compression. The modeling strategy adopted is based on (1) the elasticviscoelastic correspondence principle, (2) continuum damage mechanics to account for the effect of microcracking on the constitutive behavior, (3) a time and stressdependent viscoplastic model to account for the plastic and viscoplastic behavior, and (4) the timetemperature superposition (TTS) principle with growing damage to describe the effect of temperature on the constitutive behavior. The resulting models are integrated by the strain decomposition approach to form the VEPCD model. The VEPCD model in tension is developed for four asphaltaggregate mixtures with three of the four mixtures modified by polymers. The model is shown to accurately predict the material behavior in tension over a range of conditions different from those used to characterize the model, including the results from the thermal strain restrained specimen tensile (TSRST) tests at several different cooling rates. Finally, a brief discussion on the VEPCD model in compression and the finite element implementation of the model is given.
Introduction Developing a realistic mathematical model of the mechanical behavior of asphalt concrete with growing damage is a complicated problem. The complexity is attributed to the viscoelastic hereditary effects of the binder, the complex nature of describing the damage
163 Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.
164
Chapter Seven evolution, plastic and viscoplastic flow of the binder, friction among aggregate particles, and the coupling among these mechanisms. Additional difficulties arise from the fact that the model must be able to account for the effects of rate of loading, loading time, rest period, temperature, aging, and stress state so that the resulting model is applicable to a range of loading and environmental conditions experienced in pavements. This chapter presents a constitutive model that can describe the deformation behavior of asphaltaggregate mixtures under complex loading conditions at a wide range of temperatures. The modeling strategy adopted is based on (1) the elastic and viscoelastic behavior of asphalt concrete using the elasticviscoelastic correspondence principle based on pseudostrain, (2) the effect of microcracking on the constitutive behavior using continuum damage mechanics, (3) the plastic and viscoplastic behavior using a time and stressdependent viscoplastic model, and (4) the effect of temperature on the constitutive behavior using the TTS principle with growing damage. The resulting models are integrated by the strain decomposition approach to form the VEPCD model. The VEPCD model is then validated under various loading and temperature conditions. Finally, the viscoelastic continuum damage (VECD) model is implemented into the finite element program to model the cracking behavior of asphalt pavements. Implementation of the full VEPCD model into the finite element program is currently ongoing at North Carolina State University.
Analytical Framework The analytical framework of the model presented in this chapter is based on the strain decomposition principle suggested by Schapery (1999). In his work, Schapery demonstrated that the total strain can be decomposed into viscoelastic strain and viscoplastic strain, as follows:
ε Total = ε ve + ε vp
(71)
where eTotal = total strain eve = viscoelastic (VE) strain evp = viscoplastic (VP) strain In this formulation, the viscoelastic strain includes both linear viscoelastic (LVE) strain and strains due to microcracking, and the plastic strain is included in the viscoplastic strain. The VEPCD model adopts a stepwise approach, in which the experiment necessary for the model characterization is designed such that these strain components can be systematically evaluated from the simplest state to the state that includes more complex mechanisms. More specifically, the material’s behavior in the simplest state (i.e., LVE behavior without any cracking or permanent strain) is first modeled by the elasticviscoelastic correspondence principle. Then, the effect of microcracking damage is modeled by applying continuum damage mechanics to the experimental data from low temperatures and high strain rates where the viscoplastic strain is minimal. The strain hardening viscoplastic model is applied to the experimental data at high temperatures and slow strain rates to develop the viscoplastic model. Finally, these models are combined with the TTS principle with growing damage to allow the prediction of the material’s behavior at any temperature. The TTS principle with growing damage has
VEPCD Modeling of Asphalt Concrete with Growing Damage been proven valid for asphalt concrete by Chehab et al. (2002) for tension and by others in compression (Zhao 2002; Gibson et al. 2003; Kim et al. 2005). The underlying principles of the VEPCD model have been characterized and verified in the last 15 years through a series of research projects, and readers are referred to the reports (Kim and Lee 1997; Kim et al. 2002; Kim and Chehab 2004; Kim et al. 2005) and dissertations (Chehab 2002; Daniel 2001; Lee 1996) for theoretical details of these principles. In the following sections, these principles are briefly described.
TTS with Growing Damage It is well known that the behavior of asphalt concrete depends on time and temperature and that, when in its linear viscoelastic range, asphalt concrete is thermorheologically simple (TRS); that is, the effects of time or frequency and temperature can be expressed through one joint parameter. The viscoelastic material property as a function of time (or frequency), such as the relaxation modulus (or dynamic modulus) at various temperatures can be shifted along the horizontal log time (or log frequency) axis to form a single characteristic mastercurve. If this principle can be extended to outside of the LVE range, its impact is significant in terms of testing requirements and efficiency in modeling. The TTS with growing damage can be verified using a simple technique shown in Fig. 71. In short, stress and time are determined at a strain level from the constant crosshead rate monotonic tests at different rates and temperatures. The corresponding
FIGURE 71 Schematic representation for a single strain level of the technique used to verify timetemperature superposition with growing damage. (Underwood et al. 2006b, with permission from Association of Asphalt Paving Technologists.)
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Chapter Seven time is converted to reduced time with the timetemperature shift factors from LVE characterization (e.g., frequency and temperature sweep dynamic modulus tests) and plotted with the respective stress. If the resulting graph, for a wide range of strain levels, appears continuous, then TTS with growing damage is said to be verified. Details on the theoretical background of this technique can be found in Chehab et al. (2002). Chehab et al. (2002) demonstrated that the TRS behavior extends well beyond the LVE limits to highly damaged levels for asphalt concrete in tension. Underwood et al. (2006b) proved that TTS with growing damage is valid for various modified asphalt mixtures. Other researchers (Zhao 2002; Gibson et al. 2003; Kim et al. 2005) also found that asphalt concrete in compression is TRS at high strain levels with damage. In the interest of brevity, the representative verification case is shown in Figs. 72 and 73 for the SBSmodified mixture used in the Federal Highway Administration’s Accelerated Loading Facility (FHWA ALF) study. Figure 72 presents the stressstrain curves at varying strain rates. Also presented are the strain levels examined for the TTS analysis. The stress versus reduced time curves for the strain levels noted in Fig. 72 are presented in Fig. 73. Note that each data point in this plot represents the results from a single test. It is observed that by using the timetemperature shift factors from linear viscoelastic characterization to obtain reduced time (Fig. 73), continuous curves are obtained at all strain levels. The importance of this finding lies in the reduction of testing conditions required for modeling purposes. Once the behavior at a given temperature is known, the behavior at any other temperature can be predicted using the LVE shift factors. In terms of the VEPCD model, this principle is considered by replacing physical times with reduced times, generally calculated from Eq. (72), or more specifically by Eq. (73) if temperature does not change with time:
ξ=
t
dt
0
T
∫a
(72)
FIGURE 72 Stressstrain curves indicating strain levels for timetemperature superposition analysis for the SBS mixture. (Underwood et al. 2006b, with permission from Association of Asphalt Paving Technologists.)
VEPCD Modeling of Asphalt Concrete with Growing Damage
FIGURE 73 Stress mastercurves for the SBS mixture at (a) 0.0001, (b) 0.0005, (c) 0.001, (d) 0.0022, (e) 0.004, and (f) 0.005 e levels. (Underwood et al. 2006b, with permission from Association of Asphalt Paving Technologists.)
ξ=
t aT
(73)
For the remainder of this chapter, the models are formulated using the reduced time (x) instead of the physical time (t), based on the TTS principle with growing damage.
The VECD Model As mentioned earlier, the viscoelastic strain in this chapter covers both linear viscoelasticity and the damage due to microcracking. The VECD model forms the basis for the viscoelastic strain. This model is based on two principles: the elasticviscoelastic correspondence principle based on pseudostrain and the continuum damage mechanicsbased work potential theory.
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Chapter Seven
ElasticViscoelastic Correspondence Principle The stressstrain relationships for many viscoelastic materials can be represented by elasticlike equations through the use of socalled pseudovariables. This simplifying feature enables a class of extended correspondence principles to be established and applied to linear as well as some nonlinear analyses of viscoelastic deformation and fracture behavior (Schapery 1984). Using these correspondence principles, one can obtain viscoelastic solutions from their elastic counterparts through a simple conversion procedure. The usual Laplace transformbased correspondence principle is limited to LVE behavior with timevarying boundary conditions, whereas the correspondence principles based on pseudovariables are applicable to both linear and nonlinear behavior of a class of viscoelastic materials with stationary or timedependent boundary conditions. Also, the latter does not require a transform inversion step to obtain the viscoelastic solutions but rather requires a convolution integral which is much easier to handle than the inversion step. Consider a stressstrain equation for linear viscoelastic materials, ξ
σ ij = ∫ Eijkl (ξ − τ ) o
where
∂ε kl dτ ∂τ
(74)
sij, ekl = stress and strain tensors Eijkl(t) = the relaxation modulus matrix ξ = t aT = reduced time T = physical time aT = the timetemperature shift factor t = the integration variable
Equation (74) can be written as
σ ij = E Rε klR
or
ε klR =
σ ij ER
(75)
if we define ξ
ε klR =
∂ε 1 Eijkl (ξ − τ ) kl dτ ∫ ER o ∂τ
(76)
where ER is termed the reference modulus, which is a constant and has the same dimension as the relaxation modulus Eijkl(t). The usefulness of Eq. (75) is that a correspondence can be found between Eq. (75) and the linear elastic stressstrain relationship. That is, the equations in Eq. (75) take the form of elastic stressstrain equations even though they are actually viscoelastic stressstrain equations. The ε klR is called the pseudostrain. The pseudostrain accounts for all the hereditary effects of the material through the convolution integral. The reference modulus ER is introduced here because it is a useful parameter in discussing special material behaviors and introducing dimensionless variables. For example, if we take Eijkl(t) = ER in Eq. (76), we obtain ε klR = ε kl , and Eq. (75) reduces to the linear elastic equation σ ij = Eijkl ε kl or ε kl = σ ij Eijkl . If we take ER = 1 in Eq. (76), pseudostrains are simply the linear viscoelastic stress response to a particular strain input. For the remainder of this chapter, ER is set to one. These observations suggest that if the hysteretic behavior of asphalt concrete is due to linear viscoelasticity only, the presentation of the hysteretic data in terms of stress
VEPCD Modeling of Asphalt Concrete with Growing Damage
FIGURE 74 (a) Stressstrain behavior for mixture under LVE cyclic loading and (b) Stresspseudostrain behavior for same data. (Daniel and Kim 2002, with permission from Association of Asphalt Paving Technologists.)
and pseudostrain (instead of physical strain) would make the hysteretic behavior appear to be the same as linear elastic behavior. This observation is illustrated using the experimental data in Fig. 74. Figure 74(a) shows the stressstrain behavior for controlledstress cyclic loading within the material’s LVE range (such as for a complex modulus test). Because the material is being tested in its LVE range, no damage is induced and the hysteretic behavior and accumulating strain are due to viscoelasticity only. Figure 74(b) shows the same stress data plotted against the pseudostrains calculated from Eq. (76) with ER = 1. As can be seen from Fig. 74(b), hysteretic behavior due to both loadingunloading and repetitive loading has disappeared using the pseudostrains. It is also noted that the stresspseudostrain behavior in Fig. 74(b) is linear with a slope of one (i.e., following the line of equality). Another illustration is given in Fig. 75 using the stressstrain data from constant crosshead rate monotonic tests at two different rates of loading. The behavior during initial loading is shown as an inset in these figures. In stressstrain space, as seen in
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170
Chapter Seven
FIGURE 75 Constant crosshead rate test results in (a) Stressstrain space and (b) stresspseudostrain space. (Underwood et al. 2006b, with permission from Association of Asphalt Paving Technologists.)
Fig. 75(a), nonlinearity appears in the initial stage of loading, which suggests that damage commences from the outset. However, the nonlinearity in this zone is related only to the time effects of the material. When these time effects are removed using pseudostrain, seen in Fig. 75(b), damage does not commence at the outset of loading and, in fact, does not begin until the stress level reaches approximately 500 kPa. These two examples demonstrate the benefit of using pseudostrain; that is, pseudostrain essentially accounts for the viscoelasticity of the material and reduces the viscoelastic problem into a corresponding elastic problem, making the modeling of the
VEPCD Modeling of Asphalt Concrete with Growing Damage complex hysteretic behavior of asphalt concrete much easier. The results from experimental verification of the correspondence principle have been documented by Kim and Little (1990), Kim et al. (1995), and Lee and Kim (1998a) using uniaxial monotonic and cyclic data of asphaltic materials under a wide range of test conditions. The material property needed to calculate the pseudostrain in Eq. (76) is the relaxation modulus. Typically, the relaxation modulus test is not easy to perform due to the large amount of stress that develops at the beginning of the test from the step input of displacement. Therefore, the relaxation modulus is determined from the complex modulus using the theoretical inversion process. This approach is explained in detail in Chap. 9. Because the complex modulus tests are performed on a separate set of representative specimens for the mixture in question, the relaxation modulus determined from the complex modulus tests may not be the same as the one for specimens used in damage testing due to sampletosample variability. In that case, the initial pseudostiffness in Fig. 74(b) may not follow the line of equality (LOE). In order to minimize the effect of the sampletosample variability, the initial secant pseudostiffness I is introduced. Thus, the governing constitutive equation in uniaxial mode becomes (77)
σ = Iε R where ξ
εR =
1 ∂ε E(ξ − τ ) dτ ER ∫o ∂τ
(78)
In most cases, the I value remains between 0.9 and 1.1. When the I value is significantly out of this range, a reexamination of the data (both the relaxation modulus and test results) is necessary.
Pseudostrain Calculation The definition of pseudostrain shown in Eq. (78) naturally yields a solution through a linear piecewise technique, as shown below: t t t ⎤ dε dε dε 1 ⎡ ⎢ ∫ E(t − τ ) 1 dτ + ∫ E(t − τ ) 2 dτ + + ∫ E(t − τ ) n dτ ⎥ ER ⎢ 0 dτ d τ d τ ⎥⎦ t t ⎣ 1
εR =
2
n
1
n− 1
(79)
Such a technique, though fundamentally sound, is profoundly inefficient when analyzing large amounts of data. The source of the inefficiency lies in the need to analyze all the time steps that precede the time step of interest, thus resulting in exponentially increasing analysis time for increasing data amounts. To overcome this shortfall, a method commonly used in computational mechanics, the state variable approach, is utilized. The goal of the state variable approach is to transform the process of convolution into an algebraic operation. Theoretical details of state variable techniques can be found in the literature (Simo and Hughes 1998). In a physical sense, though, the state variable approach assigns a variable to each Maxwell element in the Prony representation of the relaxation modulus, as shown below: m
E(t) = E∞ + ∑ Ei e i=1
−t
ρi
(710)
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Chapter Seven This variable then tracks the behavior, or state, of the given element throughout loading. The formulation used in this research is shown as
ε R ( n+ 1 ) =
1 ER
⎡ n+ 1 m n+ 1 ⎤ ⎢ η0 + ∑ ηi ⎥ i=1 ⎣ ⎦
(711)
where h0 and hi are internal state variables for the elastic response and for the specific Maxwell element i at time step n + 1, respectively. Definitions of these variables are given by Eqs. (712) and (713), respectively:
η0n+1 = E∞ (ε n+1 − ε 0 ) ηin+1 = e
−Δt
ρi
ηin + Ei e
−Δt
2 ρi
and
(712)
(ε n+1 − ε n )
(713)
Equation (711) is a remarkably efficient solution technique for pseudostrains. For comparative purposes, a data set of 4000 points requires approximately 100 seconds to analyze using Eq. (79), but requires only 1.5 seconds if analyzed by Eq. (711).
Work Potential Theory Figure 76(a) shows typical stresspseudostrain hysteresis loops at different numbers of cycles in the controlledstress cyclic test. Relatively highstress amplitude is used to induce significant damage in the specimen. Unlike the negligible damage case seen in Fig. 74(b), change in the slope of each σ − ε R cycle (i.e., reduction in the pseudostiffness of the material) can be observed from this figure due to the damage incurred in the specimens. The effect of damage on pseudostiffness can also be seen in the monotonic data shown in Fig. 76(b). In this figure, the stresspseudostrain curves deviate from the line of equality as damage grows. Also, the onset of this deviation occurs at different times as the rate of loading changes, indicating the presence of the ratedependent damage mechanism. Based on these observations, the following uniaxial versions of constitutive equations are presented for linear elastic and linear viscoelastic bodies with and without damage. They also show how models of different complexity may evolve from simpler ones. Elastic body without damage: Elastic body with damage: Viscoelastic body without damage: Viscoelastic body with damage:
σ = Eε
(714)
σ = C(Sm )Eε
(715)
σ = ER ε R
(716)
σ = C(Sm )ER ε R
(717)
where C(Sm) indicates that C is a function of damage parameters Sm. The function C(Sm) represents the changing stiffness of the material due to growing damage. Equation (717) results from Eqs. (714), (715), and (716). The form of Eq. (717) is also supported by the observations made in Fig. 76; that is, the pseudostiffness changes as the damage grows. To determine an analytical representation of the damage function C, the work potential theory, the continuum damage mechanics principle developed by Schapery (1990), is adopted.
VEPCD Modeling of Asphalt Concrete with Growing Damage
FIGURE 76 Stresspseudosstrain behavior of asphalt concrete in (a) controlledstress cyclic test and (b) constantstrainrate monotonic test (different symbols represent different strain rates).
In studying the constitutive behavior of a material with damage, two general approaches are usually considered: a micromechanical approach and a continuum approach. Initiated by Kachanov (1958), continuum damage mechanics has been extensively investigated and applied to various engineering materials by many researchers (e.g., Lemaitre 1984; Kachanov 1986; Krajcinovic 1984, 1989; Bazant 1986). In continuum damage mechanics, the damaged body can be viewed as a homogeneous continuum on a macroscopic scale, and the influence of damage is typically reflected in terms of reduction in stiffness or strength of the material. The state of damage can be
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Chapter Seven quantified by a set of parameters often referred to as internal state variables or damage parameters in the context of thermodynamics of irreversible processes. The growth of damage is governed by an appropriate damage (or internal state) evolution law. The stiffness of the material, which varies with the extent of damage, is determined as a function of the internal state variables by fitting the theoretical model to available experimental data. The mechanical behavior of an elastic medium with constant material properties (i.e., without damage growth) can usually be described using an appropriate thermodynamic potential (e.g., the Helmholtz free energy for isothermal processes or the Gibbs free energy for isentropic processes). These potentials are point functions of thermodynamic state variables. When thermal effects are not considered, both the Helmholtz free energy and the Gibbs free energy potentials are identified with the socalled strain energy and represent the energy stored in the system which is algebraically equal to the work done on the system by external loading. However, when damage occurs due to external loading, the work done on the body is not entirely stored as strain energy; part of it is consumed in causing damage to the body. The amount of energy required to produce a given extent of damage is expressed as a function of internal state variables. The total work input to the body during the processes in which damage occurs depends, in general, on the path of loading. However, it has been observed that, for certain processes in which damage occurs, the work input is independent of the path of loading (Schapery 1987a; Lamborn and Schapery 1988, 1993). Schapery (1990) applied the method of thermodynamics of irreversible processes and the observed phenomenon of path independence of work in damageinducing processes to develop the work potential theory so that it may be applicable to describing the mechanical behavior of elastic media with growing damage and other structural changes. The theory is general enough to allow for strong nonlinearities and coupling between the internal state variables and to describe a variety of mechanisms including micro and macrocrack growth in monolithic and composite materials. Sicking (1992) applied the theory to model the damagerelated material nonlinearity in graphiteepoxy laminates, and Lamborn and Schapery (1993) showed the existence of a work potential for suitably limited deformation paths using experimental data from axial and torsional deformation tests on angleply fiberreinforced plastic laminates. The elements of work potential theory in terms of a strain energy formulation may be represented as follows: Strain energy density function: Stressstrain relationships: Damage evolution laws:
W = W (ε ij , Sm )
(718)
∂W ∂ε ij
(719)
∂W ∂WS = ∂Sm ∂Sm
(720)
σ ij = −
where sij = stresses eij = strains Sm = internal state variables (or damage parameters) WS = WS(Sm) = dissipated energy due to damage growth The internal state variables, Sm (m = 1, 2,..., M), account for the effects of damage, and the number of internal state variables (i.e., M) is typically determined by the number
VEPCD Modeling of Asphalt Concrete with Growing Damage of different mechanisms governing the damage growth. Equation is similar to a crack growth equation (e.g., G = Gc, where G is the energy release rate and Gc is the fracture toughness) and, in fact, Eq. (720) is used to find Sm as functions of eij. The lefthand side of Eq. (720) is the available thermodynamic force, while the righthand side is the required force for damage growth. Based on the elasticviscoelastic correspondence principle, the strains eij that appear in the elastic damage model, Eqs. (718) to (720), are replaced with corresponding pseudostrains ε ijR defined by Eq. (76). Then, according to the correspondence principle, the set of equations written in terms of pseudostrains now governs the corresponding viscoelastic damage problem. It was found from experimental studies (e.g., Park 1994) that the damage evolution laws for elastic materials cannot be translated directly into evolution laws for viscoelastic materials through the correspondence principle. Not only is the available force for growth in Sm rate dependent, but the resistance against the growth of Sm is rate dependent for most viscoelastic materials. The following evolution laws, which are similar in form to the wellknown powerlaw crack growth laws for viscoelastic materials (Schapery 1975), are adopted in this study as they can reasonably represent the actual damage evolution processes in many viscoelastic materials: ⎛ ∂W R ⎞ S m = ⎜ − ⎝ ∂Sm ⎟⎠
αm
(721)
where WR = WR ( ε ijR, Sm) = pseudostrain energy density function S m = damage evolution rate am = materialdependent constants related to the viscoelasticity of the material Equation (721) is similar to the crack propagation rate equation. The same form of evolution laws has been used successfully in describing the behavior of a filled elastomer with growing damage (Park 1994). Park et al. (1996) also adopted the work potential theory in modeling the ratedependent behavior of asphaltaggregate mixtures under constant strain rate monotonic loading. Finally, the work potential theory applied to viscoelastic media with the rate type damage evolution law is represented by the following three components for the uniaxial loading condition: Pseudostrain energy density function: Stressstrain relationship:
Damage evolution law:
W R = W R (ε R , Sm )
σ=
∂W R ∂ε R
⎛ ∂W R ⎞ S m = ⎜ − ⎝ ∂Sm ⎟⎠
(722) (723)
αm
(724)
Determination of S The work potential theory specifies an internal state variable Sm to quantify damage, which is defined as any microstructural changes that result in an observed stiffness reduction. For asphalt concrete in tension, this variable is related primarily to the
175
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Chapter Seven microcracking phenomenon. Therefore, only one internal state variable (i.e., S) is used to model the damage growth in tension. The method used to solve the damage evolution law in Eq. (724) is a matter of preference and, as such, two different solutions are hereby proposed. The first, proposed by Park et al. (1996), transforms the original form of the equation to an integrated form, assumes a >> 1 and defines a new parameter Sˆ . Equation (725) presents, in discrete form, the method proposed by Park et al.: 1
⎡ ⎛ 1 ⎞ ⎤ 1+ 1 α S = ⎢ Sˆ ⎜ 1 + ⎟ ⎥ α⎠⎦ ⎣ ⎝
(725)
where Sˆ is given by Eq. (726) 1 1 Sˆi+1 = Sˆi − (Ci − Ci−1 )(ε iR )2 t α 2
(726)
Lee and Kim (1998a, b) also propose a solution that utilizes the chain rule and makes no assumption regarding a. The solution of these researchers is presented in Eq. (727). It is noted that both methods have been successfully applied in asphalt concrete research (Park et al. 1996; Daniel and Kim 2002; Chehab et al. 2003). α
1 ⎡ 1 ⎤ 1+α Si+1 = Si + ⎢ − (Ci − Ci−1 )(ε iR )2 ⎥ Δt 1+α ⎣ 2 ⎦
(727)
To reconcile the approximations of these methods, an iterative refinement technique is incorporated into this research. In short, this method assumes that the rate of change in damage is constant over some discrete time step. This rate of change is determined at a point near the current value of damage (Si + dS) where the extrapolation error is minimized. The method begins with an initial calculation of S by either of the approximate methods, both of which require results from constant crosshead rate tests for the stresspseudostrain relationship. The initial S values are plotted with the pseudostiffness values C, calculated from the following relationship, which is obtained from Eq. (717): C=
σ I × εR
(728)
The C and initial S values are then fit to some analytical form, such as the one presented in Eq. (729), where a and b are fitting parameters: C = e aS
b
(729)
Returning to the damage evolution law, and noting that the increments of time are generally small, one can write the rate of change in damage as dS ΔS = dt Δt
(730)
VEPCD Modeling of Asphalt Concrete with Growing Damage Substituting this expression into Eq. (724), and rearranging and writing in the discrete form, one finds the following equation: ⎛ ( δ WdR ) ⎞ i Si+1 = Si + Δt ⎜⎝ − ⎟ δS ⎠
α
(731)
It must be observed that for the uniaxial case, the work function (WR) is given by WR =
1 C(S)ε R 2
(732)
Substituting Eq. (732) into (731) and simplifying, one arrives at Si+1
(δ C)i ⎞ ⎛ 1 = Si + Δt ⎜ − (ε R )2 δ S ⎟⎠ ⎝ 2
α
(733)
In Eq. (733), it is assumed that before loading occurs, S and C are zero and one, respectively. Further, dS must be specified and should be significantly less than the change in damage over a time step (typically, 0.1 is used). After calculating the value of damage (Si) and the incremental damage (Si + dS) at a given time step, the corresponding values of C are found by Eq. (729). The difference between these values (dC) is then used to calculate damage at the next time step. The process is repeated until all data points are processed. After completing this first iteration, the new values of S are plotted against the original pseudostiffness values, and a new analytical relationship is found. The entire process is repeated until the change in successive iterations is small. In this research, eight such iterations were performed, but it was noted that very little improvement was made after the third or fourth iteration. Figures 77 and 78 present the initial S calculated by both approximate techniques along with results from the refinement process. From these figures it is seen that the
FIGURE 77 Comparison of reﬁned and approximate damage calculation techniques. (Underwood et al. 2006b, with permission from Association of Asphalt Paving Technologists.)
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Chapter Seven
FIGURE 78 Comparison of reﬁned and approximate damage characteristic relationship. (Underwood et al. 2006b, with permission from Association of Asphalt Paving Technologists.)
refinement process results in S values that fall between the two approximate methods. Note that in these figures the seed values for the refinement process are obtained by the chain rule method. However, trials show that regardless of the method used to find the seed values, iterations collapse to the same curve. Details on this refinement process can be found in the work of Kim and Chehab (2004).
Damage Characteristic Relationship Daniel and Kim (2002) studied the relationship between damage parameter (S) and the normalized pseudo secant modulus (C) under varying loading conditions. The most significant finding from their study is that a unique damage characteristic relationship exists between C and S, regardless of loading type (monotonic versus cyclic), loading rate, and stress/strain amplitude. In addition, the application of the TTS principle with growing damage to the C versus S relationships at varying temperatures yields the same damage characteristic curve in the reduced time scale. The only condition that must be met in order to produce the damage characteristic relationship is that the test temperature and loading rate combination must be such that only the elastic and viscoelastic behaviors prevail with negligible, if any, viscoplasticity. When the test temperature is too high or the loading rate is too slow, it was found that the C versus S curve deviates from the characteristic curve. To ensure that the test temperature is low enough and the loading rate is fast enough not to induce any significant viscoplastic strains, the tests are performed at a low temperature (typically 5°C) with varying loading rates. If the C versus S curves at different rates overlap to form a unique relationship, the combinations of the temperature and loading rate are sufficiently satisfactory to develop the damage characteristic relationship. Finally, the VECD model is
σ = C(S)ε R
(734)
VEPCD Modeling of Asphalt Concrete with Growing Damage or ⎛ σ ⎞ d⎜ ⎝ C(S) ⎟⎠ ε ve = ER ∫ D(ξ − τ ) dτ dτ 0 ξ
(735)
by converting Eq. (734) to predict the viscoelastic strain. Note that ER in Eq. (735) is set to one and that the initial secant pseudostiffness I is not used in Eq. (734). I is only necessary in calibrating the model using the experimental data from several replicate specimens. The major advantage of the damage characteristic relationship is that it allows a reduction in testing requirements. Since the same relationship exists in monotonic and cyclic tests, the material behavior under cyclic loading can be predicted from the damage characteristic curve characterized from the much simpler monotonic tests. Daniel and Kim (2002) have verified that this approach can predict the fatigue life of asphalt concrete within the sampletosample variation.
StrainHardening Viscoplastic Model Viscoplastic strain is assumed to follow the strainhardening model (Uzan 1996; Seibi et al. 2001) of the form
ε VP =
g(σ ) ηvp
(736)
where ε vp is the viscoplastic strain rate g(0) = 0, and ηvp is the material’s coefficient of viscosity. Assuming that η is a power law in strain (Perl et al. 1983; Kim et al. 1997; Schapery 1999), Eq. (736) becomes g(σ ) ε VP = p Aε vp
(737)
where A and p are model coefficients. Rearranging and integrating g(σ ) × dt A
p dε vp × ε vp =
and
p+1 g(σ )dt A ∫0
(738)
t
p+1 ε vp =
(739)
Raising both sides of Eq. (739) to the (1/p + 1) power yields
ε vp
⎛ p + 1⎞ =⎜ ⎝ A ⎟⎠
1
p+1
⎞ ⎛t ⎜ ∫ g(σ )dt ⎟ ⎠ ⎝0
1
p+1
(740)
Letting g( σ ) = B σ 1q (Uzan 1996; Perl et al. 1983; Kim et al. 1997) and coupling coefficients A and B into coefficient Y, Eq. (740) becomes
ε vp
⎛ p + 1⎞ =⎜ ⎝ Y ⎟⎠
1
p+1
⎛t q ⎞ ⎜ ∫ σ dt ⎟ ⎠ ⎝0
1
p+1
(741)
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Chapter Seven Replacing time in Eq. (742) with reduced time yields
ε vp
⎛ p + 1⎞ =⎜ ⎝ Y ⎟⎠
1
p+1
⎛ξ q ⎞ ⎜ ∫ σ dξ ⎟ ⎝0 ⎠
1
p+1
(742)
Observe that the plastic strain is zero in this model because the viscoplastic strain vanishes at x = 0. It was found that such simplicity does indeed exist in asphalt concrete (Chehab et al. 2003).
VEPCD Model The VEPCD model is constructed, as follows, based on the strain decomposition principle in Eq. (71), the VECD model in Eq. (735), and the viscoplastic (VP) model in Eq. (742): ⎛ σ ⎞ 1 1 d⎜ ξ ⎞ p+1 ⎝ C(S) ⎟⎠ ⎛ p + 1 ⎞ p+1 ⎛ q ε T = ER ∫ D ( ξ − ξ182 dξ ′ + ⎜ ′ ) ⎜ ∫ σ dξ ⎟ dξ ′ ⎝ Y ⎟⎠ ⎝0 ⎠ 0 ξ
(743)
where x′ is the integration variable. In the following sections, the VEPCD model is calibrated using the experimental results.
Calibration of the VEPCD Model in Tension Materials and Testing System In this section, the VEPCD model is calibrated for various asphalt mixtures using the principles described earlier. Since the objective of this chapter is to describe the VEPCD modeling technique rather than compare the different mixtures, the details of the mixture properties are not presented. Readers are referred to Kim and Chehab (2004) and Kim et al. (2005) for those details. Also, a detailed comparison of the behavior of different mixtures is given in Underwood et al. (2006b). Data from five mixtures are presented in this chapter: two conventional Superpave mixtures and three modified asphalt mixtures. The two conventional mixtures include the Maryland 12.5mm Superpave mixture used as the control mixture in the NCHRP 919 project and the 12.5mm Superpave mixture used as the control mixture in the FHWA ALF study. The Maryland mixture is composed of 100% crushed limestone and an unmodified PG 6422 binder, and the ALF mixture is composed of granite aggregate and PG 7022 binder. The modified mixtures are the ones used in the ALF study and have the same aggregate and gradation as the ALF control mixture. The modified binders used in these mixtures include SBSmodified binder with PG 7028, Crumb Rubber Terminal Blend with PG 7628, and Ethylene Terpolymer with PG 7028. Superpave gyratory compacted (SGC) specimens with a 150mm diameter and 180mm height were fabricated using the Australian Superpave gyratory compactor, ServoPac. Cores of a 75mm diameter and 150mm height were obtained from the SGC specimens. The target air void content was 4% with a tolerance of ±0.5%. The MTS810 testing system with a 100kN capacity was utilized in this research. This system consists of a servohydraulic closed loop testing machine, a 16bit National Instruments data acquisition board, and a set of LabView programs for data collection
VEPCD Modeling of Asphalt Concrete with Growing Damage and analysis. Displacements were measured using loosecore LVDTs, two with a 75mm gauge length and two with a 100mm gauge length attached to the middle section of the specimen at equal distances from the ends. Using two different gauge lengths enables the determination of the onset of localization because the opening of the major cracks that start to form in the asphalt matrix between the gauge points would be numerically divided by two different gauge lengths, thus leading to two different strain values. Hence, the divergence of the strains corresponding to the different gauge lengths indicates the onset of localization and macrocracking.
Calibration Test Program The major strength of the VEPCD model is the simplicity of the calibration testing requirement. The calibration testing program is composed of three phases: (1) LVE characterization, (2) VECD characterization, and (3) viscoplastic (VP) characterization. Complex modulus testing at varying temperatures and frequencies is used for the LVE characterization. The dynamic moduli and the timetemperature shift factors are determined using the procedure given in the AASHTO TP 6203 and then used in the VECD and VP modeling. The dynamic modulus is converted to the relaxation modulus using the algorithm presented in Chap. 6. For the VECD and VP characterizations, constant crosshead rate monotonic tests are used. Instead of testing several replicates at a limited set of rates and temperatures, these tests are conducted with few or just one replicate at a wider range of loading rates. Typically, four different rates are used at 5 and 40°C to calibrate the VECD and VP models, respectively.
Linear Viscoelastic Characterization Figures 79 and 710 present the replicate averaged dynamic modulus and phase angle mastercurves for all the FHWA ALF mixtures. These figures show that at higher reduced frequencies (lower temperatures) the unmodified mixture shows substantially greater stiffness and more elasticity (evidenced by lower phase angles) than any of the other mixtures. It is also observed that the SBS and CRTB mixtures show approximately the same stiffness and elasticity under these conditions. Also, the SBS and CRTB mastercurves have smaller slopes during the transition period than the mastercurves of the control and Terpolymer mixtures. Finally, at the lower reduced frequencies (higher temperatures), the CRTB and SBS mixtures are the stiffest, which is the opposite of that observed at the higher reduced frequencies. These complicated time and temperature dependencies of different mixtures are captured in the dynamic modulus and phase angle mastercurves. The conversion technique presented in Chap. 9 is applied to these LVE properties to obtain relaxation moduli of the mixtures, which will then be used in the pseudostrain calculation. The converted relaxation modulus is fit to the Prony series representation in Eq. (710). Another important material property characterized from the complex modulus test is the timetemperature shift factor. This property is used in converting the physical time to reduced time in the VEPCD modeling approach.
VECD Characterization The VECD model calibration requires stressstrain data with negligible viscoplasticity; therefore, the constant crosshead rate monotonic test results at 5°C are used. The calibration procedure starts from the calculation of pseudostrains using Eqs. (711) to (713) and the relaxation modulus in the Prony series. Once the pseudostrains are
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FIGURE 79 Dynamic modulus mastercurves for control, CRTB, SBS, and Terpolymer mixtures in (a) semilog space and (b) loglog space. (Underwood et al. 2006b, with permission from Association of Asphalt Paving Technologists.)
calculated, the S values are determined using the iterative refinement technique presented in Eq. (733). The C values are also calculated from Eq. (728) using the measured stresses, pseudostrains, and the initial secant pseudostiffness (I) determined from the early linear portion (normally pseudostrains less than 500) of the stresspseudostrain data. The C values and S values are then crossplotted for multiple crosshead rates to check the collapse of the curves, which indicates the negligible viscoplasticity in the stressstrain data. The resulting C versus S relationship is the damage characteristic relationship and forms the basis of the VECD model. The procedure to ensure that negligible viscoplastic strain develops during testing requires a trial and error approach. Tests are performed with increasing strain rates until failure occurs in a brittle fashion during the unloading portion of the stressstrain curve. It is known from previous research (Kim and Chehab 2004) that under such conditions viscoelastic damage mechanisms dominate the material behavior. Once this
VEPCD Modeling of Asphalt Concrete with Growing Damage
FIGURE 710 Phase angle mastercurves for control, CRTB, SBS, and Terpolymer mixtures. (Underwood et al. 2006b, with permission from Association of Asphalt Paving Technologists.)
reference rate is identified, the C versus S curves from adjacent rates are compared to the curve from the reference rate to establish the damage characteristic curve. Figure 711 presents the damage characteristic curves for all four ALF mixtures. From this figure it is clearly seen that the CRTB and control mixtures show the most favorable damage characteristics, followed by the SBS mixture and, finally, the Terpolymer mixture. However, one must be careful in assessing the fatigue performance of different mixtures from this plot alone because the resistance of asphalt concrete to fatigue cracking must be quantified by considering both the resistance to deformation and resistance to damage. Also, under loading conditions where other mechanisms,
FIGURE 711 Damage characteristic curves for control, CRTB, SBS, and Terpolymer mixtures. (Underwood et al. 2006b, with permission from Association of Asphalt Paving Technologists.)
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Chapter Seven such as viscoplasticity, begin to contribute significantly, the performance ranking could change.
VP Characterization Calibration of the VP model in Eq. (742) first requires the determination of the viscoplastic strain from the total strain measured from the monotonic data. In cyclic loading with rest periods, the permanent strains after the rest periods can be used as the viscoplastic strain. However, in tension where the specimen is glued to the loading plates, it is difficult to maintain zero stress during the rest periods. In monotonic loading, although the stressstrain data at high temperatures and slow loading rates would have greater proportions of viscoplastic strain in the measured strain, it is unclear how high the temperature and slow the loading rate must be in order to consider the measured total strain as the viscoplastic strain. This difficulty is overcome using the strain decomposition principle in Eq. (71). Knowing the damage characteristic curve of the material from the 5°C monotonic testing, the viscoelastic strain can be predicted for high temperatures using Eq. (735). The viscoelastic strain is then subtracted from the total measured strain to determine the viscoplastic strain. An optimization algorithm, such as the genetic algorithm, is used to determine the VP model coefficients (p, q, and Y) in Eq. (742) from the extracted viscoplastic strain and corresponding stress and time. It is worthwhile, given the strain decomposition nature of the VEPCD model, to examine the effect of strain rate and temperature on the viscoelastic and viscoplastic characteristics of the mixtures. Figure 712(a) presents the influence of viscoelastic and viscoplastic effects on the behavior of the Maryland mixture during the constant crosshead rate tests. This figure shows the percentage of total strain attributed to viscoelastic and viscoplastic effects as a function of the reduced strain rate at a reference of 25°C. Each data point in this plot represents the results from a single test conducted at a particular strain rate and temperature and is obtained at the peak stress. From this figure one can observe a generally decreasing significance of viscoplastic strain with an increased reduced strain rate. As observed from Fig. 712(a), after a reduced strain rate of 4 e/s, seen in Region C, the total strain consists solely of viscoelastic response. In Region B, where the reduced strain rate ranges from 0.01 to 4 e/s, the viscoelastic strain constitutes about 95% of the total response. As for Region A, viscoelastic and viscoplastic behavior are both present with their proportions being equal at a reduced crosshead strain rate of 0.0001 e/s. Now that the composition percentage of component strains can be known for a particular loading condition, the conditions required for modeling each strain separately can be more accurately selected. Effects of temperature and strain rate on the viscoplastic characteristic of various mixtures are shown in Fig. 712(b). The reference temperature is 5°C. The figure illustrates that the CRTB mixture shows the least significant viscoplastic behavior at lower reduced rates. This behavior is somewhat expected knowing that the high temperature PG grade of the CRTB binder is 76°C, which is higher than all the other binders used in this study. It is also observed that the control mixture shows, by percentage, less viscoplasticity than the Terpolymer and SBS mixtures over all the tested conditions. However, the unmodified mixture has a much steeper slope than the modified mixtures and is expected to show more viscoplasticity at ranges outside those tested. Due to this increased slope and based on the results of Fig. 710, it appears that
VEPCD Modeling of Asphalt Concrete with Growing Damage
FIGURE 712 (a) Percentage of viscoelastic and viscoplastic strains as a function of reduced strain rate at 25°C (Maryland Superpave mixture). (Chehab et al. 2002, with permission from Association of Asphalt Paving Technologists, and (b) Inﬂuence of viscoplasticity on the behavior of various mixtures. (Underwood et al. 2006b, with permission from Association of Asphalt Paving Technologists.)
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Chapter Seven the real elastic benefit of modified mixtures in the field (i.e., under loading rates experienced in the field) occurs at high temperatures. Figure 712 reinforces the importance of both rate and temperature when conceptualizing asphalt concrete behavior. Recall that the reduced strain rate is dependent on both the physical strain rate and the temperature. According to this concept it is possible, then, that viscoplastic mechanisms can significantly impact material behavior at even very low temperatures if the strain rate is slow enough. Stated more generally, viscoplasticity can affect material behavior at any temperature if the input condition is slow. Conversely, any input condition can generate viscoplastic effects if the temperature is high enough. This duality is critically important to the proper understanding of the material behavior.
Validation of the VEPCD Model in Tension In this section, the VEPCD model calibrated for different mixtures is validated using loading histories drastically different from those used in the calibration process. The validation tests presented in this chapter include a random cyclic loading test, where frequency, amplitude, and number of cycles are randomly assigned for each loading group, and the TSRST strength test. The validation results of other loading histories can be found in Kim and Chehab (2004) and Kim et al. (2005). It is important to note that all graphs in this section are presented until localization. Localization is the point at which a single macrocrack begins to dominate the material behavior. At this point, strain measurements do not accurately represent the material behavior over the measurement length. For the constant crosshead tests, it is observed that localization occurs after the peak and at the point when the stress has reached about 90% of the peak stress. For the random load tests, localization is defined by measuring deformations with two different gauge lengths (100 mm and 75 mm) over the middle of the specimen. When these two measurements diverge, localization occurs (Fig. 713).
FIGURE 713 Strength mastercurves for control, CRTB, SBS, and Terpolymer mixtures. (Underwood et al. 2006b, with permission from Association of Asphalt Paving Technologists.)
VEPCD Modeling of Asphalt Concrete with Growing Damage
FIGURE 714 Random load history used in validation of the VEPCD model. (Underwood et al. 2006a, with permission from Association of Asphalt Paving Technologists.)
The input loading history for the random load test is shown in Fig. 714. In this loading history, loading frequency, stress amplitude, and number of cycles in each loading group are randomly changed. The test temperature for all cases is 25°C, and the same loading input is used for each of the mixtures. Figure 715 shows the measured and modeled responses, separated by the components, viscoelastic damage and viscoplastic, for each of the mixtures. Note that the graphs are presented only until localization. An examination of these figures shows that the measured and modeled behaviors closely agree. As the specimen approaches localization, the model tends to underpredict the measured data; however, this difference is less than 15% in the most extreme case (SBS) and could be related to the specimentospecimen variability.
TSRST Validation In this section, the VEPCD model developed for the Maryland Superpave mixture is verified using TSRST tests. The TSRST tests were performed at the FHWA TurnerFairbanks Highway Research Center according to AASHTO TP1093. Asphalt concrete beam specimens, 250 mm in length and 50 by 50 mm in cross section with an airvoid content of 4 ± 0.5%, were prepared. Compaction was done in a testing machine using a stresscontrolled sinusoidal load, and specimens were later cut using a watercooled double saw. The TSRST tests were conducted using an automated, closedloop system that cools the specimen at a constant rate while restraining it from contraction. Target cooling rates were 5°C/h, 10°C/h, and 20°C/h, resulting in actual nominal cooling rates of 4.4°C/h, 8.6°C/h, and 17.7°C/h. Time, stress, and specimen surface temperatures were recorded for six replicates tested at each of the lowest two rates and four replicates at the highest. For a specimen subjected to mechanical loading under a nonisothermal condition, the total strain (eTotal) is the sum of the mechanical strain (eMechanical) and thermal strain (eThermal). As the TSRST beam specimen cools, it has a tendency to contract. However, when the specimen is restrained at the top and bottom, no deformation is allowed (i.e., eTotal = 0). Consequently, thermally induced stresses, equivalent to those developed by
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Chapter Seven
FIGURE 715 Random load prediction results for ALF mixtures: (a) Control, (b) CRTB, (c) SBS, and (d) Terpolymer. (Underwood et al. 2006a, with permission from Association of Asphalt Paving Technologists.)
mechanically induced deformations had the specimen been free to contract, develop, and gradually increase as the specimen is cooled until fracture. Therefore, in the TSRST test, ε Mechanical − ε Thermal = αΔT, where T is the temperature drop and a is the coefficient of the thermal contraction of the mixture. The thermal coefficient of contraction for the Maryland mixture is reported as 2.055 × 10−5/°C, above the glass transition temperature Tg (Superpave Performance Models Report 2002). This measurement is in line with typical values reported in the literature for mixes with similar material properties (Fwa et al. 1995). The TSRST verification is particularly important not only because the TSRST data were not used in the model development, but also because the stresses in the TSRST tests are induced from thermal loading, unlike the stresses used in the model development which are induced by mechanical loading. The input data used for the predictions include specimen dimensions, initial temperature and load, and cooling rates. Predicted responses include stresstime history and stresstemperature history, in addition to stress, time, and temperature at failure. These responses were obtained from three models, including the LVE model, the VECD model, and the VEPCD model. The LVE model is the uniaxial form of Eq. (74), and the VECD and VEPCD models are presented in Eqs. (734) and (743), respectively. Since the temperature varies with time in the TSRST test, Eq. (72) is used to calculate the timetemperature shift factor. The predicted responses were compared against the measured values to determine the accuracy of the models. In the following, these comparisons are presented. For more details, see Chehab and Kim (2005).
VEPCD Modeling of Asphalt Concrete with Growing Damage
Prediction of Thermal Stress History For notational brevity, the thermal stress and strain are identified by s and e, respectively. Stresses predicted for the three cooling rates using the three models are plotted as a function of time in Fig. 716. Also plotted are the average measured stresses from all replicates tested at each rate. As apparent by visual inspection, the stresses predicted using the LVE model are greater than the measured, with the difference increasing as time increases and the cooling rate decreases. This discrepancy is due to the fact that the LVE model does not account for the stress relaxation due to microcracking. The error between the VECDpredicted stresses and the measured is much smaller than that of the LVE case for all cooling rates. Moreover, the error reduces with an increase in time and decrease in cooling rate. The VEPCDpredicted stresses match the measured very well, with discrepancies being the greatest at the slowest cooling rate. From comparisons of the predicted stresses among each other, it is evident that the VEPCD model yields the most accurate predictions, slightly better than the VECD model. Another important observation is that the rate of increase in VECDpredicted stresses with time deviates from corresponding to the measured and the other predictions. The significant effect of the viscoplasticity on the thermal stress prediction at low temperatures where thermal cracking is of concern should not be surprising, because the constitutive behavior of asphalt concrete is dependent not only on the temperature but also on the rate of loading. As a matter of fact, the typical cooling rates that asphalt pavements experience in the northern United States range from 0.5°C/h to 1°C/h and a maximum of 2.7°C/h in Canada (Jung and Vinson 1994). These rates are much slower than the cooling rates used in this study and, therefore, the overall global significance
FIGURE 716 Average measured and predicted stress histories for different material models and cooling rates. (Chehab and Kim 2005, ASCE.)
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Chapter Seven of the viscoplastic model may be even greater than the research presented in this chapter may suggest.
Prediction of Fracture Point As the specimen is cooled, the thermal stress increases until it equals the tensile strength of the asphalt mix, ultimately leading to the fracture of the specimen. While mechanistic material characterization models allow the prediction of the thermal stress history, alone they cannot be used to determine the instance of failure. Most of the models that have been developed to predict the fracture point using the TSRST data utilize strength as a failure criterion (Jung and Vinson 1994, SHRPA357 1993). In this study, the effect of temperature and loading rate on the strength is reflected through the reduced strain rate, that is, the product of the strain rate and the timetemperature shift factor. The strain, in this sense, would not be the measured strain because it is zero. As explained in the previous sections, it is the thermal strain that would have resulted at that instance, had the specimen not been restrained. The strength versus reduced strain rate relationships for different asphalt mixtures are shown in Fig. 713. In developing the VECD and VEPCD models, monotonic tests were conducted at several temperatures and strain rates (−10°C to 40°C; 10−5 e/s to 0.1 e/s). The LVE timetemperature shift factors were used to convert the strain rate to the reduced strain rate at a reference temperature (25°C in this study), using Eq. (72). The peak stress (fracture stress for brittle failures) for each monotonic test is plotted against the corresponding reduced strain rate of that test to construct the strength failure envelope for the mix. Then, the predicted thermal stress history for a given TSRST test is plotted on the same graph. The intersection point of the thermal stress curve and the strength failure envelope from the monotonic tests is the predicted fracture point for that particular TSRST test. Figure 717 shows the strength failure envelope plotted with the stress predicted from the VEPCD model as a function of the reduced strain rate. Once failure stress is determined, the corresponding time and temperature can be determined.
FIGURE 717 Determination of fracture point using strength failure envelope: (a) −17°C/h, (b) −8.6°C/h, and (c) −4.4°C/h, reference temperature: 25°C. (Chehab and Kim 2005, ASCE.)
VEPCD Modeling of Asphalt Concrete with Growing Damage
FIGURE 718 Comparison of the TSRST parameters predicted from the VEPCD model with the measured values. (Chehab and Kim 2005, ASCE.)
Figure 718 presents a comparison of the TSRST parameters. In general, the predicted values are in good agreement with the measured ones.
VEPCD Model in Compression The same calibration procedure used in the tension modeling is applied to the compression data, except for the test procedure used for the VP modeling. Figure 719 presents the dynamic modulus mastercurves determined from the tensioncompression test and from the compression test with zero minimum stress. The data presented in Fig. 719 for the tensioncompression test are an average of multiple specimens. It is found that these two dynamic modulus mastercurves are essentially the same. Maintaining the strain amplitude below 70 microstrains is an important condition for this agreement. The timetemperature shift factors are also found to be the same between the tensioncompression and compression tests. Data from the monotonic constant crosshead rate tests at 5°C are used for the viscoelastic damage characterization. Pseudostrains are plotted against the stress in Fig. 720(a) and (b). The stresspseudostrain behavior shown in Fig. 720(b) is essentially the same as in tension; that is, the stress versus pseudostrain follows the line of equality in the early part of loading when the damage is minor and begins to deviate from the line of equality, indicating damage growth. The stresses and pseudostrains are used to calculate the C and S values. It can be seen in Fig. 720(c) that the C versus S curves overlap nicely among different loading rates, indicating that the viscoplastic strain at these testing conditions is minimal. Also
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Chapter Seven
FIGURE 719 Dynamic moduli from compression and tensioncompression tests: (a) semilog scale, (b) loglog scale.
in the same figure, the damage characteristic curves from both tension and compression are plotted together. It can be seen that the damage characteristic curve from compression is positioned higher than that from tension. In tension, the primary damage at 5°C is the microcracking in the perpendicular direction to the loading direction, whereas the primary damage in compression is the vertical cracking along the loading direction. Because the vertical cracking in compression is due to the horizontal tensile stress induced by the vertical compression loading, for the same amount of microcracking (i.e., the same value of S), the resistance of the material to the cracking (i.e., C) is greater in compression than in tension. The C values at the peak stress obtained from the different rate tension tests are found to be between 0.3 and 0.35.
VEPCD Modeling of Asphalt Concrete with Growing Damage
FIGURE 720 Stress versus pseudostrain at 5°C: (a) until failure, (b) initial loading portion, and (c) damage characteristic curves at 5°C.
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Chapter Seven For the VP modeling, repetitive creep and recovery tests with fixed time and fixed stress are performed. The permanent strains at the end of rest periods are used to determine the p, q, and Y values. The p, q, and Y values in compression are found to be 2.088, 2.482, and 6.33 × 1020, respectively.
Verification of the VEPCD Model in Compression The VEPCD model in Eq. (743), with the creep compliance converted from the complex modulus, the damage characteristic relationship (i.e., C versus S curve), and the determined p, q, and Y values, are used to predict the stressstrain behavior of the asphalt concrete subjected to repetitive creep and recovery loading. It is noted that the loading portions of the creep and recovery tests provide the data that were not used in the VP model characterization. Figure 721 shows the viscoelastic strains for the entire loading history and the viscoplastic strains at the end of the recovery period predicted from the VEPCD model.
FIGURE 721 Measured and predicted strains from the ﬁxed stress creep and recovery tests: (a) 25°C and (b) 40°C.
VEPCD Modeling of Asphalt Concrete with Growing Damage Because the permanent strain at the end of the recovery is used to determine the VP model coefficients, the predicted viscoplastic strains are in good agreement with the measured permanent strain at the end of the recovery. Also, it is found that the viscoelastic strain recovers completely after the rest period. The most noteworthy observation to be made from Fig. 721 is the overprediction of the viscoelastic strain and, therefore, the total strain. The degree of overprediction is found to increase as the temperature increases. The same observation has been made from the monotonic prediction although it is not shown in this chapter to save space. The fact that the VEPCD model predicts the material’s behavior in tension extremely well but very poorly in compression suggests that the missing mechanism in the VEPCD model is unique to compression loading. In order to investigate the difference between the behavior of asphalt concrete in tension and compression, the stress versus pseudostrain relationships are examined first. Figure 722 presents the stress versus pseudostrain curves for intermediate to high temperatures. The stress versus pseudostrain curves for 5°C are shown in Fig. 720. It can be seen from these figures that, as the temperature increases, the stress versus pseudostrain curve changes from a simple softening shape to a more complex shape. That is, at 5°C, the stress versus pseudostrain relationship starts along the line of equality (i.e., viscoelasticity dominates the behavior with minimal microcracking damage) and then changes to a softening curve, indicating the stiffness reduction due to microcracking in the vertical direction. At the peak stress or slightly over the peak stress, the localization starts, which is the beginning of the macrocrack propagation. At higher temperatures, the stress versus pseudostrain curve starts along the line of equality, and then the slope changes to an upward direction, indicating the hardening behavior. The shape of the curve changes finally to represent the softening behavior followed by the failure at the peak stress. This pattern becomes more evident as the temperature increases and the rate of loading decreases. It is noted that, in tension, this pattern was never observed. The primary mechanisms that govern the constitutive behavior of asphalt concrete in tension are viscoelasticity, the plastic flow of the binder, and cracking. In compression, it is well known that the interlocking of aggregate particles is an important factor that affects the behavior of asphalt concrete. The effect of aggregate interlocking increases as the binder viscosity decreases, which happens when the temperature increases and the rate of loading decreases. The primary characteristic of aggregate interlocking is that it stiffens and becomes more significant as the deformation of the asphalt concrete increases until the aggregate particles begin to slip. The observations made from Fig. 722 are well supported by the expected behavior of asphalt concrete due to aggregate interlocking. It must be noted that this behavior cannot be detected in the stress versus strain plots because of the mixed effects of viscoelasticity and aggregate interlocking. The benefit of using pseudostrain (i.e., eliminating the viscoelasticity from the plot) is clearly demonstrated in this figure. In order to display this hardening and softening behavior of asphalt concrete in compression more effectively, the apparent pseudo secant stiffness (CA) is calculated and presented in Fig. 723. The pseudo secant stiffness in this figure is called apparent because the true pseudo secant stiffness is calculated using the viscoelastic strain only. In this figure, the apparent pseudo secant stiffness is calculated using the total strain, which includes both viscoelastic and viscoplastic strains and the aggregate interlocking effect on these strains. The first and second numbers shown in the legends of three figures in
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FIGURE 722
Stress versus pseudostrain curves: (a) 25°C, (b) 40°C, and (c) 55°C.
Fig. 723 represent the temperature and the ranking of the loading rate (i.e., 1 being the fastest rate and 4 being the slowest), respectively. In Fig. 723(a) for 25°C, the CA decreases all the way to failure as the strain increases. As the temperature becomes higher and the strain rate becomes slower [Fig. 723(b) and (c)], the Sshape of the CA versus strain curve becomes more evident. It is noted that the peak in the CA versus strain curve
VEPCD Modeling of Asphalt Concrete with Growing Damage
FIGURE 723 Effects of temperature and loading rate on the apparent C versus strain relationship: (a) 25°C, (b) 40°C, and (c) 55°C.
occurs between 0.4 to 0.6% strain. The peak stress in the stressstrain curve is found to be between 1 and 1.6% strain, regardless of the temperature and strain rate. A comparison of Figs. 722 and 723 reveals that the behavior of asphalt concrete in compression at high temperatures can be divided into four zones, as shown in Figs. 722(c) and 723(c). In the first zone, the viscoelasticity and the flow of the binder dominate the behavior of asphalt concrete and, as a result, the stress versus pseudostrain curve shows the softening behavior. In this zone, aggregate particles
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Chapter Seven become closer and the air voids collapse, but noticeable aggregate interlocking has not yet formed. The beginning of Zone 2 indicates the onset of aggregatetoaggregate interlocking. As the aggregate particles are locked together, the stiffness of the mixture increases (i.e., the hardening behavior), as shown in Zone 2. In Zone 3, aggregate interlocking degrades slowly as the load increases. At the peak stress, the aggregate particles slip away from each other, which causes the shear failure shown by the descending stress versus pseudostrain curve in Zone 4. The mechanistic modeling of the aggregate interlocking mechanism involves triaxial testing with confining pressure followed by a rigorous formulation of viscoplasticity with the yield surface and flow rule. Chapter 15 presents one example of this more rigorous type of model.
Finite Element Implementation of the VEPCD Model The ultimate goal of developing a constitutive model (e.g., the VEPCD model) of asphalt concrete is to predict the response and performance of asphalt pavement structures in a reliable manner. The North Carolina State University research team is currently working on the implementation of the VEPCD model into the finite element code, FEP++, developed by Guddati (2001). In the following, some preliminary results from the VECDFEP++ are presented in order to shed some light on expectations once the complete system becomes available. Figs. 724 and 725 present the changes in the damage contours as repetitive loading continues on thin and thick asphalt layers, respectively. In these simulations, a loading duration of 0.03 second and a rest period of 0.97 second are selected. The thicknesses of the thin and thick asphalt layers are 3 and 12 in., respectively. The asphalt layer is represented by the VECD model, and the nonlinear stressstatedependent model is used for the aggregate base and subgrade. The pavement structure is modeled by an axisymmetric finite element model. The most important observation from Figs. 724 and 725 is the change in the location of crack initiation as a function of the asphalt layer thickness. When the thinnest layer is modeled in Fig. 724, the severe damage is found at the bottom of the layer with negligible damage at the top of the asphalt layer. However, for the thick asphalt layer in Fig. 725, damage initiates from both the bottom of the asphalt layer and directly under the tire edge, and propagates simultaneously to form a conjoined damage contour. It can be seen that the intensity of damage under the tire edge is as high as that at the bottom of the asphalt layer. This conjoined damage contour, shown in Fig. 725, supports the findings from field studies of topdown cracking (Gerritsen et al. 1987). Also, the conjoined damage contour suggests that the throughthethickness crack may develop as these bottomup and topdown microcracks propagate further and coalesce together. Gerritsen et al. (1987) report that they found field cores with topdown cracking for about 10 cm (4 in.), no cracking at all for about 5 cm (2 in.), and bottomup cracking for about 10 cm (4 in.) in the same core. The conjoined damage contour in Fig. 725 explains the reason behind this observation. This finding clearly demonstrates the problem associated with the traditional approach to fatigue performance prediction in which the tensile strain at the bottom of the asphalt layer is related to the fatigue life of the pavement. It should be noted that the VECDFEP++ model does not need to assume where the microcracks initiate. This feature of the VECDFEP++ program is quite different from
VEPCD Modeling of Asphalt Concrete with Growing Damage
FIGURE 724 Damage evolution in the thin asphalt layer predicted by the VECDFEP++ program.
FIGURE 725 Damage evolution in the thick asphalt layer predicted by the VECDFEP++ program.
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Chapter Seven typical finite element analysis based on fracture mechanics. In the fracture mechanicsbased analysis, an artificial crack must be introduced before the load is applied and critical stresses that contribute most to the macrocrack propagation are identified.
Summary and Conclusions This chapter presents the VEPCD model as the constitutive model of asphalt concrete that accounts for the time and temperature dependence, microcracking damage, and viscoplastic flow. The VEPCD model in tension is applied to four mixtures tested under the FHWA ALF study, including three polymermodified mixtures. The model is shown to accurately predict the behavior under constant crosshead rate tests, random load cyclic tests, and TSRST tests. Through the characterization and validation process it is found that the TTS principle with growing damage is applicable to mixtures with both unmodified and modified asphalt binders. This finding is significant because the testing requirements necessary to characterize the material are significantly reduced. The VEPCD modeling in compression is more complicated than in tension due to the increased contribution of aggregate particles at high temperatures and/or slow loading rates. The experimental results suggest that the stiffening effect of aggregate interlocking must be taken into account to model the compression behavior of asphalt concrete accurately. A compression model based on HiSS yield surface and Perzyna’s viscoplasticity theory is presented as an alternative in Chap. 11. Finally, it is demonstrated that the viscoelastic continuum damage model incorporated into the finite element program (VECDFEP++) may be used not only to evaluate pavement responses under repetitive loading, but also to study the complicated cracking mechanism in asphalt pavements in a realistic manner. It is shown that, as the asphalt layer thickness increases, the propensity of topdown cracking increases, which supports field observations. The NCSU research team is currently developing a threedimensional VEPCDFEP++ program that can be used in predicting asphalt pavement performance, including fatigue cracking (both topdown and bottomup), rutting, and thermal cracking.
Acknowledgment The authors would like to acknowledge the financial support provided by the Federal Highway Administration.
References American Association of State Highway and Transportation Officials (1993), “TP1093 Method for Thermal Stress Restrained Specimen Tensile Strength.” Washington, D.C. American Association of State Highway and Transportation Officials (2003), “TP62 Standard Method of Test for Determining Dynamic Modulus of HotMix Asphalt Concrete Mixtures.” Washington, D.C. Bazant, Z. P. (1986), “Mechanics of Distributed Cracking,” Applied Mechanics Reviews, ASME, Vol. 39, pp. 675–705. Chehab, G. (2002), “Characterization of Asphalt Concrete in Tension Using a Viscoelastoplastic Model,” Ph.D. dissertation, North Carolina State University, Raleigh, N.C.
VEPCD Modeling of Asphalt Concrete with Growing Damage Chehab, G. R., Y. R. Kim, R. A. Schapery, M. W. Witczak, and R. Bonaquist (2002), “TimeTemperature Superposition Principle for Asphalt Concrete Mixtures with Growing Damage in Tension State,” Journal of Association of Asphalt Paving Technologists, Vol. 71, pp. 559–593. Chehab, G. R., Y. R. Kim, R. A. Schapery, M. W. Witczak, and R. Bonaquist (2003), “Characterization of Asphalt Concrete in Uniaxial Tension Using a Viscoelastoplastic Model,” Journal of the Association of Asphalt Paving Technologists, pp. 315–355. Chehab, G. R., and Y. R. Kim (2005), “Viscoelastoplastic Continuum Damage Model Application to Thermal Cracking of Asphalt Concrete,” Journal of Materials in Civil Engineering, ASCE, Vol. 17, No. 4, pp. 384–392. Daniel, J. S. (2001), “Development of a Simplified Fatigue Test and Analysis Procedure Using a Viscoelastic, Continuum Damage Model and Its Implementation to WesTrack Mixtures,” Ph.D. dissertation, North Carolina State University, Raleigh, NC. Daniel, J. S., and Y. R. Kim (2002), “Development of a Simplified Fatigue Test and Analysis Procedure Using a Viscoelastic Continuum Damage Model,” Journal of the Association of Asphalt Paving Technologists, Vol. 71, pp. 619–650. Federal Highway Administration. PooledFund Study entitled “FullScale Accelerated Performance Testing for Superpave and Structural Validation,” FHWA TurnerFairbanks Highway Research Center. Fwa, T. F., B. F. Low, and S. A. Tan (1995), “Laboratory Determination of the Thermal Properties of Asphalt Mixtures by Transient Heat Conduction Method,” Transportation Research Record, Transportation Research Board, National Research Council, Vol. 1492. Gerritsen, A. H., C. A. P. M. Van Gurp, J. P. J. Van der Heide, A. A. A. Molenaar, and A. C. Pronk (1987), “Prediction and Prevention of Surface Cracking in Asphalt Pavements,” 6th International Conference on Structural Design and Asphalt Pavements, The University of Michigan, Ann Arbor, Michigan, pp. 378–391. Gibson, N. H., C. W. Schwartz, R. A. Schapery, and M. W. Witczak (2003), “Viscoelastic, Viscoplastic, and Damage Modeling of Asphalt Concrete in Unconfined Compression,” Transportation Research Record, No. 1860, pp. 3–15. Guddati, M. N. (2001), “FEP++: A Finite Element Program in C++, Input Manual,” Department of Civil, Construction, and Environmental Engineering, North Carolina State University. Jung, D. H., and T. S. Vinson (1994), “Prediction of Low Temperature Cracking: Test Selection,” Rep. SHRPA400, SHRP, National Research Council, Washington, D.C. Kachanov, L. M. (1958), “Time of Rupture Process under Creep Conditions,” Izvestiya Akademii Nauk SSR OtdelenieTechnicheskikh Nauk, Vol. 8, p. 26. Kachanov, L. M. (1986), Introduction to the Theory of Damage, Martinus Nijhoff, The Hague. Kim, Y. R., and D. N. Little (1990), “OneDimensional Constitutive Modeling of Asphalt Concrete,” ASCE Journal of Engineering Mechanics, Vol. 116, No. 4, pp. 751–772. Kim, Y. R., Y. C. Lee, and H. J. Lee (1995), “Correspondence Principle for Characterization of Asphalt Concrete,” Journal of Materials in Civil Engineering, ASCE, Vol. 7, No. 1, pp. 59–68. Kim, Y. R., and H. J. Lee (1997), “Healing of Microcracks in Asphalt and Asphalt Concrete,” Final report submitted to Federal Highway Administration/Western Research Institute. Kim, Y. R., J. Daniel, and H. Wen (2002), “Fatigue Performance Evaluation of WesTrack and Arizona SPS9 Asphalt Mixtures Using Viscoelastic Continuum Damage Approach,” Final report to Federal Highway Administration/North Carolina Department of Transportation, Report No. FHWA/NC/2002–004.
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Chapter Seven Kim, Y. R., and G. Chehab (2004), “Development of a Viscoelastoplastic Continuum Damage Model for AsphaltAggregate Mixtures: Final Report as Part of Tasks F and G in the NCHRP 919 Project,” National Cooperative Highway Research Program, National Research Council, Washington, D.C. Kim, Y. R., M. N. Guddati, B. S. Underwood, T. Y. Yun, V. Subramanian, and A. H. Heidari (2005), “Characterization of ALF Mixtures Using the Viscoelastoplastic Continuum Damage Model,” Final report to the Federal Highway Administration. Krajcinovic D. (1984), “Continuum Damage Mechanics,” Applied Mechanics Review, ASME, Vol. 37, pp. 397–402. Krajcinovic D. (1989), “Damage Mechanics,” Mechanics of Materials, Vol. 8, pp. 117–197. Lamborn, M. J., and R. A. Schapery (1988), “An Investigation of Deformation PathIndependence of Mechanical Work in FiberReinforced Plastics,” Proceedings of the Fourth JapanU.S. Conference on Composite Materials, Washington, D.C., pp. 991–997. Lamborn, M. J., and R. A. Schapery (1993), “An Investigation of the Existence of a Work Potential for Fiber Reinforced Plastic,” Journal of Composite Materials, Vol. 27(4), pp. 352–382. Lee, H. J. (1996), “Uniaxial Constitutive Modeling of Asphalt Concrete Using Viscoelasticity and Continuum Damage Theory,” Ph.D. dissertation, North Carolina State University, Raleigh, NC. Lee, H. J., and Y. R. Kim (1998a), “A Uniaxial Viscoelastic Constitutive Model for Asphalt Concrete under Cyclic Loading,” ASCE Journal of Engineering Mechanics, Vol. 124, No. 1, pp. 32–40. Lee, H. J., and Y. R. Kim (1998b), “A Viscoelastic Continuum Damage Model of Asphalt Concrete with Healing,” ASCE Journal of Engineering Mechanics, Vol. 124, No. 11, pp. 1224–1232. Lemaitre J. (1984), “How to Use Damage Mechanics,” Nuclear Engineering Design, Vol. 80, pp. 233–245. Park, S. W. (1994), “Development of a Nonlinear ThermoViscoelastic Constitutive Equation for Particulate Composites with Growing Damage,” Ph.D. dissertation, Texas A&M University, Tex. Park, S. W., Y. R. Kim, and R. A. Schapery (1996), “A Viscoelastic Continuum Damage Model and Its Application to Uniaxial Behavior of Asphalt Concrete,” Mechanics of Materials, Vol. 24, No. 4, pp. 241–255. Perl, M., J. Uzan, and A. Sides (1983), “ViscoElastoPlastic Consititutive Law for a Bituminous Mixture under Repeated Loading,” Transportation Research Record 911, TRB, National Research Council, Washington, D.C., pp.20–27. Schapery, R. A. (1975), “A Theory of Crack Initiation and Growth in Viscoelastic Media, Part I: Theoretical Development, Part II: Approximate Methods of Analysis, Part III: Analysis of Continuous Growth,” International Journal of Fracture, 11, pp. 141–159, 369–388, 549–562. Schapery, R. A. (1981), “On Viscoelastic Deformation and Failure Behavior of Composite Materials with Distributed Flaws,” Advances in Aerospace Structures and Materials, AD01, ASME, New York, pp. 5–20. Schapery, R. A. (1984), “Correspondence Principles and a Generalized Jintegral for Large Deformation and Fracture Analysis of Viscoelastic Media,” International Journal of Fracture, Vol. 25, pp. 195–223. Schapery, R. A. (1987a), ‘‘Deformation and Fracture Characterization of Inelastic Composite Materials Using Potentials,’’ Polymer Engineering, Vol. 27, pp. 63–76.
VEPCD Modeling of Asphalt Concrete with Growing Damage Schapery, R. A. (1987b), ‘‘Nonlinear Constitutive Equations for Solid Propellant Based on a Work Potential and Micromechanical Model,’’ Proceedings, 1987 JANNAF Structures & Mechanical Behavior Meeting, CPIA. Schapery, R. A. (1990), “A Theory of Mechanical Behavior of Elastic Media with Growing Damage and Other Changes in Structure,” Journal of the Mechanics and Physics of Solids, Vol. 38, pp. 215–253. Schapery, R. A. (1999), “Nonlinear Viscoelastic and Viscoplastic Constitutive Equations with Growing Damage,” International Journal of Fracture, Vol. 97, pp. 33–66. Seibi, C., G. Sharma, Ali Galal, and J. Kenis (2001), “Constitutive Relations for Asphalt Concrete under High Rates of Loading,” Transportation Research Record 1767, TRB, National Research Council, Washington, D.C., pp. 111–119. SHRPA357 (1993), “Development and Validation of Performance Prediction Models and Specifications for Asphalt Binders and Paving Mixes,” Strategic Highway Research Program, Washington, D.C. Sicking, D. L. (1992), “Mechanical Characterization of Nonlinear Laminated Composites with Traverse Crack Growth,” Ph.D. dissertation, Texas A&M University, Tex. Simo, J. C., and T. J. R. Hughes (1998), Computational Inelasticity. SpringerVerlag, New York. Underwood, B. S., Y. R. Kim, and G. R. Chehab (2006a), “A Viscoelastoplastic Continuum Damage Model of Asphalt Concrete in Tension,” Proceedings of the 10th International Conference of Asphalt Pavements. Underwood, B. S., Y. R. Kim, and M. N. Guddati (2006b), “Characterization and Performance Prediction of ALF Mixtures Using a Viscoelastoplastic Continuum Damage Model,” Journal of Association of Asphalt Paving Technologists, AAPT, Vol. 75, pp. 577–636. Uzan, J. (1996), “Asphalt Concrete Characterization for Pavement Performance Prediction,” Asphalt Paving Technology, AAPT, Vol. 65, pp 573–607. Zhao, Y. (2002), “Permanent Deformation Characterization of Asphalt Concrete Using a Viscoelastoplastic Model,” Ph.D. dissertation, North Carolina State University.
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Unified Disturbed State Constitutive Modeling of Asphalt Concrete Chandrakant S. Desai
Abstract Although the need for mechanistic and unified constitutive models for pavement materials has been identified, such models are yet not readily available. Some unified approaches have been proposed; however, they are often based on ad hoc combinations of models for specific properties such as elastic, plastic, creep, and fracture. Such approaches do not provide appropriate connections to coupled responses in the bound and unbound materials. They often involve larger number parameters, sometimes without physical meanings. The disturbed state concept (DSC), presented in this chapter, provides a modeling procedure that includes the coupled response including such factors as elastic, plastic, and creep stains, microcracking and fracture, softening and healing under mechanical and environmental (thermal, moisture, etc.) loadings. It is based on a single unified and coupled framework, and can be applied to both solids (bound and unbound materials) and interfaces and joints. A brief review of various available approaches is presented and the differences and advantages of the DSC are identified. The DSC has been validated and applied to a wide range of materials such as soils, rocks, concrete, asphalt concrete, ceramics, alloys (solders), and silicon; in this chapter, it is directed toward modeling of asphalt concrete. The DSC allows evaluation of various distresses such as permanent deformation (rutting), microcracking and fracture, reflection cracking, thermal cracking, and healing. It has been implemented in two and threedimensional (2D and 3D) finite element (FE) procedures, which allow static, repetitive, and dynamic loadings. In the hierarchical scheme, DSC allows selection of various versions such as elastic, plastic, creep, microcracking leading to fracture and failure, depending on the need of the user. A number of examples are solved for various distresses considering flexible (asphalt) pavements; however, the DSC model is applicable to rigid (concrete) pavements also. It is believed that the DSC can provide a unified and
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Chapter Eight versatile approach for constitutive modeling of materials in the pavement structures, and with the nonlinear FE codes can provide a novel approach for analysis and design in pavement engineering. Further research could involve detailed (laboratory) testing of asphalt concrete, concrete, and interfaces in the pavements, together with measurements and validations of simulated (in the laboratory) and field problems.
Introduction The need for improved mechanistic procedures for design, maintenance, and rehabilitation of highway and airport pavements has been recognized for many years now. Mechanistic procedures are based on the principles of mechanics in contrast to ad hoc and empirical procedures that are often used. Accurate predictions for the response of pavements under mechanical and environmental loads require consideration of important factors such as multidimensional geometry, realistic loading, and appropriate constitutive models. A vital ingredient that influences the response is the nonlinear behavior of materials in pavements. Elastic, plastic, and creep strains, microcracking, softening, and healing under repetitive mechanical and environmental loading, and initial or in situ stress conditions, are among the important characteristics that need to be considered in modeling and testing for the nonlinear behavior. In order to incorporate the nonlinear material response and multidimensional effects in the solution procedures for design, maintenance, and rehabilitation, it becomes necessary to invoke the basic principles of mechanics so as to develop unified and mechanistic procedures.
Scope The scope of this chapter includes (1) brief review of some existing models for materials, (2) discussion of the limitations of existing procedures based mainly on empirical and/ or empiricalmechanistic approaches, (3) description of the unified modeling approach called the disturbed state concept (DSC) that provides a mechanistic model for the significant factors that influence behavior of pavement materials, (4) brief description of two and threedimensional nonlinear finite element computer procedure in the implementation of the DSC models, and (5) description of the capabilities of the DSC procedures to handle major distresses: permanent deformation (rutting), microcracking and fracture, and reflection cracking under mechanical and thermal loading, and typical application and validations. This chapter is based on various previous publications, for example, on Desai (2001), Desai and Ma (1992), and Desai et al. (1986) for development of constitutive and computer models, and on Desai (2002, 2007) for their application for pavement analysis. The major emphasis in this chapter is given to asphalt concrete.
Approaches for Pavement Analysis and Design Figure 81 shows a schematic of various approaches for design, maintenance, and rehabilitation of pavements. The empirical (E) approach is based on experience and/or knowledge of certain index properties such as California bearing ratio (CBR), limiting shear failure and limiting deflections (Huang 1993). The index and empirical models do not include effects of multidimensional geometry, loading, material behavior and
Unified Disturbed State Constitutive Modeling of Asphalt Concrete
FIGURE 81
Methods for pavement analysis and design.
spatial distribution of displacements, stresses and strains in the multilayered pavement systems. Hence, such empirical approaches are considered to possess only limited capabilities. The mechanisticempirical (ME) approach is based on the limited use of the principles of mechanics such as elasticity, plasticity, and viscoelasticity. It involves two steps. In the first step, the layered pavement system is analyzed by using a mechanistic
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Chapter Eight model such as the layered elastic theory and FE procedure that includes elastic, nonlinear elastic [e.g., resilient modulus (RM) model] or elastoplastic models. The last may involve such classical plasticity models as von Mises, MohrCoulomb, and hardening or continuous yielding (Desai 2001; Schofield and Wroth 1968; Vermeer 1982). The stresses and strains, usually under total or incremental application of the wheel load, are computed and used in empirical formulas for calculation of rutting, damage, cracking under mechanical and temperature load, and cycles to failure. Very often, uniaxial quantities such as the tensile strain, et, at the bottom of the asphalt layer, vertical compressive strain ec at the top of the subgrade layer, vertical stress sy under the wheel load, and tensile stress st at the bottom of the asphalt layer, are used to compute various distresses with the empirical formulas (Huang 1993). The ME approach can lead to improved design compared to the empirical approach. However, it does not allow for realistic material behavior as affected by elastic, plastic, and creep responses under mechanical and environmental loading. Furthermore, evaluation of distresses based on the uniaxial quantities in empirical formulas may not provide accurate predictions of the distresses as affected by the multidimensional geometry, nonhomogeneities, anisotropy, and nonlinear material response, which is dependent on stress, strain, time, and load repetitions. The full mechanistic (M) approach allows for geometry, nonhomogeneities, anisotropy, and nonlinear material properties of all layers in a unified manner. As a result, the distresses are evaluated as a part of the solution (e.g., finite element) procedure, without the need of the empirical formulas. The AASHTO Design Guides (e.g., 1986, 1993) are often used for pavement design. The most recent Design Guide (NCHRP 2004) includes the ME approach. The Strategic Highway Research Program (SHRP) (Lytton et al. 1993) and ongoing Superpave research attempt to develop the general and unified material models. However, such unified models are very often based on combinations of models for specific material properties such as linear elastic creep, viscoplastic creep, damage and fracture (Kim et al. 1997; Rowe and Brown 1997; Schapery 1965, 1990, 1999; Secor and Monismith 1962). Although such models have been used commonly in the pavement engineering area, often ad hoc combination may not be suitable for the realistic behavior of materials in which the elastic, plastic, creep, damage, fracture, and healing occur simultaneously under the applied loading. The combination of models suffers from some limitations: The component models may not be consistently integrated, they can be relatively complex, and the material parameters involved can be large. Some of the parameters do not have physical meanings, they may not relate to the specific states during deformation, and hence, they need to be determined by using mainly curve fitting and least square procedures. On the other hand, unified and concise models that can overcome many of the above limitations are available in other engineering areas such as mechanics, geomechanics, and mechanical engineering. In the early 1980s, the author conducted a research project (Desai et al. 1983) where constitutive models available at that time were developed and implemented in mechanistic 2D and 3D finite element procedures for track support and pavement systems. The models were calibrated by using comprehensive material tests and the computer codes were verified with respect to field observations. Indeed, there exists a need for advanced and unified mechanistic models. One such model based on the hierarchical single surface (HISS) plasticity model by Desai and coworkers (Desai et al. 1986; Desai et al. 1993; Desai 2001) has been also used for pavement analysis by Scarpas et al. (1997). The HISS model has been used successfully for unbound materials
Unified Disturbed State Constitutive Modeling of Asphalt Concrete by Bonaquist (1996) and Bonaquist and Witczak (1997). This chapter contains the DSC model, which includes the HISS model as a special case. It is considered to be unified and economical compared to other models currently available.
Objective The main objective of this chapter is to present a full mechanistic approach with the powerful and unified constitutive models based on the disturbed state concept (DSC) that has been already developed, and can be readily used for modeling asphalt concrete and pavement applications. Before the presentation of the DSC models and associated two and threedimensional codes for the treatment of various distresses, a brief review of some of the available models is first presented.
Review of RM Approach The resilient modulus (RM) approach has been used extensively in pavement engineering (Witczak and Uzan 1988; Barksdale et al. 1990; Huang 1993). Although it can provide satisfactory predictions of elastic uniaxial displacements, it is not capable of predicting the foregoing multidimensional effects such as rutting, microcracking, and fracture. The RM approach relies on the test behavior of pavement materials in which it is observed that after a certain critical number of loading cycles Nc (Fig. 82), the material reaches the socalled resilient state, in which the material is considered to be approximately elastic (Huang 1993). Hence, it is possible to compute uniaxial (vertical) strains and displacements in the resilient state by using the resilient modulus MR. However, the material can experience microcracking growth at a cycle much before the resilient cycle Nc. The microcracks, would initiate, grow, coalesce and, at the critical cycle Nc or critical disturbance Dc, as described later, which may occur before or after Nc, fracture may take place. It may not be possible to predict the fracture behavior by using the RM approach.
FIGURE 82
Resilient condition, repetitive load.
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Chapter Eight Often the RM is used simply as a parameter (modulus) to evaluate, for example, the vertical deflection in a pavement. The RM can be used to characterize the stressstrain behavior of a material in the resilient state by using mathematical functions such as hyperbola, parabola, and (combination of) exponential functions (Witczak and Uzan 1988; Huang 1993; Lytton et al. 1993; Desai 2001). Then, it can be implemented as a nonlinear or piecewise linear elastic model in a solution (finite element) procedure, which can yield stresses and strains. Such quantities can be used in empirical formulas for various distresses; for example, in the ME approach, Fig. 81(b). A finite element procedure with the piecewise nonlinear RM approach with interface and infinite elements is developed by Desai (2000a) for incorporation in the NCHRP 137A MechanisticEmpirical Pavement Design Guide (NCHRP 2004). Details of RM are given in various publications and other chapters. Some comments are given below.
Comments 1. Poisson’s Ratio. When MR is used in the context of nonlinear elasticity, it can take the place of the traditional tangent elastic modulus Et. Then for isotropic materials, Poisson’s ratio n can be assumed constant or can also be expressed as a function of stress (Lytton et al. 1993). In the nonlinear elastic formulation, the behavior of the material is still treated as elastic during each increment of loading. Hence, in the context of the theory of elasticity, the value of Poisson’s ratio needs to be less than 0.5; otherwise, the formulation will collapse due to the singularity in the stressstrain matrix (Desai et al. 1984; Desai and Kundu 2001). 2. Efforts have been made to express strain ratio (lateral strain e3 to the axial strain e1), the value of which may be greater than 0.5. Indeed, such a ratio can be termed as Poisson’s ratio only up to the behavior when nt < 0.5, that is, only during the contractive (volume) state and before dilation. Such formulations in the context of linear elastic theory may not be realistic. Theories such as plasticity can be used to accommodate the (dilative) behavior. More details regarding the Poisson’s ratio of asphalt concrete are presented in Chap. 3. 3. Since MR is defined usually based on uniaxial (triaxial) tests, it is valid mainly for the calculation of uniaxial (vertical) strains and displacements. 4. When MR is employed in the incremental nonlinear analysis, it represents a piecewise linear elastic model. Hence, it is not capable of accounting for the plastic or irreversible deformations. 5. When MR is used as an elastic modulus, it is tacitly assumed that the material is isotropic. In other words, MR may not allow for anisotropic behavior. 6. The resilient modulus is evaluated from the tests that involve only one stress path; for example, conventional triaxial compressions (CTC) in triaxial testing (Fig. 83). However, a real pavement can experience different stress paths under the wheel load. Hence, the RM approach is valid only for one stress path, and it will need different sets of material parameters for different stress paths. Hence, it may not provide accurate predictions for the real situations. 7. The bound (asphalt, concrete) and unbound (subbase, base, and subgrade) geologic materials experience volume change response under shear stresses caused by
Unified Disturbed State Constitutive Modeling of Asphalt Concrete
FIGURE 83
Various stress paths, for example, CTC (s1 > s2 = s 3).
wheel loads. Although a nonlinear elastic model such as the resilient modulus can allow for a part of the volume change response, it cannot predict the total volume change that can affect deformations and microcracking in the pavement. 8. The relative particle motions (translations, sliding, rotation, etc.) lead to plastic or irreversible motions, which are often expressed in terms of such internal variables as total plastic strains (trajectory) and plastic work or dissipated energy. These irreversible strains are a major factor in causing microcracking that leads to fracture and failure. The RM approach, which yields mainly elastic strains, is not capable of accounting for microcracking and fracture.
211
212
Chapter Eight 9. Although a material under repetitive loading after the resilient conditions may experience mainly elastic strains, Fig. 82, the concomitant microcracking and fracture before and after the resilient condition can be affected by the history of plastic strains or work that accumulate during the preresilient states. Hence, a mechanistic model should be able to incorporate the accumulated plastic strains or work on the subsequent response involving permanent strains (rutting), microcracking, and fracture. 10. In the pavement literature (e.g., Witczak and Uzan 1988), it is claimed that the RM approach can characterize the behavior of geologic (unbound) materials. In fact, it has been noted by many investigators that such nonlinear elastic models cannot represent realistic behavior of unbound (geologic) materials because they are affected by factors such as plastic deformations, stress path, volume change, type of loading, and in situ conditions (Desai and Siriwardane 1984; Desai 2001).
Other Models In addition to the RM approach, a number of other (semi) empirical approaches are used in pavement analysis for the evaluation of important distresses such as rutting, damage, and fracture. These approaches are based on the computed stresses and strains at selected locations in the pavement, often obtained by using layered elastic analysis or nonlinear elastic finite element procedures. In conjunction with empirical factors based on (field) observations, such approaches can sometimes provide reasonable predictions. However, they may not be considered mechanistic because they do not involve calculation of distresses based on evolving stresses and strains in the multidimensional pavements, as affected by nonlinear material response. In addition to the linear and nonlinear elastic models, plasticity models have often been used for pavement materials. Plasticity models can include classical (e.g., von Mises, MohrCoulomb, and DruckerPrager) and enhanced (e.g., continuous hardening or yielding: critical state and Cap, Vermeer and hierarchical single surface—HISS) models. Although these models can provide improvements, particularly regarding prediction of permanent deformations, they are not directly capable of handling other important factors such as microcracking and fracture. The SHRP project (Lytton et al. 1993) employed the viscoelastic models used commonly for pavement materials such as asphalt (Schapery 1965, 1999), and other models for plasticity, fracture, and damage. Each of these models was essentially an independent one, and the overall model for the combined (elastic, plastic, creep, damage, and fracture) responses may be considered to represent a combination of models. As a consequence, the overall model can be complex and involve a large number of parameters, many of which did not have physical meanings. In other words, they were not related to specific states of material behavior, and their determination involved mainly curve fitting procedures. In short, the overall model used may be unrealistic for rational and realistic modeling of the unified behavior of bound and unbound (geologic) materials. It was suggested that the RM model can be used for the unbound materials. As explained before, this may not be realistic because previous research for geologic materials showed that nonlinear elastic (resilient modulus) models cannot represent the actual behavior to allow for factors such as plastic strains, volume change, stress paths, and repetitive loading (Desai 1998b, 2001). It is believed that the approaches based on the combination
Unified Disturbed State Constitutive Modeling of Asphalt Concrete of separate viscoelasticity with plasticity, damage, and fracture mechanics models may not lead to unified and economical models for pavement materials.
Unified Model Hence, although continuing improvements have occurred for pavement distress analysis, no unified mechanistic models have yet been developed and validated for design, maintenance, and rehabilitation. A unified model should be able to characterize all significant material responses in a single framework. This chapter presents an integrated methodology based on the unified constitutive model called the disturbed state concept (DSC) for modeling of pavement materials, interfaces, and joints. It is believed that the DSC with two and threedimensional computer FE procedures provides a fully mechanistic approach considered to be desirable in pavement engineering. It can provide a unified model that is considered to be superior to other available models including the ad hoc combinations described before.
Factors in Mechanistic Unified Model The basic issue is the prediction of the performance of a pavement under repetitive mechanical and environmental (thermal, fluid, etc.) loadings. The mechanical loading is due mainly to the repeated application of trafficwheel load. The thermalloading arises form the variation of temperature with time—daily and seasonal. The fluid in pavement materials can be due to the ingress of water, which may lead to full or partial saturation of the materials. In some conventional procedures, the materials in the pavement are assumed to be linearly elastic and isotropic; then, the models such as elastic layered theory are used to predict displacements, stresses, and strains (Huang 1993). However, both the bound and unbound materials in the pavement exhibit nonlinear behavior, which is affected by factors such as the state of stress and strain; initial or in situ conditions like stress, pore water pressure, and inhomogeneities; irreversible (plastic) deformations; viscous or creep response; stress path; volume change; anisotropy; temperature; fluid and type of loading. Hence, although the assumption of elastic behavior may yield satisfactory results, their validity is highly limited. For a full mechanistic characterization, it is necessary to use constitutive or material models that allow for the foregoing factors.
Disturbed State Models The descriptions and statements of following topics are included in mechanistic DSC approach: (1) brief description of the unified and hierarchical DSC constitutive model, (2) capabilities of the DSC for various pavement distresses such as (a) permanent deformations; (b) microcracking, fracture, and reflection cracking; and (c) thermal cracking, as affected by plastic and creep strains under mechanical and environmental loading, (3) identification of parameters in the DSC model and their determination from laboratory tests, (4) validation of laboratory test data using the DSC model, (5) implementation of DSC in two and threedimensional finite element procedures, (6) statement of validation for a number of laboratory simulated and field problems in geotechnical and pavement engineering, and analysis of both 2D and 3D pavement problems, and (7) unified methodology with DSC for design, maintenance, and rehabilitation of pavement structures.
213
214
Chapter Eight
Disturbed State Concept The DSC is based on the idea that the behavior of a deforming material (element) can be expressed in terms of the behavior of the relative intact (RI) or continuum part and the microcracked (or healed) part called the fully adjusted (FA) part. During the deformation, the (initial) RI material transforms continuously into the FA part and at the limiting condition (load), the entire material element approaches the FA state. Figure 84 shows a schematic of the RI and FA states. The disturbance, which acts as the coupling and interpolation mechanism, allows definition of the actual or observed behavior in terms of those of the material in the RI and FA states. The transformation of the material from the RI to the FA state occurs due to the microstructural changes caused by relative motions such as translation, rotation, and interpenetration of the particles and softening or healing at the microlevel. The disturbance expresses such microstructural motions, and hence, it is not necessary to define material response at the particle level, as is done in the micromechanical
FIGURE 84
Schematic representation of disturbed state concept.
Unified Disturbed State Constitutive Modeling of Asphalt Concrete approaches. In fact, the DSC allows for the interaction and coupling between the RI and FA parts and avoids the need for the definition of particle level response (as in the micromechanics approach), which is difficult or impossible to measure at this time.
Equations Based on the equilibrium of forces on a material element, the incremental constitutive equations are derived as (Desai 2001): d σ a = (1 − D) d σ i + Dd σ c + dD(σ c − σ i ) ~
~
~
~
(81a)
~
d σ a = (1 − D) Ci d ε i + D Cc d ε c + dD (σ i − σ i )
or
~
~
~
~
a
dσ = C
or
~
DSC
~
~
~
~
dε ~
(81b) (81c)
where a, i, and c = observed RI and FA responses, respectively σ and ε = stress and strain vectors, respectively ~ ~ C = constitutive or stressstrain matrix ~
D = disturbance dD = increment or rate of D As a simplification, D is assumed to be a scalar in a weighted sense. It can, however, be expressed as a tensor, Dij, if the test data to define the directional values of D is available (Desai 2001).
Capabilities and Hierarchical Options Figure 85 shows a summary of the capabilities of the DSC model. In the single framework, the DSC approach is capable of allowing for elastic, plastic, and creep strains, microcracking, fracture and disturbance (damage), and stiffening under mechanical and environmental loading. This is considered to be a unique advantage compared to other available models. A major advantage of the DSC is that various specialized versions, such as elasticity, plasticity, creep, microcracking, degradation or softening, and healing or stiffening can be obtained from Eq. (81). If there is no disturbance (damage) due to microcracking and fracture, D = 0 and Eq. (81) reduces to the classical incremental equations as i
i
dσ = C d ε ~
~
~
i
(82)
where C can represent elastic, elastoplastic, or elastoviscoplastic response. If D ≠ 0, ~ the model can include damage and softening, and healing, as shown in Fig. 86. The user can choose an appropriate option for a given pavement material and needs to input only parameters relevant to that option. For instance, the bound asphalt material can be characterized by using the DSC with elastoviscoplastic model, whereas the unbound materials can be treated as elastoplastic. i
215
216
Chapter Eight
FIGURE 85
DSC capabilities.
Unified Disturbed State Constitutive Modeling of Asphalt Concrete
FIGURE 86
Schematic of softening and healing (stiffening) response in DSC.
If it is assumed that the damaged part carries no stress at all, Eq. (81) reduces to the conventional continuum damage model (Kachanov 1986). However, such a model does not allow for the interaction between the damaged and undamaged parts and can suffer from deficiencies like spurious mesh dependence. Many workers have introduced external “enhancements” in the damage model (e.g., Bazant 1994; Bazant and Cedolin 1991) to allow for the interaction. The DSC model allows the interaction implicitly, and possesses a number of advantages compared with the other (enhanced) models (Desai 2001).
217
218
Chapter Eight
Plasticity Models In the general DSC equations, Eq. (81), C i represents the behavior of the RI material. It ~ can be characterized as elastic, nonlinear elastic, or elastoplastic hardening; for the latter, the unified hierarchical single surface (HISS) plasticity model can be used. The yield function F in the HISS model for isotropic hardening is given by Desai (1980) and Desai et al. (1986) as follows: F = J 2 D − (−α J1n + γ J12 ) (1 − β Sr )−0.5 = 0 = J 2 D − F1 ⋅ F2 = 0 where
(83a) (83b)
J 2 D = J 2 D / pa2 = nondimensionalized second invariant (J2D) of the deviatoric stress tensor Sij pa = atmospheric pressure constant J1 = ( J1 + 3R) / pa = nondimensionalized first invariant of the total stress tensor σ ij R = parameter proportional to cohesion −
3
Sr = stress ratio J 3 D ⋅ J 2 D2 J3D = third invariant of Sij n = parameter associated with the phase change from contractive to dilative response g and b are associated with the ultimate yield surface, Fig. 87, and a is the hardening or growth function given by
α=
a1 ξη
(84)
1
where ξ = ∫(dε ijp ⋅ dε ijp )1/2 is the trajectory of total strains and a1 and η1 are the hardening parameters. x can be decomposed as
ξ = ξv + ξD ξv =
1 3
ε ijp
(85)
ξD = ∫(dEijp ⋅ dEijp )1/2 ε ijp , Eijp , and ε iip are the total, deviatoric, and volumetric plastic strains, respectively. Schematics of F in various stress spaces are shown in Fig. 87. The yield function F in Eq. (83) is found to be suitable for compression or tension (yielding) response, which can be shown in the positive quadrant of J1 − J 2 D plot, Fig. 87. For predominantly (yielding) compressive behavior, the possibility of tensile condition can be accounted for approximately by using an ad hoc scheme such as the stress transfer method when the material enters the tensile regime. Erkens et al. (2002) have discussed such a need for asphalt concrete, and modified functions with an ad hoc procedure have been proposed. For some materials, F in Eq. (83) may need to be
Unified Disturbed State Constitutive Modeling of Asphalt Concrete
FIGURE 87
Plots of yield surface F in stress spaces.
modified to account for significant yielding (or hardening) in both tension and compression. However, it may be difficult to develop such a continuous yield function because the parameters for the compressive and tensile responses are usually different (Desai 2007). Various classical and other plasticity models such as von Mises, DruckerPrager, MohrCoulomb, critical state, Vermeer, and Cap models are included as special cases of
219
220
Chapter Eight
FIGURE 88
Cyclic loading and disturbance.
F in Eq. (83) (Desai 2001). Thus, the general DSC model can provide a selection of the foregoing plasticity models as specialized versions.
Disturbance The disturbance (D), which can be expressed in terms of x or (plastic) work w, can characterize microcracking leading to fracture and degradation or healing. A simple expression for D is given by D = Du (1 − e − Aξ ) z D
(86)
where Du, A, and Z are the disturbance parameters. Figures 84(c) and 88 show test data form quasistatic or cyclic tests with the disturbance. Such test data can be obtained from uniaxial, shear, or triaxial tests.
Creep Behavior The DSC model, Eq. (83), allows incorporation of the creep behavior through the use of the hierarchical multicomponent DSC system; the rheological model including both elastic and plastic creep is depicted in Fig. 89. This general model allows a choice of elastic, Maxwell, viscoelastic, elastoviscoplastic (Perzyna 1966) and viscoelasticviscoplastic models depending upon the user’s need. Details of this model are given by Desai 2001; Desai et al. 1995. The major advantages of the multicomponent DSC (Desai 2001) and overlay (Pande et al. 1977) models are that they are consistent with the finite element analysis, their material parameters are similar to those for elastic, plastic, and viscoplastic models, and
Unified Disturbed State Constitutive Modeling of Asphalt Concrete
FIGURE 89
Multicomponent DSC model.
they can be determined from standard laboratory tests. These are important advantages compared to other available models. For the viscoplastic version of the multicomponent DSC, there are parameter Γ and N given in the following equations (Perzyna 1966): d ε
up
~
=Γ φ
∂F ∂σ
(87a)
~
⎛ F⎞ φ=⎜ ⎟ ⎝ Fo ⎠
N
(87b)
where ε~ is the vector of the viscoplastic strain, Fo is the reference value of F. up
Thermal Effects The thermal effects involve responses due to the temperature change (ΔT) and the dependence of material parameters on the temperature. The former is obtained by expressing the incremental strain vector as (Desai 2001; William and Shoukry 2001): t
e
p
d ε (T ) = d ε (T ) + d ε (T ) + d ε (T ) ~
where
~
~
~
(88)
ε~ = strain vector t, e, and p = total, elastic, and plastic strains, respectively T = temperature dependence d ε~ (T ) = strain vector due to the temperature change ΔT
The parameters in F, Eqs. (83) and (86), are expressed as the function of T. The temperature dependence is expressed by using a single function (Desai et al. 1997) ⎛ T⎞ p(T ) = p(Tr ) ⎜ ⎟ ⎝ Tr ⎠
λ
(89)
221
222
Chapter Eight where p = parameter (elastic, plastic, creep) Tr = reference temperature, for example, room temperature (= 27oC or 300 K) l = parameter The values of p(Tr) and l are found from laboratory test data at different temperatures.
Rate Effect The rate effect can be included by expressing the parameters as function of strain (ε ) or displacement (δ ) rate: P (T , ε ) = P (Tr , ε r ) ⋅ f
(810)
where f is an appropriate function that includes both rate and temperature dependence (Desai 2001; Scarpas et al. 1997).
Fracture In the classical fracture mechanics approach (e.g., Lytton et al. 1993), it is usually necessary to introduce, in advance, a crack(s) of arbitrary dimensions at a selected location in the pavement, say, at the interface between the asphalt pavement and base. Then the equations from fracture mechanics are used to evaluate the initiation and propagation of the fracture. This approach is considered to be limited because the selection of the initial crack and its location is arbitrary, and the crack(s) may initiate at other locations in the pavement depending on the geometry, material properties, and existence of initial cracks. Also, the fracture theories do not allow adequately for the nonlinear (elastoplastic and creep) behavior of the materials. On the other hand, the DSC allows evaluation of the initiation and location of microcracks and their growth depending on the geometry, nonlinear properties and loading conditions; Fig. 810(a) shows a stressstrain curve with the representation of initiation and growth of cracks. Here, the initiation of (micro)cracks is identified by critical disturbance Dcm and the final fracture by the critical disturbance Dc. The values of Dcm and Dc are determined from laboratory tests under quasistatic or cyclic loading that exhibit softening or healing response, Fig. 86. In the computer analysis, the initiation of microcracking in the elements is identified on the basis of the given value of Dcm. The microcracks coalesce and grow, and at the given value of Dc, the final fracture occurs. Values of disturbance greater than Dc, indicates growth of fractures leading to failure at D = Df, Fig. 810(b). The analysis provides a complete picture of the growth of microcracks and fracture in the elements with (cyclic) loading. Then the DSC provides a natural and holistic progression of fracture without the need for arbitrary selection of the location and geometry of cracks, as is required in the fracture mechanics approach.
Healing or Stiffening Both softening and healing have been incorporated in the DSC by modifying the disturbance function D Eq. (86), to include the effect of healing or stiffening depicted after the residual state (b) in Fig. 86. The approach has successfully been applied for temperatureinduced stiffening in dislocated silicon with impurities (Desai et al. 1998). It can easily be applied for healing in pavement when the test results to characterize the healing are available.
Unified Disturbed State Constitutive Modeling of Asphalt Concrete
FIGURE 810
Initiation of microcracking to fracture and disturbance.
Interfaces and Joints A significant attribute of the DSC is that the above framework can be applied to model the behavior of interfaces and joints (Desai et al. 1984; Desai and Ma 1992; Desai 2001).
Material Parameters The general DSC model with microcracking, fracture, and softening involves the following parameters:
223
224
Chapter Eight 1. Elastic: Young’s modulus E Poisson’s ratio n These parameters can be treated as variable functions of stress such as mean pressure p and shear stress J 2 D . The resilient modulus is a special case of such nonlinear model. 2. Elastoplastic: Elastic parameters E and n Plasticity parameters: von Mises: cohesive or yield stress, c (sy ) or, MohrCoulomb: cohesion c and angle of friction, f or, DruckerPrager: two parameters or, hierarchical single surface plasticity (HISS) as follows: Ultimate: g and b Hardening: a1 and h1 Phase change: n Proportional to cohesion: 3R 3. Creep: Here, four overlay options are available: Elastic: E, n Viscoplastic: Γ, N Viscoelastic, viscoelasticviscoplastic parameters depending on the model (Fig. 89) described in (Desai 2001). 4. Disturbance: Ultimate disturbance Du Parameters A and Z 5. Thermal Effects: Parameter l, Eq. (89) It is important to note that only the parameters for a specific option are needed. The foregoing parameters for the general DSC model are needed only if elastoviscoplastic and disturbance (microcracking, fracture, softening) characterization is desired. It may be noted that for the general and all significant capabilities provided by the DSC model, the number of above parameters is not large. For similar capabilities, other models used for pavement often entail a greater number of parameters (Desai 2001); for example, in SHRP/SUPERPAVE approach. Also, the above parameters have physical meanings; in other words, most are related to specific states during the material behavior.
Determination of Parameters The foregoing parameters can be estimated on the basis of standard uniaxial, shear, and/or triaxial tests on specimens of materials. In general, three triaxial tests under different confining pressures, temperature, and rates (if needed) are required to find the parameters. The procedures for finding the parameters are straightforward and given in Desai and Ma (1992); Desai et al. (1986), and Desai (2001).
Unified Disturbed State Constitutive Modeling of Asphalt Concrete
Validation for Laboratory Tests The DSC model and its specialized versions have been validated with respect to laboratory test data for a wide range of materials, for example, soils, rocks, concrete, asphalt, metal (alloys), silicon, and interfaces and joints (Desai 2001). The validations include test data used to determine the parameters, and independent tests not used in the determination; the latter provides a rigorous validation of the model. A validation for asphalt concrete is given below.
Validation for Asphalt Concrete Table 81 shows the material parameters obtained from the triaxial tests under various confining pressures and temperatures on asphalt concrete reported by Monismith and Secor (1962), Desai and Cohen (2000), and Desai et al. (2001). These tests were comprehensive; however, they usually did not exhibit the softening region. Hence, the parameters were found for only the HISS model. Scarpas et al. (1997) reported uniaxial tests for asphalt concrete in which the prepeak and postpeak (softening) behavior was observed. These tests were used to evaluate the parameters for disturbance D Eq. (86) (Simon and Desai 2001). The validation of the stressstrain curves was obtained in two ways: by integrating the constitutive Eq. (81c), and by using the finite element analysis. Figure 811(a) to (c) shows comparisons between the observed and predicted stressstrain curves for strain rate = 1 in/min at three typical temperatures (T) and confining pressures (s 3): (a) T = 40 oF, s 3 = 43.8 psi, (b) T = 77 oF, s 3 = 0.0 psi, (c) T = 140 oF, s 3 = 250 psi (Monismith and Secor 1962). Figure 812 shows typical comparison between predicted and observed typical stressstrain curve for T = 25 oC and displacement rate = 5 mm/s (Scarpas et al. 1997).
Parameter
Asphalt Concrete
Base
Subbase
Subgrade
Concrete
E
500,000 psi
56,532.85 psi
24,798.49 psi
10,013.17 psi
3 × 106 psi
n
0.3
0.33
0.24
0.24
0.25
g
0.1294
0.0633
0.0383
0.0296
0.0678
b
0.0
0.7
0.7
0.7
0.755
n
2.4
5.24
4.63
5.26
5.24
3R
121 psi
7.40 psi
21.05 psi
29.00 psi
8 × 103 psi
a1
1.23E6
2.0E8
3.6E6
1.2E6
0.46 × 10−10
h1
1.944
1.231
0.532
0.778
0.826
Du
1
0.875
A
5.176
668.0
Z
0.9397
1.502
TABLE 81
Material Parameters for 2D and 3D Analysis of Asphalt Pavements and Reflection Cracking (1 psi = 6.89 kPa)
225
226
Chapter Eight
FIGURE 811 Comparison between DSC predictions and test data for different temperatures and conﬁning stresses (1 psi = 6.89 kPa).
The above comparison shows that the DSC model can provide very good simulation of the behavior of asphalt concrete. Validations for a wide range of other materials: concrete, geologic, and metal alloys are given by Desai (2001).
Computer Implementation The DSC model has been implemented in two and threedimensional computer (finite element) procedures (Desai 1998a, 2000b; Desai 2001). The computer codes allow for the nonlinear material behavior, in situ or initial stresses, static, repetitive and dynamic
Unified Disturbed State Constitutive Modeling of Asphalt Concrete
FIGURE 812 Comparison between predicted and observed test data for T = 25°C, displacement rate = 5 mm/s.
loading, thermal and fluid effects. They include computation of displacements, strain (elastic, plastic, creep), stresses, pore water pressures and disturbance during the incremental and transient loading. Specifications of critical values of disturbance D permit identification of the initiation of microcracking leading to fracture and softening, and cycles to fatigue failure. Plots of the growth of disturbance, that is, microcracking to fracture, are obtained as a part of the computation. Accumulated plastic strains lead to the evaluation of the growth of permanent deformations and rutting.
Loading The codes allow for quasistatic and dynamic loading for dry and saturated materials. The repetitive loading on pavement can involve a large number of cycles. An approximate procedure is described below.
Repetitive Loading: Accelerated Procedure Computer analysis for 3 and 2D idealizations can be time consuming and expensive, especially when significantly greater number of cycles of loading need to be considered. Therefore, approximate and accelerated analysis procedures have been developed from a wide range of problems in civil (pavements) (Huang 1993; Lytton et al. 1993), mechanical engineering, and electronic packaging (Desai et al. 1997). Here, the computer analysis is performed for only a selected initial cycles (say, 10, 20), and then the growth of plastic strains is estimated on the basis of empirical relation between plastic strains and number of cycles obtained from laboratory test data. A general procedure with some new factors has been developed (Desai and Whitenack 2001). This procedure is modified for pavement analysis and is described below.
227
228
Chapter Eight
FIGURE 813 analysis.
Accumulated plastic strain versus number of cycles for approximate accelerated
From experimental cyclic tests on various engineering materials, the relation between plastic strain (in the case of DSC, the deviatoric plastic strain trajectory, xD), Eq. (85), and the number of loading cycles can be expressed as ⎛ N ⎞ ξD ( N ) = ξD ( N r ) ⎜ ⎟ ⎝ Nr ⎠
b
(811)
where Nr = reference cycle, and b is a parameter, depicted in Fig. 813. The disturbance Eq. (86) can be written as D = Du [1 − exp ( − A {ξD (N )Z }]
(812)
Substitution of xD(N) from Eq. (811) in Eq. (812) leads to 1/Z ⎡ 1 ⎧1 ⎛ Du ⎞ ⎫ ⎤ N = Nr ⎢ n ⎨ ⎜⎝ D − D ⎟⎠ ⎬ ⎥ u ⎢⎣ ξD (N r ) ⎩ A ⎭ ⎥⎦
1/b
(813)
Now, Eq. (813) can be used to find the cycles to failure Nf for chosen critical value of disturbance = Dc (say, 0.50, 0.75, 0.80). The accelerated approximate procedure for repetitive load is based on the assumption that during the repeated load applications, there is no inertia due to dynamic effects in loading. The inertia and time dependence can be analyzed by using the 3D and 2D procedures; however, for millions of cycles, it can be highly time consuming. Hence, applications of repeated load in the approximate procedure involve the following steps: 1. Perform full 2D or 3D FE analysis for cycles up to Nr, and evaluate the values of xD(Nr) in all elements (or at Gauss points). 2. Compute xD(N) at a selected cycle in all elements using Eq. (811).
Unified Disturbed State Constitutive Modeling of Asphalt Concrete 3. Compute disturbance in all elements using Eq. (812). 4. Compute cycles to failure Nf by using Eq. (813), for the chosen value of Dc. The above value of disturbance allows plot of contours of D in the finite element mesh, and based on the adopted value of Dc, it is possible to evaluate extent of fracture and Nf .
LoadingUnloadingReloading Special procedures are integrated in the codes to allow for loading, unloading, and reloading during the repetitive loads; details are given in Desai (2001).
Validations and Applications The DSC 2D and 3D procedures have been used to predict the laboratory and/or field behavior of a wide range of engineering problems, for example, static and dynamicsoilstructure interaction, dams and embankments, reinforced earth, tunnels, composites in chipsubstrate systems in electronic packaging (Desai 2001) and microstructure instability or liquefaction (Desai 2000c). In the area of multicomponent systems like rail tracks and pavements, its specialized versions have been used to predict the field behavior (Desai and Siriwardane 1982; Desai et al.1983; Desai et al. 1993). It is believed that the DSC approach can be applied successfully for 2D and 3D nonlinear response of both the bound and unbound materials in rigid and flexible pavement (Desai et al. 2001; Desai 2002). The DSC models are applied herein to illustrate the capabilities of 2D and 3D analyses of flexible pavements subjected to monotonic and repetitive loading including permanent deformations, fracture, and reflection cracking; in view of the length limitation, application for thermal cracking is not included at this time. Also, only the DSC model with HISS plasticity as RI behavior is used. The DSC multicomponent model is capable of including the viscoelastic and viscoplastic creep; this will be included in future work based on available test data.
Material Parameters Table 81 shows the DSC parameters for pavement and unbound materials for the following 2D and 3D analyses. The parameters for the asphalt concrete were determined from the comprehensive triaxial tests reported by Monismith and Secor (1962); the quasistatic and creep tests were conducted under various confining pressures and temperatures. The disturbance parameters were also evaluated on the basis of uniaxial tests on asphalt concrete reported by Scarpas et al. (1997). The unbound materials were characterized as elastoplastic by using the HISS model. Their parameters were determined from the triaxial tests reported by Bonaquist (1996). The parameters for concrete are adopted from Desai (2001). Typical applications involving pavement geometrics and loading are given below.
Example 1: Multilayered Asphalt Pavement Linear and nonlinear elastic models such as the resilient modulus have limited capabilities, and in particular, they cannot account for plastic deformations or rutting and fracture. Hence, it is necessary to use models that can allow for plastic deformations. This example is intended to illustrate the difference between the results from the elasticity and plasticity models.
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FIGURE 814
Finite element mesh.
Figure 814 shows the finite element mesh for a fourlayer system with asphalt (flexible) layer. The elastic, plastic (HISS model) and DSC properties are shown in Table 81. The wheel load equal to 200 psi (1.4 MPa) is applied near the center with axisymmetric idealization. The load is applied in increments and the incremental iterative procedure is used (Desai 2001). Figure 815 shows the displacements of the surface at the final load of 200 psi. It can be seen that the deformations with all layers to be elastic are much lower than those with all layers to be elasticplastic (HISSd0 model). Since rutting and microcracking leading to fracture are dependent on the plastic deformations, it is essential to use models that allow for plastic or irreversible deformations. The displacements with the DSC model for the asphalt concrete are not significantly different from the HISS model for all layers. One reason could be that the load of 200 psi does not cause sufficient plastic strains to cause microcracking and disturbance (damage). As will be seen later, the similar load for repetitive case can cause higher plastic strains and disturbance at higher cycles.
Example 2: 2D and 3D Analyses Generally, the pavement problem and wheel loading would require threedimensional analysis, particularly to predict microcracking and fracture response. However, for an
Unified Disturbed State Constitutive Modeling of Asphalt Concrete
FIGURE 815 Computed surface displacements from elasticity, plasticity, and DSC Models (1 in = 2.54 cm).
economical analysis, a twodimensional procedure can provide satisfactory but approximate solutions for certain applications. Figure 816 shows a problem that was idealized as two and threedimensional; the former involved plane strain assumption with unit thickness in the ydirection. The asphalt concrete layer was simulated by using the DSC (HISS) model and the unbound layers were simulated by using the plasticity (HISSd0 model). The total wheel load, which is applied incrementally, is 200 psi. The material properties of the four layers are shown in Table 81. Figure 817(a) and (b) shows load versus displacement curves at the central node, and stress (sz) versus load in the element near the top center for the 2D and 3D analysis, respectively. It can be seen that the computed stresses from both analyses do not show significant difference, while the displacements show difference of the order of 20%. However, for some practical problems, such a difference may be acceptable, particularly because the 3D analysis consumes more time and effort.
Example 3: 3D Analysis of Flexible Pavement Figure 818(a) and (b) shows the 3D problem and front mesh (xz plane), respectively. The material properties used are shown in Table 81.
Loading Analyses are performed by applying linearly monotonic loading and repetitive loading. The monotonic loading was applied up to 200 psi (1400 kPa) in 50 increments.
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FIGURE 816
Fourlayered system: 2D and 3D analyses.
Unified Disturbed State Constitutive Modeling of Asphalt Concrete 200 DispZ_2D DispZ_3D
Load (psi)
160 120 80 40 0 0.000
0.050
0.100 Displacement (in)
0.150
0.200
(a) LoadDisplacement Curve at Center
200.00
StrsZ_2D StrsZ_3D
Stress (psi)
160.00 120.00 80.00 40.00 0.00 0
050
100 Load (psi)
150
200
(b) Stress (σz) vs. Load in Element at Center
FIGURE 817
Computed results from 2D and 3D analyses.
For the repetitive loading, Fig. 819, the load amplitude (P) was equal to 200 psi (1400 kPa). As discussed before, the cyclic load (loading, unloading, reloading) was applied sequentially; however, time dependence was not included at this time. Full finite element analysis was performed for each load amplitude up to Nr = 20 cycles. Then the deviatoric plastic strain trajectory (xD) at the given cycle (N) was computed for the subsequent cycles using Eq. (811). The disturbance D was computed at the given cycles using Eq. (812). This allowed analysis of disturbance with cycles and computation of the cycles to failure Nf depending upon the chosen criteria for critical disturbance Dc [Eq. (813)]. Figure 820 shows surface permanent deformation for load = 100 and 200 psi, respectively. Figure 821 shows the contours of disturbance at load = 200 psi. It can be seen that under the monotonic load of 200 psi, the maximum disturbance is of the order to 0.024. In other words, no microcracking and fracture occurs. However, for repetitive loads with similar amplitude, fracture would occur at higher cycles (Desai 2002).
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FIGURE 818
Flexible pavement for 3D static and repetitive loading analyses.
Unified Disturbed State Constitutive Modeling of Asphalt Concrete
FIGURE 819
Schematic of repetitive loading.
(a) Permanent Displacement (x10) after Steps = 25 and Load = 100 psi
(b) Permanent Displacement (x10) after Steps = 50 and Load = 100 psi
FIGURE 820
Permanent displacements at various steps of loading.
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0.0024 0.0048 0.0072 0.0096 0.012 0.0144 0.0168 0.0192 0.0216 0.024
FIGURE 821
Contours of disturbance after 50 steps (load = 200 psi).
Figure 822(a) to (c) shows contours plots of disturbance after 10, 1000, and 20,000 cycles, respectively, with load amplitude = 70 psi and b = 1.0. After about 20,000 cycles for Dc = 0.8, a portion of pavement has experienced fracture.
Reflection Cracking It is often appropriate to place an overlay on the existing pavements (Molenaar 1983; Huang 1993; FHWA 1987; Kilareski and Bionda 1990). Often, cracks develop into the overlay directly above cracks in the existing pavement under static and repetitive loading. The DSC model is capable of predicting such “reflection” cracking. Figure 823 shows a fourlayered flexible pavement system with introduction of (three) cracks at different locations in the asphalt. The concrete overlay and asphalt were modeled by the DSC while the unbound layers were modeled by the HISSd0 plasticity model. The material parameters are given in Table 81. For the repetitive load behavior, the value of b was adopted as 0.80 for concrete and 0.30 for asphalt. The pavement with the overlay was subjected to the repetitive load amplitude = 5.0 MPa. Figure 824(a) to (d) shows the contours of disturbance around the cracks for N = 20, 100, 1000, and 106 cycles. It can be seen that at lower number of cycles (Nr = 20), the pavement does not experience fracture because D ≤ 0.80. However, the maximum disturbance which is greater than about 0.6 around the cracks attracts relatively higher disturbance (≈ 0.40) above the cracks. As the cycles increase, the trend continues and around N = 1000 (and 106) cycles, the disturbance around the cracks and above them show fracture (Dc ≥ 0.8). Indeed, the value of the amplitude of the repetitive load used here is quite high; for a lower amplitude (say, 0.69 MPa), the number of cycles will be higher.
Disturbance
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011 (a) N = 10 Cycles Disburbance
0.059 0.118 0.177 0.236 0.295 0.354 0.413 0.472 0.513 0.59
(b) N = 1000 Cycles Disburbance
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
(c) N = 20,000 Cycles (D)≥ 0.8 inside white curve
FIGURE 822 Contours of disturbance at various cycles: load amplitude = 70 psi, b = 1.0 (1 psi = 6.89 kPa).
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FIGURE 823
Reﬂection cracking: layered system with three existing cracks in asphalt.
Unified Methodology Although (other) efforts have been and are being made to develop unified models for pavement engineering, it is believed that they have not yet been developed to characterize all significant responses in a single framework. However, based on the mechanistic considerations, and successful application of the DSC models and computer programs (e.g., DSCSST2D, DSCDYN2D, DSCSST3D) for pavement and other engineering disciplines, it is now possible, as desired by the pavement community, to evolve DSC unified models for computation and design under various distresses. This chapter presents a number of applications for distress predictions. In view of the length limitations, applications for other factors such as thermal cracking, moisture effect, and healing are not included. However, they are included in the framework of the DSC capabilities, Fig. 85. Indeed, additional developments including comparisons with analytical and field or simulated pavement performance are considered desirable for application in analysis, design and rehabilitation.
Unified Disturbed State Constitutive Modeling of Asphalt Concrete
Contour Leve 1 < 0.1 0.1 0.2 0.3 0.4 0.5 0.6 >0.6
(a) Nr = 20 Cycles
Contour Leve 1 >0.12 0.12 0.24 0.36 0.48 0.6 0.72 >0.72
(b) N = 100 Cycles
FIGURE 824 Reﬂection cracking: contours of disturbance at various cycles. Note: Scales are different in Figs. 824(a) to (d).
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Contour Leve 1 0.8
(c) N = 1000 Cycles
Contour Leve 1 0.9
(d) N = 106 Cycles
FIGURE 824
(Continued)
Unified Disturbed State Constitutive Modeling of Asphalt Concrete
Conclusions A unified constitutive model called the disturbed state concept (DSC) is developed and used to characterize the elastic, plastic, creep, fracture, softening, and healing behavior of materials in both rigid and flexible pavements. The model has been implemented in 2D and 3D nonlinear finite element procedures and used for predicting test behavior of a wide range of problems in civil, mechanical, and electrical engineering. It is applied here to perform 2D and 3D analyses for the important design aspects for pavements such as rutting (permanent deformations), fracture, and reflection cracking. It is believed that the DSC approach possesses significant advantages over other available models. Its concept is simple, it involves fewer parameters compared to any other model with similar capability, its parameters have physical meanings, and it is suitable for application for analysis, design, and maintenance. The models and programs have been validated with respect to field and laboratory tests for a wide range of problems in civil and mechanical engineering. They have been also verified for limited problems in pavement engineering. However, it will be desirable to use detailed and carefully conducted (field) tests for pavements to further validate the proposed models.
Acknowledgments The development of the constitutive models and applications has been supported by grants from various government and private agencies such as National Science Foundation and Department of Transportation. The computer results reported herein were obtained with assistance from Dr. R. Whitenack, Dr. H. B. Li , Ms. A. Bozorgzadeh, Dr. Z. Wang, and Mr. D. Cohen—their help is gratefully acknowledged. Useful comments and suggestions by Dr. Tom Scarpas are much appreciated.
References American Association of State Highway Officials (AASHTO) (1986, 1993), Guide for Design of Pavement Structures, Washington, D.C. Barksdale, R. D., Rix, G. J., Itani, S., Khosla, P. N., Kim, R., Lamb, P. C., and Rahman, M. S. (1990), Laboratory Determination of Resilient Modulus for Flexible Pavement Design, NCHRP Report 128, Georgia Inst. of Technology. Bazant, Z. P. (1994), Nonlocal Damage Theory Based on Micromechanics of Crack Interactions, J. Eng. Mech., ASCE, 120, pp. 593–617. Bazant, Z. P., and Cedolin, L. (1991), Stability of Structures, Oxford University Press, New York. Bonaquist, R. J. (1996), Development and Application of a Comprehensive Constitutive Model for Granular Materials in Flexible Pavement Structures, Doctoral dissertation, University of Maryland, College Park, Md. Bonaquist, R. F., and Witczak, M. W. (1997), A Comprehensive Constitutive Model for Granular Materials in Flexible Pavement Structures, Proc. 4th Int. Conf. on Asphalt Pavements, Seattle, Wash., pp. 783–802. Desai, C. S. (1980), A General Basis for Yield, Failure and Potential Functions in Plasticity, Int. J. Num. & Analyt. Methods in Geomech., 4, 4, pp. 36–37.
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Chapter Eight Desai, C. S. (1998a), DSCSST2D Code for TwoDimensional Static, Repetitive and Dynamic Analysis: User’s Manual I to III, Tucson, Ariz. Desai, C. S. (1998b), Application of Unified Constitutive Model for Pavement Materials Based on Hierarchical Disturbed State Concept, Report submitted to SUPERPAVE: University of Maryland, College Park, Md. Desai, C. S. (2000a), Finite Element Code (DSC2D) for 2002 Design Guide, Report submitted to AASHTO 2002 Design Guide, Arizona State University, Tempe, Ariz. Desai, C. S. (2000b), DSCSST3D Code for ThreeDimensional Coupled Static, Repetitive and Dynamic Analysis: User’s Manual I to III, Tucson, Ariz. Desai, C. S. (2000c), Evaluation of Liquefaction Using Disturbed State and Energy Approaches, J. Geotech. & Environ. Eng., ASCE, 126, 7, pp. 618–631. Desai, C. S. (2001), Mechanics of Materials and Interfaces: The Disturbed State Concept. CRC Press, Boca Raton, Fla. Desai, C. S. (2002), Mechanistic Pavement Analysis and Design Using Unified Material and Computer Models. Keynote Paper, Proceedings, Third Int. Symp. on 3D Finite Element Modeling of Pavement Anaysis, Design and Research. Scarpas, A. and Shoukry, S. N. (Editors), Amsterdam, The Netherlands. Desai, C. S. (2007), Unified DSC Constitutive Model for Pavement Materials with Numerical Implementation, Int. J. of Geomechanics, 7, 2, 83–101. Desai, C. S., Bozorgzadeh, A., and Whitenack, R. (2001), Finite Element Analysis of Distresses in (Rigid) Pavement Systems, Report, University of Arizona, Tucson, Ariz. Desai, C. S., Chia, J., Kundu, T., and Prince, J. (1997), Thermomechanical Response of Materials and Interfaces in Electronic Packaging: Parts I and II, J. Elect. Packaging, ASME, 119, 4, pp. 294–300, 301–309. Desai, C. S., and Cohen, D. (2000), Determination of DSC Parameters for Asphalt Concrete, Report, University of Arizona, Tucson, Ariz. Desai, C. S., Dishongh, T., and Deneke, P. (1998), Disturbed State Constitutive Model for Thermomechanical Behavior of Dislocated Silicon with Impurities, J. Appl. Physics, 84, 11, pp. 5977–5984. Desai, C. S., and Kundu, T. (2001), Introductory Finite Element Method, CRC Press, Boca Raton, Florida. Desai, C. S., and Ma, Y. (1992), Modelling of Joints and Interfaces Using the Disturbed State Concept, Int. Jnl. Num. and Anlayt. Methods in Geomechs., 16, pp. 623–653. Desai, C. S., Rigby, D. B., and Samavedam, G. (1993), Unified Constitutive Model for Materials and Interfaces in Airport Pavements, Proc. ASCE Specialty Conf. on Airport Pavement Innovations—Theory to Practice, Vicksburg, Mississippi. Desai, C. S., Samtani, N. C., and Vulliet, L. (1995), Constitutive Modelling and Analysis of Creeping Slopes, J. Geotech. Eng., ASCE, 121, pp. 43–56. Desai, C. S., and Siriwardane, H. J. (1982), Numerical Models for Track Support Structures, J. Geotech. Eng. Div., ASCE, 108, GT3, pp. 461–480. Desai, C. S., Siriwardane, H. J., and Janardhanam, R. (1983), Interaction and Load Transfer through Track Support Systems, Parts 1 and 2, Final Report, DOT/RSPA/DMA50/83/12, Office of University Research, Dept. of Transportation, Washington, D.C. Desai, C. S., and Siriwardane, H. J. (1984), Constitutive Laws for Engineering Materials, PrenticeHall, Englewood Cliffs, N.J. Desai, C. S., Somasundaram, S., and Frantziskonis, G. (1986), A Hierarchical Approach for Constitutive Modelling of Geologic Materials, Int. J. Num. Analyt. Meth. Geomech., 10, pp. 225–257.
Unified Disturbed State Constitutive Modeling of Asphalt Concrete Desai, C. S., and Whitenack, R. (2001), Review of Models and the Disturbed State Concept for Thermomechanical Analysis in Electronic Packaging, Jnl. Elect. Packaging, ASME, 123, pp. 1–15. Desai, C. S., Zaman, M. M., Lightner, J. G., and Siriwardane, H. J. (1984), ThinLayer Element for Interfaces and Joints, Int. J. Num. Analyt. Meth Geomech., 8, 1, pp. 19–43. Erkens, S. J. J. G., Liu, X., Scarpas, A., and Kasbergen, C. (2002), Issues in the Constitutive Modeling of Asphalt Concrete, Proc., Third Int. Symp. on 3D Finite Element Modeling of Pavement Analysis, Design and Research. Scarpas, A. and Shoukry, S. N. (Editors), Amsterdam, The Netherlands. FHWA (1987), Crack and Seat Performance. Review Report, Demonstration Projects and Pavement Divisions, Federal Highway Administration, Washington, D. C. Huang, Y. H. (1993), Pavement Analysis and Design. PrenticeHall, Englewood Cliffs, N. J. Kachanov, L. M. (1986), Introduction to Continuum Damage Mechanics, Martinus Nijhoft Publishers, Dordrecht, The Netherlands. Kilareski, W. P., and Bionda, R. A. (1990), Structural Overlays Strategies for Jointed Concrete Pavements, Vol. 1, Sawing and Sealing of Joints in AC Overlay of Concrete Pavements, Report No. FHWARD89142, Federal Highway Administration, Washington, D. C. Kim, Y. R., Lee, H. J., Kim, Y., and Little, D. N. (1997), Mechanistic Evaluation of Fatigue Damage Growth and Healing of Asphalt Concrete: Laboratory and Field Experiments, Proc. 8th Int. Conf. on Asphalt Pavements, University of Washington, Seattle, Wash., pp. 1089–1107. Lytton, R. L. et al. (1993), Asphalt Concrete Pavement Distrss Prediction: Laboratory Testing, Analysis, Calibration and Validation, Report No. A357, Project SHRP RF. 71572, Texas A&M University, College Station, Tex. Molenaar, A. A. A. (1983), “Structural Performance and Design of Flexible Road Constructions and Asphalt Concrete Overlays,” Ph.D. dissertation, Delft University of Technology, The Netherlands. Monismith, C. L., and Secor, K. E. (1962), Viscoelastic Behavior of Asphalt Concrete Pavements, Proc. Conf. Association of Asphalt Pavings and Technologists. NCHRP (2004), “MechanisticEmpirical Design of New and Rehabilitated Pavement Structures,” NCHRP Project 137A Draft Final Report, Transportation Research Board, National Research Council, Washington, D.C. Pande, G. N., Owen, D. R. J., and Zienkiewicz, O. C. (1977), Overlay Models in Time Dependent Nonlinear Material Analysis, Computer and Structures, 7, pp. 435–443. Perzyna, P. (1966), Fundamental Problems in Viscoplasticity, Adv. Appl. Mech., 9, pp. 243–277. Rowe, G. M., and Brown, S. F. (1997), Fatigue Life Prediction Using ViscoElastic Analysis, Proc. 8th Int. Conf. on Asphalt Pavements, University of Washington, Seattle, Wash., pp. 1109–1122. Scarpas, A., AlKhoury, R., Van Gurp, C. A. P. M., and Erkens, S. M. J. G. (1997), Finite Element Simulation of Damage Development in Asphalt Concrete Pavements, Proc. 8th Int. Conf. on Asphalt Pavements, University of Washington, Seattle, Wash., pp. 673–692. Schapery, R. A. (1965), A Method of Viscoelastic Stress Analysis Using Elastic Solutions, J. Franklin Inst., 279, 4, pp. 268–289.
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Chapter Eight Schapery, R. A. (1990), A Theory of Mechanical Behavior of Elastic Media with Growing Damages and Other Changes in Structure, J. Mech. Phys. Solids, 28, pp. 215–253. Schapery, R. A. (1999), Nonlinear Viscoelastic and Viscoplastic Constitutive Equations with Growing Damage, Int. J. Fracture, 97, pp. 33–66. Schofield, A. N., and Wroth, C. P. (1968), Critical State Soil Mechanics, McGrawHill, London, United Kingdom. Secor, K. E., and Monismith, C. L. (1962), Viscoelastic Properties of Asphalt Concrete. Proc. 41st Annual Meeting, Highway Research Board, Washington, D.C. Simon, B., and Desai, C. S. (2001), Analysis of Distresses in Flexible Pavements Using the Disturbed State Concept. Report, Department of Civil Engineering and Engineering Mechanics, The University of Arizona, Tucson, Ariz. Vermeer, P. A. (1982), A FiveConstant Model Unifying WellEstablished Concepts. Proc. Int. Workshop on Constitutive Relations for Soils, Grenoble France, pp. 175–197. William, G. W., and Shoukry, J. N. (2001), 3D Finite Element Analysis of TemperatureInduced Stresses in Dowel Jointed Concrete Pavements, Int. J. Geomechanics, 3, 3, pp. 291–307. Witczak, M. W., and Uzan, J. (1988), The Universal Airport Pavement Design System. Granulat Material Characterization Reports I to IV, University of Maryland, College Park, Md.
CHAPTER
9
DBN Law for the ThermoViscoElastoPlastic Behavior of Asphalt Concrete Hervé Di Benedetto and François Olard
Abstract Asphalt mix behavior is complex. Following applied loading, very different types of properties can be observed, such as linear viscoelasticity (LVE) for very small strain amplitudes, nonlinearity for larger strain amplitudes, fatigue or rutting for a great number of applied cycles: brittle or ductile failure can occur, while thixotropy and healing may exist following the considered temperature and loading history. These different “typical” kinds of behavior are introduced. Then, the proposed DBN (Di Benedetto and Neifar) law is presented. This law is very versatile and provides an effective means of describing the different types of mix behavior by using the same formalism. An emphasis is especially placed on explaining how the law can cope with different typical behaviors. In particular, some comparisons between experimental results and numerical simulations based on the DBN law are presented. The complexity of the description and the number of constants can be very different according to the desired sophistication for the modeling. A threedimensional generalization is also introduced. Some of the presented aspects are still part of ongoing research work.
Introduction Roadway structures are subjected to complex phenomena as a result of external solicitations. Mechanical, thermal, physical, and chemical phenomena, for example, can be observed. Moreover, a coupling between these effects often appears. Faced with the complexity of the observed problems, developed design methods are often semiempirical. These methods are all the more limited as new types of
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Chapter Nine structures and more powerful materials are used and as the traffic is in constant growth. Thus the use of more rational methods seems a needed evolution. Chapter 8 presented arguments for the development of a mechanisticempirical methodology and subsequent chapters will delve into this issue in more detail. Work has already been done in this direction, like, for example, the developments of the French method (SETRALCPC 1997), applied since the 1970s, and, more recently, efforts engaged by the United States within the framework of the Strategic Highway Research Program (1988–93, continued after 1996) (SHRP 1997) which allowed the development of the Superpave method (Superpave 1994) and the NCHRP 137A MechanisticEmpirical Pavement Design Guide. However, there is still a gap between the results obtained from the research on the behavior of bituminous materials and the introduction of this knowledge in design methods. First of all, the solicitations to which the bituminous roadways are subjected are presented. From the analysis of these loading paths, different aspects of the thermomechanical behavior of bituminous mixes are considered. The description of these different kinds of behavior is introduced in the following paragraphs. The recent results analyzed in this chapter show that some improvements are still possible, even necessary, in the design methods of road structures.
Solicitations in Bituminous Roadways General The two main solicitations are related to the passage of vehicles (traffic effect) and to the effects created by climatic changes, mainly because of the temperature variations (thermal effects). These two main solicitations are considered as more important than the other types of solicitations such as degradations related to the moisture or the ageing of materials. These two major effects are presented in the following paragraphs.
Traffic Influence Each roadway layer is subjected to a complex bending and compression state due to the effect of the traffic (Fig. 91). The amplitudes of cyclic loadings remain low (strain level
FIGURE 91
Schematic of the trafﬁcinduced solicitations.
DBN Law of the order of 10−4 m/m) which can explain why the calculation of strains and stresses, for each cycle, is traditionally made by considering a linear elastic multilayer model. As the materials are isotropic, this requires, initially, the determination of Young’s modulus and Poisson’s ratio values. It can be underlined that because of the particular properties brought by the bitumen, the bituminous mixes have a ratedependent behavior (thus a ratedependent modulus) (and also a temperature dependent behavior as shown in the next paragraph). The assumption of an elastic behavior thus corresponds to an approximation which can turn out to be a very rough hypothesis. In particular, the effects of nonlinearities and the irreversibilities accumulate with the number of cycles, which can reach several millions in a roadway life. Moreover, the small repeated tensions at the base of the layers, under the effect of the passage of vehicles, create microdegradations which accumulate. This is a fatigue phenomenon which is also observed for many other materials. This accumulation generally leads to the formation and propagation of cracks through the roadway and can involve the ruin of the structure. Fatigue is dealt with in much more detail in Part 5 of this book. The repeated compressions under the passage of the load can also create permanent deformation which induces rutting on the road surface. This rutting can be due to the compression of asphalt layers but also to the deformations of lower noncohesive granular layers. More details on this distress are given in the discussions presented in Part 6 of this book.
Temperature Influence The temperature has two main mechanical effects: 1. A stiffness (modulus) change of the material. A bituminous mixture becomes “softer” at higher temperatures. More generally, bituminous mixtures have a complex temperaturesensitive behavior. Their viscoplastic behavior changes according to the considered temperature. 2. The development of stresses and strains within the material because of the thermal dilatation and contraction effects during the temperature changes (Fig. 92). The first effect is, in general, characterized by the influence of temperature upon the stiffness modulus. It is noteworthy that the modulus is also ratedependent on account of the viscous behavior of the bituminous mixtures. The second effect is very harmful: • When temperature cools down, some cracks may appear and then propagate during (daily and annual) thermal cycles. • When a cementbound base exists in the considered pavement (semirigid structure), cracking may occur resulting from cementbound base shrinkage, thermal movements, desiccation, or movements of the joints between concrete slabs. In general, the overlaid bituminous layer does not have the deformation ability to bridge living cracks without damage. This being so, the cracks then propagate upward through the bituminous wearing course from the existing cracks in the concrete. This cracking phenomenon corresponds to the socalled “reflective cracking.”
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FIGURE 92
Schematic of the temperatureinduced solicitations.
Behavior of Bituminous Materials in Pavement Structures Apart from the very thin surface layers, the different bituminous layers have a structural effect. In order to characterize this effect and its evolution with time, mechanical properties of bituminous mixtures have to be modeled considering the following aspects: • Stiffness and stiffness evolution with time • Fatigue and damage law evolution • Permanent deformations and accumulation of these deformations • Crack initiation and crack propagation, in particular at low temperatures
FIGURE 93 Typical bituminous mix behavior domains. (e strain amplitude—N number of applied cycles).
DBN Law These four properties are of primary importance when modeling road behavior. The first one, observed for very small strains, corresponds to the linear viscoelastic behavior of mixes. From analytical or numerical methods, the linear viscoelastic law allows to obtain the stress and strain fields within the pavement. The three other aspects are respectively at the origin of major distresses: degradation by fatigue, rutting, and crack propagation. Each of these properties or distresses appears for a given domain of loading and corresponds to a specific type of mix behavior. Figure 93 highlights the domains corresponding to the different aspects introduced earlier and the corresponding typical types of mix behavior according to the strain amplitude and the number of applied cyclic loadings (Di Benedetto 1990).
Presentation of the DBN Law A general constitutive law developed at the Département Génie Civile et Bâtiment (DGCB) of Ecole Nationale des TPE (ENTPE), called DBN law (Di Benedetto, Neifar), which describes, with one formalism, the different kinds of mix behavior following the considered loading domain (Fig. 93) is hereafter proposed. The primary goal is to propose a basic formulation which can be simplified or adapted according to the studied properties. The law can then be very simple and easy to use (linear viscoelastic or even elastic) or more complicated (introduction of nonlinearity, permanent deformation, or fatigue, etc.). Considering the thermosensitivity of bituminous materials, the temperature influence is always considered. First, a uniaxial formulation is presented. We then explain how nonlinearity, brittle and flow failure, permanent deformation and fatigue can be modeled with the DBN law. Using the same formalism, the introduction of thixotropy and healing, which sometimes seem to play an important role, is also proposed. Finally, a rapid explanation of some extension for the threedimensional case is given. Due to the lack of space, only a rapid description is given hereafter. For more details on the general formulation of the law, the reader is referred to the following references: e.g., Neifar (1997), Di Benedetto and Neifar (2000), Neifar and Di Benedetto (2001), Olard (2003). The generalized analogical body presenting the law is drawn in Fig. 94(a). The EPi type bodies are nonviscous models and translate any nonviscous behavior (i.e., rateindependent response). The Vi type bodies are purely viscous models and introduce the viscous irreversibility. As is underlined further, the number n of elementary bodies can be arbitrarily chosen. This choice does not change the number of constants for the model. If n is high, the calibration can be closer to the experimental data, but the simulation calculus of complex stress path loadings becomes more time consuming. An extension of the law for n tending toward infinity has also been developed (discrete spectrum to continuous
FIGURE 94 (a) Generalized analogical body, (b) Viscoplastic DBN model for bituminous mixtures. (Di Benedetto 1987)
249
250
Chapter Nine spectrum) (Neifar 1997, Neifar and Di Benedetto 2001, Di Benedetto et al. 2001). This extension does not increase the number of needed constants for the model calibration.
Description of the EPi Body Behavior Each EPi element translates a behavior qualitatively identical to that of granular noncohesive materials. The incremental formalism of interpolation type, developed by Di Benedetto (1981) for sands, can thus be directly used to qualify EPi bodies. For the considered uniaxial case, the behavior of the body is described by a relation between the stress increase (Δs) and strain increase (Δe) starting from the last point of loading reversal : Δσ = f (Δε )
(91)
The function f can be obtained from the 2 first loading curves, f+ with an asymptote s for loading case, and f− with an asymptote s− for unloading case. A cyclic rule has to be introduced. Figure 95 gives an example of a simple cyclic rule, which is a generalization of the Masing rule. In the proposed simulations, f is hyperbolic and only three constants (s+i, s−i, and Ei) are needed for each EPi body (Fig. 95), except for EP0, which is a spring of rigidity E0: +
• Ei is the slope at the origin. • si+ is the asymptote of plasticity in compression (flow in compression) and si− represents the asymptote of plasticity in tension (flow in tension). The equation of the virgin curve f+i is as follows: fi+ (ε i ) = σ i =
Eiσ i Eσ 1+ i+ i si
where the index i is relative to the body EPi.
FIGURE 95
Characterization of the EPi behavior, example of cycling sequences.
(92)
DBN Law As regards the equation of the virgin curve f−, si+ must be replaced by si− in Eq. (92). • The definition of a viscoplastic failure criterion [sfailure = F( ε )] makes it possible to + − calculate the values of si and si . Through tension/compression tests at constant strain rate, the ratio k between si+ and si− can be considered the same for all EPi bodies (Di Benedetto and Yan 1994). Then, f− can be deduced from f+ by using the following expression (−k = si+/si− for every i = 1 to n): ⎛ Δε ⎞ f − (Δε ) = − kf + ⎜ − ⎝ k ⎟⎠
(93)
It can be seen that, for very small cyclic amplitudes, the EPi body is equivalent to a spring of rigidity Ei. As is explained hereafter, the DBN law can correctly translate the fatigue or the rutting behavior of the mix, provided that the EPi body is somewhat modified by new ingredients.
Description of the Vi Body Behavior The adapted structure for bituminous mixtures suppose that the Vi bodies are linear dashpot characterized by their viscosity hi(T) which is only a function of the temperature T. As can be shown theoretically, if all the hi have a similar dependence with T, the timetemperature superposition principle (TTSP) is verified (i.e., the material is thermorheologically simple). Then the uniaxial model used for the simulations of mix behavior has the structure presented in Fig. 94. The viscous element V∞, normally in series, is not present for the bituminous mixes, which are considered as solid type bodies.
Brittle Behavior of the Spring of Rigidity, E0 The spring of rigidity E0 gives the instantaneous deformations of the mix. This special brittle spring also simulates the brittle failure of the mix (see further section). As soon as the tensile stress reaches the brittle tensile strength obtained experimentally at low temperatures, the spring “breaks.” This tensile strength increases monotonically with the temperature, as shown in Fig. 96 (Olard 2003; Olard et al. 2004a, 2004b, 2004c).
FIGURE 96 Inﬂuence of both the temperature (T) and the strain rate (ε ) on the brittle/ductile behavior for tension (or compression) tests at constant strain rate on mixes (schematic representation).
251
252
Chapter Nine
How the Same Formalism Can Be Adapted to Describe the Different Typical Kinds of Mix Behavior As shown in Fig. 93, the bituminous mixtures present different typical kinds of behavior according to the considered domain of loading. By general formalism of the DBN law (Fig. 94) these different types of behavior can be described but the EPi and Vi behaviors need to be explained and calibrated. It is clear that the more general the description, the more complex the rheological function of each body. It is also practical to consider, for a given property, only the simplified form of the model that allows to model that property. Considering these two remarks, we propose in the next paragraphs different calibration processes. Each one is adapted to a considered domain (or type) of behavior (Fig. 93) and takes into account the experimental observed “typical” type of behavior. For a considered kind of behavior, only the obtained asymptotic expression of the law can be considered. It will give the same result as the general formulation by using fewer constants and simpler equations. The law is then very versatile and depending on required accuracy, simple or more complex equations can be considered. The evolution from one type of behavior to another remains continuous (no “switch” boundary) when considering the general expression of the law.
Current Developments for the DBN Law Small Strains and Small Number of Cycles: Linear Viscoelastic (LVE) Behavior The small strain domain corresponds to the domain where the mix behavior can be considered as linear. Charif (1991), Doubbaneh (1995), Airey et al. (2002, 2003), among others, have experimentally found the limit of the linear domain to be approximately 10−5 m/m for bituminous mixtures. In the small strain domain (strain amplitude < some 10−5) and for a small number of cycles, the mix behavior is linear viscoelastic and respects the TTSP. Then, the calibration of the law gives the value of Ei (Fig. 95) and hi [Fig. 94(b)]. In this domain, the EPi elements have a linear elastic behavior and can be replaced by springs of rigidity Ei. The selected discrete model becomes an assembly in series of KelvinVoïgt bodies [Fig. 97(a)], whose complex modulus is given by n ⎞ ⎛ 1 1 E∗DBN (ω , T ) = ⎜ +∑ ⎝ E0 i =1 Ei + jωηi (T ) ⎟⎠
−1
(94)
where j = complex number that j2 = −1 w = pulsation (w = 2πf, f = frequency) T = temperature The obtained asymptotic analogical body is represented in Fig. 97(a). The calibration process corresponds to an optimization in the frequency domain with the results obtained from the 2S2P1D model (the 2S2P1D model has a continuous spectrum and consists of a generalization of the HuetSayegh model, see e.g., Olard 2003,
DBN Law
FIGURE 97 (a) Analogical asymptotic form of the law in the linear viscoelastic (LVE) domain, (b) 2S2P1D model. (Olard and Di Benedetto 2003)
Olard and Di Benedetto 2003, Di Benedetto et al. 2004b) whose complex modulus has the following expression: E∗2S2P1D (ωτ ) = E(0) +
E(∞) − E(0) 1 + δ ( jωτ )− k + ( jωτ )− h + ( jωβτ )−1
(95)
t is a function of the temperature (T) which allows to take into account the TTSP:
τ = τ 0 aT (T ) = τ 0 10
−
C1 (T −Ts ) C2 +T −Ts
(96)
t0 is a constant to be determined at the arbitrarily chosen reference temperature Ts, and aT corresponds to the shift factor given by William, Landel, and Ferry (Ferry 1980) (C1 and C2 are two constants to be determined). The 2n+1 parameters (Ei, hi) of the discrete DBN model [cf. Fig. 97(a) where EPi is a spring of rigidity Ei] are obtained from an optimization process using only the 7 constants of the 2S2P1D model and the 3 constants of the William, Landel, and Ferry (WLF) equation (Ferry 1980) whatever the number of elements n. This optimization process consists in minimizing the sum of the distances between the complex modulus of the two models (2S2P1D model and DBN model) at N points of pulsation wk. This minimization is made at the reference temperature (Ts) by using the Solver feature of MS Excel, as follows:
∑ ( ⎡⎣ E
k=N
minimization of
k =1
2S2P1D 1
2
(ω k ) − E1DBN (ω k ) ⎤⎦ + ⎡⎣ E22S2P1D (ω k ) − E2DBN (ω k ) ⎤⎦
2
)
(97)
E12S2P1D and E22S2P1D are, respectively, the real part and the imaginary part of the complex modulus E∗2S2P1D given by Eq. (95). E1DBN and E2DBN are, respectively, the real part and the imaginary part of the complex modulus E∗DBN given by Eq. (94). Note that the “glassy modulus” E(∞) in Eq. (95) corresponds to the rigidity E0 of the discrete DBN model. The “static modulus” E(0) in Eq. (95) corresponds to the mix modulus when the pulsation w tends toward zero. Then, the following relation must be checked: ⎛ n 1⎞ E(0) = ⎜ ∑ ⎟ ⎝ i= 0 Ei ⎠
−1
(98)
253
254
Chapter Nine
FIGURE 98 Example of experimental and modeled complex modulus, left: master curve and, right: complex ColeCole plane.
Material
d
k
h
E0 (MPa)
E (MPa)
b
LOG (t) at 10°C
50/70 mix
2.5
0.20
0.56
200
45,000
700
−0.523
TABLE 91
Constant Values of the 2S2P1D Model [Eq. (95) and Fig. 97] for the 50/70 Mix Presented in Fig. 98
Figure 98 presents an example of calibration in the small strain domain for 15 elements (n = 15). The values of the constants are given in Tables 91 and 92.
Large Strains and Small Number of Cycles: Nonlinearity and Viscoplastic Flow The compression and tension tests at constant strain rate on bituminous mixes show that the stress cannot exceed a threshold sp for moderate and high temperature • and/or low strain rates. This flow stress is a function of the strain rate (ε ) and of the temperature (T). The viscoplastic criterion proposed by Di Benedetto (1987) and Di Benedetto and Yan (1994) was used. A triaxial representation of this criterion is given hereafter. As for the considered uniaxial case, the expression of the criterion is given by ⎛ i ⎞ ε+ d σ p = β ln ⎜ i ⎟ + γ ⎜⎝ ε ⎟⎠ 0 i
(99)
where ε 0 is 1% / mn ; β , d, and γ are three constants of the material. The threshold in tension si− is calculated in order to obtain viscoplastic flow stress close to that given by the criterion (e.g., Olard 2003). The criterion constants are fitted from experimental results. The considered tests consist in tension and compression at different strain rates and different temperatures. The threshold values in compression si+ are obtained from the values si− by using the constant ratio parameter k (−k = si+/si− for every i = 1 to n) (Di Benedetto and Yan 1994) [cf. Eqs. (92) and (93)].
DBN Law
Element Number
Ej(MPa)
0
45,000
1
τj =
hj(MPa∗s)
ηj Ej
—
—
381
80000
2.09E+02
2
1,200
20000
1.67E+01
3
3,700
10000
2.70E+01
4
10,500
8000
7.62E−01
5
13,500
4000
2.96E−01
6
17,300
500
2.89E−02
7
47,400
100
2.11E−03
8
130,000
30
2.31E−04
9
160,000
9
5.63E−05
10
175,000
0.6
3.43E−06
11
375,210
0.05
1.33E−07
12
640,000
0.005
7.81E−09
13
1,300,000
0.0005
3.85E−10
14
1,830,000
0.00005
2.73E−11
15
2,960,000
2.90E–06
9.81E−13
(s)
TABLE 92 Values of Ej and hj of a 15Element DBN Model Obtained, at T = 15°C, by Optimisation of the 2S2P1D Model Constants Given in Table 91
These 2n (si+, si−) values are obtained from the three constants of the viscoplastic criterion [b, d, and g of Eq. (99) by an optimization process]. The increase of the number of elements n does not introduce more constants. From the general calibration procedure, it can be concluded that whatever the chosen number of elementary bodies n, the number of basic constants remains the same. This number of constants is 13, which consist of • Three for the WLF equation (TTSP) • Seven for the linear viscoelastic (LVE) domain • Three for the viscoplastic flow A comparison of plastic flow stress values (sp) obtained at 15°C with the 15element DBN model and the values given by the criterion of Di Benedetto and Yan (1994) is presented in Fig. 99 according to the strain rate. The calibration is made for the same 50/70 mix as presented in Fig. 910 for the LVE domain. Besides, Fig. 910 presents a simulation example for a restrained cooling test with a temperature rate variation of −5°C/h. In order to show the effect of nonlinearity and
255
256
Chapter Nine
FIGURE 99 Strain rate according to the viscoplastic ﬂow stress (sp) obtained at 15°C given by a 15element DBN model and by the criterion of Di Benedetto and Yan (1994), for the same mix as Fig. 98.
FIGURE 910 Thermal stress variation with temperature for different types of rheological behavior for bodies EPj (Fig. 94) at a constant cooling rate of 5°C/h. (Neifar 1997.)
DBN Law plastic flow, three thermal stress calculated curves from three types of EPi bodies, are plotted: 1. The first calculation shows the response of the DBN model. 2. The second calculation is made for a discrete linear viscoelastic model: the chosen EPi bodies are springs of rigidities Ei. 3. The third calculation (discrete viscoelastic perfectly viscoplastic model) was carried out by considering an elastic perfectly plastic behavior for each EPi which consists of an assembly in series of a spring of rigidity Ei and a slider having a threshold si+ in compression and si− in tension. Flow (sliding) of EPi bodies are indicated in Fig. 910 for the viscoelastic perfectly viscoplastic model. Figure 910 highlights the influence of the assumptions taken for the EPi behavior. For example, a difference, which can reach 50%, is noted between the linear viscoelastic behavior and the viscoplastic behavior described by the viscoplastic model. It is clear according to these curves that nonlinearity has to be taken into account to simulate the restrained cooling tests correctly. It has to be noted that failure is not considered in the simulations presented in Fig. 910. The next section deals with the brittle failure observed at low temperatures and/or high strain rates.
Large Strains at Low Temperatures: Brittle Failure At high temperatures, bituminous mixes have a purely ductile behavior (highly nonlinear stressstrain diagram with a viscoplastic flow), whereas at very low temperatures their behavior is purely brittle (linear stressstrain diagram with a catastrophic failure). At intermediate temperatures, their behavior slowly switches from ductile (high temperature) to brittle (low temperature), when decreasing the temperature (Fig. 96). Experimental results show that, as expected from the viscoplastic criterion formulation presented for instance in Eq. (99), the stress at mix failure is very dependent on the stain rate in the ductile domain at intermediate and high temperatures (Fig. 911). Furthermore, Olard et al. (2003, 2004a, 2004b) have previously established that, in a first approximation, the strain rate influence upon the determination of the mix tensile strength appears as negligible in the brittle domain, at low temperatures (Fig. 911).
FIGURE 911 Inﬂuence of both the temperature and the strain rate on the mix tensile strength for a 50–70 pen bitumen mix , the same mix as Fig. 98. (Olard et al. 2003.)
257
258
Chapter Nine
FIGURE 912 Comparison between the experimental results for tension tests at constant strain rate and numerical simulations using the DBN model, for a 50–70 pen bitumen mix (Olard et al. 2003). The symbol ⊗ indicates the lowtemperature brittle failure of the EP0 element (same mix as Fig. 98). (Olard et al. 2003.)
Under these circumstances, the brittle lowtemperature failure can be modeled if it is introduced in the body EP0 [Figure 94(b)], which then becomes elastic and brittle with a failure limit sc(T), depending only on the temperature (T). The brittle stress at failure can be modeled by a function of the temperature with only two constants. Practically, at a given temperature T, the numerical simulation using the DBN law is stopped when the calculated stress s exceeds the experimentally obtained tensile strength curve sc(T) at low temperatures (Figs. 912 and 9.13).
Developments in Progress for the DBN Law Small Strains and Great Number of Cycles: Fatigue Damage Law The fatigue phenomenon of bituminous mixtures is observed for strain amplitudes from around 10−4 m/m and some hundred thousand cycles (Fig. 913). A rational approach was developed at the ENTPE (Di Benedetto et al. 1996 and 97; Ashayer 1998; Baaj 2002; Di Benedetto et al. 2004a). This approach is based on damage theory, with a correction of the experimental artifact effects existing during traditional fatigue tests.
DBN Law
FIGURE 913 Comparison between the experimental restrained cooling test at –10°C per hour (also called TSRST) (temperature measured at the surface of the mix sample) and numerical simulations carried out by using either a linear viscoelastic model or the developed DBN model, with a thermal contraction coefﬁcient (alpha) of 23 μm/m/°C (same mix as Fig. 98). (Olard et al. 2004.)
The developments allow consideration of either stress or strain control tests, which give the same analyzed results. An example of global nonlinear damage simulation for stress controlled fatigue test is presented in Fig. 914. The modeling of this damage behavior with the DBN law can be made when considering that the initial moduli of each body EPi (Ei in Fig. 95) take into account fatigue by a damage law of the following kind: d(DiN)/dN = functionE(eiN, DiN)
(910)
where DiN is the damage of the EPi body at cycle N (= 1−EiN/Ei) and EiN and eiN are respectively the modulus and the strain amplitude at cycle N. The curve of the fatigue results in the Black diagram, that is, in the modulus of the complex modulus—phase angle (E∗−f) plot, [Fig. 914(b)] shows that the viscosities hi are also damaged. An equation of the following type could be considered: d(DhiN)/dN = functionh(eiN, hiN)
( 911)
where hiN is the viscosity of the dashpoti [Figure 94(b)] at cycle N and DhiN is the damage of this viscosity at cycle N (=1−hiN/hi). The introduction of the global damage law at the level of each EPi and Vi bodies (of the DBN law) is still the subject of ongoing research.
Cyclic Loading from Stress Control Tests: Accumulation of Permanent Deformation The permanent deformations, which create rutting on the road, have two different origins: 1. The first one can be observed during creep test. 2. The second one is purely linked with the cycling effect, which creates a specific reorganization of the granular skeleton.
259
260
Chapter Nine
FIGURE 914 (a) Experimental and modeled fatigue stress control test at 0.9 MPa, 10°C, 10 Hz, (b) modulus of the complex modulus—phase angle (E∗ −f) plot and comparison with complex modulus test results. (Baaj 2002, Di Benedetto et al. 2004a.)
An adapted experiment was specially developed to quantify these two contributions (Neifar et al. 2002). Modeling of the first effect is possible when considering • vp classical viscoplastic law of the type: ε = f (σ ) . Even linear or nonlinear viscoelastic laws are able to translate this tendency within certain limits. The second phenomenon, which is more complex to characterize, can be taken into account with our DBN model if an accumulated nonviscous deformation is introduced in the EPi model. This is done by the choice of a function f [Eq. (91)], which allows the accumulation of irreversible strain with the number of cycles, as explained in Fig. 915.
Thixotropy Continuous cyclic loading test results show a rapid decrease of the complex modulus at the beginning of the tests. This decrease is partially due to heating (local and global) created by the dissipated energy induced by the viscous effects. Some analyses (Ashayer 1998; Di Benedetto et al. 1999; De la Roche 1996) tend to show that a rheological
DBN Law
FIGURE 915 Schematic representation of the permanent deformation created by cycling effect on each EPi body.
phenomenon, which could be thixotropy, can explain the other part of this rapid decrease. This phenomenon is also observed for binders. In our rational procedure for analyzing the fatigue test results, the thixotropy effect is linked with the dissipated energy (Di Benedetto et al. 1999). As the complex modulus evolution at the beginning of the test in the ColeCole or Black diagram is situated very close to the linear viscoelastic “master” curve (as can be observed in Fig. 914), the introduction of thixotropy in the DBN law can be done only at the level of the viscosities hi by an equation as follows : DhiN thixo = functionthixo
(dissipated energy, dissipated energyN, eiN)
(9.12)
where DhiN thixo is the damage of the viscosity i at cycle N created by thixotropy. It has to be underlined that this damage is totally recoverable as can be observed experimentally.
Healing It is known that bituminous materials have the specific property of healing. Modeling of this characteristic is rather complex and few rational attempts are proposed in the literature. This property could be taken into account with the DBN law by introducing a function of recovery of damage with time [and not with N as in Eq. (911)]: d(DhiN)/dt = functionhealing
(loading parameters, DiN, DhiN)
(9.13)
ThreeDimensional Generalization The generalization to threedimensional case of the previous developments needs the introduction of tensors instead of scalars for the different parameters (Di Benedetto et al. 2007a and b). This generalization is needed to model the road behavior correctly. It implies that the experiment should not only be analyzed in one direction. The volume variation and/or stress or strain evolution in all the directions have then also to be measured. In particular, a viscoplastic flow failure criterion was developed (Di Benedetto et al. 1994). It is represented in Fig. 916. An example of volume evolution during haversine stress loading, for different stress amplitudes (tension and compression), temperatures, and frequencies, is given in
261
262
Chapter Nine
FIGURE 916 3D failure criterion developed for viscoplastic ﬂow: shape and data. (Di Benedetto et al. 1994.)
FIGURE 917 Volume variation versus deviatoric (e axial−e radial) irreversible strain for haversine stress tests at different maximum stresses, temperatures, and frequencies. (Neifar et al. 2003.)
Fig. 917 (Neifar et al. 2003). The contractancy and dilatancy domains for permanent deformation, can be observed. These kinds of results are of primary importance for a correct description of the complex phenomena existing in the bituminous mixes.
Conclusions Different aspects of the complex behavior of bituminous mixtures are presented. Typical types of behavior can be identified when considering the strain amplitude and the number of applied cycles (Fig. 93). A correct interpretation of the laboratory tests as well as a rational approach for the road design requires that the different facets of the behavior of the bituminous mixtures be correctly modeled.
DBN Law This chapter briefly presents the global law (DBN law) developed in DGCB laboratory. It explains how the different types of behavior can be modeled using the same formalism but considering or not simplified formulation adapted to the required property. The law is very versatile and can be considered to model a wide range of characteristics such as • Linear viscoelasticity • Nonlinearity • Viscoplastic flow • Brittle failure • Accumulation of permanent deformation • Fatigue • Thixotropy • Healing The model formalism gives the possibility to consider any number of elementary bodies, without changing the number of basic constants. It is then easy to choose a more powerful description by increasing the number of elementary bodies. This general DBN law thus consists in a very effective alternative to the widespread procedures which are based only on the linear viscoelastic (or even elastic) properties of the bituminous materials. This law allows to simulate the response for any imposed strain or stress path. Following the quality and the range of required description, the expression of the law can be chosen as more or less complex. Asymptotic behaviors associated with simpler expressions are obtained in a continuous way when the loading conditions give a typical type of behavior. Threedimension generalization of the law is introduced. Experimental results integrating the whole directions are of main importance and needed to allow this generalization. Some of the presented expressions show the general principle and need more additional research developments.
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Chapter Nine Di Benedetto, H., 1981, Etude du comportement cyclique des sables en cinématique rotationnelle. Thèse de Doctorat, ENTPEUSMG, p. 170. [In French]. Di Benedetto, H., 1987, Modélisation du comportement des géomatériaux : Application aux enrobés bitumineux et aux bitumes. Doctorat ès sciences, INP Grenoble. p. 252. [In French]. Di Benedetto, H., 1990, Nouvelle approche du comportement des enrobés bitumineux: résultats expérimentaux et formulation rhéologique. Proceeding of the 4th International RILEM Symposium MTBM, Budapest, pp. 387–400. [In French]. Di Benedetto, H., and X. Yan., 1994, Comportement mécanique des enrobés bitumineux et modélisation de la contrainte maximale. Materials and Structures, k . Nbs. pp 539–47. [In French]. Di Benedetto, H., M. A. Ashayer Soltani, and P. Chaverot, 1996, Fatigue damage for bituminous mixtures: a pertinent approach. Journal of the Association of Asphalt Paving Technologist, p. N65. Di Benedetto, H., M. A. Ashayer Soltani, P. Chaverot, 1997, Fatigue damage for bituminous mixtures. Proceeding of the fifth International RILEM Symposium MTBM, Lyon. Di Benedetto H., M. A. Ashayer Soltani, and P. Chaverot, May 1999, Etude rationnelle de la fatigue des enrobés : annulation des effets parasites de premier et second ordre. Intternational Eurobitume Workshop, Performance Related Properties for Bituminous Binders. [In French]. Di Benedetto, H. and M. Neifar, 2000, Loi thermoviscoplastique pour les enrobés bitumineux. 2nd Eurasphalt and Eurobitume Congress, Barcelone, p. 9 [In French]. Di Benedetto, H., M. Neifar, B. Dongmo, and F. Olard, 2001, Loi thermoviscoplastique pour les mélanges bitumineux: simulation de la perte de linéarité et du retrait empêché, 36ème Colloque Annuel du Groupe Français de Rhéologie, MarnelaVallée. [In French]. Di Benedetto, H., C. de la Roche, H. Baaj, A. Pronk, and R. Lundstrom, April 2004a, Fatigue of bituminous mixes, Materials and Structures, Vol. 37, pp. 202–216. Di Benedetto, H., F. Olard, C. Sauzeat, and B. Delaporte, 2004b, Linear viscoelastic behavior of bituminous materials: from binders to mixes, Special Issue of the International Journal Road Materials and Pavement Design 1st EATA, Vol. 4, pp. 163–202. Di Benedetto, H., B. Delaporte, and C. Sauzeat, 2007a, Threedimensional linear behavior of bituminous materials: experiments and modeling, ASCE International Journal of Geomechanics, Vol. 7(2), pp. 149–157. Di Benedetto, H., M. Neifar, C. Sauzeat, F. Olard, 2007b, Threedimensional thermoviscoplastic behaviour of bituminous materials: the DBN model, International Journal Road Materials and Pavement Design, Vol. 8(2), pp. 285–316. Doubbaneh, E., 1995, Comportement mécanique des enrobés bitumineux en “petites” et “moyennes” déformations. Ph.D. thesis, ENTPEINSA. p. 219. [In French]. Ferry, J. D., 1980, Viscoelastic Properties of Polymers, 3rd ed., New York: John Wiley and Sons, 1980. NCHRP 137A Research Team, 2004, Guide for MechanisticEmpirical Design of New and Rehabilitated Pavement Structures, Final Report. NCHRP 137A, ARA, Inc. and ERES Consultants Division. Neifar, M., 1997, Comportement thermomécanique des enrobés bitumineux: expérimentation et modélisation. Ph.D. thesis, ENTPEINSA. p. 289. [In French]. Neifar, M., and H. Di Benedetto, 2001, ThermoViscoplastic Law for Bituminous Mixes. International Journal Road Materials and Pavement Design. Vol. 2. (1), pp. 71–95.
DBN Law Neifar, M., H. Di Benedetto, J. M. Piau, and H. Odeon, 2002, Permanent deformation of bituminous mixes: monotonous and cyclic contributions. 6th international conference on the Bearing Capacity of Roads, Railways and Airfields, Lisbon. Neifar, M., H. Di Benedetto, and B. Dongmo, April 2003, Permanent deformation and complex modulus: two different characteristics from a unique test. 6th International RILEM Symposium on Performance Testing and Evaluation of Bituminous Materials, PTEBM´03, Zurich. Olard, F., and H. Di Benedetto, 2003, General “2S2P1D” model and relation between the linear viscoelastic behaviors of bituminous binders and mixes. International Journaf Road Materials and Pavement Design, Vol. 4, Issue 2. Olard, F., H. Di Benedetto, A. Dony, JC. Vaniscote, 2003, Properties of bituminous mixtures at low temperatures and relations with binder characteristics. 6th International RILEM Symposium on Performance Testing and Evaluation of Bituminous Materials, Zurich. Olard, F., 2003, Comportement thermomécanique des enrobés bitumineux à basses températures. Relations entre les propriétés du liant et de l’enrobé. Ph.D. thesis, ENTPEINSA. [In French]. Olard, F., H. Di Benedetto, H. B. Eckmann, and JC. Vaniscote, 2004a, Lowtemperature failure behavior of bituminous binders and mixes. Annual Meeting of Transportation Research Board, Washinton, D.C. Olard, F., H. Di Benedetto, M. Mazé, and JP. Triquigneaux, 2004b, Thermal cracking of bituminous mixtures: experimentation and modeling. Proceeding of the 3rd Eurobitume and Eurasphalt Congress, Vienna. Olard, F., H. Di Benedetto, JC. Vaniscote, and B. Eckmann, 2004c, Failure behavior of bituminous binders and mixes at low temperatures. Proceeding of the 3rd Eurobitume and Eurasphalt Congress, Vienna. SETRALCPC, May 1997, Guide Technique. French Design Manual for Pavement Structures. Ed LCPC et SETRA, Paris. SHRP 1997, From Research to Reality: Assessing the Results of the Strategic Highway Research Program, Publication No. FHWASA98008, Federal Highway Administration, Washington, D.C., 1997 Superpave 1994, Background of Superpave Asphalt Binder Test Methods. Publication Number FHWASA94069, Federal Highway Administration, Washington, D.C.
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PART
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Models for Rutting CHAPTER 10 Rutting Characterization of Asphalt Concrete Using Simple Shear Tests
CHAPTER 11 Permanent Deformation Assessment for Asphalt Concrete Pavement and Mixture Design
Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.
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CHAPTER
10
Rutting Characterization of Asphalt Concrete Using Simple Shear Tests John T. Harvey, Shmuel L. Weissman, and Carl L. Monismith
Abstract This chapter discusses the use of the Strategic Highway Research Program (SHRP) developed shear test for the permanent deformation (rutting) characterization of hot mix asphalt (HMA). Applications of the results of this test in mix design and pavement performance analyses are described. Included are the following: (1) discussion of the mechanics of permanent deformation in HMA; (2) factors to be considered in test selection for rutting evaluation including— (a) specimen size relative to boundary effects and maximum aggregate size as related to the representative volume element (RVE), (b) nonlinear response effects, (c) static (creep) versus dynamic (repeated) load evaluation, and (d) effects of laboratory HMA compaction method on permanent deformation response; (3) a brief description of the shear test equipment and the range of tests which are performed; and (4) examples of the use of shear test data for mix design and analysis including performance prediction. The chapter concludes with a discussion of the formulation of a constitutive relationship for HMA behavior at elevated temperatures based on available information. This model is composed of a viscoelastic component in parallel with a rateindependent elastoplastic part and is formulated within the framework of finite deformations.
Introduction Rutting in asphalt concrete mixes under traffic loading occurs predominantly at elevated temperatures. Theoretical and laboratory studies (Sousa et al. 1994; Weissman 1997; Weissman et al. 1999) as well as field data (Brown et al. 1989; Epps et al. 1999) and the results of accelerated performance tests conducted at the University of California, Berkeley using the Heavy Vehicle Simulator (HVS) as a part of the Caltrans Accelerated Pavement Testing (CAL/APT) project (Harvey et al. 2000) suggest the following:
269 Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.
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Chapter Ten • Shape distortion (shear) is the main contributor to permanent deformations in the asphaltbound layer, compared to volume change (densification). • Accumulation of permanent deformations in this layer is very sensitive to the layer’s resistance to shape distortion, and is relatively insensitive to resistance to volume change. • The nonlinear nature of asphaltaggregate mixes, particularly at higher temperatures, requires a direct test to measure resistance to shape distortion; any indirect tests may contain hidden errors (which may be large) introduced through the assumptions required to convert the measured data to a property related to shape distortion. • Tirepavement contact stress distribution plays an important role in the development of permanent deformations in the asphaltbound layer (de Beer et al. 1997). • Because of the high shear stress under the edge of the tire and just below the pavement surface coupled with the higher pavement temperatures occurring at and near the surface of the HMA (hotmix asphalt) layer, the rutting in HMA is limited to the top 75 to 100 mm (3 to 4 in.). Based on these considerations, a shear test was selected as the methodology to evaluate the propensity for rutting in binderaggregate mixes (HMA).∗ This chapter provides a discussion of (1) the mechanics of permanent deformation in HMA; (2) factors to be considered in test selection for rutting evaluation including (a) specimen size relative to boundary effects and maximum aggregate size as related to the RVE, (b) nonlinear response effects, (c) static (creep) versus dynamic (repeated) load evaluation, and (d) effects of laboratory HMA compaction method on permanent deformation response; (3) a brief description of the test equipment and the range of tests which are performed; and (4) examples of the use of shear test data for mix design and analysis including performance prediction. The chapter concludes with a discussion of the formulation of a constitutive relationship for HMA behavior at elevated temperatures based on available information. This model is composed of a viscoelastic component in parallel with a rateindependent elastoplastic part and is formulated within the framework of finite deformations.
Mechanics of Permanent Deformation This section summarizes the results of a study to identify HMA material properties critical to the characterization of permanent deformation in pavement structures (Weissman 1997). Included are considerations of (1) volume change versus shape distortion and their relative importance in pavement structures, (2) representative volume element and laboratory test specimen size considerations, and (3) nonlinear response characteristics of HMA.
∗ The term binder is used here to represent a more encompassing term which includes both conventional asphalt cements and those containing modifiers to enhance mix performance. Subsequently, the term HMA (hot mix asphalt) will be used to represent mixes containing either conventional asphalt cements or modified binders with densegraded aggregates.
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Volume Change versus Shape Distortion At elevated temperatures, AC mixes exhibit markedly different volumechange and shapedistortion deformation modes in terms of their sensitivity to temperature, rate of loading, and residual permanent deformation. Volume change can be defined as deformation during which the change in all three principal strains is equal. Resistance to volume change is referred to as the bulk modulus K. Shape distortion is volumepreserving deformation; resistance to this form of deformation is referred to as the shear modulus G. Both forms of deformation are illustrated schematically in Fig. 101. Data, reported in Sousa et al. (1994) for 15 different mixes from two types of tests, simple shear creep tests with a shear stress of 69 kPa (10 psi) and hydrostatic pressure tests with a pressure 690 kPa (100 psi), provide some indication of the relative behavior of HMA in the two deformation modes. Both types of tests were performed at 50°C (122°F). In both tests, the load was increased at a steady rate from 0 to the full value in 10 seconds, maintained for another 100 seconds, and reduced to 0 at a constant rate over a period of 10 seconds. Measurements continued for an additional 120 seconds after the load was removed, thus making the total test time 240 seconds. Figure 102 shows the average strain history for these 15 mixes in the two tests. The effective bulk modulus K from the hydrostatic test is roughly 25 times larger than the effective shear modulus G from the simple shear test. It can be seen that the hydrostatic tests (approximating volumechange tests) exhibit significantly less creep than the simple shear tests (approximating shapedistortion tests). The same average curves are also shown in Fig. 103, in which each of the two curves is normalized with respect to its strain value just before the load is removed. This figure illustrates that volumechange tests recover a larger percentage of the total strain present prior to unloading. This specific set of data shows that, on the average, the mixes recovered only about 18 percent in the shapedistortion test, while the same mixes recovered about 42 percent in the volumechange tests. The largest contribution to volume loss in HMA mixes is likely the reduction in the volume of air since aggregate and binder are nearly incompressible. The data shown in Fig. 102 and Fig. 103 were obtained using newly fabricated specimens and thus, represent untrafficked materials. Half of the specimens were compacted to a target airvoid content of 4 percent and the other half to 8 percent. It would be expected that the percentage of strain recovery in volumechange tests would increase with additional load cycles. Therefore, in reality, the difference in the percentage of recovered strain between the two deformation modes may actually be larger than that indicated by the data presented in the two figures. To analyze the relative importance of the two deformation modes on the behavior of pavement systems, threedimensional (3D) finite element simulations of three
(a)
FIGURE 101
(b)
(c)
Schematic representations of volume change (a) and shape distortion (b, c).
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FIGURE 102
Average strain history for 15 different mixes. (Sousa et al. 1994).
FIGURE 103 Normalized curves for average strain history for 15 mixes shown in Fig. 102.
different pavement structures were examined (Weissman 1997). Results from an analysis for one of the structures are discussed in this section. The layer thicknesses and material properties for this pavement structure are shown in Table 101. Since the study focused on the HMA layer, the rest of the layers were assumed to exhibit elastic behavior to simplify computations.
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Material Properties
Layer
Layer Thickness mm (in.)
K MPa (ksi)
G MPa (ksi)
Asphalt concrete
212 (8.3)
1726 (250)
207 (30)
Granular base
250 (10)
345 (50)
74 (10.7)
Granular subbase
250 (10)
230 (33.3)
49 (7.1)
Subgrade
1000 (40)*
92 (13.3)
20 (2.9)
*The subgrade layer, although semiinfinite in reality, is represented in the finite element model with a 1000mmthick layer.
TABLE 101 Pavement Crosssection and Material Properties, Section 2
In the opinion of the authors of this chapter, no constitutive law currently available as of June 2003 provides a good approximation of the behavior of HMA mixes at the higher temperatures relative to rutting. Thus, for this study, a nonlinear elastic constitutive relationship was used that provided for different behavior in volume decrease and volume increase while allowing for temperature dependence of the shear modulus G. This model, discussed in more detail in Weissman (1997), permitted the evaluation of the relative contribution of the two deformation modes. The selected model has two limitations. First, being an elastic model, it does not account for residual permanent deformation. This does not restrict the main purpose of these simulations, however, which was to obtain an indication of the relative importance of change in volume versus shape distortion. The elastic model can be used to represent the effective properties of the different materials during the first loading sequence. Thus, for the first load cycle, the model can produce good indications of actual pavement behavior. In addition, if the results are combined with laboratory test data, it is possible to arrive at conclusions regarding permanent deformation in the AC layer. In view of the difference in recovery during unloading, (e.g., Fig. 103), the elastic simulation underestimates the contribution of shapedistortion relative to that of volume change. The second limitation is due to the missing rate effect in the HMA model. To offset this, the material constants for the HMA were selected based on laboratory tests conducted at the relevant rate of loading. Therefore, this limitation did not constrain the ability to draw conclusions. A dual tire configuration was used for loading. The tirepavement contactstress distribution, reported in de Beer et al. (1997), corresponded to that for a Goodyear G159A, 11R 22.5 with a load of 25 kN (5.6 kips) per tire and a tire pressure of 690 kPa (100 psi). All three stress components—vertical, longitudinal, and lateral—of the tirepavement contact stress distribution were used in the analyses. Three temperature conditions were utilized: ranging from 40 to 60°C in the HMA layer. Details of the 3D finite element simulations are described in Weissman (1997). One of the studies involved varying G for the HMA while maintaining K constant; G was first reduced by an order of magnitude and then increased by an order of magnitude for the first case where the temperature at the surface was 60°C and 40°C at the bottom. The simulations were repeated with similar variations in K while keeping G constant. The resulting vertical displacements in the HMA layer under a dual wheel are shown in Fig. 104. This figure shows the first 500 mm (20 in.) on the right side of the
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FIGURE 104 Vertical deformation in the AC layer ﬁrst (500 mm from the symmetry plane); effect of variation in G while maintaining K constant, or vice versa.
symmetry plane between the two tires; the tire is on the left side of the image centered over the fine mesh area and only the HMA layer is shown. Plots of the maximum vertical displacement versus change in G and K are shown in Fig. 105. As can be seen, the structural response is more sensitive to changes in the resistance of the AC layer to shape distortion (G) than to change in its resistance to volume change (K).
FIGURE 105 Variation of the maximum vertical displacement in the AC layer with changes in material properties.
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s In Fig. 104, it will be noted that the maximum deformations are concentrated near the surface under the tire. Similar results were obtained for other pavement structures analyzed. If these concentrations are combined with the assumption that the residual deformations (i.e., deformation remaining after the load is removed) represent a certain percentage of the deformation during the load applications, then it can be concluded that the permanent deformations or ruts in the HMA mix layers of comparable pavement structures are predominantly confined to the top 75 mm to 100 mm (3 to 4 in.) of the pavement, a result supported by field observations (Brown et al. 1989, Epps et al. 1999). Volume loss (densification) in the top 50 or 75 mm (2 or 3 in.) of HMA layers can account for 1 to 2 mm (0.04 to 0.08 in.) of the rut at most, for airvoid content changes of about 5 percent. However, rut depths of 15 mm (0.6 in.) or more are observed in inservice pavements. It can therefore be concluded that shape distortion is the dominant contributor to rutting. Because rutting occurs in the upper section of the HMA layer, the tirepavement contactstress distribution plays an important role in the development of permanent deformation in the asphalt layer. The analyses also showed that the rutting in the HMA layer is more significant at higher temperatures.
Representative Volume Element and Laboratory Test Specimen Size Considerations Laboratory specimen size is an important factor to be considered in material testing. Laboratory tests are usually developed around theories (e.g., continuum mechanics and constitutive relationships), and are designed to identify specific parameters associated with the models. An important question is whether the theory stipulated is applicable to a specific test. In particular, many models in common use in the field of mechanics of materials are based on homogenization of properties across heterogeneous media. Thus, it is important to have enough material for the homogenizing process to provide “reasonable” properties. For HMA mixes, the question of scale between particle size and the dimensions of the specimen is important because the maximum aggregate size may not be much smaller than the specimen size. Thus, it is important to verify at what minimum specimen dimensions continuum mechanics, or any theory based on homogenization, becomes applicable. This specimen dimension is referred to as the representative volume element (RVE), defined as the smallest volume large enough so that the global characteristics of the material remain constant, regardless of the location of the RVE. When specimens smaller than the RVE are tested, much variability is observed. Consequently, the mean value of the results must be obtained from a large number of test specimens to arrive at a statistically meaningful value (Hashin 1983). On the other hand, it is likely that less variable test results will be obtained when specimens larger than the RVE are used. Using specimens smaller than the RVE has two major disadvantages. First, a large number of specimens may be required. Second, an averaging process ignores any bias in the test procedure which may result in large errors. Bias might occur, for example, because of the mix compaction method and selection of test specimens from a specific part of the compacted mix or from the properties of a component of the specimen (for example, a large aggregate dominating the properties of the other components in the specimen). In view of these limitations, the use of specimens larger than the RVE is recommended. However, in some cases, the use of specimens smaller than the RVE may
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Chapter Ten be unavoidable, for example, in a mix containing larger aggregates. In such cases, statistically meaningful results can be obtained by testing a large number of replicates, although the above limitations should be noted. Results of typical laboratory tests of HMA mixes show the classic indications noted in the literature for specimens smaller than the RVE, (e.g., Weissman et al. 1999; Harvey et al. 1999). A few of the results of simulations of virtual axial tests on mixes containing two different aggregates have been included to support this discussion. (Weissman 1997) Twodimensional, plane strain, finiteelement simulations were performed to investigate the effects of the RVE on specimen response. The finite element meshes used in the analysis were obtained from digitization of photographs of aggregate and mastic in the cut planes of actual specimens. Two mixes were used; nominal maximum aggregate size for both mixes was 19 mm. In the simulations, both the aggregate and mastic were assumed to be linear elastic. The stiffness of the aggregate was taken as E = 100 MPa (14,500 psi) and Poisson’s ratio was set n = 0.35. To assess the effect of temperature on the RVE, the material properties for the mastic, defined as the binder and aggregate smaller than 1 mm, were E = 100, 10, and 1 MPa (14,500; 1450; and 145 psi) and n = 0.49, where the value of 100 MPa (14,500 psi) was used to represent lower temperature and the value of 1 MPa (145 psi) to represent higher temperature. For the virtual axial tests, boundary conditions assumed one edge fixed in the normal direction with a uniform displacement imposed on the opposing edge in the direction of the constrained edge (i.e., axial compression in the horizontal direction in Fig. 106). Virtual linear variable displacement transducers (LVDTs) of various lengths from zero to the full length of the specimen were placed along three lines on the section,
FIGURE 106 Contours of axial displacements (y direction): Pleasanton aggregate, Eaggregate = 100 MPa and Emastic = 100 MPa.
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FIGURE 107 Contours of axial displacements (y direction): Pleasanton aggregate, Eaggregate = 100 MPa and Emastic = 1 MPa.
and strains were measured around the center line. Results of the finite element simulations included distributions of axial deformation in the virtual specimens and effective stiffness moduli. The virtual specimen length was 150 mm (5.9 in.). Figure 107 illustrates the distribution of axial deformations for the case were Eaggregate = Emastic = 100 MPa (14,500 psi), and Fig. 106 presents results for Eaggregate = 100 MPa (14,500 psi) and Emastic = 1 MPa (145 psi), where E represents Young’s modulus of the material. The axial deformation in Fig. 107 is relatively uniform, whereas large variations are present in Fig. 106 where there is a 100:1 ratio of the aggregate to mastic stiffness. Figures 108 and 109 shows the deformations obtained by the three lines of various lengths of virtual LVDTs for one of the two mixes evaluated. The imposed average strain for all specimens was 1 percent. These plots indicate the variation in the axial strain measured by these LVDTs. As expected, the figures associated with Emastic = 100 MPa (14,500 psi) indicate that the RVE at low temperatures can be relatively small. The plot of results for Emastic = 1 MPa (145 psi) clearly show large oscillations, even for gage lengths larger than 100 mm (4.0 in.). Also, some of the results appear to show bias: the predicted strain converges from one side—either above or below. Actual physical tests may show considerably more bias, thus hampering the use of specimens with dimensions less than those of the RVE. Figures 107 and 109 indicate a band of ± 20 percent about the imposed average strain at high temperature for LVDT lengths about 3.5 to 5 times the nominal maximum aggregate size (NMAS) (about 75 to 100 mm for 19mm NMAS). Current laboratory procedures typically use only two to four replicates. Thus, if specimens are smaller than the RVE, there is no guarantee that the average result
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FIGURE 108 Effect of gage length on measured axial strain (x direction): Pleasanton aggregate, Eaggregate = 100 MPa and Emastic = 100 MPa.
FIGURE 109 Effect of gage length on measured axial strain (x direction): Eaggregate = 100 MPa and Emastic = 1 MPa.
obtained from two to four replicates predicts a statistically meaningful value of the material property. Weissman et al. (1999) and Harvey et al. (1999) provide data to support the above discussion. Dimensions of the RVE are dependent on aggregate size, shape, and orientation. Accordingly, the RVE for mixes containing different aggregates with the same nominalsized aggregate may differ. Finally, because of aggregate shape and construction
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s procedure, the dimensions of the RVE may differ in the three principal dimensions, particularly at higher temperatures (Weissman et al. 1999). The RVE dimension also depends on temperature and rate of loading. This is due to the rate of loading and temperature dependence of the material properties of the mastic (asphalt and fine aggregate), whereas the aggregate properties are relatively insensitive to these effects. As a result, at low temperatures the properties of the two components are closer, whereas at elevated temperatures the aggregate may be orders of magnitude stiffer than the mastic. Thus, larger size specimens are required at high temperatures than for tests on the same mix at lower temperatures. Additionally, dynamic tests may require smaller specimens than static (creep) tests because the properties of the aggregate and mastic are closer at higher frequencies of loading. For example, smaller specimens may be used for the repeated simple shear test at constant height with a loading time of 0.1 seconds than are required for a shear creep test.
Nonlinear Response Characteristics of Asphalt Aggregate Mixes At higher temperatures, asphaltaggregate mixes exhibit nonlinear response characteristics (Harvey et al. 1999). Thus, when making comparisons of responses determined from different tests it is important that these comparisons be made at the same strain level. The effects of frequency of loading, strain level, and temperature on shear stiffness of AC were studied using the simple shear test (AASHTO TP794 with modifications) (AASHTO 1994, PRC 1999). This study is reported in Harvey et al. (1999) and Harvey et al. (2000). The results summarized in Fig. 1010 show that the complex shear modulus (G∗) increases with decreasing temperature, increasing frequency, and smaller shear strains. At low temperatures and high frequencies, the shear stiffness G∗ should be about onethird of the Young’s modulus E∗, at corresponding conditions. For example, the value of G∗ at a temperature of 20°C (68°F), a frequency of 10 Hz at 0.01 and 0.05 percent strain is about 2.2 GPa (320,000 psi), Fig. 1010. The complex modulus, E∗, determined in flexure for the same mix at the same temperature and frequency and at 0.015 percent strain is about 6.5 GPa (943,000 psi) suggesting that the linear elasticity assumption is reasonable for these conditions. That is, the following relationship can be used: G∗ =
E∗ 2(1 + ν )
(101)
where ν is Poisson’s ratio. However, as the temperature and strain amplitude increase, this assumption becomes less valid. Another study (Alavi 1992) also shows the danger in converting data from indirect tests. The study reports the results of tests on HMA configured as hollow cylinders loaded in axial, torsion, and a combination of both. Poisson’s ratios were predicted using Eq. (101) from values of G∗ and E∗ both of which had been measured directly. Poisson’s ratio values as high as about 5.5 were calculated. Direct determinations of Poisson’s ratio were also made using strain gage data. Values of Poisson’s ratio varied between 0.15 and 0.4. At the low temperature of 4°C (40°F), the directly measured and computed values were closer to each other than for the other conditions. As the temperature increased, the difference between the computed and measured values increased and at the higher
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FIGURE 1010 Averaged shear frequency sweep results of three replicate specimens for each strain and temperature.
temperatures the values differ significantly. Thus, the use of Eq. (101) to deduce G∗ from E∗ or vice versa is reasonable only at low temperatures and high frequencies. Data presented in Harvey et al. (1999) and Harvey et al. (2000) illustrate the nonlinear behavior of shear modulus as a function of temperature and strain level. While the data at a particular strain level can be used to construct mastercurves of the complex shear modulus G∗ versus frequency using the concept of interchangeability of time and temperature, it must be emphasized that such relationships cannot be used to directly determine the correct values of the complex Young’s modulus E∗ because the material exhibits nonlinear response. This, of course, also applies when calculating the complex shear modulus from the complex Young’s modulus.
The Laboratory Simple Shear Test to Characterize Permanent Deformation Based on the concept that permanent deformation (rutting) for “reasonably” compacted mixes results primarily from deviatoric strains, a simple shear test was developed during the SHRP to measure this response.
Test Equipment and Procedures The prototype equipment is illustrated schematically in Fig. 1011 and a photograph is shown in Fig. 1012. This equipment is capable of testing specimens ranging in size from 150 mm (6 in.) in diameter by 50 mm (2 in.) high to 200 mm (8 in.) in diameter by
FIGURE 1011 Schematic representation of simple shear tester. (Sousa et al. 1994.)
FIGURE 1012 Prototype simple shear test equipment. (Sousa et al. 1994.)
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FIGURE 1013 Recently developed simple shear test equipment. (Cox and Son 2000.)
75 mm (3 in.) high over the temperature range –10 to +70°C and at confining pressures up to 690 kPa (100 psi). A newer version of the simple shear equipment is shown in Fig. 1013. In contrast to the earlier shear device, this equipment permits testing of specimens in the temperature range ambient to 70°C and does not have the provision for lateral pressure application. It does have the capability of testing circular and large rectangular specimens the maximum size of the latter being 350 mm (14 in.) long by 150 to 200 mm (6 to 8 in.) wide by 100 mm (4 in.) high. The cost of the newer equipment is approximately $80,000 to $120,000, considerably less than the prototype equipment. Shear and vertical loads are applied by servohydraulic actuators which are computer controlled, Fig. 1011. Both shear test units can perform the following types of tests: 1. Creep 2. Frequency sweeps (0.01 to 20 Hz) 3. Repeated loading at constant height using a haversine wave form (termed RSSTCH) 4. Stress relaxation (e.g., stepped relaxation tests) Using the original equipment, the tests can be performed either with unconfined or confining pressure. For the equipment shown in Fig. 1013 a small cell surrounding the test specimen is required if a confined test is performed. Shear deformations are measured by linear variable displacement transducers, which are mounted on the cap and base to which each specimen is bounded. The two most commonly performed tests with the equipment are (1) the RSSTCH, at a specific temperature, and (2) frequency sweeps over a range in temperatures. Procedures for these tests are contained in AASHTO TP794 (AASHTO 1994). Test specimens are prepared by rolling wheel compaction, the reason for which will be discussed subsequently. Either circular specimens (obtained by coring) or
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FIGURE 1014 Frequency sweep tests in shear at strain of 0.000100 mm/mm (in./in.) illustrating curve shifting to 40°C.
prismatic specimens (obtained by sawing) are cut from the rolling wheel compacted slabs. Parallel surfaces between the top and bottom surfaces of each specimen are obtained by using a doublebladed saw set at a dimension corresponding to the desired specimen height. To minimize specimen variability, the direction of compaction is noted on the surface of each specimen. This is done both for specimens compacted in the laboratory and for specimens obtained from slabs or cores obtained from field constructed pavements (either for Heavy Vehicle Simulator tests or for actual traffic loading). By testing specimens with all cut surfaces, one aspect of specimen variability is minimized. Frequency sweeps are performed at small shear strains 100 × 10−6 mm/mm (or in./in.). An example of test results from this type of test is shown in Fig. 1014. Similar results had been presented earlier in Fig. 1010 illustrating the nonlinearity in response when larger strains are used in this test. The majority of RSSTCH test data have been obtained using a shear stress of 10 psi (69 kPa). Shear loading is applied in the form of haversine with a time of loading of 0.1 second and a time interval between load applications of 0.6 seconds. This combination of stress level and time of loading was selected based on experience gained in mix analysis and design studies for highway loading conditions. Performance for a range in traffic loading has shown these test conditions to be reasonable (Sousa et al. 1994). The tests are conducted to at least 5000 stress repetitions or to a permanent shear strain of 5 percent, whichever occurs first. Figure 1015 illustrates an example of the relationship between permanent shear strain gp and stress repetitions N obtained in this test. Each curve is adjusted by defining the intercept of gp at N = 0 based on the first 10 repetitions and subtracting this value from all measurements of gp. An equation of the form
γ p = aN b
(102)
283
284
Chapter Ten
FIGURE 1015
Permanent shear strain g p versus load repetitions N.
is then fit to the data for the values of N greater than 100 or 1000 repetitions (the portion of the curve linear after loglog transformation of gp and N) depending on mix behavior.∗ In this expression, the coefficients a and b result from the regression analysis. A shear modulus is also determined from the recoverable shear strain measured at N = 100 repetitions, that is, G=
τ γ recov
=
shear stress [69Kpa(10 psi)] recoverable shear strain at N = 100
(103)
For a given mix, the number of repetitions to 5 percent permanent shear strain is a function of its airvoid content, increasing as the airvoid content is decreased to a value between 3 and 2 percent. Below an airvoid content of about 2 percent, with further decrease in airvoid content, the number of repetitions again decreases. For mix design using conventional binders, it is suggested that the test be performed at a single temperature, representative of the critical temperature at the pavement site (Sousa et al. 1994). The critical temperature is defined as the temperature at a 50 mm (2 in.) depth at which the maximum permanent deformation occurs.† Shear creep tests have been used in a limited number of cases. An example of the application of the test is described in Vallerga et al. (1996). In this example, the longterm behavior of an asphalt concrete barrier used to encapsulate a concrete tank containing radioactive waste was evaluated. Shear creep tests at elevated temperatures provided information for use in finite element analyses of the system to determine ∗
For very stiff mixes, the extrapolation of data beyond N = 10,000 may be utilized. Determination of critical temperatures by the procedure described in Deacon et al. (1994) is based on the assumption that the traffic is applied at a uniform rate throughout the year. If there were a concentration of traffic in a warmer period, the critical temperature would be somewhat higher than obtained by assumption of the uniform annual rate. The procedure described in Deacon et al. (1994) can be modified to account for this difference.
†
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s potentially damaging thermal stresses resulting from elevated temperatures in the radioactive waste. Thus far, the use of the shear test to define stress relaxation characteristics of HMA has been limited to the development of an improved constitutive relation to estimate the accumulation of rutting due to repeated trafficking (e.g., Weissman et al. 2000).
Test Variability and Reliability Considerations In the previous section it was demonstrated that if the specimen size is less than the RVE, tests from a large number of specimens may be required to define a representative measure of permanent deformation response, for a specific temperature and time of loading. Sousa et al. (1994) contains the results of a study to define variability in the RSSTCH and to develop a parameter which could be used in mix design and analysis to provide a specific level of reliability; where reliability is considered to be the probability that the mix will provide satisfactory performance for the design period. The approach adopted for mix design purposes has been to define a multiplier greater than one to be applied to the traffic demand such that the ability of the mix to carry the traffic results in a rut depth that does not exceed some prescribed value. This can be stated as follows: N supply ≥ M ⋅ N demand
(104)
where Nsupply = estimated repetitions to a limiting prescribed rut depth [e.g., 12.5 mm (0.5 in.)] Ndemand = applied traffic demand M = reliability multiplier (greater than 1) whose magnitude is dependent upon variability of the estimated repetitions to a prescribed rut depth and the traffic demand, together with the desired reliability of the design The reliability can be determined from the following: ln( M) = ZR ⋅ {var[ln(N supply )] + var[ln( N demand )]}0.5
(105)
where ZR is a function of the reliability level, which assumes values of 0.253, 0.841, 1.28, and 1.64 for reliability levels of 60, 80, 90, and 95 percent, respectively var[ln(N supply )] = variance of the natural logarithm of Nsupply var[ln(N demand )] = variance of the natural logarithm of Ndemand The var[ln(N supply )] was determined from the results of 31 tests using the RSSTCH on 150mm (6 in.) diameter × 50mm (2 in.) high specimens at 50°C and a shear stress of 70 kPa (10 psi). The test data represented one HMA prepared by rolling wheel compaction for a range in asphalt contents [4.5 to 6.0 percent (aggregate basis)] and airvoid contents (approximately 3 to 8 percent). Table 102 contains a listing of values for M for a range in var{ln(Ndemand)}, number of specimens tested, and the desired reliability level. Since some of the examples to be presented subsequently are based on results of tests using 150mm (6 in.) diameter specimens, reference will be made to this table.
285
286
Chapter Ten
Sample Size
Variance of ln(Ndemand)
60 Percent Reliability (ZR = 0.253)
80 Percent Reliability (ZR = 0.841)
90 Percent Reliability (ZR = 1.28)
1
0.2
1.349
2.704
4.545
6.957
0.4
1.377
2.896
5.046
7.955
0.6
1.404
3.09
5.567
9.022
1
1.455
3.48
6.673
11.381
0.2
1.304
2.416
3.83
5.587
0.4
1.334
2.609
4.305
6.49
0.6
1.363
2.802
4.797
7.456
1
1.417
3.188
5.839
9.592
0.2
1.28
2.27
3.482
4.945
0.4
1.312
2.464
3.946
5.805
0.6
1.342
2.657
4.425
6.723
1
1.397
3.042
5.437
8.754
0.2
1.267
2.197
3.313
4.64
0.4
1.3
2.392
3.772
5.479
0.6
1.331
2.585
4.245
6.375
1
1.338
2.97
5.243
8.356
2
3
4
95 Percent Reliability (ZR = 1.64)
Sousa et al. 1994.
TABLE 102 Reliability Multiplier
Test Specimen Size and Preparation Considerations It should be noted that, as with other tests, there are some limitations in the simple shear test. For example, the shear test does not measure the deviatoric component of normal strains. The major imperfection in this test, however, comes from missing tractions on the leading and trailing edges of the specimen, as indicated in Fig. 1016. This introduces boundary layers near these edges that may affect the solution. Fortunately, the width of this boundary layer is independent of the specimen length and instead depends on the specimen height. Therefore, the relative contribution of these boundary layers can be diminished if the lengthtoheight ratio of the specimen is increased. To demonstrate the effect of the lengthtoheight ratio (length of specimen/height of specimen), a series of threedimensional finite element simulations were conducted. (Sousa et al. 1994) In these simulations, a 50mm (2.0in.) high and 100mm (4.0in.) wide specimen, with a length that varied between 25 and 500 mm (1.0 and 20.0 in.), was used. A nonlinear elastic material model was used; and the results are shown in Fig. 1017. Figure 1017 highlights two important findings. First, 10 percent error, or less, in the predicted shear modulus (G) can be expected for specimens with a lengthtoheight ratio greater than three. Second, Gmeasured converges to G monotonically from below. This
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s
FIGURE 1016
Simple shear test, traction represented by dotted arrows is not applied.
FIGURE 1017
Convergence of measured G to actual G with increased lengthtoheight ratio.
provides a conservative value for G. In general these results indicate that the level of error can be reduced by increasing the lengthtoheight ratio. Thus, assuming the specimen height is prescribed by the RVE requirements, it is possible to select a specimen length that would result in an error level smaller than a specified value. Planestrain finite element simulations of the simple shear test reported in Weissman (1997) and Weissman et al. (1999) support the results of the analyses shown in Fig. 1017. The results of these analyses also suggest that the reliability of the test could be improved by using rectangular parallel sided rather than cylindrical specimens. For a lengthtoheight ratio of three, the critical dimension for RVE is the height. Using the same criteria as applied to the triaxial test, the height of the shear specimens should probably be about 75 to 100 mm for a 19mm nominal size aggregate. With most compaction methods, the nominal maximum size of the aggregate will be oriented in the horizontal direction (i.e., the longer side will be oriented parallel to the surface) and the aggregate length to height ratio is not important. When preparing specimens for permanent deformation evaluation in laboratory tests, it is important that the aggregate structure of the laboratorycompacted mix be about the same as that of the mix compacted in the field. Hveem was one of the first asphalt researchers to recognize this, resulting in the development of the Triaxial Institute kneading compactor. (Vallerga 1951; Endersby et al. 1952; Monismith et al. 1956)
287
288
Chapter Ten The Laboratoire Central Ponts et Chasusses (LCPC) conducted a study of specimens prepared by a number of different compaction procedures soon after the introduction of their gyratory compactor developed to evaluate the compactibility of mixes. It was observed that the rolling wheel compactor developed by the LCPC produced specimens that best reflected performance on comparable specimens compacted in situ (van Grevenynghe 1986). The LCPC does not use gyratorycompacted specimens for permanent deformation evaluation; rather, they use a form of rolling wheel compaction. (Bonnot 1986) During the Strategic Highway Research Program an extensive study was conducted of the influence of compaction method on the permanent deformation response of mixes. The compaction procedures included a mechanized version of the Texas gyratory compactor [150mm (6in.) diameter mold], the Triaxial Institute kneading compactor and a form of rolling wheel compaction (Sousa et al. 1991). Results of the study supported the work of the LCPC suggesting that some form of rolling wheel compaction was most suitable for laboratory specimen preparation. Additional work confirmed these findings and extended them to the SHRP gyratory compactor. (Harvey et al. 1994; PRC 1999) Recently, through the CAL/APT program it was possible to compare the permanent deformation characteristics of cores obtained from two overlay pavements, containing a conventional densegraded aggregate with an AR4000 asphalt cement and a gapgraded material with an asphalt rubber asphalt cement, constructed according to Caltrans procedures, with the same mixes compacted by laboratory rolling wheel compaction (Harvey et al. 1999) and the Superpave gyratory compactor. Simple shear tests (RSSTCH) were performed according to AASHTO TP794 at 40, 50, and 60°C (104, 122, and 140°F). Results of these tests are summarized in Fig. 1018 for the asphalt rubber mix at 50 and 60°C (122 and 140°F). Similar results were observed for the conventional densegraded AC. The specimens prepared by the SHRP gyratory compactor exhibit greater resistance to permanent deformation than the field cores. This difference is due, in a large part, to the difference in aggregate structure created by the SHRP gyratory compactor as compared to rolling wheel compaction (Harvey et al. 1999; Harvey et al. 2001). Also, the data suggest that the specimens prepared by rolling wheel compaction are similar in response to the field cores.
Mix Design and Analysis, Performance Evaluation Experience in mix design (PRC 1999; Harvey et al. 1995), performance evaluation (Monismith et al. 1999), and pavement analysis (Monismith et al. 2001) considering pavement deformation, have been developed using the simple shear test in the RSSTCH mode. Results from these investigations are briefly summarized in this section. Some data are included suggesting that the stiffness modulus of binderaggregate mix should be used with caution as a sole criterion for mix design purposes to mitigate rutting. Also, a discussion is included of the use of creep versus repeated loading to define permanent deformation response.
Mix Design An approach to mix design suggested in Sousa et al. (1994) is to select a binder content for a particular aggregate and gradation to limit rutting to some predetermined level, for example, 12 to 13 mm (0.5 in.). Essentially, this consists of testing a mix over a range in binder contents and selecting the highest binder content, which will permit the mix
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s
FIGURE 1018 Inﬂuence of compaction method on behavior of mixes in the RSSTCH at 50°C and 60°C; gapgraded asphaltrubber hot mix.
to accommodate the design traffic at a critical temperature∗ without exceeding the limiting rut depth. When the design binder content has been selected, the performance of this mix in a selected structural section is evaluated to ensure that the anticipated traffic for the design period can be carried so that the level of fatigue cracking will not exceed some prescribed level such as 10 percent in the wheel paths (Sousa et al. 1994). ∗
The critical temperature is defined as the temperature at a 50mm (2in.) depth at which the maximum permanent deformation occurs, assuming in this case that the truck traffic is applied at a uniform rate throughout the year, as noted earlier.
289
290
Chapter Ten
FIGURE 1019
Permanent deformation system.
The approach for rutting is illustrated in Fig. 1019, and the repeated simple shear test at constant height (RSSTCH) is utilized. As noted earlier, a prescribed shear stress (e.g., 70 kPa or 10 psi) is repeatedly applied for a specific loading time and a time interval between load applications at a specified test temperature. Tests are normally conducted for about 5000 repetitions or to some prescribed value of limiting shear strain and the repetitions corresponding to a fixed level of strain are determined. This value, as seen in Fig. 1019, is termed Nsupply. In Fig. 1019 an expression relating rut depth to the plastic strain gp is shown, that is, RD = K ⋅ γ p
(106)
The parameter K was developed from a series of finite element analyses described in Sousa et al. (1994). The results of these analyses are shown in Fig. 1020. Also shown in this figure are the results of a more recent analysis (Long et al. 2002) showing that K
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s
K Value
1000
100 Dual radial, 40°C Dual redial, 50°C Dual redial, 60°C Widebase, 50°C Widebase, 60°C K SHRP Data K=254
10
FIGURE 1020 below.
0
50
100
150 200 250 Thickness (mm)
300
350
400
K value versus asphalt concrete layer thickness assuming a very stable layer
may be dependent on tire type and configuration (dual versus single). For thicker pavements a value of K = 254 (rut depth in millimeters) or 10 (rut depth in inches) has been used. This appears to be a conservative value as seen in Fig. 1020. This value was used in the example in Fig. 1015 to determine Nsupply for gp = 5 percent corresponding to 12.5mm rut depth. The anticipated traffic must be converted to its laboratory equivalent termed Ndemand so that Eq. (104) described earlier, is satisfied: N supply ≥ M ⋅ N demand
(104)
Ndemand is determined from the estimate of the design ESALs, a temperature conversion factor (TCF) which converts the traffic year round to an equivalent number applied at the critical temperature (Deacon et al. 1994), and a shift factor (SF), which converts the repetitions applied in the field to an equivalent number in the laboratory (Sousa et al. 1994), that is, N demand = design ESAL s × TCF × SF
(107)
Values for the reliability multiplier M which reflects the test variance and the estimated variance in the ln(ESALs) for a specified level of reliability are shown in Table 102 which provides values which have been used to date. (Sousa et al. 1994) Two examples for the use of the RSSTCH for mix design are presented in this section: (1) mixes for a warrantied overlay pavement on Interstate5 in Northern California (Harvey et al. 1995), and (2) mixes for a longlife pavement rehabilitation on Interstate710 in Long Beach, California (PRC 1999). Both examples make use of the procedure illustrated in Fig. 1019.
291
Chapter Ten
Warrantied Pavement, I5 The project, an overlay consisting of two lifts of asphalt concrete, each approximately 50 mm (2 in.) in thickness, was constructed on Interstate 5 near Redding, California on an existing concrete pavement which was cracked and seated prior to the placement of the overlay. The lower lift contained a conventional dense graded aggregate with a PBA6 binder (mix designation—DGAC) while the upper lift contained a gapgraded aggregate with a gapgraded asphalt rubber binder (mix designation—ARHMGG). The contractor was required to warranty the performance of the overlay for a period of 5 years in this case according to the requirement that rutting could not exceed 12 to 13 mm (0.5 in.). (Harvey et al. 1995) The consultant to the contractor elected to use the RSSTCH test for binder content selection for both mixes. Results of tests performed at 45°C (113°F), the critical temperature estimated for the site, are summarized in Table 103 and are plotted in Fig. 1021. The value for Ndemand (corresponding to an estimated traffic of 10 × 106 ESALs) was determined to be 229,000 repetitions at a level of reliability of 95 percent. From the mix design information, values of binder content of 5.2 percent and 7.5 percent (aggregate basis) were recommended for the DGAC and ARHM, respectively. It will be noted in Table 103 that a higher value of shear strain was selected for the ARHMGG since its layer thickness is only about 50 mm (2 in.) and the higher stability DGAC is below. The basis for this decision is illustrated in Fig. 1020. That is, with a fixed value for rut depth, K decreases as pavement thickness decreases, as seen in Fig. 1020, hence the higher value of shear strain. At the end of the 5year period, the mix performed as expected and the contractor received final payment for the project in accordance with the warranty.
1.0E+07
DGAC ARHMGG
N@ γp = 5% or N@ γp = 8.5%
292
1.0E+06
229,000 repetitions (95% reliability)
1.0E+05
1.0E+04 4.5
Temperature = 45°C
5
5.5
6
6.5
7
7.5
8
Binder Content (percent by weight of aggregate)
FIGURE 1021
Relationship between binder content and repetitions to a prescribed shear strain.
8.5
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s
Binder Content Percent (aggregate basis)
Void Percent
N @ gp = 0.05 Shear Strain × 103*
5.0
3.7
4973
5.5
2.4
154
6.0
3.5
75
DGAC
ARHMGG N @ γp = 0.085 Shear Strain × 103 6.0
3.7
2600
7.0
3.3
520
8.0
3.1
34
∗
Average of 3 or 4 specimens.
TABLE 103 Test Data for Mixes Used in Warrantied Project
Long Life Pavement Rehabilitation, I710 Two mixes were evaluated for this pavement rehabilitation project in Long Beach, California (PRC 1999). While one aggregate and grading was used for both mixes, one contained a PBA6a∗ (PG6440)∗ binder and the other an AR8000 asphalt cement (PG6416). As noted earlier, the usual procedure for measuring Nsupply in the RSSTCH is to carry the test to 5000 to 10,000 repetitions or a permanent shear strain of 5 percent (RD = 12.5 mm for k = 254), whichever is larger. (For the PBA6a∗ mix tested at 4.7 percent binder content, by weight of aggregate, the RSSTCH was continued to about 40,000 repetitions and the curve extrapolated to 5 percent strain.) Results of tests at 50°C for both mixes for binder contents in the range 4.2 to 5.2 percent (aggregate basis) are shown in Fig. 1022. Based on the requirements for Ndemand for the project, that is, 660,000 repetitions,† the mix containing the PBA6a* binder at a binder content of 4.7 percent was selected for use in the upper 75 mm (3 in.) of the pavement structure. For the next 150 mm (6 in.), the mix containing the AR8000 binder was selected. Since the pavement will likely be subjected to the traffic prior to the placement of the PBA6a∗ mix, a design binder content of 4.7 percent was also selected for this layer based on a conservative estimate for Ndemand of 146,000 repetitions. To evaluate the proposed mix design for the PBA6a∗ mix, a test section for rutting evaluation by the HVS was constructed using the same binder and aggregate as used in the mix design and 75 mm of PBA6a∗ mix on top of 75 mm of AR8000 mix on PCC. Results of the HVS test conducted on this mix are shown in Fig. 1023. The test was conducted at same temperature as the RSSTCH, that is, 50°C at 50 mm depth. Also The material is a modified binder containing an elastomeric component. The ∗ indicates that this includes a higher proportion of elastomeric material than the PBA6a binder. † Ndemand was determined for a design traffic of 30 × 106 ESALs, a TCF = 0.11, a SF = 0.04, and a reliability multiplier, M = 5.0. ∗
293
Chapter Ten 10,000,000 Temperature = 50°C PBA 6A
N@ γp = 5%
1,000,000
AR 8000 660,000 repetitions
100,000
10,000
1000 3.0
3.5 4.0 4.5 5.0 5.5 Asphalt Content (percent by weight of aggregate)
6.0
FIGURE 1022 Repetitions to a permanent shear strain of 5 percent versus binder content; tests performed at 50°C (122°F). 30 38mm ARAMGG
25
Rut depth, mm
294
62mm ARAMGG 75mm ARAMGG 76mm ARAMGG
20 15
10
5
0
FIGURE 1023
50,000
100,000 HVS Load Applications
150,000
200,000
Rut depth versus HVS load applications with 40kN load on dual tires at 50°C.
shown in this figure are test results for mixes containing an AR4000 asphalt cement and an asphalt rubber binder, both of which meet California DOT requirements for freeway traffic (Monismith et al. 2001). These mixes exhibited Nsupply values at 5 percent shear strain generally less than 104 repetitions whereas the PBA6a∗ mixes had values of N > 6 × 105 repetitions for comparable airvoid contents. These data lend support to the use of the RSSTCH for mix evaluation since a discernible difference in rutting performance of the various mixes was observed.
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s
Mix Evaluations The WesTrack experiment (Monismith et al. 2000) provided an opportunity to assess the applicability of the RSSTCH to evaluate the performance of Superpave mixes subjected up to 5 × 106 ESALs; in addition, it provided the opportunity to develop performance equations considering both RSSTCH results and other mix properties which can be used in performancerelated specifications. Three general mix types were utilized termed coarse, fine, and fine plus. The coarse mixes were replaced during the project with mixes with essentially the same grading but with a different aggregate type. Epps et al. (1999) contains details of these gradations as well as the Superpave mix designs. To arrive at the stress repetitions in the RSSTCH laboratory tests at which to select the strains for comparative purposes, Eqs. (10.2) and (106) were used. For the original sections, in order to use all of the results, the comparisons are based on rutting data obtained at about 1.5 × 106 ESALs. Use of Eq. (108), with a TCF of 0.116 and an SF of 0.04, provided a value of N = 7000 repetitions. For the replacement sections, the comparisons are made at about 0.6 × 106 ESALs or N = 2700 repetitions. Results of these analyses are presented in Figs. 1024 and 1025 for the original fine and replacement mixes, respectively. These figures contain plots of downward measured rut depth versus the plastic strain corresponding to either 7000 or 2700 repetitions in the RSSTCH. The shear test results were obtained from tests at 50°C (122°F) and a shear stress of 70 kPa (10 psi). For the original mixes, each data point represents the average of two tests on cores obtained prior to trafficking (referred to as t = 0). For the replacement sections, in addition to the t = 0 test data, results of tests on cores after the trafficking was complete (postmortem) are also shown. In general, for each set of mixes there is a reasonable relationship between rut depth and plastic strain as measured in the RSSTCH test at 50°C, indicating that the test can
FIGURE 1024 Downward rut depth versus g p at N = 7000 repetitions in RSSTCH at 50°C; ﬁne mixes, original WesTrack test sections.
295
296
Chapter Ten
FIGURE 1025 Downward rut depth versus g p at N = 2700 repetitions in RSSTCH at 50°C; coarse mixes, WesTrack replacement sections. [Note: Test data from both t = 0 cores and cores obtained at the conclusion of trafﬁc (t = postmortem) are included.]
rank rutting performance in situ. The slope of this relationship is, however, different for each mix.
Development of Mix Design Criteria for Taxiway Subjected to Heavy Aircraft Loading In August 1995, shoving and rutting were observed in the asphaltconcrete turn areas of the taxiway adjacent and leading to the international terminal of the San Francisco International Airport (SFIA). The distress occurred when the air temperature was about 35°C (95°F) and was attributed to slowmoving and sharpturning Boeing 747400 aircraft. Because the wing gear of this aircraft do not turn, a significant shear force, an action termed sluing, is exerted by each of the tires at the pavement surface. In October 1995, rutting distortions (termed dimpling) were observed under stopandgo aircraft movements on another taxiway attributed primarily to the Boeing 747400 in queue awaiting takeoff. To correct these as well as other rutting problems resulting from the Boeing 747400 operations, a trial mix was selected to be used as a potential model for establishing a specification for a High Stability mix. It should be noted at the outset that the mixes which exhibited the rutting distress met current Federal Aviation Administration specifications (Monismith et al. 1999). This sequence of events, including the introduction of the High Stability mix, provided an opportunity to evaluate the applicability of the RSSTCH for designing and evaluating asphalt (binder)aggregate mixes for airfield pavements subjected to large, heavily loaded aircraft. Materials used at the SFIA when the rutting failures occurred consisted of an AR4000 asphalt cement and aggregate obtained from the nearby Brisbane quarry. The materials used for the High Stability mix consisted of an AR16,000 asphalt and an allcrushed granite aggregate from a quarry at Logan, California.
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s The RSSTCH was used to evaluate the permanent deformation characteristics of the cores obtained when the rehabilitation actions were taken. To evaluate field rutting performance of the High Stability mix placed on Taxiway B in August 1996, use was made of the laser profilometer, developed by the CSIR of South Africa and a part of the instrumentation used to evaluate pavement response during HVS testing. Results of the RSSTCH are presented in Figs. 1026 through 1028. Figure 1026 contains the data for the High Stability mixes from Taxiways A and B while Fig. 1027 contains the data for mixes containing the AR4000 asphalt cement from Taxiways A, B, and M. For a given mix, the number of repetitions to 5 percent permanent shear strain is a function of its airvoid content, increasing as the airvoid content is decreased to a value
FIGURE 1026 Shear stress repetitions to 5 percent shear versus airvoid content at 50°C—high stability mixes. 1,000,000 Taxiway A (Aug. ’95) Taxiway B (Aug. ’95) Taxiway B (June. ’96)
N at γp = 0.05, T = 50°C
100,000
Taxiway M (Sept. ’96) Reference
10000
1000
100
10
0
2
4
6 8 Air void content (percent)
10
12
FIGURE 1027 Shear stress repetition to 5 percent shear strain versus airvoid content at 50°C—mixes containing AR4000 asphalt cement.
297
298
Chapter Ten
FIGURE 1028
Mix stiffness measured in shear versus airvoid content.
between 3 and 2 percent. Below an airvoid content of about 2 percent, the number of repetitions again decreases. In Fig. 1026, a line has been drawn and designated as a “reference curve” in order to delineate the area above which the results of the shear test data on the High Stability mix are generally situated (24 of 28 data points). When this line is placed on Fig. 1027, it will be noted that with few exceptions, the mixes that have exhibited rutting and shoving fall below the line (21 of 27 data points). An additional concern is whether one can differentiate between the mixes based on stiffness (in this case, shear stiffness) as measured in the simple shear equipment. To answer this concern, the available stiffness data for the various mixes are plotted in Fig. 1028. Note that there is no differentiation between the mixes containing the AR4000 asphalt with Brisbane aggregate and the mixes containing the AR16,000 material with Logan granite. These findings underscore the need to perform a repeated load test to evaluate permanent deformation resistance of mixes. Data presented in Figs. 1026 and 1027 serve as the basis for new mix design criteria. One approach is to use the reference curve as the basis for mix evaluation. Although the criteria were developed from the available mixes, the results are applicable to similar mixes and environmental conditions as those found at SFIA. The criteria can be validated and modified with data for additional mixes. If a specimen is compacted to an airvoid content of about 3 percent and it provides a value of N = 25,000 repetitions at 5 percent strain in the RSSTCH, it should be capable of sustaining traffic of the type described herein since this requirement represents the behavior of the High Stability mix used at SFIA. While the reference curve of Figs. 1026 and 1027 was established initially by engineering judgment, a subsequent analysis using a logistic regression model was performed to analyze improved criteria. The results are shown in Fig. 1029. In this
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s
FIGURE 1029 Shear stress repetitions to 5 percent shear strain versus airvoid content at 50°C—all mixes.
regression, the probability of failure is given in Eq. (108). Failure is defined as the mix not being able to withstand the traffic. Probability of failure =
1 1 + exp(−11.8119 + 0.6002 ⋅ AV + 0.9995 ⋅ ln N )
(108)
where AV is percent airvoid content, and N is repetitions to 5 percent shear strain. Figure 1029 shows isolines of probability of failure corresponding to values of 20, 40, 60, and 80 percent. The reference curve of Figs. 1026 and 1027 corresponds to a probability of failure of 50 percent. Thus, based on Fig. 1029, tentative criteria for mix design using the shear test (RSSTCH) can be established for different probabilities of rutting failure. Allowable repetitions to failure in the RSSTCH at 50°C for the conditions of loading described herein for probabilities of failure of 20 to 50 percent, at an airvoid content of 3 percent are as follows: Probability of Failure
Np at gp = 5 percent
50
25,000
40
35,000
20
100,000
This information can be used as described earlier for mix design purposes.
Creep versus Repeated Loading Creep and repeated loading have been used to define permanent deformation response characteristics of HMA in both the triaxial compression shear tests. Some evidence will be presented in this section to suggest that a form of repeated loading is required to better characterize permanent response of mixes containing a range in binder types.
299
300
Chapter Ten In the Shell pavement design procedure, provision has been made to compare the permanent deformation characteristics of different mixes measured in creep and to select a mix in which the estimated rutting for specific traffic loading and environmental conditions would not exceed some predetermined value. The rut depth at the pavement surface due only to permanent deformations in the asphaltbound layer is a function of the mix stiffness (Smix) (Shell 1985). The value of Smix is determined from creep loading over a range in loading times. While this approach has worked satisfactorily for mixes containing conventional asphalt cements, Valkering et al. reported that when the procedure was used for mixes containing nonconventional binders, “… a correction is required to take account of the different relationship between rutting and binder viscosity. The dynamic creep test has shown potential for a more universal applicability, extending to include those asphalts based on a modified binder… The greater suitability of the dynamic test for rating the effect of the binder modification is ascribed to the recovery effects of the test.” (Valkering et al. 1990.) In another publication (Lizenga 1997), Shell investigators suggest that the use of creep test data may overpredict rutting for mixes containing some modified binders. The research of Tanco on the permanent deformation response of conventional and modified asphaltaggregate mixes under simple and compound loading conditions supports the work of the Shell investigators (Tanco 1992). He found that repeated load tests were more responsive to the presence of modified binders in AC mixes than static constant load (creep) tests. Similar research had been reported by Tayebali (Tayebali 1990). This is illustrated with data obtained from tests on mixes containing a conventional AR8000 (PG 6416) asphalt and a PBA6a∗ (PG 6440) binder for the Interstate Route 710 project (Monismith et al. 2001). Table 104 contains a summary of Hveem Stabilometer test results obtained for both mixes. These results suggest that mixes containing the AR8000 binder should have greater permanent deformation resistance than those containing the PBA6a* binder. Test results using the repeated simple shear test at constant height (RSSTCH), however, indicate that the mix with the PBA6a* will sustain more traffic to a fixed rut depth [~12.5 mm (0.5 in.)] than the mix with the AR8000 asphalt as seen in Fig. 1023. These results further stress the importance of using a “dynamic” rather than a static test to measure permanent deformation response, particularly when evaluating mixes containing modified binders.
Stabilometer S Values Asphalt Content—percent by weight of aggregate Mix AR8000 ∗
PBA6a
4.2
4.7
5.2
5.7
36
39
40
34
—
26
35
26
Monismith et al. 2001.
TABLE 104
Stabilometer S Values for Mixes Containing AR8000 and PBA6a∗ Binders
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s
Recursive Rut Depth Prediction In this approach, which is a recursive extension of the approach used on projects previously presented in this chapter, the pavement is assumed to behave as a multilayered elastic system. An idealization of a specific asphalt pavement, in this case the WesTrack structure, is shown in Fig. 1030 together with key parameters used to estimate rut depth development with traffic, that is, t, g e, and ev where t, g e, are elastic shear stress and strain at a depth of 50 mm (2 in.) below the outside edge of tire and ev is elastic vertical compressive strain at the subgrade surface. These three parameters can be determined on an hourbyhour basis and a program like the Integrated climate model (ICM) can be used to define temperature distributions both with time and depth in the HMA to permit estimates of the mix stiffnesses. For convenience, if a program like ELSYM5 (5 layers) is used, it is recommended that the HMA layer be subdivided into three layers with thicknesses from top to bottom of 25 mm (1 in.), 50 mm (2 in.), and the remaining HMA thickness as the third layer to simulate the effects of temperature gradients on mix stiffness. In the computations, a constant Poisson’s ratio of 0.35 is suggested. If programs are available with capability of treating more than 5 layers, the third HMA layer can be further subdivided to produce a more representative stiffness distribution in the HMA layer. Moduli of the underlying layers can also be varied to reflect seasonal influences on the stiffness moduli of those layers. Poisson’s ratios in the range 0.35 to 0.4 are recommended for untreated granular layers and 0.4 to 0.45 for untreated finegrained (subgrade) soils.
FIGURE 1030 WesTrack pavement representation for mechanisticempirical modeling for rutting.
301
302
Chapter Ten In this approach, rutting in the asphalt concrete is assumed to be controlled by shear deformations. Accordingly, the computed values for t and g e at a depth of 50 mm (2 in.) beneath the edge of the tire are used for the rutting estimates, as shown in Fig. 1030. Densification of the asphalt concrete is excluded in these estimates since it has a comparatively small influence on surface rutting, particularly for mixes with airvoid contents less than about 8 percent. In simple loading, permanent shear strain in the HMA is assumed to accumulate according to the following expression:
γ i = a exp(bτ )γ e nc where
(109)
g i = permanent (inelastic) shear strain at 50 mm (2 in.) depth t = shear stress or shear stress normalized to shear stress at a reference temperature determined at this depth using elastic analysis g e = corresponding elastic shear strain n = number of axle load repetitions a, b, c = regression coefficients
The timehardening principle is used to estimate the accumulation of inelastic strains in the asphalt concrete under in situ conditions. The resulting equations are as follows: ai = a exp(bτ )γ ej
γ 1i = a1 ⎡⎣ Δn1 ⎤⎦
cc
⎡ i ⎛⎜⎝ 1c ⎞⎟⎠ ⎤ ⎢ ⎛ γ j−1 ⎞ ⎥ i γ j = a ⎢⎜ Δ n + j ⎥ a j ⎟⎠ ⎝ ⎢⎣ ⎥⎦ where
(1010) (1011) c
(1012)
j = jth hour of trafficking g ej = elastic shear strain at the jth hour Δnj = number of axle load repetitions applied during the jth hour
The concept is illustrated schematically in Fig. 1031. Rutting in the AC layer due to the shear deformation is determined from the following: rdAC = Kγ ij
(1013)
As seen in Fig. 1020, K ranges from about 5.5 for a 150mm (6 in.) layer to 10 for a 305mm (12 in.) thick AC layer when the rut depth rdAC is expressed in inches (Monismith et al. 2000). To estimate the contribution to rutting from base and subgrade deformations, a modification to the Asphalt Institute subgrade strain criteria (Shook et al. 1982) is utilized. The equation expressing the criterion for 12.5 mm (0.5 in) of surface rutting is n = 1.05 × 10−9 ε v −4.484
(1014)
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s
FIGURE 1031 Timehardening procedure for inelastic strain accumulation under stress repetitions of different magnitudes in compound loading.
where n is the allowable number of repetitions and ev is the compressive strain at the top of the subgrade. Since these criteria do not address rutting accumulation in the pavement structure, rut depth (rd) contributed by the unbound layers was assumed to accumulate as follows: rd = dne
(1015)
where d, e are experimentally determined coefficients. Least squares analyses for the WesTrack data suggest that the value for d in Eq. (1012) using the Asphalt Institute criteria is d=
f ⎡⎣ 1.05 × 10−9 ε v −4.484 ⎤⎦
(1016)
where f is 3.548 and e is 0.372. Using the timehardening principle, as was used for the asphalt concrete, rut depth accumulation can be expressed in a form similar to Eq. (1012), that is, 1 ⎡ ⎤ ⎛ rd j−1 ⎞ 0.372 ⎢ ⎥ + n rd j = d j ⎜ Δ j ⎥ ⎢ ⎝ d j ⎟⎠ ⎢⎣ ⎥⎦
0.372
(1017)
The framework for rutdepth estimation, using Eqs. (1012), (1013), and (1017), is illustrated in Fig. 1032. This approach has a distinct advantage over the direct regression
303
304
Chapter Ten
FIGURE 1032 Framework for rut depth estimates.
approach in that it permits prediction of rut depth as a function of traffic and environment as well as a function of the mix parameters. For WesTrack, 13 sections were used to calibrate the coefficients of Eqs. (1010) and (1015). Initially, a value for b = 0.0487 in Eq. (1010) was used based on the results of RSSTCH tests on laboratory mixed and compacted (LMLC) specimens. Subsequently, a value of b = 0.071 (10.28 in metric units) was determined to provide better correspondence between measured and computed rut depths. Using the procedure illustrated in Fig. 1031, least squares regression provided values of a and c for each of the 23 WesTrack sections where rutting without observed fatigue cracking was obtained. These are summarized in Table 105. It should be noted
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s
Section
c
RMSE in. (mm)
1
5.41658
0.022521
0.027 (0686)
4
0.01392
0.66306
0.037 (0.940)
7
0.01509
0.77181
0.102 (2.590)
9
0.00410
0.83989
0.076 (1.930)
11
1.64235
0.29677
0.040 (1.016)
12
1.05802
0.33734
0.087 (2.210)
13
0.01186
0.75472
0.078 (1.981)
14
6.15197
0.25614
0.050 (1.270)
15
7.30191
0.20716
0.001 (0.254)
18
0.39160
0.41493
0.035 (0.889)
19
3.86629
0.29245
0.044 (1.118)
20
7.03048
0.26222
0.050 (1.270)
21
0.00973
0.81183
0.118 (2.997)
22
29.32602
0.10116
0.042 (1.067)
23
0.59761
0.43650
0.050 (1.270)
24
0.49708
0.49941
0.084 (2.134)
25
0.05564
0.67400
0.102 (2.591)
35
52.77398
0.12388
0.024 (0.610)
37
12.04868
0.26447
0.032 (0.813)
38
23.14986
0.13996
0.024 (0.610)
39
13.73983
0.18501
0.030 (0.762)
54
51.08506
0.12941
0.012 (0.305)
55
3.22487
0.35783
0.032 (0.813)
Average
a
0.051 (1.295)
TABLE 105 Calibration Results for 23 Sections, Conventional Analysis
that the average root mean square error (RMSE) for rut depth for the 24 sections is 0.051 in. Figs. 1033 to 1036 illustrate comparisons between computed and measured rut depths for Section 4 (fine), Section 19 (fineplus), Section 7 (coarse), and Section 38 (replacement, coarse). To be able to use mixes like those at WesTrack, but in different traffic and temperature environments, in this mechanistic—empirical procedure requires values of a and c dependent on mix properties. Calibrations using the 23 sections of Table 105 resulted in regressions of the following form: ln(field a) = a 0 + a1 Pwasp + a2Vair + a3 Fine + a4 Coarse + a5 ln(field c)
(1018)
ln(field c) = b0 + b1 Pwasp + b2 Fine − plus + b3 Coarse
(1019)
305
306
Chapter Ten
FIGURE 1033 Comparison between computed and measured rut depth versus time; Section 4.
FIGURE 1034 Comparison between computed and measured rut depth versus time; Section 7.
where Pwasp = asphalt content by mass of mix, percent Vair = airvoid content, percent Fine, Fineplus, Coarse = indicator variables representing the three gradings used at WesTrack a0,…, an, b0,…, bn = regression coefﬁcients For wider applications, however, it is desirable to have relationships for a and c that are not limited to the mix types used at WesTrack.
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s
FIGURE 1035 Comparison between computed and measured rut depth versus time; Section 19.
FIGURE 1036 Comparison between computed and measured rut depth versus time; Section 38.
An approach recommended at this time makes use of the results of the laboratory RSSTCH test and the mix variables—asphalt content and airvoid content—to determine field a and c values. A series of regressions were performed for 23 WesTrack sections shown in Table 105 considering these variables. Results of the calibrations are shown in Table 106 for ln (field a) and Table 107 for field c. In these tables, lab a is from the expression:
γ i = anb
307
308
Chapter Ten
Regr. 1
Regr. 2
Regr. 3
Regr. 4
Regr. 5
Regr. 6
Constant
14.9116
24.7107
24.3317
24.9718
25.3649
20.4844
Pwasp
−3.67001
−5.02990
−5.04342
−5.23716
−5.71438
−5.12624 0.313875
Vair Pwasp·Vair
0.0823738
RSST05 lab a R
2
Sections Omitted
6.219E05
9.699E05
1301.81
1622.41
1745.07
1858.91
2472.96
2264.05
0.611
0.629
0.684
0.752
0.888
0.951
None
14
14, 15
1, 14, 15
1, 14, 15, 19
1, 4, 14, 15, 19
TABLE 106 Calibration of Equations for Simulating ln(field a) Based on Mix and RSST Variables
Regr. 1
Regr. 2
Regr. 3
Regr. 4
Regr. 5
Regr. 6
Constant
−0.944102
−1.75309
−1.72144
−1.77798
−1.83917
−1.49931
Pwasp
0.312598
0.426673
0.427803
0.444915
0.493348
0.452398 −0.0217923
Vair Pwasp·Vair
−0.0064968
RSST05 lab a
−6.216E06
−8.575E06
−87.5258
−113.452
−123.693
−133.748
−190.11
−175.759
R
0.556
0.591
0.648
0.728
0.890
0.936
Sections Omitted
None
14
14, 15
1, 14, 15
1, 14, 15, 19
1, 4, 14, 15, 19
2
TABLE 107 Calibration of Equations for Simulating field c Based on Mix and RSST Variables
obtained from analysis of the RSSTCH results as shown in Fig. 1015. The term RSST 5 corresponds to the repetitions for a value of g i = 5 percent, also illustrated in Fig. 1015. From the analyses, regression 6 in both Tables 106 and 107 is recommended for use to define a and c for use in the mechanisticempirical procedure discussed above. It must be emphasized, however, that the use of these regression equations should be limited to the range of values for the parameters used in their development. For example, values of RSST 5 at g i = 5 percent should be less than about 50,000 repetitions. An example is provided in Deacon et al. (2002) in which the direct application of this approach was applied to the HVS test results shown in Fig. 1023 for the PB6a∗ mix. The results of this comparison are shown in Fig. 1037.
Constitutive Relationship Formulation for HMA Behavior at Elevated Temperatures Currently (Weissman, et al. 2003) a new constitutive relationship describing the mechanical behavior of HMA at elevated temperatures is under development. The
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s
FIGURE 1037 Rut depth versus unidirectional HVS load applications: pavement temperature at 50 mm (2 in.) depth = 50°C (122°F).
relationship utilizes an unconventional viscoplastic model in the sense that it is composed of a viscoelastic component in parallel with a rateindependent elastoplastic part (i.e., the two components act independently of each other and are coupled only by sharing the deformation). The model attempts to incorporate some of the following critical properties of HMA behavior at elevated temperatures, some of which have been discussed earlier: (1) rate dependence, (2) temperature dependence, (3) volumetricdeviatoric coupling, (4) markedly different properties in tension and compression, (5) anisotropic behavior, (6) large residual stains (as a percentage of the total strain before unloading), (7) significantly different behavior in pure volumetric and pure shapedistortion deformations, (8) dependence on the air void content, and (9) creep tests below a certain stress threshold, show bounded flow. The choice is motivated by the following observations: • The amount of residual rut induced by a moving load is inversely proportional to the velocity of the load. Thus, a fast moving load is primarily shouldered by the viscoelastic component and, therefore, only a small amount of residual deformation develops. When a slowmoving load is encountered, the viscoelastic component “relaxes,” and plastic flow ensues. Upon unloading, part of the plastic flow remains locked in. • A viscoelastic model alone cannot be made to match both the loading and unloading histories because of the large residual deformations, which appear also in creep tests at low stress levels. As a result, some form of a damaged model must also be included. In the present work a plastic component is included, which can be viewed as a special form of damage.
309
310
Chapter Ten • A classical viscoplastic model is not suitable because the volumetricdeviatoric coupling (property 3) strongly depends on temperature and rate of loading. At elevated temperatures and slow rates of loading the coupling is strong, while at low temperatures and for fast rates of loading the coupling is weak. Moreover, test results suggest that the coupling is primarily elastic (Sousa et al. 1994). Therefore, laboratory data suggests that the elastic response associated with the viscous part is different from that related to the plastic part which includes volumetricdeviatoric coupling. This is different from classical viscoplastic models where the viscous and plastic models are in parallel and in sequence with a single elastic model. • In creep tests where the applied stress is above a mixspecific stress threshold, asphalt concrete exhibits unbounded flow. A model consisting of a viscoelastic fluid in parallel with an elastoplastic component, which only above a certain stress threshold exhibits perfect plastic flow, can explain such behavior. In addition to the above observations, it should be noted that • According to the proposed model, upon removal of the load the plastic flow induces “relaxation” in the viscoelastic part. This nonequilibrium stress is equilibrated by an opposing stress in the elastoplastic component. Therefore, this model includes possible plastic flow during unloading. • If there is no plastic flow, all of the viscoelastic flow is unrecoverable. Thus, the viscoelastic component regulates the plastic flow. An important feature that distinguishes the model under development from previous models proposed for HMA is that it is formulated within the framework of finite deformations. Previous models have been formulated within the framework of small deformation theory. The finite deformation approach adopted for this model results from measurements taken from pavement test sections subjected to Heavy Vehicle Simulator (HVS) loading (Harvey et al. 2000) These measurements show that, for fully developed ruts (e.g., about 12 mm) the average vertical strain in the top 75 mm of the asphalt concrete surface layer may reach 10 percent or 15 percent, and rotations at the lateral edges of ruts range between 15 and 45 degrees. These measurements exceed the range for which infinitesimal theories are intended. For example, if the standard small strain measure is employed to evaluate the strain in a body undergoing a rigid rotation of 30 degrees (the median rotation measured at the rut edges), then compressive strains of 13 percent are predicted (the true strain is, of course, zero). This suggests that models based on small deformation theories are appropriate to model only the rut initiation phase. To go beyond that to the postrut initiation phase requires the use of models based on finite deformation theory. Because the rut initiation phase is essentially over a few load repetitions, and pavements are typically considered serviceable for quite a few more millions of load repetitions, the ability to predict the behavior in this important phase requires a model of the type being developed. The model described in Weissman et al. (2000) incorporates the following features (discussed above) for HMA behavior at elevated temperatures: (1) rate and temperature dependency, (2) volumetricdeviatoric coupling, (3) different properties in tension and compression, (4) possible plastic flow during unloading (Bauschinger effect), and (5) the ability to retain upon unloading a large residual deformation (as a percentage of the deformation during the loading period).
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s Laboratory tests are currently underway. These tests are divided into two sets. The first is used for material identification purposes; it attempts to isolate different components of the model in order to simplify the property extraction process. In particular, the proposed tests seek to separate the ratedependent component of the model (viscoelastic) from the rateindependent part (elastoplastic) by employing a combination of stepped relaxation tests and stress controlled frequency sweeps. Also, the volumetric and deviatoric components of the motion are coupled by conducting independent simple shear at constant height and hydrostatic pressure tests.∗ The second set of tests is intended for validation purposes. Therefore, a test that coupled all effects is optimal for this purpose. Such a test is offered by the axial test (with controlled lateral pressure). This test, with different load histories, therefore has been selected for this purpose. The main focus of this effect thus far has been theoretical. The next step will require transforming the first proposed set of laboratory tests into actual tests, and the establishment of procedures to reliably extract the material constraints from the test data. Following that phase, the proposed model must be employed to predict the results of the validation tests. This process is necessary to tune and modify the proposed constitutive law until a good agreement between the predictions and the validation test results is achieved.† Thus, a potentially interactive phase where the constitutive law and the laboratory tests and extraction procedures are modified will be undertaken. When‡ predictions achieve a satisfactory level of agreement with the validation tests, the model will be used to predict rut development in pavements subjected to HVS loading under controlled environmental conditions. These simulations can then form the basis for an empiricalanalytical procedure that will tie the results of some simple laboratory tests to rutting in actual pavements. Finally some simplifications will no doubt be required to permit these developments to be used for mix design and analysis purposes. It is likely that these simplifications will follow a procedure similar to that described in previous sections in which the RSSTCH test has been used for mix design and rut depth prediction.
Summary The information presented in this chapter illustrates that shear deformation contributes a significantly greater portion of total permanent deformation (rutting) in AC mixes than volume change so long as airvoid contents are less than about 8 or 9 percent of the completion of construction. This has been demonstrated using the results of laboratory tests and simulations of a representative pavement section. Results are supported by observed rutting in inservice pavements. Shear deformations leading to rutting are limited to the upper portion of the HMA layer. Thus a laboratory test which measures, primarily, shear deformations appears to be the most effective way to define the propensity of a mix for rutting. To make this evaluation, use of a specimen representative of the upper 75 to 100 mm (3 to 4 in.) of the HMA layer should be used; this specimen should have the same aggregate structure and anticipated equilibrium airvoid content. ∗
Note that the proposed tests seek to uncouple the deviatoric and volumetric components of the motion not of the stress. † At no point should the validation tests be used for material identification or the selection of any parameter in the model. Rather, failure to predict certain behavior is an indication that the constitutive law needs to be amended to account for additional effects.
311
312
Chapter Ten In the shear test the mean square error used to arrive at the M value in Eq. (105) is 0.602. This value, which is somewhat high, is at least partly attributable to the size of the specimen used relative to the maximum aggregate size and the fact that the representative volume element at elevated temperatures (>40°C) may be larger than the 150mm (6in.) diameter by 50mm (2in.) high cores for nominal 19mm (0.75in.) maximum size aggregate. Largersized specimens will reduce this variability. For example, it is likely, for a mix containing a 19mm (3/4in.) nominal maximum size aggregate, that a specimen 100 mm (4 in.) high by 300 mm (12 in.) long should be used. It has been shown that asphalt mixes are nonlinear in terms of applied stress, particularly at higher temperatures and comparatively slow rates of loading. This nonlinearity negates the possibility of deducing shear response from an axial test unless a configuration for both specimen size and loading of the type noted are used. Similarly it would be difficult to deduce axial response from a test which measures primarily shear response. In addition to the above items pertaining to specimen size and shape to define the propensity of mixes to rutting, information has also been presented on the effects of mode of loading (creep versus dynamic/repeated) and aggregate structure as influenced by method of compaction. For mixes containing modified binders, available data indicate that it is important to use dynamic (repeated) loading rather than creep loading. For mix design purposes it is important that specimens be prepared by rolling wheel compaction to ensure that the aggregate structure is comparable to that obtained in situ. This is especially important for permanent deformation evaluations. Also, the advantage of preparing specimens by this compaction process is that the resulting slabs provide the necessary cores (for permanent deformation) with all cut (sawed) faces. This should reduce variability among specimen. The data presented illustrate the efficacy of the SHRPdeveloped simple shear test, performed in the repeated load, constant height mode, for mix design and performance evaluation. The data suggest that this type of test provides a better evaluation of the permanent deformation response of mixes containing modified binders when compared to conventional tests such as the Hveem Stabilometer. When investigations are conducted on specimens obtained from existing pavements it is important to note the direction of traffic on the resulting specimens. This is necessary to ensure that the orientation of the specimen in the test equipment is the same as its orientation relative to traffic loading (and compaction direction). For testing at one temperature, it is recommended that the critical temperature for permanent deformation be determined by the procedure reported in Deacon et al. (1994). At the present time a shift factor of 0.04 for use in Eq. (107) is suggested. However, this can be modified as experience is gained for specific environment and traffic conditions. A recursive procedure is provided, based on a mechanisticempirical approach, for rutting prediction in asphalt concrete pavements. The approach is based on a timehardening procedure for the accumulation of permanent strain in the asphaltbound layer as a function of both traffic loading and environment. It combines multilayerelastic analysis to determine key stress and strain values to define key parameters for use in determining permanent strain at a depth of 50 mm (2 in.) at the outside edge of the tire as a function of stress, strain, and load repetitions. Using an SHRPdeveloped procedure, the permanent strain, accumulated for a specific loading history, is related to rut depth.
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s The examples presented using the analysis procedure for four different WesTrack sections have provided reasonable comparisons between observed and measured surface rut depths given a range in pavement temperatures for the WesTrack traffic loading. In addition, direct use of RSSTCH data for a mix containing a modified binder in the analytically based procedure provided a reasonable comparison with measured rut depths in a controlled HVS test conducted at high temperature (50°C) conditions.
References Alavi, S., 1992, “Viscoelastic and Permanent Deformation Characteristics of AsphaltAggregate Mixes Tested as Hollow Cylinders and Subjected to Dynamic Axial and Shear Loads,” Ph.D. thesis, University of California, Berkeley, Calif. American Association of State Highway and Transportation Officials (AASHTO), 1994, “Test Method for Determining the Permanent Deformation and Fatigue Cracking Characteristics of Hot Mix Asphalt (HMA) Using the Simple Shear Test (SST) Device, AASHTO TP794,” AASHTO Provisional Standards, pp. 164–186. Bonnot, J., 1986, “Asphalt Aggregate Mixtures,” Transportation Research Record 1096. Transportation Research Board, Washington, D.C., pp. 42–51. Brown, E. R., and S. A. Cross, 1989, “A Study of InPlace Rutting of Asphalt Pavements,” Asphalt Paving Technology, Association of Asphalt Paving Technologists, Vol. 58, pp.1–39. de Beer, M., and C. Fisher, 1997, “Contact Stresses of Pneumatic Tires Measured with the VehicleRoad Surface Pressure Transducer Array (VRSPTA) System for the University of California at Berkeley (UCB) and the Nevada Automotive Test Center (NATC).” Transportek, CSIR, South Africa, Vols. 1 and 2. Deacon, J. A., J. T. Harvey, I. Guada, L. Popescu, and C. L. Monismith, 2002, “An AnalyticallyBased Approach to Rutting Prediction,”—Transportation Research Record No. 1806, Transportation Research Board, Washington, D.C., pp. 9–18 Deacon, J., J. Coplantz, A. Tayebali, and C. Monismith, 1994, “Temperature Considerations in AsphaltAggregate Mixture Analysis and Design,” Transportation Research Record 1454, Transportation Research Board, National Research Council, Washington, D.C., pp. 97–112. Endersby, V. A., and B. A. Vallerga, 1952, “Laboratory Compaction Methods and Their Effects on Mechanical Stability Tests for Asphaltic Pavements,” Proceedings, Association of Asphalt Paving Technologists, Vol. 21, pp. 298–348. Epps, J. A., R. B. Leahy, T. Mitchell, C. Ashmore, S. Seeds, S. Alavi, and C. L. Monismith, 1999, “WesTrack—The Road to PerformanceRelated Specifications,” Proceedings, International Conference on Accelerated Pavement Testing, Reno, Nev., (Available on CDROM from University of Nevada, Reno, Technology Transfer Center). Harvey, J. T., B. A. Vallerga, and C. L. Monismith, 1995, “Mix Design Methodology for a Warrentied Pavement: Case Study,” Transportation Research Record No. 1492, Transportation Research Board, Washington D.C., pp. 184–192. Harvey, J. T., C. Monismith, and J. Sousa, 1994, “A Comparison of Field and LaboratoryCompacted AsphaltRubber, SMA, Recycled and Conventional AsphaltConcrete Mixes Using SHRP A003A Equipment,” Journal of the Association of Asphalt Paving Technologists, Vol. 63, pp. 511–560. Harvey, J. T., et al., 2000, “Effects of Material Properties, Specimen Geometry, and Specimen Preparation Variables on Asphalt Concrete Tests for Rutting,” Journal of the Association of Asphalt Paving Technologists, Vol. 69, pp. 236–280.
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Chapter Ten Harvey, J. T. et al., 2001, “Laboratory Shear Tests for Rutting of Caltrans Asphalt Concrete and AsphaltRubber Hot Mix and Comparison with HVS Results,” Report to the California Department of Transportation Studies, University of California, Berkeley, Calif. Harvey, J. T., I. Guada, and F. Long, 1999, “Effect of Material Properties, Specimen Geometry, and Specimen Preparation Variables on Asphalt Concrete Tests for Rutting,” Pavement Research Center, University of California, Berkeley, Report to Office of Technology Applications, FHWA, Pavement Research Center, University of California, Berkeley, Calif., pp. 83. Harvey, J. T., J. Roesler, N. F. Coetzee, and C. L. Monismith, 2000, “Caltrans Accelerated Pavement Test Program—Summary Report Six Year Period: 1994–2000,” Report to the California Department of Transportation, Pavement Research Center, University of California, Berkeley, Calif., pp. 112. Hashin, Z., 1983, “Analysis of Composite Materials—A Survey,” Journal of Applied Mechanics, Vol. 50, pp. 481–505. Lizenga, J., 1997, “On the Prediction of Pavement Retting in the Shell Pavement Design Method,” 2nd European Symposium on Performance of Bituminous Materials, Leeds. Long, F., S. Govindjee, and C. L. Monismith, 2002, “Permanent Deformation of Asphalt Concrete Pavements: Development of a Nonlinear Viscoelastic Model for Mix Design and Analyses,” Proceedings, Ninth International Conference on Asphalt Pavements, Copenhagen, Section 1:64, pp. 14. Monismith, C. L., and B.A. Vallerga, 1956, “Relationship between Density and Stability of Asphaltic Paving Mixtures,” Proceedings, Association of Asphalt Paving Technologists, Vol. 25, pp. 88–108. Monismith, C. L., J. T. Harvey, I. Guada, F. Long, and B. A. Vallerga, 1999, “Asphalt Mix Studies, San Francisco International Airport,” Report to B. A. Vallerga, Inc., Pavement Research Center, University of California, Berkeley, pp. 40. (plus appendices). Monismith, C. L., F. Long, and J. T. Harvey, 2001, “California’s Interstate710 Rehabilitation: Mix and Structural Section Designs, Construction Specifications,” Journal of the Association of Asphalt Paving Technologists, Vol. 70, pp. 762–799. Monismith, C. L., J. A. Deacon, and J. T. Harvey, 2000, “WesTrack: Performance Models for Permanent Deformations and Fatigue, Pavement Research Center,” Report to Nichols Consulting Engineers, Chtd., University of California, Berkeley, Calif. p. 373. Pavement Research Center (PRC), 1999, “Revision to AASHTO TP7 (TP799).” University of California, Berkeley, Submitted to AASHTO for consideration. Shell International Petroleum Company, Limited, 1978, “Shell Pavement Design Manual, and, 1985,” Addendum to the Shell Pavement Design Manual, Shell International Petroleum Company, Limited, London. Shook, J. F., F., N. Finn, M. W. Witczak, and C. L. Monismith, 1982, “Thickness Design of Asphalt Pavements, The Asphalt Institute Method,” Proceedings, Fifth International Conference on the Structural Design of Asphalt Pavement, University of Michigan and Delft University of Technology, Vol. 2, pp. 17–44. Pavement Research Center, 1999, “Mix Design and Analysis and Structural Section Design for Full Depth Pavement for Interstate Route 710.” TMUCBPRC992, Pavement Research Center, Richmond, Calif. Sousa, J. B., J. A. Deacon, S. L. Weissman, R. B. Leahy, J. T. Harvey, G. Paulsen, J. S. Coplantz, and C. L. Monismith, 1994, “Permanent Deformation Response of Asphalt
R u t t i n g C h a r a c t e r i z a t i o n o f A s p h a l t C o n c r e t e U s i n g S i m p l e S h e a r Te s t s Aggregate Mixes.” Report SHRPA415, Strategic Highway Research Program, National Research Council, Washington D.C., p. 437. Sousa, J. B., J. A. Deacon, and C. L. Monismith, 1991, “Effect of Laboratory Compaction Method on the Permanent Deformation Characteristics of AsphaltAggregate Mixes,” Journal of the Association of the Asphalt Paving Technologists, Vol. 60, pp. 533–585. Tanco, A. J., 1992, “Permanent Deformation Response of Conventional and Modified AsphaltAggregate Mixes under Simple and Compound Shear Loading Conditions,” Ph.D. thesis, University of California, Berkeley, p. 273. Tayebali, A. A., 1990, “Influence of Rheological Properties of Modified Asphalt Binders on the LoadDeformation Characteristics of the BinderAggregate Mixtures,” Ph.D. thesis, University of California, Berkeley, pp. 420. Vallerga, B. A., 1951, “Recent Laboratory Compaction Studies of Bituminous Paving Mexitures,” Proceedings, Association of Asphalt Paving Technologists, Vol. 20, pp. 117–153. Vallerga, B. A., A. A. Tayebali, S. L. Weissman, and C. L. Monismith, 1996, “Mechanical Properties Characterization of Asphalt Concrete Barrier for Radioactive Nuclear Waste Vaults,” Materials for the New Millennium, ASCE, Washington D.C., pp 1288–1297. van Grevenynghe, M. P., 1986, “Influence des modes de préparation et du prélèvement des éprovettes sur les caractéristiques structurelles et méchaniques des enrobés.” Internal RILEM Report, Laboratoire des Ponts et Chausses, Nantes, France. Valkering, C. P., D. J. L. Lancon, E. de Hiltster, and D. A. Stoker, 1990, “Rutting Resistance of Asphalt Mixes Containing NonConventional and PolymerModified Binders,” Journal of the Association of Asphalt Paving Technologists, Vol. 59, pp. 590–609. Weissman, S. L., 1997, The Mechanics of Permanent Deformation in AsphaltAggregate Mixtures: A Guide to Laboratory Test Selection, Symplectic Engineering Corp., Berkeley, Calif., p. 55. Weissman, S. L., and J. L. Sackman, 2000, “A Viscoplastic Constitutive Law for Asphalt Concrete Mixes at Elevated Temperatures; a Finite Deformation Formulation,” Symplectic Engineering, Berkeley, Calif., p. 61. Weissman S. L., J. T. Harvey, J. L. Sackman, and F. Long, 1999, “Selection of Laboratory Test Specimen Dimension for Permanent Deformation of Asphalt Concrete Pavements,” Transportation Research Record 1681, Transportation Research Boards, Washington, D.C., pp. 113–120. Weissman S. L., and J. L. Sackman, 2003, “A Finite Strain Constitutive Law for Asphalt Concrete Mixtures at Elevated Temperatures Based on the Multiplicative Decomposition of the Deformation Gradient,” Symplectic Engineering, Berkeley, Calif., p. 89.
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CHAPTER
11
Permanent Deformation Assessment for Asphalt Concrete Pavement and Mixture Design Charles W. Schwartz and Kamil E. Kaloush
Abstract This chapter addresses three current areas of research into rutting of asphalt mixtures: (a) a review of mechanisticempirical modeling approaches, and in particular the model adopted for the proposed new national pavement design methodology; (b) an overview of advanced constitutive modeling approaches to the rutting problem, with particular emphasis on viscoplasticity and continuum damage; and (c) a description of recent work toward developing a simple performance test to identify the rutting potential of mixtures during design based on measurement of fundamental engineering response and properties.
Introduction Permanent deformations are often the controlling loadassociated distress governing the performance of hot mix asphalt (HMA) concrete pavements. They typically manifest as rutting that appears as longitudinal depressions in the wheel paths accompanied by small upheavals to the sides. The width and depth of the rutting is highly dependent upon the pavement structure (layer thicknesses and material properties), traffic volume and distribution, and site environment conditions. Rutting develops gradually over the life of a welldesigned pavement, of the pavement as incremental, volumetric, and shear permanent strains accumulate. The incremental permanent strains will be functions of the stiffness, permanent deformation resistance, and induced stresses in the asphalt layer. In a poorly designed pavement or mixture, excessive rutting may develop very quickly as a consequence of a shear failure within the asphalt.
317 Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.
318
Chapter Eleven In order to assess the permanent deformation resistance of an asphalt mixture, material response parameters obtained from laboratory testing are used to evaluate and predict the mixture performance. These permanent deformation parameters will be dependent on the temperature, loading rate, stress state, aggregate and binder characteristics, mixture volumetrics, and other mixture variables. Often, the objective of the laboratory testing will be a simple screening to identify mixtures having a high potential for premature failure. Permanent deformations have been the focus of much research in pavement engineering over the past 50 years. Most of the available permanent deformation models in the literature are empirical or semimechanistic with limited fundamental material characterization other than through elastic or quasielastic properties. Poor correlations with actual field performance are the common result. Some of the empirical models lack robustness because they are derived from limited sets of materials and environmental conditions and thus are not transferable to other conditions. This chapter addresses three current areas of research into rutting of asphalt mixtures. Specifically, the chapter includes (a) a review of mechanisticempirical modeling approaches, and in particular the model adopted for the performance prediction and design methodology developed in National Cooperative Highway Research Program (NCHRP) Project 137A; (b) an overview of advanced constitutive modeling approaches to the rutting problem, with particular emphasis on viscoplasticity and continuum damage; and (c) a description of recent work toward developing a simple performance test to identify the rutting potential of mixtures during design based on measurement of fundamental engineering response and properties. Assessments of likely nearterm advances in the current state of the art are also included.
MechanisticEmpirical Rutting Models Mechanisticempirical approaches to rutting prediction couple mechanistic computations of pavement stresses and strains with empirical predictions of the consequent rutting. The empirical rutting prediction models, which sometimes also include other parameters related to mixture characteristics and/or site environmental conditions, must be calibrated against observed field performance data. In most cases, the available field performance data are quite limited.
Subgrade Rutting Models It is interesting to remember that the earliest mechanisticempirical rutting models explicitly considered only the strains in the subgrade (e.g., the Shell method described in Claessen et al. 1977; the Asphalt Institute method described in Shook et al. 1982), Chen et al. (1994), and Pidwerbesky et al. (1997) provide concise summaries of the evolution of early models for predicting the number of cycles Nd to permanent deformation failure as a function of the vertical compressive strain ec at the top of the subgrade: Nd = f4ec−f
5
(111)
where f4 and f5 are model calibration parameters. Values recommended by various agencies for f4 and f5 are summarized in Table 111. It should be recognized that the implicit permanent deformation limits underlying the values in Table 111 are different for each approach. The Asphalt Institute values, for example, are based on a total rut depth at the pavement surface of 0.5 in. while the TRRL procedure is based on a total
Permanent Deformation Assessment
Agency
f4
f5 –9
Asphalt Institute (1982)
1.365 × 10
4.477
50% reliability
6.15 × 10–7
4.0
85% reliability
–7
1.94 × 10
4.0
95% reliability
1.05 × 10–7
4.0
6.18 × 10–8
3.95
–9
4.35
Shell (1978)
TRRL (Powell et al. 1984) 85% reliability Belgian (Verstraeten et al. 1982)
3.05 × 10
Source: Chen et al. 1994, ASCE.
TABLE 111
Rutting Parameters f4 and f5 for Various Agencies
rut depth of 0.4 in. at a reliability of 85%. The implicit design limit for subgrade vertical compressive strain is on the order of 0.001 (1000 microstrains). Timm and Newcomb (2003) adapted a model of the form of Eq. (111) for predicting asphalt rutting at the MnRoad project. In their approach, ec is taken as the horizontal tensile strain at the bottom of the asphalt layer. They found average values of 7.0 × 10−15 and 3.909 for the calibration coefficients f4 and f5.
Permanent Strain Models The mechanisticempirical models in this category relate the permanent vertical compressive strain ep at the midthickness of an asphalt sublayer to the number of load cycles N, temperature T, induced stress level, and other parameters. One of the earliest permanent strain models was that implemented in the VESYS program by Kenis and colleagues (Kenis 1977; Kenis et al. 1982; Kenis 1988; Kenis and Wang 1997): Δep(N) = emNa
(112)
where Δep(N) is the incremental permanent strain caused by the Nth load cycle, e is the mechanistically computed peak total strain, and m and a are material properties determined from laboratory repeated load permanent deformation tests. The values for m and a are functions of mixture type, temperature, and stress state. Rauhut (1980) discusses some of the quantitative influences of these factors on m and a. Qi and Witczak (1998) provide additional insight into the influence of load and rest time on permanent deformations characterized using a model form similar to Eq. (112). For a fixed cyclic load magnitude, the peak total strain e can be assumed to be approximately constant; Eq. (112) can then be integrated to determine the cumulative plastic strain ep(N) after N load cycles: ⎛ μ ⎞ ( 1−α ) ε p (N ) = ∫ εμ N −α dN = ε ⎜ N ⎝ 1 − α ⎟⎠ 0 N
(113)
319
320
Chapter Eleven Because of the variation of stresses, strains, and temperatures through the asphalt thickness, the asphalt layer is usually divided into sublayers and Eq. (113) applied to each sublayer individually. The overall rut depth for the layer for a given season is then computed as the sum of the permanent deformations for the individual sublayers: n
PDj = ∑ ε pi hi
(114)
i=1
where PDj = pavement permanent deformation for season j n = number of sublayers epi = permanent strain in sublayer i hi = thickness of sublayer i These calculations are repeated for each load level and season over the analysis period. Some type of timehardening scheme is commonly employed when accumulating permanent deformations over multiple load levels and/or seasons. The usual approach is illustrated in Fig. 111 for a permanent deformation model of the form: ep(N) = f(e, T, N)
(115)
where e is the mechanistically computed strain (usually the resilient elastic strain) for a given load magnitude and T is temperature. At point A in Fig. 111, the total permanent strain ep, i–1 at time ti–1 corresponds to a total number of traffic repetitions Nti–1. Over the next time interval i characterized by layer temperature T1 and resilient elastic strain ei for a given load level, there is an equivalent number of traffic repetitions Ntequivi that is associated with the total deformation at time ti−1 but corresponding to the temperature and load conditions T1 and ei prevailing over the new time interval (point B in Fig. 111). By adding the incremental traffic repetitions Ni for time interval i to the total equivalent T1, εi
εp Ni
T2, εi
εp.i1
T3, εi
C
εp.i
A
B
Ntequivi
Nti
Nti1
T4, εi
N
FIGURE 111 Timehardening scheme for accumulating permanent deformations over multiple seasons. (ElBasyouny 2004.)
Permanent Deformation Assessment number of repetitions Ntequivi at the start of the interval, the total permanent strain at the end of time interval i can be estimated (point C in Fig. 111). The permanent strain model developed by the Asphalt Institute includes the effect of mixture variables on rutting (May and Witczak 1992): logε p = −14.97 + 0.408log(N ) + 6.865log(T ) + 1.107 log(σ d ) −0.117 log(η) + 1.908logVbeff + 0.971(Va ) where
(116)
T = temperature, °F sd = mechanistically determined deviator stress (psi) in the asphalt layer h = the binder viscosity at 70°F, 106 poise Vbeff = effective asphalt volume, % Va = volume of air voids, %
The other terms are as defined previously. Baladi used regional and Long Term Pavement Performance (LTPP) test sections and the MICHPAVE program to develop the following mechanisticempirical equation for predicting rut depth (Baladi 1989): logRD = −1.6 + 067Va − 1.4logH HMA + 0.07Tavg − 0.000434ηKV + 0.15logN ESAL − 0.4logMR(Soil)
(117)
− 0.50logMR(Base) + 0.1logδ 0 + 0.01logε c(HMA) − 0.7 logH Base,Equiv + 0.09log(50 − H HMA − H Base,Equiv ) where
RD = rut depth, inches NESAL = number of 80kN (18kip) ESALs Tavg = average annual temperature, °F HHMA = thickness of the HMA layer, inches HBase, Equiv = equivalent thickness of base material, inches MR(Base) = resilient modulus of the base material, psi d0 = surface deflection, inches Va = air voids in the mix, % MR(Soil) = resilient modulus of the subgrade, psi ec(HMA) = compressive strain at the bottom of the HMA layer hKV = kinematic viscosity of binder at 135°C (275°F), centistokes
In this model, the two mechanistically computed parameters are the surface deflection d0 and the compressive strain ec(HMA) at the bottom of the HMA layer. Deacon et al. (2002) used data from the Westrack field experiment to formulate a mechanisticempirical model for asphalt rutting based on permanent shear strain: gp = aebtgeNc
(118)
321
322
Chapter Eleven where gp is the permanent shear strain at a depth of 2 in. (50 mm) below the surface of the asphalt layer and t and ge are the mechanistically determined elastic shear stress and strain at the same location. The parameters a, b, and c are material properties determined from repeated load permanent deformation tests in shear using the Superpave shear tester (SST); these properties will be functions of mixture characteristics, temperature, and stress level. A timehardening principle is used to estimate the accumulation of permanent strains in the asphalt under varying site conditions: gp,1 = a1[ΔN1]c
γ p ,t
1 ⎡ ⎤ c γ ⎛ ⎞ p , t −1 = at ⎢ + ΔN t ⎥ ⎢⎣ ⎜⎝ at ⎟⎠ ⎥⎦
(119) c
(1110)
where at = aebtge,t, ge,t and gp,t are the elastic and permanent shear strains for the tth hour of loading and DNt is the number of load applications during the tth hour. The mechanistically computed elastic shear strain ge,t varies over time in response to traffic variations and the influence of temperature on the asphalt stiffness. The total rutting in the HMA layer RDHMA expressed in inches is then estimated from the permanent shear strain using the following relation: RDHMA = Krgp,t
(1111)
where Kr is a coefficient relating rut depth to permanent strain and Kr is a function of the thickness of the asphalt layer; values determined from finite element analyses of representative pavement structures range from about 5.5 for a 6in. layer to 10 for a 12in.thick layer.
Permanent to Resilient Strain Ratio Models The two most significant factors governing permanent strain accumulation for a given asphalt mixture are temperature and stress level. Although some of the empirical permanent strain models explicitly consider temperature and/or stress level [e.g., Eq. (116)], most incorporate these effects implicitly through the material parameter terms [e.g., Eqs. (112) and (118)]. For this latter category, full material characterization requires a factorial of tests spanning a range of temperatures and stress levels. The rationale for the permanent to resilient strain ratio models in essence is to consolidate some of the influences of temperature and stress level. Both of these parameters influence the resilient elastic strains as well as the permanent strains. Normalizing the permanent strains with the elastic strains should therefore capture most of the temperature and stress effects. This concept is the basis for the asphalt rutting model implemented in the NCHRP Project 137A mechanisticempirical design methodology (NCHRP 2004). The model has its origins in an extensive laboratory study by Leahy (1989) of the repeated load permanent deformation response of over 250 asphalt concrete specimens encompassing
Permanent Deformation Assessment two aggregate types, two binder types, three binder contents, three stress levels, and three temperatures: ⎛ εp ⎞ log ⎜ ⎟ = −6.631 + 0.435 log N + 2.767 log T + 0.110 log σ d ⎝ εr ⎠ + 0.118 log η + 0.930 log Vbeff + 0.501 log Va
( R 2 = 0.76 )
(1112)
where er is the resilient (elastic) strain and the other terms are as defined for the Asphalt Institute model in Eq. (116). Sensitivity studies found that temperature is by far the most important parameter in Eq. (1112); the model is much less sensitive to the stress magnitude, material types, and other mix parameters. Ayres (1997) reanalyzed Leahy’s original data plus additional laboratory test data and recommended a model of the form: ⎛ εp ⎞ log ⎜ ⎟ = −4.8066 + 0.4296 log N + 2.5816 log T ⎝ εr ⎠
(R 2 = 0.72)
(1113)
The slightly lower R2 for this model is the consequence of removing the four mixrelated parameters from the Leahy model in Eq. (1112). The drop in R2 from 0.76 to 0.725 confirms the relatively small importance of these parameters as compared to temperature and number of load repetitions. In particular, the small drop in R2 after removing the deviatoric stress level term sd confirms that most of the stress influence on permanent deformation behavior is captured by the resilient strain normalizing term in the strain ratio approach. Kaloush (2001) improved the robustness of the rutting model even further by combining Leahy’s original data with the very large number of repeated load permanent deformation test results from NCHRP Project 919, yielding a revised model of the form: ⎛ εp ⎞ log ⎜ ⎟ = −3.1555 + 0.3994 log N + 1.7340 log T ⎝ εr ⎠
(R 2 = 0.64)
(1114)
The lower R2 for this model as compared to Ayres’ results is attributable to the much broader and more diverse data set analyzed by Kaloush. Equation (111) is the basis for the asphalt rutting model implemented in the NCHRP 137A design methodology (NCHRP 2004; ElBasyouny 2004). The final model form including field calibration coefficients is expressed as ⎛ εp ⎞ log ⎜ ⎟ = Bσ 3 [a1 β r 1 + a2 β r 2 log(T ) + a3 β r 3 log(N )] ⎝ εr ⎠
(1115)
where er = resilient elastic strain calculated at the middepth of an HMA sublayer at temperature T N = number of axle loads over time interval for a specific axle type T = temperature of the HMA at middepth, °F
323
Chapter Eleven Bs3 = adjustment factor for lateral confinement ai = nonlinear regression coefficients bir = regional calibration factors The database underlying empirical Eqs. (1112) through (1115) is all based on unconfined conditions. The actual horizontal stresses in an asphalt layer are different, however, ranging from compression at the top to tension at the bottom. The Bs3 term in Eq. (1115) is an adjustment based on trench data from the MnRoad test sections to account for these variations in confining stresses with depth in the field: Bs3 = (C1 + C2z) × 0.3282z
(1116)
with C1 = −0.1039H2HMA + 2.4868HHMA−17.342 C2 = 0.0172H2HMA + 1.7331HHMA+ 27.428
(1117)
where HHMA is thickness of the HMA layer and z is depth within the HMA layer. The ai values in Eq. (1115) were determined via a global calibration using data from 387 field sections in the LTPP database; the final calibrated values were a1 = −3.4488, a2 = 1.5606, and a3 = 0.4791 for bri = 1. These are assumed to be mixture independent in the NCHRP 137A modeling approach. A comparison of predicted versus measured HMA rutting from the calibrated NCHRP 137A models is given in Fig. 112. Note that the actual measured rutting in the LTPP sections is the total surface rutting; estimation of the portion of this total surface rutting attributable to the HMA rutting requires some additional assumptions. The R2 value of 0.648 is thus considered quite respectable, especially given the diversity of field conditions included in the calibration data set.
1 R2 = 0.648 N = 387 Se = 0.063 SSe = 1.883
0.9 Predicted HMA Rutting (in)
324
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Average Estimated Measured HMA Rutting (in) Predicted vs Ave. estimated measured HMA rutting
Equality line
FIGURE 112 Nationally calibrated predicted versus estimated measured asphalt rutting for the NCHRP 137A HMA rutting model. (NCHRP 2004).
Permanent Deformation Assessment As already noted, strain ratio models of the form of Eq. (1115) imply that the resilient strain normalizing term captures all stress level/stress state influences on permanent deformation. There is some evidence that this assumption is not always appropriate. Uzan (2004) has recently implemented into the Israeli mechanisticempirical pavement design procedure the following stress and temperaturedependent strain ratio model for HMA permanent deformation:
εp ⎡ ⎛ N⎞⎤ ⎛N⎞ log = a0 (θ ) + a1 (θ )log ⎜ ⎟ + a2 (θ ) ⎢ log ⎜ ⎟ ⎥ εr ⎝ aT ⎠ ⎥⎦ ⎝ aT ⎠ ⎢⎣ log aT = −
2
c1 [T − T0 − f ( p)] c20 + c21σ 3 + T − T0 − f ( p)
f (p) = c30 + c31p
(1118)
(1119) (1120)
where the ai terms are linear functions of bulk stress q, cij are material constants, s3 is the horizontal stress in the triaxial repeated load permanent deformation test, and p is the experimental pressure. Uzan notes that the temperature shift function aT for permanent deformations is not the same as the shift function for dynamic modulus.
Regression Models The models in this category are similar to the permanent strain and strain ratio models in that they usually have some mechanistic content such as a computed strain or deflection level. However, many other terms are also included to account for mixture characteristics, environmental variables, and other factors. These other terms are only indirectly related to the mechanistic response of the pavement. Perhaps the most widely known of the regression approaches are the Highway Development and Management ModelIII (HDMIII) rutting performance models developed by the World Bank (Paterson 1987; Kannemeyer and Visser 1995). In conceptual terms, the models predict the incremental changes in the mean and standard deviation of the rut depth ΔmRD and ΔsRD:
where
ΔmRD = Kr × f(t, ESALs, SN, compaction, deflection, precipitation)
(1121)
ΔsRD = Kr × f(mRD, SN, ESALs, compaction)
(1122)
t = time ESALs = number of standard axle loads SN = structural number Kr = local calibration factor
The mechanistic input to the HDMIII model is the computed deflection in Eq. (1121) for the incremental change in the mean rut depth. Since the emphasis of the HDMIII models is on prediction of roughness, however, the mean rut depth model is not used directly but only as an input for predicting rut depth standard deviation, which contributes directly to pavement roughness.
325
326
Chapter Eleven
Summary As described in this section, a wide variety of mechanisticempirical models have been proposed for predicting rutting in HMA layers. No systematic comparisons of the predictions from these various models for a common set of conditions have been performed to date. Most of the models have been calibrated against only very limited sets of laboratory and/or field data. The NCHRP 137A model is arguably the leading mechanisticempirical model for HMA rutting at present, both because it builds upon many of the earlier efforts and because of its robust national calibration against nearly 400 field test sections from the LTPP database. The NCHRP 137A model also includes an explicit option for regional calibration to improve the accuracy of its predictions. Enhancement of the NCHRP 137A rutting model will be the focus of the forthcoming NCHRP Project 930A.
Advanced Constitutive Models for Rutting Mechanisticempirical methods for pavement performance prediction are a step forward from current purely empirical procedures because they tackle at least part of the problem— that is, the pavement stress and strain response due to traffic loads and environmental conditions—using sound, mechanicsbased techniques. The overall accuracy and robustness of the mechanisticempirical approach nonetheless still rely heavily upon the quantity and quality of the empirical data used for calibrating the empirical distress model component. The next step beyond mechanisticempirical approaches is fully mechanistic distress prediction. This requires much more sophisticated constitutive models for asphalt concrete behavior that capture not just the stiffness of the material— for example, as required for the multilayer elastic response analyses in most mechanisticempirical approaches—but also its degradation response (e.g., the consequent permanent deformations, cracking, and other distresses). Arguments for such an approach have been presented in more detail in previous chapters. Correct computation of permanent deformations and rutting in asphalt concrete generally requires a viscoplastic constitutive model. Linear elastic models by definition are incapable of producing permanent deformations. Conventional plasticity models can effectively simulate the first cycle plastic strains for uniform stress repeated loading but are usually incapable of predicting the additional incremental plastic strains from the subsequent load cycles. Even with strain hardening, the yield surface will fully expand during the first load cycles; subsequent load cycles at the same load magnitude will just touch this expanded yield surface but cause no additional plastic flow. Viscoelastic models have also been attempted (e.g., Collop et al. 1995; Hopman et al. 1997; Long and Monismith 2002; Collop et al. 2003), but these require (either explicitly or implicitly) a Maxwelltype model with a final dashpot in series in order to produce permanent deformations. Although this type of viscoelastic model is appropriate for viscoelastic liquids (e.g., asphalt binder), it does not realistically capture the behavior of viscoelastic solids (e.g., asphalt concrete mixtures) under a general state of triaxial stress (e.g., confined compression). The major components of an elasticviscoplastic model include a flow surface that defines the instantaneous boundary (usually in terms of stress state) between viscoplastic flow and no flow conditions; a flow rule that defines the viscoplastic strain rates for stress states outside the instantaneous flow surface; and a hardening (or softening) rule
Permanent Deformation Assessment that describes the evolution of the flow surface as a consequence of the material response. Note that for asphalt concrete the initial flow surface is usually quite small— for example, viscoplastic flow begins even at very small stress levels. Gibson et al. (2003a, 2003b), Gibson (2006) and Schwartz et al. (2004) describe one approach toward viscoplastic modeling of asphalt concrete in compression in combination with a Schaperytype viscoelastic continuum damage model; Chehab et al. (2003) provide a companion study under uniaxial tension. Other researchers who have applied the Schapery model to various aspects of asphalt concrete behavior over recent years include Park et al. (1996), Kim et al. (1997), Lee and Kim (1998a, 1998b), Daniel et al. (2002), Chehab et al. (2004), and Lundstrom and Isacsson (2004). The starting point for the extended Schapery formulation is a standard partitioning of total strain et into viscoelastic eve and viscoplastic evp components: et = eve + evp
(1123)
Although both eve and evp can include contributions from microstructural damage, the Schapery model incorporates damage effects in the viscoelastic strains only. The specific model formulations for the linear viscoelastic, viscoplastic, and damage components are described in the following subsections.
Linear Viscoelasticity The linear viscoelastic properties of the asphalt concrete are described by a relaxation modulus mastercurve as determined from small strain dynamic modulus tests in compression at multiple frequencies and temperatures as discussed in detail in Chap. 4 of this book. Figure 113 summarizes the measured storage modulus mastercurve for a 12.5mm dense graded Superpave mixture having limestone aggregate and an
FIGURE 113 Storage modulus mastercurve from small strain dynamic modulus tests in compression on a 12.5mm dense graded Superpave mixture.
327
328
Chapter Eleven unmodified PG 6422 binder (Superpave Models Team 1999). Interconversion techniques described in Chap. 6 are used to convert this storage modulus mastercurve to the relaxation modulus mastercurve. Timetemperature superposition concepts are employed to combine actual time and temperature into a single value of reduced time at both small and large strain levels (Schwartz et al. 2002). As presented in Chap. 6, the relaxation modulus mastercurve is represented by a Prony series corresponding to a generalized melement Maxwell viscoelastic model (without a final dashpot in series): m
E(t) = Eo + ∑ Ei e
−t ρi
i=1
(1124)
where Ei and ri = spring constant and dashpot relaxation time for each of the parallel Maxwell elements E0 = longterm equilibrium modulus t = time Table 112 summarizes the relaxation modulus Prony series terms corresponding to the storage modulus data in Fig. 113.
Viscoplasticity Viscoplastic strains are predicted using a Perzynatype flow rule for the viscoplastic strain rates (Perzyna 1966): dε ijvp dt
= Γ f (G)
∂F ∂σ ij
Ei (MPa)
ri (s)
E0 = 412.8
—
1
1.430E+01
1.500E+07
2
3.210E+01
8.005E+05
3
7.420E+01
4.272E+04
4
1.796E+02
2.280E+03
5
4.588E+02
1.217E+02
6
1.232E+03
6.493E+00
7
2.956E+03
3.465E−01
8
5.286E+03
1.849E−02
9
6.531E+03
9.869E−04
10
5.727E+03
5.267E−05
11
3.848E+03
2.811E−06
12
2.160E+03
1.500E−07
i
TABLE 112
Prony Series Terms for Relaxation Modulus of a 12.5mm Dense Graded Superpave Mixture
(1125)
Permanent Deformation Assessment In Eq. (1125), Γ is a fluidity parameter that is conceptually similar to a viscosity term. The function 〈 f(G)〉 is an overstress function that governs the magnitude of viscoplastic flow; G is termed the flow surface G(sij) = 0 in multidimensional stress space, and the 〈 〉 brackets denote a value of zero if f(G) ≤ 0 and a value equal to f(G) when f(G) > 0. The term F(sij) = 0 is a potential surface in multidimensional stress space; the (∂F)/(∂sij) gradient term requires that the viscoplastic strain increments are normal to the potential surface. Generalized viscoplasticity models make a distinction between the flow surface G(sij) = 0 and the potential surface F(sij) = 0. For simplicity, the same surface formulation is often used for both G and F, resulting in what is known as associated flow viscoplasticity. Simply stated, Eq. (1125) specifies that viscoplastic strains develop only when the applied stress state lies outside the flow surface, and the magnitude of the strain rate is proportional to how far the stress state is outside the flow surface. Materials such as metals may be realistically represented by flow surfaces that depend only on shear stresses, for example, VonMises theory. Asphaltic materials are more akin to granular geomaterials, which gain strength from confining pressure and exhibit dilation. Realistic modeling of these behavior aspects requires flow surfaces that depend on both shear and confining stresses, for example, DruckerPrager or generalized MohrCoulomb theories. Erkens (2002), Erkens et al. (2003) provide a more indepth discussion of flow surfaces versus observed behavior. Strain hardening (and softening) materials like asphalt concrete can be simulated with these theories by changing the size and shape of the flow surface as a function of internal state variables that track deformation history. The largest variety in the application of the viscoplasticity or rate dependent plasticity lies in various researchers’ implementation of flow surfaces, flow rules, and hardening. Examples can be found in work by Erkens (2002), Erkens et al. (2003), Huang et al. (2002), Levenberg and Uzan (2004), Tashman et al. (2003, 2004), and Park et al. (2004). The hierarchical single surface (HiSS) model (Desai and Zhang 1987) was adopted as the flow surface for the present formulation work. Employing the extended timetemperature superposition concepts described previously, Eq. (1125) can be rewritten in terms of the reduced viscoplastic strain rate for associated flow conditions (i.e., F = G) as dε ijvp dtR
= Γ f (F )
∂F ∂σ ij
(1126)
where the reduced time tR is given as tR =
t a(T )
(1127)
and where a(T) is the temperature shift function as determined from either large strain multirate and multitemperature constant strain rate tests or, more simply, from small strain dynamic modulus tests (Schwartz et al. 2002; Chehab et al. 2002). The HiSS flow surface F can be expressed compactly in terms of stress invariants as F = 0 = J2D − [g (I1 + R(x))2 − a(x)(I1 + R(x))n]
(1128)
329
330
Chapter Eleven in which • J2D and I1 are the usual shear and bulk stress invariants. • g and n are fixed parameters that govern the size and shape of the growing flow surface. • x = e1vp + e2vp + e3vp is the accumulated viscoplastic strain trajectory, which is a very simple quantification of deformation used as the hardening internal state variable. • R(x) and a(x) are parameters that govern the size and nature of the capped surface. They are adjusted as hardening accumulates, thereby reducing the potential for viscoplastic flow. These parameters are assumed to follow simple power law or exponential relationships R(x) = R0 + RA x k2 and a(x) = a0e xk1. A small initial undisturbed surface must be assumed as a starting point in this formulation. Through trial and error the optimal functional form for f(F) in Eq. (1126) was determined as ⎛ F ⎞ f (F ) = A ⎜ − 1⎟ ⎝ F0′ ⎠
N
(1129)
where F is taken as the distance in principal stress space from the applied stress to the hydrostatic axis in the direction normal to the current flow surface, and F′0 is taken as the portion of that distance from the hydrostatic axis to the current flow surface. Both are determined at the beginning of the current time step. The terms A and N are material parameters to be determined from calibration testing. As can be seen from the schematic shown in Fig. 114, the potential for viscoplastic flow diminishes asymptotically as the flow surface expands due to hardening. This is precisely what is observed in experiments such as constant creep loading.
Principal stress, Pi (P1, P2, P3) ∂Fi+1 ∂sj
Flow surface@ ti+1
F ∂Fi ∂sj
Flow surface@ ti
F′o Hydrostatic
FIGURE 114 Schematic of ﬂow surface hardening.
Permanent Deformation Assessment
Cycle
Deviator Stress (kPa)
1
19
2
35
3
62
4
106
5
179
6
303
7
495
8
818
9
1354
Confining Stress (kPa)
0, 250, 500
Temperature (°C)
35
Creep Time (sec)
10
Rest Time (sec)
100
TABLE 113 Cyclic Creep and Recovery Tests for Viscoplastic Calibration
The model was calibrated using cyclic creep and recovery tests in which the deviator stress magnitude increased from one cycle to the next. Table 113 summarizes the deviator and confining stresses and loading histories for these tests, all of which were conducted at a temperature of 35oC where significant viscoplasticity is expected. Three replicates were tested for each condition. The nonrecoverable viscoplastic strains were directly measured at the end of each recovery period. Using nonlinear optimization techniques and a set of initial estimates for the model parameters in Eqs. (1128) and (1129), the model parameters were iteratively adjusted until the summed squared errors between measured and predicted axial and radial strains were minimized. During calibration, it was determined that A parameter in Eq. (1129) needed to be modified to enhance the confinementinduced suppression of viscoplasticity. The A parameter was therefore redefined as the following function: ⎛ θ ⎞ A=⎜ ⎝ θ REF ⎟⎠
k3
(1130)
The q angles can be interpreted as the inclination of the current stress vector in I1versus J 2 D space; the reference angle qREF corresponds to a uniaxial stress path angle and has a value of 0.528 radians. The goodnessoffit statistics for the predicted versus measured strains from the calibration tests were quite good, with R2 = 0.97, 0.93, and 0.96 for the axial viscoplastic strains at 0, 250. and 500 kPa confined tests respectively and R2 = 0.97, 0.98, and 0.82 for the corresponding radial viscoplastic strains. The calibrated model parameters are summarized below in Table 114. The multidimensional viscoplasticity model was validated against confined cyclic creep and recovery tests conducted at 35°C and 250 kPa confinement. These test conditions are different from those used for the model calibration. Accumulated viscoplastic strains were measured at the end of each recovery period. As shown in
331
332
Chapter Eleven
Parameter
G −7.5190
g
n
N
a0
Value
10
0.039525
2.25982
2.5533
0.0055485
Parameter
k1
R0
RA
K2
k3
Value
−38.5093
23.0031
3756.6
0.54361
4.7736
TABLE 114
Calibrated Viscoplastic Model Parameters
FIGURE 115 Predicted and measured accumulated viscoplastic strains from cyclic creep and recovery validation tests (250 kPa conﬁnement, 35 ο C test temperature).
Fig. 115, the predicted accumulated viscoplastic strains agree extremely well with the measured values in both the axial and radial directions.
Continuum Damage Nonlinear effects due to material damage from microcracks are captured using the continuum damage mechanics approach developed by Ha and Schapery (1998). The basic concepts of the Schapery continuum damage model are also described in Chap. 7 of this volume. First an elasticviscoelastic correspondence principle is used to compute a pseudostrain via a convolution integral of the relaxation modulus: tR
εR =
∂ε 1 E(t − t ′ ) ve dt ′ ER ∫0 R R ∂tR′ R
(1131)
where E(tR) = relaxation modulus ER = arbitrary reference modulus (typically taken as unity) tR = reduced time A pseudostrain energy density function WR is defined in terms of pseudostrain and damage: WR =
2 1 C(S) ( ε R ) 2
(1132)
Permanent Deformation Assessment where C(S) is a damage function defined in terms of an internal state variable S. The damage function C(S) ranges between 1 for intact material to 0 for a completely damaged material (for ER = 1). Stressstrain relationships are generated from the strain energy density:
σ≡
∂W R = C(S)ε R ∂ε R
(1133)
Note that when C = 1 and the reference modulus is unity, Eq. (1133) reduces to the stressstrain relations for a linear viscoelastic material without damage. A damage evolution law governs the development of the damage internal state variable S: ∂S ⎛ ∂W R ⎞ S = = − ∂tR ⎜⎝ ∂S ⎟⎠
α
(1134)
where a is a material constant, a(T) is the temperature shift relationship from the timetemperature superposition of the relaxation modulus mastercurve, and dtR =
dt a(T )
(1135)
The uniaxial continuum damage model parameters are calibrated using constant rate of strain tests to failure. Tests are conducted in compression at multiple strain rates and at low temperatures. Low temperature tests are used to minimize viscoplasticity effects. Even at 5°C some minimal viscoplastic strains develop, but the calibrated viscoplasticity model can be used to predict the viscoplastic component that must be subtracted from the total measured strains. Calibration of the damage function C(S) and a value is achieved through a nonlinear optimization procedure as detailed in the Gibson et al. (2003); the calibrated results for the 12.5mm dense graded Superpave mixture are summarized in Fig. 116. The damage function can be represented in series form as 6
C(S) = ∑ i=1
1 −a S e i+3 i
(1136)
The calibrated ai values in Eq. (1136) for the 12.5mm dense graded Superpave mixture are summarized in Table 115. The uniaxial form for the damage model can be generalized to confined triaxial conditions using the approach by Ha and Schapery, 1998. Starting from a dual pseudostrain energy density formulation in terms of the axial pseudostrain eR1 and confining pressure p WDR =
1 1 C (S)(ε 1R )2 + C12 (S)ε 1R p + C22 (S)p 2 2 11 2
(1137)
333
334
Chapter Eleven
FIGURE 116
TABLE 115
Calibrated damage function for 12.5mm dense graded Superpave mixture.
i
ai
1
1.649E−03
2
4.610E−06
3
1.853E−05
4
1.766E−05
5
1.243E−04
6
2.489E−05
Damage Function Terms for 12.5mm Dense Graded Superpave Mixture
The constitutive relations for a damaged linear viscoelastic material can then be expressed as Δσ ≡
∂WDR = C11 (S)ε 1R + C12 (S)p ∂ε 1R
(1138)
ε vR ≡
∂WDR = C12 (S)ε 1R + C22 (S)p ∂p
(1139)
where WRD = dual energy density Δs = deviator stress (tension positive) p = confining pressure (compression positive, p = −s3 for triaxial loading conditions) eR1 = axial pseudostrain (tension positive) eRv = volumetric pseudostrain (expansion positive)
Permanent Deformation Assessment Cij(S) = damage functions S = damage internal state variable The pseudostrain quantities are calculated using the correspondence principle via the hereditary convolution integrals: tR
ε 1R =
∂ε 1 E(t −τ ) 1 dτ ER ∫0 R R ∂τ R R
ε vR =
∂ε 1 K (tR −τ R ) v dτ R K R ∫0 ∂τ R
(1140)
tR
(1141)
where E(tR) and K(tR) are the Young’s and bulk relaxation moduli at reduced time tR; the reference Young’s modulus ER is taken as unity and omitted from the equations; the reference bulk modulus KR corresponds to ER = 1. The limiting case for Eqs. (1138) and (1139) at S → 0 corresponds to an undamaged linearly viscoelastic state. For isotropic conditions • C11(S) → ER as S → 0, where ER is the reference modulus in the stress versus pseudostrain relation (taken as unity in the present work) • C12(S) → (1 − 2n0 ) as S → 0, where n0 is the initial Poisson’s ratio •
C22 (S) → −
2 (1 − 2ν 0 )(1 + ν 0 ) as S → 0 ER
The most direct way to determine the C12(S) relation is to apply Eq. (1139) to the unconfined constant rate compression tests for which p = 0, yielding: C12 (S) =
ε vR ε 1R
(1142)
Once both C11(S) and C12(S) are known, the remaining C22(S) damage function can be determined using the confined constant strain rate compression tests and Eq. (1139). The assumed functional forms for C11(S) and C12(S) and the calibrated coefficients for the 12.5mm Superpave dense graded mix are summarized in Table 116 and Table 117, respectively; details of the calibration procedure can be found in Gibson (2006).
C12 (S) = c1 +
c2 1 + e c (log S+ c )+ c 3
4
c1
−0.262
c2
162.634
c3
1.304
c4
753.676
c5
1.530
5
TABLE 116 Calibrated C12(S) Damage Function and Constants a = 1.75
335
336
Chapter Eleven C22(S) = a + bS + cS2 a
−1.3700E+00
b
−3.3085E−04
c
4.6979E−10
TABLE 117 Calibrated C22(S) Damage Function and Constants, a = 1.75
FIGURE 117 Measured and predicted axial and radial strain versus deviatoric stress from 250 kPa conﬁned constant strain rate tests at 10oC.
The multidimensional damage model formulation was validated against confined controlled strain rate to failure tests conducted at 10°C and 250 kPa confinement. These test conditions are different from those used for the model calibration. At this temperature and confinement, viscoplasticity is effectively suppressed and thus the measured strains correspond to just the linear viscoelastic and continuum damage components. The results for the axial and radial directions are shown in Fig. 117. The predicted deviatoric stress versus axial and radial strain agree very closely with the measured response.
Model Validation The final combined model (viscoelasticity plus viscoplasticity plus damage) was validated against laboratory tests different from those used for model calibration. An example of the model validation is provided in Fig. 118 and Fig. 119, which summarize predicted versus measured total strains from unconfined constant strain rate to failure tests at 25°C and 40°C. At these temperatures, which are significantly different from any used to calibrate any of the material parameters in the model, all of the strain components (linear viscoelastic, damage, and viscoplastic) are of significant magnitudes. As shown in the figures, there is good agreement between the predicted and measured total responses. Total strain is slightly underpredicted initially, but the predictions improve in the peak and near postpeak regions. Divergence between predictions and measurements grow in the far postpeak region, but this is because of
Permanent Deformation Assessment
FIGURE 118 Measured stress versus predicted and measured total strain for controlled strain rate tests at 25oC.
FIGURE 119 Measured stress versus predicted and measured total strain for controlled strain rate tests at 40oC.
damage localization and the development of macrocracks for which continuum damage theories are no longer applicable. In summary, although fully mechanistic prediction of flexible pavement rutting is still at or beyond the frontiers of the current state of the art, great strides forward have been taken in recent years, particularly with regard to robust constitutive modeling for asphalt concrete. This, combined with everincreasing computational power, suggests that the development of fully mechanistic procedures, at least as research tools, is only a matter of time.
337
338
Chapter Eleven
Simple Performance Test for Mixture Rutting One of the objectives of the NCHRP Project 919 was to recommend a simple performance test to complement the Superpave volumetric mixture design procedure. The need for a simple performance test arose from the concern that the Superpave mixture design procedure was based entirely upon volumetric proportioning of the asphalt mixture and did not include any direct test method to evaluate permanent deformation resistance of the mix. Asphalt mixture laboratory testing can be divided into three general categories: empirical, performancerelated, and performancebased. Empirical tests like Marshall stability are often of limited usefulness because the property measured in the test does not relate directly to performance. Performance related tests like compressive strength, on the other hand, measure engineering properties that have been found to be roughly correlated to mixture performance; however, these properties by themselves are usually insufficient as the basis of a fundamental performance prediction model over wide varieties of mixture types. Performancebased tests measure material properties that can be used in models to predict mixture response to a wide range of load and environmental conditions. Performancebased tests are clearly the best candidates for a simple performance test. Performancebased test methods can be categorized by the type of the test, type of load application, and type of load pulse as summarized in Table 118. In order to provide accurate and realistic relationships between laboratory measured strains in asphalt mixtures and pavement deformations in the field, it is important to conduct the laboratory tests under stress and environmental conditions similar to those in the field. Three fundamental factors are very important to consider: • Climatic conditions (e.g., pavement temperature) at the given geographic site • Traffic level (i.e., number of repetitions) expected during the pavement service life, including the rate of loading • Stress levels expected within the asphalt layer for a given pavement structure Any “simple” performance test must be sensitive to the influences of these fundamental factors.
Type of Test or Test Geometry
Type of Load Application
Type of Load Pulse
• Uniaxial or Triaxial Compressive • Indirect Tension • Direct Tension • Simple Shear • Direct Shear • Hydrostatic • Torsional or Rotational • Flexural Beam
• Static or Creep • Constant Deformation Rate • Repeated Load or Cyclic • Dynamic Loading
• None • Square • Haversine • Sinusoidal • Triangular
TABLE 118 General Elements of PerformanceBased Test Methods
Permanent Deformation Assessment A wide range of performancerelated and performancebased tests were evaluated during NCHRP Project 919 for their suitability as a simple performance test for permanent deformation and for fatigue cracking. The focus of the present chapter is on permanent deformation, and three tests were eventually identified as the most fruitful for assessing the permanent deformation resistance of asphalt mixtures: the dynamic modulus test, the repeated load permanent deformation test, and the static creep permanent deformation test. The use of dynamic modulus as a simple performance indicator for asphalt mixture permanent deformation resistance is described in Chap. 4 of this volume (see also Pellinen 2001). The present discussion will focus on the permanent deformation tests.
Static Creep Permanent Deformation Test In a static creep test, the total strain versus time relationship for a mixture is obtained experimentally in the lab under constant stress conditions (typically, constant uniaxial stress). While the creep test has been in use in pavement engineering for many decades, the onset of tertiary deformation—defined as the flow time—was identified by Hafez (1997) as a property strongly correlated with the permanent deformation resistance of a mixture. Hafez’s results, although successful and encouraging, were limited in that they were derived only from unconfined tests on mixtures have a single aggregate gradation (dense grading) and were measured with relatively unsophisticated instrumentation techniques. Subsequent work by Kaloush (2001) focused on enhancements in these areas.
Evaluation of Flow Time Figure 1110 shows typical test results between the calculated compliance D(t) = e(t)/s and time. It can be seen from this figure that the compliance can be divided into three major phases: primary creep, secondary creep, and tertiary flow. For constant stress loading conditions, the creep strain rate and compliance changes decrease with time during the primary creep phase, are approximately constant during the secondary creep phase, and increase during the tertiary flow phase. At low stress levels, asphalt concrete mainly exhibits primary creep, that is, the creep rate slowly decreases to zero as the total strain reaches an asymptotic limit. This also suggests that the creep rate in the secondary phase may approach zero at low stresses. At higher stress levels, the constant secondary creep rate phase will depend on the magnitude of the applied stress.
FIGURE 1110
Typical compliance versus time during constant stress creep.
339
340
Chapter Eleven The large increase in compliance within the tertiary flow zone generally occurs at constant volume. The flow time Ft is therefore defined as the time of onset of shear deformation under constant volume. The flow time can also be viewed as the minimum point in the relationship of rate of change of compliance versus loading time. The approach used by Mirza and Witczak (1994) to determine the flow time from creep test data is as follows. Ten sample points are taken from every log decade of time at approximately equal intervals. Then, at a specific time ti, a quadratic polynomial is fitted to the nearest five sample points (two forward and two backward about ti) in order to smooth out the effects of measurement noise and variability: D(t)i = a + bt + ct2 where
(1143)
t = time D(t)i = “smoothed” compliance within interval about time ti a, b, c = regression coefficients
Taking the derivative d(D(t)i ) = b + 2ct dt
(1144)
Therefore, the rate of change in the smoothed compliance at time ti is equal to b + 2cti. One can obtain the rate of change in compliance over the entire time range of interest by repeating the above procedure at multiple data points. The time at which the rate of change in compliance is zero—that is, the time at which the change in compliance begins to increase—is defined as the flow time Ft.
Repeated Load Permanent Deformation Test Another common approach to evaluate the permanent deformation characteristics of paving materials is to apply a repeated load for several thousand repetitions and record the cumulative permanent deformation as a function of the number of cycles [e.g., Monismith et al. (1975) and Witczak and Kaloush (1998) for uniaxial loading; Brown and Cooper (1984) for confined conditions]. Typically, a haversine pulse load of 0.1 s and 0.9 s dwell (rest time) is applied over a test duration of approximately 3 h. This loading history results in approximately 10,000 load cycles applied to the specimen. Like the creep test, the plot of cumulative permanent strain versus number of load repetitions (in loglog space) is generally defined by three zones: primary, secondary, and tertiary. Similarly, the load cycle at the onset of tertiary flow is termed the flow number FN. The cumulative permanent strain ep versus number of load cycles N is typically characterized using a powerlaw relationship: ep = aNb
(1145)
in which a and b are related to the intercept and slope in transformed log ε p = log a + b log N space. It must be emphasized that the a and b parameters are derived from the linear (in loglog space) secondary portion of the cumulative plastic strain versus load cycle curve and, therefore, ignore the initial primary transient
Permanent Deformation Assessment response and any final tertiary instability. An alternative form of the mathematical model can be used to characterize the incremental plastic strain per load repetition epn: dε p d(aN b ) = ε pn = dN dN
(1146)
epn = abN(b1)
(1147)
or
The resilient strain er is generally assumed to be independent of the number of load cycles N. As a consequence, the ratio of plastic to resilient strains can be expressed as
ε pn ⎛ ab ⎞ b 1 = N ε r ⎜⎝ ε r ⎟⎠ Letting μ =
(1148)
ab and a = 1 − b one obtains: εr
ε pn = μ N −α εr
(1149)
In the above equation, epn is the incremental plastic strain due to the Nth load cycle; m represents the permanent to resilient strain at N = 1; and a governs the rate of decrease in incremental permanent deformation with increasing load cycles. As described previously, this is the approach adopted in the VESYS program. Figure 1111 illustrates typical test data following the relationship in Eq. (1149) and the flow point where the incremental permanent strain per cycle begins to increase.
AlphaMU Parameters for permanent deformation Project : SP task F (Regression cycles = 50 to 2000) Mix ID # : AAA034 10000 Flow Pt. = 1.62E+ (cycles) Mu (μ) = 1.046 Alpha (α) = 0.607
1000 εp(n) εr (x 1000)
100
10 1.0E+00
FIGURE 1111
1.0E+01
1.0E+02 Loading Cycles
Determination of the ﬂow number.
1.0E+03
1.0E+04
341
342
Chapter Eleven Unconfined repeated load tests may not always provide a true indication of the relative performance of asphalt mixtures (e.g., Brown and Snaith 1974; Brown and Cooper 1984). The intercept and slope parameters obtained from a triaxial repeated load permanent deformation test are functions of the vertical and confining stresses and on temperature. The systematic influence of these factors on the flow number or flow time is unknown at this point.
Testing Issues An extensive study by Witczak et al. (2000) addressed specimen instrumentation and geometry issues important for creep and repeated load permanent deformation testing. The influence of specimen aspect ratio and size relative to the nominal maximum aggregate size were of particular interest in this study, since practical laboratory specimens are rarely as large as ideal. The specimen must be sufficiently large that the measured response is independent of the specimen size. Witczak et al. (2000) concluded that minimum specimen diameter of 100 mm and a minimum height to diameter ratio of 1.5 were to accurately characterize both the permanent deformation in the secondary zone and the onset of tertiary flow for mixtures having nominal aggregate sizes up to 37.5 mm. These recommendations assume specimens having sawed parallel ends that are fully lubricated to minimize end restraint. The recommended LVDT mounting system is based on studs glued directly to the surface of the specimen as opposed to the clamps used historically by other researchers.
Experimental Program The first phase of the simple performance test experimental program utilized laboratoryfabricated specimens using original construction materials from three experimental sites: The Minnesota Test Road (MnRoad), The Accelerated Loading Facility (ALF) test sections at the Federal Highway (FHWA) TurnerFairbank research facility in Virginia, and the Westrack full scale test facility near Reno, Nevada. Each of these experimental sites was constructed to study the influence of a wide variety of pavement structural and material properties upon pavement performance. Static creep and repeated load permanent tests were conducted on at least two replicate test specimens for each mixture. The cylindrical test specimens were 100 mm (4 in.) in diameter and 150 mm (6 in.) in height. Tests were conducted at two temperatures of 37.8°C (100°F) and 54.4°C (130°F). Unconfined tests were conducted at deviator stress levels of 69, 138, and 207 kPa (10, 20, and 30 psi). Confined tests were conducted typically at 138 kPa (20 psi) confining pressure and 828 kPa (120 psi) deviator stress level. Extensive statistical analyses were performed to identify and evaluate all measured laboratory responses with regard to their correlation with the field rut depth measurements. Summaries of the goodness of fit statistics and rationality of the trends for each test parameter were presented in Kaloush (2001). The flow number FN and flow time Ft were found to best discriminate the performance among the different mixtures. Examples for the MnRoad sites are shown in Fig. 1112 and Fig. 1113; results from the other test sites are similar. Large flow values (either FN or Ft)
Permanent Deformation Assessment
FIGURE 1112
FIGURE 1113
Rut depth versus unconﬁned ﬂow number of repetitions, MnRoad mixtures.
Rut depth versus unconﬁned ﬂow time, MnRoad mixtures.
correlate with better rutting resistance and stability for similar temperature, stress, and traffic conditions. The results from these repeated load permanent deformation tests were the basis for the mechanisticempirical rutting model given previously in Eq. (1114). Based on statistical analyses of the ep/er ratio at the onset of tertiary flow—that is, at N = FN —for both confined and unconfined tests corresponded to a mean value of 55 with a standard deviation of 20 at failure. These data were also used to develop an empirical
343
344
Chapter Eleven relationship between FN and mixture volumetric properties, binder type, temperature, and stress level: FN = (4.3237 × 108)T−2.2150 m0.3120Vbeff−2.604Va−0.1525
(R2 = 0.72)
(1150)
where T = temperature, °F m = binder viscosity at 70°F (106 poise) Vbeff effective binder content by volume, % Va = air voids, % The correlation between the flow number and flow time was also investigated as part of the study. This correlation would provide a practical link between the two flow parameter tests and would be of great benefit in the development of appropriate design criteria. The initial study by Kaloush (2001) showed encouraging results for this relationship. Qayoum (2004) combined confined and unconfined test results from over 64 additional mixtures at 7 sites to determine the following correlation between FN and Ft: log(FN) = 1.904 log(Ft)0.5101
(R2 = 0.71)
(1151)
Mix Design Criteria Ideally, the results shown in Fig. 1112 (or Fig. 1113) would be used to develop criteria for the design of rut resistant asphalt mixtures. Kaloush (2001) contributed an initial conceptual framework; additional developments to incorporate temperature and traffic level effects were provided by Qayoum (2004). The starting point is the observation that rut depth RD for a mixture can be related to a reduced flow number FNr via a powerlaw relationship of the form: RD = a (FNr)b
(1152)
where a and b are parameters that are functions of mixture properties and traffic. Figure 1114 shows the hypothesized relationship between rut depth, traffic, and flow number as described by Eq. (1152). The reduced flow number FNr in Eq. (1152) is formulated using standard timetemperature superposition techniques: FNr =
FN a(T )
(1153)
where FNr is a “reduced” flow number (analogous to reduced time or reduced frequency in dynamic modulus testing) and a(T) is a temperature shift factor. Findings by Qayoum (2004) suggest that a(T) values determined from dynamic modulus testing are adequate for shifting the flow number. Qayoum’s analyses of the MnRoad, Westrack, and FHWA ALF sites also suggested that the influence of traffic on the a and b parameters in Eq. (1152) can be captured via the relations: a = mN−n
(1154)
b = k log(N)−l
(1155)
Permanent Deformation Assessment
FIGURE 1114
Relationship between ﬁeld rut depth, trafﬁc, and ﬂow number.
where m, n, k, and l are constants and N is the traffic level in ESALs. Equation (1153) can then be expressed as RD = mN−n(FNr)klog(N)−l
(1156)
Figure 1115 summarizes the predictions from Eq. (1156) for seven MnRoad sections at four levels of traffic; the R2 value of 0.92 indicates excellent agreement between predicted and measured asphalt rutting. As summarized in Table 119, the predictions for the other test sites showed similarly good agreement with measured rutting. The results in Fig. 1115 and Table 119 are all for individual mixtures at individual field sections. The next step is the development of a global model that can be applied to any mixture and/or site. Work to this end is currently underway. 1.00 Se/sy = 0.307 2 R = 0.92
0.90
Predicted rut depth (in)
0.80 0.70 0.60 Cell 1
0.50
Cell 3
0.40
Cell 4 Cell 14
0.30
Cell 15 0.20
Cell 19 Cell 21
0.10 0.00 0.00
Equality line 0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
100
Measured rut depth (in)
FIGURE 1115 Measured versus predicted rut depth for MnRoad plant mixes using all cells. (Qayoum 2004.)
345
346
Chapter Eleven
R2
Se/Sy
U
Rd = 1.0989∗(N)
−0.0115∗Log(N)−0.3201
0.97
0.20
ALFField Cores
C
Rd = 1.2552∗(N)1.6972∗(Fn)−02429∗Log(N)−0.25154
0.89
0.40
ALFLab Blend
U
Rd = 1.4646∗(N)0.2718∗(Fn)0.1885∗Log(N)−0.4629
0.56
0.78
MnRoadPlant Mix
U
Rd = 1.2483∗(N)
0.93
0.29
MnRoadPlant Mix
C
Rd = 1.20326∗(N)0.21255∗(Fn)0.28828∗Log(N)−1.58221
0.93
0.28
WestrackPlant Mix
U
Rd = 0.00341∗(N)0.36446∗(Fn)0.00304∗Log(N)−0.04683
0.89
0.35
WestrackPlant Mix
C
Rd = 0.00042∗(N)0.60852∗(Fn)0.05482∗Log(N)−0.18035
0.95
0.24
Test Site
Test Type
Rutting Model
ALFField Cores
∗(Fn)
0.3848
∗(Fn)
0.0425
0.1649∗Log(N)−1.3745
Source: Qayoum 2004.
TABLE 119
Final Rutting Models for the Individual Test Sites
Summary Three current areas of research into rutting of asphalt mixtures have been briefly addressed: (a) mechanisticempirical modeling approaches, and, in particular, the model adopted for the performance prediction and design methodology developed in NCHRP Project 137A; (b) advanced constitutive modeling approaches to the rutting problem, with particular emphasis on viscoplasticity and continuum damage; and (c) development of a simple performance test to identify the rutting potential of mixtures during design based on measured fundamental engineering properties and response. Mechanisticempirical approaches to rutting prediction couple mechanistic computations of pavement stresses and strains with empirical predictions of the consequent rutting. Many mechanisticempirical rutting models have been proposed over the past decades, but most have been calibrated against only very limited sets of laboratory and/or field data and no systematic comparisons of their predictions for a common set of conditions have been made to date. The NCHRP 137A model is arguably the leading mechanisticempirical model for HMA rutting at present, both because it builds upon many of the earlier efforts and because it has been calibrated against nearly 400 field test sections. The NCHRP 137A model also includes an explicit option for regional calibration to improve the accuracy of its predictions. Although the full potential of mechanisticempirical rutting models has yet to be fully tapped, their overall accuracy and robustness will always rely upon the quantity and quality of the empirical data used for their calibration. Fully mechanistic distress prediction bypasses this limitation, but this requires much more sophisticated constitutive models for asphalt concrete behavior. Great strides have been made in recent years on material models that capture the viscoelastic, viscoplastic, and damage response components needed to simulate the behavior of asphalt concrete over its full
Permanent Deformation Assessment range of temperatures, loading rates, and stress conditions. These models are now being implemented into threedimensional nonlinear finite element codes and applied to realistic test and field scenarios. Screening of asphalt mixtures for rut susceptibility during mix design is another important element in the design of rutresistant pavements. Recent progress toward a simple performance test was motivated by the concern that the Superpave mixture design procedure was based entirely upon volumetric proportioning of the asphalt mixture and did not include any direct test method to evaluate permanent deformation resistance of the mix. The time to tertiary flow failure appears to be a good indicator of the rutting resistance of a given mixture. This can be quantified either via the flow time as measured in a static creep test or the flow number as measured in a repeated load permanent deformation test. Conceptual guidelines for establishing the minimum flow time or flow number are currently being formalized and validated.
Acknowledgments Portions of the work described in this chapter were performed as part of NCHRP Project 137A “Development of the 2002 Guide for the Design of New and Rehabilitated Pavement Structures” and NCHRP Project 919 “Superpave Support and Performance Models Management.” A. Hanna was the NCHRP program manager for Project 137A; J. Hallen of Applied Research Associates was the principal investigator and M. W. Witczak of Arizona State was the head of the flexible pavement team. E. Harrigan was the Program Manager for Project 919; M. W. Witczak of Arizona State University was the principal investigator, and C. W. Schwartz of the University of Maryland and H. L. Von Quintus of Fugro/BRE (now with Applied Research Associates) were the coprincipal investigators. All opinions expressed in this chapter are solely those of the authors.
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Permanent Deformation Assessment Gibson, N. H., C. W. Schwartz, R. A. Schapery, and M. W. Witczak (2003b), “Confining Pressure Effects on Viscoelasticity, Viscoplasticity, and Damage in Asphalt Concrete,” Proceedings, 16th ASCE Engineering Mechanics Conference, Seattle, Wash., July. Ha, K., and R. A. Schapery (1998), “A ThreeDimensional Viscoelastic Constitutive Model for Particulate Composites with Growing Damage and Its Experimental Verification,” International Journal of Solids and Structures, Vol. 35, No. 26–27, pp. 3497–3517. Hafez, I. (1997), “Development of a Simplified Asphalt Mix Stability Procedure for Use in Superpave Volumetric Mix Design,” Ph.D. dissertation, University of Maryland, College Park, Md. Hopman, P., R. Nilsson, and A. Pronk (1997), “Theory, Validation, and Application of the ViscoElastic Multilayer Program VEROAD,” Proceedings, 8th International Conference on Asphalt Pavement, Seattle, Wash., Vol. I, pp. 693–706. Huang, B., L. Mohammad, W. Wathugula, and H. Paul (2002), “Development of a ThermoViscoplastic Constitutive Model for HMA Mixtures,” Journal of the Association of Asphalt Paving Technologists, Vol. 71, pp. 594–618. Kaloush, K. E. (2001), “Simple Performance Test for Permanent Deformation of Asphalt Mixtures,” Ph.D. dissertation, Arizona State University, Tempe, Ariz. Kannemeyer, L., and A. T. Visser (1995), “Calibration of HDMIII Performance Models for Use in Pavement Management of South African National Roads,” Transportation Research Record, No. 1508, Washington, D.C., pp. 31–38. Kenis, W. J. (1977), “Predictive Design Procedures: A Design Method for Flexible Pavements Using the VESYS Structural Subsystem,” Proceedings, 4th International Conference on the Structural Design of Asphalt Pavements, Ann Arbor, Mich., Vol. 1. Kenis, W. J., J. A. Sherwood, and T. F. McMahon (1982), “Verification and Application of the VESYS Structural Subsystem,” Proceedings, 5th International Conference on the Structural Design of Asphalt Pavements, Delft, The Netherlands, Vol. 1, pp. 333–346. Kenis, W. J. (1988), “The Rutting Models of VESYS,” Proceedings, 25th Conference on Paving and Transportation, 25th Conference on Paving and Transportation, Albequerque, N.Mex., January. Kenis, W. J., and W. Wang (1997), “Calibrating Mechanistic Flexible Pavement Rutting Models from Full Scale Accelerated Tests,” Proceedings, 8th International Conference on Asphalt Pavements, Seattle, Wash., Vol. I, pp. 663–672. Kim, Y. R., H. J. Lee, and D. N. Little (1997), “Fatigue Characterization of Asphalt Concrete Using Viscoelasticity and Continuum Damage Theory,” Journal of the Association of Asphalt Paving Technologists, Vol. 66, pp. 520–569. Leahy, R. B. (1989), “Permanent Deformation Characteristics of Asphalt Concrete,” Ph.D. dissertation, University of Maryland, College Park, Md. Lee, H. J., and Y. R. Kim (1998a), “Viscoelastic Constitutive Model for Asphalt Concrete under Cyclic Loading,” Journal of Engineering Mechanics, ASCE, Vol. 124, No. 1, January, pp. 32–40. Lee, H. J., and Y. R. Kim, (1998b), “Viscoelastic Continuum Damage Model of Asphalt Concrete with Healing,” Journal of Engineering Mechanics, ASCE, Vol. 124, No. 11, November, pp. 1224–1232. Levenberg, E., and J. Uzan (2004), “Triaxial SmallStrain ViscoelasticViscoplastic Modeling of Asphalt Aggregate Mixes,” Mechanics of Time Dependent Materials, Vol. 8, No. 4, pp. 365–384. Long, F., and C. L. Monismith (2002), “Use of a Nonlinear Viscoelastic Constitutive Model for Permanent Deformation in Asphalt Concrete Pavements,” 3D Finite Element Modeling of Pavement Structures, in: A. Scarpas and S. N. Shoukry, eds., Proceedings,
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Chapter Eleven 3rd International Syposium on 3D Finite Element for Pavement Analysis, Design, and Research, Amsterdam, The Netherlands, pp. 91–110. Lundstrom, R., and U. Isacsson (2004), “An Investigation of the Applicability of Schapery’s Work Potential Model for Characterization of Asphalt Fatigue Behavior,” Journal of the Association of Asphalt Paving Technologists, Vol. 73, pp. 657–687. May, R. W., and M. W. Witczak (1992), “An Automated Asphalt Concrete Mix Analysis System,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 61, Charleston, S.C. Mirza, M. W., and M. W. Witczak (1994), Bituminous Mix Dynamic Material Characterization Data Acquisition and Analysis Programs Using 458.20 MTS Controller, Program Background and Users Guide, University of Maryland, College Park, Md. Monismith, C. L., N. Ogawa, and C. Freeme (1975), “Permanent Deformation of Subgrade Soils due to Repeated Loadings,” Transportation Research Record, No. 537, Washington, D.C., pp. 1–17. NCHRP (2004), “MechanisticEmpirical Design of New and Rehabilitated Pavement Structures,” NCHRP Project 137A Draft Final Report, Transportation Research Board, National Research Council, Washington, D.C. Park, S. W., Y. R. Kim, and R. A. Schapery (1996), “A Viscoelastic Continuum Damage Model and Its Application to Uniaxial Behavior of Asphalt Concrete,” Mechanics of Materials, Vol. 24, pp. 241–255. Park, D. W., A. EppsMartin, and E. Masad (2004), “Simulation of Permanent Deformation Using an ElasticViscoplastic Constitutive Relation,” Proceedings of the Second International Conference on Accelerated Pavement, Minneapolis, Minn. Paterson, W. D. O. (1987), Road Deterioration and Maintenance Effects: Models for Planning and Management, The World Bank, Washington, D.C. Pellinen, T. K. (2001), “Investigation of the Use of Dynamic Modulus as Indicator of HotMix Asphalt Performance,” Ph.D. dissertation, Arizona State University, Tempe, Ariz. Perzyna, P. (1966), “Fundamental Problems in Viscoplasticity,” Advances in Applied Mechanics., 9, pp. 243–377. Pidwerbesky, B. D., B. D. Steven, and G. Arnold (1997), “Subgrade Strain Criterion for Limiting Rutting in Asphalt Pavements,” Proceedings, 8th International Conference on Asphalt Pavement, Seattle, Wash, Vol. II, pp. 1529–1544. Powell, W. D., J. F. Potter, H. C. Mayhew, M. E. Nunn (1984), “The Structural Design of Bituminous Pavements,” TRRL Laboratory Report 1132, Transport and Road Research Laboratory, U.K. Qayoum, M. M. (2004), “Investigations for Using the Repeated Load Permanent Deformation Test in a Design Criteria for Asphalt Mixtures,” M. S. thesis, Arizona State University, Tempe, Ariz. Qi, X., and M. W. Witczak (1998), “TimeDependent Permanent Deformation Models for Asphaltic Mixtures,” Transportation Research Record, No. 1639, Washington, D.C., pp. 8993. Rauhut, B. J. (1980), “Permanent Deformation Characterization of Bituminous Mixtures for Predicting Pavement Rutting,” Transportation Research Record, No. 777, Washington, D.C., pp. 9–14. Shell International Petroleum Co., Ltd. (1978), Shell Pavement Design Manual, Shell International Petroleum Co., Ltd., London. Schwartz, C. W., N. H. Gibson, R. A. Schapery, and M. W. Witczak (2002), “TimeTemperature Superposition for Asphalt Concrete at Large Compressive Strains,” Transportation Research Record, No. 1789, Washington, D.C., pp. 101–112.
Permanent Deformation Assessment Schwartz, C. W., N. H. Gibson, R. A. Schapery, and M. W. Witczak (2004), “Viscoplasticity Modeling of Asphalt Concrete Behavior,” Recent Advances in Materials Characterization and Modeling of Pavement Systems, in: E. Tutumluer, Y. M. Najjar, and E. Masad, eds., Geotechnical Special Publication No. 123, ASCE, Reston, Va., pp. 144–159. Shook, J. F., F. N. Finn, M. W. Witczak, and C. L. Monismith (1982), “Thickness Design of Asphalt Pavements—The Asphalt Institute Method,” Proceedings, 5th International Conference on the Structural Design of Asphalt Pavements, Delft University of Technology, Delft, The Netherlands, Vol. I, pp. 17–44. Superpave Models Team (1999), Volumetric Design of Standard Mixtures Used by the University of Maryland Models Team, Internal Team Report. SUPERPAVE Support and Performance Models Management, NCHRP Project 919, Department of Civil Engineering, University of Maryland, College Park, Md. Tashman, L., E. Masad, H. Zbib, D. Little, and K. Kaloush (2003), “Anisotropic Viscoplastic Continuum Damage Model for Asphalt Mixes,” Recent Advances in Materials Characterization and Modeling of Pavement Systems, in: E. Tutumluer, Y. M. Najjar, and E. Masad, eds., Geotechnical Special Publication No. 123, ASCE, Reston, Va., pp. 111–125. Tashman, L., E. Masad, H. Zbib, D. Little, and K. Kaloush (2004), “Anisotropic Viscoplastic Continuum Damage Model for Asphalt Mixes,” Recent Advances in Materials Characterization and Modeling of Pavement Systems, ASCE Geotechnical Special Publication, No. 123, pp. 111–125. Timm, D. H., and D. E. Newcomb (2003), “Calibration of Flexible Pavement Performance Equations for Minnesota Road Research Project,” Transportation Research Record, No. 1853, National Research Council, Washington, D.C. Uzan, J. (2004), “Permanent Deformation in Flexible Pavements,” Journal of Transportation Engineering, ASCE, Vol. 130, No. 1, January/February, pp. 6–13. Verstraeten, J., V. Veverka, and L. Francken (1982), “Rational and Practical Designs of Asphalt Pavements to Avoid Cracking and Rutting,” Proceedings, 5th International Conference on the Structural Design of Asphalt Pavements, Delft, The Netherlands. Witczak, M. W., and K. E. Kaloush (1998), “Performance Evaluation of Asphalt Modified Mixtures Using Superpave and P401 Mix Gradings,” Technical Report to the Maryland Department of Transportation, Maryland Port Administration, Baltimore, Md. Witczak, M. W., R. Bonaquist, H. Von Quintus, and K. Kaloush (2000), “Specimen Geometry and Aggregate Size Effects in Uniaxial Compression and Constant Height Shear Tests,” Journal of the Association of Asphalt Paving Technologists, Vol. 69, pp. 733–793.
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PART
Models for Fatigue Cracking and Moisture Damage CHAPTER 12 Micromechanics Modeling of Performance of Asphalt Concrete Based on Surface Energy
CHAPTER 13 Field Evaluation of Moisture Damage in Asphalt Concrete
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CHAPTER
12
Micromechanics Modeling of Performance of Asphalt Concrete Based on Surface Energy Dallas N. Little, Amit Bhasin, and Robert L. Lytton
Abstract Surface energies of asphalt cements and aggregates were precisely measured using the Wilhelmy plate method and the universal sorption device (USD), respectively. Surface energies have two primary components; a polar acidbase component and a nonpolar Lifshitzvan der Waals component; that are shown to be fundamentally related to fracture and fracture healing. These components of surface energy can then be used in micromechanical models to predict the fatigue behavior of asphalt concrete mixtures. Surface energies of asphalt cement and aggregates are used to calculate cohesive strength of the asphalt binder and the adhesive bond between the asphalt binder and the aggregate. Adhesive bond strength calculations were used to accurately predict moisture damage.
Introduction Little et al. (2001), Lytton et al. (2001), Kim (1988), and Kim et al. (1994, 1997) developed a series of mechanistic theories to model the fatigue life of the asphalt pavements by considering both microfracture and healing based upon the first principles of fracture and healing, viscoelastic correspondence principles, and continuum damage theories. Lytton et al. (2001) calculated that the healing effect accounts for a large portion of the shift factor between laboratory and field fatigue models, and this shift factor can range between about 3 to over 100. Little et al. (1993) showed that the ability of asphalt cement to heal varies considerably and is affected by the composition and chemistry of the
355 Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.
356
Chapter Twelve asphalt cement. Kim et al. (1994) successfully validated that healing is a measurable and repeatable property, both in the laboratory and in the field. Little et al. (1999) also found that the fatigue damage and healing in asphalt pavements are directly related to the surface energy characteristics of the asphaltaggregate system. The surface energy of a solid or liquid, by definition, is the energy needed to create a unit area of a new element in a vacuum. A more accurate term for this definition is surface free energy. However, throughout this chapter the term surface energy is used for brevity. The theory of surface energy can be found in many surface physical chemistry books and has been widely used by the colloid, lubrication, adhesive coating, and painting industries. A variety of methods are used to measure surface energies, such as the maximum bubble pressure method, Wilhelmy plate, pendant drop, and sessile drop method (Adamson 1997). Two methods, universal gas sorption and the Wilhelmy plate, are used to measure surface energies of aggregate and asphalt, respectively. The universal gas sorption method utilizes the characteristic adsorption behavior of a particular probe vapor onto the surface of an aggregate to indirectly determine the surface energy of the aggregate. This method can accommodate the peculiarity of the irregular shape, size, mineralogy, and rough surface texture of the aggregate. The Wilhelmy plate method measures the advancing and receding contact angles between the surface of an asphalt binder and a probe liquid from which the advancing and receding surface energy can be computed, respectively (Cheng 2002 and Cheng et al. 2002a, 2002b). Surface energy theory is also used to model the moisture damage in asphalt concrete (Chen 1997). Moisture damage is directly related to cohesive and adhesive fracture process within an asphaltaggregate system in the presence of water. The cohesive and adhesive fracture processes are dictated by the surface energy characteristics of the asphalt and aggregate. Surface energies of asphalt binder and aggregate can be measured and used to calculate cohesive bond strength of the asphalt binder (or mastic) as well as the adhesive bond strength between the asphalt binder (or mastic) and the aggregate. Surface energies are fundamentally related to fracture and healing. The thermodynamic potential for water to displace asphalt binder from its interface with the aggregate can be determined using the surface energies of these materials. This thermodynamic potential is fundamentally related to moisture damage due to stripping in asphalt mixtures. Therefore, surface energy, as a material property is an indispensable component of micromechanics modeling. Furthermore, research at the Texas A&M University demonstrates that it is possible to measure the rate of diffusion of water into asphalt cement as well as the waterholding capacity of asphalt binder using the universal sorption device. These properties can be used to define the rheological changes induced by water migration into the asphalt binder, which is of key importance when assessing the effect of moisture on mixture rheology and ultimately to assess moisture damage. In order to verify the effect of surface energy on the fatigue model, including the healing effect, and the moisture damage model, a series of mechanistic based experiments (Kim et al. 2002) were conducted on the fine aggregate matrix (FAM) portion of asphalt mixtures (asphalt binder with aggregates smaller than 1.18mm sieve) as well as full asphalt concrete mixtures. The performance modeling results based on the surface energy theories were in agreement with the results of the mechanistic tests conducted using the FAM and asphalt mixtures.
Surface Energy and Performance of Asphalt
Fundamental Theories Schapery’s Fundamental Law of Viscoelastic Fracture Mechanics The fundamental relationship in cohesive fracture mechanics as stated by Schapery (1984) was derived from the first principles of fracture: 2γ f = ER D f (td ) J v where
(121)
γ f = the surface energy of a crack surface (units: FL−1) ER = reference modulus Df (td) = tensile creep compliance at time of loading, td, that is required for a crack to move through a distance equal to the length of the process zone ahead of the crack tip Jv = the viscoelastic Jintegral, which is the change in dissipated pseudostrain energy per unit of crack area from one tensile load cycle to the next
Equation (121) obeys the first law of thermodynamics. The energy input described on the righthand side of the equation is transferred to newly created crack surfaces. This energy is represented by the lefthand side of the equation. It is logical to write a corollary statement [to Eq. (121)] in which the surface energy of healing (wetting) 2γ h is related to the energy of healing [ER Dh (tu )H v ]. In this energy term, Dh is compressive compliance, which is associated with pushing crack faces back together; and Hv is the healing corollary to the Jintegral for fracture.
Surface Energy The surface energy of a solid (asphalt or aggregate) can be considered to be comprised of a nonpolar component and an acidbase component (Good and van Oss 1991 and Good 1992). Equation (122) is used to describe the total surface energy and its components:
γ = γ LW + γ AB where
(122)
g = surface energy of an asphalt or aggregate (units: FL/L2) g = Lifshitzvan der Waals component of the surface energy (units: FL/L2) g AB = acidbase component of the surface energy (units: FL/L2) LW
The Lifshitzvan der Waals force contains at least three components: London dispersion forces, Debye induction forces, and Keesom orientation forces. The London dispersion force is the attraction between neighboring electronic shells. It is an induced dipole to induced dipole interaction. The Debye induction force is produced by a dipole inducing a dipole in a neighboring molecule. The Keesom orientation force is the interaction of two dipoles which orient themselves in relation to each other (Maugis 1999). The acidbase interaction includes all interactions of electron donor (proton acceptor)—electron acceptor (proton donor) type bonds including hydrogen bonds. To quantitatively predict the acidbase interaction, Good and van Oss (1991) postulated a resolution of the acidbase term g AB into a Lewis acidic surface
357
358
Chapter Twelve parameter and a Lewis basic surface parameter. The relation among g AB and its components is shown in Eq. (123):
γ AB = 2 γ + γ −
(123)
where g + is Lewis acid component of surface interaction; and g – is Lewis base component of surface interaction. Several methods are proposed in the literature to measure the surface energy of the asphaltaggregate system. Elphingstone (1997) and Cheng et al. (2001) determined the surface energy components of different types of asphalt binders using the Wilhelmy plate technique. Li (1997) measured the surface energies of a variety of European aggregates. Cheng et al. (2001) measured surface energies of some widely used aggregates in the southern United States using the USD. Little and coworkers made several improvements to the analytical and experimental techniques to determine the surface energy components of asphalt binders and aggregates using the Wilhelmy plate technique and USD, respectively (Little and Bhasin 2006; Hefer et al. 2006, Bhasin and Little 2006). This improved methodology was used to determine the surface energy components of several different asphalt binders and aggregates from different parts of the United States (Little and Bhasin 2006; Bhasin et al. 2007).
Adhesion and Cohesion Pavement cracks occur either at the asphaltaggregate interface or within the asphalt mastic (asphalt binder with filler particles finer than 75 μm). The interfacial strength between asphalt cement and aggregate is called adhesion. The strength within the mastic is called cohesion. From a thermodynamics point of view, the cohesive bond energy ΔGic is the energy required to create a unit area crack within the binder or mastic in a vacuum. The relationship between cohesive bond energy and surface energy is (Good 1992 and 1977): ΔGiC = 2γ i
(124)
Similar to the surface energy, cohesive bond energy has two components: the Lifshitzvan der Waals part, ΔGicLW, and the acidbase part, ΔGicAB, as shown in Eq. (125). ΔGiC = ΔGicLW + ΔGicAB
(125)
The adhesive bond energy corresponds to the creation of a unit crack area at the interface between two dissimilar bodies (masticaggregate) in a vacuum. Adhesive bond energy is defined by Eqs. (126) and (127): ΔGija = γ i + γ j − γ ij
(126)
ΔGija = ΔGijaLW + ΔGijaAB
(127)
where gij is the interfacial energy between any two materials i and j. The two components of interfacial energy are described in Eq. (128).
γ ij = γ ijLW + γ ijAB
(128)
Surface Energy and Performance of Asphalt The Berthelot geometric mean is used to calculate the Lifshitzvan der Waals component of surface energy as follows (Good 1992).
γ ijLW =
(
γ iLW − γ LW j
)
2
(129)
ΔGijaLW = 2 γ iLW γ LW = ΔGicLW ΔG cLW j j
(1210)
Equations (1211) and (1212) define the acidbase component of surface energy due to the complementary nature of the acidbase interaction.
γ ijAB = 2
(
γ i+ − γ +j
)(
γ i− − γ −j
)
ΔGijaAB = 2 γ i+ γ −j + 2 γ i− γ +j
(1211) (1212)
The lower magnitude of energy of cohesion or adhesion dictates the likely mode of fracture.
Micromechanics Modeling of Fatigue and Healing of Asphalt Mixtures Fatigue Cracking A cohesive fracture hypothesis was developed (Little et al. 1997 and 2001) for cyclic fatigue loading based on the Schapery’s fundamental law of fracture mechanics (Schapery 1984 and 1989) and Lytton’s healing model (Lytton et al. 2001). It has the general form of Paris’ law:
dc = dN
where
( Δt ) f
∫ 0
1
K f α (D1 f ER J v ) m (ΔG f − D0 f ER J v )
f
1 mf
dt −
dh dN
(1213)
ΔGf = the fracture energy density (cohesive or adhesive) of a crack surface a = the length of the fracture process zone preceding the crack surface ER = reference modulus used, if necessary, in representing a nonlinear viscoelastic material as an equivalent nonlinear elastic material D0f , D1f, mf = creep compliance coefficients of the power law: D f (t) = D0 f + D1 f t m Kf = constants that depend on the value of m, the slope of the log creep compliance versus log time curve; a common value is onethird Jv = the viscoelastic Jintegral, which is the change of dissipated energy per unit of crack growth area from one tensile load cycle to the next f
It is assumed that the Jintegral, dissipated energy, is applied according to the normalized waveform with time w(t) as follows: J v = J v 0 w(t)
(1214)
359
360
Chapter Twelve where Jv0 = the maximum value of Jv during the time interval (Δt) and if there is no healing period, then the following familiar form of Paris’s law is obtained. dc = A[ J v 0 ]n dN
(1215)
n = 1/mf
(1216)
In Eq. (1215)
and the coefficient A is Δt f
A=
∫ 0
K f α (D1ER )n w(t)n dt (ΔG f − D0 ER J v )n
(1217)
In an asphalt concrete compliance test, D0 is normally much smaller than D1, and one can obtain the following, simplified equation by neglecting the contribution of D0: ( Δt ) f mf
A = [K f D1ER ] n
∫ 0
α w(t)n dt (ΔG f )n
(1218)
Equation (1218) was further simplified by incorporating other material properties into the parameter Kf and rewriting it as follows (Masad et al. 2007): ⎡DE ⎤ A = Kf ⎢ 1 R ⎥ ⎣ ΔG f ⎦
n
(1219)
Furthermore, based on the definition of Jintegral, Eq. (1215) can be rewritten as (Masad et al. 2007): ⎡ ∂WR ⎤ dc ∂N ⎥ = A⎢ ⎢ ∂csa ⎥ dN ∂N ⎥⎦ ⎢⎣
n
(1220)
where WR = rate of energy dissipated due to damage, usually determined by fitting dissipated pseudostrain energy measured from a fatigue test to the form WR = a + b ln(N ) csa = crack surface area which can be considered to be 2π r 2 for a circular crack with equivalent radius of r N = load cycle Integrating Eq. (1220) and combining with Eq. (1219), the fracture equation for equivalent crack radius at any cycle N is given as follows (Masad et al. 2007): n ⎡ ⎤ ⎛ DE b⎞ r (N ) R(N ) = 1/2 n+1 = ⎢ (2n + 1)n+1 ⎜ 1 R ⎟ N ⎥ π Δ 4 G K ⎝ f ⎠ ⎢⎣ ⎥⎦
1/2 n+ 1
(1221)
Surface Energy and Performance of Asphalt The initial level of damage in the specimen prior to the start of the test should be accommodated in the expression for crack growth. Therefore, the fracture Eq. (1221) is written, more correctly, in the form a crack growth index as follows: n ⎡ ⎤ ⎛ DE b⎞ ΔR(N ) = R(N ) − R(1) = ⎢ (2n + 1)n+1 ⎜ 1 R ⎟ N ⎥ π 4 Δ G ⎝ f ⎠ ⎢⎣ ⎥⎦
1/2 n+ 1
(1222)
Masad et al. 2007 present a more detailed derivation for Eq. (1222) as well as alternative forms of this equation when the dissipated pseudostrain energy determined from a fatigue test does not follow the form assumed before, that is, WR = a + b ln(N ) . From Eq. (1222) it is evident that the fatigue cracking or fracture process can be modeled using the creep compliance properties of the material (D1 and n), the rate of crack growth or energy dissipation (b), and the fracture energy of the material that is derived based on the surfaceenergy components (ΔGf).
Healing When a material can heal and has time to heal between load applications, the following modified form of the Paris’ law explains the rate of crack growth as influenced by healing: dc dh = A[ J v 0 ]n − dN dN
(1223)
dh The rate of crack healing per load cycle dN is governed by Eq. (1223) (Lytton 2001): i
i
i (h1 − h2 )(Δt)h dh = h2 (Δt)h + i i dN h1 − h 2 1+ (Δt)h hβ
where
(1224)
h = the actual healing i i i i h1 , h2 = the healing rates generated by the nonpolar (h1 ) and polar (h2 ) surface energies hb = a factor that varies between zero and one and represents the maximum degree of healing that can be achieved by the asphalt binder
Schapery (1989) derived a relationship between healing rate and several material properties including surface energy. Schapery’s relationship (1989) was used to explain longterm healing (healing that continues throughout the rest period). This relationship is 1
⎡ 2γ m ER2 D1hγ ⎤m h2 = ⎢ 1/m 2 ⎥ β c H 1 − ν ( ) m v ⎦ ⎣ i
AB
h
(1225)
h
n = Poisson’s ratio g AB = acidbase part of the cohesive surface energy of healing cm, gm = functions of m as defined in Schapery (1989) (cm varies between 1 and 1.5, and gm varies between 1 and 2/3 as m varies between 0 and 1)
where
361
362
Chapter Twelve b = size of the crack healing zone D1h = creep compliance coefficients of power law for healing process Hv = healing form of the Jintegral representing the mechanical strain energy needed to heal a fractured surface area Lytton et al. (1998) proposed that the nonpolar part or Lifshitzvan der Waals components of the cohesive surface energy actually inhibits shortterm healing: 1
⎡ K E D H ⎤m h1 = ⎢ h R LW1h v ⎥ β 2γ ⎣ ⎦ i
(1226)
h
Direct tensile, controlled—strain fatigue experiments were conducted at Texas A&M University (Little et al. 1997) on a limestone mixture with four different asphalt binders. Healing rates were determined based on the recovery of dissipated pseudostrain energy (DPSE) following rest periods which varied in length from 30 seconds to 45 minutes. The results showed that as the Lifshitzvan der Waals components of surface energy of • the binder• increase, h1 decreases; while as the acidbase components of the binder increase, h2 increases (Little et al. 1997). An alternative approach to model the contribution of healing during rest periods is to identify the mechanisms responsible for healing and measure the material properties that are required to model these mechanisms. Little and Bhasin (2007) describe the healing mechanism as comprised of two main processes. The first is the wetting or crack closing process followed by intrinsic healing or gain of strength across the wetted crack surfaces. Following the broad methodology originally described by Wool and O’Connor (1981), the effective healing of a crack surface is modeled using the following convolution form (Little and Bhasin, 2007): τ =t
R=
∫
τ =−∞
where
Rh (t − τ )
dφ(τ , X ) dτ dτ
(1227)
R = net macroscopic healing Rh(t) = intrinsic healing function of the material f(t, X) = wetting function t = time variable
The wetting distribution function f(t, X) defines wetting at the contact of the two crack surfaces on a domain X over time t. The intrinsic healing function, Rh(t), defines the rate at which two crack faces that are in complete contact with each other (“wet”) regain strength due to interdiffusion and randomization of the molecules from one face to the other. From a material property point of view, an asphalt binder that has a molecular structure that favors interdiffusion of molecules will promote healing. The wetting distribution function f(t, X) in Eq. (1227) can be described using Schapery’s (1989) relationship between rate of healing and material properties such as the work of cohesion due to surface energy, creep compliance properties of the binder, and effective crack healing length as follows: ⎡ 1 ⎧ πΔGc ⎫⎤ dφ(t , X ) = a b = β ⎢ ⎨ 4(1 − ν 2 )σ 2 β − D0 ⎬ ⎥ dt D k m 1 b ⎩ ⎭⎦ ⎣
−1m
(1228)
Surface Energy and Performance of Asphalt where
ΔGc = the work of cohesion as described before ν = the Poisson’s ratio D0, D1, and m = creep compliance parameters obtained by fitting D(t) = D0 + D1t m km = material constant that can be computed from m sb = bonding stress at the interface b = crack healing length a b = speed of crack healing
Little and Bhasin (2007) present more details related to the application of this equation with fatigue cracking tests. The intrinsic healing function Rh(t) in Eq. (1227) represents the sum effect of (i) instantaneous strength gain due to interfacial cohesion at the crack interface, and (ii) timedependent strength gain due to interdiffusion of molecules between the crack surfaces. A sigmoid function is used to represent the net effect of the two processes as the intrinsic healing function as follows: Rh (t) = R0 + p(1 − e − qt ) r
(1229)
Rh(t) = timedependent dimensionless function that represents the increase in any mechanical property of the wetted crack interface as a fraction or percentage of the same property for the intact material R0 = instantaneous healing due to cohesion at the crack interface, whose magnitude is proportional to the work of cohesion or surface free energy of the material p, q, and r = material parameters that quantify the effect of healing due to the interdiffusion of molecules between the crack surfaces
where
Moisture Damage Prediction Model Moisture is a key factor in the deterioration of the asphalt pavement. Terrel (1994) identified three mechanisms of moisture damage in the: (a) loss of cohesion (strength) and stiffness of the asphalt film; (b) failure of the adhesive bond between the aggregate, and asphalt cement (often called stripping); and (c) degradation or fracture of the aggregate, particularly when the mixture is subjected to freezing. Based on observations (a) and (b), it is important to consider both the effect of moisture, which migrates into the asphalt or mastic, as well as the effect of moisture on the adhesive bond between asphalt and aggregate. The diffusion of water into the asphalt mastic affects the rheology of the mastic. The clear effect of moisture in the mixture can be seen by measuring the modulus or compliance of the mixture at various levels of saturation. One typically sees a substantial decrease in modulus as the moisture content increases and a corresponding increase in modulus as the moisture content is decreased. This cyclical effect is due, at least partially, to the weakening effect of moisture penetration or permeation on the asphalt cement or mastic. Of course, when water diffuses through the mastic or moves via cracks in the mastic to the asphaltaggregate interface, debonding or stripping can also occur.
Model of Moisture Movement in the Mastic Moisture in the mastic has a significant impact on its rheology and engineering properties. It is necessary to evaluate the impact of moisture in the mastic when
363
364
Chapter Twelve
FIGURE 121
The onedimensional consolidation procedure of soil.
evaluating cohesive strength of the mastic and the adhesive bond between the asphalt cement and/or the mastic and the aggregate. The USD can be used to measure the amount of water vapor absorbed within an asphalt film. The USD will be discussed in detail later, but the basis for USD analysis is measurement of the sorption of various solvents on the surface or within the mass of a liquid, solid, or semisolid. The difficulty involved with measuring the absorption or rate of absorption of water within an asphalt or mastic film using the USD is to separate adsorbed water (on the surface of the asphalt film) from absorbed water (within the asphalt film). To address this problem, the authors developed an absorption model based on diffusion theory to differentiate between adsorption and absorption. The diffusion of water vapor into the asphalt film is similar to onedimensional consolidation of soil presented in Fig. 121. The consolidation process shown in this figure is similar to the nature of absorption of water vapor into asphalt. Figure 121 is a plot of percent consolidation completed versus a time parameter T. As consolidation occurs in saturated soils, excess pore water pressure u is dissipated. In the case of absorption of water vapor into an asphalt film, we may consider percent of consolidation completed to be analogous to the degree of moisture absorbed. The term T, in Fig. 121 is used as the nondimensional time parameter for modeling the consolidation of soil. This is replaced by Dt/ 2 when we consider diffusion rather than consolidation; where D is diffusivity, t is time, and is the thickness of the asphalt or mastic layer. The sorption of water into asphalt or mastic as monitored by the USD test occurs in two stages. In the first stage, both adsorption at the asphalt surface and absorption within the asphalt occur simultaneously. In the second stage, adsorption on the surface of the asphalt comes to equilibrium, but absorption (diffusion) continues and eventually becomes constant. Figure 122 illustrates water vapor sorption into a thin film of asphalt cement. The upper curve of the plot is the sorption curve from the USD. The lower curve is the adsorption curve for water onto the asphalt film surface in the USD test as a function of time. The difference between the two is the absorption of water vapor into the asphalt film. During the second stage of USD testing, only absorption occurs as described by Eq. (1229).
(
w = w100 1 − e
−
3 Dt l2
)+C
(1230)
Surface Energy and Performance of Asphalt
FIGURE 122
The sorption procedure of moisture to asphalt thin ﬁlm. (Cheng 2002.)
where w100 is the maximum absorption of asphalt thin film w is the measured water mass C is the absorption constant at a certain vapor pressure level, and other parameters are defined previously By transforming Eq. (1230), we arrive at ln
( )
dw ⎛ 3Dw100 ⎞ ⎛ 3D ⎞ t = ln ⎜⎝ ⎟− dt l2 ⎠ ⎝ l2 ⎠
(1231)
Parameters D and w100 are determined by performing a linear regression of USD data obtained during the second stage of testing. Since both adsorption and absorption occur during the first stage of USD testing, the model for water content accumulation becomes
(
w = w100 1 − e
−
3 Dt l2
) + w (1 − e a
−α t
)
(1232)
where wa is the maximum adsorption on the surface of asphalt thin film for each vapor pressure stage; a is adsorption time constant which determines the speed of the adsorption; and other parameters are previously defined. The following equation can be derived from the first stage of the equation:
( )
3w100 D − 3lDt ⎡ dw e ln ⎢ − dt l2 ⎣ 2
⎤ ⎥ = ln(α w a ) − α t ⎦
(1233)
Parameters wa and w100 are obtained from a regression analysis of first stage USD lab testing results.
365
366
Chapter Twelve
Adhesive Failure Model Based on Surface Energy Stripping of the asphalt film from the aggregate surface due to adhesive failure in the presence of water may occur either at the pavement surface or internally within the mixture (White 1987). Adhesive strength is influenced by the surface energies of asphalt and aggregate, the surface texture of aggregate, and the presence of water. Equation (126) is used to calculate the surface energy of adhesion between two different materials. For the general case, the surface energy of adhesion for two different materials in contact a , is explained by the following equation: within a third medium, ΔG132 a = γ 13 + γ 23 − γ 12 ΔG132
= 2γ 3LW + 2 γ 1LW γ 2LW − 2 γ 1LW γ 3LW − 2 γ 2LW γ 3LW + 4 γ 3+ γ 3− − 2 γ 3+
(
)
γ 1− + γ 2− − 2 γ 3−
(
γ 1+ + γ 2+
)
(1234)
+ 2 γ 1+ γ 2− + 2 γ 1− γ 2+ Equation (1234) can be used to calculate the adhesion between asphalt cement and aggregate in the presence of water, where subscripts 1, 2, and 3 represent asphalt, aggregate, and water, respectively. The adhesive bond between asphalt and aggregate can also be presented by using Gibbs free energy per unit mass of aggregate as expressed by Eq. (1235), in which the specific surface area (SSA) per unit mass of aggregate also reflects the effect of surface texture of the aggregate. The Gibbs free energy per unit mass of aggregate is the energy needed to pull the asphalt cement film from (detach) the unit mass of aggregate (Curtis et al. 1992). ΔG
ergs ergs cm 2 =γ 2 × SSA g g cm
(1235)
Measurement of Surface Energy Surface Energy of Asphalt Binder Principle of the Wilhelmy Plate Method Applied to Determine Dynamic Contact Angle The contact angle between asphalt cement and a liquid can be measured using the Wilhelmy plate method. This method is based on kinetic force equilibrium of a thin plate immersed or withdrawn from a liquid at very slow and constant rate (Adamson 1997; Maugis 1999). As illustrated in Fig. 123, the dynamic contact angle between asphalt and a probe liquid measured during the immersing process is called the advancing contact angle; while the dynamic contact angle during the withdrawal process is called the receding contact angle. When a plate is suspended in air, Eq. (1236) is valid: F = Wtplate + Wtasphalt – V.rair.g
(1236)
Surface Energy and Performance of Asphalt
FIGURE 123
where
Schematic illustration of Wilhelmy plate method. (Cheng 2002.)
F = force from the DCAA (dynamic contact angle analyzer) balance, which is also the force required to hold the plate Wtplate and Wtasphalt = weight of the glass plate and weight of the coated asphalt film, respectively V = volume of the asphalt plate rair = density of the air g = local acceleration of gravity
When the plate is partially immersed in a fluid, the balance measures the force as Eq. (1237): F = Wtplate + Wtasphalt + Pt gL cosq − Vim rL g − (V−Vim)rair g
(1237)
where Pt = perimeter of the asphalt coated plate gL = total surface energy of the liquid q = dynamic contact angle between asphalt and the liquid Vim = volume of the immersed plate By subtracting Eq. (1235) from Eq. (1237), Eq. (1238) is obtained: ΔF = Pt gL cos q − Vim rL g + Vim rair g
(1238)
Equation (1239) is obtained by rearranging terms in Eq. (1237), and the contact angle can be calculated from all the parameters on the righthand side which are determined during testing (Cheng et al. 2002a and 2002b). cos q = (ΔF + Vim (rL − rair)g)/(Pt gL)
(1239)
367
368
Chapter Twelve −
Surface Energy
gL
gLLW
gL+
g
Water
72.6
21.6
25.5
25.5
51
Glycerol
64
34
3.6
57.4
30
Formamide
58
39
2.28
39.6
19
TABLE 121
L
gLAB
Surface Energies of Probe Liquids (ergs/cm2)
Calculation of the Surface Energy from Dynamic Contact Angles Based on the YoungDupre equation and the assumption that equilibrium film pressure is neglected for asphalt, Eq. (1240) is obtained (Good 1992):
γ Li (1 + cosθ i ) = 2 γ SLW γ LiLW + 2 γ S− γ Li+ + 2 γ S+ γ Li−
(1240)
In Eq. (1240), gLi, g +Li,, and g −Li are surface energy components of the liquid. The term, qi can be measured using the Wilhelmy plate method. There are three unknowns for the asphalt semisolid in this equation: g LWs, g −s, and g +s. These unknowns are the three components of asphalt surface energy: Lifshitzvan der Waals, Lewis base, and Lewis acid, respectively. To solve for the above parameters, at least three probe liquids whose surface energies are known must be used to produce three simultaneous equations. The results reported in this chapter are based on the use of the following three probe liquids: glycerol, formamide, and distilled water. These liquids were selected as probe liquids because of their relatively large surface energies, immiscibility with asphalt, and range of surface energy components. The surface energy components of the three liquids are listed in Table 121. By assuming yi (x) = 1 + cosθi a1i = 2
γ LiLW γ Li+ γ Li− ; and ; a2 i = 2 ; a3 i = 2 γ Li γ Li γ Li
(1241)
x1 = γ SLW ; x2 = γ S− ; x3 = γ S+ The following matrix form of linear simultaneous equations is established: ⎡ a11 ⎢a ⎢ 21 ⎣ a31
a12 a22 a32
a13 ⎤ ⎡ x1 ⎤ a23 ⎥ ⎢ x2 ⎥ = ⎥⎢ ⎥ a33 ⎦ ⎣ x3 ⎦
⎡ y1 ⎤ ⎢ y2 ⎥ ⎢ ⎥ ⎣ y3 ⎦
(1242)
Equation (1242) is easy to solve, and its solution provides the surface energy components of solid asphalt described in Eq. (1241). In more recent work, Little and Bhasin (2006) and Hefer et al. (2006) recommend the use of more than three liquids to improve the reliability and accuracy of the surface energy components that are back calculated based on the measured contact angles with these probe liquids.
Wilhelmy Plate Testing Protocol for Asphalt Surface Energy The methodology for using the Wilhelmy plate to measure the surface energy of asphalt employs the following steps:
Surface Energy and Performance of Asphalt 1. Select three probe liquids with known surface energy components. The results provided in this chapter are based on the use of formamide, glycerol, and distilled water, probe liquids. Recent studies recommend the use of at least five probe liquids (methylene iodide and ethylene glycol in addition to the three mentioned here). 2. Prepare 12 glass microcover slides (50 mm by 24 mm by 0.15 mm) for each asphalt being evaluated. Thoroughly clean and dry the glass plates. 3. Coat a thin asphalt film onto the glass cover slides. Heat the asphalt in the oven to between 90 and 135°C, depending on the viscosity of the asphalt sample. Dip the glass plate into the liquid asphalt to a depth of about 15 mm, and then suspend the slide in the oven. Allow the excess asphalt to drain from the plate surface until only a very thin layer of asphalt remains on the plate. Hold the asphalt plate in an upside position in the oven for a few minutes until a smooth asphalt film surface is observed over the width of the plate and with a vertical coverage of about 10 mm (from the top of the asphalt coating the plate to the bottom of the plate). Remove the plate from the oven. Measure the dimensions of the asphalt sample plate and then place the plates in a desiccator overnight. 4. Measure the contact angle between the asphalt and each probe liquid using at least three different asphalt coated plates. Each plate may only be used once. Use WinDCA software to acquire force data from the balance and the computed advancing and receding contact angles. 5. Check the probe liquid surface energy by the Du Nouy Ring method and test the clean glass plate between sample measurements. This step is to make sure the surface energy components of the liquid do not change with time. Du Nouy is generally credited with the development of a rather widely used method to obtain the surface energy of a liquid through the determination of the force to detach a ring or loop of (platinum) wire from the surface of a liquid. The first approximation of the detachment force is obtained by multiplying the surface tension by the periphery of the surface detached: F = 4πRg, where F, R, and g represent the detachment force, radius of ring, and the surface energy of the liquid, respectively. 6. Use the measured contact angle with the different probe liquids to generate a set of equations using Eq. (1239). Solve for the three unknown surface energy components of the asphalt.
Surface Energy Testing Results of Asphalts As an illustration, Fig. 124 shows a typical contact angle measurement using the Wilhelmy plate method. The lower curve in the figure represents the advancing force measurement during the immersion process, and the upper curve represents the receding or dewetting process. Each curve is unique for each asphalt and liquid pair. Normally, the energy calculated from the receding (fracture) process is higher than that calculated during advancing (healing) process. Ten asphalts were tested using the Wilhelmy plate method at room temperature. Asphalts AAA, AAD, and AAM are from the Strategic Highway Research Program (SHRP) designated Materials Reference Library. In addition, a PG7028 asphalt was tested together with high cure rubber (HCR), and 12 percent rubber blended asphalt (12Rubber) prepared by the Chemical Engineering Department at Texas A&M University.
369
Chapter Twelve DAAMG3.DCD  Cycle #1 of 2  auto analysis  Thu Jul 05 13:10:37 2001 300 Advancing (0.0405, 87.68°, 0.7885) Receding (0.8418, 32.67°, 0.8387)
250 200 Force, mg
370
150 100 50 0 0
1
2
3
4
5
6
7
Position, mm
FIGURE 124
Wilhelmy plate test results for AAM with glycerol. (Cheng et al., 2002b.)
The HCR is a soft base asphalt that is airblown and combined with 12 percent of No. 20 mesh rubber at 260oC for 3 hours and blended at 1600 rpm. HCR3month, HCR6month, 12Rubber3month, and 12Rubber6month represent the HCR and 12Rubber aged in the lab for 3 months and 6 months, respectively. Tables 122 and 123 present the surface energy results of the asphalts tested. Standard deviations were calculated from at least three replicates and are very small (coefficients of variation ranged between 3 and 10 percent) showing that the results are repeatable. Different asphalts have very different surface energies. For the surface energy of wetting, the order is PG7028 > 12Rubber6month > 12Rubber3month > HCR3month > AAD > HCRunaged > AAA > HCR6month > 12RbuuberUnaged > AAM (where the symbol > means that surface energy of the preceding binder is greater than that of the following binder). The maximum surface energy, which occurs in PG7028, is 19.85 ergs/cm2; and the minimum for asphalt AAM, is 10.00 ergs/cm2. For the surface energy of dewetting, the order is HCRunaged > AAA > PG7028 > HCR3month > AAM > 12RubberUnaged > 12Bubber3month > 12Rubber6month > HCR6month > AAD. The maximum value, which occurs in HCRunaged, is 55.51 ergs/cm2; and the minimum value for asphalt AAD is 27.14 ergs/cm2. Asphalt Name
g LW
g–
g+
g AB
g Total
AAA
11.52
1.84
1.02
2.73
14.25
AAD
14.73
2.57
0.00
0.00
14.73
AAM
4.00
1.39
6.57
6.00
10.00
18.23
3.59
0.19
1.63
19.85
HCR
6.51
4.17
3.44
7.56
14.07
12Rubber
8.82
4.90
2.00
6.24
15.06
PG7028
Source: Cheng et al., 2002a.
TABLE 122
Surface Energies of Asphalt Binders from Advancing Contact Angles (Units: ergs/cm2)
Surface Energy and Performance of Asphalt
Asphalt Name
g LW
g –
AAA
15.98
41.69
AAD
8.59
AAM PG7028 HCR 12Rubber
g+
g AB
g Total
8.87
38.46
54.44
14.48
5.96
18.55
27.14
9.33
21.40
17.79
39.02
48.35
7.34
28.76
18.38
45.98
53.31
13.22
39.51
11.31
42.28
55.51
3.34
37.82
15.69
48.71
52.05
Source: Cheng et al., 2002a.
TABLE 123 Surface Energies of Asphalt Binders from Receding Contact Angles (Units: ergs/cm2)
g LW
Asphalt Name
g–
g+
g AB
g Total
HCRunaged
6.51
4.17
3.44
7.56
14.07
HCR3month
8.82
4.90
2.00
6.24
15.06
HCR6month
8.94
1.21
3.62
4.18
13.11
12RubberUnaged
8.80
1.52
2.81
4.13
12.93
12Rubber3month
12.75
1.27
1.18
2.44
15.19
12Ribber6month
14.91
1.74
1.07
2.64
17.55
Source: Cheng et al., 2002a.
TABLE 124
Surface Energies of Asphalt Binders at Different Aging Stages from Advancing Contact Angles (Units: ergs/cm2)
Asphalt Name
g LW
g–
g+
g AB
g Total
HCRunaged
13.22
39.51
11.31
42.28
55.51
HCR3month
3.34
37.82
15.69
48.71
52.05
HCR6month
10.62
15.42
8.62
23.05
33.67
12RubberUnaged
13.62
18.87
10.52
28.18
41.80
12Rubber3month
8.32
18.51
13.12
31.16
39.48
12Ribber6month
6.76
15.28
15.03
30.30
37.06
Source: Cheng et al., 2002a.
TABLE 125 Surface Energies of Asphalt Binders at Different Aging Stages from Receding Contact Angles (Units: ergs/cm2)
Tables 124 and 125 show the aging effect on the surface energy asphalt. Aging can significantly affect the surface energy characteristics. HCR originally has a surface energy of wetting of 14.07 ergs/cm2 and surface energy of dewetting of 55.51 ergs/cm2. After aging for 6 months in the lab, the surface energy of wetting reduces to 13.11 ergs/cm2, and
371
372
Chapter Twelve the surface energy of dewetting reduces to 33.67 ergs/cm2. The surface energy of dewetting fell almost 30 percent. Unaged 12Rubber has a surface energy of wetting of 12.93 ergs/cm2, and surface energy of dewetting of 41.80 ergs/cm2. After 6 months of labaging, the surface energy of wetting increased to 17.55 ergs/cm2, and the surface energy of dewetting fell to 37.06 ergs/cm2. Bhasin et al. (2007) report similar results for several different base and polymer modified asphalt binders subjected to artificial aging using the Pressure Aging Vessel (PAV).
Surface Energy of Aggregate Development of USD Testing Protocol The USD was developed to test the surface energy of aggregate. The USD is comprised of a Rubotherm magnetic suspension balance system, computer, Messpro (computer software), temperature control, high quality vacuum, vacuum regulator, pressure transducer, probe liquid container, and a vacuum dissector. The schematic USD setup diagram is shown in Fig. 125. The Rubotherm magnetic suspension balance has the ability to measure a sample mass of up to 200 g with accuracy of 10−5g, which is quite sufficient for precise measurement of the surface free energy of aggregate. The size of aggregate tested is that which passes the No. 4 (4.75mm) sieve but is retained on the No. 8 sieve (2.36mm). The aggregate sample holder as shown in Fig. 126 is made of a
FIGURE 125 Simpliﬁed USD setup diagram. (Cheng et al., 2002b.)
FIGURE 126
Aggregate surface energy testing sample holder.
Surface Energy and Performance of Asphalt
Probe Liquid
g
g LW
g AB
g+
g–
nhexane
18.4
18.4
0
0
0
MPK
24.7
24.7
0
0
19.6
Water
72.8
21.8
51.0
25.5
25.5
TABLE 126 Surface Energies of the Probe Liquids Used with the Aggregates (Units: ergs/cm2)
fine aluminum screen. The surface free energy of aggregate will not be affected by the size of the aggregate because size is accounted for during the calculation process. The total and component surface energies of the selected probe liquids: nHexane, Methyl Propyl Ketone (MPK), and distilled water at 25o C, were obtained from the literature (Good, 1992) and are listed in Table 126.
USD Testing Protocol for Aggregate Surface Energy The methodology by which the USD is used to measure surface free energies of aggregate includes the following steps: 1. Select three probe liquids with known surface energy components. The results reported in this chapter are based on the following three probe liquids: nhexane (nonpolar), MPK (methyl propyl ketone/2pentanone, monopolar), and water (bipolar) were selected. 2. Degas the adsorption cell to achieve a vacuum with absolute cell pressure below 5 millitorr. Use the pressure controller to achieve a vapor pressure of about 10 percent of the maximum vapor pressure of the selected probe. Allow the mass of the aggregate to come to equilibrium. Measure the specific amount of probe vapor adsorbed on the surface of the aggregate. 3. Repeat step 2 by increasing the probe vapor pressure in intervals of 10 percent of its maximum vapor pressure and record the equilibrium amount of vapor adsorbed for each increment until saturation vapor pressure is reached. 4. Correct the adsorption data for solvent vapor buoyancy by using the generalized Pitzer correlation (Smith et al. 1996). 5. Calculate the specific surface area of aggregate by using the BET Eq. (1243). P 1 ⎛ c − 1⎞ P = + n(P0 − P) ⎜⎝ nm c ⎟⎠ P0 nm c
(1243)
where nm = monolayer capacity of the aggregate surface c = BET constant n = specific amount of vapor adsorbed on the aggregate surface P = partial vapor pressure of the probe vapor P0 = saturated vapor pressure of the probe vapor 6. Calculate the spreading pressure at saturation vapor pressure, πe, for each solvent using Gibbs adsorption Eq. (1244).
πe =
RT A
Po
n
∫ P dP 0
(1244)
373
374
Chapter Twelve where πe = spreading pressure at saturation vapor pressure of the solvent, ergs/cm2 R = universal gas constant, 83.14 cm3 psi/gK T = absolute temperature, K A = specific surface area of absorbent, m2 P0 = saturated vapor pressure of solvent, psi n = specific amount adsorbed on the surface of the absorbent, mg P = vapor pressure, psi 7. Express the work of adhesion of a liquid on a solid WA in terms of the surface tension (surface energy) of the liquid gl and the equilibrium spreading pressure of adsorbed vapor on the solid surface πe as shown in Eq. (1245) (Zettlemoyer 1969), (1246), and (1247): WA = π e + 2γ l = ΔGsl
(1245)
ΔGsl = ΔGslLW + ΔGslAB = 2 γ sLW γ lLW + 2 γ s+ γ l− + 2 γ s− γ l+
π e + 2γ = 2 γ sLW γ lLW + 2 γ s+ γ l− + 2 γ s− γ l+
(1246) (1247)
where subscripts s and l refer to solid and liquid, respectively. 8. Calculate the surface energy of the aggregate a. Use Eq. (1247) to calculate gsLW of the surface from a nonpolar solvent on the surface of the asphalt or aggregate:
γ sLW =
(π e + 2γ l )2 4γ lLW
(1248)
b. If a known monopolar basic liquid vapor (subscript m) and a known bipolar liquid vapor (subscript b) are selected, use Eqs. (1249) and (1250) to calculate the values of g +s and g −s:
γ
+ s
(π =
γ
− s
(π =
e
LW + 2γ lm − γ sLW γ lm − 4γ lm
e
)
2
+ 2γ lb − 2 γ sLW γ lbLW − 2 γ s+ γ lb− 4γ lb+
(1249)
)
2
(1250)
c. Calculate the total surface energy of the aggregate gs using Eq. (1251):
γ s = γ sLW + 2 γ + γ −
(1251)
Little and Bhasin (2006) and Bhasin and Little (2006) present a more detailed description and an improvised version of this methodology to determine the surface energy components of aggregates using the USD.
Aggregate Testing Three aggregates; Texas limestone, Colorado limestone, and Georgia granite; were selected for surface energy measurement. After wet sieving, about 150 g of each
Surface Energy and Performance of Asphalt aggregate were collected. The aggregates were washed again using distilled water and then put into 120°C oven and dried for at least 8 hours. The aggregate sample container shown in Fig. 126 was washed carefully with distilled water and acetone and dried in an oven at 120°C for 1 hour and allowed to come back to room temperature. The aggregate samples were then tested using the procedure described above with water, nhexane and MPK probe vapors.
Surface Energy of Standard Glass Balls Uniformly sized standard glass balls (4 mm in diameter) were prepared and tested in order to verify the precision of the USD measuring system. Both adsorption and desorption tests with three vapors—distilled water, MPK, and nHexane—were performed using the USD system. Figure 127 shows six adsorption test results for distilled water on glass balls. Curves 7 and 10 represent adsorption tests in which the adsorbed mass was measured at different vapor pressures while the vapor was incrementally increased from vacuum to saturation vapor pressure into the sample chamber of the USD. Curves 6, 8, and 9 represent desorption tests in which the desorbed mass was measured at different vapor pressures while the vapor was incrementally reduced from saturation vapor pressure to vacuum in the sample chamber of the USD. Curve 11 shows an adsorption test followed by a desorption test on the same sample. From these testing results, one observes that (a) all curves are grouped very closely, and (b) the desorption curves are not significantly different from the adsorption curves. The testing results are repeatable, and the adsorptiondesorption hysteresis phenomenon is insignificant. Based on the test results of distilled water, nhexane, and MPK, the surface energy and its components for the glass balls were calculated and are listed in Table 127. These results also demonstrate that the measurements of surface energy using the USD are precise and repeatable and that adsorptiondesorption hysteresis is not significant.
FIGURE 127 The adsorption and desorption isotherms of water vapor onto glass ball samples. (Notes: 6, 8, 9: desorption; 7, 10: adsorption; 11: adsorption and desorption.) (Cheng et al., 2002a.)
375
376
Chapter Twelve
Glass Balls
g LW
g–
g+
g AB
g Total
Mean Value
190.0
10.6
852.5
189.5
379.6
Standard Deviation
0.2
0.0
101.5
11.7
11.6
Coefficient of Variation
0.1%
0.5%
11.9%
6.2%
3.1%
TABLE 127 Surface Energy and Its Components of the Glass Balls (Units: ergs/cm2)
Surface Energy of Aggregates The BET theory was used to calculate the specific surface areas (SSA) of aggregates, which is the surface areas per unit mass of absorbent. The specific surface areas of the selected samples calculated using the adsorption isotherm of the three probe vapors are listed in Table 128. The SSAs in descending order are Texas limestone, Colorado limestone, and Georgia granite. Each aggregate sample had approximately the same size gradation. The SSAs of the limestones are considerably greater than those of Georgia granite. In other words, the two limestones have a relatively rougher surface texture than the Georgia granite. This finding was verified by visual examination using a scanning electron microscope. The saturated spreading pressures of the probe liquids onto the aggregate surface were calculated using Gibb’s equation and the results are listed in Table 129. Surface energy measurement results are summarized in Table 1210. Based on the surface free energy data from Table 1210, the following observations are made: • Surface energy is a material property of the aggregate. A large difference in surface energies exists among aggregates of different mineralogical types. The Georgia granite shows the highest acid component of surface energy while the limestone shows the highest base component of surface energy. Readers are cautioned that the acidbase scale of surface energy components is relative. Therefore, it is permissible to compare or rank different materials based on the relative magnitudes of their acid or base components. However, one cannot compare or make absolute appraisals based on magnitudes of the acid or base component of a given material (Little and Bhasin 2006). For example, different aggregates may be compared and ranked according to the acidic (or basic) character of surface energy. However, for any given aggregate, the acid Aggregate
Specific Surface Areas (m2/g)
Georgia granite #4#8 1
0.10
Georgia granite #4#8 2
0.11
Texas limestone #4#81
0.44
Texas limestone #4#82
0.43
Colorado limestone #4#81
0.31
Colorado limestone #4#82
0.26
Source: Cheng et al., 2002b.
TABLE 128 Specific Surface Area for Two Replicate Measurements of the Three Selected Aggregates
Surface Energy and Performance of Asphalt
Spreading pressure pe (ergs/cm2)
Aggregates
nHexane
MPK
Georgia granite #4#8 1
73.02
22.90
98.66
Georgia granite #4#82
50.08
23.62
115.53
Texas limestone #4#81
44.16
44.77
102.89
Texas limestone #4#82
41.81
34.24
139.57
Colorado limestone #4#81
38.73
40.71
93.15
Colorado limestone #4#82
41.05
38.89
88.66
*
Water
*The size of the aggregate. Source: Cheng et al., 2002b.
TABLE 129 Equilibrium Spreading Pressure of Probe Vapors for Two Replicates of Each Aggregate Type
Aggregate
g–
g+
g AB
g Total
SSA m2/g
Georgia granite
96.0
73.3
133.2
206.5
0.1
Texas limestone
285.5
16.1
86.5
102.6
0.4
Colorado limestone
206.5
7.3
79.9
87.3
0.3
Source: Cheng et al., 2002b.
TABLE 1210 Sample Mean of Surface Free Energies (Units: ergs/cm2) and Specific Surface Areas
component cannot be compared with its base component to conclude that the aggregate surface has more acidic or basic character. • The difference between duplicate samples was larger than expected. These differences were most notable among replicates of the granite samples and are most probably due to lower specific surface area of the aggregate and consequently a reduced precision in the adsorbed mass. • The adsorption and desorption characteristics of probe vapors were found to be very similar for glass beads. Based on this observation, surface energy components for the aggregates were determined using only adsorption tests which allowed better control than desorption tests.
Validation of the Fatigue and Healing Models Cohesive and Adhesive Bond Energies Cohesive fracture energy is two times the surface energy according to Eq. (124). The higher the total surface energy, the stronger the cohesive strength of the material is. The order of cohesive strength among the 10 asphalts is the same as the order of the surface energy of dewetting.
377
378
Chapter Twelve
Aggregate
Texas Limestone1
Georgia Granite1
Asphalt
ΔGLW
ΔGAB
ΔGTotal
ΔGLW
ΔGAB
ΔGTotal
AAM
56.8
148.1
204.9
70.5
128.1
198.6
PG7028
50.4
151.3
201.6
62.5
136.7
199.2
HCRunaged
67.6
121.2
188.8
83.9
127.6
211.6
12Rubber6month
48.4
135.7
184.0
60.0
114.4
174.4
12RubberUnaged
68.7
114.8
183.4
85.2
106.2
191.4
AAA
74.4
108.4
182.7
92.3
121.8
214.1
12Rubber3month
53.7
127.5
181.2
66.6
113.2
179.8
HCR3month
34.0
141.2
175.2
42.2
138.0
180.2
HCR6month
60.6
103.9
164.5
75.2
96.1
171.3
AAD
54.5
87.0
141.5
67.6
85.2
152.8
Source: Cheng et al., 2002a.
TABLE 1211 Adhesion between Asphalt and Aggregate (Units: ergs/cm2)
Adhesion between the asphalt mastic and aggregate can be obtained from Eqs. (127), (1210), and (1212). Texas limestone and Georgia granite were selected for adhesion analysis. The surface energies of Texas limestone and Georgia granite were measured by the USD method. Adhesion strengths are shown in Table 1211. The adhesive strength depends not only on the surface energy of asphalt, but also on the surface energy of aggregate. Normally, the adhesion between the asphalt and aggregate is higher than the cohesion of the binder. For Georgia granite, the order of adhesion with asphalt is AAA > HCRunaged > PG7028 > AAM > 12RubberUnaged > HCR3month > 12Rubber3month > 12Rubber6month > HCR6month > AAD, where the symbol > means greater than. Aging reduces the adhesion between Georgia granite and both HCR6month and 12Rubber6month. The rank order of adhesion between Texas limestone and asphalts is different from that of Georgia granite. The order of the adhesion between the Texas limestone and asphalt is AAM > PG7028 > HCRunaged > 12Rubber6month > 12RubberUnaged > AAA > 12Rubber3month > HCR3month > HCR6month > AAD. Aging reduces the adhesion of HCR with Texas limestone as it does with the granite. But for 12Rubber with Texas limestone, aging reduced the Lifshitzvan der Waals part of adhesion and yet increased the acidbase part of adhesion. The total adhesion between the 12Rubber and the Texas limestone did not change with aging. Based on the above analysis, the strength of adhesion is determined by both asphalt and aggregate characteristics. Analysis based on the order of adhesive bond can help to select the best and most compatible asphaltaggregate pairings in terms of adhesive strength.
Cohesive Fatigue and Healing It is evident from Eqs. (1215) through (1222) that surface energy is an important parameter in the micromechanical fatigue model. Because the value for asphalt mixes of mf is between 0 to 1, a higher value of surface energy of dewetting yields a lower the value of A [Eq. (1217)] and thus a longer fatigue life for a given set of values of all other parameters and loading
Surface Energy and Performance of Asphalt
FIGURE 128
Total surface energy of dewetting of asphalts. (Cheng et al., 2002a.)
conditions. Figure 128 shows the surface energies for the 10 asphalts evaluated in this study. In this case the receding contact angle or surface energy of dewetting was used. Hefer et al. (2006) present a more detailed critique on the differences and recommendations for use of wetting and dewetting surface energies (based on advancing and receding contact angles, respectively). If only the surface energy of asphalt is considered, the order of the fatigue life (high to low) is HCRunaged > AAA > PG7028 > AAM > 12RubberUnaged > AAD, where > refers to greater than. Aging will reduce the fatigue life of the same asphalt. Aging changes the rank order of fatigue lives to HCRunaged > HCR3month > HCR6month, and 12RubberUnaged > 12Rubber3month > 12Rubber6month. In a mixture, the fatigue also depends on the other parameters, such as creep compliance or relaxation modulus. See Eqs. (1217) and (1222). According to previous research (Little et al. 1997, 1999; Lytton et al. 1998) and Eqs. (12.22), (12.28), and (12.29), the surface energy of the asphalt directly affects fatigue cracking and healing process. Figures 129 and 1210 show the Lifshitzvan der Waals components and acidbase components of surface energy of wetting, respectively. Asphalts AAD and PG7028 have relatively higher Lifshitzvan der Waals components and lower acidbase components. Asphalts AAM and HCRunaged have relatively lower Lifshitzvan der Waals components and higher acidbase components. Thus AAM and HCR should have a better healing capability than AAD and PG7028 based on Eqs. (12.24) through (12.26). Aging increases the Lifshitzvan der Waals component and decreases the acidbase component, Fig. 1211. Thus, the healing ability of the binder should drop because of the aging of the binder. Figures 1212 (Si 2001) and 1213 (Kim et al. 2002) validate these findings. In Si’s work, the extension index is the percent increase in fatigue life due to rest periods. Fatigue lives of samples subjected to 10 rest periods of 2 minutes each were compared to fatigue lives of identical samples without rest periods. In Fig. 1213, Kim et al. (2002) used a healing potential index (HPI), which is based on the rate of damage, and the extension in the load cycles to failure (FLI) to monitor the effects of healing. The data in Figs. 1212 and 1213 are consistent with the hypothesis and models presented earlier and show that asphalt AAM is a better healer than asphalt AAD.
379
380
Chapter Twelve
FIGURE 129 Lifshitzvan der Waals components of surface energy of wetting for asphalts. (Cheng et al., 2002a.)
F IGURE 1210 Acidbase components of surface energy of wetting for asphalts. (Cheng et al., 2002a.)
Another example can be clearly seen in Figs. 1214 and 1215 (Kim et al. 2002). In the figures, RP refers to ten 30s rest periods, and ΔNf represents the increase in the number of cycles to fatigue failure due to the rest periods. Furthermore, HCR is the best healer and healing potential drops significantly with aging.
Adhesive Fatigue and Healing In asphalt pavements, fracture occurs not only within the asphalt mastic, but also at the interface of asphalt binder and aggregate. The analysis of the fatigue life of a pavement should consider both cohesive fracture and adhesive fracture. Adhesion is calculated from the surface energy (based on receding contact angles) and is listed in Table 1212. The higher the adhesive bond energy, ΔG, the higher is the fatigue life of pavement material. For adhesive failure, the work of adhesion can be used for ΔGf in Eq. (1222).
FIGURE 1211 Aging effect on the components of surface energy of wetting. (Cheng et al., 2002a.)
FIGURE 1212 Extension fatigue life (from direct tensile fatigue testing due to healing for AAD, AAM, and HCR with Brazos gravel (BG) and limestone (LS) aggregates). (Cheng et al., 2002a.)
FIGURE 1213 Increased fatigue life due to healing from dynamic mechanical analyzer (DMA) tests on AAD1, AAM1, and HCR1. (Kim et al., 2002.)
381
382
Chapter Twelve
FIGURE 1214 DMA (dynamic mechanic analyzer) fatigue and healing test on asphalt AAM. (Kim et al., 2002.)
FIGURE 1215 DMA (dynamic mechanic analyzer) fatigue and healing test on asphalt AAD. (Kim et al., 2002.)
Moisture Damage Modeling and Its Mechanistic Validation To validate the hypothesis of the effect of surface energy and adhesive bonding and moisture diffusion on moisture damage of asphalt mixtures, four asphalt binders; AAD, AAM, HCRunaged, and HCR6month and three aggregates; Georgia granite, Texas limestone, and Colorado limestone were selected. Using Eqs. (124) and (125), the cohesion of asphalt binders and/or mastics were calculated and data are presented in Table 1213. Asphalt AAM has the highest cohesion, and AAD has the least based on surface energy. Aging substantially reduces the cohesion of high cure rubber asphalt (HCR). The adhesive bond energy in the asphalt mixtures was calculated using Eqs. (127), (1210), and (1212). The results are shown in Table 1214. From Table 1214, several conclusions are drawn:
Surface Energy and Performance of Asphalt
Texas Limestone1
Aggregates
Georgia Granite1
Asphalt
2 γ 1LW γ 2LW
2 γ 1+ γ 2− + 2 γ 1− γ 2+
2 γ 1LW γ 2LW
2 γ 1+ γ 2− + 2 γ 1− γ 2+
AAM
37.2
88.0
46.2
61.8
HCRunaged
47.5
65.1
58.9
56.4
12RubberUnaged
55.2
58.2
68.5
45.0
HCR3month
55.2
50.4
68.5
49.4
HCR6month
55.6
65.6
69.0
48.1
12Rubber3month
66.4
38.1
82.4
32.4
12Rubber6month
71.8
36.5
89.1
33.2
AAA
63.1
35.7
78.3
33.1
PG7028
79.4
17.0
98.5
27.1
AAD
71.4
2.0
88.6
15.8
Source: Cheng et al., 2002a.
TABLE 1212 Surface Energies of Adhesion for Healing (Units: ergs/cm2)
Cohesion
ΔG LW
ΔG AB
ΔG Total
AAD1
17.2
37.1
54.3
AAM1
18.7
78.0
96.7
HCRunaged1
27.2
56.4
83.6
HCR6month1
13.5
60.6
74.1
Source: Cheng et al., 2002b.
TABLE 1213 Cohesion of Binders Calculated from Surface Energy (Units: ergs/cm2)
Georgia Granite
Texas Limestone
Colorado Limestone
AAD1
152.8
141.5
125
AAM1
198.6
204.9
178.8
HCRunaged1
211.6
188.8
165.7
HCR6month1
171.3
164.5
145.1
H2O
256.3
263.5
231.2
Source: Cheng et al., 2002b.
TABLE 1214 Adhesion of AsphaltAggregate or WaterAggregate (Units: ergs/cm2)
383
384
Chapter Twelve • Aging has a significant impact on cohesion and adhesion. Both the cohesion of HCR and the adhesion between the HCR and all three aggregates decreased because of aging of the HCR asphalt. • At temperatures near 25°C, the descending order of free energy of adhesion (high surface energy to low) for Georgia granite is water > HCRunaged1 > AAM1 > HCR6month1 > AAD1; the order with Texas limestone is water > AAM1 > HCRunaged1 > HCR6month1 > AAD1; and the order with Colorado limestone is water > AAM1 > HCRunaged1 > HCR6month1 > AAD. Water is always highest in terms of adhesion. This means that the three aggregates tested are hydrophilic. This gives theoretical support to the common knowledge that water disrupts the asphalt aggregate bond due to its favorable affinity to bond to the aggregate surface. Furthermore, different asphalt binders have different adhesion characteristics, which influence the moisture damage potential of the pavement. AAM1 and HCR have relatively high adhesion potentials in the presence of water. • The adhesion potentials between asphalt AAM and aggregates are ranked from highest to lowest as Texas limestone > Colorado limestone > Georgia granite, and the adhesion potentials between asphalt AAD and aggregates are ranked from highest to lowest as Georgia granite > Texas limestone > Colorado limestone. However, the limestones tested on average have three times more surface area than granite. This results in a larger effective area of adhesion. For a given gradation, this idea can be better explained by using Gibbs free energy per unit mass of aggregate material. Equation is used to calculate the Gibbs free energy per unit mass of aggregates. The results are listed in Table 1215. The results show that for a unit amount of aggregate, the Gibbs free energy in the order of highest to lowest is: Texas limestone > Colorado limestone > Georgia granite.
Stripping Analysis Stripping as an adhesion failure mechanism may occur either at the pavement surface or internally within the mixture (White 1987). At the surface, stripping begins at weak points such as joints, areas of poor quality control or areas of high airvoid content due to low compaction. Gradually, the mastic bond with the surface of the coarse aggregates degrades. The asphalt film may separate from the aggregate by emulsification. In SHRPA403 (Terrel 1994), the following factors that appear to affect adhesion were described: • Surface tension of the asphalt cement and aggregate • Chemical composition of the asphalt and aggregate
Georgia Granite
Texas Limestone
Colorado Limestone
AAD1
1.58E+05
6.14E+05
3.75E+05
AAM1
2.06E+05
8.89E+05
5.36E+05
HCRunaged1
2.19E+05
8.19E+05
4.97E+05
HCR6month1
1.78E+05
7.14E+05
4.35E+05
Source: Cheng et al., 2002b.
TABLE 1215
Gibbs Free Energy per Unit Amount of Aggregate (Units: ergs/g)
Surface Energy and Performance of Asphalt • Asphalt viscosity • Surface texture of the aggregate • Aggregate porosity • Aggregate cleanliness • Aggregate moisture content and temperature at the time of mixing with asphalt cement All of the above factors impact adhesion. Curtis et al. (1993) stated that both siliceous and calcareous aggregate can be strippers, but both can also be nonstrippers. The siliceous aggregate has slick, smooth areas, which may give rise to stripping; and different aggregates may have very different textures. Adhesion between asphalt and aggregate can be directly calculated using thermodynamic theory, which more directly explains why striping occurs in some asphalt aggregate systems while other systems are not prone to stripping. The adhesion calculation actually considers all the factors listed above. Adhesion uses surface energy to reflect the physical and chemical interaction of the surface. Specific surface area calculated by using BET theory accounts for the surface texture of aggregate. It can be used to explain why some granites are more likely to strip than limestone. Stripping may occur in both asphalt granite systems and asphalt limestone systems. According to the adhesion results in Table 1214, the interaction between water and aggregate (granite or limestone) is stronger than between asphalts (AAD1, AAM1, and HCR) and aggregate (granite or limestone). However, water must be able to move to the interface between asphalt and aggregate in order to disrupt the asphaltaggregate interface. Granite has a much smoother surface texture than the two limestones according to Table 1210. At the same gradation, limestone has three times more surface area and almost three times the Gibbs free energy of granite according to Table 1215. Thus, it is much easier to induce adhesive fracture at the asphaltgranite interface than at the asphaltlimestone interface under a given wheel load and under given environmental stresses. Both adhesion and cohesion forces were lowered with binder aging for the HCR mixtures as shown by the results of Table 1213 and Table 1214. An asphalt pavement composed of the aggregates evaluated in this study and HCR binder will, therefore, be more likely to strip following aging of the asphalt binder. If a new interface caused by fracture is in contact with water, it will not heal according to the results of Table 1214. Gradually, the larger areas of adhesive bond will fail resulting in overall mixture failure or stripping.
Accelerated Moisture Damage Testing on AsphaltAggregate Mixtures Asphalt mixtures were prepared with two different asphalt binders (AAD1 and AAM1) and two different aggregates (Texas limestone and Georgia granite). The mixtures were tested in repeated load permanent deformation testing in both the dry and wet condition. In the wet conditioning process, the samples were first placed in a vacuum and soaked to achieve between 85 and 90 percent saturation. Then the samples were subjected to repeated compressive loading while under water. The test results agree well with adhesion theory predictions. As an illustration, the testing results of limestone with AAD1 and AAM1 in dry and wet testing conditions are plotted in Table 1216. Based on the surface energy, the adhesion properties in dry and in the presence of water are calculated and listed in Table 1216. From Fig. 1216, it can be seen that in the dry testing condition, the AAMlimestone mixture is more resistant to permanent deformation than the AADlimestone mixture. From Table 1216, the calculated adhesion between AAM and Texas limestone is higher
385
386
Chapter Twelve Dry Condition ΔG12∗
Wet Condition ΔG132∗
AAMLimestone
204.9
−30.9
AADLimestone
141.5
−66.8
AAMGranite
198.6
−30.0
AADGranite
152.8
−48.3
∗
Asphalt = 1, aggregate = 2, and water = 3 Source: Cheng et al., 2002b.
TABLE 1216
Adhesion of AsphaltAggregate Mixtures in the Dry and Wet Conditions Calculated Using the Surface Energies (Units: ergs/cm2)
FIGURE 1216 Accelerated permanent deformation experiments for AADlimestone and AAMlimestone in the dry and wet conditions.
than that between AAD and Texas limestone. Figure 1216 shows that the AADlimestone mixture develops more moisture damage than the AAMlimestone mixture does in the presence of water (wet testing condition). Table 1216 shows that the AADlimestone adhesive bond is more negative in the presence of water than the AAMlimestone adhesive bond. The more negative value of adhesion means more sensitivity to moisture damage or stripping. Thus, the testing results are consistent with the adhesion theory predictions. Diffusion of moisture into the asphalt film during the moisture conditioning of the asphalt concrete samples is another important reason for the moisture damage. The USD was used to measure the moisture sorption characteristic of distilled water into the asphalt thin film AAD and AAM. By using the diffusion model, the parameters D and W100 (defined in the previous discussion of diffusion) were calculated at the water vapor pressure, p, at which the p/p0 is equal to 0.75, where p0 is saturated vapor pressure of water vapor. The results are presented in Table 1217. This analysis shows that more water can permeate through the AAD binder than through the AAM binder. This could mean that more water can permeate through a mastic comprised of AAD than AAM and thus begin the mechanism of stripping at the interface. Although this certainly cannot be proven with the sparse data at hand, it is consistent with the observed data from Fig. 1216.
Surface Energy and Performance of Asphalt
AAD
AAM
D
0.0008
0.0029
W100
1.53E3
1.14E3
W100 ratio
AAD/AAM =
1.34
TABLE 1217 The Results Computed from the USD by Diffusion Model
Conclusions 1. Surface energies of asphaltaggregate systems, previously referred to by Schapery (1984) as surface energy density, can be measured using the Wilhelmy plate method and the USD. In the case of aggregates, the peculiarity of sample size, irregular shape, mineralogy, and surface texture are accommodated in the analytical methods used with the USD. 2. The surface energies of asphalt cement can be measured in both wetting (simulated crack reformation or healing) and dewetting (simulating crack formation or fracture) modes. Normally, the surface energy of dewetting is higher than that of wetting based on contact angles measured using the Wilhelmy plate method. Aging reduced the surface energies of high cure rubber asphalt for both wetting and dewetting modes. Intuitively, the effect of aging is logical because one would expect aging to render a mixture more susceptible to fracture fatigue. However, the fact that aging decreased surface energies may be counterintuitive as one might expect oxidative aging to increase polarity. This complex subject deserves more study, but it should be remembered that the surface energy characteristics may be considerably different than the bulk chemistry or properties of the asphalt or mastic. 3. The surface interaction properties of aggregates vary widely and are affected by surface area as well as surface energy per unit of mass. Both characteristics must be considered in assessing bond strength and durability in the presence of moisture. The surface energy of aggregate is of a higher magnitude than that of asphalt cement. Aggregate surface energies vary greatly depending on the type of aggregate and its source. Limestone has a (relatively) higher basic component of surface energy, while granite has a (relatively) higher acidic component of surface energy. Furthermore, based on the vapor adsorption test data obtained from the USD, the specific surface area of limestone is much larger than the specific surface area of granite. 4. The interfacial bond strength between an asphalt binder and an aggregate depends on the surface energies of both materials. Georgia granite develops the strongest adhesion with asphalt AAD, but Texas limestone develops the strongest adhesion with asphalt AAM. Determination of the adhesive bond energy will help to select the best asphaltaggregate combinations that assure improved performance. Bhasin et al. (2007) characterized the dry and wet bond strength of combinations of 14 asphalt binders and 10 aggregates. Their work demonstrated the efficacy of screening for and cataloging the potential of moisture damage by using this approach. 5. The surface energy is a fundamental parameter in the fatigue and healing models developed based on Schapery’s fundamental law of fracture mechanics for
387
388
Chapter Twelve viscoelastic materials. Both surface energies of cohesion and adhesion are used in the model. Based on cohesive properties, HCR and AAM have the potential for a longer fatigue life than AAD and 12Rubber asphalt. Aging reduced the fatigue life of HCR and the 12Rubber asphalt. Based on greater work of adhesion values derived from surface energy measurements, HCR and AAM mixtures were expected to have longer fatigue lives than AAD. This was verified by mixture fatigue testing. 6. The rankings of asphalt binders AAD, AAM, and the HCR binders (both unaged and aged) that use the Lifshitzvan der Waals and acidbase components of surface energy to predict efficiency of fracture healing are consistent in every case with previous determinations of fracture healing by Little et al. (1999). 7. Stripping analyses based on surface energy demonstrates that the adhesive bond energy of aggregates with water is higher than that of the aggregate with asphalt at nominal service temperatures. For a similar gradation, the limestone tested in this study has a rougher surface texture resulting in a higher specific surface area and much higher Gibbs free energy per unit of mass of the aggregate than the granite as demonstrated in Table 1215. The granite used in this study was reported to experience stripping related problems. The small specific surface area of this aggregate as well as its typically lower work of adhesion with different asphalt binders explains this observed behavior. Accelerated mechanical mixture testing results validate these predictions based on the adhesive bond energies. 8. Aged pavements are more vulnerable to stripping which can partly be explained by the fact that the work of cohesion and adhesion is typically lowered upon aging. 9. In recent studies, the methodology to obtain surface energy components using the Wilhelmy plate device and USD has been significantly improved (Little and Bhasin, 2006). Due to these improvements, the results reported by Little and Bhasin (2006) differ slightly in magnitude from the results reported in this chapter. However, the general trends remain very similar. 10. For adhesive failure to occur, water must penetrate the asphalt binder or mastic in order to get to the asphalt binder–aggregate interface. The diffusion model developed based on the USD can be used to measure the amount of moisture absorbed by an asphalt film, which can, in turn, be used in moisture damage analysis. Asphalt AAD absorbs more water than AAM and was more susceptible to moisture damage in the mixtures tested in this study. Both the weakening effect of moisture diffusion into the asphalt as well as adhesive bond strength must be considered when modeling moisture damage of asphalt mixtures.
Acknowledgments The authors are very grateful to the Western Research Institute and the Federal Highway Administration. This project is sponsored by Federal Highway Administration and under subcontract to the Western Research Institute. We are particularly grateful to Dr. Ernest Bastian of the Federal Highway Administration (FHWA) for his guidance and encouragement.
Surface Energy and Performance of Asphalt
References Adamson, A. W., and Gast, A. P., (1997), “Physical Chemistry of Surfaces,” 6th ed., John Wiley and Sons, New York. Bhasin, A., Howson, J. E., Masad, E., Little, D. N., and Lytton, R. L., (2007), “Effect of Modification Processes on Bond Energy of Asphalt Binders.” Transportation Research Record: Journal of the Transportation Research Board, Vol. 1998, pp. 29–37. Bhasin, A., and Little, D. N., (2006), “Characterization of Aggregate Surface Energy Using the Universal Sorption Device.” Journal of Materials in Civil Engineering (ASCE), Vol. 19, No. 8, pp. 634–641. Chen, C. W., (1997), “Mechanistic Approach to the Evaluation of Microdamage in Asphalt Mixes,” Ph.D. dissertation, Civil Engineering, Texas A&M University. Cheng, D., (2002), “Surface Free Energy of AsphaltAggregate System and Performance Analysis of Asphalt Concrete,” Texas A&M University, College Station, Tex. Cheng, D., Little, D. N., Lytton, R. L., and Holste, J. C., (2001), “Surface Free Energy Measurement of Aggregates and Its Application on Adhesion and Moisture Damage of AsphaltAggregate System,” 9th Annual Symposium Proceedings of International Center for Aggregate Research, Austin, Tx. Cheng, D., Little, D. N., Lytton, R. L., and Holste, J. C. (2002a), “Surface Energy Measurement of Asphalt and Its Application to Predicting Fatigue and Healing in Asphalt Mixtures,” Transportation Research Records, No. 1810, pp. 44–53. Cheng, D., Little, D. N., Lytton, R. L., and Holste, J. C., (2002b), “Use of Surface Free Energy of AsphaltAggregate System to Predict Moisture Damage Potential,” Journal of the Association of Asphalt Paving Technologists, Vol. 71, pp. 59–88. Curtis, C. W., Lytton, R. L., and Brannan, C. J., (1992), “Influence of Aggregate Chemistry on the Adsorption and Desorption of Asphalt,” Transportation Research Record, No. 1362, pp. 1–9. Elphingstone, G. M., (1997), “Adhesion and Cohesion in AsphaltAggregate Systems,” Ph.D. dissertation, Texas A&M University. Curtis, C. W., Ensley, K., and Epps, J., (1993), “Fundamental Properties of AsphaltAggregate Interactions Including Adhesion and Absorption,” Strategic Highway Research Program Report No. SHRPA341, 8, National Research Council, Washington, D.C. Good, R. J., (1977), “Surface Free Energy of Solids and Liquids: Thermodynamics, Molecular Forces, and Structure,” Journal of Colloid and Interface Science, Vol. 59, No. 3, p. 398. Good, R. J., and van Oss, C. J., (1991), “The Modern Theory of Contact Angles and the Hydrogen Bond Components of Surface Energies,” Plenum Press, New York. Good, R. J., (1992), “ContactAngle, Wetting, and Adhesion: A Critical Review,” Journal of Adhesion Science and Technology, Vol. 6, No. 12, pp. 1269–1302. Hefer, A. W., Bhasin, A., and Little, D. N. (2006), “Bitumen Surface Energy Characterization Using a Contact Angle Approach,” Journal of Materials in Civil Engineering (ASCE), Vol. 18, No. 6, pp. 759–767. Kim, Y. R., (1988), “Evaluation of Healing and Constitutive Modeling of Asphalt Concrete by Means of the Theory of Nonlinear Viscoelasticity and Damage Mechanics,” Ph.D. dissertation, Texas A&M University. Kim, Y. R., Whitmoyer, S. L., and Little, D. N., (1994), “Healing in Asphalt Concrete Pavements: Is It Real?” Transportation Research Record, No. 1454, Transportation Research Board, Washington, D.C., pp. 89–96. Kim, Y. R., Lee, H. J., and Little, D. N., (1997), “Fatigue Characterization of Asphalt Concrete Using Viscoelasticity and Continuum Damage Theory,” Journal of the Association of Asphalt Paving Technologists, Vol. 66, pp. 520–569.
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Chapter Twelve Kim, Y., Little, D. N., and Lytton, R. L., (2002), “Use of Dynamic Mechanical Analysis (DMA) to Evaluate the Fatigue and Healing Potential of Asphalt Binders in Sand Asphalts Mixtures,” Journal of the Association of Asphalt Paving Technologists, Vol. 71, pp. 176–206. Li, W., (1997), “The Measurement of Surface Energy for SHRP Aggregate RB,” Final Report of Cahn Balance Thermogravimetry Gas Adsorption Experiments, Texas A&M University, College Station, Tex. Little, D. N., Prapnnachari, D., Letton, A., and Kim, Y. R., (1993), “Investigation of the Microstructural Mechanisms of Relaxation and Fracture Healing in Asphalt,” Air Force Office of Scientific Research, Final Report No. AFOSR890520. Little, D. N., Lytton, R. L., Williams, D., and Chen, C. W., (2001), “Fundamental Properties of Asphalts and Modified Asphalts—Volume 1: Microdamage and Microdamage Healing,” Federal Highway Administration Final Report—No. FHWARD98141, Washington D.C. Little, D. N., Lytton, R. L., Williams D., and Kim, Y. R., (1997), “Propagation and Healing of Microcracks in Asphalt Concrete and Their Contributions to Fatigue,” Asphalt Science and Technology, pp. 149–195. Little, D. N., Lytton, R. L., Williams, D., and Kim, Y. Richard, (1999), “An Analysis of the Mechanism of Microdamage Healing Based on the Application of Micromechanics First Principles of Fracture and Healing,” Journal of the Association of Asphalt Paving Technologists, Vol. 68. Little, D. N., and Bhasin, A., (2006), “Using Surface Energy Measurements to Select Materials for Asphalt Pavement,” Final Report for Project 937, National Cooperative Highway Research Program, Transportation Research Board, Washington, D.C. Little, D. N., and Bhasin, A., (2007), “Exploring Mechanisms of Healing in Asphalt Mixtures and Quantifying Its Impact,” In: Self Healing Materials, S. van der Zwaag, ed., Springer, Dordrecht, The Netherlands, 205–218. Lytton, R. L., Chen, C. W. and Little, D. N., (2001), “Fundamental Properties of Asphalts and Modified Asphalts, Vol. III: A Micromechanics Fracture and Healing Model for Asphalt Concrete,” Federal Highway Administration Final Report—No. FHWARD98143, Washington, D.C. Masad, E., Branco, V. C., Little, D. N., and Lytton, R. L., (2007), “A Unified Method for the Dynamic Mechanical Analysis of Sand Asphalt Mixtures,” International Journal of Pavement Engineering, In Press. Maugis, D., (1999), Contact, Adhesion and Rupture of Elastic Solids, Springer, Heidelberg, pp. 3–12. Schapery, R. A., (1984), “Correspondence Principles and a Generalized Jintegral for Large Deformation and Fracture Analysis of Viscoelastic Media,” International Journal of Fracture, Vol. 25, pp. 195–223. Schapery, R. A., (1989), “On the Mechanics of Crack Closing and Bonding in Linear Viscoelastic Media,” International Journal of Fracture, Vol. 39, pp. 163–189. Smith, J. M., van Ness, H. C., and Abbott, M. M., (1996), Introduction to Chemical Engineering Thermodynamics, McGrawHill Companies, Inc. 5th ed., New York, NY, p. 538. Si, Z., (2001), “Characterization of Microdamage and Healing of Asphalt Concrete Mixtures,” Ph.D. dissertation, Texas A&M University, p. 136. Terrel, R. L., (1994), “Water Sensitivity of AsphaltAggregate Mixes,” Strategic Highway Research Program Report No. SHRPA403, Nation al Research Council, Washington, D.C. White, T. D., (1987), “Stripping in HMA Pavements,” Hot Mix Asphalt Technology, pp. 18–20. Wool, R. P., and O’ Connor, K. M., (1981), “A Theory of Crack Healing in Polymers,” Journal of Applied Physics, Vol. 52, No. 10, pp. 5953–5963. Zettlemoyer, A. C., (1969), Hydrophobic Surfaces, Academic Press, New York and London, Vol. 8.
CHAPTER
13
Field Evaluation of Moisture Damage in Asphalt Concrete G. W. Maupin, Jr.
Abstract Failure of asphalt pavement caused by moisture damage, also known as stripping, accounts for a considerable expenditure of funds for repair and rehabilitation every year. Stripping involves the loss of adhesion between the aggregate and asphalt cement and/or emulsification of the asphalt cement. Several theories are associated with the stripping phenomenon, such as mechanical adhesion, chemical reaction and molecular orientation, and surface energy. Even though the failure mechanism is not well understood, stripping failures in the pavement must be analyzed so that the correct remedial action may be taken. Several types of field evaluation techniques are available to determine the extent of stripping damage in an asphalt pavement. They all involve removing samples from the pavement. The cores that are usually used for testing should be sampled in a controlled, systematic manner. The techniques used in such testing are visual estimation, image analysis, and strength analysis. The degree of stripping in a sample can be determined by visual estimation or image analysis, each of which has strengths and weaknesses. Strength analysis of cores can also be used to estimate the present and predicted future condition of the pavement. A pseudostrength deterioration curve can be used to picture the deterioration of the asphalt layer from initial construction to a particular time in the future and to determine the need to remove and replace the stripped material.
Introduction Moisture damage of asphalt concrete, also known as stripping, is responsible for many pavement failures and the expenditure of funds for repair and rehabilitation each year in the United States. Stripping can occur as a result of two basic failure mechanisms, adhesion and cohesion, and it is important to be able to recognize these mechanisms when stripping failures are investigated. Adhesion failure is evidenced by a complete separation of the asphalt film from the aggregate surface. Bare aggregate is visible when the pavement is
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392
Chapter Thirteen broken apart, and attempts to extract cores are futile because the asphalt mixture is disintegrated. Cohesion failure manifests as a softening of the asphalt binder in an emulsification process as water penetrates the binder. A cohesive failure probably will not result in the display of bare aggregate, but the asphalt mixture will have low strength. Moisture damage can also result from a combination of these two failure mechanisms, and distinguishing the relative contribution of each one to the failure is difficult. A great deal of money is spent annually on the maintenance and rehabilitation of roads and streets by state departments of transportation, cities, and local governments. Because of the lack of adequate resources and personnel, agencies often merely make an educated guess to decide the type of repair to make. However, pavement failures must be investigated thoroughly in order to apply the correct and most costeffective repair. This chapter discusses some field investigations and techniques of asphalt pavement examination used by the author to determine the severity of stripping.
Mechanisms of Damage Many studies of moisture damage have led to several postulates concerning the mechanisms of damage involved (Hicks 1991). Probably none of these theories alone explains the attraction problems between aggregate and liquid asphalt, but aspects of several theories may act together. Three main theories provide a basic understanding of what may be happening to pavements as failure attributable to stripping occurs: (1) the mechanical theory, (2) the chemical reaction and molecular orientation theory, and (3) the surface energy theory.
Mechanical Theory Mechanical adhesion is affected by aggregate properties such as texture, porosity, surface area, particle shape, and surface coatings. A rough surface texture should promote an interlock between the aggregate surface and liquid asphalt. Smooth glassy surfaces, such as those with quartz, provide less interlock and are more prone to strip than aggregates with a fine rough texture such as basalt. If an aggregate has some porosity, it will absorb enough asphalt to form a mechanical connection. On the other hand, if the aggregate is too porous, it may absorb so much asphalt that the surface coating becomes thin and is easily penetrated by water. Therefore, an optimum degree of absorption may exist that results in mechanical interlock and adequate film thickness, thereby producing the minimum potential for moisture damage. The surface area of the aggregate has an indirect effect on film thickness, and the finer portion of the aggregate structure affects the surface area more per unit weight than the coarse portion. If the asphalt content is held constant, the thickness of the film on the coated aggregate with a large surface area is less than with the same weight of coated aggregate with a small surface area. The lessthick film creates a better opportunity for water to penetrate the film and contact the aggregate surface.
Chemical Reaction and Molecular Orientation Theory Acidic aggregates such as granite and gravel are usually more prone to strip than basic aggregates such as limestone. Particular chemical polar compounds of asphalt are absorbed by aggregates, and the absorption is aggregate dependent. Some of these polar materials are desorbed by water more easily than others depending on the
Field Evaluation of Moisture Damage in Asphalt Concrete particular asphaltaggregate combination. Therefore, stripping propensity may depend on the amount and type of asphalt compounds, such as sulfoxides and carboxylic acids, which are attracted and easily absorbed on the aggregate surface. Aging seems to have a beneficial effect on particular mixtures by forming particular polar compounds in the asphalt that migrate to the aggregate surface and increase the viscosity of the asphalt.
Surface Energy Theory Surface energy has been measured by various methods to study the attraction of asphalt and water to aggregate surfaces. Water can displace particular asphalt compounds from the aggregate surface. Although the testing procedure for surface energy has been used only in research studies, its use could conceivably extend to approval of asphaltaggregate sources or determination of materials that might cause problems. The previous chapter dealt with this theoretical framework exclusively.
Typical Failures Severe failures often initially show the migration of asphalt to the surface followed by the formation of potholes or rutting resulting from disintegration of the asphalt mixture. Figure 131 shows typical examples of these types of failures in the early stages. Latter stages of stripping sometimes require complete removal and replacement of the asphalt layer that was causing the problem. Although more recent types of cracking distresses caused by stripping usually do not progress to catastrophic failures, the cracks allow water to enter and can develop into rough pavement and more serious distresses. A field survey of Virginia’s asphalt surface mixtures was done in 1996 to determine if stripping continued to be a problem (Maupin 1997). Approximately 1500 cores were taken, half were visually examined for stripping, and half underwent air void analysis. Stripping severity levels were arbitrarily set at various percentage ranges for both coarse and fine aggregate to facilitate analysis and discussion of the results (Table 131). Since the same degree of stripping is generally thought to be more damaging in the fine aggregate, the levels were slightly different for the coarse and fineaggregate portions. For the coarse aggregate, slight, moderate, moderately severe, and severe stripping was observed at approximately 20, 30, 40, and 10 percent, respectively, of the sites. For the fine aggregate, slight, moderate, moderately severe, and severe stripping was observed at
(a)
FIGURE 131 Typical pavement stripping failures.
(b)
393
394
Chapter Thirteen
Severity Level
% Coarse Aggregate Stripped
% Fine Aggregate Stripped
Slight
0–14
0–9
Moderate
15–29
10–24
Moderately severe
30–49
25–39
Severe
> 50
> 40
Source: Maupin 1997, adapted with permission from Transportation Research Board.
TABLE 131
Defined Stripping Severity Levels
approximately 20, 50, 20, and 5 percent, respectively, of the locations. Therefore, a fairly high degree of stripping was observed at many of the sites examined. Virginia has had pavement distress problems over the last several decades that are believed to be caused by stripping. When stripping was first recognized in the state, many of the failures were severe, but more recently distresses that might be attributed to stripping have been moderately severe. This change can probably be attributed to the use of improved antistripping additives and more quality control stripping testing during production of the mixtures.
Evaluation of Damage A thorough pavement evaluation should be performed prior to specifying the type of repair or treatment that is necessary. Field strength measuring equipment, such as the falling weight deflectometer (FWD), can be used to determine the structural capacity of the pavement and to indicate the thickness of the additional asphalt concrete that must be applied. However, in addition to the strength of the overall pavement structure, the condition of the various layers should be assessed in order to decide whether the material can remain in place or should be removed. Material soundness depends on several factors including asphalt aging and stripping. Of course, aging can accelerate various types of cracking but the factor of interest in this discussion is the effect of stripping.
Sampling The first step of a field evaluation is to develop a sampling plan that will result in the correct estimation of the degree of stripping in the section of pavement of interest. A general evaluation of a section of pavement containing distresses that are generally distributed and minor to moderate in severity requires that samples be taken randomly. Cores are usually used to evaluate stripping, but sawed samples may be used for visual evaluation. In fact, drysawed samples may be preferable in some instances, such as if the pavement is severely damaged and cores cannot be extracted intact. The samples should be randomly selected both longitudinally and transversely in the roadway lane. An ASTM Practice for Random Sampling of Construction Materials, ASTM D 3665 is one guide that can be used to locate sample locations (American Society of Testing and Materials, 2002). However, if moderatetosevere distresses are pervasive, most of the samples should be located in these distressed areas. The reason for this is that the highly distressed pavement will control the type of rehabilitation required for the entire section.
Field Evaluation of Moisture Damage in Asphalt Concrete
Types of Field Evaluation The author has used three types of field evaluation: (1) visual inspection of split cores, (2) image analysis of split cores, and (3) strength measurements of cores. Each method has strengths and weaknesses.
Visual Inspection of Split Cores Strength evaluations sometimes project initial and future constructed strengths through treating the samples with drying and accelerated conditioning, respectively. However, only one condition is of interest in the visual inspection of samples: the inplace present condition. Samples should be taken from the pavement and protected by tight wrapping in plastic wrap or placement in a plastic bag until split apart. If the samples are taken and allowed to dry, the stripping may tend to heal and the test will give false results. Generally, two scales can be used to measure the stripped aggregate in a sample. As discussed previously, one method is to use a scale based on percentage of stripped area. This method can be further complicated by the separate estimations of the stripping in the coarse and fine aggregate. To do this, the area of the sample composed of unstripped coarse and fine aggregate must be visualized. Most users simply estimate the percentage of the projected crosssectional area that has exposed bare aggregate. Knowing the degree of stripping in the coarse and fine aggregate separately can be advantageous since stripping of the fine aggregate is generally believed to contribute more to pavement damage than does stripping of the coarse aggregate. The other scale involves simply rating the stripping by arbitrary degrees such as slight, moderate, and severe. Visual inspection is subjective, and the results can be quite variable, particularly between two or more evaluators. Some aggregates appear to be easier to evaluate, with similar results between operators. Lightcolored nonabsorptive aggregates are usually the easiest to evaluate. Dull, dark aggregates are particularly difficult to evaluate because differentiating between a bare aggregate surface and a partially coated surface is difficult. In addition, for one or more evaluators to evaluate and agree on the degree of stripping of absorptive aggregates that absorb the light oils of the asphalt cement is more difficult. The author’s experience indicates that average differences between evaluators can typically be as much as 30 percent for coarse aggregate and 40 percent for fine aggregate. However, for some aggregates, the agreement between evaluators has been as low as 5 percent for both coarse and fine aggregates. The agreement between evaluators can be improved. One way is for an experienced person to show how the mixtures should be evaluated. Sample mixtures with various degrees of stripping are evaluated by the person and an explanation is given concerning why the particular values were assigned, particularly for the difficult mixtures. Some agencies have developed a board with samples or pictures of samples that are identified by degree of stripping. This method is particularly useful because it can be kept and used as a handy reference to refresh the evaluator’s mind and all evaluators will tend to work from the same baseline.
Image Analysis of Split Cores Image analysis with a digital camera, a computer, and special image software can be used to determine a quantitative measure of visual stripping. Image digitization converts the image into a numeric form that can be used by the computer for various analyses. The image is divided into very small blocks called pixels that can be sampled, and the brightness can be measured and quantified with various options. This quantity
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FIGURE 132 Stripped particles selected by image analysis software.
is stored in the corresponding pixel of the computer’s image bitmap. Color images require more bits to store the information; however, most asphalt specimen images can be stored in black and white. ImagePro Plus software has adjustments such as contrast and brightness that can be changed to select stripped particles in an asphalt specimen. Once the selection intensity is fixed, the software can select stripped aggregate particles of a designated size. The particles selected will be identified visually by an outline around each particle or group of adjacent particles, as shown in Fig. 132. The program can count the particles of a particular size and measure the area of the sample containing such particles. Therefore, the image analysis technique ideally can remove the subjectivity that is a weakness of the visual technique. Some weaknesses of the image analysis technique must be considered. Since asphalt often has shiny surfaces that tend to reflect light, filters may be required on the camera lens and light source to eliminate the reflections in the digitized camera image. Reflections may be recorded as light aggregate particles (stripped) and may affect the accuracy of the results. Another weakness is the difficulty of differentiating between stripped particles and unstripped particles if the aggregate is dark in color; this is especially true if the asphalt binder surface is dull in appearance.
Strength Measurements of Cores Inplace field measurements are often made of the pavement structure to assess strength and determine how much additional overlay is needed for the traffic loading. In some cases, very weak layers may need to be removed; however, deciding which layers to remove based on strength measurements such as those measured by the FWD is difficult. Although backcalculation techniques are used to calculate the stiffness of layers, determining the stiffness of different asphalt layers is difficult. This is where the types of evaluations described in this chapter are useful and strength measurements of samples removed from the roadway may especially be helpful. Measuring the indirect tensile strength is one method used to evaluate new asphalt mixtures for potential stripping, and it can also be used to assess cores removed from the pavement. Lottman (1986) found that pavement that strips usually gains strength because of aging for a short time and then begins to weaken because of stripping, as indicated in the lower curve of Fig. 133. To determine how much damage has been done, the strength of
Field Evaluation of Moisture Damage in Asphalt Concrete
FIGURE 133 Strength versus age of asphalt pavement layer. (Maupin 1989, adapted with permission from Transportation Research Board.)
the material in its present condition and the strength of the material if stripping had not occurred must be known. The ratio of the present strength to the unstripped strength (tensile strength ratio, TSR) gives an indication of the damage that has occurred due to stripping. The unstripped strength can be measured by two methods, both of which are only estimates. The first method is to take a set of cores and dry them in an attempt to heal the stripping. The second method is to take cores, heat them, remold them, and test them. Neither method gives the exact strength of unstripped material. The first method tends to underestimate the unstripped strength because the stripping detachment cannot heal completely. Figure 134 shows a core where drying was attempted, but the center of the core failed to dry and heal and stripping was still present. If the cores are heated and
FIGURE 134 Center of dried core showing lack of healing.
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FIGURE 135 (a) Deterioration curve and (b) development of deterioration curve. (Maupin 1989, adapted with permission from Transportation Research Board.)
molded into new specimens and then tested, the remolding process may change particular properties of the material. The heating and remolding process will likely harden the asphalt cement and increase the strength, and the increased strength will overestimate the unstripped strength. The correct unstripped strength is likely somewhere between the strengths determined by drying and remolding. A pseudostrength curve shown in Fig. 135(a) may be developed to analyze the degree of damage to the pavement layers. Three strength measurements are necessary to develop the curve; two of these, unstripped and present condition, have already been discussed. The third measurement involves a measure of the predicted minimum strength of the asphalt at some time in the future. This value is estimated by testing a set of cores that have been conditioned to simulate the additional stripping that is anticipated in the future. Figure 135(b) illustrates the development of the three strengths: present, unstripped, and future. An example of a typical strength examination on an actual project is given here for illustrative purposes. The section of interstate pavement was composed of 230 mm of select material (a processed aggregate), 150 mm of crushed stone base, 190 mm of asphalt base mixture, 30 mm of intermediate mixture, 23 mm of surface mixture, and an overlay of 19 mm of opengraded porous friction course. The pavement had undergone random cracking and had potholes and was due for repair or rehabilitation. Pavement
Field Evaluation of Moisture Damage in Asphalt Concrete strength measurements were also made with a device that applied a dynamic loading to the pavement to determine its structural capacity. Approximately fifty 102mm cores were obtained by wet drilling and then grouped as follows and tested: 1. Present (as soon after removal as practical) 2. Dried (dried until moisture loss ceased) 3. Conditioned (RootTunnicliff procedure, ASTM D 4867) (Tunnicliff and Root 1984; American Society of Testing and Materials 2002) 4. Remolded Ten 102mm cores were selected randomly for each group. The “present cores” were wrapped in plastic wrap to prevent moisture from escaping, transported to the lab, and tested as soon as possible. The cores were taken with a wetcore drill, and the author does not believe that this process influenced the strength substantially. The “dry cores” were dried in the lab until the moisture weight loss ceased, but healing of the stripped surfaces evidently did not occur. The “conditioned cores” were vacuum saturated with water and soaked in a 60oC water bath for 24 hours. The cores that were “remolded” were heated, remixed, and compacted into specimens 102 mm in diameter by 62.5 mm to the average field voids of the cores using Marshall compaction equipment. All cores were tested in indirect tension using the apparatus designated in ASTM 4867 at a temperature of 25oC and a deformation rate of 51 mm/min. Four asphalt layers were investigated: a quartzite surface mixture, a limestone base mixture, and two distinct layers of intermediate mixture. One of the intermediate mixtures contained quartzite aggregate, and the other intermediate mixture contained limestone aggregate. Figures 136 and 137 illustrate the estimated deterioration of the various mixtures. The deterioration of both intermediate mixtures was very similar and thus is shown with the unstripped strength being approximately 1300 kPa, the present strength approximately 600 kPa, and the predicted future strength approximately 500 kPa. The quartzite surface and intermediate mixtures were not predicted to lose much additional
FIGURE 136 Deterioration curves for example project. (Maupin 1989, adapted with permission from Transportation Research Board.)
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FIGURE 137 Deterioration curves for example project. (Maupin 1989, adapted with permission from Transportation Research Board.)
strength from the present inplace strength because of stripping. In addition, the surface mixture had the highest unstripped strength, which is logical, since the surface was exposed more to oxidative aging than the other asphalt layers. The strength of the unstripped base mix was the lowest of the mixtures tested, possibly because it was tested only in a dry condition (not remolded). As was discussed previously, the use of specimens that are dried usually leads to an underestimation of the unstripped strength. The dried strength values for the mixtures were always less than the remolded strength values. What can be concluded from the example project? Two values can be analyzed to determine pavement serviceability from the perspective of stripping: strength and TSR. Table 132 lists the strength and TSR values for the specific mixtures presented in the example. The TSR is obtained by dividing the strength in the present or future condition by the unstripped strength. The present TSR values were greater than 0.30, which the author recommends as a minimum value that may necessitate removal of the layer. The TSR of the base mixture was predicted possibly to decrease below 0.30 in the future, but it was borderline. Even though the arbitrary minimum 0.30 value is somewhat less than values generally used as a rejection value for testing new mixtures, the author’s experience was used to set this value. The minimum value may be somewhat different for regions associated with different environmental factors and materials and should be
Present
Future
Unstripped Strength, kPa
Strength, kPa
TSR
Strength, kPa
TSR
Quartzite surface
1580
590
0.37
530
0.34
Quartzite intermediate
1270
560
0.44
570
0.45
Limestone intermediate
1210
610
0.50
500
0.41
Limestone base
854
330
0.39
230
0.27
Mixture
TABLE 132
Strength and TSR Values for an Example Project
Field Evaluation of Moisture Damage in Asphalt Concrete adjusted to match historical performance. Strength can also be used as a factor in determining if the layer should be removed. Georgia used 275 kPa as the minimum acceptable tensile strength (telephone conversation with Ronald Collins on July 28, 1987). Therefore, none of the values in the example was below this value except for the base layer in the “future,” which coincides with the observations concerning TSR. As mentioned previously, inplace strength measurements were also made with a Dynaflect device that applies dynamic loads to the pavement and measures the resultant deflection basin. The structural capacity of the pavement structure may be estimated from this information. The structural capacity of the combined asphalt layers was estimated to be 29 percent of that for new asphalt. A weighted average of the present TSR measurements according to pavement layer thickness indicated that the strength was 44 percent of that of new asphalt, which was comparable to the value obtained with the Dynaflect device. This comparison adds credibility to the strength analysis approach described herein.
References American Society for Testing and Materials. (2002), Annual Book of ASTM Standards, Vol. 04.03. Philadelphia, Pa. Hicks, R. G. (1991), “Moisture Damage in Asphalt Concrete.” National Cooperative Highway Research Program Synthesis of Highway Practice 175. Transportation Research Board, Washington, D.C., pp. 47. Lottman, R. P. (1986), “Predicting MoistureInduced Damage to Asphaltic Concrete: TenYear Field Evaluation.” National Cooperative Highway Research Program. Transportation Research Board, Washington, D.C., unpublished manuscript. Maupin, G. W., Jr. (1989), Assessment of Stripped Asphalt Pavement. Transportation Research Record 1228. Transportation Research Board, Washington, D.C., pp. 17–21. Maupin, G. W., Jr. (1997), Followup Field Investigation of the Effectiveness of Antistripping Additives in Virginia. Virginia Transportation Research Council, Charlottesville. Tunnicliff, D. G., and R. E. Root. (1984), “Use of Antistripping Additives in Asphaltic Concrete Mixtures, Laboratory Phase.” NCHRP Report 274. National Cooperative Highway Research Program, Transportation Research Board, Washington, D.C.
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PART
Models for LowTemperature Cracking CHAPTER 14 Prediction of Thermal Cracking with TCMODEL
CHAPTER 15 LowTemperature Fracture in Asphalt Binders, Mastics, and Mixtures
Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.
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CHAPTER
14
Prediction of Thermal Cracking with TCMODEL William G. Buttlar, Reynaldo Roque, and Dennis R. Hiltunen
Abstract Thermal cracking is a significant form of asphalt pavement deterioration that can occur in cold climates or where large daily temperature cycles occur. Thermal cracks form when thermally induced strains produce localized stresses which exceed the fracture resistance of the pavement. While asphalt binder specifications can be very useful in minimizing the possibility of thermal cracking by controlling creep and fracture properties of the binder, ultimately thermal cracking is driven by properties of the asphalt concrete mixture. This chapter provides a detailed overview of a mixturebased thermal cracking prediction model called TCMODEL, which was originally developed under the Strategic Highway Research Program (SHRP) and later revised and updated for inclusion in the AASHTO MechanisticEmpirical Design Guide (MEPDG) software. The mechanisticempirical model predicts amount of thermal cracking versus time based upon mixture creep, strength, and thermal contraction properties, along with climatic inputs and information related to the overall pavement structure.
Introduction Structural components or systems subjected to diurnal temperature cycling are often prone to a deterioration mechanism commonly known as thermal cracking, particularly when significant restraints against contraction are present. Thermal cracking of asphalt pavements (Fig. 141) is a very serious pavement distress, since it is usually irreversible and often expensive to repair. Historically, thermal cracking has been linked to asphalt binder properties at low temperatures and therefore asphalt binder tests and specifications have been used in the control of thermal cracking. However, binder specifications alone cannot consider potentially important parameters such as asphalt mixture mechanical properties (creep compliance, fracture properties), thermal properties (thermal coefficient), or pavement configuration (layer types and thicknesses).
405 Copyright © 2009 by the American Society of Civil Engineers. Click here for terms of use.
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Chapter Fourteen
FIGURE 141 Thermal crack in an asphalt pavement extending through pavement shoulders.
The purpose of this chapter is to provide a detailed description of the mechanicsbased thermal cracking performance model (TCMODEL) that was developed as part of the Strategic Highway Research Program (SHRP) to evaluate, and eventually, to supplement the PerformanceGraded (PG) binder specification based upon the aforementioned motivation. More recently the model has been enhanced, recalibrated, and incorporated into the MEPDG software, as will be discussed in the latter portion of this chapter. TCMODEL predicts the amount (or frequency) of thermal cracking that will develop in a pavement as a function of time. Inputs to the model include asphalt mixture mechanical and thermomechanical properties, such as the creep compliance mastercurve, mixture tensile strength (both measured), and thermal coefficient of contraction (estimated); pavement structure, and hourly pavement temperature as a function of depth. Since the modeling system can determine whether or not a particular mixture, based upon its measured and estimated material properties, will meet specific thermal cracking performance requirements, the system provides the basis for a true performancebased mixture specification for thermal cracking.
Thermal Cracking Mechanism As described by Haas et al. (1987), Roque et al. (1993), and others, the primary mechanism generally associated with temperatureinduced thermal cracking is a “topdown” propagating, transverse pavement crack, as schematically described in Fig. 142. Contraction strains induced by pavement cooling lead to thermal tensile stress development in the restrained surface layer. Thermal stress development is greatest in the longitudinal direction of the pavement (the spacing S on Fig. 142 is large compared to pavement width until significant cracking occurs). Also, thermal stresses are greatest at the surface of the pavement because pavement temperature is lowest at the surface and because temperature changes are highest there. For very severe cooling cycles (very low temperatures and/or very fast cooling rates) transverse thermal cracks may develop at specific locations within the pavement under one or very few cooling cycles. This is generally referred to as lowtemperature cracking or singleevent thermal cracking. Additional cracks will develop at different
Prediction of Thermal Cracking with TCMODEL
Thermal crack
Surface layer thickness, D
FIGURE 142
Nonuniform thermal stress
Crack spacing, S
Schematic of physical model of pavement section.
locations as the pavement is exposed to subsequent cooling cycles. For milder cooling conditions, cracks may advance and develop at a slower rate, such that it may take several cooling cycles for cracks to propagate completely through the surface layer. This is generally referred to as thermal fatigue cracking. Both phenomena are typically classified under the general category of thermal cracking in pavement engineering.
Previous Models At the start of the SHRP project, there were no existing models to predict thermal cracking performance (amount of cracking versus time) using fundamental, lowtemperature mixture properties. Empirical models had been developed (Fromm and Phang 1972; Haas et al. 1987) to predict the number of cracks or crack spacing, but these models did not include time as a variable, and they were primarily based upon asphalt cement properties rather than mixture properties. Other existing models predicted mixture cracking potential (COLD, Finn et al. 1986; CRACK3, Roque and Ruth 1990), but did not predict thermal cracking performance in terms of amount of cracking versus time. A model called THERM, developed by Lytton, et al. (1983) provided thermal cracking predictions as a function of time, but relied on estimated mixture properties rather than mixture properties directly measured at low temperatures. Therefore, SHRP A005 researchers undertook the development of a new model to predict thermal cracking performance (amount of cracking versus time) using measured mixture properties, along with sitespecific environmental and structural information. The PCbased thermal cracking model that resulted was called TCMODEL, which was developed with the following features: • Mixture characterization that included the measured time and temperaturedependent (linear viscoelastic) behavior of the mixture. • Pavement temperatures that were computed on an hourly basis throughout the life of the pavement. • Thermal stress predictions that accounted for timedependent relaxation and nonlinear cooling rates. • Stress predictions as a function of depth. • Amount of cracking versus time using a mechanicsbased approach. Because significant advances in both testing and modeling were needed, the development of this modeling system represented a major undertaking, and required several years to develop and calibrate. Though the majority of TCMODEL’s original
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Chapter Fourteen framework has remained unchanged since its original completion in 1993, significant efforts in the decade that followed led to improved model inputs and produced additional data for model calibration. A detailed description of TCMODEL, including its original and subsequent development and an example of typical results generated with the model is presented in the following sections.
Physical Model The physical representation of the actual pavement structure assumed in TCMODEL is shown in Figs. 142 and 143. In Fig. 142 an asphalt concrete surface layer of thickness D is shown to be subjected to a tensile stress distribution with depth. Stresses develop due to contraction of the asphaltic concrete material during cooling. Stresses are not uniform with depth because of a thermal gradient, that is, the pavement temperatures vary with depth. It is assumed that within the surface layer there are potential crack sites uniformly spaced at a distance S. At each of these crack sites the induced thermal stresses can potentially cause a crack to propagate through the surface layer (Fig. 143), at which time it is assumed that a transverse crack will be visible on the pavement surface. It is assumed that each of these cracks can propagate at different rates due to spatial variation of the relevant material properties within the surface layer, and as will be shown later, only a single crack site is modeled in TCMODEL. A calibrated statistical model is then used to estimate a population of thermal cracks with varying depths.
Model Components The major components and subcomponents of TCMODEL are • Inputs module • Hourly pavement surface layer temperature profile • Mechanical and thermal properties of asphalt surface mixture • Pavement response model • Viscoelastic interconversion algorithm • Stress prediction via hereditary integral • Pavement distress model • Stress intensity model • Crack growth prediction via Paris model • Probabilistic crack amount model Detailed information for each of the model components is presented in the following sections:
Co
ΔC
FIGURE 143 Schematic of crack depth fracture model.
Prediction of Thermal Cracking with TCMODEL
Inputs Module The required input data for TCMODEL includes pavement structure information (layer types and thicknesses), pavement material properties, and sitespecific environmental data.
Environmental Effects Model Since hourly pavement temperature versus depth is not normally available, with the exception of instrumented pavement test sections, it is typically necessary to first run a pavement temperature prediction model before running TCMODEL. The model used during SHRP research was a modified version of the Enhanced Integrated Climatic Model program developed at the Texas Transportation Institute for the Federal Highway Administration (Lytton et al. 1989). This program has subsequently been debugged and improved for inclusion in the MEPDG software. This comprehensive heat and moisture flow model requires a significant number of inputs, including: minimum and maximum daily air temperatures recorded at sitespecific weather stations, surface shortwave absorptivity, surface emissivity factor, conductivity, heat capacity, convection coefficient, windspeed, percent sunshine, depth to water table, layer thicknesses, layer types (asphaltic concrete, stabilized base, or AASHTO classification for unbound granular materials and soils), latitude of site, average monthly wind velocity, average monthly sunshine, and the like. Fortunately, the MEPDG software simplifies the process by providing typical values for many of these inputs for hundreds of locations across the United States. The EICM simulation results required by TCMODEL are hourly asphalt pavement temperatures at the surface and at every 51.2mm (2 in.) interval of depth.
Thermal Properties of Mixture In addition to pavement structure and temperature data, TCMODEL requires mechanical and thermal properties of the asphalt mixture. The linear coefficient of thermal contraction for the asphalt mixture is computed using the following relationship, which is a modified version of the relationship proposed by Jones et al. (1968): B MIX = where
VMA × B AC + V AGG × B AGG 3 × V TOTAL
(141)
BMIX = linear coefficient of thermal contraction of the asphalt mixture (1/°C) BAC = volumetric coefficient of thermal contraction of the asphalt cement in the solid state (1/°C) BAGG = volumetric coefficient of thermal contraction of the aggregate (1/°C) VMA = percent volume of voids in the mineral aggregate (equals percent volume of air voids plus percent volume of asphalt cement minus percent volume of absorbed asphalt cement) VAGG = percent volume of aggregate in the mixture VTOTAL = 100 percent
Given that the coefficient of thermal contraction of asphalt cement and aggregate is not measured as part of routine mixture design, an average value of volumetric coefficient of thermal contraction of 3.45 × 10−4/°C is used for the binder input property for the model. Thus, the predicted thermal coefficient for the mix will be dependent upon the aggregate thermal coefficient and mixture volumetrics. A sensitivity analysis revealed that the thermal coefficient for a typical mixture varied by a maximum of 5 percent when the contraction coefficient of the binder was varied by the range reported
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Chapter Fourteen for the SHRP MRL binders. Alternatively, measured values of mixture coefficient of thermal contraction can be used in the MEPDG. More information on the measurement and estimation of mixture thermal coefficient can be found in Stoffels and Kwanda (1996), Mehta et al. (1999), and Nam and Bahia (2004).
Mechanical Properties of the Asphalt Surface Mixture If one considers the primary mechanism of thermal cracking shown in Fig. 142 (thermal stress development and crack propagation), the primary material properties controlling this mechanism are the viscoelastic properties, which control thermal stress development, and the fracture properties, which control the rate of crack development. These are the mixture mechanical properties that are required in TCMODEL. TCMODEL was developed and calibrated assuming that the user would conduct creep tests and obtain a measure of creep compliance at three test temperatures and for a duration of at least 100 seconds at each temperature. The creep compliance is simply the time dependent strain divided by the constant stress used in a creep test. During the SHRP program, the standard indirect tension testing approach commonly used to test cylindrical specimens was modified for lowtemperature testing, as documented by Roque and Buttlar (1992), Buttlar and Roque (1994), Buttlar et al. (1996) and as specified in AASHTO T322 (2003). By mounting sensors to the interior portion of the flat faces of the specimen away from loading heads, accurate measures of creep compliance can be obtained. Correction factors, based upon threedimensional finite element analysis, are used to enhance the accuracy of measurement interpretation. TCMODEL requires the creep compliance mastercurve (discussed in Chaps. 4 and 6) to be represented with a rheodictic type (Tschoegl 1989) generalized VoightKelvin model (Fig. 144). Rheodictic material behavior is typified by the presence of viscous flow at long loading times and is represented by an isolated dashpot placed in series with other springdashpot elements in a rheological model. This isolated dashpot can be observed in Fig. 144 and is represented as the last term in Eq. (142). Although asphalt concrete
D0 η1
D1
D2
η2
D3
η3
D4
η4
ηv
FIGURE 144 Generalized VoightKelvin model used to represent creep compliance master curve.
Prediction of Thermal Cracking with TCMODEL mixtures are relatively stiff at low temperatures, the rheodictic form provided the best fit of mastercurve data measured in the SHRP A005 study. N
D(ξ ) = D(0) + ∑ Di (1 − e −ξ / τ ) + i
i=1
where
ξ ηv
(142)
D(x) = creep compliance at reduced time x x = reduced time D(0), Di, ti, hv = model parameters for compliance, retardation times, and rheodictic flow, respectively
Besides the close fit to experimental data afforded by the generalized VoightKelvin (VK) model, this model was chosen to represent the master creep compliance curve for two other reasons: • The VK model simplifies the transformation of the master creep compliance curve to the master relaxation modulus curve (described in the next section). • The VK model simplifies the solution of the viscoelastic constitutive model used to calculate pavement stress. Within the thermal cracking model, the shift factortemperature relationship is modeled as piecewise linear between shift factors determined at the specified test temperatures, assuming a semilog relationship between the log of shift factors versus temperature (Hiltunen and Roque 1994). Linear interpolation is used to estimate shift factors at temperatures other than those used in mastercurve development. VoightKelvin model parameters and shift factors are obtained by performing creep compliance tests at multiple temperatures and horizontally “shifting” data from different temperatures to establish one smooth, continuous curve at a chosen reference temperature. The resulting curve is called the master creep compliance mastercurve. Additional details on how individual compliance curves are shifted to obtain the mastercurve have been presented in Chaps. 4 and 6 and published extensively in the literature (Hiltunen and Roque 1994; Buttlar et al. 1998; Buttlar and Roque 1997). Therefore, this procedure is not repeated herein in the interest of brevity. The use of timetemperature superposition implies that the asphalt mixture is assumed to behave as a thermorheologically simple material. The calibration of TCMODEL performed in the SHRP program involved forming mastercurves from creep compliance data obtained at 0, −10, and −20°C, each for a duration of 1000 seconds. At the end of the SHRP project, it was recommended that future revisions of the model should consider 100second creep testing to reduce testing time. This recommendation was carried out in the AASHTO T322 protocol for creep testing of asphalt mixtures at low temperatures. Subsequently, 100second creep test results were used in the recalibration of TCMODEL conducted in the NCHRP 137A project with good success.
Pavement Response Model Governing Constitutive Equation for Stress Prediction and Relaxation Modulus The viscoelastic properties of the asphaltic concrete mixture control the level of stress development during cooling. A pavement response model within TCMODEL is used to predict the farfield stresses within the pavement system (overall pavement thermal
411
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Chapter Fourteen stresses before crack tip stress intensity factors are applied). This model uses the material properties, pavement structure information, and the hourly pavement temperature predictions from the environmental effects model. A onedimensional constitutive model is applied at various depths in the pavement (quasi twodimensional approach) to approximate the temperatureinduced farfield tensile stresses. Although a full twodimensional or even threedimensional modeling approach would produce improved stress predictions, their use was deemed as impractical given that an hourly analysis period was selected. The quasi twodimensional modeling approach produces reasonable results under the current simplifying model assumptions, for example, that traffic loads can be ignored (accounted for with model calibration), and that crack spacing is suitably large so that crack interaction can be ignored. The model offers an improvement over some earlier thermal cracking models, which based their stress predictions solely on pavement temperatures at the surface. The model used for thermal stress predictions within the asphalt layer is a hereditary integral of the form ξ
σ (ξ ) = ∫E(ξ − ξ ′) 0
where
dε dξ ′ dξ ′
(143)
s(x) = stress at reduced time x E(x−x′) = relaxation modulus at reduced time x−x′ e = strain at reduced time x (= a(T(x′)−T0)) a = linear coefficient of thermal contraction T(x′) = pavement temperature at reduced time x′ T0 = pavement temperature when s = 0 x′ = variable of integration
A generalized Maxwell model was selected to represent the viscoelastic properties of the asphaltic concrete mixture in relaxation. A schematic representation of the model is shown in Fig. 145. Mathematically, the relaxation modulus for a generalized Maxwell model can be expressed according to the following Prony series, which differs from the similar model presented in Chap. 6 by the removal of the long time modulus: N+1
E(ξ ) = ∑ E i e −ξ / λ
(144)
i
i=1
where E(x) is relaxation modulus at reduced time x, and Ei and li are Prony series parameters for master relaxation modulus curve (moduli and relaxation times for the Maxwell elements).
E1
E2
E3
E4
E5
l1
l2
l3
l4
l5
FIGURE 145 Generalized Maxwell model for relaxation modulus.
Prediction of Thermal Cracking with TCMODEL This function describes the relaxation modulus as a function of time at a single temperature, which is generally known as the reference temperature. The function defined at the reference temperature is called the master relaxation modulus curve. Relaxation moduli at other temperatures are determined by using the method of reduced variables (timetemperature superposition), which assumes that the mixture behaves as a thermorheologically simple material. Relaxation moduli at other temperatures are determined by replacing real time (i.e., time corresponding to the temperature of interest) with reduced time (i.e., time corresponding to the temperature at which the relaxation modulus is defined) according to the following equation: t ξ = aT
(145)
where x = reduced time t = real time aT = temperature shift factor (determined from creep compliance data)
Viscoelastic Interconversion: Creep Compliance to Relaxation Modulus As discussed above and conceptually in Chap. 6, the constitutive model form required for stress prediction in TCMODEL is the relaxation modulus and associated shift factors. The conversion details presented in Chap. 6 show that the relationship between the creep compliance and the relaxation modulus is governed by a convolution integral: ∞
∫D(t − τ ) 0
dE(τ ) dτ = 1 dτ
(146)
Taking the Laplace transformation of each side results in L[D(t)] × L[E(t)] =
1 s2
(147)
where L[D(t)] = Laplace transformation of the creep compliance D(t) L[E(t)] = Laplace transformation of the relaxation modulus E(t) S = Laplace parameter (the transformed time variable) t = time (for the mastercurve, the reduced time variable x is used) TCMODEL includes an algorithm to solve this equation for the master relaxation modulus E(x) given the master creep compliance D(x). Model development and verification have been documented in detail by Hiltunen and Roque (1994). The temperature shift factors for the master relaxation modulus curve are as determined from the creep compliance data, that is, the same shift factors are applicable to both the creep and relaxation data.
Numerical Methods for Applying Hereditary Integral: Hourly Stress Predictions The convolution integral equation was initially expressed in terms of reduced time x because timetemperature superposition was being used to represent the creep compliance and relaxation modulus curves. With a change of variables, the equation can be written in terms of real time t as follows: t
σ (t) = ∫E[ξ(t) − ξ ′(t)] 0
dε d ′t d ′t
(148)
413
414
Chapter Fourteen Using the Prony series representation of E(x) (Eq. 144), the following finite difference solution to the above equation has been developed (Soules et al. 1987): N+1
σ (t) = ∑σ i(t)
(149)
i=1
where
σ i(t) = e − Δξ / λ σ i(t − Δt) + Δε E i i
λi ( 1 − e − Δξ / λ ) Δξ i
(1410)
and where Δe and Δx are the changes in strain and reduced time, respectively, over time t−Δt to t, and all other variables are as previously defined. This approach is similar to the state variable approach presented in Chap. 7. The pavement response model performs the following sequence of computations: • Temperatures are predicted at multiple depths (nodes) within the asphaltic concrete layer using the environmental effects model. The nodes are located at 2in. intervals. • Temperatureinduced strains are computed at each of the nodes. • The onedimensional model presented earlier is used to predict stresses at each node, thus establishing an approximate stress distribution with depth. • The predicted stress distribution is used as input to the crack depth (fracture) model. The fracture model uses the stress at the current location of the crack tip to estimate the crack advancement.
Pavement Distress Model The pavement distress model consists of three primary parts: the stress intensity factor model, the crack depth (fracture) model, and the crack amount model. The stress intensity factor model predicts the stress at the tip of a local vertical crack using the farfield stress computed by the pavement response model and the pavement structure and material properties. Based upon the stress at the tip of the crack, the crack depth (fracture) model predicts the amount of crack propagation due to the imposed stress. Finally, the crack amount model predicts the number (or frequency) of thermal cracks per unit length of pavement from the depth of the local vertical crack and the probabilistic crack distribution model.
Stress Intensity Factor Model The stress intensity factor model (CRACKTIP) is a twodimensional finite element (FEM) program that models a single vertical crack in the asphaltic concrete layer via a crack tip element. The CRACKTIP program was developed at the Texas Transportation Institute (Chang et al. 1976). Suitable finite element meshes were identified and sidebyside comparisons of the CRACKTIP finite element program with the ANSYS program and with standard handbook solutions were performed in order to verify the accuracy of the CRACKTIP program for use in the thermal cracking model (Lytton et al. 1993). It was determined that computer run times would be excessive if the CRACKTIP finite element model were to be incorporated directly into the thermal cracking model. Therefore, an investigation was conducted to determine if a simplified equation could be developed to predict the results of the CRACKTIP program. The approach was to
Prediction of Thermal Cracking with TCMODEL presolve the CRACKTIP program for a broad range of conditions and determine whether a simple relationship could be developed to obtain a reasonable estimate of the stress intensity factors predicted by the model. The following regression equation was determined to provide reasonably accurate estimates of stress intensity factors: K = σ (0.45 + 1.99C 0.56 0 )
(1411)
where K = stress intensity factor s = farfield stress from pavement response model at depth of crack tip C0 = current crack length Equation is used in the thermal cracking model in lieu of the CRACKTIP program.
Crack Growth Prediction Model The amount of stable crack propagation induced by a given thermal cooling cycle is estimated in TCMODEL using the simple phenomenological relationship introduced by Paris et al. (1961): ΔC = AΔK n where
(1412)
ΔC = change in the crack depth due to a cooling cycle ΔK = change in the stress intensity factor due to a cooling cycle A, n = fracture parameters
The change in crack depth (ΔC) is computed and accumulated on an hourly basis to determine the total crack depth as a function of time. Thus, the material properties that directly influence the amount of cracking that will develop in a pavement subjected to specified levels of thermal stress are the fracture parameters A and n. Since it was not deemed practical by SHRP researchers to perform fracture tests for mixture specification purposes, it was decided that fracture parameters A and n should be determined on the basis of material properties measured as part of the specification tests, along with theoretical or experimental relationships between measured properties and fracture parameters A and n. Schapery’s theory of crack propagation in nonlinear viscoelastic materials (Schapery 1973) indicates that the fracture parameters A and n are theoretically related to • The slope of the linear portion of the log compliancelog time mastercurve determined from creep tests (m) • The tensile strength of the mixture • The fracture energy density of the mixture determined experimentally by monitoring the energy release through crack propagation Equation (1413) presents the model used to obtain the mvalue from slope of the master creep compliance curve at long loading times. D(ξ ) = D0 + D1ξ m
(1413)
The mixture tensile strength is obtained by performing a strength test at a fairly rapid, constant rate of crosshead deformation (12.5 mm/min) in the IDT test.
415
416
Chapter Fourteen Experiments by Molenaar (1984) led to the following relationship: log A = 4.389 − 2.52 ∗ log(E∗ σ m∗ n)
(1414)
where E is mixture stiffness and sm is mixture strength. Molenaar measured all material properties to develop this relationship. Experiments conducted by Lytton et al. (1983) led to the following relationship:
(
n = 0.8 × 1 +
1 m
)
(1415)
These findings agree with Schapery's theoretical development for nonlinear viscoelastic materials, where Schapery proved that both A and n are related to m, and that A is also a function of the fracture energy density of the material. Molenaar’s equation suggests that the material strength and stiffness are suitable surrogates for the fracture energy density in determining the parameter A. Since the meaning of mixture stiffness included in Molenaar’s relationship in this application is unclear, particularly when considering variable temperature conditions during thermal stress development, it was decided that this value should be determined as a calibration coefficient as part of the field calibration efforts (see Model Calibration section below). The following equation was determined after field calibration: log A = 4.389 − 2.52 × log(k × σ m × n)
(1416)
where k (= 10,000) is the coefficient determined through field calibration and sm is the undamaged mixture strength. Therefore, the two measured properties used to obtain the fracture parameters are • The mvalue, which is the slope of the linear portion of the log compliancelog time relationship determined from creep tests • The tensile strength of the mixture The tensile strength at −10°C was selected as the standard input to the fracture model. One of the main reasons for this decision was that it is well known that asphalt mixture strength increases with decreasing temperature until a maximum value is reached at a specific temperature, below which the strength decreases with decreasing temperature. The temperature at which the strength peaks varies from mixture to mixture, but the threshold is typically lower than −10°C. The strength reduction at lower temperatures is probably a result of the embrittlement of the asphalt binder and by the internal damage caused by stresses induced by differential contraction between aggregate and binder when laboratory specimens are cooled to testing temperatures. Although temperaturedependent tensile strengths were considered in the SHRP study, it was determined that mixture tensile strength at a single temperature of −10°C provided sufficient information for the selected fracture model.
Probabilistic Crack Amount Model In order to predict the amount of cracking per unit length of pavement section from the average crack depth and the distribution of crack depths within the section, the following assumptions were made:
Prediction of Thermal Cracking with TCMODEL • Within a given pavement section there is a maximum number of thermal cracks that can occur and these cracks are uniformly spaced throughout the section (or conversely, there exists a minimum crack spacing beyond which no further cracks will develop). This assumption appears rational because, below a certain crack spacing, insufficient friction exists to develop the stresses required to advance another crack. Initially, each of these potential cracks starts out as a very small local vertical crack (or flaw, fissure, etc.) at the surface of the asphaltic concrete layer. Detailed analytical investigations on thermal crack spacing can be found in Timm and Voller (2003) and Yin et al. (2007). • A crack is not counted (or observed) as a crack until the local vertical crack propagates through the entire depth of the asphalt concrete surface layer. In other words, no contribution is made to the amount of global thermal cracking until the local vertical crack breaks through the surface layer. • For a given pavement section at a given point in time, each of the local vertical cracks defined above has potentially propagated a different amount through the surface layer because of the fact that the material properties of the pavement vary spatially throughout the section. This spatial distribution of crack depths is assumed to be normally distributed. The mean of the distribution is assumed to be equal to the crack depth computed from the mechanistic model described above using the material properties measured in the laboratory. The variance of the distribution is unknown, and was included in the model as a coefficient to be estimated during the calibration efforts. The variance was assumed to be constant across all pavement sections. Based upon the above assumptions, the model shown in Fig. 146 was developed between the amount of cracking for the pavement section and the proportion of the maximum number of vertical cracks that have actually broken through the surface layer. Essentially, the amount of cracking is a function of the probability that the crack depth is equal to or greater than the thickness of the surface layer. As shown in the figure, this probability is determined by assuming that the logarithm of the depth of cracks in the
Thermal crack
Surface layer thickness, D
P(log C > log D)
log Co log D log C ~ N (log Co, s 2)
FIGURE 146 Crack amount model: crack depth distribution.
417
418
Chapter Fourteen pavement is normally distributed with mean equal to log C0 (the crack depth predicted by the model), and a variance of s2. The amount of cracking is computed as follows: AC = β 1 × P (log C > log D)
(1417)
⎛ log C/D ⎞ AC = β 1 × N ⎝ ⎠ σ
(1418)
or
where AC = observed amount of thermal cracking b1 = regression coefficient determined through field calibration P () = probability that () is true N () = standard normal distribution evaluated at () s = standard deviation of the log of the depth of cracks in the pavement C = crack depth D = thickness of surface layer This particular model, which is based on the logarithm of C and D, was selected for the following reasons: • As seen in the equations presented above, use of the logarithm form implies that the amount of cracking is proportional to the ratio of C/D, which has the effect of normalizing the crack depth with respect to the surface layer thickness. • The use of log C0 also implies that the variance of the crack depth increases as the crack depth increases. This appeared to be a rational effect. It should be noted that this model does not predict any more than 50% of the total possible amount of cracking that can develop in the pavement. This corresponds to the instant when the average crack depth is equal to the thickness of the surface layer, which implies that 50 percent of all cracks in the pavement have penetrated the entire thickness of the surface layer. Based on field observations and calibrations conducted as part of this investigation, a maximum amount of cracking of 400 m of transverse cracking per 500 m of pavement was selected as the maximum amount of thermal cracking that would typically develop in a pavement. Assuming a traffic lane width of 4 m, this corresponds to a crack frequency of approximately 1 crack per 5 m of pavement. Thus, the model will continue running until a predicted amount of cracking of about 200 m per 500 m of pavement (i.e., 1 crack per 10 m) is reached. Therefore, once calibrated, the model is capable of predicting the proper rate of thermal cracking for a given pavement, but the model will cease execution when the aforementioned limit is reached. It was decided that higher levels of thermal cracking frequency (Deme and Young 1987) would not be considered in TCMODEL, since closer crack spacing would necessitate more complicated response and fracture models due to crack interaction and the proper consideration of available frictional restraint of the asphalt layer in contact with the underlying pavement layer needed to develop thermal stresses once other cracks have formed. However, given that a pavement with a crack every 10 m is generally considered severely cracked, the current limitations made in the model were considered to be sufficient for engineering design and analysis.
Prediction of Thermal Cracking with TCMODEL
Model Calibration and Sample Output As with any other mechanisticempirical pavement prediction model, it is necessary to calibrate TCMODEL in order to enhance model accuracy. Because of the detailed inputs required for TCMODEL, including material, structural, and climatic parameters, calibration of TCMODEL can be a timeconsuming task. At the present time, TCMODEL has been calibrated twice: (1) during SHRP research using data from the SHRP LongTerm Pavement Performance (LTPP) General Pavement Sections (GPS), and (2) during NCHRP 137A research, using the aforementioned GPS data, plus additional data from Canadian Strategic Highway Research Program (CSHRP) sections and Minnesota Road Research (Mn/ROAD) sections (from both high traffic volume and low traffic volume areas). The AASHTO MEPDG software was developed under the National Cooperative Highway Research Program (NCHRP) project 137A. Both calibrations can be considered as “national” calibration efforts, that is, involving data from many geographical regions (United States and Canada in this case). Other calibration data sets should be considered in the future, such as regional, state, local, and the like, depending upon the needs and available resources of the user and/or agency.
Calibration Methods and Results The calibration parameters selected for the model were k, which is a stiffness term used in one of the parameters of the fracture model [Eq. (1416)], and b1 and s from the statistical crack distribution model. A nonlinear regression routine, called the “press” procedure (Roque et al. 1993) was used to minimize the differences between measured and predicted thermal cracking. For the original calibration effort conducted during NCHRP 137A, the press procedure produced the following parameters: k = 10,000; b1 = 353.5, and s = 0.769, which produced a very good model fit (R2 = 0.88). Table 141 presents a comparison of predicted and observed thermal cracking for this set of calibration parameters. As explained earlier, the nature of the probabilistic crack distribution model is such that the maximum amount of cracking that can be predicted by the model under the current assumptions is onehalf of b1, or about 177 m of transverse cracking per every 500 m of pavement.
Model Changes Implemented under NCHRP 137A At the time of printing of this book, the AASHTO MEPDG software program developed under NCHRP 137A has just been completed. This section describes some important modifications and extensions to TCMODEL, which were implemented in the MEPDG.
Use of Three Input Levels The AASHTO MEPDG software allows the user to choose various levels of detail with regards to model inputs. For instance, the user with very limited resources for laboratory testing can elect to run one or more prediction models within the software package using typical values provided by the software. The following analysis levels were used: • Level 1—This involves the full suite of inputs to TCMODEL, as described in this chapter. This would therefore require creep compliance data at three test temperatures (usually 0, −10, and −20°C), and tensile strength data at −10°C. This produces results with the highest degree of reliability.
419
420
Chapter Fourteen
Project
Strategic Highway Research Program (SHRP) General Pavement Sections (GPS)
Canadian Strategic Highway Research Program (CSHRP)
Section
Location
Observed Cracking (m/500 m)
404086
Chickasaw, Okla.
96
75
041022
Hackberry, Ariz.
0
11
322027
Oasis, Nev.
≥ 177
≥ 177
201005
Ottawa, Kan.
≥ 177
≥ 177
161010
Idaho Falls, Idaho
≥ 177
176
161001
Coeur D’Alene, Idaho
0
13
311030
Edison, Neb.
36
8
491008
Marysvale, Utah
≥ 177
174
561007
Cody, Wyo.
≥ 177
≥ 177
081047
Rangley, Colo.
≥ 177
≥ 177
241634
Berlin, Md.
0
0
451008
Salem, S.C.
96
≥ 177
341011
Trenton, N.J.
36
0
291010
Waynesville, Mo.
120
≥ 177
181028
Huntington, Ind.
12
5
231026
Farmington, Maine
12
0
271087
Farmington, Minn.
132
≥ 177
271028
Frazee, Minn.
≥ 177
≥ 177
Lamont 1
Lamont, Alberta
110
≥ 177
Lamont 2
Lamont, Alberta
≥ 177
≥ 177
Lamont 3
Lamont, Alberta
0
0
Lamont 5
Lamont, Alberta
24
0
Lamont 6
Lamont, Alberta
0
1
Lamont 7
Lamont, Alberta
0
0
Sherbrooke A
Sherbrooke, Quebec
0
0
Sherbrooke B
Sherbrooke, Quebec
0
4
Sherbrooke C
Sherbrooke, Quebec
0
0
Sherbrooke D
Sherbrooke, Quebec
0
0
Hearst 1
Hearst, Ontario
0
0
Hearst 2
Hearst, Ontario
0
0
TABLE 141 Comparison of Observed and Predicted Thermal Cracking Using TCMODEL
Predicted Cracking (m/500 m)
Prediction of Thermal Cracking with TCMODEL
Project
Minnesota Road Research Program (Mn/ROAD)
Section
Location
Observed Cracking (m/500 m)
Predicted Cracking (m/500 m)
Cell 16
Ostego, Minn.
≥ 177
≥ 177
Cell 17
Ostego, Minn.
≥ 177
≥ 177
Cell 26
Ostego, Minn.
0
14
Cell 27
Ostego, Minn.
≥ 177
≥ 177
Cell 30
Ostego, Minn.
108
130
TABLE 141 (Continued )
• Level 2—This involves laboratory testing only at −10°C. To obtain the required mastercurve input for TCMODEL, the data at −10°C is extrapolated to shorter and longer loading times using a simple power law model [Eq. (1413)]. Temperature shift factors are obtained using a regression equation, which was developed from the data base generated during model calibration. The equation is a function of the D0 term in the power model. This produces results with intermediate reliability. • Level 3—This level does not require laboratory testing of the asphalt mixture. Instead, regression equations were developed at Arizona State University to predict creep compliance and tensile strength as a function of the asphalt binder properties and mixture parameters such as air voids, voids in the mineral aggregate (VMA), effective binder volume, and the like. For brevity, these empirical material models are not reported herein, but can be found in the documentation which accompanies the MEPDG software. Obviously, this level is much simpler and quicker to perform than levels 1 and 2, but at the expense of somewhat lower prediction reliability. Regional or local calibration is recommended to compensate for this added layer of model empiricism.
Modified Calibration Approach The final stage of TCMODEL calibration under NCHRP 137A involved checking the initial calibration using the most uptodate design guide software version. In particular, it was anticipated that the difference between the pavement temperatures input files used in the initial TCMODEL calibration and the latest climatic engine in the design guide software could be significant. The version of the climatic model software used to generate pavement temperature profile histories used in the initial calibration of TCMODEL under NCHRP 137A was over 10 years old, and a number of revisions had taken place during the 137A project. Thus, it was deemed necessary to check, and if necessary, recalibrate TCMODEL in light of these changes. As expected, it was determined that the changes in the climatic model warranted a recalibration of TCMODEL, since a systematic underprediction of thermal cracking was noted when the initial calibration factors were used with the new temperature files.
421
422
Chapter Fourteen In conjunction with the recalibration efforts, the parameter b1 was set to 400. The change in b1 value from the previous calibration effort was somewhat arbitrary, although the rationale for taking b1 as 400 to set the maximum possible cracking level in TCMODEL to 400/2 or 200 meters of thermal cracking per 500meter section. This is consistent with the model cutoff level originally planned in the SHRP program, and as a matter of practicality, makes intuitively more sense than the cutoff level of 176.7 associated with previous model calibration efforts. A new calibration parameter b2 was introduced, which leads to the computation of Acal, which is then used in place of Eq. (1416), as follows: Acal = β 2 × 10 ^ (4.389 − 2.52 × log(E × σ m × n))
(1419)
Based upon a limited number of model runs, it appears that a value of b2 of approximately 5 will produce suitably conservative thermal cracking predictions for a level 1 analysis. Local calibration of TCMODEL is strongly encouraged, which will have the added benefit of using mixtures from mix design, sampled at the hotmix plant, or from field cores taken at the time of construction. Up to this point, the available data being used for calibration of TCMODEL involves a broad sampling of fieldaged cores across the United States and Canada. More importantly, these cores were taken at the time of the thermal cracking observation (greater than 10 years in service in some cases), while for design purposes, one would usually have available materials with shortterm aging only. In procuring field samples from the SHRP GPS, CSHRP, and Mn/ROAD section, cores were fabricated in the lab to produce indirect tension test specimens that had materials no closer than 50 mm from the pavement surface to minimize field aging effects. However, in the case of thin surface lifts, it was necessary to utilize materials as close as 12.5 mm from the surface of the pavement. Local calibration efforts should utilize shortterm aged materials to the extent possible, although this approach presents a dilemma, for example, it would require that existing virgin materials were suitably stored away from older projects and/or would require a number of years to collect thermal cracking performance if new projects were selected and tested.
Sample Output Figures 147 through 149 present a sensitivity analysis performed using TCMODEL. From these analysis results, one can gauge the relative sensitivity of TCMODEL to model input parameters such as thermal coefficient (Fig. 147), mixture tensile strength (Fig. 148), and binder grade used in the mixture (Fig. 149). The case study selected was one of the SHRP GPS locations used in the calibration of TCMODEL: Edison, Nebraska. In this relatively cold, Midwest location, it is obvious that critical cooling events are the primary source of predicted thermal cracking, that is, singleevent thermal cracking. This explains the rational of the “jumps” observed on the cracking versus time plots. This is in contrast to fatigue and rutting predictions shown in earlier chapters, which exhibited more gradual distress versus time relationships.
Prediction of Thermal Cracking with TCMODEL 160 Key: Coefficient of thermal Expansion (mm/mm/°C):
Amount of thermal cracking (m/500 m)
140
0.0000108 0.0000144
120
0.0000180 0.0000216
100
0.0000252 80 60 40 20 0 0
FIGURE 147
19
24
36
48 Time (months)
60
72
84
96
TCMODEL output for Edison, Neb. sensitivity study: effect of thermal coefﬁcient.
160
Key: Mixture tensile Strength at − 10°C (MPa):
Amount of thermal cracking (m/500 m)
140 2.07 2.58
120
3.10 100
3.62 4.13
80 60 40 20 0 0
12
24
36
48 Time (months)
60
72
84
96
FIGURE 148 TCMODEL output for Edison, Neb. sensitivity study: effect of mixture tensile strength.
423
Chapter Fourteen 200
Key: Asphalt binder Grade (AASHTO MP1):
180
Amount of thermal cracking (m/500 m)
424
PG 5822 160
PG 5228
140
PG 6422
120 100 80 60 40 20 0 0
12
24
36
48
60
72
84
96
Time (months)
FIGURE 149
TCMODEL output for Edison, Neb. sensitivity study: effect of asphalt binder grade.
Summary and Future Work TCMODEL represents a comprehensive mechanisticempirical performance prediction model for thermal cracking in asphalt pavements. The model is unique in how it combines fundamental lowtemperature mixture properties, comprehensive pavement temperature profile histories, a historysensitive viscoelastic response model, a mechanisticempirical crack propagation model, and an innovative probabilistic crack distribution model. In its present form, the model appears to be capable of producing accurate thermal cracking predictions, as evidenced by comparison to actual performance of SHRP GPS, CSHRP, and Mn/ROAD test pavements. That notwithstanding, there are still many potential areas of improvement for TCMODEL that, if pursued, could lead to additional benefits such as enhanced prediction accuracy and/or reduced calibration demand. One area worthy of study is the effect of mixture aging in the field on thermal cracking development. Currently, TCMODEL does not directly account for mixture aging with time. A second area where TCMODEL can be improved in the future is in fracture testing and modeling. Currently, TCMODEL utilizes mixture tensile strength at a single test temperature of −10°C, and then develops fracture parameters by taking this value as an estimate of the undamaged tensile strength of the mixture, as described earlier, along with a slope parameter from the master compliance curve (mvalue). In the future, TCMODEL can be improved once a better understanding of mixture fracture behavior is obtained, and when more rigorous fracture models can be readily incorporated. The Paris law approach is not a true fracture mechanics approach, and its reliance on the change in stress concentration regardless of crack length arguably renders the approach as semiempirical, or phenomenological. Recent advances in asphalt mixture fracture testing methods and fracture mechanics modeling should be incorporated in subsequent TCMODEL versions (Wagoner et al. 2005; Song et al. 2006).
Prediction of Thermal Cracking with TCMODEL A third area where TCMODEL can be improved in the future is in the area of response modeling. Currently a quasi twodimensional pavement response model is used, which accounts only for temperatureinduced stresses and strains. Recent work has indicated that traffic loads applied during critical cooling events can increase tensile stresses by more than 50 percent (Waldhoff et al. 2000). Although TCMODEL is calibrated to field performance and therefore indirectly accounts for average traffic effects an improved response model should be incorporated into TCMODEL as threedimensional finite element modeling becomes more computationally efficient. Finally, further model calibration and validation is recommended, particularly with an emphasis on including pavements utilizing polymermodified asphalt binders, which were not included in earlier calibration sections. For agencies wishing to make major design or policy decisions based upon TCMODEL, it is highly recommended that regional and/or local calibration and validation studies should first be conducted.
References American Association of State Highway and Transportation Officials (AASHTO) Designation T32203, “Standard Method of Test for Determining the Creep Compliance and Strength of HotMix Asphalt (HMA) Using the Indirect Tensile Test Device,” Standard Specifications for Transportation Materials and Methods of Sampling and Testing, Part 2B: Tests, 22nd ed., 2003. ANSYS Finite Element Computer Program (PCLinear 2), Swanson Analysis System, Houston, Pa., 1991. Buttlar, W. G., and R. Roque, “Development and Evaluation of the Strategic Highway Research Program Measurement and Analysis System for Indirect Tensile Testing at Low Temperatures,” Transportation Research Record No. 1454, Transportation Research Board, Washington, D.C., pp. 163–171, 1994. Buttlar, W. G., and R. Roque, “Effect of Asphalt Mixture Master Compliance Modeling Technique on Thermal Cracking Pavement Performance,” Proceedings of the 8th International Conference on Asphalt Pavements, International Society for Asphalt Pavements, Seattle, Wash., Vol. 2, pp. 1659–1669, 1997. Buttlar, W. G., R. Roque, and N. Kim, “Accurate Asphalt Mixture Tensile Strength,” Proceedings of the fourth Materials Engineering. Conference, Materials for the New Millennium, American Society of Civil Engineers, Washington, D.C., pp. 163–172, 1996. Buttlar, W. G., R. Roque, and B. Reid, “An Automated Procedure for Generation of the Creep Compliance Master Curve for Asphalt Mixtures,” Transportation Research Record, No. 1630, National Research Council, National Academy Press, Washington, D.C., pp. 28–36, 1998. Chang, H. S., R. L. Lytton, and S. H. Carpenter, “Prediction of Thermal Reflection Cracking in West Texas,” Research Report No. TTI2873183, Texas Transportation Institute, Texas A&M University, College Station, Tex., March 1996. Deme, I. J., and F. D. Young, “Ste. Anne Test Road Revisited Twenty Years Later,” Proceedings of the Canadian Technical Asphalt Association, Vol. XXXI, 1987. Finn, F., C. L. Saraf, R. Kulkarni, K. Nair, W. Smith, and A. Abdullah, “Development of Pavement Structural Subsystems,” NCHRP Report 291, Transportation Research Board, Washington, D. C., December, p. 59, 1986. Fromm, H. J., and W. A. Phang, “A Study of Transverse Cracking of Bituminous Pavements,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 41, pp. 383–418, 1972.
425
426
Chapter Fourteen Haas, R., F. Meyer, G. Assaf, and H. Lee, “A Comprehensive Study of Cold Climate Airport Pavement Cracking,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 56, pp. 198–245, 1987. Hiltunen, D. R., and R. Roque, “The Use of TimeTemperature Superposition to Fundamentally Characterize Asphalt Concrete Mixtures at Low Temperatures,” Engineering Properties of Asphalt Mixtures and the Relationship to Performance, ASTM STP 1265, Gerald A. Huber, Gerald A., and Dale S. Decker, eds., American Society for Testing and Materials, Philadelphia, 1994. Jones, G. M., M. I. Darter, and G. Littlefield, “Thermal ExpansionContraction of Asphaltic Concrete,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 37, pp. 56–97, 1968. Lytton, R. L., D. E. Pufahl, C. H. Michalak, H. S. Liang, and B. J. Dempsey,” An Integrated Model of the Climatic Effects on Pavements,” Report No. FHWARD90033, Federal Highway Administration, Washington, D.C., November, 1989. Lytton, R. L., R. Roque, J. Uzan, D. R. Hiltunen, E. Fernando, and S. M. Stoffels, “Performance Models and Validation of Test Results,” Strategic Highway Research Program Report A357, Project A005, Washington, D.C., 1993. Lytton, R. L., U. Shanmugham, and B. D. Garrett, “Design of Asphalt Pavements for Thermal Fatigue Cracking, Research Report No. FHWA/TX83/06+2844, Texas Transportation Institute, Texas A&M University, College Station, Texas, January 1983. Mehta, Y., S. Stoffels, and D. Christensen, “Determination of Thermal Contraction of Asphalt Concrete Using Indirect Tensile Test Hardware,” Journal of the Association of Asphalt Paving Technologists, Vol. 68, pp. 349–368, 1999. Molenaar, A. A. A., “Fatigue and Reflection Cracking due to Traffic Loads,” Proceedings of the Association of Asphalt Paving Technologists, Vol. 53, pp. 440–474, 1984. Nam, K., H. U. Bahia, “Effect of Binder and Mixture Variables on Glass Transition Behavior of Asphalt Mixtures,” Journal of the Association of Asphalt Paving Technologists, Vol. 73, pp. 89–120, 2004. Paris, P. C., M. P. Gomez, and W. E. Anderson, “A Rational Analytical Theory of Fatigue,” The Trend in Engineering, Vol. 13, No. 1, January, 1961. Roque, R., and W. G. Buttlar, “Development of a Measurement and Analysis System to Accurately Determine Asphalt Concrete Properties Using the Indirect Tensile Test,” Journal of the Association of Asphalt Paving Technologists, Vol. 61, pp. 304–332, 1992. Roque, R., D. R. Hiltunen, and S. M. Stoffels, “Field Validation of SHRP Asphalt