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Springer Handbook of Experimental Solid Mechanics

Springer Handbooks provide a concise compilation of approved key information on methods of research, general principles, and functional relationships in physical sciences and engineering. The world’s leading experts in the fields of physics and engineering will be assigned by one or several renowned editors to write the chapters comprising each volume. The content is selected by these experts from Springer sources (books, journals, online content) and other systematic and approved recent publications of physical and technical information. The volumes are designed to be useful as readable desk reference books to give a fast and comprehensive overview and easy retrieval of essential reliable key information, including tables, graphs, and bibliographies. References to extensive sources are provided.

Springer

Handbook of Experimental Solid Mechanics Sharpe (Ed.) With DVD-ROM, 874 Figures, 58 in four color and 50 Tables

123

Editor: Professor William N. Sharpe, Jr. Department of Mechanical Engineering Room 126, Latrobe Hall The Johns Hopkins University 3400 North Charles Street Baltimore, MD 21218-2681, USA [email protected]

Library of Congress Control Number:

ISBN: 978-0-387-26883-5

2008920731

e-ISBN: 978-0-387-30877-7

c 2008, Springer Science+Business Media, LLC New York All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC New York, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. The use of designations, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Product liability: The publisher cannot guarantee the accuracy of any information about dosage and application contained in this book. In every individual case the user must check such information by consulting the relevant literature. Production and typesetting: le-tex publishing services oHG, Leipzig Senior Manager Springer Handbook: Dr. W. Skolaut, Heidelberg Typography and layout: schreiberVIS, Seeheim Illustrations: Hippmann GbR, Schwarzenbruck Cover design: eStudio Calamar Steinen, Barcelona Cover production: WMXDesign GmbH, Heidelberg Printing and binding: Stürtz GmbH, Würzburg Printed on acid free paper SPIN 11510079

9085/3820/YL

543210

V

Preface to Handbook on Experimental Stress Analysis

Experimental science does not receive Truth from superior Sciences: she is the Mistress and the other sciences are her servants ROGER BACON: Opus Tertium. Stress analysis has been regarded for some time as a distinct professional branch of engineering, the object of which is the determination and improvement of the mechanical strength of structures and machines. Experimental stress analysis strives to achieve these aims by experimental means. In doing so it does not remain, however, a mere counterpart of theoretical methods of stress analysis but encompasses those, utilizing all the conclusions reached by theoretical considerations, and goes far beyond them in maintaining direct contact with the true physical characteristics of the problems under considerations. Many factors make the experimental approach indispensable, and often the only means of access, in the investigation of problems of mechanical strength. At our present state of knowledge it is remarkable how quickly we can reach the limit of applicability of mathematical methods of stress analysis, and there is a multitude of comparatively simple, and in practice frequently occurring, stress problems for which no theoretical solutions have yet been obtained. In addition to this, theoretical considerations are usually based on simplifying assumptions which imply certain detachment from reality, and it can be decided only by experimentation whether such idealization has not resulted in an undue distortion of the essential features of the problem. No such doubt needs to enter experimental stress analysis, especially if it is done under actual service conditions, where all the factors due to the properties of the employed materials, the methods of manufacture, and the conditions of operation are fully represented. The advantage of the experimental approach becomes especially obvious if we consider that it is possible to determine experimentally the stress distribution in a machine part in actual operation without knowing the nature of the forces acting on the part under these circumstances, which proposition is clearly inaccessible to any theoretical method of analysis. To these major advantages we may add one more, from the point of view of the average practicing engineer, whose mathematical preparation is not likely to enable him to

deal theoretically with some of the complex strength problems which he, nevertheless, is expected to settle satisfactorily. To these men experimental methods constitute a recourse that is more readily accessible and that, with proper care and perseverance, is most likely to furnish the needed information. Several principal methods and literally hundreds of individual tools and artifices constitute the “arsenal” of the experimental stress analyst. It is interesting to observe, however, that each of these devices, no matter how peculiar it sometimes appears to be, has its characteristic feature and, with it, some unique advantage that may render this tool most suitable for the investigation of a particular problem. The stress analyst cannot afford, therefore, to ignore any of these possibilities. This circumstance, together with the ever-increasing demand on mechanical strength, will always tend to keep experimental stress analysis a distinct entity in the field of technical sciences. There has been a long-felt need of a comprehensive reference book of this nature, but, at the same time, it was recognized that no one person could possibly write with authority on all the major experimental procedures that are being used at present in the investigation of mechanical strength. It was proposed therefore that the problem could be solved only by a concerted effort which might be initiated most suitably under the aegis of the Society for Experimental Stress Analysis, and the writer was appointed as editor with complete freedom to proceed with the organization of this undertaking. Invitations were sent to thirty eminent engineers and scientists who were best known for their outstanding contributions in one or more of the specific branches of experimental stress analysis. It was most impressive to witness the readiness and understanding with which these men, many of them not even associated with the Society, responded to the request and joined the editor in contributing their work, without remuneration, to the furtherance of the aims of the Society, which thus becomes the sole recipient of all royalties from this publication. This being the first comprehensive publication in its field, it may be of general interest to say a few words about the method used in the planning and coordina-

VI

tion of the material. In inviting the contributors, I first briefly out-lined the subject to be covered requesting, in return, from each author a more detailed outline of what he would propose on his respective subject. These authors’ outlines were subsequently collected in a booklet, a copy of which was sent to each participant, thus informing him in advance of projected contents of all the other parts of the book. This scheme proved of considerable help in assuring adequate coverage of all matters of interest, without undue overlaps, repetition, or need of frequent cross references. In the final plan, as seen in the table of contents, the main body of the book was divided into 18 chapters, each dealing with either a principal method, from mechanical gages to x-ray analysis, or a major topic of interest, such as residual stresses, interpretation of service fractures, or analogies. In addition to these, an appendix was devoted to the discussion of three theoretical subjects which are of fundamental importance in the planning and interpretation of experimental stress work. In the final outcome, not only the book as a whole but also most of the individual chap-

ters turned out to be pioneering ventures in their own rights, often constituting the first systematic exposition of their respective subject matter. Another innovation was undertaken in the treatment of bibliographical references, where an effort was made to review briefly the contents of each entry, since it was found that the mere titles of technical articles seldom convey a satisfactory picture of their respective contents. Despite all precautions the book is bound to have errors and shortcomings, and it is the sincere hope of the editor that users of the book will not hesitate to inform him of possibilities of improvement which may be incorporated in a later edition. In the course of this work the editor was greatly aided by advice from numerous friends and colleagues, among whom he wishes to acknowledge in particular the invaluable help received from B. F. Langer, R. D. Mindlin, W. M. Murray, R. E. Peterson, and G. Pickett. Evanston, Illinois April 1950

M. Hetenyi

VII

Preface to the Handbook on Experimental Mechanics, First Edition

The Handbook on Experimental Stress Analysis, which was published under the aegis of the Society for Experimental Stress Analysis in 1950, has been the comprehensive and authoritative reference in our field for more than thirty years. Under the able editorship of the late M. Herenyi, 31 authors contributed without compensation 18 chapters and 3 appendices to this handbook. It received international acclaim and brought considerable income to the Society for Experimental Mechanics. Since 1950, new experimental techniques, such as holography, laser speckle interferometry, geometric moire, moire interferometry, optical heterodyning, and modal analysis, have emerged as practical tools in the broader field of experimental mechanics. The emergence of new materials and new disciplines, such as composite materials and fracture mechanics, resulted in the evolution of traditional experimental techniques to new fields such as orthotropic photoelasticity and experimental fracture mechanics. These new developments, together with the explosive uses of on/off-line computers for rapid data processing and the combined use of experimental and numerical techniques, have expanded the capabilities of experimental mechanics far beyond those of the 1950s. Sensing the need to update the handbook, H. F. Brinson initiated the lengthy process of revising the

handbook during his 1978-79 presidency of the Society. Since M. Hetenyi could not undertake the contemplated revision at that time, the decision was made to publish a new handbook under a new editor. Opinions ranging from topical coverage to potential contributors were solicited from various SEM members, and after a short respite I was chosen as editor by the ad hoc Handbook Committee chaired by J. B. Ligon. Despite the enormous responsibility, our task was made easier by inheriting the legacy of the Herenyi Handbook and the numerous suggestions that were collected by H. Brinson. The new handbook, appropriately entitled Handbook on Experimental Mechanics, is dedicated to Dr. Hetenyi. Twenty-five authors have contributed 21 chapters that include, among others, the new disciplines and developments that are mentioned above. The handbook emphasizes the principles of the experimental techniques and de-emphasizes the procedures that evolve with time. I am grateful to the contributors, who devoted many late afterhours in order to meet the manuscript deadlines and to J. B. Ligon who readily provided welcomed assistance during the trying times associated with this editorship.

Albert S. Kobayashi 1987

VIII

Preface to the Handbook on Experimental Mechanics, Second Edition

Since the publication of the first edition, considerable progress has been made in automated image processing, greatly reducing the heretofore laborious task of evaluating photoelastic and moire fringe patterns. It is therefore appropriate to add Chapter 21: “Digital Image Processing” before the final chapter, “Statistical Analysis of Experimental Data.” Apart from the new chapter, this second edition is essentially same as the first edition with minor corrections and updating. Exceptions to this are the addition of a section on optical fiber sensors in Chapter 2: “Strain Gages,” and extensive additions to

Chapter 14, which is retitled “Thermal Stress Analysis,” and to Chapter 16: “Experimental Modal Analysis.” To reiterate, the purpose of this handbook is to document the principles involved in experimental mechanics rather than the procedures and hardware, which evolve over time. To that extent, we, the twenty-seven authors, judging from the many appreciative comments which were received upon the publication of the first edition, have succeeded. Albert S. Kobayashi April 1993

IX

Preface

This handbook is a revision and expansion of the Handbook on Experimental Mechanics published by the Society for Experimental Mechanics in 1987 with a second edition in 1993 – both edited by Albert Kobayashi. All three of these trace a direct lineage to the seminal Handbook of Experimental Stress Analysis conceived and edited by Miklós Hetényi in 1950 and they encapsulate the history of the field. In 1950, the capability of measuring strains on models and structures was just becoming widely available. Engineers were still making their own wire resistance foil gages, and photoelasticity measurements required film processing. Conversion of these measurements to stresses relied on slide rules and graph paper. Now, foil resistance gages are combined with automatic data acquisition, and photoelasticity is just as automated. Input from both experimental methods is combined with finite element analysis to present stress variations in color on a computer screen. The focus then was on large structures such as airframes; in fact, the efforts of the Society for Experimental Stress Analysis (founded in 1938) were crucial to the rapid development of aircraft in the 1940s. While measurements on large structures continue to be important, researchers today also measure the mechanical properties of specimens smaller than a human hair. The field is completely different now. Experimental techniques and applications have expanded (or contracted if you prefer) from stress analysis of large structures to include the electromechanical analysis of micron-sized sensors and actuators. Those changes – occurring gradually over the early years but now more rapidly – led to a change in the society name to the Society for Experimental Mechanics. Those changes also have led to the expansion of the current volume with the deletion of some topics and the addition of others

in order to address these emerging topics in the “micro world”. This volume presents experimental solid mechanics as it is practiced in the early part of the 21st century. It is a field that is important as a technology and rich in research opportunities. A striking feature of this handbook is that 20 of the 36 chapters are on topics that have arisen or matured in the 15 years since the last edition; and, in most cases, these have been written by relatively young researchers and practitioners. Consider microelectromechanical systems (MEMS), for example. That technology, originated by electrical engineers only 25 years ago, now permeates our lives. It was soon learned that designers and manufacturers needed better understanding of the mechanical properties of the new materials involved, and experimental mechanists became involved only 15 years ago. That is just one example; several of the chapters speak to it as well as similar completely new topics. The reader will find in this volume not only information on the traditional areas of experimental solid mechanics, but on new and emerging topics as well. This revision was initiated by the Executive Board of the Society and managed by the very capable staff at Springer, in particular Elaine Tham, Werner Skolaut, and Lauren Danahy. Sound advice was provided over the course of the effort by Jim Dally and Tom Proulx. However, the real work was done by the authors. Each chapter was written by authors, who are not only experts, but who volunteered to contribute to this Handbook. Although they are thoroughly familiar with the technical details, it still required a major effort on their part to prepare a chapter. On behalf of the Society and Springer, I acknowledge and thank them. Baltimore June 2008

William N. Sharpe, Jr.

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About the Editor

William N. Sharpe, Jr. holds the Alonzo G. Decker Chair in Mechanical Engineering at the Johns Hopkins University where he served as department chairman for 11-1/2 years. His early research focused on laser-based strain measurement over very short gage lengths to study plasticity and fracture behavior. The past fifteen years have been devoted to development and use of experimental methods to measure the mechanical properties of materials used in microelectromechanical systems. He is a Fellow and Past President of the Society for Experimental Mechanics from which he received the Murray Medal. The American Society of Mechanical Engineers awarded him the Nadai Award as well as Fellow grade. He has received an Alexander von Humboldt Award along with the Roe Award from the American Society of Engineering Education.

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List of Authors Jonathan D. Almer Argonne National Laboratory 9700 South Cass Avenue Argonne, IL 60439, USA e-mail: [email protected] Archie A.T. Andonian The Goodyear Tire & Rubber Co. D/410F 142 Goodyear Boulevard Akron, OH 44305, USA e-mail: [email protected] Satya N. Atluri University of California Department of Mechanical & Aerospace Engineering, Center for Aeorspace Research & Education 5251 California Avenue, Suite 140 Irvine, CA 92612, USA David F. Bahr Washington State University Mechanical and Materials Engineering Pullman, WA 99164, USA e-mail: [email protected] Chris S. Baldwin Aither Engineering, Inc. 4865 Walden Lane Lanham, MD 20706, USA e-mail: [email protected] Stephen M. Belkoff Johns Hopkins University International Center for Orthopaedic Advancement, Department of Orthopaedic Surgery, Bayview Medical Center 5210 Eastern Avenue Baltimore, MD 21224, USA e-mail: [email protected] Hugh Bruck University of Maryland Department of Mechanical Engineering College Park, MD 20742, USA e-mail: [email protected]

Ioannis Chasiotis University of Illinois at Urbana-Champaign Aerospace Engineering Talbot Lab, 104 South Wright Street Urbana, IL 61801, USA e-mail: [email protected] Gary Cloud Michigan State University Mechanical Engineering Department East Lansing, MI 48824, USA e-mail: [email protected] Wendy C. Crone University of Wisconsin Department of Engineering Physics 1500 Engineering Drive Madison, WI 53706, USA e-mail: [email protected] James W. Dally University of Maryland 5713 Glen Cove Drive Knoxville, TN 37919, USA e-mail: [email protected] James F. Doyle Purdue University School of Aeronautics & Astronautics West Lafayette, IN 47907, USA e-mail: [email protected] Igor Emri University of Ljubljana Center for Experimental Mechanics Cesta na brdo 85 Lubljana, SI-1125, Slovenia e-mail: [email protected] Yimin Gan Universität GH Kassel Fachbereich 15 – Maschinenbau Mönchebergstr. 7 34109 Kassel, Germany e-mail: [email protected]

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List of Authors

Ashok Kumar Ghosh New Mexico Tech Mechanical Engineering and Civil Engineering Socorro, NM 87801, USA e-mail: [email protected]

Peter G. Ifju University of Florida Mechanical and Aerospace Engineering Gainesville, FL 32611, USA e-mail: [email protected]

Richard J. Greene The University of Sheffield Department of Mechanical Engineering Mappin Street Sheffield, S1 3JD, UK e-mail: [email protected]

Wolfgang G. Knauss California Institute of Technology-GALCIT 105-50 1201 East California Boulevard Pasadena, CA 91125, USA e-mail: [email protected]

Bongtae Han University of Maryland Mechanical Engineering Department College Park, MD 20742, USA e-mail: [email protected]

Albert S. Kobayashi University of Washington Department of Mechanical Engineering Seattle, Washington 98195-2600, USA e-mail: [email protected]

M. Amanul Haque Pennsylvania State University Department of Mechanical Engineering 317A Leonhard Building University Park, PA 16802, USA e-mail: [email protected]

Sridhar Krishnaswamy Northwestern University Center for Quality Engineering & Failure Prevention Evanston, IL 60208-3020, USA e-mail: [email protected]

Craig S. Hartley El Arroyo Enterprises LLC 231 Arroyo Sienna Drive Sedona, AZ 86332, USA e-mail: [email protected] Roger C. Haut Michigan State University College of Osteopathic Medicine, Orthopaedic Biomechanics Laboratories A407 East Fee Hall East Lansing, MI 48824, USA e-mail: [email protected] Jay D. Humphrey Texas A&M University Department of Biomedical Engineering 335L Zachry Engineering Center, 3120 TAMU College Station, TX 77843-3120, USA e-mail: [email protected]

Yuri F. Kudryavtsev Integrity Testing Laboratory Inc. PreStress Engineering Division 80 Esna Park Drive Markham, Ontario L3R 2R7, Canada e-mail: [email protected] Pradeep Lall Auburn University Department of Mechanical Engineering Center for Advanced Vehicle Electronics 201 Ross Hall Auburn, AL 36849-5341, USA e-mail: [email protected] Kenneth M. Liechti University of Texas Aerospace Engineering and Engineering Mechanics Austin, TX 78712, USA e-mail: [email protected]

List of Authors

Hongbing Lu Oklahoma State University School of Mechanical and Aerospace Engineering 218 Engineering North Stillwater, OK 74078, USA e-mail: [email protected] Ian McEnteggart Instron Coronation Road, High Wycombe Buckinghamshire, HP12 3SY, UK Dylan J. Morris National Institute of Standards and Technology Materials Science and Engineering Laboratory Gaithersburg, MD 20877, USA e-mail: [email protected] Sia Nemat-Nasser University of California Department of Mechanical and Aerospace Engineering 9500 Gilman Drive La Jolla, CA 92093-0416, USA e-mail: [email protected] Wolfgang Osten Universität Stuttgart Institut für Technische Optik Pfaffenwaldring 9 70569 Stuttgart, Germany e-mail: [email protected] Eann A. Patterson Michigan State University Department of Mechanical Engineering 2555 Engineering Building East Lansing, MI 48824-1226, USA e-mail: [email protected] Daniel Post Virginia Polytechnic Institute and State University (Virginia Tech) Department of Engineering Science and Mechanics Blacksburg, VA 24061, USA e-mail: [email protected]

Ryszard J. Pryputniewicz Worcester Polytechnic Institute NEST – NanoEngineering, Science, and Technology CHSLT – Center for Holographic Studies and Laser Micro-Mechatronics Worcester, MA 01609, USA e-mail: [email protected] Kaliat T. Ramesh Johns Hopkins University Department of Mechanical Engineering 3400 North Charles Street Baltimore, MD 21218, USA e-mail: [email protected] Krishnamurthi Ramesh Indian Institute of Technology Madras Department of Applied Mechanics Chennai, 600 036, India e-mail: [email protected] Krishnaswamy Ravi-Chandar University of Texas at Austin 1 University Station, C0600 Austin, TX 78712-0235, USA e-mail: [email protected] Guruswami Ravichandran California Institute of Technology Graduate Aeronautical Laboratories Pasadena, CA 91125, USA e-mail: [email protected] Robert E. Rowlands University of Wisconsin Department of Mechanical Engineering 1415 Engineering Drive Madison, WI 53706, USA e-mail: [email protected] Taher Saif University of Illinois at Urbana-Champaign Micro and Nanotechnology Laboratory, 2101D Mechanical Engineering Laboratory 1206 West Green Street Urbana, IL 61801, USA e-mail: [email protected] Wolfgang Steinchen (deceased)

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Jeffrey C. Suhling Auburn University Department of Mechanical Engineering 201 Ross Hall Auburn, AL 36849-5341, USA e-mail: [email protected] Michael A. Sutton University of South Carolina Center for Mechanics, Materials and NDE Department of Mechanical Engineering 300 South Main Street Columbia, SC 29208, USA e-mail: [email protected]

Robert B. Watson Vishay Micro-Measurements Sensors Engineering Department Raleigh, NC, USA e-mail: [email protected] Robert A. Winholtz University of Missouri Department of Mechanical and Aerospace Engineering E 3410 Lafferre Hall Columbia, MO 65211, USA e-mail: [email protected]

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Contents

List of Abbreviations ................................................................................. XXVII

Part A Solid Mechanics Topics 1 Analytical Mechanics of Solids Albert S. Kobayashi, Satya N. Atluri .......................................................... 1.1 Elementary Theories of Material Responses ..................................... 1.2 Boundary Value Problems in Elasticity ............................................ 1.3 Summary ...................................................................................... References ..............................................................................................

3 4 11 14 14

2 Materials Science for the Experimental Mechanist Craig S. Hartley ........................................................................................ 2.1 Structure of Materials .................................................................... 2.2 Properties of Materials................................................................... References ..............................................................................................

17 17 33 47

3 Mechanics of Polymers: Viscoelasticity Wolfgang G. Knauss, Igor Emri, Hongbing Lu ............................................ 3.1 Historical Background.................................................................... 3.2 Linear Viscoelasticity ..................................................................... 3.3 Measurements and Methods .......................................................... 3.4 Nonlinearly Viscoelastic Material Characterization ........................... 3.5 Closing Remarks ............................................................................ 3.6 Recognizing Viscoelastic Solutions if the Elastic Solution is Known ... References ..............................................................................................

49 49 51 69 84 89 90 92

4 Composite Materials Peter G. Ifju ............................................................................................. 4.1 Strain Gage Applications ................................................................ 4.2 Material Property Testing ............................................................... 4.3 Micromechanics ............................................................................ 4.4 Interlaminar Testing ...................................................................... 4.5 Textile Composite Materials............................................................ 4.6 Residual Stresses in Composites ..................................................... 4.7 Future Challenges.......................................................................... References ..............................................................................................

97 98 102 107 111 114 117 121 121

5 Fracture Mechanics Krishnaswamy Ravi-Chandar ................................................................... 5.1 Fracture Mechanics Based on Energy Balance ................................. 5.2 Linearly Elastic Fracture Mechanics .................................................

125 126 128

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Contents

5.3 Elastic–Plastic Fracture Mechanics .................................................. 5.4 Dynamic Fracture Mechanics .......................................................... 5.5 Subcritical Crack Growth ................................................................ 5.6 Experimental Methods................................................................... References ..............................................................................................

132 137 140 140 156

6 Active Materials Guruswami Ravichandran ........................................................................ 6.1 Background .................................................................................. 6.2 Piezoelectrics ................................................................................ 6.3 Ferroelectrics ................................................................................ 6.4 Ferromagnets ................................................................................ References ..............................................................................................

159 159 161 162 166 167

7 Biological Soft Tissues Jay D. Humphrey ..................................................................................... 7.1 Constitutive Formulations – Overview ............................................ 7.2 Traditional Constitutive Relations ................................................... 7.3 Growth and Remodeling – A New Frontier ...................................... 7.4 Closure ......................................................................................... 7.5 Further Reading ............................................................................ References ..............................................................................................

169 171 172 178 182 182 183

8 Electrochemomechanics of Ionic Polymer–Metal Composites Sia Nemat-Nasser .................................................................................... 8.1 Microstructure and Actuation ......................................................... 8.2 Stiffness Versus Solvation............................................................... 8.3 Voltage-Induced Cation Distribution .............................................. 8.4 Nanomechanics of Actuation ......................................................... 8.5 Experimental Verification .............................................................. 8.6 Potential Applications ................................................................... References ..............................................................................................

187 188 191 193 195 197 199 199

9 A Brief Introduction to MEMS and NEMS Wendy C. Crone ........................................................................................ 9.1 Background .................................................................................. 9.2 MEMS/NEMS Fabrication ................................................................. 9.3 Common MEMS/NEMS Materials and Their Properties ....................... 9.4 Bulk Micromachining versus Surface Micromachining ...................... 9.5 Wafer Bonding .............................................................................. 9.6 Soft Fabrication Techniques ........................................................... 9.7 Experimental Mechanics Applied to MEMS/NEMS.............................. 9.8 The Influence of Scale.................................................................... 9.9 Mechanics Issues in MEMS/NEMS ..................................................... 9.10 Conclusion .................................................................................... References ..............................................................................................

203 203 206 206 213 214 215 217 217 221 224 225

Contents

10 Hybrid Methods James F. Doyle ......................................................................................... 10.1 Basic Theory of Inverse Methods .................................................... 10.2 Parameter Identification Problems ................................................. 10.3 Force Identification Problems ........................................................ 10.4 Some Nonlinear Force Identification Problems ................................ 10.5 Discussion of Parameterizing the Unknowns ................................... References ..............................................................................................

229 231 235 240 246 255 257

11 Statistical Analysis of Experimental Data James W. Dally ........................................................................................ 11.1 Characterizing Statistical Distributions ............................................ 11.2 Statistical Distribution Functions .................................................... 11.3 Confidence Intervals for Predictions ............................................... 11.4 Comparison of Means .................................................................... 11.5 Statistical Safety Factor .................................................................. 11.6 Statistical Conditioning of Data ...................................................... 11.7 Regression Analysis ....................................................................... 11.8 Chi-Square Testing ........................................................................ 11.9 Error Propagation .......................................................................... References ..............................................................................................

259 260 263 267 270 271 272 272 277 278 279

Part B Contact Methods 12 Bonded Electrical Resistance Strain Gages Robert B. Watson ..................................................................................... 12.1 Standardized Strain-Gage Test Methods ......................................... 12.2 Strain and Its Measurement ........................................................... 12.3 Strain-Gage Circuits....................................................................... 12.4 The Bonded Foil Strain Gage .......................................................... 12.5 Semiconductor Strain Gages ........................................................... References ..............................................................................................

283 284 284 285 291 325 332

13 Extensometers Ian McEnteggart ...................................................................................... 13.1 General Characteristics of Extensometers ........................................ 13.2 Transducer Types and Signal Conditioning ...................................... 13.3 Ambient-Temperature Contacting Extensometers............................ 13.4 High-Temperature Contacting Extensometers ................................. 13.5 Noncontact Extensometers............................................................. 13.6 Contacting versus Noncontacting Extensometers ............................. 13.7 Conclusions ................................................................................... References ..............................................................................................

335 336 337 338 341 343 345 346 346

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Contents

14 Optical Fiber Strain Gages Chris S. Baldwin ...................................................................................... 14.1 Optical Fiber Basics........................................................................ 14.2 General Fiber Optic Sensing Systems ............................................... 14.3 Interferometry .............................................................................. 14.4 Scattering ..................................................................................... 14.5 Fiber Bragg Grating Sensors ........................................................... 14.6 Applications of Fiber Optic Sensors ................................................. 14.7 Summary ...................................................................................... References ..............................................................................................

347 348 351 354 359 361 367 368 369

15 Residual Stress Yuri F. Kudryavtsev .................................................................................. 15.1 Importance of Residual Stress ........................................................ 15.2 Residual Stress Measurement ......................................................... 15.3 Residual Stress in Fatigue Analysis ................................................. 15.4 Residual Stress Modification .......................................................... 15.5 Summary ...................................................................................... References ..............................................................................................

371 371 373 381 383 386 386

16 Nanoindentation: Localized Probes of Mechanical Behavior

of Materials David F. Bahr, Dylan J. Morris .................................................................. 16.1 Hardness Testing: Macroscopic Beginnings...................................... 16.2 Extraction of Basic Materials Properties from Instrumented Indentation ..................................................... 16.3 Plastic Deformation at Indentations ............................................... 16.4 Measurement of Fracture Using Indentation ................................... 16.5 Probing Small Volumes to Determine Fundamental Deformation Mechanisms ..................... 16.6 Summary ...................................................................................... References ..............................................................................................

402 404 404

17 Atomic Force Microscopy in Solid Mechanics Ioannis Chasiotis ..................................................................................... 17.1 Tip–Sample Force Interactions in Scanning Force Microscopy ........... 17.2 Instrumentation for Atomic Force Microscopy.................................. 17.3 Imaging Modes by an Atomic Force Microscope ............................... 17.4 Quantitative Measurements in Solid Mechanics with an AFM ........... 17.5 Closing Remarks ............................................................................ 17.6 Bibliography ................................................................................. References ..............................................................................................

409 411 412 423 432 438 439 440

389 389 392 396 399

Contents

Part C Noncontact Methods 18 Basics of Optics Gary Cloud ............................................................................................... 18.1 Nature and Description of Light ..................................................... 18.2 Interference of Light Waves ........................................................... 18.3 Path Length and the Generic Interferometer................................... 18.4 Oblique Interference and Fringe Patterns ....................................... 18.5 Classical Interferometry ................................................................. 18.6 Colored Interferometry Fringes ....................................................... 18.7 Optical Doppler Interferometry ....................................................... 18.8 The Diffraction Problem and Examples ........................................... 18.9 Complex Amplitude ....................................................................... 18.10 Fraunhofer Solution of the Diffraction Problem ............................... 18.11 Diffraction at a Clear Aperture ........................................................ 18.12 Fourier Optical Processing .............................................................. 18.13 Further Reading ............................................................................ References .............................................................................................. 19 Digital Image Processing for Optical Metrology Wolfgang Osten ....................................................................................... 19.1 Basics of Digital Image Processing .................................................. 19.2 Techniques for the Quantitative Evaluation of Image Data in Optical Metrology ...................................................................... 19.3 Techniques for the Qualitative Evaluation of Image Data in Optical Metrology ...................................................................... References ..............................................................................................

447 448 449 451 453 455 461 464 468 470 472 474 476 479 479

481 483 485 545 557

20 Digital Image Correlation for Shape

and Deformation Measurements Michael A. Sutton .................................................................................... 20.1 20.2 20.3 20.4 20.5 20.6 20.7 20.8 20.9 20.10

Background .................................................................................. Essential Concepts in Digital Image Correlation ............................... Pinhole Projection Imaging Model.................................................. Image Digitization ......................................................................... Intensity Interpolation .................................................................. Subset-Based Image Displacements ............................................... Pattern Development and Application ............................................ Two-Dimensional Image Correlation (2-D DIC)................................. Three-Dimensional Digital Image Correlation.................................. Two-Dimensional Application: Heterogeneous Material Property Measurements............................. 20.11 Three-Dimensional Application: Tension Torsion Loading of Flawed Specimen.................................. 20.12 Three-Dimensional Measurements – Impact Tension Torsion Loading of Single-Edge-Cracked Specimen ..................................... 20.13 Closing Remarks ............................................................................

565 566 568 569 573 573 575 577 579 581 585 588 593 597

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Contents

20.14 Further Reading ............................................................................ References ..............................................................................................

597 599

21 Geometric Moiré Bongtae Han, Daniel Post ........................................................................ 21.1 Basic Features of Moiré .................................................................. 21.2 In-Plane Displacements ................................................................ 21.3 Out-Of-Plane Displacements: Shadow Moiré .................................. 21.4 Shadow Moiré Using the Nonzero Talbot Distance (SM-NT) ............... 21.5 Increased Sensitivity...................................................................... References ..............................................................................................

601 601 607 611 617 623 626

22 Moiré Interferometry Daniel Post, Bongtae Han ........................................................................ 22.1 Current Practice ............................................................................. 22.2 Important Concepts ....................................................................... 22.3 Challenges .................................................................................... 22.4 Characterization of Moiré Interferometry ........................................ 22.5 Moiré Interferometry in the Microelectronics Industry ..................... References ..............................................................................................

627 630 634 644 645 646 652

23 Speckle Methods Yimin Gan, Wolfgang Steinchen (deceased)............................................... 23.1 Laser Speckle ................................................................................ 23.2 Speckle Metrology ......................................................................... 23.3 Applications .................................................................................. 23.4 Bibliography ................................................................................. References ..............................................................................................

655 655 658 668 672 672

24 Holography Ryszard J. Pryputniewicz .......................................................................... 24.1 Historical Development.................................................................. 24.2 Fundamentals of Holography ......................................................... 24.3 Techniques of Hologram Interferometry.......................................... 24.4 Representative Applications of Holography ..................................... 24.5 Conclusions and Future Work ......................................................... References ..............................................................................................

675 676 677 679 685 696 697

25 Photoelasticity Krishnamurthi Ramesh ............................................................................ 25.1 Preliminaries ................................................................................ 25.2 Transmission Photoelasticity .......................................................... 25.3 Variants of Photoelasticity ............................................................. 25.4 Digital Photoelasticity.................................................................... 25.5 Fusion of Digital Photoelasticity Rapid Prototyping and Finite Element Analysis ........................................................... 25.6 Interpretation of Photoelasticity Results .........................................

701 704 705 710 719 732 734

Contents

25.7 Stress Separation Techniques ......................................................... 25.8 Closure ......................................................................................... 25.9 Further Reading ............................................................................ 25.A Appendix ...................................................................................... References ..............................................................................................

735 737 737 738 740

26 Thermoelastic Stress Analysis Richard J. Greene, Eann A. Patterson, Robert E. Rowlands ......................... 26.1 History and Theoretical Foundations .............................................. 26.2 Equipment .................................................................................... 26.3 Test Materials and Methods ........................................................... 26.4 Calibration .................................................................................... 26.5 Experimental Considerations.......................................................... 26.6 Applications .................................................................................. 26.7 Summary ...................................................................................... 26.A Analytical Foundation of Thermoelastic Stress Analysis .................... 26.B List of Symbols .............................................................................. References ..............................................................................................

743 744 745 747 749 749 753 759 760 762 763

27 Photoacoustic Characterization of Materials Sridhar Krishnaswamy ............................................................................. 27.1 Elastic Wave Propagation in Solids ................................................. 27.2 Photoacoustic Generation .............................................................. 27.3 Optical Detection of Ultrasound...................................................... 27.4 Applications of Photoacoustics ....................................................... 27.5 Closing Remarks ............................................................................ References ..............................................................................................

769 770 777 783 789 798 798

28 X-Ray Stress Analysis Jonathan D. Almer, Robert A. Winholtz ..................................................... 28.1 Relevant Properties of X-Rays ........................................................ 28.2 Methodology ................................................................................. 28.3 Micromechanics of Multiphase Materials ........................................ 28.4 Instrumentation ............................................................................ 28.5 Experimental Uncertainties ............................................................ 28.6 Case Studies .................................................................................. 28.7 Summary ...................................................................................... 28.8 Further Reading ............................................................................ References ..............................................................................................

801 802 804 807 809 810 813 817 818 818

Part D Applications 29 Optical Methods Archie A.T. Andonian ............................................................................... 29.1 Photoelasticity .............................................................................. 29.2 Electronic Speckle Pattern Interferometry .......................................

823 824 828

XXIII

XXIV

Contents

29.3 Shearography and Digital Shearography ......................................... 29.4 Point Laser Triangulation ............................................................... 29.5 Digital Image Correlation ............................................................... 29.6 Laser Doppler Vibrometry ............................................................... 29.7 Closing Remarks ............................................................................ 29.8 Further Reading ............................................................................ References ..............................................................................................

830 831 832 834 835 836 836

30 Mechanical Testing at the Micro/Nanoscale M. Amanul Haque, Taher Saif ................................................................... 30.1 Evolution of Micro/Nanomechanical Testing .................................... 30.2 Novel Materials and Challenges...................................................... 30.3 Micro/Nanomechanical Testing Techniques ..................................... 30.4 Biomaterial Testing Techniques ...................................................... 30.5 Discussions and Future Directions .................................................. 30.6 Further Reading ............................................................................ References ..............................................................................................

839 840 841 842 856 859 862 862

31 Experimental Methods in Biological Tissue Testing Stephen M. Belkoff, Roger C. Haut ............................................................ 31.1 General Precautions ...................................................................... 31.2 Connective Tissue Overview ............................................................ 31.3 Experimental Methods on Ligaments and Tendons.......................... 31.4 Experimental Methods in the Mechanical Testing of Articular Cartilage ...................................................................... 31.5 Bone ............................................................................................ 31.6 Skin Testing .................................................................................. References ..............................................................................................

871 871 872 873 876 878 883 884

32 Implantable Biomedical Devices

and Biologically Inspired Materials Hugh Bruck ............................................................................................. 32.1 Overview....................................................................................... 32.2 Implantable Biomedical Devices..................................................... 32.3 Biologically Inspired Materials and Systems .................................... 32.4 Conclusions ................................................................................... 32.5 Further Reading ............................................................................ References ..............................................................................................

891 892 899 909 923 924 924

33 High Rates and Impact Experiments Kaliat T. Ramesh...................................................................................... 33.1 High Strain Rate Experiments ......................................................... 33.2 Wave Propagation Experiments ...................................................... 33.3 Taylor Impact Experiments ............................................................. 33.4 Dynamic Failure Experiments ......................................................... 33.5 Further Reading ............................................................................ References ..............................................................................................

929 930 945 949 949 953 954

Contents

34 Delamination Mechanics Kenneth M. Liechti ................................................................................... 34.1 Theoretical Background ................................................................. 34.2 Delamination Phenomena ............................................................. 34.3 Conclusions ................................................................................... References ..............................................................................................

961 962 968 980 980

35 Structural Testing Applications Ashok Kumar Ghosh ................................................................................. 985 35.1 Past, Present, and Future of Structural Testing ................................ 987 35.2 Management Approach to Structural Testing ................................... 990 35.3 Case Studies .................................................................................. 997 35.4 Future Trends ................................................................................ 1012 References .............................................................................................. 1013 36 Electronic Packaging Applications Jeffrey C. Suhling, Pradeep Lall ................................................................. 36.1 Electronic Packaging...................................................................... 36.2 Experimental Mechanics in the Field of Electronic Packaging ........... 36.3 Detection of Delaminations ........................................................... 36.4 Stress Measurements in Silicon Chips and Wafers ............................ 36.5 Solder Joint Deformations and Strains ............................................ 36.6 Warpage and Flatness Measurements for Substrates, Components, and MEMS .......................................... 36.7 Transient Behavior of Electronics During Shock/Drop ....................... 36.8 Mechanical Characterization of Packaging Materials ........................ References ..............................................................................................

Acknowledgements ................................................................................... About the Authors ..................................................................................... Detailed Contents...................................................................................... Subject Index.............................................................................................

1015 1017 1019 1022 1024 1031 1036 1039 1041 1042 1045 1049 1059 1083

XXV

XXVII

List of Abbreviations

A AASHTO ABS ACES AC AFM AFNOR AISC ALT AREA ASTM

Association of State Highway Transportation Officials American Bureau of Shipping analytical, computational, and experimental solution alternating current atomic force microscopy Association Française de National American Institute of Steel Construction accelerated lift testing American Railway Engineering Association American Society for Testing and Materials

B bcc BGA BHN BIL BSI BS BrUTS

body-centered cubic ball grid array Brinell hardness number boundary, initial, and loading British Standards Institute beam-splitter bromoundecyltrichlorosilane

C CAD CAE CAM CAR CBGA CCD CCS CDC CD CGS CHS CMC CMM CMOS CMP CMTD CNC CNT COD COI CP

computer-aided design computer-aided engineering computer-aided manufacturing collision avoidance radar ceramic ball grid array charge-coupled device camera coordinate system Centers for Disease Control compact disc coherent gradient sensing coefficient of hygroscopic swelling ceramic matrix composite coordinate measuring machine complementary metal–oxide-semiconductor chemical–mechanical polishing composite materials technical division computer numerical control carbon nanotube crack opening displacement crack opening interferometry conductive polymer

CSK CSM CTE CTOD CVD CVFE CW CZM

cytoskeleton continuous stiffness module coefficient of thermal expansion crack-tip opening displacement chemical vapor deposition cohesive-volumetric finite elements continuous wave cohesive zone model

D DBS DCB DCPD DC DEC DGL DIC DIN DMA DMD DMT DNP DPH DPN DPV DRIE DR DSC DSPI DSPSI DSP DTA DTS DTS

directional beam-splitter double-cantilever beam dicyclopentadiene direct current diffraction elastic constant Devonshire–Ginzburg–Landau digital image correlation Deutsches Institut für Normung dynamical mechanical analyzers digital micromirror device Derjaguin–Muller–Toporov distance from the neutral point diamond pyramid hardness dip-pen nanolithography Doppler picture velocimetry deep reactive-ion etching deviation ratio digital speckle correlation digital speckle-pattern interferometry digital speckle-pattern shearing interferometry digital speckle photography differential thermal analyzer distributed temperature sensing dodecyltrichlorosilane

E EAP ECM ECO EDM EDP EFM EFPI EMI ENF EPFM EPL

electroactive polymer extracellular matrix poly(ethylene carbon monoxide) copolymer electric-discharge machining ethylenediamine pyrochatechol electrostatic force microscopy extrinsic Fabry–Pérot interferometer electromagnetic interference end-notched flexure elastic–plastic fracture mechanics electron-beam projection lithography

XXVIII

List of Abbreviations

ESIS ESPI ESSP ES ET EW EXAFS

European Structural Integrity Society electronic speckle pattern interferometry electrostatically stricted polymer expert system emerging technologies equivalent weight extended x-ray absorption fine structure

IEC IFM IMU IPG IPMC IP IR-GFP ISDG ISO

focal adhesion complex fine-pitch ball grid array fiber Bragg grating flip-chip ball grid array face-centered cubic flip-chip plastic ball grid array Food and Drug Administration finite element modeling finite element fast Fourier transform full field of view functionally graded material focused ion beam force modulation microscopy Fillers–Moonan–Tschoegl frequency modulation friction stir welds Fourier-transform method field of view

ISTS IT

F FAC FBGA FBG FC-BGA fcc FC-PBGA FDA FEM FE FFT FFV FGM FIB FMM FMT FM FSW FTE FoV

G GF GMR GPS GRP GUM G&R

growth factor giant magnetoresistance global positioning sensing glass-reinforced plastic Guide for the Expression of Uncertainty in Measurement growth and remodeling

H HAZ hcp HDI HEMA HFP HNDE HRR HSRPS HTS

heat-affected zone hexagonal close-packed high-density interconnect 2-hydroxyethyl methacrylate half-fringe photoelasticity holographic nondestructive evaluation Hutchinson–Rice–Rosengreen high strain rate pressure shear high-temperature storage

I IBC IC

International Building Code integrated circuits

ion exchange capacity interfacial force microscope inertial measurement unit ionic polymer gel ionomeric polymer–metal composite image processing infrared grey-field polariscope interferometric strain/displacement gage International Organization for Standardization impulsive stimulated thermal scattering interferometer

J JIS

Japanese Industrial Standards

L LAN LASIK LCE LDV LED LEFM LFM LIGA LMO LORD LPCVD LPG LVDT LVDT

local area network laser-based corneal reshaping liquid crystal elastomer laser Doppler vibrometry light-emitting diode linear elastic fracture mechanics lateral-force AFM lithography galvanoforming molding long-working-distance microscope laser occlusive radius detector low-pressure CVD long-period grating linear variable differential transformer linear variable displacement transducer

M MBS MEMS MFM micro-CAT MITI MLF MMC MOEMS MOSFET MO M-TIR μPIV MPB MPCS MPODM MW(C)NT

model-based simulation micro-electromechanical system magnetic force microscopy micro computerized axial tomography Ministry of International Trade and Industry metal leadframe package metal matrix composites microoptoelectromechanical systems metal-oxide semiconductor field-effect transistor microscope objective modified total internal reflection microparticle image velocimetry morphotropic phase boundary most probable characteristics strength multipoint overdeterministic method multiwalled (carbon) nanotubes

List of Abbreviations

N NA NC-AFM NCOD NC NDE NDT/NDE NEMS NEPA NHDP NIST NMR NSF NSOM NVI

numerical aperture noncontact AFM normal crack opening displacement nanocrystalline nondestructive evaluation nondestructive testing/evaluation nanoelectromechanical system National Environmental Policy Act The National Highways Development Project National Institute of Standard and Technology nuclear magnetic resonance National Science Foundation near-field scanning optical microscopy normal velocity interferometer

OFDR OIML OIM OMC ONDT OPD OPS OSHA

optical/digital fringe multiplication object coordinate system orientation distribution function optoelectronic holography optoelectronic laser interferometric microscope optical frequency-domain reflectometry Organisation Internationale de Metrologie Legale orientation imaging microscopy organic matrix composites optical nondestructive testing optical path difference operations per second Occupational Safety and Health Administration

P PBG PBS PCB PCF PDMS PECVD PE PID PIN PIV PLA PLD PLZT PL PM fiber PMC

photonic-bandgap phosphate-buffered saline printed circuit board photonic-crystal fiber polydimethylsiloxane plasma-enhanced CVD pulse-echo proportional-integral-derivative p-type intrinsic n-type particle image velocimetry polylactic acid path length difference Pb(La,Zr,Ti)O3 path length polarization maintaining fiber polymer matrix composites

phase-measurement interferometry polymethyl methacrylate Pb(Mgx Nb1−x )O3 polarization maintaining power meter plastic quad flat package photorefractive crystal point spread function phase shifting technique plated-through holes polyvinyl chloride polyvinylidene fluoride physical vapor deposition phase velocity scanning Pb(Zr,Ti)O3

Q QC QFP QLV

O O/DFM OCS ODF OEH OELIM

PMI PMMA PMN PM PM PQFP PRC PSF PST PTH PVC PVDF PVD PVS PZT

quality-control quad flat pack quasilinear viscoelastic theory

R RCC RCRA RF RIBS RMS RPT RSG RSM RS RVE R&D

reinforced cement concrete Resource Conservation and Recovery Act radiofrequency replamineform inspired bone structures root-mean-square regularized phase tracking resistance strain gage residual stress management residual stress representative volume element research and development

S S-T-C S/N SAC SAM SAM SAR SAW SCF SCS SC SDF SD SEM SEM SHPB SIF SI SLC

self-temperature compensation signal to noise Sn–Ag–Cu scanning acoustic microscopy self-assembled monolayer synthetic-aperture radar surface acoustic wave stress concentration factor sensor coordinate system speckle correlation structure data files standard deviation scanning electron microscopy Society for Experimental Mechanics split-Hopkinson pressure bar stress intensity factor speckle interferometry surface laminar circuit

XXIX

XXX

List of Abbreviations

SLL SMA SNL SNR SOI SOTA SPATE SPB SPM SPSI SPS SP SQUID SRM SSM STC STM SThM ST SW(C)NT

surface laminar layer shape-memory alloy Sandia National Laboratories signal-to-noise ratio silicon on insulator state-of-the-art stress pattern analysis by thermal emissions space–bandwidth product scanning probe microscope speckle pattern shearing interferometry spatial phase shifting speckle photography superconducting quantum interface device sensitivity response method sacrificial surface micromachining self-temperature compensated scanning tunneling microscope scanning thermal microscopy structural test single-walled (carbon) nanotubes

TIG TIM TIR TMAH TM TPS TSA TS TWSME Terfenol-D TrFE

thermal barrier coatings temperature coefficient of expansion temperature coefficient of resistance transverse displacement interferometer time division multiplexing thermoelectric cooler transmission electron microscopy thermal evaluation for residual stress analysis tetrafluoroethylene

VLE

T TBC TCE TCR TDI TDM TEC TEM TERSA TFE

tungsten inert gas thermal interface materials total internal reflection tetramethylammonium hydroxide thermomechanical temporal phase shifting thermoelastic stress analysis through scan two-way shape memory effect Tb0.3 Dy0.7 Fe2 trifluoroethylene

U UCC UDL UHV ULE UP

ultrasonic computerized complex uniformly distributed load ultrahigh vacuum ultralow expansion ultrasonic peening

V VCCT VDA VISAR

virtual crack closure technique video dimension analysis velocity interferometer system for any reflector vector loop equations

W WCS WDM WFT WLF WM WTC

world coordinate system wavelength-domain multiplexing windowed Fourier transform Williams–Landel–Ferry wavelength meter World Trade Center

1

Part A

Solid Mec Part A Solid Mechanics Topics

1 Analytical Mechanics of Solids Albert S. Kobayashi, Seattle, USA Satya N. Atluri, Irvine, USA 2 Materials Science for the Experimental Mechanist Craig S. Hartley, Sedona, USA 3 Mechanics of Polymers: Viscoelasticity Wolfgang G. Knauss, Pasadena, USA Igor Emri, Lubljana, Slovenia Hongbing Lu, Stillwater, USA 4 Composite Materials Peter G. Ifju, Gainesville, USA 5 Fracture Mechanics Krishnaswamy Ravi-Chandar, Austin, USA 6 Active Materials Guruswami Ravichandran, Pasadena, USA

7 Biological Soft Tissues Jay D. Humphrey, College Station, USA 8 Electrochemomechanics of Ionic Polymer–Metal Composites Sia Nemat-Nasser, La Jolla, USA 9 A Brief Introduction to MEMS and NEMS Wendy C. Crone, Madison, USA 10 Hybrid Methods James F. Doyle, West Lafayette, USA 11 Statistical Analysis of Experimental Data James W. Dally, Knoxville, USA

3

Albert S. Kobayashi, Satya N. Atluri

In this chapter we consider certain useful fundamental topics from the vast panorama of the analytical mechanics of solids, which, by itself, has been the subject of several handbooks. The specific topics that are briefly summarized include: elementary theories of material response such as elasticity, dynamic elasticity, viscoelasticity, plasticity, viscoplasticity, and creep; and some useful analytical results for boundary value problems in elasticity.

1.1

Elementary Theories of Material Responses ........................... 1.1.1 Elasticity ..................................... 1.1.2 Viscoelasticity ..............................

1.1.3 Plasticity ..................................... 1.1.4 Viscoplasticity and Creep ...............

7 9

Boundary Value Problems in Elasticity .... 1.2.1 Basic Field Equations .................... 1.2.2 Plane Theory of Elasticity............... 1.2.3 Basic Field Equations for the State of Plane Strain ............................. 1.2.4 Basic Field Equations for the State of Plane Stress ............................. 1.2.5 Infinite Plate with a Circular Hole... 1.2.6 Point Load on a Semi-Infinite Plate

11 11 12

Summary .............................................

14

References ..................................................

14

1.2

4 4 6

1.3

12 12 13 13

Herein, we employ Cartesian coordinates exclusively. We use a fixed Cartesian system with base vectors ei (i = 1, 2, 3). The coordinates of a material particle before and after deformation are xi and yi , respectively. The deformation gradient, denoted as Fij , is defined to be ∂yi ≡ yi, j . (1.1) ∂x j

A wide variety of other strain measures may be derived [1.1–4]. Let ( da) be a differential area in the deformed body, and let n i be direction cosines of a unit outward normal to ( da). If the differential force acting on this area is d f i , the true stress or Cauchy stress τij is defined from the relation

The displacement components will be denoted by u i (= yi − xi ), such that

Thus τij is the stress per unit area in the deformed body. The nominal stress (or the transpose of the so-called first Piola–Kirchhoff stress) tij and the second Piola– Kirchhoff stress Sij are defined through the relations

Fij = δij + u i, j ,

(1.2)

where δij is a Kronecker delta. The Green–Lagrange strain tensor εij is given by 1 1 εij = (Fki Fk j − δij ) ≡ (u i, j + u j,i + u k,i u k, j ) . 2 2 (1.3)

When displacements and their gradients are infinitesimal, (1.3) may be approximated as 1 (1.4) εij = (u i, j + u j,i ) ≡ u (i, j) . 2

d f i = ( da)n j τij .

d f i = ( da)N j t ji = ( dA)N j S jk yi,k ,

(1.5)

(1.6) (1.7)

where ( dA)N j is the image in the undeformed configuration, of the oriented vector area ( da)n j in the deformed configuration. Note that both t ji and S ji are stresses per unit area in the undeformed configuration, and t ji is unsymmetric, while S ji is, by definition, symmetric [1.3,4]. It should also be noted that a wide variety of other stress measures may be derived [1.3, 4].

Part A 1

Analytical Me 1. Analytical Mechanics of Solids

4

Part A

Solid Mechanics Topics

Part A 1.1

From the geometric theory of deformation [1.5], it follows that ∂xi (1.8) , ( da)n j = (J)( dA)Nk ∂y j where ρ0 dv (1.9) = . J= d∀ ρ In the above dv is a differential volume in the deformed body, and d∀ is its image in the undeformed body. From (1.5) through (1.9), it follows that ∂x j ∂xi ∂xi τm j and Sij = J τmn . (1.10) tij = J ∂ym ∂ym ∂yn

Another useful stress tensor is the so-called Kirchhoff stress tensor, denoted by σij and defined as σij = Jτij .

(1.11)

When displacements and their gradients are infinitesimal, J ≈ 1, ∂xi /∂yk = δik and so on, and thus the distinction between all the stress measures largely disappears. Hence, in an infinitesimal deformation theory, one may speak of the stress tensor σij . For more on finite deformation mechanism of solids see [1.6, 7].

1.1 Elementary Theories of Material Responses The mathematical characterization of the behavior of solids is one of the most complex aspects of solid mechanics. Most of the time, the general behavior of a material defies our mathematical ability to characterize it. The theories discussed below must be viewed simply as idealizations of regimes of material response under specific types of loading and/or environmental conditions.

invariants of εij . These invariants may be defined as

1.1.1 Elasticity

where eijk is equal to 1 if (ijk) take on values 1, 2, 3 in a cyclic order, and equal to −1 if in anticyclic order, and is zero if two of the indices take on identical values. Sometimes, invariants J1 , J2 , and J3 , defined as

In this idealization, the underlying assumption is that stress is a single-valued function of strain and is independent of the history of straining. Also, for such materials, one may define a potential for stress in terms of strain, in the form of a strain-energy density function, denoted here by W. It is customary [1.4] to measure W per unit of the undeformed volume. In the general case of finite deformations, different stress measures are related to the derivative of W with respect to specific strain measures, labeled as conjugate strain measures, i. e., strains conjugate to the appropriate form of stress. Thus it may be shown [1.3] that tij =

∂W , ∂F ji

Sij =

∂W . ∂εij

(1.12)

Note that, for finite deformations, the Cauchy stress τij does not have a simple conjugate strain measure. When W does not depend on the location of the material particle (in the undeformed conjugation), the material is said to be homogeneous. A material is said to be isotropic if W depends on εij only through the basic

I1 = 3 + 2εkk , I2 = 3 + 4εkk + 2(εkk εmm − εkm εkm ) , and I3 = det|δmn + 2εmn | ≡ 1 + 2εkk + 2(εkk εmm − εkm εkm ) ,

J1 = (I1 − 3), J2 = (I2 − 2I1 + 3) , J3 = (I3 − I2 + I1 − 1)

(1.13)

(1.14)

are also used. When the material is isotropic, the Kirchhoff stress tensor may be shown to be the derivative of W with respect to a certain logarithmic strain measure [1.4]. Also, by decomposing the deformation gradient Fij into pure stretch and rigid rotation [1.4, 8], one may derive certain other useful stress measures, such as the Biot–Lure stress, Jaumann stress, and so on [1.4]. An isotropic nonlinearly elastic material may be characterized, in its behavior at finite deformations, by W=

∞

Crst (I1 − 3)r (I2 − 3)s (I3 − 1)t ,

r,s,t=0

C000 = 0 .

(1.15)

The ratio of volume change due to deformation, dv/ d∀, is given, for finite deformations, by I3 . Thus, for

Analytical Mechanics of Solids

¯ = W(εij ) + p(I3 − 1) , W

(1.16)

that is, tij =

∂W ∂I3 +p , ∂F ji ∂F ji

Sij =

∂W ∂I3 +p . (1.17) ∂εij ∂εij

For isotropic, incompressible, elastic materials, W(εij ) = W(I1 , I2 ) .

(1.18)

Thus (1.17) and (1.18) yield, for instance, ∂W ∂W δij + 4[δij (1 + εmm ) − δim δ jn εmn ] ∂I1 ∂I2 + p[δij (1 + 2εmm ) − 2δim δ jn εmn + 2eimn e jrs εmr εns ] . (1.19)

Sij = 2

A well-known representation of (1.18) is due to Mooney [1.8], where W(I1 , I2 ) = C1 (I1 − 3) + C2 (I2 − 3) .

(1.20)

So far, we have discussed isotropic materials. In general, for a homogeneous solid, one may write W = E ij εij + 12 E ijmn εij εmn + 13 E ijmnrs εij εmn εrs + · · · .

(1.21)

We use, for convenience of presentation, Sij and εij as conjugate measures of stress and strain. Since Sij and εij are both symmetric, one must have E ij = E ji , E ijmn = E jinm = E ijnm = E mnij , E ijmnrs = E jimnrs = E ijnmrs = E ijmnsr = · · · = Ersijmn = · · · . (1.22) Thus Sij = E ij + E ijmn εmn + E ijmnrs εmn εrs + · · · . (1.23) Henceforth, we will consider the case when deformations are infinitesimal. Thus εij ≈ (1/2)(u i, j + u j,i ). Further, the differences in the definitions of various stress measures disappear, and one may speak of the stress σij . Thus (1.23) may be rewritten as σij = E ij + E ijmn εmn + E ijmnrs εmn εrs + · · · . (1.24)

A material is said to be linearly elastic if a linear approximation of (1.24) is valid for the magnitude of strains under consideration. For such a material, σij = E ij + E ijmn εmn .

(1.25)

The stress at zero strain (i. e., E ij ) most commonly is due to temperature variation from a reference state. The simplest assumption in thermal problems is to set E ij = −βij ΔT where ΔT [= (T − T0 )] is the temperature increment from the reference value T0 . For an anisotropic linearly elastic solid, in view of the symmetries in (1.23), one has 21 independent elastic constants E ijkl and six constants βij . In the case of isotropic linearly elastic materials, an examination of (1.13) through (1.15) reveals that the number of independent elastic constants E ijkl is reduced to two, and the number of independent βs to one. Thus, for an isotropic elastic material, σij = λεkk δij + 2μεij − βΔT δij ,

(1.26a)

where λ and μ are Lamé parameters, which are related to the Young’s modulus E and the Poisson’s ratio ν through E Eν , μ= . λ= (1 + ν)(1 − 2ν) 2(1 + ν) The bulk modulus K is defined as 3λ + 2μ . K= 3 The inverse of (1.26a) is ν 1+ν (1.26b) σij + αΔT δij , εij = − σmn δij + E E where β and α are related through Eα β= (1.26c) 1 − 2ν and α is the linear coefficient of thermal expansion. The state of plane strain is characterized by the conditions that u r = u r (X s ), r, s = 1, 2, and u 3 = 0. Thus ε3i = 0, i = 1, 2, 3. In plane strain, 1 − ν2 ν σ22 + α(1 + ν)ΔT , σ11 − ε11 = E 1−ν (1.27a)

1 − ν2 ν ε22 = σ22 − σ11 + α(1 + ν)ΔT , E 1−ν

(1.27b)

1+ν σ12 , E σ33 = ν(σ11 + σ) − αEΔT . ε12 =

(1.27c) (1.27d)

5

Part A 1.1

incompressible materials, I3 = 1. For incompressible materials, stress is determined from strain only to within a scalar quantity (function of material coordinates) called the hydrostatic pressure. For such materials, one ¯ in may define a modified strain-energy function, say W, which the incompressibility condition, I3 = 1, is introduced as a constraint through the Lagrange multiplier p. Thus

1.1 Elementary Theories of Material Responses

6

Part A

Solid Mechanics Topics

Part A 1.1

The state of plane stress is characterized by the conditions that σ3k = 0, k = 1, 2, 3. Here one has 1 (1.28a) ε11 = (σ11 − νσ22 ) + αΔT , E 1 ε22 = (σ22 − νσ11 ) + αΔT , (1.28b) E 1+ν ε12 = (1.28c) σ12 , E ν ε33 = − (σ11 + σ22 ) + αΔT . (1.28d) E Note that in (1.27c) and (1.28c), ε12 is the tensor component of strain. Sometimes it is customary to use the engineering strain component γ12 = 2ε12 . Note also that, in the case of a linearly elastic material, the strainenergy density W is given by W= =

σij (t) = εkl (0+ )E ijkl +

t E ijkl (t − τ) 0

∂εkl dτ ∂τ (1.33a)

= E kl (0+ )εijkl +

t εijkl (t − τ) 0

∂E kl dτ . ∂τ (1.33b)

1 2 σij εij 1 2 (σ11 ε11 + σ22 ε22 + σ33 ε33

In the above, it has been assumed that σkl = εkl = 0 for t < 0 and that εij (t) and E ij (t) are piecewise continuous. E ijkl (t) is called the relaxation tensor for an anisotropic material. Conversely, one may write

+ 2ε12 σ12 + 2ε13 σ13 + 2ε23 σ23 ) ≡ 12 (σ11 ε11 + σ22 ε22 + σ33 ε33 + γ12 σ12 + γ23 σ23 + γ13 σ13 ) .

(1.29)

From (1.26b) it is seen that, for linearly elastic isotropic materials, 1 − 2ν σmm σmm + 3αΔT ≡ + 3αΔT . (1.30) εkk = E 3k When the bulk modulus k → ∞ (or ν → 12 ), it is seen that εkk → 3αΔT and is independent of the mean stress. Note also from (1.26c) that β → ∞ as ν → 12 . For such materials, the mean stress is indeterminate from deformation alone. In this case, the relation (1.26a) is replaced by σij = −ρδij + 2μεij

(1.31a)

with the constraint εkk = 3αΔT ,

(1.31b)

εij

where ρ is the hydrostatic pressure and is the deviator of the strain. Note that the strain-energy density of a linearly elastic incompressible material is W = μεij εij − p(εkk − 3αΔT ) ,

materials are those for which the current deformation is a function of the entire history of loading, and conversely, the current stress is a function of the entire history of straining. Linearly viscoelastic materials are those for which the hereditary relations are expressed in terms of linear superposition integrals, which, for infinitesimal strains, take the forms

(1.32)

wherein p acts as a Lagrange multiplier to enforce (1.31b).

1.1.2 Viscoelasticity A linearly elastic solid, by definition, is one that has the memory of only its unstrained state. Viscoelastic

+

t

εij (t) = σkl (0 )Cijkl +

Cijkl (t − τ) 0

∂σkl dτ , ∂τ (1.34)

where Cijkl (t) is called the creep compliance tensor. For isotropic linearly viscoelastic materials, E ijkl = μ(t)(δik δ jl + δlk δ jk ) + λ(t)δij δkl ,

(1.35)

where μ(t) is the shear relaxation modulus and B(t) ≡ [3α(t) + 2μ(t)]/3 is the bulk relaxation modulus. It is often assumed that B(t) is a constant; so that the material is assumed to have purely elastic volumetric change. In viscoelasticity, a Poisson function corresponding to the strain ratio in elasticity does not exists. However, for every deformation history there is computable Poisson contraction or expansion behavior. For instance, in a uniaxial tension test, let the stress be σ11 , the longitudinal strain ε11 , and the lateral strain ε22 . For creep at constant stress, the ratio of lateral contraction, denoted by νc (t), is νc (t) = −ε22 (t)/ε11 (t). On the other hand, under relaxation at constant strain, ε11 , the lateral contraction ratio is ν R (t) = −ε22 (t)/ε11 . It is often convenient, though not physically correct, to assume that Poisson’s ratio is a constant, which renders B(t) proportional to μ(t). A constant bulk modulus provides a much better and simple approximation for the material behavior than a constant Poisson’s ratio when properties over the whole time range are needed.

Analytical Mechanics of Solids

σ ij ( p) = pE ijkl ( p)εkl ( p)

(1.36a)

εij ( p) = pC ijkl ( p)σ kl ( p) ,

(1.36b)

and

deformation is assumed to be insensitive to hydrostatic pressure, the yield function is assumed, in general, to depend on the stress deviator, σij = σij − 13 σmm δij . The commonly used yield functions are von Mises

where (·) is the Laplace transform of (·) and p is the Laplace variable. From (1.36a) and (1.36b), it follows that p E ijkl C klmn = δim δn j . 2

M

μm exp(−μm t) ,

m=1 M

B(t) = B0 +

Bm exp(−βm t) .

m=1

1.1.3 Plasticity Most structural metals behave elastically for only very small values of strain, after which the materials yield. During yielding, the apparent instantaneous tangent modulus of the material is reduced from those in the prior elastic state. Removal of load causes the material to unload elastically with the initial elastic modulus. Such materials are usually labeled as elastic–plastic. Observed phenomena in the behavior of such materials include the so-called Bauschinger effect (a specimen initially loaded in tension often yields at a much reduced stress when reloaded in compression), cyclic hardening, and so on [1.9, 10]. (When a specimen a specimen is subjected to cyclic straining of amplitude −ε to +ε, the stress for the same value of tensile strain ε, prior to unloading, increases monotonically with the number of cycles and eventually saturates.) Various levels of sophistication of elastic-plastic constitutive theories are necessary to incorporate some or all of these observed phenomena. Here we give a rather cursory review of this still burgeoning literature. In most theories of metal plasticity, it is assumed that plastic deformations are entirely distortional in nature, and that volumetric strain is purely elastic. The elastic limit of the material is assumed to be specified by a yield function, which is a function of stress (or of strain, but most commonly of stress). Since plastic

J2 = 12 σij σij ,

(1.38)

Tresca f (σij ) = (σ1 − σ2 )2 − 4k2 (σ2 − σ3 )2 − 4k2 (1.39) × (σ1 − σ3 )2 − 4k2 = 0 .

(1.37)

It is also customary to represent the relaxation moduli, μ(t) and B(t), in series form, as μ(t) = μ0 +

f (σij ) = J2 − k2 ,

In (1.38) and (1.39), k may be a function of the plastic strain. Both (1.38) and (1.39) represent a surface, which is defined as the yield surface, in the stress space. The two equations also imply the equality of the tensile and compressive yield stresses at all times – so-called isotropic hardening. The yield surface expands while its center remains fixed in the stress space.√ The relation of k to test data follows: in (1.38), k = σ¯ / 3, where σ¯ is the yield stress in uniaxial tension, which may be a function of plas√ tic strain for strain-hardening materials, or k = τ¯ / 2, where τ¯ is the yield stress in pure shear; or in (1.39), k = σ¯ /2 or τ¯ . Experimental data appear to favor the use of the Mises condition [1.11, 12]. To account for the Bauschinger effect, one may use the representation of the yield surface f (σij − αij ) = 0 = 12 σij − αij σij − αij − 13 σ¯ 2 =0,

(1.40)

αij

where represents the center of the yield surface in the deviatoric stress space. The evolution equations suggested for αij by Prager [1.13] and Ziegler [1.14], respectively, are that the incremental dαij is proporp tional to the incremental plastic strain dεij or dαij = c dεij

(1.41)

dαij = dμ(σij − αij ) .

(1.42)

p

and

(·)

In the above, denotes the deviatoric part of the second-order tensor (·) where an additive decomposition of differential strain into elastic and plastic parts p (i. e., dεij = dεije + dεij ) was used. Elastic processes (with no increase in plastic strain) and plastic processes (with increase in plastic strain) are defined [1.12] as

7

Part A 1.1

The Laplace transforms of (1.33) and (1.34) may be written as

1.1 Elementary Theories of Material Responses

8

Part A

Solid Mechanics Topics

Part A 1.1

elastic process: f < 0 or

f = 0 and

∂f dσij ≤ 0 , ∂σij

(1.43)

plastic process: ∂f dσij > 0 . (1.44) ∂σij The flow rule for strain-hardening materials, arising out of consideration of stress working in a cyclic process and stability of the process, often referred to as Drucker’s postulates [1.15], is given by f = 0 and

p dεij

∂f = dλ . ∂σij

(1.45)

The scalar dλ is determined from the fact that d f = 0 during a plastic process, the so-called consistency condition. Using the isotropic-hardening (J2 flow) theory for which f is given in (1.38), this consistency condition leads to 9 σ dσmn p (1.46) dεij = σij mn 2 , 4 H σ¯ where H is the slope of the curve of stress versus plastic strain in uniaxial tension (or, more correctly, the slope of the curve of true stress versus logarithmic strain in pure tension). On the other hand, for Prager’s linearly kinematic hardening rules, given in (1.40) and (1.41), the consistency condition leads to 3 p σmn − αmn dσmn σij − αij . (1.47) dεij = 2cσ¯ 2 For pressure-insensitive plasticity, the stress–strain laws may be written as dσmn = (3λ + 2μ) dεmn , p dσij = 2μ dεij − dεij .

(1.48a) (1.48b) p

Choosing a parameter ζ such that ζ = 1 when dεij = 0 p and α = 0 when dεij = 0, we have dσ 9 σmn mn dσij = 2μ dεij − σij α (1.49) 4 H σ¯ 2 for isotropic hardening, and

3 dσij = 2μ dεij − (σ − αmn ) 2cσ¯ 2 mn × dσmn (σij − αij )α

(1.50)

for Prager’s linearly kinematic hardening. Taking the tensor product of both sides of (1.49) with σij (and

dσ noting that σmn dσmn = σmn mn by definition), we have 3α σ dσ (1.51a) dσij σij = 2μ dεij σij − 2H mn mn or 2μH dσij σij = dε σ , when α = 1 . (1.51b) H + 3μ ij ij Use of (1.51b) in (1.49) results in

9α dσij = 2μ dεij − 2μ σ σ dε . 4(H + 3μ)σ¯ 2 ij mn mn

(1.52)

Combining (1.48a) and (1.52), one may write the isotropic-hardening elastic-plastic constitutive law in differential form as dσij = 2μδim δ jn + λδij δmn 9αμ − 2μ σ σ (1.53) dεmn , (2H + 6μ)σ¯ 2 ij mn dε ≡ σ dε wherein σmn mn has been noted. Simimn mn larly, by taking the tensor product of (1.50) with (σij − αij ) and repeating steps analogous to those in (1.51) through (1.53), one may write the kinematic-hardening elastoplastic constitutive law as

3μ dσij = 2δim δ jn + λδij δmn − 2μ (c + 2μ)σ¯ 2 × (σij − αij )(σmn − αmn ) dεmn . (1.54) Note that all the developments above are restricted to the infinitesimal strain and small-deformation case. Discussion of finite-deformation plasticity is beyond the scope of this summary. (Even in small deformation plasticity, if the current tangent modulus of the stress–strain relation are of the same order of magnitude as the current stress, one must use an objective stress rate, instead of the material rate dσij , in (1.54).) Here the objectivity of the stress–strain relation plays an important role. We refer the reader to [1.4, 16, 17]. We now briefly examine the elastic-plastic stress– strain relations in the isotropic-hardening case, for plane strain and plane stress, leaving it to the reader to derive similar relations for kinematic hardening. In the planestrain case, dε3n = 0, n = 1, 2, 3. Using this in (1.53), we have

dσαβ = 2μδαθ δβν + λδαβ δθν 9αμ − 2μ σ σ (1.55) dεθν (2H + 6μ)σ¯ 2 αβ θυ

Analytical Mechanics of Solids

dσ33 = λ dεθθ − 2μ

9αμ σ σ dεϑν , (2H + 6μ)σ¯ 2 33 θυ

α, β, θ, ν = 1, 2 . (1.56) Note that, in the plane-strain case, σ33 , as integrated from (1.56), enters the yield condition. In the planestress case the stress–strain relation is somewhat tedious to derive. Noting that in the plane-stress case, dεαβ = dεeαβ + p dεαβ one may, using the elastic strain–stress relations as given in (1.33), write p

dεαβ = dεeαβ + dεαβ

1 ν p = dσαβ − dσθθ δαβ + dεαβ (1.57) 2μ 1+ν

1 ν dσαβ − = dσθθ δαβ 2μ 1+ν 9 , (1.58) + σαβ (σθυ dσθυ ) 4H σ¯ 2 wherein (1.16) has been used. Equation (1.58) may be inverted to obtain dσαβ in terms of dεθν . This 3 × 3 matrix inversion may be carried out, leading to the result [1.18] 2 Q ) + 2P dε11 dσ11 = (σ22 E + (−σ11 σ22 + 2ν P) dε22 σ11 + νσ22 − 2 dε12 , σ12 1+ν Q σ22 + 2ν P) dε11 dσ22 = (−σ11 E 2 + (σ11 ) + 2P dε22 σ + σ11 2 − 22 dε12 , σ12 1+ν σ11 + νσ22 Q dσ12 = − σ12 dε11 E 1+ν σ + νσ11 − 22 σ12 dε22

1+ν R 2H + + (1 − ν)σ¯ dε12 , 2(1 + ν) 9E where σ2 2H 2 P= σ¯ + 12 , Q = R + 2(1 − ν2 )P , 9H 1+ν (1.59)

and 2 + 2νσ11 σ22 + σ22 2. R = σ11

(1.60)

As noted earlier, the classical plasticity theory has several limitations. Intense research is underway to im-

prove constitutive modeling in cyclic plasticity, and so on, some notable avenues of current research being multisurface plasticity model, endochronic theories, and related internal variable theories (see, e.g., [1.19–21]).

1.1.4 Viscoplasticity and Creep A viscoplastic solid is similar to a viscous fluid, except that the former can resist shear stress even in a rest configuration; but when the stresses reach critical values as specified by a yield function, the material flows. Consider, for instance, the loading case of simple shear with the only applied stress being σ12 . Restricting ourselves to infinitesimal deformations and strains, let the shearstrain rate be ε˙ 12 = ( dε12 / dt). Then ε˙ 12 = 0 until the magnitude of σ12 reaches a value k, called the yield stress. When |σ12 | > k, ε˙ 12 , by definition for a simple viscoplastic material, is proportional to |σ12 | − k and has the same sign as σ12 . Thus, defining a function F 1 for this one-dimensional problem as |σ12 | (1.61) −1 , k the viscoplastic property may be characterized by the equation F1 =

2η˙ε12 = k(F 1 )σ12 , where

(1.62)

F1

must have the property ⎧ ⎨0 if F 1 < 0 , F 1 = ⎩ F 1 if F 1 ≥ 0 ,

(1.63)

and where η is the coefficient of viscosity. The above relation for simple shear is due to Bing2 for simple shear, ham [1.22]. Recognizing that J = σ12 a generalization of the above for three-dimensional case was given by Hohenemser and Prager [1.23] as νp

η˙εij = 2k

F 1

∂F , ∂σij

(1.64a)

where

1/2 1/2 σij σij /2 J2 F= −1 = −1 , k k

(1.64b)

and the specific function F is defined similar to F 1 of (1.63). For an elasto-viscoplastic solid undergoing infinitesimal straining, one may use the additive decomposition p

ε˙ ij = ε˙ ije + ε˙ ij

(1.65)

9

Part A 1.1

and

1.1 Elementary Theories of Material Responses

10

Part A

Solid Mechanics Topics

Part A 1.1

and the stress–strain rate relation ν p σ˙ ij = E ijkl ε˙ kl − ε˙ kl ,

(1.66)

where E ijkl are the instantaneous elastic moduli. Note that the viscoplastic strains in (1.64a) are purely deviatoric, since ∂F/∂σij = σij /2 is deviatoric. Thus, for an isotropic solid, (1.66) may be written as σ˙ mn = (3λ + 2μ) ε˙ mn and

σ˙ ij = 2μ ε˙ ij − ε˙ ij .

(1.68)

(1.69a)

(1.69b)

The expression (1.69a) is often referred to as time hardening and (1.69b) as strain hardening. In as much as (1.68), (1.69a), and (1.69b), are valid for constant stress, (1.69a) and (1.69b), when integrated for variable stress histories, do not necessarily give the same results. Usually, strain hardening leads to better agreement with experimental findings for variable stresses. In the study of creep at a given temperature and for long times, called steady-state creep, the creep strain rate in uniaxial loading is usually expressed as ε˙ c = f (σ, T ) ,

(1.70)

where T is the temperature. Assuming that the effects of σ and T are separable, the relation ε˙ c = f 1 (σ) f 2 (T ) = Aσ n f 2 (T ) = Bσ n

3 σ σ 2 ij ij

and ε˙ ceq =

2 ε˙ ij ε˙ ij 3

1/2 (1.73)

(1.67b)

or, equivalently, ε˙ c = g(σ, εc ) .

(1.72)

where the subscript “eq” denotes an “equivalent” quantity, defined, analogous to the case of plasticity, as σ˙ eq =

where σ is the uniaxial stress and t is the time. The creep rate may be written as ε˙ c = f (σ, t) ,

n , ε˙ ceq = Bσeq

(1.67a)

On the other hand, for metals operating at elevated temperatures, the strain in uniaxial tension is known to be a function of time, for a constant stress of magnitude even below the conventional elastic limit. Most often, based on extensive experimental data [1.24], the creep strain under constant stress, in uniaxial tests, is expressed as ε˙ c = Aσ n t m ,

involve no volume change. Thus, in the multiaxial case, ε˙ ijc is a deviatoric tensor. The relation (1.71) may be generalized to the multiaxial case as

(1.71a) (1.71b)

is usually employed, with B denoting a function of the temperature. The steady-state creep strains are associated largely with plastic deformations and are usually observed to

such that σeq ε˙ ceq = σij ε˙ ijc . Thus (1.72) implies that ε˙ ijc =

3 B(σeq )n−1 σij . 2

(1.74)

For the elastic-creeping solid, one may again write ε˙ ij = ε˙ ije + ε˙ ijc

(1.75)

and once again, the stress–strain rate relation may be written as (1.76) σ˙ ij = E ijkl ε˙ kl − ε˙ ckl . In the above, the applied stress level has been assumed to be such that the material remains within the elastic limit. If the applied loads are of such a magnitude as to cause the material to exceed its yield limit, one must account for plastic or viscoplastic strains. An interesting unified viscoplastic/plastic/creep constitutive law has been proposed by Perzyna [1.25]. Under multiaxial conditions, the relation for inelastic strain rate suggested in [1.26] is ε˙ ija = Aψ( f )

∂q , ∂σij

(1.77)

where A is the fluidity parameter, the superscript “a” denotes an elastic strain rate, and f is a loading function, expressed, analogous to the plasticity case, as f (σij , k) = ϕ(σij ) − k = 0 ,

(1.78)

q is a viscoplastic potential defined as q = q(σij ) , and ψ( f ) is a specific function such that ⎧ ⎨0 if f < 0 ψ( f ) = ⎩ψ( f ) if f ≥ 0 .

(1.79)

(1.80)

Analytical Mechanics of Solids

ψ( f ) = f n and

f =

3 σ σ 2 ij ij

(1.81a)

1/2 − σ¯ = σeq − σ¯ .

(1.81b)

By letting σ¯ = 0 and q = f , one may easily verify that ε˙ ija of (1.77) tends to the creep strain rate ε˙ ijc of (1.74). Letting σ¯ be a specified constant value and q = f , we obtain, using (1.81b) in (1.77), that ε˙ ija =

σij 3 A(σeq − σ¯ )n 2 σeq

for f > 0 (i. e., σeq > σ¯ ) .

(1.82)

The equivalent inelastic strain may be written as 2 a a 1/2 = A(σeq − σ¯ )n (1.83a) ε˙ ij ε˙ ij ε˙ aeq = 3 or 1 a 1/2 . (1.83b) ε˙ eq σeq − σ¯ = A Thus, if a stationary solution of the present inelastic model (i. e., when ε˙ aeq → 0) is obtained, it

is seen that σeq → σ. ¯ Thus a classical inviscid plasticity solution is obtained. This fact has been utilized in obtaining classical rate-independent plasticity solutions from the general model of (1.77), by Zienkiewicz and Corneau [1.26]. An alternative way of obtaining an inviscid plastic solution from Perzyna’s model is to let A → ∞. This concept has been implemented numerically by Argyris and Kleiber [1.27]. Also, as seen from (1.83b), σeq or equivalently the size of the yield surface is governed by isotropic work-hardening effects as characterized by the dependence on viscoplastic work, and the strain-rate effect as q characterized by the term (˙εeq )1/n . Thus rate-sensitive plastic problems may also be treated by Perzyna’s model [1.25]. Thus, by appropriate modifications, the general relation (1.77) may be used to model creep, rate-sensitive plasticity, and rate-insensitive plasticity. By a linear combination of strain rates resulting from these individual types of behavior, combined creep, plasticity, and viscoplasticity may be modeled. However, such a model is more or less formalistic and does not lead to any physical insights into the problem of interactive effects between creep, plasticity, and viscoplasticity. Modeling of such interactions is the subject of a large number of current research studies.

1.2 Boundary Value Problems in Elasticity 1.2.1 Basic Field Equations When deformations are finite and the material is nonlinear, the field equations governing the motion of a solid may become quite complicated. When the constitutive equation is of a differential form, such as in plasticity, viscoplasticity, and so on, it is often convenient to express the field equations in rate form as well. On the other hand, when stress is a single-valued function of strain as in nonlinear elasticity, the field equations may be written in a total form. In general, when numerical procedures are employed to solve boundary/initial value problems for arbitrary-shaped bodies, it is often convenient to write the field equations in rate form, for arbitrary deformations and general constitutive laws. A wide variety of equivalent but alternative forms of these equations is possible, since one may use a wide variety of stress and strain measures, a wide variety of rates of stress and strain, and a variety of coordinate

systems, such as those in the initial undeformed configuration (total Lagrangian), the currently deformed configuration (updated Lagrangian), or any other known intermediate configuration. For a detailed discussion see [1.3, 4]. Each of the alternative forms may offer advantages in specific applications. It is beyond the scope of this chapter to discuss the foregoing alternative forms. Here we state, for a finitely deformed nonlinearly elastic solid, the relevant field equations governing stress, strain, and deformation when the solid undergoes dynamic motion. For this purpose, let x j denote the Cartesian coordinates of a material particle in the undeformed solid. Let u k (xi ) be the arbitrary displacement of a material particle from the undeformed to the deformed configuration. Let Sij be the second Piola–Kirchhoff stress in the finitely deformed solid. Note that S jj is measured per unit area in the undeformed configuration. Let the Green–Lagrange strain tensor, which is work-conjugate to Sij [1.3],

11

Part A 1.2

If q ≡ f , one has the so-called associative law, and if q = f , one has a nonassociative law. Perzyna [1.25] suggests a fairly general form for ψ as

1.2 Boundary Value Problems in Elasticity

12

Part A

Solid Mechanics Topics

Part A 1.2

be εij . Let ρ0 be the mass density in the undeformed solid; b j be body forces per unit mass; ti be tractions measured per unit area in the undeformed solid, prescribed at surface St , of the undeformed solid; and let u i be prescribed displacements at Su . The field equations are [1.4]: linear momentum balance Sik (δ jk + u j,k ) + ρ0 b j = ρ0 u¨ j ,

(1.84)

angular momentum balance Sij = S ji ,

(1.85)

strain displacement relation εij = 12 (u i, j + u j,i + u k,i u k, j ) , constitutive law

(1.86)

∂W , ∂εij

(1.87)

at Si ,

(1.88)

Sij = traction boundary condition n j Sik (δ jk + u j,k ) = t j

displacement boundary condition u j = u¯ i at Su . (1.89) In the above, (·)k denotes ∂(·)/∂xk ; (¨·) denotes ∂ 2 (·)/∂t 2 ; n i are components of a unit normal to St ; and W is the strain-energy density, measured per unit volume in the undeformed body. When the deformations and strains are infinitesimal, the differences in the alternate stress and strain measures disappear. Further, considering only an isothermal linearly elastic solid, the equations above simplify as σij,i + ρ0 b j = ρ0 u¨ j , σij = σ ji , 1 εij = (u i, j + u j,i ) , 2 σij = E ijkl εkl , n i σij = t¯i at St , u i = u¯ i

at Su ,

(1.90) (1.91) (1.92) (1.93) (1.94) (1.95)

and the initial condition, u i (xk , 0) = u ∗ (xk ), u˙ i (xk , 0) = u˙ i∗ (xk , 0) at t = 0 . (1.96)

problems, unfortunately, are few and are limited to semifinite or finite domains; homogeneous and isotropic materials; and often relatively simple boundary conditions. In practice, however, it is common to encounter problems with finite but complex shape in which the material is neither homogeneous nor homogeneous and the boundary conditions are complex. The rapid development and easy accessibility of large-scale numerical codes in recent years are now providing engineers with numerical tools to analyze these practical problems, which are mostly three dimensional in nature and that commonly occur in engineering. Often, however, some three-dimensional problems can be reduced, as a first approximation, to two-dimensional problems for which analytical solutions exist. The utility of such twodimensional solutions lies not in their elegant analysis but in their use for physically understanding certain classes of problems and, more recently, as benchmarks for validating numerical modeling and computational procedures. In the following, we will reformulate the basic equations in two-dimensional Cartesian coordinates and provide analytical example solutions to two simple problems in terms of a polar coordinate system.

1.2.3 Basic Field Equations for the State of Plane Strain A plane state of strain is defined as the situation with zero displacement, say u 3 = 0. Equation (1.90) through (1.96) recast with this definition are σ11,1 + σ21,2 + ρ0 b1 = ρ0 u¨ 1 σ12,1 + σ22,2 + ρ0 b2 = ρ0 u¨ 2 , σ1,2 = σ2,1 , ε11 = u 1,1 ,

ε22 = u 2,2 ,

1 − ν2

(1.97) (1.98)

ε12 =

1 2 (u 1,2 + u 2,1 ) ,

ν σ22 , E 1−ν 1 − ν2 ν ε22 = σ22 − σ11 , E 1−ν 1+ν ε12 = σ12 , E σ33 = ν(σ11 + σ22 ) . ε11 =

σ11 −

(1.99) (1.100a) (1.100b) (1.100c) (1.100d)

1.2.2 Plane Theory of Elasticity The basic field equations and boundary conditions for a three-dimensional boundary/initial value problem in linear elasticity are given in (1.90) through (1.96). Analytical (exact) solutions to idealized three-dimensional

1.2.4 Basic Field Equations for the State of Plane Stress The plane state of stress is defined with zero surface traction, say σ33 = σ31 = σ32 = 0, parallel to the

Analytical Mechanics of Solids

1 (1.101a) (σ11 − νσ22 ) , E 1 ε22 = (σ22 − νσ11 ) , (1.101b) E 1+ν ε12 = (1.101c) σ12 , E ν ε33 = − (σ11 + σ22 ) . (1.102) E In the following, we cite three classical analytical solutions to the linear boundary value problem in Eqs. (1.97) through (1.102) for specific cases that are often of interest. ε11 =

1.2.5 Infinite Plate with a Circular Hole Consider a plane problem of an infinite linearly elastic isotropic body containing a hole of radius a. Let the body be subjected to uniaxial tension, say o∞ 11 , along the x1 axis. Let the Cartesian coordinate system be located at the center of the hole and let r and θ be the corresponding polar coordinates with θ being the angle measured from the x1 axis. The state of stress near the hole is given by [1.28] ∞ σ11 a2 σrr = 1− 2 2 r ∞ σ a4 a2 + 11 1 − 4 2 + 3 4 cos 2θ , (1.103a) 2 r r σ∞ σ∞ a2 a4 σθθ = 11 1 + 2 − 11 1 + 3 4 cos 2θ , 2 2 r r σrθ = −

∞ σ11

2

1+2

a2 r2

−3

a4 r4

(1.103b)

sin 2θ .

(1.103c)

The solution for a biaxial stress state may be obtained by superposition. For a compendium of solutions of holes in isotropic and anisotropic bodies, and for shapes of holes other than circular, such as elliptical holes, see Savin [1.28].

A basic problem for a heterogeneous medium such as a composite, is that of an inclusion (or inclusions). Consider then a rigid inclusion of radius a and assume a perfect bonding between the medium and the inclusion. The solution for stresses near the inclusion due to a far-field uniaxial tension, say o∞ 11 , are σ∞ a2 σrr = 11 1 − ν 2 2 r ∞ σ a4 a2 + 11 1 − 2β 2 − 3δ 4 cos 2θ , (1.104a) 2 r r ∞ ∞ σ11 σ11 a2 a4 σθθ = 1+ν 2 − 1 − 3δ 4 cos 2θ , 2 2 r r σrθ = −

∞ σ11

a2

a4

(1.104b)

(1.104c) 1 + β 2 + 3δ 4 sin 2θ , 2 r r 2(λ + μ) μ λ+μ β=− ν=− δ= , λ + 3μ λ+μ λ + 3μ (1.104d)

where λ and μ are Lamé constants.

1.2.6 Point Load on a Semi-Infinite Plate Consider a concentrated vertical force P acting on a horizontal straight edge of a semi-infinitely large plate. The origin of a Cartesian coordinate is at the location of load application with x1 in the direction of the force. Consider again a polar coordinate of r and θ with θ being the angle measured from the x1 axis, positive in the counterclockwise direction. The state of stress is a very simple one given by 2P cos θ π r σθθ = σrθ = 0 . σrr = −

(1.105a) (1.105b)

This state of stress satisfies the natural boundary conditions as all three stress components vanish on the straight boundary, i. e., θ = π/2, except at the origin, where σrr → ∞ as r → 0. A contour integration of the vertical component of σrr over a semicircular arc from the origin yields the statically equivalent applied force P and thus all boundary conditions are satisfied. By using stress equations of transformations, which can be found in textbooks on solid mechanics, the stresses in polar coordinates can be converted into Cartesian coordinates as 2P (1.106a) cos4 θ , σ11 = σrr cos2 θ = − πa

13

Part A 1.2

x1 –x2 plane. This state shares the same stress equations of equilibrium and strain-displacement relations with the state of plane strain, i. e. (1.97) through (1.99). Equations (1.97) and (1.98) are necessary and sufficient to solve a two-dimensional elastostatic boundary value problem when only tractions are prescribed on the boundary. Thus the stresses of the plane-stress and plane-strain solutions coincide while the strains differ. The stress–strain relations for the state of plane stress are

1.2 Boundary Value Problems in Elasticity

14

Part A

Solid Mechanics Topics

Part A 1

2P (1.106b) sin2 θ cos2 θ , πa 2P σ12 = σrr sin θ cos θ = − sin θ cos3 θ . (1.106c) πa σ22 = σrr sin2 θ = −

Since strain is what is actually being measured, the corresponding strains can be computed by (1.100) for the plane-strain state or by (1.101) for the plane-stress state.

1.3 Summary It is obviously impossible in this extremely brief review even to mention all of the important subjects and recent developments in the theories of elasticity, plasticity, viscoelasticity, and viscoplasticity. For further details,

readers are referred to the many excellent books and survey articles (see for example [1.29] and [1.30]), many of which are referenced in the succeeding chapters, in each of the disciplines.

References 1.1

1.2

1.3

1.4

1.5 1.6 1.7 1.8 1.9

1.10 1.11 1.12 1.13

1.14

C. Truesdell (Ed.): Mechanics of Solids. In: Encyclopedia of Physics, Vol. VIa/2 (Springer, Berlin, Heidelberg 1972) C. Truesdell, W. Noll: The Nonlinear Field Theories of Mechanics. In: Encyclopedia of Physics, Vol. III/3, ed. by S. Flügge (Springer, Berlin, Heidelberg 1965) S.N. Atluri: Alternate stress and conjugate strain measures, and mixed foundations involving rigid rotations for computational analysis of finitely deformed solids, with application to plates and shells. I- Theory, Comput. Struct. 18(1), 93–116 (1986) S.N. Atluri: On some new general and complementary energy theorems for rate problems in finite strain, classical elastoplasticty, J. Struct. Mech. 8(1), 61–92 (1980) A.C. Eringen: Nonlinear Theory of Continuous Media (McGraw-Hill, New York 1962) R.W. Ogden: Nonlinear Elastic Deformation (Dover, New York 2001) Y.C. Fung, P. Tong: Classical and Computational Solid Mechanics (World Scientific, Singapore 2001) M. Mooney: A theory of large elastic deformation, J. Appl. Phys. 11, 582–592 (1940) A. Abel, R.H. Ham: The cyclic strain behavior of crystal aluminum-4% copper, the Bauschinger effect, Acta Metallur. 14, 1489–1494 (1966) A. Abel, H. Muir: The Bauschinger effect and stacking fault energy, Phil. Mag. 27, 585–594 (1972) G.I. Taylor, H. Quinney: The plastic deformation of metals, Phil. Trans. A 230, 323–362 (1931) R. Hill: The Mathematical Theory of Plasticity (Oxford Univ. Press, New York 1950) W. Prager: A new method of analyzing stress and strains in work-hardening plastic solids, J. Appl. Mech. 23, 493–496 (1956) H. Ziegler: A modification of Prager’s hardening rule, Q. Appl. Math. 17, 55–65 (1959)

1.15

1.16

1.17

1.18

1.19

1.20

1.21

1.22 1.23

1.24 1.25

D.C. Drucker: A more fundamental approach to plane stress-strain relations, Proc. 1st U.S. Nat. Congr. Appl. Mech. (1951) pp. 487–491 S. Nemat-Nasser: Continuum bases for consistent numerical foundations of finite strains in elastic and inelastic structures. In: Finite Element Analysis of Transient Nonlinear Structural Behavior, AMD, Vol. 14, ed. by T. Belytschko, J.R. Osias, P.V. Marcal (ASME, New York 1975) pp. 85–98 S.N. Atluri: On constitutive relations in finite strain hypoelasticity and elastoplasticity with isotropic or kinematic hardening, Comput. Meth. Appl. Mech. Eng. 43, 137–171 (1984) Y. Yamada, N. Yoshimura, T. Sakurai: Plastic stressstrain matrix and its application to the solution of elastic-plastic problems by the finite element method, Int. J. Mech. Sci. 10, 343–354 (1968) K. Valanis: Fundamental consequences of a new intrinsic tune measure plasticity as a limit of the endochronic theory, Arch. Mech. 32(2), 171–191 (1980) Z. Mroz: An attempt to describe the behavior of metals under cyclic loads using a more general workhardening model, Acta Mech. 7, 199–212 (1969) O. Watanabe, S.N. Atluri: Constitutive modeling of cyclic plasticity and creep using an internal time concept, Int. J. Plast. 2(2), 107–134 (1986) E.C. Bingham: Fluidity and Plasticity (McGraw-Hill, New York 1922) K. Hohenemser, W. Prager: Über die Ansätze der Mechanik isotroper Kontinua, Z. Angew. Math. Mech. 12, 216–226 (1932) I. Finnie, W.R. Heller: Creep of Engineering Materials (McGraw-Hill, New York 1959) P. Perzyna: The constitutive equations for rate sensitive plastic materials, Quart. Appl. Mech. XX(4), 321–332 (1963)

Analytical Mechanics of Solids

1.27

O.C. Zienkiewicz, C. Corneau: Visco-plasticity, plasticity and creep in elastic solids – a unified numerical solution approach, Int. J. Numer. Meth. Eng. 8, 821–845 (1974) J.H. Argyris, M. Keibler: Incremental formulation in nonlinear mechanics and large strain elastoplasticity – natural approach – Part I, Comput. Methods Appl. Mech. Eng. 11, 215–247 (1977)

1.28 1.29 1.30

G.N. Savin: Stress Concentration Around Holes (Pergamon, Elmsford, 1961) A.S. Argon: Constitutive Equations in Plasticity (MIT Press, Cambridge, 1975) A.L. Anand: Constitutive equations for rate independent, isotropicelastic-plastic solid exhibitive pressure sensitive yielding and plastic dilatancy, J. Appl. Mech. 47, 439–441 (1980)

15

Part A 1

1.26

References

17

Materials Scie

2. Materials Science for the Experimental Mechanist

This chapter presents selected principles of materials science and engineering relevant to the interpretation of structure–property relationships. Following a brief introduction, the first section describes the atomic basis for the description of structure at various size levels. Types of atomic bonds form a basis for a classification scheme of materials as well as for the distinction between amorphous and crystalline materials. Crystal structures of elements and compounds are described. The second section presents the thermodynamic and kinetic basis for the formation of microstructures and describes the use of phase diagrams for determining the nature and quantity of equilibrium phases present in materials. Principal methods for the observation and determination of structure are described. The structural foundations for phenomenological descriptions of equilibrium, dissipative, and transport properties are described. The chapter includes examples of the relationships among physical phenomena responsible for various mechanical properties and the values of

2.1

Structure of Materials ........................... 2.1.1 Atomic Bonding ........................... 2.1.2 Classification of Materials .............. 2.1.3 Atomic Order ............................... 2.1.4 Equilibrium and Kinetics ............... 2.1.5 Observation and Characterization of Structure .................................

17 18 21 22 28

Properties of Materials .......................... 2.2.1 The Continuum Approximation ...... 2.2.2 Equilibrium Properties .................. 2.2.3 Dissipative Properties ................... 2.2.4 Transport Properties of Materials .... 2.2.5 Measurement Principles for Material Properties ..................

33 34 35 38 43

References ..................................................

47

2.2

31

46

these properties. In conclusion the chapter presents several useful principles for experimental mechanists to consider when measuring and applying values of material properties.

2.1 Structure of Materials Engineering components consist of materials having properties that enable the items to perform the functions for which they are designed. Measurements of the behavior of engineering components under various conditions of service are major objectives of experimental mechanics. Validation and verification of analytical models used in design require such measurements. All models employ mathematical relationships that require knowledge of the behavior of materials under a variety of conditions. Assumptions such as isotropy, homogeneity, and uniformity of materials affect both analytical calculations and the interpretation of experimental results. Regardless of the scale or purpose of the measurements, properties of materials that comprise the

components affect both the choice of experimental techniques and the interpretation of results. In measuring static behavior, it is important to know whether relevant properties of the constituent materials are independent of time. Similarly, measurements of dynamic behavior require information on the dynamic and dissipative properties of the materials. At best, the fundamental nature of materials, which is the ultimate determinant of their behavior, forms the basis of these models. The extent to which such assumptions represent the actual physical situation limits the accuracy and significance of results. The primary axiom of materials science and engineering states that the properties and performance of

Part A 2

Craig S. Hartley

18

Part A

Solid Mechanics Topics

Part A 2.1

a material depend on its structure at one or more levels, which in turn is determined by the composition and the processing, or thermomechanical history of the material. The meaning of structure as employed in materials science and engineering depends on the scale of reference. Atomic structure refers to the number and arrangement of the electrons, protons, and neutrons that compose each type of atom in a material. Nanostructure refers to the arrangement of atoms over distances of the order of 10−9 m. Analysis of the scattering of electrons, neutrons, or x-rays is the principal tool for measurements of structure at this scale. Microstructure refers to the spatial arrangement of groups of similarly oriented atoms as viewed by an optical or electron microscope at resolutions in the range 10−6 –10−3 m. Macrostructure refers to arrangements of groups of microstructural features in the range of 10−3 m or greater, which can be viewed by the unaided eye or under low-power optical magnification. Structure-insensitive properties, such as density and melting point, depend principally on composition, or the relative number and types of atoms present in a material. Structure-sensitive properties, such as yield strength, depend on both composition and structure, principally at the microscale. This survey will acquaint the experimental mechanist with some important concepts of materials science and engineering in order to provide a basis for informed selections and interpretations of experiments. The chapter consists of a description of the principal factors that determine the structure of materials, including techniques for quantitative measurements of structure, followed by a phenomenological description of representative material properties with selected examples of physically based models of the properties. A brief statement of some principles of measurement that acknowledge the influence of material structure on properties concludes the chapter. Additional information on many of the topics covered in the first two sections can be found in several standard introductory texts on materials science and engineering for engineers [2.1–4]. Since this introduction can only briefly survey the complex field of structure–property relationships, each section includes additional representative references on specific topics.

2.1.1 Atomic Bonding The Periodic Table The realization that all matter is composed of a finite number of elements, each consisting of atoms with a characteristic arrangement of elementary particles, be-

came widespread among scientists in the 19th and 20th century. Atomic theory of matter led to the discovery of primitive units of matter known as electrons, protons, and neutrons and laws that govern their behavior. Although discoveries through research in high-energy physics constantly reveal more detail about the structure of the atom, the planetary model proposed in 1915 by Niels Bohr, with some modifications due to later discoveries of quantum mechanics, suffices to explain most of the important aspects of engineering materials. In this model, atoms consist of a nucleus, containing protons, which have a positive electrical charge, and an approximately equal number of electrically neutral neutrons, each of which has nearly the same mass as a proton. Surrounding this nucleus is an assembly of electrons, which are highly mobile regions of concentrated negative charge each having substantially smaller mass than a proton or neutron. The number of electrons is equal to the number of protons in the nucleus, so each atom is electrically neutral. Elements differ from one another primarily through the atomic number, or number of protons in the nucleus. However, many elements form isotopes, which are atoms having identical atomic numbers but different numbers of neutrons. If the number of neutrons differs excessively from the number of protons, the isotope is unstable and either decays by the emission of neutrons and electromagnetic radiation to form a more stable isotope or fissions, emitting electromagnetic radiation, neutrons, and assemblies of protons and neutrons that form nuclei of other elements. The Periodic Table, shown in Fig. 2.1, classifies elements based on increasing atomic number and a periodic grouping of elements having similar chemical characteristics. The manner in which elements interact chemically varies periodically depending on the energy distribution of electrons in the atom. The basis for this grouping is the manner in which additional electrons join the atom as the atomic numbers of the elements increase. Quantum-mechanical laws that govern the behavior of electrons require that they reside in the vicinity of the nucleus in discrete spatial regions called orbitals. Each orbital corresponds to a specific energy state for electrons and is capable of accommodating two electrons. Electron orbitals can have a variety of spatial orientations, which gives a characteristic symmetry to the atom. Four quantum numbers, arising from solutions to the Schrödinger wave equation, governs the behavior of the electrons: the principal quantum number n, which can have any integer value from 1 to infinity; the azimuthal quantum number , which can have any integer

Materials Science for the Experimental Mechanist

value from 0 to (n − 1); the magnetic quantum number m , which can have any integer value between − and +; and the spin quantum number m s which has values ±1/2. The Pauli exclusion principle states that no two electrons in a system can have the same four quantum numbers. As the number of electrons increases with increasing atomic number, orbitals are filled beginning with those having the lowest electron energy states and proceeding to the higher energy states. Elements with electrons in full, stable orbitals are chemically inert gases, which occupy the extreme right column of the periodic table (group 8). Electronegative elements, which occupy columns towards the right of

2.1 Structure of Materials

Part A 2.1

the periodic table, have nearly full orbitals and tend to interact with other atoms by accepting electrons to form a negatively charged entity called an anion. The negative charge arises since electrons join the originally neutral atom. Electropositive elements occupy columns towards the left on the periodic table and ionize by yielding electrons from their outer orbitals to form positively charged cations. Broadly speaking, elements are metals, metalloids, and nonmetals. The classification proceeds from the most electropositive elements on the left of the periodic table to the most electronegative elements on the right. A metal is a pure element. A metal that incorpo8A

1A 1

2

H

H

1s1 hydrogen

1s2 helium

1.008

2A

3A

4A

5A

6A

7A

3

4

5

6

7

8

9

4.003

10

Li

Be

B

C

N

O

F

Ne

[He]2s1 lithium

[He]2s2 beryllium

[He]2s22p1 boron

[He]2s22p2 carbon

[He]2s22p3 nitrogen

[He]2s22p4 oxygen

[He]2s22p5 fluorine

[He]2s22p6 neon

6.941

9.012

10.81

12.01

14.01

16.00

19.00

20.18

11

12

13

14

16

16

17

18

Na

Mg

Al

Si

P

S

Cl

Ar

[Ne]3s1 sodium

[Ne]3s2 magnesium

22.99

24.31

3B

4B

5B

6B

7B

19

20

21

22

23

24

25

26

27

K

Ca

Sc

Ti

V

Cr

Mn

Fe

[Ar]4s1 pollassium

[Ar]4s2 calcium

[Ar]4s23d1 scandium

[Ar]4s23d2 titanium

[Ar]4s23d3 vanadium

[Ar]4s13d5 chromium

[Ar]4s23d5 manganese

39.10

40.08

44.96

47.88

50.94

52.00

55.94

37

38

39

40

41

42

Rb

Sr

Y

Zr

Nb

[Kr]5s1 nubidium

[Kr]5s2 strontium

[Kr]5s24d1 yttrium

[Kr]5s24d2 zirconium

85.47

87.62

88.91

91.22

55

57

57

Cs

Ba

1

[Xe]6s casium

132.9

87

[Ne]3s23p1 aluminum

[Ne]3s23p2 silicon

[Ne]3s23p3 phosphorus

[Ne]3s23p4 sulfur

[Ne]3s23p5 chlorine

[Ne]3s23p6 argon

11B

12B

26.98

28.09

30.97

32.07

35.45

39.95

28

29

30

31

32

33

34

35

36

Co

Ni

Cu

Zn

Ga

Ge

As

Se

Br

Kr

[Ar]4s23d6 iron

[Ar]4s23d7 cobalt

[Ar]4s23d8 nickel

[Ar]4s13d10 copper

[Ar]4s23d10 zinc

55.85

58.93

58.69

63.55

65.39

69.72

72.58

74.92

78.96

79.90

43

44

45

46

47

48

49

50

51

52

53

52

Mo

Tc

Ru

Rh

Pd

Ag

Cd

In

Sn

Sb

Te

I

Xe

[Kr]5s14d4 niobium

[Kr]5s14d5 molybdenum

[Kr]5s24d5 technetium

[Kr]5s14d7 ruthenium

[Kr]5s14d8 rhodium

[Kr]4d10 palladium

[Kr]5s14d10 silver

[Kr]5s24d10 cadmium

92.91

95.94

(98)

101.1

102.9

106.4

107.9

112.4

114.8

118.7

121.8

127.6

126.9

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

La*

Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

Tl

Pb

Bi

Po

At

Rn

[Xe]6s2 barium

[Xe]6s25d1 lanthanum

[Xe]6s24f145d2 hafnium

[Xe]6s24f145d3 tantalum

[Xe]6s24f145d4 tungsten

[Xe]6s24f145d5 rhenium

[Xe]6s24f145d6 osmium

[Xe]6s24f145d7 iridium

[Xe]6s14f145d9 platinum

[Xe]6s14f145d10

[Xe]6s24f145d10

[Xe]6s24f145d106p1

[Xe]6s24f145d106p2

[Xe]6s24f145d106p3

[Xe]6s24f145d106p4

[Xe]6s24f145d106p5

[Xe]6s24f145d106p6

gold

mercury

thallium

lead

bismuth

polonium

astatine

radon

137.3

138.9

178.5

180.9

183.9

186.2

190.2

190.2

195.1

197.0

200.5

204.4

207.2

208.9

(209)

(210)

(222)

8B

[Ar]4s23d104p1 [Ar]4s23d104p2 [Ar]4s23d104p3 [Ar]4s23d104p4 [Ar]4s23d104p5 [Ar]4s23d104p6 gallium germanium arsenic selenium bromine krypton

[Kr]5s24d105p1 [Kr]5s24d105p2 indium tin

83.80

[Kr]5s24d105p3 [Kr]5s24d105p4 [Kr]5s24d105p5 [Kr]5s24d105p6 antimony tellurium iodine xenon

131.3

88

89

104

105

106

107

108

109

110

111

112

114

116

118

Ra

Ac ~

Rf

Db

Sg

Bh

Hs

Mt

Ds

Uuu

Uub

Uuq

Uuh

Uuo

1

[Rn] 7s francium

[Rn]7s2 radium

[Rn]7s26d1 actinium

(223)

(226)

(227)

(272)

(277)

(296)

(298)

(?)

Fr

58 Lanthanide series*

Actinide series ~

[Rn]7s25f146d2 [Rn]7s25f146d3 [Rn]7s25f146d4 [Rn]7s25f146d5 [Rn]7s25f146d6 [Rn]7s25f146d7 [Rn]7s15f146d9 rutherfordium dubnium seaborgium bohrium hassium meitnerium darmstadtium

(257)

59

(260)

60

(263)

61

(262)

62

(265)

63

(266)

64

(271)

65

66

67

68

69

70

71

Ce

Pr

Nd

Pm

Sm

Eu

Gd

Tb

Dy

Ho

Er

Tm

Yb

Lu

[Xe]6s24f15d1 cerium

[Xe]6s24f3 praseodymium

[Xe]6s24f4 neodymium

[Xe]6s24f5 promethium

[Xe]6s24f6 samarium

[Xe]6s24f7 europium

[Xe]6s24f75d1 gadolinium

[Xe]6s24f9 terbium

[Xe]6s24f10 dysprosium

[Xe]6s24f11 holmium

[Xe]6s24f12 erbium

[Xe]6s24f13 thulium

[Xe]6s24f14 ytterbium

[Xe]6s24f145d1 lutetium

140.1

140.9

144.2

(147)

(150.4)

152.0

157.3

158.9

162.5

164.9

167.3

168.9

173.0

175.0

90

91

92

93

94

95

96

97

98

99

100

101

102

103

Th

Pa

U

Np

Pu

Am

Cm

Bk

Cf

Es

Fm

Md

No

Lr

[Rn]7s25f46d1 neptunium

[Rn]7s25f6 plutonium

[Rn]7s25f7 americium

[Rn]7s25f76d1 curium

[Rn]7s25f9 berkelium

[Rn]7s25f10 californium

[Xe]6s24f11 einsteinium

[Rn]7s25f12 fermium

[Rn]7s25f13 mendelevium

[Rn]7s25f14 nobelium

[Rn]7s25f146d1 lawrencium

(237)

(242)

(43)

(247)

(247)

(249)

(254)

(253)

(256)

(254)

(257)

[Rn]7s26d2 thorium

232.0

[Rn]7s25f26d1 [Rn]7s25f36d1 protactinium uranium

(231)

(238)

Liquids at room temperature

Gases at room temperature

Solids at room temperature

Fig. 2.1 The Periodic Table of the elements. Elements named in blue are liquids at room temperature. Elements named in red are

gases at room temperature. Elements named in black are solids at room temperature

19

20

Part A

Solid Mechanics Topics

Part A 2.1

rates atoms of other elements into its structure without changing its essential metallic character forms an alloy, which is not a metal since it is not a pure element. The major differences in materials have their origins in the nature of the bonds formed between atoms, which are determined by the manner in which electrons in the highest-energy orbitals interact with one another and by whether the centers of positive and negative charge of the atoms coincide. The work required to remove an ion from the substance in which it resides is a measure of the strength of these bonds. At suitable temperatures and pressures, all elements can exist in all states of matter, although in some cases this is very difficult to achieve experimentally. At ambient temperature and pressure, most elements are solids, some are gases, and a few are liquid. Primary Bonds Primary bonds are the strongest bonds that form among atoms. The manner in which electrons in the highest energy levels interact produces differences in the kinds of primary bonds. Valence electrons occupy the highest energy levels of atoms, called the valence levels. Valence electrons exhibit three basic types of behavior: atoms of electropositive elements yield their valence electrons relatively easily; atoms of electronegative elements readily accept electrons to fill their valence levels; and elements between these extremes can share electrons with neighboring atoms. The valence of an ion is the number of electrons yielded, accepted or shared by each atom in forming the ion. Valence is positive or negative according to whether the ion has a positive (cation) or negative (anion) charge. The behavior of valence electrons gives rise to three types of primary bonds: ionic, covalent and metallic. Ionic bonds occur between ions of strongly electropositive elements and strongly electronegative elements. Each atom of the electropositive element surrenders one or more electrons to one or more atoms of the electronegative element to form oppositely charged ions, which attract one another by the Coulomb force between opposite electrical charges. This exchange of electrons occurs in such a manner that the overall structure remains electrically neutral. To a good approximation, ions involved in ionic bonds behave as charged, essentially incompressible, spheres, which have no characteristic directionality. In contrast, covalent bonds involve sharing of valence electrons between neighboring atoms. This type of bonding occurs when the valence energy levels of the atoms are partially full, corresponding to valences in the vicinity of 4. These bonds

are strongly directional since the orbitals involved are typically nonspherical. In both ionic and covalent bonds nearest-neighbor ions are most strongly involved and the valence electrons are highly localized. Metallic bonds occur in strongly electropositive elements, which surrender their valence electrons to form a negatively charged electron gas or distribution of highly nonlocalized electrons that moves relatively freely throughout the substance. The positively charged ions repel one another but remain relatively stationary because the electron gas acting as glue holds them together. Metallic bonds are relatively nondirectional and the ions are approximately spherical. A major difference between the metallic bond and the ionic and covalent bonds is that it does not involve an exchange or sharing of electrons with nearest neighbors. The bonds in many substances closely approximate the pure bond types described above. However, mixtures of these archetypes occur frequently in nature, and a substance can show bonding characteristics that resemble more than one type. This hybrid bond situation occurs most often in substances that exhibit some characteristics of directional covalent bonds along with nondirectional metallic or ionic bonds. Secondary Bonds Some substances are composed of electrically neutral clusters of ions called molecules. Secondary bonds exist between molecules and are weaker than primary bonds. One type of secondary bond, the van der Waals bond, is due to the weak electrostatic interaction between molecules in which the instantaneous centers of positive and negative charge do not coincide. A molecule consisting of a single ion of an electropositive element and a single ion of an electronegative element, such as a molecule of HCl gas, is a simple example of a diatomic molecule. The center of negative charge of the system coincides with the nucleus of the chlorine ion, but the center of positive charge is displaced from the center of the ion because of the presence of the smaller, positively charged hydrogen ion (a single proton), which resides near the outer orbital of the chlorine ion. This results in the formation of an electrical dipole, which has a short-range attraction to similar dipoles, such as other HCl molecules, at distances of the order of the molecular dimensions. However, no long-range attraction exists since the overall charge of the molecule is zero. The example given is a permanent dipole formed by a spatial separation of centers of charge. A temporary dipole can occur when the instantaneous centers of charge separate because of the motion of electrons.

Materials Science for the Experimental Mechanist

2.1.2 Classification of Materials It is useful to categorize engineering materials in terms either of their functionality or the dominant type of atomic bonding present in the material. Since most materials perform several functions in a component, the classification scheme described in the following sections takes the latter approach. The nature and strength of atomic bonding influences not only the arrangement of atoms in space but also many physical properties such as electrical conductivity, thermal conductivity, and damping capacity. Ceramics Ceramic materials possess bonding that is primarily ionic with varying amounts of metallic or covalent character. The dominant features on the atomic scale are the localization of electrons in the vicinity of the ions and the relative incompressibility of atoms, leading to structures that are characterized by the packing of rigid spheres of various sizes. These materials typically have high melting points (> 1500 K), low thermal and electrical conductivities, high resistance to atmospheric corrosion, and low damping capacity. Mechanical properties of ceramics include high moduli of elasticity, high yield strength, high notch sensitivity, low ductility, low impact resistance, intermediate to low thermal shock resistance, and low fracture toughness. Applications that require resistance to extreme thermal, electrical or chemical environments, with the ability to absorb mechanical energy without catastrophic failure a secondary issue, typically employ ceramics. Metals Metallic materials include pure metals (elements) and alloys that exhibit primarily metallic bonding. The

free electron gas that permeates the lattice of ions causes these materials to exhibit high electrical and thermal conductivity. In addition they possess relatively high yield strengths, high moduli of elasticity, and melting points ranging from nearly room temperature to > 3200 K. Although generally malleable and ductile, they can exhibit extreme brittleness, depending on structure and temperature. One of the most useful features of metallic materials is their ability to be formed into complex shapes using a variety of thermomechanical processes, including melting and casting, hot working in the solid state, and a combination of cold working and annealing. All of these processes produce characteristic microstructures that lead to different combinations of physical and mechanical properties. Applications that require complex shapes having both strength and fracture resistance with moderate resistance to environmental degradation employ metallic materials. Metalloids Metalloids are elements in groups III–V of the periodic table and compounds formed from these elements. Covalent bonding dominates both the elements and compounds in this category. The name arises from the fact that they exhibit behavior intermediate between metals and ceramics. Many are semiconductors, that is, they exhibit an electrical conductivity lower than metals, but useable, which increases rather than decreases with temperature like metals. These materials exhibit high elastic moduli, relatively high melting points, low ductility, and poor formability. Commercially useful forms of these materials require processing by solidification directly from the molten state followed by solid-state treatments that do not involve significant deformation. Metalloids are useful in a variety of applications where sensitivity and response to electromagnetic radiation are important. Polymers Polymeric materials, also generically called plastics, are assemblies of complex molecules consisting of molecular structural units called mers that have a characteristic chemical composition and, often, a variety of spatial configurations. The assemblies of molecules generally take the form of long chains of mers held together by hydrogen bonds or networks of interconnected mers. Most structural polymers are made of mers with an organic basis, i. e., they contain carbon. They are characterized by relatively low strength, low thermal and electrical conductivity, low melting points, often high

21

Part A 2.1

The resulting attraction forms a weak bond at small distances and is typical of van der Waals bonds. The other secondary bond, the hydrogen bond, involves the single valence electron of hydrogen. In materials science and engineering, the most important type of hydrogen bond is that which occurs in polymers, which consist of long chains or networks of chemically identical units called mers. When the composition of a mer includes hydrogen, it is possible for the hydrogen atom to share its valence electron with identical mers in neighboring chains, so that the hydrogen atom is partly in one chain and partly in another. This sharing of the hydrogen atom creates a hydrogen bond between the chains. The bond is relatively weak but is an important factor in the behavior of polymeric materials.

2.1 Structure of Materials

22

Part A

Solid Mechanics Topics

Part A 2.1

ductility, and high formability by a variety of techniques. These materials are popular as electrical and thermal insulators and for structural applications that do not require high strength or exposure to high temperatures. Their principal advantages are relatively low cost, high formability, and resistance to most forms of atmospheric degradation. Composites Composite materials consist of those formed by intimate combinations of the other classes. Composites combine the advantages of two or more material classes by forming a hybrid material that exhibits certain desirable features of the constituents. Generally, one type of material predominates, forming a matrix containing a distribution of one or more other types on a microscale. A familiar example is glass-reinforced plastic (GRP), known by the commercial name of R . In this material, the high elastic modulus Fibreglass of the glass fibers (a ceramic) reinforces the toughness and formability of the polymeric matrix. Other classes of composites have metal matrices with ceramic dispersions (metal matrix composites, MMC), ceramic matrices with various types of additions (ceramic matrix composites, CMC), and polymeric matrices with metallic or ceramic additions. The latter, generically known as organic matrix composites (OMC) or polymer matrix composites (PMC), are important structural materials for aerospace applications.

2.1.3 Atomic Order Crystalline and Amorphous Materials The structure of materials at the atomic level can be highly ordered or nearly random, depending on the nature of the bonding and the thermomechanical history. Pure elements that exist in the solid state at ambient temperature and pressure always exhibit at least one form that is highly ordered in the sense that the surroundings of each atom are identical. Crystalline materials exhibit this locally ordered arrangement over large distances, creating long-range order. The formal definition of a crystal is a substance in which the structure surrounding each basis unit, an atom or molecule, is identical. That is, if one were able to observe the atomic or molecular arrangement from the vantage point of a single structural unit, the view would not depend on the location or orientation of the structural unit within the material. All metals and ceramic compounds and some polymeric materials have crystalline forms. Some ma-

terials can exhibit more than one crystalline form, called allotropes, depending on the temperature and pressure. It is this property of iron with small amounts of carbon dissolved that is the basis for the heat treatment of steel, which provides a wide range of properties. At the other extreme of atomic arrangement are amorphous materials. These materials can exhibit local order of structural units, but the arrangement a large number of such units is haphazard or random. There are two principal categories of amorphous structures: network structures and chain structures. The molecules of network structures lie at the nodes of an irregular network, like a badly constructed jungle gym. Nevertheless, the network has a high degree of connectivity and if the molecules are not particularly mobile, the network can be very stable. This type of structure is characteristic of most glasses. Materials possessing this structure possess a relatively rigid mechanical response at low temperatures, but become more fluid and deformable at elevated temperatures. Frequently the transition between the relatively rigid, low-temperature form and the more fluid high-temperature form occurs over a narrow temperature range. By convention the midpoint of this transition range defines the glasstransition temperature. Linear chain structures are characteristic of polymeric materials made of long chains of mers. Relatively weak hydrogen bonds and/or van der Waals bonds hold these chains together. The chains can move past one another with varying degrees of difficulty depending on the geometry of the molecular arrangement along the chain and the temperature. An individual chain can possess short-range order, but the collections of chains that comprise the substance sprawl haphazardly, like a bowl of spaghetti. Under certain conditions of formation, however, the chains can arrange themselves into a pattern with long-range order, giving rise to crystalline forms of polymeric materials. In addition, some elements, specific to the particular polymer, can bond with adjacent chains, creating a three-dimensional network structure. The addition of sulfur to natural rubber in the R is an example. process called Vulcanizing Materials that can exist in both the crystalline and amorphous states can also have intermediate, metastable structures in which these states coexist. Glass that has devitrified has microscopic crystalline regions dispersed in an amorphous network matrix. Combinations of heat treatment and mechanical deformation can alter the relative amounts of these structures, and the overall properties of the material.

Materials Science for the Experimental Mechanist

metric compound contains ions exactly in the ratios that produce electrical neutrality of the substance. In binary (two-component) compounds, the ratio of the number of ions of each kind present is the inverse of the ratio of the absolute values of their valences. For example, Na2 O has two sodium atoms for each oxygen atom. Since the valence of sodium is +1 and that of oxygen is −2, the 2 : 1 ratio of sodium to oxygen ions produces electrical neutrality of the structure.

Cubic

Simple

Body-centered

Tetragonal

Simple

Face-centered

Monoclinic

Body-centered

Simple

End-centered

Orthohombic

Simple Rhombohedral

Body-centered Hexagonal

Fig. 2.2 Bravais lattices and crystal systems

Face-centered

End-centered Triclinic

23

Part A 2.1

Crystal Structures of Elements and Compounds Because ionic bonds require ions of at least two elements, either metallic or covalent bonds join ions of pure elements in the solid state, although the condensed forms of highly electronegative elements and the inert gases exhibit weak short-range bonding typical of van der Waals bonds. Chemical compounds, which can exhibit ionic bonding as well as the other types of strong bonds, form when atoms of two or more elements combine in specific ratios. A stoichio-

2.1 Structure of Materials

24

Part A

Solid Mechanics Topics

Part A 2.1

In the solid state, patterns of atoms and molecules form lattices, which are three-dimensional arrays of points having the property that the surroundings of each lattice point are identical to those of any other lattice point. There are only 14 unique lattices, the Bravais lattices, shown in Fig. 2.2. Each lattice possesses three non-coplanar, non-collinear axes and a characteristic, unique array of lattice points occupied by structural units, which can be individual atoms or identical clusters of atoms, depending on the nature of the substance. The relative lengths of the repeat distance of lattice points along each axis and the angles that the axes make with one another define the seven crystal systems. Figure 2.2 also shows the crystal system for each of the Bravais lattices. Each of the illustrations in Fig. 2.2 represents the unit cell for the lattice, which is the smallest arrangement of lattice points that possesses the geometric characteristics of the extended structure. Repeating one of the figures in Fig. 2.2 indefinitely throughout space with an appropriately chosen structural unit at each lattice point defines a crystal structure. Lattice parameters include the angles between coordinate axes, if variable, and the dimensions of the unit cell, which contains one or more lattice points. To determine the number of points associated with a unit cell, count 1/8 for each corner point, 1/2 for each point on a face, and 1 for each point entirely within the cell. A primitive unit cell contains only one lattice point (one at each corner). The coordination number Z is the number of nearest neighbors to a lattice point. One of the most important characteristics of crystal lattices is symmetry, the property by which certain rigid-body motions bring the lattice into an equivalent configuration indistinguishable from the initial configuration. Symmetry operations occur by rotations about an axis, reflections across a plane or a combination of rotations, and translations along an axis. For example, a plane across which the structure is a mirror image of that on the opposite side is a mirror plane. An axis about which a rotation of 2π/n brings the lattice into coincidence forms an n-fold axis of symmetry. This characteristic of crystals has profound implications on certain physical properties. The geometry of the lattice provides a natural coordinate system for describing directions and planes using the axes of the unit cell as coordinate axes and the lattice parameters as units of measure. Principal crystallographic axes and directions are those parallel to the edges of the unit cell. A vector connecting two lattice points defines a lattice direction. A set of

three integers having no common factor that are in the same ratio as the direction cosines, relative to the coordinate axes, of such a vector characterizes the lattice direction. Square brackets, e.g., [100], denote specific crystallographic directions, while the same three integers enclosed by carats, e.g., 100, describe families of directions. Directions are crystallographically equivalent if they possess an identical arrangement of lattice points. Families of directions in the cubic crystal system are crystallographically equivalent, but those in noncubic crystals may not be because of differences in the lattice parameters. The Miller indices, another set of three integers determined in a different manner, specify crystallographic planes. The notation arose from the observation by 19th century crystallographers on naturally occurring crystals that the reciprocals of the intercepts of crystal faces with the principal crystallographic axes occurred in the ratios of small, whole numbers. To determine the Miller indices of a plane, first obtain the intercepts of the plane with each of the principal crystallographic directions. Then take the reciprocals of these intercepts and find the three smallest integers with no common factor that have the same ratios to one another as the reciprocals of the intercepts. Enclosed in parentheses, these are the Miller indices of the plane. For example, the (120) plane has intercepts of 1, 1/2, and ∞, in units of the lattice parameters, along the three principal crystallographic directions. Families of planes are those having the same three integers in different permutations, including negatives, as their Miller indices. Braces enclose the Miller indices of families, e.g., {120}. Crystallographically equivalent planes have the same density and distribution of lattice points. In cubic crystals, families of planes are crystallographically equivalent. Although there are examples of all of the crystal structures in naturally occurring materials, a relatively few suffice to describe common engineering materials. All metals are either body-centered cubic (bcc), facecentered cubic (fcc) or hexagonal close-packed (hcp). The latter two structures consist of different stacking sequences of closely packed planes containing identical spheres or ellipsoids, representing the positive ions in the metallic lattice. Figure 2.3 shows a plane of spheres packed as closely as possible in a plane. An identical plane fitted as compactly as possible on top of or below this plane occupies one of two possible locations, corresponding to the depressions between the spheres. These locations correspond to the upright and inverted triangular spaces between spheres in the figure. The same option exists when placing a third

Materials Science for the Experimental Mechanist

B-layer C-layer

Fig. 2.3 Plane of close-packed spheres

c

a

Fig. 2.4 Hexagonal close-packed unit cell

which is not a close-packed structure. Figure 2.6 shows the unit cell of this structure. The structure has a coordination number of eight and the unit cell contains two atoms. The density of a crystalline material follows from its crystal structure and the dimensions of its unit cell. By definition, density is mass per unit volume. For a unit cell this becomes the number of atoms in a unit cell n times the mass of the atom, divided by the cell volume Ω: ρ=

nA . Ω N0

25

Part A 2.1

identical plane on the second plane, but now two distinct situations arise depending on whether the third plane is exactly over the first or displaced from it in the other possible stacking location. In the first case, when the first and third planes are directly over one another, the stacking sequence is characteristic of hexagonal close-packed structures and is indicated ABAB. . . The close-packed, or basal, planes are normal to an axis of sixfold symmetry. Figure 2.4 shows the conventional unit cell for the hcp structure. Based on the hexagonal cell of the Bravais lattice, this unit cell contains two atoms. The c/a ratio is the height of the cell divided by the length of the side of the regular hexagon forming the base.√If the ions are perfect spheres, this ratio is 1.6333 = (8/3). In this instance, the coordination number of the structure is 12. However, most metals that exhibit this structure have c/a ratios different from this ideal value, indicating that oblate or prolate ellipsoids are more accurate than spheres as models for the atoms. Consequently, the coordination number is a hybrid quantity consisting of six atoms in the basal plane and six atoms at nearly the same distances in adjacent basal planes. Nevertheless, the conventional value for the coordination number of the hcp structure is 12 regardless of the c/a ratio. When the third plane in a close-packed structure occurs in an orientation that is not directly above the first, the stacking produces a face-centered cubic (fcc) structure. The sequence ABCABC. . . represents this stacking. The {111} planes are close-packed in this structure, the coordination number is 12, and the unit cell contains four atoms, as shown in Fig. 2.5. The third crystal structure typical of metallic elements and alloys is the body-centered cubic structure,

2.1 Structure of Materials

(2.1)

The mass of an atom is the atomic weight, A, divided by Avogadro’s number, N0 = 6.023 × 1023 , which is the number of atoms or molecules in one gram-atomic or gram-molecular, respectively, weight of a substance. The (8 − N) rule classifies crystal structures of elements that bond principally by covalent bonds, where N (≥ 4) is the number of the element’s group in the Periodic Table. The rule states that the element forms a crystal structure characterized by a coordination number of (8 − N). Thus, silicon in group 4 forms a crystal

Fig. 2.5 Face-centered cubic unit cell and {111} plane

26

Part A

Solid Mechanics Topics

Part A 2.1

Fig. 2.6 Unit cell of bcc structure

that has four nearest neighbors. Crystal structures based on this rule can be quite complex. Ionic compounds are composed of cations and anions formed from two or more kinds of atoms. These compounds form crystal structures based on two principles: (1) the entire structure must be electrically neutral and (2) the ions can be regarded as relatively rigid spheres of differing sizes forming a three-dimensional structure based on the efficient packing of different size spheres. The radius ratio rule is the principle that describes the crystal structures of many ionic compounds. This rule specifies the coordination number Z which gives the most efficient packing for spheres whose radius ratios are in a particular range. The radius ratio R employed for this calculation is the radius of the cation to that of the anion. Table 2.1 illustrates the relationship of this ratio to the resulting lattice geometry. Defects in Crystals The structures described in the previous sections are idealized descriptions that accurately characterize the arrangement of the vast majority of atomic and molecular arrangements in materials. Within any substance there will exist irregularities or defects in the structure that profoundly influence many of the properties of the material. These defects can exist on the electronic, atomic or molecular scale, depending on the substance. A common classification scheme for atomic and molecular defects in materials utilizes their dimensionality. Point defects have dimensions comparable to an atom or molecule. Line defects have an appreciable

extent along a linear path in the material, but essentially atomic dimensions in directions normal to that path. Surface defects have appreciable extent on a surface in two dimensions, but essentially atomic extent normal to the surface. Surface defects are regions of high local atomic disorder. This characteristic makes them prone to higher chemical activity than the regions that they bound. In addition, the local disorder associated with the boundaries has different mechanical properties than the bulk. These two features of surface defects cause them to be extremely important in affecting the mechanical and often the chemical properties of materials. Volume defects occur over volumes of several tens to several millions of atoms. The scale of these defects is generally at the mesoscale and above. The simplest kind of point defect occurs when an atom of a pure substance is missing from a lattice site. The vacant lattice site is a vacancy. Atoms of an element that leave their lattice sites and attempt to share a lattice site with another atom form interstitialcies. Clearly, interstitialcies are closely associated with vacancies. This combination typically occurs under conditions of neutron radiation, and the associated damage is a limiting factor to the use of materials in such environments. A lattice site occupied by an atom different from the host is a substitutional impurity, creating a substitutional solid solution with the host as solvent and the impurity as solute. The concentration of such impurities that a host element can tolerate is dependent in large part on the relative sizes of the impurity and the host atoms. Typically, solid solubility is extremely limited if the atomic sizes of solute and solvent differ by more than 15%. Interstitial solid solutions occur when smaller solvent atoms occupy some of the spaces (interstices) between solute atoms. Only five elements – C, N, O, H, and B – can be interstitial solutes in metals, principally because of the relative sizes of the atoms involved. The systematic or random insertion into linear polymers of mers having different chemical compositions forms copolymers. Since the substituent mers constitute a disruption in the basic polymer chain, they are substi-

Table 2.1 Radius ratios and crystal lattice geometry Z

R

2 3 4 6 8

0.000–0.155 0.155–0.225 0.225–0.414 0.414–0.732 0.732–1.000

Exact range √ 0 to 23 3 − 1 √ √ 2 1 3 √3 − 1 to √ 2 6−1 1 2−1 2 6 − 1 to √ √ 2 − 1 to 3 − 1 √ 3 − 1 to 1

Lattice geometry Linear Trigonal planar Tetrahedral Octahedral Cubic

Materials Science for the Experimental Mechanist

the region of the crystal that has experienced slip from that which has not. This boundary forms a linear crystal defect called a dislocation. As a dislocation passes over its slip plane, one part of the crystal moves relative to the other by a lattice vector, the Burgers vector, which is generally, but not always, one atomic spacing in the direction of the highest density of atoms. Dislocations are an important source of internal stress in crystalline materials and their motion is the principal mechanism of permanent deformation. Further information on the behavior of dislocations can be found in standard references [2.5, 6]. Homogeneous crystalline materials generally consist of an aggregate of grains, which are microscopic crystalline regions having differing orientations. That is, the principal crystallographic directions in each grain have different orientations relative to an external coordinate system than corresponding directions in neighboring grains. Grain boundaries, which are surfaces across which grain orientations change discontinuously, separate the grains. Since the grain boundaries contain most of the atomic disorder that accommodates the orientation change, they are surface defects. Grain boundaries permeate the material, forming a three-dimensional network that has a shape and topology determined by its thermomechanical history. Close-packed crystal structures, such as facecentered cubic and hexagonal close-packed, exhibit another form of surface defect called a stacking fault. This occurs when local conditions of deformation or crystal growth produce a stacking sequence that is locally different from that for the crystal structure. For example, suppose the stacking sequence of closepacked planes in the sequence ABC|ABC. . . is locally disrupted across the surface indicated by the vertical line so that to the left of the line the crystal is shifted to cause A planes to assume B stacking, B planes to assume C stacking and C planes to assume A stacking. Then the stacking sequence would be ABC|BCA. . . This is topologically equivalent to the removal of a plane of A stacking, creating a three-layer hcp structure while preserving the face-centered cubic structure on either side of the fault plane. Such a configuration is an intrinsic stacking fault and is associated with the formation and motion of certain kinds of dislocations in face-centered cubic crystals. Materials can also consist of an aggregate of regions having differing chemical composition and crystal structure from one another. Each region that is chemically and physically distinct and separated from the rest of the system by a boundary is called a phase. Each

27

Part A 2.1

tutional point defects in the same sense as solute atoms in a solid solution. The semiconducting properties of metalloids are due to the thermal excitation of electrons from filled valence bands into sparsely populated conduction bands. In a pure substance, this behavior is intrinsic semiconductivity. The addition of small amounts of impurities (of the order of one impurity atom per million host atoms) that have a different chemical valence significantly modifies the electrical conductivity of many such substances. These substitutional impurities, which are electronic defects as well as lattice defects, contribute additional electronic energy states that can contribute to conductivity. This is extrinsic semiconductivity. A donor impurity has more valence electrons than the host does, and contributes electron energy states. Acceptor impurities have a lower valence than the host, leaving holes, which can contribute to the transport of electrical charge, in the valence energy band. The holes are defects in the electronic structure having equal and opposite charges to the electrons. Point defects in ionic compounds must maintain electrical neutrality. One or more defects possessing equal and opposite charges must be present to cancel any local charge caused by the addition or removal of ions from lattice sites. One such defect is the Frenkel defect, a cation vacancy associated with a nearby cation interstitial. The excess charge created by the interstitial cation balances the charge deficiency due to the vacancy. In principle, such defects are possible with anions replacing cations, but the relatively large size of anions generally precludes the existence of anion interstitials. The other important type of point defect in ionic materials is the Schottky defect, which is a cation vacancy–anion vacancy pair where the ions have equal and opposite charges. Clearly, the removal of an electrically neutral unit maintains local electrical neutrality. Both types of defects can affect electrical and mechanical properties of the substance. Permanent deformation that preserves the number of lattice sites in crystalline materials generally occurs by slip, also called glide, which is the motion of one part of the crystal relative to the other across a densely populated atomic plane, the slip plane, in a densely populated atomic direction, the slip direction. For energetic reasons this motion does not occur simultaneously across the entire crystal, but commences at a free surface or region of high internal stress and propagates over the atomic plane until it either passes out of the crystal or encounters a barrier to further motion. At any instant during this propagation, a boundary separates

2.1 Structure of Materials

28

Part A

Solid Mechanics Topics

Part A 2.1

phase can contain a network of grain boundaries. However, the interphase boundaries that separate the phases are surface defects that differ from grain boundaries, since the regions they separate differ in not only spatial orientation and chemical composition but also, in most cases, in crystal structure. A multiphase material has a system of interphase boundaries whose extent and topology depend not only on the thermomechanical history of the material but also on the relative amounts of the phases present. Cracks and voids caused by solidification are examples of volume defects. These often arise during processing and can be significant sources of mechanical weakness and vulnerability to environmental attack of the material. Gases formed during fission of nuclear fuel can also form internal voids during operation, causing swelling and distortion on a macroscopic scale.

2.1.4 Equilibrium and Kinetics Thermodynamic Principles The overall composition and processing history determine the arrangements of groups of atoms in the microstructure. The resulting structure of the material determines its response to many conditions of measurement. Concepts of thermodynamic equilibrium, the rate of approach to equilibrium, and the response of the material to its physical and mechanical environment govern relationships that connect composition, processing, and microstructure. A thermodynamic system is any collection of items separated from their surroundings by a real or imaginary interface, which can be permeable or impermeable to the exchange of energy. Thermodynamics is the study of the laws that govern

1. the exchange of energy between the system and its surroundings 2. the energy content of such a system 3. the capacity of the system to do work and 4. the direction of heat flow in the system. The sum of the kinetic and potential energies of all components of the system is the internal energy U, which measures the energy content of a system. The first law of thermodynamics relates the rate of change of internal energy of a system to the rate of heat exchange with the surroundings and the rate of work done by external forces U˙ = Q˙ − P ,

(2.2)

where Q˙ represents the rate of heat exchanged with the surroundings, P is the power exerted by external forces, and the superposed dot indicates the derivative with respect to time. The second law of thermodynamics states that the rate of production of entropy S at an absolute temperature T , defined as Q˙ (2.3) , T must be ≥ 0. For a real system, changes in S refer to changes from a standard temperature and pressure. The internal energy less the product of the volume of the system V and the pressure on the system, P, is the heat content or enthalpy H. The relationship connecting these thermodynamic state variables is S˙ =

H = U + PV .

(2.4)

The definition of free energy, which is the capacity of the system to do work, differs when dealing with gaseous and condensed systems of matter. The Gibbs free energy G employed for condensed systems is related to other thermodynamic variables by G = H − TS .

(2.5)

The following discussion designates free energy by F without reference to the state of the system. Changes in the free energy of a system occur when variables that define the state of the system do work on the system. Gradients in the free energy of a system with respect to these state variables are generalized thermodynamic forces characterized by adjectives that describe the nature of the variable – chemical, thermal, mechanical, electrical, etc. – giving rise to the force. Absolute equilibrium exists when these forces sum vectorially to zero, i. e., the free energy of the system is an absolute minimum. A system in equilibrium with one or more type of force, while not being in equilibrium with others, is in a state of partial equilibrium. Equilibria can be stable, metastable or unstable. If small changes in the thermodynamic forces tend to alter the state of the system from its equilibrium condition, the equilibrium is unstable. If such changes tend to restore the system to its equilibrium state, the equilibrium is stable if the original state represents an absolute minimum in the free energy of the system or metastable if the minimum is local. These concepts are illustrated in Fig. 2.7, which depicts various states of mechanical equilibrium. Although materials are rarely in thermodynamic equilibrium with their surroundings during engineering applications, the extent of deviation from its equilibrium state determines the propensity of a structure to change

Materials Science for the Experimental Mechanist

with time. Since changes in structure cause changes in properties, the suitability of a material for a particular engineering application may also change.

2.1 Structure of Materials

29

Force ΔF *

P = exp(−ΔF ∗ /kT ) ,

(2.6)

when ΔF ∗ is expressed per atom or mole of the substance. In (2.6) k is Boltzmann’s constant, the universal gas constant per atom or molecule. The Arrhenius equation expresses the rate of change of many chemical reactions and solid-state processes: r = ν0 exp(−ΔF ∗ /kT ,

m n (a)

Phase Diagrams The structure of engineering materials at the microscale generally consists of regions that are physically and chemically distinct from one another, separated from the rest of the system by interfaces. These regions are called phases. The spatial distribution and, to a limited extent, the chemical composition of phases can be altered by thermomechanical processing prior to the end use of the material. Sometimes service conditions

m y

(b)

l

m

(c)

x

Fig. 2.7 (a) Metastable, (b) unstable, and (c) stable mechanical equilibrium states for a brick (after Barrett et al. [2.7])

can cause changes in the microstructure of a material, usually with accompanying changes in properties. Knowledge of the phases present in a material at various temperatures, pressures, and compositions is essential to the prediction of its engineering properties and to assessing its propensity for change under service conditions. The phase diagram, a map of the thermodynamically stable phases present at various temperatures, pressures, and compositions, presents this information in a graphical form. The number of degrees of freedom f (≥ 0) of a system refers to the number of environmental and composition variables that can be changed without changing the number of phases in equilibrium. The phase rule relates f to the number of distinct chemical species or components C and the number of phases present Φ: f = C −Φ+2 ,

(2.7)

in which the rate of change r is the product of the frequency of attempts at change ν0 and P is the probability of a successful attempt. Figure 2.8 illustrates the process schematically for the case of a thermally activated process assisted by external stress.

ΔF

l

(2.8)

ΔF (1-3) ΔF * (1) ΔF

(1-2)

(3)

(2) A*

A(1-2)

A0

Fig. 2.8 Free energy (F) versus slipped area ( A0 ) for deformation processes

Part A 2.1

Rate of Approach to Equilibrium In order for a system that is in metastable equilibrium in its current state (state 1) to attain the (stable or metastable) equilibrium state with the next lowest free energy (state 2), passage through an unstable equilibrium state (state 3) whose free energy is higher than that of states 1 or 2 is generally required. State 3 is an activated state and constitutes a barrier to the transition of the system from state 1 to the lower free energy of state 2. The free energy difference between states 1 and state 3 ΔF (1−3) must be supplied by internal and external sources in order to effect the transition from the metastable state to the activated state, from which subsequent transition to the next lower energy state is assumed to be spontaneous. The nature of the barrier determines the free energy difference. The free energy ΔF (1−3) contains both thermal energy and work done by external fields during activation. The difference between ΔF (1−3) and the work done by external sources is called the free energy of activation ΔF ∗ . The Boltzmann factor P is the probability that the system will change its state from state 1 to state 3

30

Part A

Solid Mechanics Topics

Part A 2.1

where 2 refers to temperature and pressure. The degrees of freedom in phase diagrams of condensed systems at constant pressure (isobaric diagrams) is given by a form of equation (2.6) in which 2 is replaced by 1, since the pressure is fixed. Either the weight percent or atomic percent of the components specifies the composition of a multicomponent system. In many engineering applications, weight percent is preferred as a more practical guide to the relative amounts of components. In scientific studies of the nature of diagrams formed from elements having similar chemical characteristics, atomic fraction is preferred. For an ncomponent system, the weight percent of component i can be calculated from its atomic percent by the relation (a/o)i (at.wt.)i × 100% , (2.9) (w/o)i = n (a/o) j (at.wt.) j j=1

where (a/o) and (w/o) are the atomic percent and weight percent, respectively, and at.wt. refers to the atomic weight of the subscripted substance. The summation extends over all components. A similar expression relates the atomic percent of a component to the weight percents and atomic weights of the components. It is impossible to represent a phase diagram in three dimensions for a material consisting of more than three components. More commonly, diagrams that show the stable phases for a two-component (binary) system at one atmosphere pressure and various temperatures are used as guides to predict the stable phases. This permits a two-dimensional map with temperature as the ordinate and composition as the abscissa to represent the possible structures. ASM International offers an extensive compilation of binary phase diagrams of metals [2.8]. A similar compilation is available for ceramic systems [2.9]. Three-component (ternary) phase diagrams are more difficult to represent since they require two independent compositional variables in addition to temperature. The most common three-dimensional representation is a prism in which the component compositions are plotted on an equilateral triangle base and the temperature on an axis normal to the base. Ternary diagrams of many metallic systems have been determined are available as a compilation similar to that for the binary systems [2.10]. Phase diagrams not only give information on the number and composition of phases present in equilibrium, but also provide the data necessary to calculate the relative amounts of phases present at any temperature. Figure 2.9 illustrates this principle, the inverse lever law, using the silver–copper binary diagram as an example.

From this diagram we note that an alloy of 28.1 w/o Cu solidifies at 779.1 ◦ C into a solid consisting of two solid phases: a silver-rich phase α containing 8.8 w/o Cu and a copper-rich phase β containing 92.0 w/o Cu. This alloy, which passes directly from the liquid state to the solid state without going through a region of solid and liquid in equilibrium, is a eutectic. To determine how much of each phase is present in the eutectic alloy at the solidification temperature, we note that, while the overall composition of the alloy c0 is 28.1 w/o Cu, it is a mixture of copper- and silver-rich phases, each having a known composition. Performing a mass balance on the amount of copper in the alloy yields the result: cβ − c0 × 100% (%)α = cβ − cα (92 − 28.1) × 100% = 76.8% (2.10) = (92 − 8.8) for the weight percentage of the silver-rich phase in the alloy. The inverse lever law gets its name from the fact that the numerator in (2.10) is the distance on the composition axis from the overall alloy composition to the composition of the copper-rich phase cβ on the opposite side of the diagram from the silver-rich phase whose percentage is to be determined. The denominator is the entire distance between the compositions of the copperrich and silver-rich phases. The calculation is analogous to balancing the relative amounts of the phases on a lever whose fulcrum is at the overall alloy composition. While the numerical result shown illustrates the calculation for the eutectic alloy and temperature, the principle applies to any constant temperature as long as we choose the appropriate overall alloy composition Temperature (°C) 1200 1100 Liquid 1000 961.93 °C 900 800 8.8 % 28.1 % 700 (Ag) 600 500 400 300 200 Ag 10 20 30 40 50

1084.87 °C

(Cu) 779.1°C 92 %

60

70

80 90 Cu wt % copper

Fig. 2.9 Silver–copper phase diagram at 1 atm pressure

Materials Science for the Experimental Mechanist

and the compositions of the phases in equilibrium at that temperature.

2.1.5 Observation and Characterization of Structure

Metallographic techniques Metallographic approaches to studying the structure of materials are either surface techniques or transmission techniques. Surface techniques rely on images formed by the reflection of a source of illumination by a prepared surface. The wavelength of the illuminating radiation and the optical properties of the observation system determine the resolution of the technique. Optical metallography employs a reflecting microscope and illumination by visible light to examine the polished and etched surface of a specimen either cut from bulk material or prepared on the surface of a sample, taking care not to introduce damage during the preparation. Consequently, the examination is generally a destructive technique. Magnifications up to approximately 1200 × are possible with careful specimen preparation. Properly chosen etches reveal the arrangement of grain boundaries, phases and defects intersected by the plane of polish. Contrast arises from the differing reflectivities of the constituents of the microstructure. Interpretation of micrographs to relate the observed structure to the properties and behavior of the material requires experience and knowledge of the composition and thermomechanical history of the material. Scanning electron microscopy (SEM) employs a beam of electrons as an illuminating source [2.11]. The de Broglie wavelength of an electron is λ = h/ p, where h is Planck’s constant and p is the momentum of the electron. The de Broglie wavelength of electrons varies with the accelerating voltage V since for an electron eV = p2 /2m, where e and m are the charge and mass, respectively, of the electron. Electromagnetic lenses use the charge on the electron beam to condense, focus, and magnify it. Magnifications ob-

tained by this technique range from those characteristic of optical microscopy to nearly 100 times the best resolution obtained from optical techniques. Contrast among microconstituents arises because of their differing scattering powers for electrons. Since electron optics permits a much greater depth of field than is possible with visible light, SEM is the observation tool of choice for examining surfaces with a high degree of spatial irregularity, such as those produced by fracture. While the images are nearly always two-dimensional sections through three-dimensional structures, stereo microscopy is widely used in the examination of fracture surfaces. Although this method can employ an appreciable range of wavelengths, it is necessary to operate in a vacuum and to provide a means of dissipating any electrical charge on the specimen induced by the electron beam. In recent years, the development of special environmental chambers has permitted the observation of nonconductive materials at atmospheric pressure. Transmission metallographic techniques produce images that are the projection of the content of the irradiated volume on the plane of observation. Clearly, such techniques rely on the transparency of the material to the illumination. This requires the preparation of thin sections of the material, followed by treatments to improve the contrast of structural features, then observed by transmission microscopy. Visible light generally suffices as the illuminating source for biological materials. Since most engineering materials are not transparent to visible light, observation by transmitted radiation is less common for these materials. The following section discusses both of the principal exceptions, transmission electron microscopy (TEM) and x-ray radiography. Information obtained from metallographic observations serves a variety of purposes: to determine the microstructural state of a material at various stages of processing, to reveal the condition of material in a failed engineering part, and to obtain a semiquantitative estimate of some types of material behavior. All cases where the images represent plane sections require analysis by stereological techniques to obtain quantitative information that for relating structure to properties [2.12]. For a microstructure containing two or more microconstituents, such as phases, inclusions or internal cavities, the simplest parametric characterization is the volume fraction VV (n) of the n-th microconstituent. It is easily shown that this is also equal to the area fraction A A (n) of the n-th constituent on an observation plane, which can easily be measured on a polished metallographic specimen [2.12].

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Part A 2.1

In order to implement in the selection and design of materials and processes the axiom that structure at various levels determines the properties of materials, it is necessary to have means of examining, measuring, and describing quantitatively the structure at various scales. The information so obtained then leads to the development, validation, and verification of models that describe the behavior of engineering structures. The following sections describe two types of characterization techniques that are useful for this purpose.

2.1 Structure of Materials

32

Part A

Solid Mechanics Topics

Part A 2.1

Another common measurement of microstructure is the mean linear intercept Λ(t) or its reciprocal PL (t), the intercept density, of a test line parallel to the unit vector t with the traces of internal boundary surfaces on an observation plane. The intercept density is the number of intersections per unit length of the test line with the feature of interest. Since microstructures are generally anisotropic PL (t) depends on the orientation of the test line. The volume average of all measurements is an average intercept density, PL ; its reciprocal is the mean linear intercept Λ. The grain size refers to the mean linear intercept of test lines with grain boundaries. When employing this terminology, remember that the measurement is actually the mean distance between grain boundaries in the plane of measurement and may not apply to the three-dimensional structure. In addition, the concept of size implies an assumption of shape, which is not included in the measurement. Nevertheless Λ remains the most widely used quantitative characterization of microstructure in the materials literature. Similar measurements can be made of the intersections of test lines with interphase boundaries and internal cavities, or pores. The associated mean linear intercepts are phase diameters or pore diameters, respectively. The intercept density for a particular class of boundaries also measures the total boundary area per unit volume SV through the fundamental equation of stereology: SV = 2 PL .

(2.11)

This quantity is important when studying properties influenced by phenomena that occur at internal boundaries. Diffraction Techniques In an irradiated array of atoms, each atom acts as a scattering center, both absorbing and re-radiating the incident radiation as spherical wavelets having the same wavelength as the incident wave. When the wavelength of the incident radiation is comparable to the spacing between atoms, interference effects can occur among the scattered wavelets. A crystalline material acts as a regular array of scattering centers, and produces a pattern of scattered radiation that is characteristic of the array and the wavelength of the radiation. Amorphous materials also produce scattering, but the lack of long-range atomic order precludes the development of patterns of diffracted beams. Diffracted rays occur by the constructive interference of radiation reflected from atomic planes. The model proposed by W. H. Bragg and W. L. Bragg, illustrated in Fig. 2.10, describes the phenomenon.

Radiation of wavelength λ is incident at an angle θ on parallel atomic planes separated by a distance d. Scattered radiation from adjacent planes interferes constructively, producing a scattered beam of radiation, when the path difference between rays scattered from adjacent planes is an integral number n of wavelengths. The relationship describing this phenomenon is Bragg’s law, nλ = 2d · sin θ ,

(2.12)

which is the basis for all measurements of the atomic dimensions of crystals. X-rays, a highly energetic electromagnetic radiation having a wavelength the same order of magnitude as the spacing between atoms in crystals, inspired the original derivation of (2.12) [2.13, 14]. If the crystal is sufficiently thin to permit diffracted beams to penetrate it, transmission diffraction patterns occur on the side of the crystal opposite to the incident radiation. However, even if the diffracted beams cannot penetrate the crystal, back-reflection diffraction patterns occur on the same side as the incident beam. Crystal structure analysis employs both types of patterns. One method of identifying unknown materials employs analysis of diffraction patterns formed by radiation of known wavelength to determine the lattice parameter and crystal structure of an unknown substance, which is then compared with a database of known substances for identification [2.15, 16]. Both incident and diffracted X-radiation can penetrate substantial thicknesses of many materials, rendering this diffraction technique useful for determining the condition of material in the interior of an engineering part. Lattice distortion giving rise to internal stress, averaged over the irradiated volume, can be determined by comparing the interplanar spacings of a material containing internal stress with those of a material free of internal stress [2.17]. This technique of x-ray stress analysis is extremely valuable in deter-

θ

θ θ d

Fig. 2.10 Derivation of Bragg’s law

Materials Science for the Experimental Mechanist

trons and neutrons, which have de Broglie wavelengths comparable to the interatomic spacing of the crystals. Electron beams are employed in both transmission and back-reflection modes to reveal information on the crystal structure and defect content of materials. The fact that electron beams possess electrical charge makes it possible to employ electromagnetic lenses to condense, focus, and magnify them. Transmission electron microscopy (TEM) uses diffracted electron beams transmitted through material and subsequently magnified to reveal features of atomic dimensions [2.21]. Since electron beams are highly absorbed by most materials, specimens examined in TEM are only a few thousand atoms thick. Nevertheless, this technique reveals much useful information about the nature and behavior of dislocations, grain boundaries and other atomic-scale defects. Orientation imaging microscopy (OIM) [2.22] employs back-reflection electron diffraction patterns to form maps called pole figures by determining the orientation of individual grains. These maps describe the orientation distribution of grains in a material, which is a major cause of the anisotropy of many bulk properties. The greater penetrating power of neutron beams reveals information from greater thicknesses of material than x-rays [2.23]. Since these beams are not electrically charged, their magnification by electromagnetic lenses is not possible. Therefore, their principal use is for measurements of changes in lattice spacings due to internal stresses, and not for forming images. In addition to revealing the structure of materials, many advanced techniques reveal not only the structure but also the chemistry of a substance at the microscale. While techniques such as the three-dimensional (3-D) atom probe, Auger electron spectroscopy and imaging, Z-contrast microscopy, and many others complement those for structural examination and often employ similar physical principles; a complete survey is beyond the scope of this review.

2.2 Properties of Materials Material properties are attributes that relate applied fields to response fields induced in the material. Applied and induced fields can be of any tensorial rank [2.24, 25]. Fields are conjugate to one another if the applied field does an incremental amount of work on the system by causing an incremental change in the induced field. External fields and the associated field variables include temperature T , stress τ, and electrical (E) and

magnetic (H) fields; their respective conjugate fields are entropy S, strain ε, electrical displacement D, and magnetic induction B. Cross-effects occur when changes in an applied field cause changes in induced fields that are not conjugate to the applied field. Consider a system subjected to a cyclic change of applied field variables such that each variable changes arbitrarily while the others are constant, then the pro-

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mining residual stresses in heat-treated steel parts, for example. It has the advantage that lattice strain, which is the source of internal stress, is the direct result of the measurement. This is in contrast to the calculation of stress from measurements of strain on the external surface of a material, which include components due to both lattice strain and the stress-free, permanent strain caused by dislocation motion. When stress is calculated from such surface measurements, the assumption that dislocation motion is absent or negligible is implicit [2.18]. Diffraction contrast from scattered x-rays forms the basis for x-ray topography of crystals, useful in the study of the defect structure of highly perfect single crystals used in electronic applications [2.19]. Magnified diffraction spots from crystals reveal images of the structure of the material with a resolution comparable to the wavelength of the radiation. Small changes in the orientation and spacing of atomic planes due to the presence of internal defects change the diffraction conditions locally, creating contrast in the image of the diffracted beam. The availability of high-intensity xray sources from synchrotrons has greatly increased the applicability and usefulness of this technique. Radiography also forms images by the interaction of material with x-rays, but in this case, the contrast mechanism is by the differential absorption of radiation caused by varying thicknesses of material throughout an irradiated section as well as by regions having differing densities, which affects the ability of material to absorb x-rays [2.20]. Radiography can reveal macroscopic defects and heterogeneous distributions of phases in materials. Since contrast is not due to diffraction by the structure, this process cannot provide information on its crystal structure or the state of lattice strain in the material. The principles of diffraction apply not only to electromagnetic radiation but also to the scattering of highly energetic subatomic particles, such as elec-

2.2 Properties of Materials

34

Part A

Solid Mechanics Topics

Part A 2.2

cess reverses until the variables have their initial values. If such a closed cycle occurs under conditions such that the system is continuously in equilibrium with its surroundings, the thermodynamic states at the beginning and the end of the process will be the same and no net work is done. Properties that relate conjugate fields in this case are equilibrium properties and depend only on the thermodynamic state of a system, not its history. A caloric equation of state relates conjugate fields, which permits the description of the relationship between applied and induced fields in terms of appropriate derivatives of a thermodynamic potential. If processes that occur during the cycle dissipate energy, they produce entropy and the system is not in the same state in the initial and final conditions. Although the applied field variables are the same at the beginning and end of the cycle, the conjugate induced variables do not return to their original values. The change of thermodynamic state occurs because of irreversible changes to the structure of the material caused, at least in part, by changes in the internal structure of the system that occur during the cyclic change. Associated material properties are dissipative and exhibit hysteresis. Dissipative properties are dependent on time and the history of the system as well as its current thermodynamic state. However, in some cases it is possible to relate these properties to appropriate derivatives of a complex dissipative potential [2.24]. In general, equilibrium properties are structure insensitive, while dissipative properties are structure sensitive. The search for additional state variables that depend on the internal structure of the material is a subject of much continuing research. Transport processes are dynamic processes that cause the movement of matter or energy from one part of the system to another. These produce a flux of matter or energy occurring in response to a conjugate thermodynamic force, defined as the gradient of a thermodynamic field [2.26]. Attributes that connect fluxes and forces are generally structure-insensitive material properties.

2.2.1 The Continuum Approximation Continuity The complex nature of real materials requires approximation of their structure by a mathematical concept that forms a basis for calculations of the behavior of engineering components. This concept, the continuous medium, replaces discrete arrays of atoms and molecules with a continuous distribution of matter. Fields and properties defined at every mathematical

point in the medium are continuous, with continuous derivatives, except at a finite number of surfaces separating regions of continuity [2.24]. The process that defines properties at a point in the medium consists of taking the limit of the volume average of the property over increasingly smaller volumes. For sufficiently large volumes the average will be independent of sample size. However, as the volume sampled decreases below a critical range, the volume average will exhibit a dependence on sample size, which is a function of the nature of the property being measured and the material. The smallest sample volume that exhibits no size dependence defines the continuum limit for the material property. The continuum approximation assigns the value of the size-independent volume average of the property to a point at the centroid of the volume defined by the continuum limit. It follows that, when the size of a sample being tested approaches the continuum limits of relevant properties, results can be quite different from those predicted by a continuum model of the material. This limit depends on the material, its structural state, and the property being measured, so that all need to be specified when assigning a value to the continuum limit. The continuum approximation employs the concept of the material particle to link the discrete nature of real materials to the continuum. By the argument above, the material attributes of such a particle must be associated with a finite volume, the local value of the continuum limit for the attributes. As noted earlier, continuum limits may differ for different properties; consequently the effective size associated with a material particle will depend on the property associated with the continuum. Coordinate systems based on material particles, in contrast to those based on the crystal lattice, must necessarily be continuous by the assumption of global continuity of the medium. Despite the inherent differences in concepts of material structure, it is possible to develop models of many material properties based on atomic or molecular characteristics that interface well with the continuum model at the scale of the continuum limit. These limits can only be determined experimentally or by the comparison of properties calculated from physically based models of atomic arrays of various sizes. Such models form essential links between continuum mechanics and the knowledge of structure provided by materials science. Homogeneity The continuous medium used for calculation of the behavior of engineering structures also possesses the

Materials Science for the Experimental Mechanist

Isotropy The usual model of a homogeneous continuous medium also assumes that properties of the material are isotropic, i. e., they are independent of the direction of measurement. This assumption generally results in the minimum number of independent parameters that describe the property and simplifies the math-

ematics involved in analysis. However, isotropy is infrequently observed in real materials, and neglect of the anisotropic nature of material properties can lead to serious errors in estimates of the behavior of engineering structures. Properties of materials possess symmetry elements related to the structure of the material. The following Gedanken experiment illustrates the meaning of symmetry of a property [2.25]. Measure the property relative to some fixed set of coordinate axes. Then operate on the material with a symmetry operation. If the value of the property is unchanged, then it possesses the symmetry of the operation. Neumann’s principle states that the symmetry elements of any physical property of a crystal must include the symmetry elements of the point group of the crystal. Notice that the principle does not require that the crystal and the property have the same symmetry elements, just that the symmetry of the property contains the symmetry elements of the crystal. Since an isotropic property has the same value regardless of the direction of measurement, it contains all possible point groups. As mentioned above, bulk properties of polycrystalline and multiphase materials are appropriate averages of the properties of their microconstituents, taking account of the distribution of composition, size, and orientation. A common source of bulk anisotropy in a material is the microscopic variation in orientation of its grains or microconstituents. An orientation distribution function (ODF) describes the orientation distribution of microscopic features relative to an external coordinate system. Depending on the feature being measured, quantitative stereology or x-ray diffraction analyses provide the information required to determine various types of ODFs. Used in conjunction with the properties of individual crystals of constituent phases, ODFs can describe the anisotropy of many bulk properties, such as the elastic anisotropy of metal sheet formed by rolling.

2.2.2 Equilibrium Properties Although the definition of equilibrium properties is phenomenological, without reference to the structure of the material, values of the properties depend on the structure of materials though models described in treatises on condensed-matter physics and materials science [2.27]. Properties described in the following sections are all first order in the sense that they arise from an assumed linear relationship between applied and induced fields. However, most properties have measurable de-

35

Part A 2.2

property of homogeneity, that is, properties at a material point do not depend on the location of the point in the medium. While this assumption is reasonably accurate for single-phase materials of uniform composition, it no longer applies to multiphase materials at the microscale. In the latter case, properties of the bulk depend on appropriate averages of the properties of the constituent phases, which include parameters such as the volume fraction of each phase, the average size of the particles of each phase, and in some cases the orientation of the phases relative to one another and to externally applied fields. Higher order models of multiphase materials treat each phase as a distinct continuum with properties distinct from other phases. Physical fields within the body are also influenced by the nature, orientation, and distribution of the phases present. Average values of induced internal fields determined from the external dimensions of a body and fields applied to the surfaces may not accurately reflect the actual conditions at points in the interior of the body due to different responses from constituent phases within the material. Material properties determined from the bulk response to these mean fields may differ significantly from similar properties measured on homogeneous specimens of the constituent phases. Measurements of local fields rely on techniques that sample small, finite volumes to determine local material response, then inferring the values of the fields producing the response. Since a quantitative determination of variables that determine the bulk properties of multiphase materials is often difficult, time-consuming, and expensive, experimental measurements on bulk specimens typically determine these properties. It is important to realize that applying the results of such measurements implicitly assumes that differences in structure of the material at the microscale have a negligible effect on the resulting properties. Structural variability due to inhomogeneity can be taken into account experimentally by determining properties of the same material with a range of structures and assigning a quantitative value to the uncertainty in the results.

2.2 Properties of Materials

36

Part A

Solid Mechanics Topics

Part A 2.2

pendencies on other applied fields. Such field-induced changes cause second-order effects, such as optical birefringence and the electro-optical effect, where firstorder properties depend on strain and electrical field, respectively. For compactness, this review will not consider these effects, although many of them can provide the basis for useful measurements in experimental mechanics. The usual description of equilibrium properties employs conjugate effects. In principle, any change in an applied field produces changes in all induced fields, although some of these effects may be quite small. Figure 2.11, adapted from Nye [2.25] and Cady [2.28], illustrates the interrelationship among first-order effects caused by applied and induced fields. Applied field variables appear at the vertices of the outer tetrahedron, while the corresponding vertices of the inner tetrahedron represent the conjugate induced field variables. Lines connecting the conjugate variables represent principal, or direct, effects. Specific Heat For a reversible change at constant volume, an incremental change in temperature dT produces a change of entropy dS: ε C (2.13) dT , dS = T

where C ε is the heat capacity at constant strain (volume) and T is the absolute temperature. To maintain constant volume during a change in temperature, it is necessary to vary the external tractions on the surface of the body (or the pressure, in the case of a gas). When measurements are made at constant stress (or pressure), the volume varies. The heat capacity under these conditions is C τ . Specific heat is an extensive, isotropic property.

τ H B

D E

ε

S T

Fig. 2.11 Relationships among applied and induced fields

Electrical Permittivity A change in electrical field dE produces a change in electrical displacement dD:

dDi = κij dE j ,

(2.14)

where the suffixes, which range from 1 to 3, represent the components of the vectors D and E, represented by bold symbols, along reference coordinate axes and summation of repeated suffixes over the range is implied here and subsequently. The second-rank tensor κ is the electrical permittivity, which is in general anisotropic. Magnetic Permeability A change in magnetic field dH produces a corresponding change in magnetic induction or flux density dB:

dBi = μij dH j ,

(2.15)

where μ is the second-rank magnetic permeability tensor, which is in general anisotropic. Equations (2.14) and (2.15) apply only to the components of D and B that are due to the applied electrical or magnetic field, respectively. Some substances possess residual electrical or magnetic moments at zero fields. The integral forms of (2.14) and (2.15) must account for these effects. Hookean Elasticity An incremental change in stress dτ produces a corresponding change in strain dε according to:

dεij = sijk dτk ,

(2.16)

where s is the elastic compliance tensor. In this case, the applied and induced fields are second-rank tensors, so the compliances form a fourth-rank tensor that exhibits the symmetry of the material. The elastic constant, or elastic stiffness, tensor c, is the inverse of s in the sense that cijmn smnkλ = 1/2(δik δ jλ + δiλ δ jk ). Equation (2.16) is Cauchy’s generalization of Hooke’s law for a general state of stress. A material that obeys (2.16) is an ideal Hookean elastic material. Most engineering calculations involving elastic behavior assume not only Hooke’s law but also isotropy of the elastic constants. This reduces the number of independent elastic constants to two, i. e. cijmn = λδij δmn + μ(δim δ jn + δin δ jm ), where λ and μ are the Lamé constants. Various combinations of these constants are also employed, such as Young’s modulus, the shear modulus, the bulk modulus and Poisson’s ratio, but only two are independent in an isotropic material. The Chapter on continuum mechanics in this work lists additional relationships among the isotropic elastic constants [2.29].

Materials Science for the Experimental Mechanist

Cross-Effects All conjugate fields can be included in the description of the thermodynamic state. Define a function Φ such that

Φ = U − τij εij − E D − Bk Hk − TS .

(2.17)

Then using the definition of work done on the system by reversible changes in the conjugate variables and the first and second laws of thermodynamics, we have dΦ = −εij dτij − D dE − Bk dHk − S dT . (2.18) Since Φ = Φ(τ, E, H, T), ∂Φ ∂Φ dτij + dE dΦ = ∂τij E,H,T ∂E τ,H,T ∂Φ ∂Φ + dHk + dT ∂Hk E,τ,T ∂T E,H,τ

the coefficients of the form 2 ∂εij ∂ Φ = − ∂τij ∂T E,H ∂T σ,E,H ∂S = = βijE,H ∂τij T,E,H

(2.22)

τ

ΔS = βij τij + (C /T )ΔT ,

(2.23)

where sT is the isothermal compliance tensor. Extension of this argument shows that the full matrix of matera)

Heat of deformation

Strain

Entropy

Elasticity

Heat capacity

(2.19)

Piezocaloric effect

Stress

it follows that ∂Φ = −εij , ∂τij E,H,T ∂Φ = −Bk , ∂Hk E,τ,T

∂Φ = −D , ∂E τ,H,T ∂Φ = −S . ∂T E,H,τ (2.20)

Further differentiation of (2.19) and (2.20) with respect to the independent variables gives relationships among

(2.21)

for the thermal expansion coefficient tensor β at constant electrical and magnetic field. Thermoelasticity illustrates cross-effects that occur among equilibrium properties. Consider the section of the tetrahedron containing only the applied field variables stress and temperature and the associated induced field variables, strain and entropy, illustrated in Fig. 2.12. In this illustration, the line connecting stress and strain represents the elastic relationships and the line connecting entropy and temperature denotes the heat capacity. The indirect, or cross-effects are the heat of deformation, the piezocaloric effect, thermal expansion, and thermal pressure, as shown in the figure. Ten independent variables specify the thermodynamic substate of the material. These can be the nine stress components and temperature, nine strain components and temperature, or either of the preceding with entropy replacing temperature. Specification of ten independent variables determines the ten dependent variables. Nye [2.25] shows that the relationships connecting the effects become: T τk + βij ΔT , εij = sijk

c s σ

Thermal expansion

Temperature

Thermal pressure

b)

f

ε α

S α

–f

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Part A 2.2

While isotropy is a convenient approximation for mathematical calculations, all real materials exhibit elastic anisotropy. Single crystals of crystalline materials display symmetry of the elastic coefficients that reflects the symmetry of the crystal lattice at the microscale [2.25]. The single-crystal elastic constants are often expressed in a contracted notation in which the two suffixes of stress and strain components become a single suffix ranging from 1 to 6. The independent terms of the corresponding elastic constant tensor become a 6 × 6 array, which is not a tensor. This notation is convenient for performing engineering calculations, but when the elastic constants are not isotropic, changes in the components of the elastic constant matrix due to changes in the coordinate system of the problem must be calculated from the full, fourth-rank tensor description of the constants [2.25]. Anisotropic crystalline materials can display a lower elastic symmetry at the mesoscale than at the single-crystal scale due to preferred orientation of the crystallites introduced during processing.

2.2 Properties of Materials

T/C C/T T

Fig. 2.12a,b Equilibrium properties relevant to thermoelastic behavior: (a) quantity and (b) symbol

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Part A

Solid Mechanics Topics

Part A 2.2

ial coefficients defined by relationships similar to (2.21) and written in the form of (2.22) is symmetric. The atomic basis for the linear relationships in (2.22) arises from the nature of the forces between atoms. Although the internal energy of a material is in reality a complex, many-body problem that is beyond the capabilities of even modern computers to solve, summation over all atoms of the bond energies of nearest-neighbor pairs offers a physically reasonable and computationally manageable alternative. The energy of an atom pair exhibits a long-range attraction and a short-range repulsion, resulting in a minimum energy for the pair at a distance equal to or somewhat greater than the interatomic spacing in the material. In the simplest case, a diatomic molecule of oppositely charged ions, the Coulomb force between unlike charges provides the attractive force and the mutual repulsion that arises when outer shells of electrons begin to overlap is the repulsive force. The minimum energy of the pair is called the dissociation energy, and is the energy required to separate the two ions to an infinite spacing at absolute zero. For the example cited the energy is radially symmetric, so that the energy–distance relationship is independent of the relative orientation of the ions, and the energy minimum occurs at the equilibrium interionic spacing. Figure 2.13 illustrates this behavior schematically. It follows from (2.16)–(2.20) that sijk = −

∂2Φ , ∂τij ∂τk

(2.24)

which can be expressed in terms of the strain relative to the equilibrium spacing by (2.16). Thus the elastic compliances are proportional to the curvature of the energy-distance curve of ionic pairs, measured at the equilibrium interionic distance. Extension of this model to condensed systems of atoms having more complex bonds leads to modification of the details of the relationships. The elastic constants of single crystals of single-phase cubic alloys can be estimated by an extension of this concept [2.30]. The energy–bond length relationship shown in Fig. 2.13 applies only at absolute zero. At finite temperatures, the energy of the pair is increased above the minimum by an amount proportional to kT . This implies that for any finite temperature T there exist two values of the interionic spacing with the same energy. The observed spacing at T is the average of these two. Most materials exhibit a curve that is asymmetric about r0 , the equilibrium spacing at T = 0 causing the mean spacing to change with temperature. Generally the na-

Lattice energy (eV) 15 12 9

Repulsive energy

6 3 0 –3 –6 Minimum energy

–9

Attractive energy

–12 –15

0

0.1

0.2

r0

0.4

0.5 0.6 0.7 0.8 Interionic spacing (nm)

Fig. 2.13 Energy–bond length relationship for an ionic

solid

ture of this asymmetry is such that the mean spacing increases with temperature, as illustrated in Fig. 2.13, leading to a positive coefficient of thermal expansion. Different values of the elastic compliances result from measurements under isothermal (ΔT = 0) or adiabatic (ΔS = 0) conditions. The adiabatic compliance tensor sS is determined by setting ΔS = 0 and solving for ΔT in the second of (2.22), eliminating ΔT from the first equation and defining sS as the resulting coefficients of σ. The relationships among the isothermal and adiabatic quantities are τ C S T − sijk = −βij βk (2.25) . sijk T Since the coefficients of the thermal expansion tensor are positive for most materials, adiabatic compliances are typically smaller than isothermal compliances. Nye [2.25] gives expressions such as (2.25) relating differences between various properties with different sets of applied field variables being constant. This reasoning emphasizes the fact that the definitions of these properties include specification of the measurement conditions.

2.2.3 Dissipative Properties Equilibrium processes are reversible and conservative, whereas nonequilibrium processes are irreversible and nonconservative. In the latter cases, work done on the system dissipates in the form of heat, generating entropy. This introduces history and rate dependence in the response of the system to external fields and influ-

Materials Science for the Experimental Mechanist

depends on the conditions of measurement. Engineering standards establish boundaries between these regimes that form the basis for design calculations. For example the limit of applicability of the equations of elasticity in engineering calculations can be defined as the stress at which strain no longer obeys Hooke’s law (proportional limit) or the more easily measured offset yield stress, defined as the stress required to produce some small but easily measurable permanent strain. The boundaries are always compromises between the accuracy with which they can be measured and the intended uses of the information. Viscoelasticity When stresses or strains are independent variables applied periodically with time, the response of the corresponding dependent variable is out of phase with the applied field. This causes a delay in response, manifested by an induced field that has the same frequency, but is out of phase with the applied field. Consider a uniaxial stress τ(t) = τ0 exp(iωt) applied with frequency ω that causes a strain ε(t) = ε0 exp[i(ωt − δ)]. The complex compliance S∗ is defined as

S∗ =

ε0 ε (t) = exp (−iδ) . τ (t) τ0

(2.26)

The storage compliance S is the real component of S∗ , and the loss compliance S

is the imaginary component. These are related to the loss angle, δ, by tan(δ) = S

/S . The complex stiffness or modulus is the reciprocal of S∗ . Applying this concept to other states of stress and strain permits determination of all components of the viscoelastic constant tensor. The complex elastic constants and the loss angle are functions of the frequency of the applied field as can be noted by analysis of the above example using the theory of damped oscillations. Consider an experiment in which the component of total strain, ε0 , in a particular direction remains constant while the corresponding stress component is measured on a plane normal to the same axis. Let instantaneous and relaxed values of stress be τ0 and τ∞ , respectively. The dependence of stress on time t has the form: τ (t) = τ∞ + (τ0 − τ∞ ) exp (−t/ξε ) ,

(2.27)

if a single relaxation mechanism dominates the process. Each mechanism of relaxation possesses a characteristic relaxation time, which is the mathematical manifestation of the physical process causing the delay in response of the induced field to the applied field. The

39

Part A 2.2

ences the phenomenological material coefficients that relate applied and induced fields. The definition of properties associated with dissipative processes is formally similar to that for equilibrium properties with the caveat that history dependence affects the properties and their values under various measurement conditions. The following sections employ mechanical properties to illustrate mathematical descriptions of dissipative processes and definitions of associated properties. While these illustrations are most germane to typical problems in experimental mechanics, the mathematical descriptions of phenomena are similar for other processes relating other pairs of conjugate variables. Naturally, the physical interpretation of associated properties will vary with the particular conjugate phenomena investigated. The idealized behavior described by (2.16), is nondissipative. Real materials typically exhibit a timedependent response of the dependent variable to the independent variable, which is generally expressed as a relationship between the time rate of change of the dependent variable and the independent variable. Since the dependent variable does not instantaneously assume its final value in response to changes in the independent variable, these are not in phase during a cyclic process, and energy is dissipated during a closed cycle of deformation. In a static experiment viscoelastic, or anelastic, behavior is characterized by an induced strain field that exhibits not only an instantaneous response to a constant applied stress field, but also continues to increase until it reaches its final, higher, value over a period of time. Similarly, the stress field induced in response to an applied strain field relaxes from an initial value to a lower, constant value after a sufficiently long time. This leads to the definition of relaxed and unrelaxed elastic coefficients in relations of the form of (2.16), depending on whether the instantaneous or final values of the induced variables are employed. Chapter 29 on the characterization of viscoelasticity materials presents a comprehensive treatment of the viscoelastic behavior of materials. Viscoplastic behavior occurs when the material does not return to its original state on the restoration of the initial boundary conditions. Classical continuum plasticity is the limiting case in which time-dependent behavior is sufficiently small to be neglected. Many comprehensive treatments of the mathematics and phenomenology of these models are available to the interested reader; see [2.24]. The transition from reversible, dissipative behavior to irreversible, dissipative behavior is difficult to establish unequivocally and

2.2 Properties of Materials

40

Part A

Solid Mechanics Topics

Part A 2.2

value of t for which (τ0 − τ∞ ) exp(−t/ξε ) reaches 1/ e of its final value defines the relaxation time at constant strain, ξε . The ratio of ε0 to τ(t) is the viscoelastic compliance S(t). Clearly, this ratio depends on time during the relaxation process. Relaxed and unrelaxed compliances S0 and S∞ follow from dividing ε0 by τ0 and τ∞ , respectively. Viscoelastic stiffnesses, relaxed and unrelaxed elastic stiffnesses, or moduli, are the reciprocals of the corresponding compliances. A similar experiment in which the applied stress is kept constant and the strain increases with time to a limiting value results in the definition of a relaxation time at constant stress ξσ which is generally different in magnitude from ξε . A single relaxation time for viscoelastic effects often refers to the geometric mean of relaxation times determined in experiments where stress and strain are, respectively, the independent variables. Zener develops the theory of superposition of a distribution of a spectrum of relaxation times to the description of material response [2.30]. While the example given applies to uniaxial normal stress and the resulting normal strain, the concept applies to other states of stress or strain to obtain components of the full elastic constant and compliance tensors. The physical phenomena that cause relaxation in a material depend on its atomic and molecular structure. For example, the individual molecular chains in a linear polymer do not respond instantaneously to an applied stress field because of their mutual interference with relative motion. Over time, thermal motion of the molecules assisted by the local stress overcomes interference and a configuration in equilibrium with the applied stress field results. Upon removal of the stress, the material relaxes towards its initial configuration. Depending on the magnitude of the deformation and the extent of mutual interference of molecules, it may not be possible for the material to achieve its original state. In this case, the residual deformation is permanent and the deformation is viscoplastic. The book by Ferry [2.31] gives more detail on the theory and data on the viscoelastic properties of polymers. Mechanisms for energy dissipation in polymeric materials occur at the scale of longchain molecules, while those in metals and ceramics occur at the scales of atoms and ions [2.30]. Consequently, relaxation times for polymers tend to be longer than for metals and ceramics, leading to higher damping capacities in the former materials. Damping phenomena are relatively more important to engineering applications of polymeric materials that of metals and ceramics.

Viscoplasticity A cyclic deformation process that does not return the material to its initial state, even after relaxation, produces permanent, or plastic, deformation. This occurs when processes in the material that respond to a change in independent variable, stress or strain, result in irreversible changes that prevent the material from assuming its initial state after restoration of the independent variable to its original value. The physical mechanisms responsible for irreversible behavior in materials are extensively studied and modeled at the microscale. The natures of all such mechanisms are such that time-dependent behavior results on a macroscale. Notwithstanding a reasonably detailed understanding of the microscopic processes involved in permanent deformation, the application of this knowledge to the description of behavior at the macroscale is one of the most challenging research areas in the mechanics of materials. Plasticity in amorphous materials involves permanent rearrangement of the linear molecular or network structure of the substance. The fundamental process of deformation is the stretching, distorting or breaking of individual atomic or molecular bonds and reforming them with different neighbors. A volume change may or may not accompany these processes. Changes typically occur in a manner that is correlated with the local stress state and often involve a combination of thermal activation with stress. Since amorphous structures do not possess long-range atomic or molecular order, it is not appropriate to describe this process in terms of the presence or propagation of structural defects. Permanent deformation of these materials is still best described in terms of phenomenological theories. Chapter 29 reviews some phenomenological approaches to viscoplasticity in which differences of behavior among materials are contained in material parameters appearing in the theories, but not specifically linked to atomic structure or mechanisms. This section presents a formally similar treatment, but introduces material parameters for crystalline materials that are based on the atomic and defect structure of the deforming material. Representative phenomenological approaches to viscoplasticity in which differences of behavior among materials are contained in material parameters appearing in the theories, but not specifically linked to atomic structure or mechanisms. As noted earlier, in crystalline materials permanent deformation at the microscale below half the absolute melting temperature occurs primarily by slip, the relative displacement of material across crystallographic

Materials Science for the Experimental Mechanist

shape coordinate system, embedded in the body, having nodes at material points and deforming congruently with the body as a whole. If no dislocations remain in the body after deformation, lattice coordinates and shape coordinates are related by a one-to-one mapping. While deformation of the shape coordinate system is always compatible (unless voids or cracks form during deformation), deformation of the lattice coordinate system is not compatible if dislocations remain in the body. Following the convention employed in Chap. 1, define a total, or shape, distortion as the deformation gradient FS which maps a reference vector in the initial state dX to its counterpart in the final state dx using shape coordinates. The lattice distortion FL , relates dx to its counterpart expressed in lattice coordinates dζ while the dislocation distortion FD relates dζ to dX [2.32]. Then dx = FL dζ = FL FD dX = FS dX, leading to the multiplicative decomposition FS = FL FD .

(2.28)

By the assumption of continuity FS is compatible, while FL and FD can be compatible or incompatible, depending on the nature of the deformation process. If either is incompatible, the other must be also and the incompatibilities must be equal and opposite. To facilitate understanding the connection between dislocation deformation and macroscale changes in shape, define a reference state consisting of a body free of external tractions and having a zero or selfequilibrating internal stress field. Consider the transition of a small volume of material δV bounded by a surface δS from the reference state to a deformed state by a two-step thought process [2.33, 34]: 1. remove δV from the reference material, replacing any tractions exerted by the rest of the body with tractions on the surface of δV so that its state remains unchanged, and give δV an arbitrary permanent change in shape δV ( dζ = FD dX), 2. replace δV in the original location of δV by applying the necessary tractions to its surface, δS , then remove the tractions, allowing δV to relax to its new, deformed configuration δV

bounded by δS

( dx = FL dζ). Two possibilities exist for step 1: dislocations may or may not be introduced into δV by the permanent change in shape. Each leads to a different situation in δV

after step 2. Case I refers to the situation where no dislocations are introduced in δV and FD is compatible within δV resulting in a stress-free distortion.

41

Part A 2.2

planes. Relative motion occurs such that the local structure of the material is preserved, that is, nearestneighbor distances are unchanged and the material remains crystalline. Motion does not occur simultaneously across an entire slip plane, but sequentially, so that the atomic distortion caused by the motion is localized near boundaries between regions where relative displacement has occurred and regions where it has yet to occur. These boundaries are dislocations, which are the principal source of internal stress in materials. For crystalline regions permanently deformed entirely by slip, dislocations pass out of the region and vanish at free surfaces or accumulate at internal surfaces that bound the region. Characterization of a dislocation requires specification of the slip vector, or Burgers vector b, which is the vector describing the relative displacement of material across the slip plane, and the line direction, a unit vector, t, tangent to the boundary, pointing in an arbitrarily chosen positive direction. Conventionally the senses of b and t are chosen such that the unit normal to the slip plane n = (b × t)/|b| points towards the region of lattice expansion associated with the dislocation. While motion of dislocations in their slip planes conserves lattice sites and produces no volume change, motion of dislocations normal to their slip planes, known as climb, creates or destroys vacant lattice sites, producing a permanent change in volume. Generally this nonconservative motion is associated with deformation at temperatures above one-half of the absolute melting point of the material, while conservative motion with no volume change dominates at lower temperatures. Permanent deformation by dislocation motion results in one of two final configurations. If dislocations pass entirely through the body, the shape is permanently changed, but no (additional) residual stress is introduced in the deformed material. In this case the total deformation is equal to the deformation due to dislocations, or dislocation deformation, both of which are compatible in the usual sense of finite deformation theory [2.24]. In the second case, dislocations remain in the material, causing additional deformation of the crystal lattice accompanied by residual stresses. Relating the latter scenario to conventional measures of deformation based on changes in the external dimensions of a body requires recognition of the fact that deformation of the crystal lattice may not be congruent with the deformation of the body as a whole. It is necessary to describe the process with reference to two coordinate systems: the lattice coordinate system, based on the crystal lattice in which dislocation deformation is measured, and the

2.2 Properties of Materials

42

Part A

Solid Mechanics Topics

Part A 2.2

Since a permanent change of shape has occurred, when δV is reintroduced into the body any misfit must be accommodated by dislocations distributed on δS

after relaxation. The resulting compatible FL within δV

is composed of a component due to these dislocations and a component due to surface tractions arising from the constraint of the rest of the body. This corresponds to Eshelby’s misfitting inclusion problem [2.33]. Case II refers to the situation where dislocations remain in δV

after step 1 and FD is not compatible. After reinsertion within δV

FL now consists of an incompatible distortion due to the dislocations within δV

in addition to the compatible distortion due to the boundary conditions on δS

. The implications of this decomposition become evident if δV

is considered as a representative volume element (RVE) or continuum limit for a continuum theory of plasticity. Including dislocations in δV

requires a decomposition of shape distortion in which both the dislocation and lattice components are incompatible. Figure 2.14 illustrates the relationships among the various components of deformation discussed above and the familiar elastic and plastic distortion components employed in continuum mechanics Chap. 1. Referring to the previous discussion of the decomposition of distortion, case I applies to the situation when the dislocation distortion coincides with the plastic distortion FP and the lattice distortion is entirely due to boundary conditions on δV

. Both distortions are compatible except on the boundary δS where dislocations have accumulated. Case II corresponds to the situation when dislocations that remain in δV cause an incompatible lattice distortion FLD that has an incompatibility equal and opposite to FD . The remaining compatible component of the lattice distortion FLC is Deformed state

Reference state FS Fp FD

FL

F LC = F e

F LD

Fig. 2.14 Decomposition of shape (total) distortion (after [2.35])

due to tractions on the boundary of δV and is formally equal to the conventional elastic distortion Fe . The multiplicative decompositions now become FS = FL FD = (FLC )(FLD FD ) = Fe Fp ,

(2.29)

where we have identified Fe , the conventional elastic distortion, with FLC and the conventional compatible plastic distortion Fp with the product FLD FD . Continuum plasticity employs the decomposition of total distortion into the two compatible distortion components on the extreme right-hand side (RHS) of (2.29), which does not permit the inclusion of dislocations explicitly in the theory and restricts its application to situations in which permanent distortion is uniform throughout the deformed region of material. Crystal plasticity modifies this model by invoking the geometry of crystalline slip to allow deformed regions of the material to experience differing amounts of distortion in response to the local stress field. By the arguments above the boundaries between these regions must contain dislocations sufficient to make the local shape distortion compatible, although the presence of these dislocations is not included explicitly in the theory. However, this approach often employs dislocation theory to construct constitutive relations at the microscale, which permits the local rate of distortion to differ throughout the material in response to the local state of stress [2.36]. For a viscoplastic material the rate of plastic distortion depends on the applied stress through a flow law of the form [2.37] ε˙ˆ ijD = Φ(σˆ ∗ , π1 · · · π K , Θ)σˆ ij∗ ,

(2.30)

in which the carat indicates the deviatoric component of the tensor Φ is a scalar function πi are internal material parameters dependent on structure and Θ is the absolute temperature. The effect of internal structure on mechanical behavior enters through its contribution to the viscoplastic potential Φ. The rate dependence of mechanical response originates with the resistance of the material to dislocation motion. Obstacles to dislocation motion are primarily the intrinsic resistance of the lattice (Peierls stress) and the stress fields of other defects and dislocations. Dislocations move over their slip planes by overcoming the resistance due to these obstacles under the influence of the local stress field and thermal activation, as illustrated schematically in Fig. 2.8. Observable deformation occurs by the collective motion of large numbers of dislocations over an obstacle array with a spectrum of strengths.

Materials Science for the Experimental Mechanist

The kinematics of deformation relate to dislocation motion through the geometry of the microscopic process. The dislocation distortion rate due to dislocation motion on N slip systems is ˙ D = LD = F

N

b(k) ⊗ (ρ(k) × v(k) ) ,

(2.31)

where the superposed dot refers to differentiation with respect to time. For the k-th slip system, ρ(k) is a vector whose length is a measure of the instantaneous length of mobile dislocations per unit volume having Burgers vector b(k) and whose direction is determined by the edge-screw character of the distribution of line directions, while v(k) is a vector normal to ρ(k) that specifies the mean velocity of the mobile dislocation configuration [2.35]. For typical deformation processes, the lattice distortion and lattice distortion rate are small compared to the dislocation distortion rate so that (2.31) is approximately equal to the total or shape distortion rate. Employing a single thermally activated deformation process due to a single dislocation mechanism, Hartley [2.35] demonstrated how the microscale parameters describing the process can lead to an interpretation of the viscoplastic potential in terms of parameters determined by the model of the dislocation–barrier interaction. Similar models can be constructed for creep, or high-temperature deformation under constant stress, where the primary mechanism of dislocation motion is climb, or motion normal to the glide plane. Constructing a model of this process that can be inserted into a continuum description of deformation includes several challenging components. First, models must be constructed of the interaction of individual dislocations with various obstacles at the scale of individual obstacles. These models must then be scaled up to include the interaction of a distribution of dislocations with a distribution of obstacles on a slip plane. The interaction of dislocation motion on the various slip planes in a crystal follows at the next scale, and finally the distribution of crystals and their associate slip planes throughout the material must be established. At each level, appropriate statistical models of the microscopic quantities must be constructed and related to the adjacent scales of structure. This process of multiscale modeling has received much attention in the past decade. Attempts to relate previous knowledge of dislocation theory and viscoplasticity have benefited considerable from advances in the computational speed and capacity of supercomputers. Present capabilities still fall short of permitting engineering properties

to be calculated from inputs based entirely on microscopic models of dislocation processes. Nevertheless, considerable progress has been made in this area and the inclusion of microscale models of deformation processes in engineering design codes promises to add a significant component of material design to the process in the future.

2.2.4 Transport Properties of Materials When a system is not in the same state throughout, gradients of free energy exist, causing fluxes of energy that tend to eliminate the gradients. The free energy gradients are thermodynamic forces and the resulting fluxes can include matter, energy or electric charge depending on the nature of the force. Although the description of the relationship between forces and fluxes is called irreversible thermodynamics [2.26], it is important to recognize that systems in which such forces occur are not in thermodynamic equilibrium [2.26]. Phenomenological equations similar to those defining equilibrium and dissipative properties define properties that relate forces and fluxes. Thermal Transport A difference in temperature between various parts of a system causes heat to flow. In this case, the thermodynamic force is the temperature gradient. A vector represents the heat flux. The magnitude of the vector is equal to the flux of heat flowing across unit area normal to the vector and the direction of the vector is the direction of heat flow. In general, the heat flux vector need not be parallel to the temperature gradient. The dependence of the heat flux on the temperature gradient defines the thermal conductivity, k, by the relationship

qi = −kij T j ,

(2.32)

where the suffix j refers to partial differentiation with respect to the spatial variable x j . The negative sign specifies that heat flows from regions of higher temperature to regions of lower temperature. The thermal resistivity tensor r is the inverse of k; both are symmetric second-rank tensors. Principal values of the conductivity tensor are positive for all known materials. The conductivity ellipsoid is the quadric surface [2.24] k () = kij i j

(2.33)

such that the thermal conductivity k() in the direction of the unit vector is the reciprocal of the square root

43

Part A 2.2

k=1

2.2 Properties of Materials

44

Part A

Solid Mechanics Topics

of the radius vector from the origin to the surface in the direction of . By Neumann’s principle, the symmetry elements of the conductivity tensor contain those of the point group of any crystalline material that they describe. Isotropic thermal conductivity tensors describe a material that is isotropic with respect to heat flow.

Part A 2.2

Electrical Charge Transport The treatment of electrical conductivity in anisotropic materials is formally identical to that for thermal conductivity. In the former case, the flux is electrical charge instead of heat and the corresponding force is the gradient of electrical potential ϕ I = −E i . The expression

ji = −σik ϕk = σik E k

(2.34)

defines the electrical conductivity tensor σ in terms of the current density j and E. Equation (2.34) is Ohm’s law for a material exhibiting general anisotropy with respect to electrical conductivity. The electrical resistivity tensor ρ is the inverse of σ. All comments in the previous section about the symmetry properties and representation of the thermal conductivity and resistivity tensors apply to the corresponding tensors of electrical properties. The preceding discussion describes the conduction of heat and electrical charge as independent processes. However, when both occur together, they are coupled in a fashion similar to that describing equilibrium properties. For a description of the phenomenon of thermoelectricity and the associated material properties, the interested reader can consult [2.25, Chap. 12]. Mass Transport Engineering materials produced by ordinary manufacturing processes are never in their equilibrium state. A typical example is the segregation of components in an alloy prepared by solidification from the liquid state. The segregation is due to nonuniform cooling and compositional differences between the liquid and the first solids that form on solidification. It is generally desirable to eliminate this segregation to form a material with more uniform properties than the as-cast structure exhibits. This modification is possible because of the tendency of the material to remove spatial variations in composition under appropriate conditions by diffusion, a solid-state mass transport process. Two types of diffusion occur in materials. Stochastic, or random, diffusion occurs because of the random thermal motion of atoms or molecules in a substance. In the absence of any force biasing the random motion, the mean position of the diffusing entity will not change

with time. However, its root-mean-square (RMS) displacement from its origin will not vanish. This means that at any instant t after measurement begins, the displacement of the entity from its starting point may not be zero. In three dimensions, the RMS displacement u(t) is √ u(t) = 6Dt , (2.35) where D is the diffusion coefficient for the diffusing species. The diffusion coefficient depends strongly on temperature and the mechanism by which the diffusing entity moves through the material. Elementary arguments based on a model of diffusion as a thermally activated process lead to the expression of the Arrhenius form for the diffusion coefficient [2.38] −ΔH ∗ (2.36) , D = D0 exp kT where ΔH ∗ is the enthalpy of activation and D0 is a factor containing the square of the distance moved in a single jump of the diffusing entity, the frequency of jumps, a geometric factor, and a term related to the entropy of activation. Self-diffusion is synonymous with stochastic diffusion. Thermodynamic forces existing in materials bias stochastic diffusion, causing chemical diffusion, a net flux of matter that tends to eliminate gradients in the chemical potential of each component. For the purpose of describing diffusion, the chemical potential μ(i) of component i in a system is the partial derivative of the Gibbs free energy of the system with respect to the number of atoms or molecules of component i present, holding constant the temperature, stress state, and number of atoms of all other components. Multicomponent systems can exhibit fluxes of each component due to gradients in the chemical potentials of all components. In chemical diffusion, the force conjugate to the flux of each component is the gradient of the chemical potential of that component. Gradients of other fields, such as temperature and electrical potential, can also cause fluxes of the components. The description of diffusion including these cross effects is formally similar to that for thermoelectricity, giving rise to the definition of the thermal diffusion coefficient, electrostatic diffusion coefficient, and related properties. The following section presents a brief treatment of the principal equations and concepts in chemical diffusion in a two-component system. Consider a binary system of components A and B in which equal and opposite fluxes of the components J (K ) , where K takes on the values A and B, occur

Materials Science for the Experimental Mechanist

parallel to a characteristic direction xi . Fick’s first law assumes that the flux of each component is proportional to the concentration gradient and occurs in the opposite direction Ji(A) = − D˜

∂NA , ∂xi

(2.37)

Property Change Through Heat Treatment One of the most useful features of many structural materials, particularly metals and alloys, is the ability to alter the microstructure so that selected physical and mechanical properties can be significantly changed without substantially affecting the external shape. This change is usually accomplished through heat treatment, which is the exposure of a material to an elevated temperature, generally somewhere between the proposed service temperature and the melting point of the material and often in a protective atmosphere to prevent adverse interaction with the environment. The extent of the exposure must be determined empirically to effect desired changes of properties without undesirable side effects. During the exposure internal stresses may be relieved by

the nucleation and growth of new, stress-free grains (recrystallization), preceded by the mutual annihilation of pre-existing dislocations (recovery); metastable phases present may transform to more stable forms and compositional variations at the microscale can be diminished by interdiffusion of components of the material. The three examples below illustrate such changes. Many structural metals and alloys are prepared for use by a sequence of processes beginning with the solidification of an ingot from a molten source, followed by several stages of substantial permanent deformation, often alternated with heat treatments, to arrive at a final form suitable for the fabrication of engineering structures. The deformation generally begins at elevated temperatures (above 0.5Tm , where Tm is the absolute melting temperature) where the deformation can be accomplished at lower loads. This process also breaks up the grain structure of the cast ingot, which is highly anisotropic. The elevated temperatures at which deformation is conducted promote mass transport of alloying elements, which promotes compositional homogeneity by diminishing local variations in the material. Since interaction with the environment, resulting in undesirable consequences such as surface roughness, accompanies deformation at elevated temperatures, final processing occurs at near-ambient temperatures. This low-temperature deformation introduces work-hardening, which may occur to such an extent that additional processing is not possible without damage to the material. In such cases, intermediate heat treatments that cause recovery and recrystallization in the material provide a means for lowering the flow stress to a point where additional cold work can be accomplished without fracture of the material. Changes in the internal structure of materials by altering the number, amount, and composition of phases present is a common result of heat treatment. Both the hardening of steel and precipitation hardening of aluminum alloys are examples of this process. The hardening of steel results from the fact that iron undergoes a transformation of crystal structure from body-centered cubic to face-centered cubic on heating from ambient temperature to above 910 ◦ C. When carbon is added, the bcc form of iron, ferrite, has a lower solubility limit than the fcc form, austenite. Thus an alloy that may be single phase at an elevated temperature will tend to become two-phase at ambient temperature as the structure transforms from austenite to ferrite. If the alloy is cooled slowly through the transforma-

45

Part A 2.2

where NA is the atomic fraction of component A, and D˜ is the coefficient of chemical diffusion, also known as the interdiffusion coefficient. The interdiffusion coefficient contains the thermodynamic factor relating the concentration to the chemical potential of component A. In many systems, experiments show that the fluxes of two species are not equal and opposite due to the accompanying flux of the point defects responsible for diffusion at the nanoscale. Although the gradient in defect concentration is generally negligible relative to that of the other diffusing species, the defect flux can be similar in magnitude to the fluxes of the other diffusing species. The phenomenological treatment of this effect describes the flux of each component in terms of a relation similar to (2.37) with the interdiffusion coefficient replaced by an intrinsic diffusion coefficient D(K) which contains the thermodynamic factor relating the chemical potential to the concentration of component K. Neglecting the defect concentration relative to those of the atomic or molecular species requires that N (A) + N (B) = 1, and the gradients are equal and opposite. Then (2.32) describes the flux of either species with the interdiffusion coefficient replaced by (N (A) D(B) + N (B) D(A) ). For additional information on experimental and theoretical treatment of this topic, the interested reader can consult the excellent book by Glicksman [2.38].

2.2 Properties of Materials

46

Part A

Solid Mechanics Topics

Part A 2.2

tion temperature range, this occurs by the formation of a second phase, Fe3 C or cementite, which appears in the form of an eutectoid, pearlite, consisting of alternating lamellae of cementite and ferrite. However if the alloy is cooled rapidly (quenched) from the higher temperature, the precipitation has insufficient time to occur and the excess carbon is trapped in the bcc lattice, distorting it to form a metastable, bodycentered tetragonal structure (martensite). Martensite is considerably harder than ferrite or pearlite, so the resulting alloy has substantially higher yield strength that a material of identical composition that has been slowly cooled from the same elevated temperature. Since martensite is often too brittle for use in the as-quenched condition, subsequent heat treatments below the transformation temperature can further alter the structure and increase toughness by reducing internal stresses without significantly affecting the yield strength. Precipitation hardening has similarities to the heat treatment of steel, but does not require a phase change of the pure element that is the base of the alloy system. This process is most commonly employed to strengthen aluminum alloys. The effect exploits the fact that the solubility of most alloying elements in a metal increases with increasing temperature. Thus, in the appropriate range of composition, an alloy that is single phase at an elevated temperature may have an equilibrium structure consisting of two or more phases at ambient temperature. Heat treatment to cause precipitation hardening consists of heating such an alloy to a temperature in the single-phase region, followed by quenching to ambient temperature to retain the singlephase structure, then reheating to a lower temperature in the multiphase region and holding for an empirically determined time (aging) to permit the structure to approach the equilibrium phase structure. In the early stage of aging metastable proto-precipitates are formed, which strain the lattice of the matrix material and increase the yield stress. As aging progresses and precipitates grow from these nuclei, internal stresses diminish, causing the yield stress to decrease, approaching the value of the as-quenched, single-phase solid solution. Processing of these materials is designed to conduct the aging treatment at temperatures and times that optimize the combination of strength, ductility, and toughness required for intended applications. Service temperatures for precipitation hardened alloys must be sufficiently below the aging temperatures so that additional unintended changes in properties do not occur during use.

2.2.5 Measurement Principles for Material Properties The measured properties of materials and the use of these measurements in various experimental mechanics applications require an appreciation of the limitations and dependencies described in the previous sections. The physical foundations of phenomenological relationships connecting structure and properties of materials comprise the preceding sections of this chapter. This connection leads to the following general principles, which call attention to the need for appropriate qualification of experimental measurements involving the properties of materials. Isolation In conducting experiments where measurements of conjugate fields provide data for the calculation of material properties, either isolate the system from additional fields that can affect the results or hold all but the conjugate fields constant. Without isolation or a constant environment of other fields, the measured properties will include cross-effects giving an incorrect value for the property associated with the conjugate fields. The same principle applies when specifically measuring cross-effects. Structure Consistency When measuring a material property or using the value of a material property in a measurement, ensure that samples to which the property applies are structurally identical. This is especially true of structure-sensitive properties, such as yield strength. However, variations in microstructure due to varying conditions of preparation can also affect other properties, such as electrical resistivity, sufficiently to cause inconsistencies in measurements with different samples. Quantitative Specification of Structure When measuring a material property or employing a material as an element in a measurement system, describe the composition and microstructural condition of the material in sufficient quantitative detail that subsequent similar experiments can employ structurally comparable material. Failure to observe this principle will introduce an unaccountable element of variability in the results of apparently comparable experiments. Consistency of Assumptions In analyzing results of measurements involving materials, ensure that the conditions of measurement

Materials Science for the Experimental Mechanist

satisfy the assumptions employed in the calculation. For example, the calculation of internal stress from measured values of strain must employ lattice strain or acknowledge the fact that the use of measured values of total strain implicitly assume that no plastic deforma-

References

tion occurs during the measurements. Only diffraction techniques measure lattice strain. Other techniques that use surface strains must satisfy the conditions under which these strains equal lattice strains, i. e., negligible dislocation motion.

2.2 2.3 2.4

2.5 2.6 2.7

2.8

2.9

2.10

2.11

2.12 2.13 2.14 2.15

2.16

2.17

J.F. Shackelford: Introduction to Materials Science for Engineers, 6th edn. (Pearson-Prentice Hall, New York 2004) D.W. Callister: Materials Science and Engineering An Introduction, 6th edn. (Wiley, New York 2003) D.R. Askeland: The Science and Engineering of Materials, 2nd edn. (Thomson, Stanford 1992) M.F. Ashby, D.R. Jones: Engineering Materials 1: An Introduction to Their Properties and Applications, 2nd edn. (Butterworth Heinemann, Oxford 2000) J.P. Hirth, J. Lothe: Theory of Dislocations, 2nd edn. (Wiley, New York 1982) F.R.N. Nabarro, J.P. Hirth (Eds.): Dislocations in Solids, Vol. 12 (Elsevier, Amsterdam 2004) C.S. Barrett, W.D. Nix, A. Tetleman: Principles of Engineering Materials (Prentice-Hall, Englewood Cliffs 1973) T.B. Massalski, H. Okamoto, P.R. Subramanian, L. Kacprzak: Binary Alloy Phase Diagrams, 2nd edn. (ASM Int., Materials Park 1990) R.S. Roth, T. Negas, L.P. Cook: Phase Diagrams for Ceramists, Vol. 5 (American Ceramic Society, Inc., 1983) P. Villers, A. Prince, H. Okamoto: Handbook of Ternary Alloy Phase Diagrams (ASM Int., Materials Park 1995) J. Goldstein, D.E. Newbury, P. Echlin, D.C. Joy, A.D. Romig Jr., C.E. Lyman, C. Fiori, E. Lifshin: Scanning Electron Microscopy and X-Ray Microanalysis, 2nd edn. (Springer, New York 1992) R.T. DeHoff, F.N. Rhines: Quantitative Microscopy (McGraw-Hill, New York 1968) B.D. Cullity: Elements of X-ray Diffraction, 2nd edn. (Addison-Wesley, Reading 1978) B.E. Warren: X-ray Diffraction (General, New York 1990) D.K. Smith, R. Jenkins: The powder diffraction file: Past, present and future, Rigaku J. 6(2), 3–14 (1989) W.I.F. David, K. Shankland, L.B. McCusker, C. Baerlocher (Eds.): Structure Determination from Powder Diffraction Data, International Union of Crystallography (Oxford Univ. Press, New York 2002) I.C. Noyan, J.B. Cohen, B. Ilschner, N.J. Grant: Residual Stress Measurement by Diffraction and Interpretation (Springer, New York 1989)

2.18 2.19

2.20

2.21

2.22

2.23

2.24

2.25 2.26 2.27 2.28

2.29 2.30 2.31 2.32 2.33 2.34

2.35

C.S. Hartley: International Conference on Experimental Mechanics, ICEM12-12th (Italy 2004) D.K. Bowen, B.K. Tanner: High Resolution X-ray Diffractometry and Topography (Taylor Francis, New York 1998) L. Cartz: Nondestructive Testing Radiography, Ultrasonics, Liquid Penetrant, Magnetic Particle, Eddy Current (ASM Int., Materials Park 1995) D.B. Williams, C.B. Carter: Transmission electron microscopy: a textbook for materials science (Kluwer Academic, New York, 1996) B.L. Adams, S.I. Wright, K. Kunze: Orientation imaging: The emergence of a new microscopy, Met. Mat. Trans. A 24, 819–831 (1993) J.A. James, J.R. Santistban, L. Edwards, M.R. Daymond: A virtual laboratory for neutron and synchrotron strain scanning, Physica B 350(1–3), 743–746 (2004) L.E. Malvern: Introduction to the Mechanics of a Continuous Medium (Prentice-Hall, Englewood Cliffs 1969) J.F. Nye: Physical Properties of Crystals (Clarendon, Oxford 1985) S.M. DeGroot: Thermodynamics of Irreversible Processes (North-Holland, Amsterdam 1951) M.P. Marder: Condensed Matter Physics (Wiley, New York 2000) W.G. Cady: Piezoelectricity: An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals (Dover, New York 1964) C.S. Hartley: Single crystal elastic moduli of disordered cubic alloys, Acta Mater. 51, 1373–1391 (2003) C. Zener: Elasticity and Anelasticity of Metals (Univ. Chicago Press, Chicago 1948) J.D. Ferry: Viscoelastic Properties of Polymers, 3rd edn. (Wiley, New York 1980) B. A. Bilby: Continuous Distributions of Dislocations, Progress in Solid Mechanics (1960) J.D. Eshelby: The continuum theory of lattice defects, Solid State Phys. 3, 79–144 (1956) E. Kröner: Continuum theory of defects. In: Les Houches, Session XXXV, 1980 – Physics of Defects, ed. by R. Balian, M. Kleman, J.-P. Poirer (NorthHolland, Amsterdam 1981) pp. 282–315 C.S. Hartley: A method for linking thermally activated dislocation mechanisms of yielding with

Part A 2

References 2.1

47

48

Part A

Solid Mechanics Topics

2.36

continuum plasticity theory, Philos. Mag. 83, 3783– 3808 (2003) R. Asaro: Crystal Plasticity, J. Appl. Mech. 50, 921–934 (1983)

2.37 2.38

J. Lubliner: Plasticity Theory (Macmillan, New York 1990) M.E. Glicksman: Diffusion in Solids (Wiley, New York 2000)

Part A 2

49

Mechanics of 3. Mechanics of Polymers: Viscoelasticity

Wolfgang G. Knauss, Igor Emri, Hongbing Lu

3.1

3.2

Historical Background ........................... 3.1.1 The Building Blocks of the Theory of Viscoelasticity ..........................

49

Linear Viscoelasticity ............................. 3.2.1 A Simple Linear Concept: Response to a Step-Function Input .............. 3.2.2 Specific Constitutive Responses (Isotropic Solids) .......................... 3.2.3 Mathematical Representation of the Relaxation and Creep Functions 3.2.4 General Constitutive Law for Linear and Isotropic Solid: Poisson Effect .. 3.2.5 Spectral and Functional Representations ...........................

51

3.2.6 Special Stress or Strain Histories Related to Material Characterization 3.2.7 Dissipation Under Cyclical Deformation............ 3.2.8 Temperature Effects ...................... 3.2.9 The Effect of Pressure on Viscoelastic Behavior of Rubbery Solids ......................... 3.2.10 The Effect of Moisture and Solvents on Viscoelastic Behavior................ 3.3

3.4

50

51

Measurements and Methods .................. 3.3.1 Laboratory Concerns ..................... 3.3.2 Volumetric (Bulk) Response ........... 3.3.3 The CEM Measuring System ............ 3.3.4 Nano/Microindentation for Measurements of Viscoelastic Properties of Small Amounts of Material......... 3.3.5 Photoviscoelasticity ...................... Nonlinearly Viscoelastic Material Characterization ................................... 3.4.1 Visual Assessment of Nonlinear Behavior................... 3.4.2 Characterization of Nonlinearly Viscoelastic Behavior Under Biaxial Stress States ............

56 63 63

68 69 69 70 71 74

76 83 84 84

85

53

3.5

Closing Remarks ...................................

89

53 55

3.6 Recognizing Viscoelastic Solutions if the Elastic Solution is Known.............. 3.6.1 Further Reading ...........................

90 90

55

References ..................................................

92

3.1 Historical Background During the past five decades the use of polymers has seen a tremendous rise in engineering applications. This growing acceptance of a variety of polymer-based de-

signs derives in part from the ease with which these materials can be formed into virtually any shape, and in part because of their generally excellent performance

Part A 3

With the heavy influx of polymers into engineering designs their special, deformation-rate-sensitive properties require particular attention. Although we often refer to them as time-dependent materials, their properties really do not depend on time, but time histories factor prominently in the responses of polymeric components or structures. Structural responses involving time-dependent materials cannot be assessed by simply substituting time-dependent modulus functions for their elastic counterparts. The outline provided here is intended to provide guidance to the experimentally inclined researcher who is not thoroughly familiar with how these materials behave, but needs to be aware of these materials because laboratory life and applications today invariably involve their use.

50

Part A

Solid Mechanics Topics

Part A 3.1

in otherwise normally corrosive environments. This recent emergence is driven by our evolving capabilities during the last seven decades to synthesize polymers in great variety and to address their processing into useful shapes. Historically polymers have played a significant role in human developments, as illustrated by the introductory comments in [3.1]. Of great consequence for the survival or dominance of tribes or nations was the development of animal-derived adhesives for the construction of high-performance bows, starting with the American Indian of the Northwest through the developments by the Tartars and leading to the extraordinary military exploits of the Turks in the latter Middle Ages [3.2]. In principle, these very old methods of producing weaponry continue to aid today in the construction of modern aerospace structures. While the current technology still uses principles exploited by our ancestors many years ago, the advent of the synthetic polymers has provided a plethora of properties available for a vast range of different engineering designs. This range of properties is, indeed, so large that empirical methods are no longer sufficient to effect reliable engineering developments but must now be supported by optimum analytical methods to aid in the design process. One characteristic of polymers is their relative sensitivity to load exposure for extended periods of time or to the rate of deformations imposed on them. This behavior is usually and widely combined under the concept of viscoelastic behavior, though it is sometimes characterized as representing fading memory of the material. These time-sensitive characteristics typically extend over many decades of the time scale and characteristically set polymers apart from the normal engineering metals. While the strain-rate sensitivity [3.3] and the time dependence of failure in metals [3.4] are recognized and creep as well as creep rupture [3.5–10] of metals is well documented, one finds that the incorporation of rate-dependent material properties into models of time-dependent crack growth – other than fatigue of intrinsically rate-insensitive materials – still stands on a relatively weak foundation. Metallic glasses (i.e., amorphous metals) are relatively newcomers to the pool of engineering materials. Their physical properties are at the beginning of exploration, but it is already becoming clear through initial studies [3.11, 12] that their amorphous structure endows them with properties many of which closely resemble those of amorphous polymers. While these developments are essentially in their infancy at this time it is

well to bear in mind that certain parts of the following exposition are also applicable to these materials. Because the emphasis in this volume is placed on experimental methods, rather than on stress analysis methods, only a cursory review of the linearized theory of viscoelasticity is included. For the reader’s educational benefit a number of books and papers have been listed in the Further Reading section, which can serve as resources for a more in-depth treatment. This review of material description and analysis is thus guided by particular deformation histories as a background for measurements addressing material characterization to be used in engineering design applications. Although the nonlinearly viscoelastic characteristic of these materials are not well understood in a general, three-dimensional setting, we include some reference to these characteristics in the hope that this understanding will assist the experimentalist with properly interpreting laboratory measurements.

3.1.1 The Building Blocks of the Theory of Viscoelasticity Forces are subject to the laws of Newtonian mechanics, and are, accordingly, governed by the classical laws of motion. While relativistic effects have been studied in connection with deforming solids, such concerns are suppressed in the present context. Many texts deal with Newtonian mechanics to various degrees of sophistication so that only a statement of the necessary terminology is required for the present purposes. In the interest of brevity we thus dispense with a detailed presentation of the analysis of stress and of the analysis of strain, except for summarizing notational conventions and defining certain variables commonly understood in the context of the linear theory of elasticity. We adhere to the common notation of the Greek letters τ and ε denoting stress and strain, respectively. Repeated indices on components imply summation; identical subscripts (e.g., τ11 ) denote normal components and different ones shear (e.g., τ12 ). The dilatational components of stress, τii , are often written as σkk , with the strain complement being εkk . Because the viscoelastic constitutive description is readily expressed in terms of deviatoric and dilatational components, it is necessary to recall the components Sij of the deviatoric stress as 1 Sij = τij − τkk · δij , 3

(3.1)

where δij denotes the Kronecker operator. Similarly, the corresponding deviatoric strain e is written in compo-

Mechanics of Polymers: Viscoelasticity

nent form as 1 eij = εij − εkk · δij . (3.2) 3 For further definitions and derivations of measures of stress or strain the reader is referred to typical texts.

3.2 Linear Viscoelasticity

51

The remaining building block of the theory consists of the constitutive behavior, which differentiates viscoelastic materials from elastic ones. The next section is devoted to a brief definition of linearly viscoelastic material behavior.

3.2 Linear Viscoelasticity face (boundary) of a viscoelastic solid. Specification of such a quantity under uniaxial relaxation is not particularly useful, except to note that in the limit of short (glassy) response its value is a limit constant, and also under long-term conditions when the equilibrium (or rubbery) modulus is effective, in which case the Poisson’s ratio is very close to 0.5 (incompressibility).

3.2.1 A Simple Linear Concept: Response to a Step-Function Input It is convenient for instructional purposes to consider that the stress can be described, so that the strain follows from the stress. The reverse may hold with equal validity. In general, of course, neither may be prescribed a priori, and a general connection relates them. The structure of the linear theory must be completely symmetric in the sense that the mathematical formulation applies to these relations regardless of which variable is considered the prescribed or the derived one. For introductory purposes we shall use, therefore, the concept of a cause c(t) (input) and an effect e(t) (output) that are connected by a functional relationship. The latter must be linear with respect to (a) the amplitude (additivity with respect to magnitude) and (b) time in the sense that they obey additivity independent of time. It is primarily a matter of convenience that the cause-and-effect relation is typically expressed with the aid of a step-function cause. Other representations are

Table 3.1 Nomenclature for viscoelastic material functions Type of loading Shear

Bulk

Uniaxial extension

μ(t) J(t) μ (ω) μ (ω) J (ω) J (ω)

K (t) M(t) K (ω) K (ω) M (ω) M (ω)

E(t) D(t) E (ω) E (ω) D (ω) D (ω)

Mode Quasistatic

Relaxation Creep Strain prescribed

Harmonic Stress prescribed

Storage Loss Storage Loss

Part A 3.2

The framework for describing linearly viscoelastic material behavior, as used effectively for engineering applications, is phenomenological. It is based mathematically on either an integral or differential formulation with the material representation described realistically in numerical (tabular) or functional form(s). The fundamental equations governing the linearized theory of viscoelasticity are the same as those for the linearized theory of elasticity, except that the generalized Hooke’s law of elasticity is replaced by a constitutive description that is sensitive to the material’s (past) history of loading or deformation. It will be the purpose of the immediately subsequent subsections to summarize this formalism of material description in preparation for various forms of material characterization. Little or no reference is made to general solution methods for viscoelastic boundary value problems. For this purpose the reader is referred to the few texts available as listed in Sect. 3.6.1. Rather than repeating the theory as already outlined closely in [3.13] we summarize below the concepts and equations most necessary for experimental work; if necessary, the reader may consult the initially cited reference(s) (Sect. 3.6.1) for a more expansive treatment. In brief, the viscoelastic material functions of first-order interest are given in Table 3.1. Note the absence of a generic viscoelastic Poisson function, because that particular response is a functional of the deformation or stress history applied to the sur-

52

Part A

Solid Mechanics Topics

Part A 3.2

feasible and we shall address a common one (steadystate harmonic) later on as a special case. For now, let E(t, t1 ) represent a time-dependent effect that results from a step cause c(t1 ) = h(t1 ) of unit amplitude imposed at time t1 ; h(t1 ) denotes the Heaviside step function applied at time t1 . For the present we are concerned only with nonaging materials, i. e. with materials, the intrinsic properties of which do not change with time. (With this definition in mind it is clear that the nomenclature timedependent materials in place of viscoelastic materials is really a misnomer; but that terminology is widely used, nevertheless.) We can assert then that for a non-aging material any linearity of operation, or relation between an effect and its cause, requires satisfaction of Postulate (a): proportionality with respect to amplitude, and Postulate (b): additivity of effects independent of the time sequence, when the corresponding causes are added, regardless of the respective application times. Condition (a) states that, if the cause c(0) elicits E(t, 0), then a cause of different amplitude, say c1 (0) ≡ α · h(0), with α a constant, elicits a response α · E(t, 0). Under the non-aging restriction this relation is to be independent of the time when the cause starts to act, so that c1 (t1 ) ≡ α · h(t1 ) → αE(t, t1 ); t > t1 also holds. This means simply that the response–effect relation shown in the upper part of Fig. 3.1 holds also for a different time t2 , which occurs later in time than t1 .

Condition (b) entails then that, if two causes c1 (t1 ) ≡ α1 · h(t1 ) and c2 (t2 ) ≡ α2 · h(t2 ), imposed at different times t1 and, t2 act jointly, then their corresponding effects α1 · E(t, t1 ) and α2 · E(t, t2 ) is their sum while observing their proper time sequence. Let the common time scale start at t = 0; then the combined effect, say e(t), is expressed by c(t) ≡ c1 (t1 ) + c2 (t2 ) → e(t) = α1 · E(t − t1 ) + α1 · E(t − t2 ) . (3.3)

Specifically, here the first response does not start until the time t1 is reached, and the response due to the second cause is not experienced until time t2 , as illustrated in Fig. 3.1. Having established the addition process for two causes and their responses, the extension to an arbitrary number of discrete step causes is clearly recognized as a corresponding sum for the collective effects e(tn ), up to time t, in the generalized form of (3.3), namely (3.4) e(t) = αn E(t − tn ) ; (tn < t) . This result may be further generalized for causes represented by a continuous cause function of time, say c(t). To this end consider a continuously varying function c(t) decomposed into an initially discrete approximation of steps of finite (small) amplitudes. With the intent of ultimately proceeding to the limit of infinitesimal steps, note that the amplitude of an individual step amplitude at, say, time τn is given by (3.5) αn → Δc(τn ) = (Δc/Δτ) Δτ , τn

Stress

which, when substituted into (3.4), leads to e(t) = E(t − tn )(Δc/Δτ) Δτ .

Strain

τn

t1

Time

Stress

t1

Time

Strain

In the limit n → ∞ (Δτ → dτ), the sum Δc e(t) = lim E(t − tn ) Δτ , Δτ τn

(3.7)

passes over into the integral 2

t

2 1

(3.6)

e(t) =

1

E(t − τ)

dc(τ) dτ . dτ

(3.8)

0

t1

t2

Time

t1

t2

Time

Fig. 3.1 Additivity of prescribed stress steps and corres-

ponding addition of responses

Inasmuch as this expression can contain the effect of a step-function contribution at zero time of magnitude c(0), this fact can be expressed explicitly through the

Mechanics of Polymers: Viscoelasticity

alternate notation

3.2 Linear Viscoelasticity

53

effect, one obtains the inverse relation(s) t

e(t) = c(0)E(t) +

E(t − τ)

0+

dc(τ) dτ , dτ

(3.9)

3.2.2 Specific Constitutive Responses (Isotropic Solids) For illustrative purposes and to keep the discussion within limits, the following considerations are limited to isotropic materials. Recalling that the stress and strain states may be decomposed into shear and dilatational contributions (deviatoric and dilatational components), we deal first with the shear response followed by the volumetric part. Thermal characterization will then be dealt with subsequently. Shear Response Let τ denote any shear stress component and ε its corresponding shear strain. Consider ε to be the cause and τ its effect. Denote the material characteristic E(t) for unit step excitation from Sect. 3.2.1 in the present shear context by μ(t). This function will be henceforth identified as the relaxation modulus in shear (for an isotropic material). It follows then from (3.8) and (3.9) that t dε(ξ) (3.10) dξ τ(t) = 2 μ(t − ξ) dξ 0

t = 2ε(0)μ(t) + 2 0+

μ(t − ξ)

dε(ξ) dξ . dξ

(3.11)

The factor of 2 in the shear response is consistent with elasticity theory, inasmuch as in the limits of short- and long-term behavior all viscoelasticity relations must revert to the elastic counterparts. If one interchanges the cause and effect by letting the shear stress represent the cause, and the strain the

t J(t − ξ)

dτ dξ dξ

(3.12)

0

t

1 1 = τ(0)J(t) + 2 2

J(t − ξ) 0+

dτ dξ , dξ

(3.13)

where now the function E ≡ J(t) is called the shear creep compliance, which represents the creep response of the material in shear under application of a step shear stress of unit magnitude as the cause. Bulk or Dilatation Response Let εii (t) represent the first strain invariant and σ jj (t) the corresponding stress invariant. The latter is recognized as three times the pressure P(t), i. e., σ jj (t) ≡ 3P(t). In completely analogous fashion to (3.12) and (3.13) the bulk behavior, governed by the bulk relaxation modulus K (t) ≡ E(t), is represented by

t σ jj = 3

K (t − ξ)

dεii (ξ) dξ dξ

(3.14)

0

t = 3εii (0)K (t) + 3

K (t − ξ)

0+

dεii (ξ) dξii . dξ (3.15)

Similarly, one writes the inverse relation as 1 εii = 3

t M(t − ξ)

dσ jj (ξ) dξ dξ

(3.16)

0

1 1 = σ jj (0)M(t) + 3 3

t M(t − ξ) 0+

dσ jj (ξ) dξii , dξ (3.17)

where the function M(t) ≡ E(t) represents now the dilatational creep compliance (or bulk creep compliance); in physical terms, this is the time-dependent fractional volume change resulting from the imposition of a unit step pressure.

3.2.3 Mathematical Representation of the Relaxation and Creep Functions Various mathematical forms have been suggested and used to represent the material property functions

Part A 3.2

where the lower integral limit 0+ merely indicates that the integration starts at infinitesimally positive time so as to exclude the discontinuity at zero. Alternatively, the same result follows from observing that for a step discontinuity in c(t) the derivative in (3.5) is represented by the Dirac delta function δ(t). In fact, this latter remark holds for any jump discontinuity in c(t) at any time, after and including any at t = 0. In mathematical terms this form is recognized as a convolution integral, which in the context of the dynamic (vibration) response of linear systems is also known as the Duhamel integral.

1 ε(t) = 2

54

Part A

Solid Mechanics Topics

Part A 3.2

analytically. Preferred forms have evolved, with precision being balanced against ease of mathematical use or a minimum number of parameters required. All viscoelastic material functions possess the common characteristic that they vary monotonically with time: relaxation functions decreasing and creep functions increasing monotonically. A second characteristic of realistic material behavior is that time is (almost) invariably measured in terms of (base 10) logarithmic units of time. Thus changes in viscoelastic response may appear to be minor when considered as a function of the real time, but substantial if viewed against a logarithmic time scale. Early representations of viscoelastic responses were closely allied with (simple) mechanical analog models (Kelvin, Voigt) or their derivatives. Without delving into the details of this evolutionary process, their generalization to broader time frames led to the spectral representation of viscoelastic properties, so that it is useful to present only the rudiments of that development. The building blocks of the analog models are the Maxwell and the Voigt models illustrated in Fig. 3.2a,b. In this modeling a mechanical force F corresponds to the shear stress τ and, similarly, a displacement/deflection δ corresponds to a strain ε. Under a stepwise applied deformation of magnitude ε0 – separating the force-application points in the Maxwell model – the stress (force) abates or relaxes by the relation τ(t) = ε0 μm exp(−t/ξ) , a)

b)

F

(3.18)

c)

F

μ μ

η

η

μ1

η1

μ2

η2

μ3

η3

F

F F

d) μ∞

F

μ1

μ2

μ3

μj

η1

η2

η3

ηj

F

where ξ = ηm /μm is called the (single) relaxation time. Similarly, applying a step stress (force) of magnitude to the Voigt element engenders a time-dependent separation (strain) of the force-application points described by τ0 [1 − exp(−t/ς)] , (3.19) ε(t) = μν where ς = ην /μν is now called the retardation time since it governs the rate of retarded or delayed motion. Note that this representation is used for illustration purposes here and that the retardation time for the Voigt material is not necessarily meant to be equal to the relaxation time of the Maxwell solid. It can also be easily shown that this is not true for a standard linear solid either. By inductive reasoning, that statement holds for arbitrarily complex analog models. The relaxation modulus and creep compliance commensurate with (3.18) (Maxwell model) and (3.19) (Voigt model) for the Wiechert and Kelvin models (Fig. 3.2c,d) are, respectively μn exp(−t/ξn ) (3.20) μ(t) = μ∞ + n

and J(t) = Jg +

Jn [1 − exp(−t/ςn )] + η0 t ,

where Jg and η0 arise from letting η1 → 0 (the first Voigt element degenerates to a spring) and μn → 0 (the last Voigt element degenerates to a dashpot). These series representations with exponentials are often referred to as Prony series. As the number of relaxation times increases indefinitely, the generalization of the expression for the shear relaxation modulus, becomes ∞ dξ (3.22) , μ(t) = μ∞ + H (ξ) exp(−t/ξ) ξ 0

where the function H (ξ) is called the distribution function of the relaxation times, or relaxation spectrum, for short; the creep counterpart presents itself with the help of the retardation spectrum L(ζ ) as ∞

μj

ηj

J(t) = Jg +

L(ζ )[1 − exp(−t/ζ )] 0

F

Fig. 3.2a–d Mechanical analogue models: (a) Maxwell, (b) Voigt, (c) Wiechert, and (d) Kelvin

(3.21)

n

dζ + ηt , ζ (3.23)

Note that although the relaxation times ξ and the retardation times ζ do not, strictly speaking, extend over the range from zero to infinity, the integration limits are so

Mechanics of Polymers: Viscoelasticity

3.2 Linear Viscoelasticity

assigned for convenience since the functions H and L can always be chosen to be zero in the corresponding part of the infinite range.

3.2.5 Spectral and Functional Representations

3.2.4 General Constitutive Law for Linear and Isotropic Solid: Poisson Effect

A discrete relaxation spectrum in the form H (ξ) = μn ξn δ(ξn ) ,

One combines the shear and bulk behavior exemplified in (3.13), (3.16) and (3.19), (3.20) into the general stress–strain relation t 2 ∂εkk K (t − τ) − μ(t − τ) dτ σij (t) = δij 3 ∂τ t +2 0−

μ(t − τ)

∂εij dτ , ∂τ

(3.24)

where δij is again the Kronecker delta. Poisson Contraction A recurring and important parameter in linear elasticity is Poisson’s ratio. It characterizes the contraction/expansion behavior of the solid in a uniaxial stress state, and is an almost essential parameter for deriving other material constants such as the Young’s, shear or bulk modulus from each other. For viscoelastic solids the equivalent behavior cannot in general be characterized by a constant; instead, the material equivalent to the elastic Poisson’s ratio is also a time-dependent function, which is a functional of the stress (strain) history imposed on a uniaxially stressed material sample. This time-dependent function covers basically the same (long) time scale as the other viscoelastic responses and is typically measured in terms of 10–20 decades of time at any one temperature. However, compared to these other functions, its value changes usually from a maximum value of 0.35 or 0.4 at the short end of the time spectrum to 0.5 for the long time frame. Several approximations are useful. In the nearglassy domain (short times) its value can be taken as a constant equal to that derived from measurements well below the glass-transition temperature. In the long time range for essentially rubber-like behavior the approximation of 0.5 is appropriate, though not if one wishes to convert shear or Young’s data to bulk behavior, in which case small deviations from this value can play a very significant role. If knowledge in the range between the (near-)glassy and (near-)rubbery domain are required, neither of the two limit constants are strictly appropriate and careful measurements are required [3.14–16].

(3.25)

where δ(ξn ) represents the Dirac delta function, clearly leads to the series representation (3.20) and can trace the modulus function arbitrarily well by choosing the number of terms in the series to be sufficiently large; a choice of numbers of terms equal to or larger than twice the number of decades of the transition is often desirable. For a history of procedures to determine the coefficients μn see the works by Hopkins and Hamming [3.17], Schapery [3.18], Clauser and Knauss [3.19], Hedstrom et al. [3.20], Emri and Tschoegl [3.21–26], and Emri et al. [3.27, 28], all of which battle the ill-conditioned nature of the numerical determination process. This fact may result in physically inadmissible, negative values (energy generation), though the overall response function may be rendered very well. A more recent development that largely circumvents such problems, is based on the trust region concept [3.29], which has been incorporated into MATLAB, thus providing a relatively fast and readily available procedure. The numerical determination of these coefficients occurs through an ill-conditioned integral or matrix and is not free of potentially large errors in the coefficients, including physically inadmissible negative values, though the overall response function may be rendered very well. Although expressions as given in (3.22) and (3.23) render complete descriptions of the relaxation or creep behavior once H (ξ) or L(ξ) are determined for any material in general, simple approximate representations can fulfill a useful purpose. Thus, the special function μ0 − μ∞ ξ0 n exp(−ξ0 /ξ) (3.26) H (ξ) = Γ (n) ξ with the four parameters μ0 , μ∞ , ξ0 , and n representing material constants, where Γ (n) is the gamma function, leads to the power-law representation for the relaxation response μ(t) = μ∞ +

μ0 − μ∞ . (1 + t/ξ0 )n

(3.27)

This equation is represented in Fig. 3.3 for the parameter values μ∞ = 102 , μ0 = 105 , ξ0 = 10−4 , and n = 0.35. It follows quickly from (3.22) and the figure that μ0 represents the modulus as t → 0, and μ∞

Part A 3.2

0−

55

56

Part A

Solid Mechanics Topics

3.2.6 Special Stress or Strain Histories Related to Material Characterization

log modulus 6

For the purposes of measuring viscoelastic properties in the laboratory we consider several examples in terms of shear states of stress and strain. Extensional or compression properties follow totally analogous descriptions.

5 4 3 2

Part A 3.2

1 –10

–5

0

5

10 log t

Fig. 3.3 Example of the power-law representation of a relaxation modulus

its behavior as μ(t → ∞); ξ0 locates the central part of the transition region and n the (negative) slope. It bears pointing out that, while this functional representation conveys the generally observed behavior of the relaxation phenomenon, it usually serves only in an approximate manner: the short- and long-term modulus limits along with the position along the log-time axis and the slope in the mid-section can be readily adjusted through the four material parameters, but it is usually a matter of luck (and rarely possible) to also represent the proper curvature in the transitions from short- and long-term behavior. Nevertheless, functions of the type (3.26) or (3.27) can be very useful in capturing the essential features of a problem. With respect to fracture Schapery draws heavily on the simplified power-law representation. An alternative representation of one-dimensional viscoelastic behavior (shear or extension), though not accessed through a distribution function of the type described above, is the so-called stretch exponential formulation; it is often used in the polymer physics community and was introduced for torsional relaxation by Kohlrausch [3.30] and reintroduced for dielectric studies by Williams and Watts [3.31]. It is, therefore, often referred to as the KWW representation. In the case of relaxation behavior it takes the form (with the addition of the long-term equilibrium modulus μ∞ ),

(3.28) μ(t) = μ∞ + μ0 exp −(t/ξ0 )β . Further observations and references relating to this representation are delineated in [3.13].

Unidimensional Stress State We call a stress or strain state unidimensional when it involves only one controlled or primary displacement or stress component, as in pure shear or unidirectional extension/compression. Typical engineering characterizations of materials occur by means of uniaxial (tension) tests. We insert here a cautionary note with respect to laboratory practices. In contrast to working with metallic specimens, clamping polymers typically introduces complications that are not necessarily totally resolvable in terms of linear viscoelasticity. For example, clamping a tensile specimen in a standard test machine with serrated compression claps introduces a nonlinear material response such that, during the course of a test, relaxation or creep may occur under the clamps. Sometimes an effort is made to alleviate this problem by gluing metal tabs to the end of specimens, only to introduce the potential of the glue line to contribute to the overall relaxation or deformation. If the contribution of the glue line to the deformation is judged to be small, an estimate of its effect may be derived with the help of linear viscoelasticity, and this should be stated in reporting the data. For rate-insensitive materials the pertinent property is Young’s modulus E. For viscoelastic solids this constant is supplanted by the uniaxial relaxation modulus E(t) and its inverse, the uniaxial creep compliance D(t). Although the general constitutive relation (3.24) can be written for the uniaxial stress state (σ11 (t) = σ0 (t), say, σ22 = σ33 = 0), the resulting relation for the uniaxial stress is an integral equation for the stress or strain ε11 (t), involving the relaxation moduli in shear and dilatation. In view of the difficulties associated with determining the bulk response, it is not customary to follow this interconversion path, but to work directly with the uniaxial relaxation modulus E(t) and/or its inverse, the uniaxial creep compliance D(t). Thus, if σ11 (t) is the uniaxial stress and ε11 (t) the corresponding strain, one writes, similar to (3.10) and (3.12),

t σ11 (t) = ε11 (0)E(t) + 0+

E(t − ξ)

dε11 (ξ) dξ dξ

Mechanics of Polymers: Viscoelasticity

ε

ε

ε0

t0

tions to generate a ramp

+

t0

t

t0

t

t

modulus (3.22) together with the convolution relation (3.10) to render, with ε˙ (t) = const = ε˙ 0 and ε(0) = 0, the general result t

0+

We insert here a cautionary note with respect to laboratory practices: In contrast to working with metallic specimens, clamping polymers typically introduces complications that are not necessarily totally resolvable in terms of linear viscoelasticity. For example, clamping a tensile specimen in a standard test machine with serrated compression clamps introduces nonlinear material response such that during the course of a test relaxation or creep may occur under the clamps. Sometimes an effort is made to alleviate this problem by gluing metal tabs to the end of specimens, only to introduce the potential of the glue-line to contribute to the overall relaxation or deformation. If the contribution of the glue-line to the deformation is judged to be small an estimate of its effect may be derived with the help of linear viscoelasticity, and such should be stated in reporting the data. Constant-Strain-Rate History. A common test method

for material characterization involves the prescription of a constant deformation rate such that the strain increases linearly with time (small deformations). Without loss of generality we make use of a shear strain history in the form ε(t) = ε˙ 0 t (≡ 0 for t ≤ 0, ε˙ 0 = const for t ≥ 0) and employ the general representation for the relaxation τ

µ (t)

Error

ε0

t0

t

t0

t

Fig. 3.5 Difference in relaxation response resulting from

step and ramp strain history

μ(t − ξ)˙ε0 dξ

τ(t) = 2 0

t = 2˙ε0

∞ μ∞ +

0

0

t − ξ dς H (ς) exp − du ς ς (3.29)

t = 2μ˙ε0

μ(u) du = 2˙ε0 t · 0

1 t

t μ(u) du 0

= 2ε(t)μ(t) (3.30) ¯ ; t −ξ ≡ u . t Here μ(t) ¯ = 1t 0 μ(u) du is recognized as the relaxation modulus averaged over the past time (the time-averaged relaxation modulus). Ramp Strain History. A recurring question in viscoelas-

tic material characterization arises when step functions are called for analytically but cannot be supplied experimentally because equipment response is too slow or dynamic (inertial) equipment vibrations disturb the input signal: In such situations one needs to determine the error if the response to a ramp history is supplied instead of a step function with the ramp time being t0 . To provide an answer, take explicit recourse to postulate (b) in Sect. 3.2.1 in connection with (3.29)/(3.30) to evaluate (additively) the latter for the strain histories shown in Fig. 3.4. To arrive at an approximate result as a quantitative guide, let us use the power-law representation (3.27) for the relaxation modulus. Making use of Taylor series approximations of the resulting functions for t 0 one arrives at (the derivation is lengthy though straightforward) n t0 /ξ0 τ(t) = μ(t) 1 + (3.31) +... 2ε0 2 (1 + t/ξ0 )

Part A 3.2

and the inverse relation as t dσ11 (ξ) ε11 (t) = σ11 (0)D(t) + D(t − ξ) dξ . dξ

ε

57

Fig. 3.4 Superposition of linear func-

ε

=

3.2 Linear Viscoelasticity

58

Part A

Solid Mechanics Topics

Part A 3.2

as long as μ∞ can be neglected relative to μ0 (usually on the order of 100–1000 times smaller). The derivation is lengthy though straightforward. The expression in the square brackets contains the time-dependent error by which the ramp response differs from the ideal relaxation modulus, as illustrated in Fig. 3.5, which tends to zero as time grows without limit beyond t0 . By way of example, if n = 1/2 and an error in the relaxation modulus of maximally 5% is acceptable, this condition can be met by recording data only for times larger than t/t0 = 5 − ς0 /t0 . Since ς0 /t0 is always positive the relaxation modulus is within about 5% of the ramp-induced measurement as long as one discounts data taken before 5t0 . To be on the safe side, one typically dismisses data for an initial time interval equal to ten times the ramp rise time. In case the time penalty for the dismissal of that time range is too severe, methods have been devised that allow for incorporation of this earlier ramp data as delineated in [3.32, 33]. On the other hand, the wide availability of computational power makes an additional data reduction scheme available: Using a Prony series (discrete spectrum) representation, one evaluates the constant-strain-rate response with the aid of (3.30), leaving the individual values of the spectral lines as unknowns. With regard to the relaxation times one has two options: (a) one leaves them also as unknowns, or (b) one fixes them such that they are one or two per decade apart over the whole range of the measurements. The second option (b) is the easier/faster one and provides essentially the same precision of representation as option (a). After this choice has been made, one fits the analytical expression with the aid of Matlab to the measurement results. Matlab will handle either cases (a) or (b). There may be issues involving possible dynamic overshoots in the rate-transition region, because a test machine is not able to (sufficiently faithfully) duplicate the rapid change in rate transition from constant to zero rate, unless the initial rate is very low. This discrepancy is, however, considerably smaller that that associated with replacing a ramp loading for a step history. Mixed Uniaxial Deformation/Stress Histories Material parameters from measured relaxation or creep data are typically extracted via Volterra integral equations of the first kind, i. e., of the type of (3.20) or (3.21). A problem arises because these equations are ill-posed in the sense that the determination of

the kernel (material) functions from modulus or creep data involving Volterra equations of the first kind can lead to sizeable errors, whether the functions are sought in closed form or chosen in spectral or discrete (Prony series) form [3.18, 19, 27]. On the other hand, Volterra equations of the second kind do not suffer from this mathematical inversion instability (well-posed problem). Accordingly, we briefly present an experimental arrangement that alleviates this inherent difficulty [3.28]. At the same time, this particular scheme also allows the simultaneous determination of both the relaxation and creep properties, thus circumventing the calculation of one from the other. In addition, the resulting data provides the possibility of a check on the linearity of the viscoelastic data through a standard evaluation of a convolution integral. Relaxation and/or creep functions can be determined from an experimental arrangement that incorporates a linearly elastic spring of spring constant ks as illustrated in Fig. 3.6, readily illustrated in terms of a tensile situation. The following is, however, subject to the assumption that the elastic deformations of the test frame and/or the load cell are small compared to those of the specimen and the deformation of the added spring. If the high stiffness of the material does not warrant that assumption it is necessary to determine the contribution of the testing machine and incorporate it into the stiffness ks . Similar relations apply for a shear stress/deformation arrangement. In the case of

lb

Δl b0 ls ls Δl Δl s0

Fig. 3.6 Arrangement for multiple material properties de-

termination via a single test

Mechanics of Polymers: Viscoelasticity

bulk/volume response the spring could be replaced by a compressible liquid, though this possibility has not been tested in the laboratory, to our knowledge. For a suddenly applied gross extension (compression) of the spring by an amount Δl = const, both the bar and the spring will change lengths according to Δlb (t) + Δls (t) = Δl ,

(3.32)

where the notation in Fig. 3.6 is employed (subscript ‘b’ refers to the bar and “s” to the spring). The correspondingly changing stress (force) in the bar is given by

(3.33)

which is also determined by Ab Fb (t) = lb

t E(t − ξ)

d [Δlb (ξ)] dξ dξ

0

+

Ab 0 Δl E(t) , lb b

(3.34)

which, together with (3.32), renders upon simple manipulation Ab Δlb (t) εb (0)E(t) + Δl ks Δl t E(t − ξ)

+

d [εb (ξ)] dξ = 1 . dξ

(3.35)

0

This is a Volterra integral equation of the second kind, as can be readily shown by the transformation of variables ξ = t − u; it is well behaved for determining the relaxation function E(t). By measuring Δlb (t) along with the other parameters in this equation, one determines the relaxation modulus E(t). Similarly, one can cast this force equilibrium equation in terms of the creep compliance of the material and the force in the spring as ks lb Fb (t) + Ab

t D(t − ξ)

d [Fb (ξ)] dξ dξ

0

Time-Harmonic Deformation A frequently employed characterization of viscoelastic materials is achieved through sinusoidal strain histories of frequency ω. Historically, this type of material characterization refers to dynamic properties, because they are measured with moving parts as opposed to methods leading to quasi-static relaxation or creep. However, in the context of mechanics dynamic is reserved for situations involving inertia (wave) effects. For this reason, we replace in the sequel the traditional dynamic (properties) with harmonic, signifying sinusoidal. Whether one asks for the response from a strain history that varies with sin(ωt) or cos(ωt) may be accomplished by dealing with the (mathematically) complex counterpart

ε(t) = ε0 exp(iωt) · h(t)

0

(3.36)

It is clear then that, if both deformations and the stress in the bar are measured, both the relaxation modulus

(3.38)

so that after the final statement has been obtained one would be interested, correspondingly in either the real or the imaginary part of the result. Here h(t) is again the Heaviside step function, according to which the real part of the strain history represents a step at zero time with amplitude ε0 . The evaluation of the appropriate response may be accomplished with the general modulus representation so that substitution of (3.22) and (3.38) into (3.12) or (3.13) renders, after an interchange in the order of integration, ∞ dς τ(t) = 2ε0 μ∞ + H (ς) exp(−t/ς) ς t t −ξ H (ς) exp − ς 0 0 dς × exp(iωξ) dξ ς ∞

0

+ Fb (0)D(t) = ks Δl .

and the creep compliance can be determined and the determination of the Prony series parameters proceeds without difficulty [3.21–26] The additional inherent characteristic of this (hybrid) experimental–computational approach is that it may be used for determining the limit of linearly viscoelastic behavior of the material. By determining the two material functions of creep and relaxation simultaneously one can examine whether the determined functions satisfy the essential linearity constraint, see (3.62)–(3.64) t D(t − ξ)E(ξ) dξ = t . (3.37)

+ 2ε0 iω

59

Part A 3.2

Fb (t) = Fs (t) = kb (t)Δls (t) = ks [Δl − Δlb (t)] ,

3.2 Linear Viscoelasticity

60

Part A

Solid Mechanics Topics

t + 2ε0 iωμ∞

exp(iωξ) dξ ,

(3.39)

0

which ultimately leads to τ(t) = 2ε0 [μ(t) − μ∞ ] ∞ iωH (ς) − 2ε0 exp(−t/ς) dς 1 + iως 0

∞ iωH (ς) dς . + 2ε(t) μ∞ + 1 + iως

(3.40)

Part A 3.2

0

The first two terms are transient in nature and (eventually) die out, while the third term represents the steady-state response. For the interpretation of measurements it is important to appreciate the influence of the transient terms on the measurements. Even though a standard linear solid, represented by the spring–dashpot analog in Fig. 3.7 does not reflect the full spectral range of engineering materials, it provides a simple demonstration for the decay of the transient terms. Its relaxation modulus (in shear, for example) is given by μ(t) = μ∞ + μs exp(−t/ζ0 ) ,

(3.41)

where ζ0 denotes the relaxation time and μ∞ and μs are modulus parameters. Using the imaginary part of (3.40) corresponding to the start-up deformation history ε(t) = ε0 sin(ωt)h(t) one finds for the corresponding stress history ωζ0 μs τ(t) =R= (cos ωt + ωζ0 sin ωt) 2μ∞ ε0 μ∞ 1 + ω2 ζ02 ωζ0 μs e−t/ζn . (3.42) − μ∞ 1 + ω2 ζ02 F

μ

η

μ0 F

The last term is the transient. An exemplary presentation with μs /μ∞ = 5, ωζ0 = 1, and ζ0 = 20 is shown in Fig. 3.8. For longer relaxation times the decay lasts longer; for shorter ones the converse is true. One readily establishes that in this example the decay is (exponentially) complete after four to five times the relaxation time. The implication for real materials with very long relaxation times deserves extended attention. The expression for the standard linear solid can be generalized by replacing (3.41) with the corresponding Prony series representation. 1 ωζn μn τ(t) = (cos ωt + ωζn sin ωt) 2μ∞ ε0 μ∞ n 1 + ω2 ζn2 1 ωζn μn −t/ζn0 e . (3.43) − μ∞ n 1 + ω2 ζn2 Upon noting that the fractions in the last term sum do not exceed μn /2 one can bound the second sum by 1 ωζn μn −t/ζn0 e μ∞ n 1 + ω2 ζn2 1 1 μ(t) ≤ μn e−t/ζn0 = −1 . (3.44) 2μ∞ n 2 μ∞ This expression tends to zero only when t → ∞, a time frame that is, from an experimental point of view, too long in most instances. For relatively short times that fall into the transition range, the ratio of moduli is not small, as it can be on the order of 10 or 100, or even larger. There are, however, situations for which this error can be managed, and these correspond to those cases when the relaxation modulus changes very slowly during the time while sinusoidal measurements are being R 6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –6 0

20

40

60

80

100

120 140 160 Normalized time

Fig. 3.7 Standard

Fig. 3.8 Transient start-up behavior of a standard linear

linear solid

solid under ε(t) = h(t) sin(ωt)

Mechanics of Polymers: Viscoelasticity

τ(t) = μ∞ + 2ε(t)

∞ 0

iωH (ς) dς . 1 + iως

Both the strain ε(t) and the right-hand side are complex numbers. One calls μ∗ (ω) ≡ μ∞ +

∞ 0

iωH (ς) dς 1 + iως

Stress:

(3.46)

τ (t) sin[ωt +Δ(ω)] 2ε0 μ(ω) Strain:

1

the complex modulus μ∗ = μ (ω) + iμ (ω) with its real and imaginary parts defined by ∞ (ως)2 μ (ω) = μ∞ + H (ς) dς (3.47) 1 + (ως)2 0

(the storage modulus) and ∞ ως μ (ω) = H (ς) dς 1 + (ως)2

(3.48)

0

(the loss modulus), respectively. Polar representation allows the shorthand notation μ∗ = μ(ω) exp[iΔ(ω)] ,

(3.49)

where

μ (ω) and μ (ω) μ(ω) ≡ |μ∗ (ω)| = [μ (ω)]2 + [μ (ω)]2 , tan Δ(ω) =

(3.50)

so that, also μ (ω) = μ(ω) cos Δ(ω) and μ (ω) = μ(ω) sin Δ(ω) .

(3.51)

The complex stress response (3.45) can then be written, using (3.50), as τ(t) = 2ε(t)μ∗ (ω) = 2ε0 exp(iωt)μ(ω) exp[iΔ(ω)] , (3.52)

(3.45)

ε (t) = sin ωt ε0

t Δω

Fig. 3.9 Illustration of the frequency-dependent phase shift between the applied strain and the resulting stress

61

which may be separated into its real or imaginary part according to τ(t) = 2ε0 μ(ω) cos[ωt + Δ(ω)] and τ(t) = 2ε0 μ(ω) sin[ωt + Δ(ω)] .

(3.53)

Thus the effect of the viscoelastic material properties is to make the strain lag behind the stress (the strain is retarded) as illustrated in Fig. 3.9. It is easy to verify that the high- and low-frequency limits of the steadystate response are given by μ∗ (ω → ∞) = μ(t → 0) = μ0 , the glassy response, and μ∗ (ω → 0) = μ(t → ∞) = μ∞ , as the long-term or rubbery response (real). An Example for a Standard Linear Solid. For the stan-

dard linear solid (Fig. 3.7) the steady-state portion of the response (3.52) simplifies to ω2 ς02 , μ (ω) = μ∞ + μs 1 + ω2 ς02 ως0 μ (ω) = μs , (3.54) 1 + ω2 ς02 ως0 tan Δ(ω) = . (3.55) μ∞ /μs + (1 + μ∞ /μs )(ως0 )2

Part A 3.2

made. This situation arises when the material is near its glassy state or when it approaches rubbery behavior. As long as the modulus ratio can be considered nearly constant in the test period, the error simply offsets the test results by additive constant values that may be subtracted from the data. Clearly, that proposition does not hold when the material interrogation occurs around the middle of the transition range. There are many measurements being made with commercially available test equipment, when frequency scans or relatively short time blocks of different frequencies are applied to a test specimen at a set temperature, or while the specimen temperature is being changed continuously. In these situations the data reduction customarily does not recognize the transient nature of the measurements and caution is required so as not to interpret the results without further examination. Because viscoelastic materials dissipate energy, prolonged sinusoidal excitation generates rises in temperature. In view of the sensitivity of these materials to temperature changes as discussed in Sects. 3.2.7 and 3.2.8, care is in order not to allow such thermal build-up to occur unintentionally or not to take such changes into account at the time of test data evaluations. Consider now only the steady-state portion of (3.40) so that

3.2 Linear Viscoelasticity

62

Part A

Solid Mechanics Topics

which, upon using the transformation t − ξ = u, yields

Log of functions 3

τ(t) = iω 2ε(t)

2

∞

μ(u) e−iωu du = μ∗ (ω) .

If one recalls that the integral represents the Fourier transform F {μ(t), t → ω} of the modulus in the integrand one may write

1 0

μ∗ (ω) = iωF {μ(t), t → ω} –1

Part A 3.2

–2 –1.5

(3.59)

−∞

(3.60)

along with the inverse, –1

–0.5

0

0.5

1

1.5 2 2.5 Log frequency

Fig. 3.10 Steady-state response of a standard linear solid to

sinusoidal excitation, (μ0 = 1, μ = 100, μ/η = ς0 = 0.1). Symbols: μ (ω); short dash: μ (ω); long dash: tan Δ(ω)

While this material model is usually not suitable for representing real solids (its time frame is far too short), this simple analog model represents all the proper limit responses possessed by a real material, in that it has short-term (μ0 + μs , glassy), long-term (μ∞ , rubbery) as well as transient response behavior as illustrated in Fig. 3.10. Note that, with only one relaxation time present, the transition time scale is on the order of at most two decades. The more general representation of the viscoelastic functions under sinusoidal excitation can also be interpreted as a Fourier transform of the relaxation or creep response. Complex Properties as Fourier Transforms. It is often

desirable to derive the harmonic properties from monotonic response behaviors (relaxation or creep). To effect this consider the strain excitation of (3.38), ε(t) = ε0 exp(iωt)h(t) ,

(3.56)

and substitute this into the convolution relation for the stress (3.11), t μ(t − ξ)

τ(t) = 2ε(0)μ(t) + 2 0−

dε(ξ) dξ , dξ

(3.57)

and restrict consideration to the steady-state response. In this case, the lower limit is at t → −∞ so that the integral may be written as t μ(t − ξ) eiωξ dξ ,

τ(t) = lim 2ε0 iω t→∞

−t

(3.58)

μ(t) =

1 2π

∞ −∞

μ∗ (ω) −iωt dω . e iω

(3.61)

Thus the relaxation modulus can be computed from the complex modulus by the last integral. Note also that because of (3.60) μ and μ are derivable from a single function, μ(t), so that they are not independent. Conversely, if one measures μ and μ in a laboratory they should obey a certain interrelation; a deviation in that respect may be construed either as unsatisfactory experimental work or as evidence of nonlinearly viscoelastic behavior. Relationships Among Properties In Sect. 3.2.2 exemplary functional representation of some properties has been described that are generic for the description of any viscoelastic property. On the other hand, the situation often arises that a particular function is determined experimentally relatively readily, but really its complementary function is needed. The particularly simple situation most often encountered is that the modulus is known, but the compliance is needed (or vice versa). This case will be dealt with first. Consider the case when the relaxation modulus (in shear), μ(t) is known, and the (shear) creep compliance J(t) is desired. Clearly, the modulus and the compliance cannot be independent material functions. In the linearly elastic case these relations lead to reciprocal relations between modulus and compliance. One refers to such relationships as inverse relations or functions. Analogous treatments hold for all other viscoelastic functions. Recall (3.10) or (3.11), which give the shear stress in terms of an arbitrary strain history. In the linearly elastic case these inverse relations lead to reciprocal relations between modulus and compliance. Recall also that the creep compliance is the strain history resulting from a step stress being imposed in a shear test. As a corollary, if the prescribed strain

Mechanics of Polymers: Viscoelasticity

history is the creep compliance, then a constant (step) stress history must evolve. Accordingly, substitution of the compliance J(t) into (3.10) must render the step stress of unit amplitude so that t h(t) = J(0)μ(t) +

μ(t − ξ)

dJ(ξ) dξ . dξ

(3.62)

0+ + → 0 , J(0+ )/μ(0+ ) = 1

t μ(t − ξ)J(ξ) dξ = t .

(3.63)

0

Note that this relation is completely symmetric in the sense that, also, t J(t − ξ)μ(ξ) dξ = t .

(3.64)

0

Similar relations hold for the uniaxial modulus E(t) and its creep compliance D(t), and for the bulk modulus K (t) and bulk compliance M(t). Interrelation for Complex Representation. Because

the so-called harmonic or complex material characterization is the result of prescribing a specific time history with the frequency as a single time-like (but constant) parameter, the interrelation between the complex modulus and the corresponding compliance is simple. It follows from equations (3.45) and (3.46) that 1 2ε0 eiωt 2ε(t) = ∗ = = J ∗ (ω) , τ(t) μ (ω) τ0 ei(ωt+Δ(ω))

(3.65)

where the function J ∗ (ω) is the complex shear compliance, with the component J (ω) and imaginary component −J (ω) related to the complex modulus by J ∗ (ω) = J (ω) − iJ (ω) = J(ω)eiγ (ω) =

1 e−iΔ(ω) = μ∗ (ω) μ(ω)

(3.66)

so that, clearly, 1 and γ (ω) = −Δ(ω) , μ(ω) with J(ω) = [J (ω)]2 + [J (ω)]2 . J(ω) =

(3.67)

63

Thus in the frequency domain of the harmonic material description the interconnection between properties is purely algebraic. Corresponding relations for the bulk behavior follow readily from here.

3.2.7 Dissipation Under Cyclical Deformation In view of the immediately following discussion of the influence of temperature on the time dependence of viscoelastic materials we point out that general experience tells us that cyclical deformations engender heat dissipation with an attendant rise in temperature [3.34, 35]. How the heat generated in a viscoelastic solid as a function of the stress or strain amplitude is described in [3.13]. Here it suffices to point out that the heat generation is proportional to the magnitude of the imaginary part of the harmonic modulus or compliance. For this reason these (magnitudes of imaginary parts of the) properties are often referred to as the loss modulus or the loss compliance. We simply quote here a typical result for the energy w dissipated per cycle and unit volume, and refer the reader to [3.13] for a quick, but more detailed exposition: ∞ 2mπ m ε2m μ (3.68) . w/cycle = π T m=1

3.2.8 Temperature Effects Temperature is one of the most important environmental variables to affect polymers in engineering use, primarily because normal use conditions are relatively close to the material characteristic called the glasstransition temperature – or glass temperature for short. In parochial terms the glass temperature signifies the temperature at which the material changes from a stiff or hard material to a soft or compliant one. The major effect of the temperature, however perceived by the user, is through its influence on the creep or relaxation time scale of the material. Solids other than polymers also possess characteristic temperatures, such as the melting temperature in metals, while the melting temperature in the polymer context signifies specifically the melting of crystallites in (semi-)crystalline variants. Also, typical amorphous solids such as silicate glasses and amorphous metals exhibit distinct glass-transition temperatures; indeed, much of our understanding of glass-transition phenomena in polymers originated in understanding related phenomena in the context of silicate glasses.

Part A 3.2

Note that, as t so that at time 0+ an elastic result prevails. Upon integrating both sides of (3.62) with respect to time – or alternatively, using the Laplace transform – one readily arrives at the equivalent result; the uniaxial counterpart has already been cited effectively in (3.37).

3.2 Linear Viscoelasticity

64

Part A

Solid Mechanics Topics

Part A 3.2

The Entropic Contribution Among the long-chain polymers, elastomers possess a molecular structure that comes closest to our idealized understanding of molecular interaction. Elastomer is an alternative name for rubber, a cross-linked polymer that possesses a glass transition temperature which is distinctly below normal environmental conditions. Molecule segments are freely mobile relative to each other except for being pinned at the cross-link sites. The classical constitutive behavior under moderate deformations (up to about 100% strain in uniaxial tension) has been formulated by Treloar [3.36]. Because this constitutive formulation involves the entropy of a deformed rubber network, this temperature effect of the properties is usually called the entropic temperature effect. In the present context it suffices to quote his results in the form of the constitutive law for an incompressible solid. Of common interest is the dependence of the stress on the material property appropriate for uniaxial tension (in the 1-direction) 1 1 2 subject to λ1 λ2 λ3 = 1 , σ = NkT λ1 − 3 λ1 (3.69)

where λ1 , λ2 , and λ3 denote the (principal) stretch ratios of the deformation illustrated in Fig. 3.11 (though not shown for the condition λ1 λ2 λ3 = 1), the multiplicative factor consists of the number of chain segments between cross-links N, k is Boltzmann’s constant, and T is absolute temperature. Since for infinitesimal deformations λ1 = 1 + ε11 , one finds that NkT must equal the elastic Young’s modulus E ∞ . Thus the (small-strain) Young’s modulus is directly proportional to the absolute temperature, and this holds also for the shear modulus because, under the restriction/assumption of incompressibility the shear modulus μ∞ of the rubber

obeys μ∞ = 13 E ∞ . Thus μ∞ /T = Nk is a material constant, from which it follows that comparative moduli obtained at temperatures T and T0 are related by T μ∞|T = μ∞|T0 or equivalently T0 T E ∞|T = E ∞|T0 . (3.70) T0 If one takes into account that temperature changes affect also the dimensions of a test specimen by changing both its cross-sectional area and length, this is taken into account by modifying (3.71) to include the density ratio according to ρT μ∞|T0 or equivalently μ∞|T = ρ0 T0 ρT E ∞|T = E ∞|T0 , (3.70a) ρ0 T0 where ρ0 is the density at the reference temperature and ρ is that for the test conditions. To generate a master curve as discussed below it is therefore necessary to first multiply modulus data by the ratio of the absolute temperature T (or ρT , if the densities at the two temperatures are sufficiently different) at which the data was acquired, and the reference temperature T0 (or ρ0 T0 ). For compliance data one multiplies by the inverse density/temperature ratio. Time–Temperature Trade-Off Phenomenon A generally much more significant influence of temperature on the viscoelastic behavior is experienced in connection with the time scales under relaxation or creep. To set the proper stage we define first the notion of the glass-transition temperature Tg . To this end consider a measurement of the specific volume as a function Volume

λ1

λ3

λ2

B A Equilibrium line

Fig. 3.11 Deformation of a cube into a parallelepiped. The

unit cube sides have been stretched (contracted) orthogonally in length to the stretch ratios λ1 , λ2 , and λ3

Tg

Temperature

Fig. 3.12 Volume–temperature relation for amorphous solids (polymers)

Mechanics of Polymers: Viscoelasticity

log G (t) Experimental window

sensitive properties, at least for polymers. For ease of presentation we ignore first the entropic temperature effect discussed. The technological evolution of metallic glasses is relatively recent, so that a limited amount of data exist in this regard. However, new data on the applicability of the time–temperature trade-off in these materials have been supplied in [3.12]. Moreover, we limit ourselves to considerations above the glass-transition temperature, with discussion of behavior around or below that temperature range reserved for later amplification. Experimental constraints usually do not allow the full time range of relaxation to be measured at any one temperature. Instead, measurements can typically be made only within the time frame of a certain experimental window, as indicated in Fig. 3.13. This figure shows several (idealized) segments as resulting from different temperature environments at a fixed (usually atmospheric) pressure. A single curve may be constructed from these segments by shifting the temperature segments along the log-time axis (indicated by arrows) with respect to one obtained at a (reference) temperature chosen arbitrarily, to construct the master curve. This master curve is then accepted as the response of the material over the extended time range at the chosen reference temperature. Because this time– temperature trade-off has been deduced from physical measurements without the benefit of a time scale of unlimited extent, the assurance that this shift process is a physically acceptable or valid scheme can be derived only from the quality with which the shifting or

P = P0 T1

T0 = T3

⎛ σ 273 ⎛ ⎜ (psi) ⎝ ε0 T ⎝

log0 ⎜ 5

T3

Temperature (°C) –30.0 –25.0 –22.5 –22.0 –17.5 –15.0 –12.5 –7.5 –5.0 –2.5 5.0

T2 T4

log aT4

4

Master curve at T3

3

T5

ε0 = 0.05

T1 < T2 < ··· ωc . Next consider the carrier load to increase at a constant rate P˙0 . Differentiation of (3.119) with respect to time guarantees a positive loading rate as long as P˙0 ≥ ΔP0 ω. Additionally, the substitution √ of this inequality into (3.91) together with a(t) = RH(t) shows that the contact area will then also not decrease during the entire indentation history. Figure 3.28 shows a comparison of these developments with an application of (3.111) [3.96] under the assumption of a constant Poisson ratio: when measurements over a relatively short time (such as ≈ 250 s used in this study) are made, the Poisson’s ratio [3.15] does not change significantly for polymers in the glassy state, such as PMMA or polycarbonate (PC) and thus introduces negligible errors in the complex compliance data. To compute the complex modulus of PC and polymethyl methacrylate (PMMA) at 75 Hz, data were acquired continuously at this frequency for ≈ 125 s. In general the data increase correctly with time, and approach a nearly constant value for each material. These constant values are considered to represent the steady state and are quoted as the storage modulus. Also shown for comparison in Fig. 3.28 are data measured with the aid of conventional dynamic mechanical analysis (DMA) for the same batch of PC and PMMA. The uniaxial storage modulus of PC measured by DMA at 0.75 Hz is 2.29 GPa. However, the storage modulus computed using (3.111) is at least 40% higher than this

3.3 Measurements and Methods

84

Part A

Solid Mechanics Topics

ing temperatures are of the same order of magnitude. Because dynamic events occur in still shorter time frames wave mechanics typically little concern in this regard. The situation is quite different when soft or elastomeric polymers serve as photoelastic model materials. In that case quasistatic environments (around room temperature) typically involve only the long-term or rubbery behavior of the material with the stiffness measured in terms of the rubbery or long-term equilibrium modulus. On the other hand, when wave propagation phenomena are part of the investigation, the longest relaxation times (relaxation times that govern the tran-

sition to purely elastic behavior for the relatively very long times) are likely to be excited so that the output of the measurements must consider the effect of viscoelastic response. It is beyond the scope of this presentation to delineate the full details of the use of viscoelastically photoelastic material behavior, especially since during the past few years investigators have shown a strong inclination to use alternative tools. However, it seems useful to include a list of references from which the evolution of this topic as well as its current status may be explored. These are listed as a separate group in Sect. 3.6.1 under References on Photoviscoelasticity.

Part A 3.4

3.4 Nonlinearly Viscoelastic Material Characterization Viscoelastic materials are often employed under conditions fostering nonlinear behavior. In contrast to the mutual independence in the dilatational and deviatoric responses in a linearly viscoelastic material, the viscoelastic responses in different directions are coupled and must be investigated in multiaxial loading conditions. Most results published in the literature are restricted to investigating the viscoelastic behavior in the uniaxial stress or uniaxial strain states [3.101, 102] and few results are reported for time-dependent multiaxial behavior [3.103–105]. Note that BauwensCrowet’s study [3.104] incorporates the effect of pressure on the viscoelastic behavior but not in the data analysis, simply as a result of using uniaxial compression deformations. Along similar lines Knauss, Emri, and collaborators [3.106–110] provided a series of studies deriving nonlinear viscoelastic behavior from changes in the dilatation (free-volume change) which correlated well with experiments and gave at least a partial physical interpretation to the Schapery scheme [3.111, 112] for shifting linearly viscoelastic data in accordance with a stress or strain state. The studies became the precursors to investigate the effect of shear stresses or strains on nonlinear behavior as described below. Thus the proper interpretation of the uniaxial data as well as their generalization to multiaxial stress or deformation states is highly questionable. This remains true regardless of the fact that such data interpretation has been incorporated into commercially available computer codes. Certainly, there are very few engineering situations where structural material use is limited to uniaxial states. In this section we describe some aspects of nonlinearly viscoelastic behavior in

the multiaxial stress state. References regarding nonlinear behavior in the context of uniaxial deformations are too numerous to list here. The reader is advised to consult the following journals: the Journal of Polymer, Applied Polymer Science, Polymer Engineering and Science, the Journal of Materials Science and Mechanics of Time-Dependent Materials, to list the most prevalent ones.

3.4.1 Visual Assessment of Nonlinear Behavior Although it is clear that even under small deformations entailing linearly viscoelastic behavior the imposition of a constant strain rate on a tensile or compression specimen results in a stress response that is not linearly related to the deformation when the relation is established in real time (not on a logarithmic time scale). Because such responses that appear nonlinear on paper are not necessarily indicative of nonlinear constitutive behavior, it warrants a brief exposition of how nonlinear behavior is unequivocally separated from the linear type. To demonstrate the nonlinear behavior of a material, consider first isochronal behavior of a linearly viscoelastic material. Although isochronal behavior can be obtained for different deformation or stress histories, consider the case of shear creep under various stress levels σn , so that the corresponding strain is εn = J(t)σn for any time t. Consider an arbitrary but fixed time t ∗ , at which time the ratio εn /σn = J(t ∗ )

(3.121)

Mechanics of Polymers: Viscoelasticity

Shear stress (MPa) 16 14 t = 10s 12 t = 104 s 10

ial. These values are likely to be different for other materials.

3.4.2 Characterization of Nonlinearly Viscoelastic Behavior Under Biaxial Stress States In the following sections, we describe several ways of measuring nonlinearly viscoelastic behavior in multiaxial situations. Hollow Cylinder under Axial/Torsional Loading A hollow cylinder under axial/torsional loading conditions provides a vehicle for investigating the nonlinearly viscoelastic behavior under multiaxial loading conditions. Figure 3.30 shows a schematic diagram of a cylinder specimen. Dimensions can be prescribed corresponding to the load and deformation range of interest as allowed by an axial/torsional buckling analysis. Two examples are given here for specimen dimensions. With the use of a specimen with outer diameter of 22.23 mm, a wall thickness of 1.59 mm, and a test length of 88.9 mm, the ratio of the wall thickness to the radius is 0.14. These specimens can reach a surface shear strain on the order of 4.0–4.5%. With the use of an outer diameter of 25.15 mm, a thickness of 3.18 mm, and a test length of 76.2 mm, the ratio of the wall thickness to the radius is 0.29. This allows a maximum shear strain prior to buckling in the range of 8.5–12.5% based on elastic analysis. The actual maximum shear strain that can be achieved prior to buckling can be slightly different due to the viscoelastic effects involved in the material. Estimates of strains leading to buckling may be achieved by considering a Young’s modulus for the material that corresponds to the lowest value achieved at the test temperature and the time period of interest for the measurements.

8

x

6

Fig. 3.30 A thin-

4 2 y 0

0

0.01

0.02

0.03 Shear strain

Fig. 3.29 Isochronal shear stress–shear strain relation of PMMA at 80 ◦ C

85

D O z

walled cylinder specimen for combined tension/compression and torsion to generate biaxial stress states [3.113]

Part A 3.4

takes on a particular property value. At that time all possible values of σn and εn are related linearly: a plot of σ versus ε at time t ∗ renders a linear relation with slope 1/J(t ∗ ) and encompassing the origin. For different times t ∗ , straight lines with different slopes result and the linearly viscoelastic material can thus be characterized by a fan of straight lines emanating from the origin, the slope of each line corresponding to a different time t ∗ . The slopes decrease monotonically as these times increase. Each straight line is called a linear isochronal stress–strain relation for the particular time t ∗ . Figure 3.29 shows such an isochronal representation for PMMA. It is seen that, at short times and when the strains are small in creep (shear strain 0.005, shear stress 8 MPa), the material response is close to linearly viscoelastic where all the data stay within the linear fan formed, in this case, by the upper line derived from the shear creep compliance at 10 s, and the lower line corresponding to the compliance in shear at 104 s. At the higher stress levels and at longer times, when the strains are larger, material nonlinearity becomes pronounced, as data points in Fig. 3.29 are outside the linear fan emanating from the origin. In this isochronal plot, the deviation from linearly viscoelastic behavior begins at approximately 0.5% strain level. Isochronal data at other temperatures indicate also that the nonlinearity occurs at ≈ 0.5% strain and a shear stress of about 7.6 MPa at all temperatures for this mater-

3.4 Nonlinearly Viscoelastic Material Characterization

86

Part A

Solid Mechanics Topics

Part A 3.4

Application of Digital Image Correlation The use of strain gages for determining surface strains on a polymer specimen is fraught with problems, since strain gages tend to be much stiffer than the polymer undergoing time-dependent deformations, and, in addition, the potential for increases in local temperature due to the currents in the strain gage complicates definitive data evaluation. Digital image correlation (DIC, Chap. 20) thus offers a perfect tool, though that method is not directly applicable to cylindrical surfaces as typically employed. While we abstain from a detailed review of this method in this context (we refer the reader to [3.114] for particulars), here it is of interest for the completeness of presentation only to summarize this special application to cylindrical surface applications. The results are presented in the form of (apparent) creep compliances defined by 2ε(t)/τ0 . We emphasize again that the creep compliance for a linear material is a function that depends only on time but not on the applied stress. This no longer holds in the nonlinearly viscoelastic regime, but we adhere to the use of this ratio as a creep compliance for convenience. To use DIC for tracking axial, circumferential, and shear strains on a cylindrical surface, a speckle pattern is projected onto the specimen surface. While the same image acquisition system can be used as for flat images special allowance needs to be made for the motion of surface speckles on a cylindrical surface, the orientation of which is also not known a priori. If the focal length of the imaging device is long compared to the radius of the cylinder, an image can be considered as a projection of a cylinder onto an observation plane. Planar deformations can be determined using digital im-

age correlation techniques [3.115, 116], and corrected for curvature to determine the axial, circumferential, and shear strains [3.114]. To assure that the motion is properly interpreted in a cylindrical coordinate system – that the camera axis is effectively very well aligned with the cylinder axis and test frame orientation – the imaging system must also establish the axis of rotation. This can be achieved through offsetting the specimen against a darker background (Fig. 3.31) so as to ensure sufficient contrast between the specimen edge and background for identification of any inclination of the cylinder axis relative to the reference axis within the image recording system. The axis orientation is then also evaluated using the principles of digital image correlation. Without such a determination the parameters identifying the projection of the cylindrical surface onto a plane lead to uncontrollable errors in the data interpretation. The details of the relevant data manipulation can be found in the cited references. Specimen Preparation Specimens can be machined from solid cylinders or from tubes, though tubular specimens tend to have a different molecular orientation because of the extrusion process. Prior to machining, the cylinders need to be annealed at a temperature near the glass-transition temperature to remove residual stresses. The thinwalled cylinder samples need to be annealed again after machining to remove or reduce residual surface stresses possibly acquired during turning. To avoid excessive gravity deformations, annealing is best conducted in an oil bath. Any possible weight gain must be monitored with a balance possessing sufficient resolution. The weight gain should be low enough to avoid any effect of the oil on the viscoelastic behavior of materials. For testing in the glassy state, specimens must have about the same aging times; and the aging times should be at least a few days so that during measurements the aging time change is not significant (on a logarithmic scale). Because physical aging is such an important topic we devote further comments to it in the next subsection. Prior to experiments samples need to be kept in an environment with a constant relative humidity that is the same as the relative humidity during measurements. The relative humidity can be generated through a saturated salt solution in an enclosed container [3.118]. Physical Aging in Specimen Preparation. When an

Fig. 3.31 A typical speckle pattern on a cylinder surface

inclined with respect to the observation axis of the imaging system

amorphous polymer is cooled (continuously) from its melt state, its volume will deviate from its equilibrium state at the glass-transition temperature (Fig. 3.12).

Mechanics of Polymers: Viscoelasticity

87

the time required to attain equilibrium after quenching within practical limits. The value of t ∗ may have to be determined prior to commencing characterization tests. If the viscoelastic properties are investigated without paying attention to the aging process, characterization of polymers and their composites are not likely to generate repeatable results. For characterization of the long-term viscoelastic behavior through accelerated testing, physical aging effects have to be considered, in addition to time–temperature superposition and other mechanisms. An Example of Nonlinearly Viscoelastic Behavior under Combined Axial/Shear Stresses Figure 3.32 shows the creep response in pure shear for PMMA at 80 ◦ C [3.113, 117]. The axial force was controlled to be zero in these measurements. The plotted shear creep compliance was converted from the relaxation modulus in shear under infinitesimal deformation, representing the creep behavior in the linearly viscoelastic regime. For a material that behaves linearly viscoelastically, all curves should be coincident in this plot. In the case of data at these stress levels, the deformations in shear at higher stress levels are accelerated relative to the behavior at infinitesimal strains. This observation constitutes another criterion for separating linear from nonlinear response. To represent the nonlinear characteristics we draw on the isochronal representation discussed above. At any given time spanned in Fig. 3.32, there are five data points from creep under a pure shear stress, giving four sets of isochronal stress/strain data. Plotting these four log (shear creep compliance) (1/MPa) –2.2 –2.4

σ = 0, τ = 16 MPa σ = 0, τ = 14.7 MPa σ = 0, τ = 12.3 MPa σ = 0, τ = 9.4 MPa

T = 80°C

–2.6 –2.8 Inversion from µ(t)

–3 –3.2 0.5

1

1.5

2

2.5

3

3.5

4 4.5 log (time) (s)

Fig. 3.32 Shear creep compliance of PMMA at several levels of shear stress at 80 ◦ C (after [3.117])

Part A 3.4

Polymers have different viscoelastic characteristics depending on whether they are below the glass-transition temperature (Tg ), in the glass-transition region (in the neighborhood of Tg ), or in the rubbery state (above Tg ). In the rubbery state a polymer is in or near thermodynamic equilibrium, where long-range cooperative motions of long-chain molecules are dominant and result in translational movements of molecules. Below the glass-transition temperature, short-range motions in the form of side-chain motions and rotations of segments of the main chain (primarily in long-term behavior) are dominant. The glass-transition range depends on the cooling rate. After cooling a polymer initially in the rubbery state to an isothermal condition in the glassy state, the polymer enters a thermodynamically nonequilibrium, or metastable, state associated with a smaller density than an optimal condition (equilibrium) would allow. In the equilibrium state the density would increase continuously to its maximum value. If the temperature in the isothermal condition is near Tg , the density increase can occur in a relatively short time, but if the temperature is far below Tg this process occurs over a long period of time, on the order of days, weeks, or months. Prior to reaching the maximum density, as time evolves, depending on how long this process has taken place, the polymer possesses a different viscoelastic response. This phenomenon is called physical aging because no chemical changes occur. The time after quenching to an isothermal condition in the glassy state is called aging time. The viscoelastic functions (e.g., bulk and shear relaxation moduli) change during aging until that process is complete within practical time limitations. The effect of physical aging is similar to a continual decrease of the temperature and results in the reduction of the free volume that provides the space for the mobility of the polymer chain segments as the chain undergoes any rearrangement. There now exists a relatively large body of information on physical aging and the reader is referred to a number of representative publications, in which references in the open literature expand on this topic. References [3.38–44,119–122] showed that physical aging leads to an aging time factor multiplying the external time, analogous to the temperature-dependent multiplier (shift factor) for thermorheologically simple solids in the context of linear viscoelasticity theory. Elaborations of this theme for various materials have been offered to a large extent by McKenna and by Gates as well as their various collaborators Effects of physical aging can be pronounced before aging time reaches the value, say, t ∗ , which is

3.4 Nonlinearly Viscoelastic Material Characterization

88

Part A

Solid Mechanics Topics

log(axial creep compliance) (1/MPa) –2.2

A–A

Clamped with bolt A – A

Glued for θ > 40°

–2.4 Tension + torsion σ = 25.3 MPa, τ = 14.5 MPa

T = 50°C

–2.6 θ

–2.8 Compression + torsion σ = 25.3 MPa, τ = 14.5 MPa

–3

Part A 3.4

–3.2 0.5

A x y A

1

1.5

2

2.5

3

3.5

4 4.5 log (time) (s)

Fig. 3.33 Axial creep compliance of PMMA under tension/torsion and compression/torsion at 50 ◦ C

data points at each of the 16 fixed times, say, gives the isochronal stress–strain relation shown in Fig. 3.29. It is clear that the creep rate increases with an increase in applied shear stress, indicating nonlinear creep behavior in shear. We note that for isochronal behavior at strains above 0.5%, there exists a fan emanating from the shear strain 0.5% and a shear stress of 7.6 MPa. The corresponding fan center is considered to be the yield point, above which the creep rate is accelerTest section 10.16 R = 10.16

22.54

22.3

22.54

44.6

15.14 30.48

Fig. 3.34 Geometry of an Arcan specimen (all dimensions are in mm, thickness is 3 mm)

Fig. 3.35 Fixture for testing Arcan specimens

ated measurably. It is of interest to note that the creep process is more pronounced (accelerated) in tension/torsion than under compression/torsion as illustrated in Fig. 3.33 for 50 ◦ C. We have already observed that thin-walled cylinders tend to buckle under sufficiently high torsion and/or compression. A cylinder with an outer diameter of 25.15 mm, a thickness of 3.18 mm, and a test length of 76.2 mm would buckle at ≈ 5% shear strain under pure torsion. The use of thicker-walled cylinders would reduce the homogeneity of the stress and strain within the cylinder wall and lead to inaccuracy in the determination of stress or strain. Other techniques, such as testing with the Arcan specimen should, therefore, be used when the nonlinearly viscoelastic behavior at larger deformations is investigated. Use of the Arcan Specimen Arcan’s specimen [3.123–125] can be used for multibiaxial test with the use of a uniaxial material test system. Figure 3.34 shows an Arcan specimen, and Fig. 3.35 a corresponding test fixture. The loading axis can form different angles with respect to the specimen axis so that biaxial stress states can be generated in the region of uniform deformation in the middle of the specimen. When the loading axis of the fixture is aligned with the major specimen axis, this configura-

Mechanics of Polymers: Viscoelasticity

Fig. 3.36 Isochronal contours of creep strains under fixed biaxial loading. Each contour corresponds to a different time between 10 s and about 105 s. Ellipses correspond to linear response characteristics

3.5 Closing Remarks

89

Normal strain 0.025 0.02 0.015 0.01 0.005 0 – 0.005 – 0.01 – 0.015 – 0.02 – 0.025 –0.03

–0.02

–0.01

0

0.01

0.02 0.03 Normal strain

show ellipses (a/b = 2) that would correspond to totally linearly viscoelastic behavior.

3.5 Closing Remarks As was stated at the very beginning, today’s laboratory and general engineering environment is bound to involve polymers, whether of the rigid or the soft variety. The difference between these two derives merely from the value of their glass-transition temperature relative to the use temperature (usually room temperature). As illustrated in this chapter the linearized theory of viscoelasticity is well understood and formulated mathematically, even though its current application in engineering designs is usually not on a par with this understanding. A considerable degree of response estimation can be achieved with this knowledge, but a serious deficiency arises from the fact that when structural failures are of concern the linearized theory soon encounters limitations as nonlinear behavior is encountered. There is, today, no counterpart nonlinear viscoelastic material description that parallels the plasticity theory for metallic solids. Because the atomic structures of metals and polymers are fundamentally different, it would seem imprudent to characterize polymer nonlinear behavior along similar lines of physical reasoning and mathematical formulation, notwithstanding the fact that in uniaxial deformations permanent deformations in metals and polymers may appear to be similar. That similarity disappears as soon as temperature or extended

time scales follow an initially nonlinear deformation history. It is becoming clear already that the superposition of dilatational stresses or volumetric strains has a greater influence on nonlinear material response of polymers than is true for metals. Consequently it would seem questionable whether uniaxial tensile or compressive behavior would be a suitable method for assessing nonlinear polymer response, since that stress state involves both shear and bulk (volumetric) components. To support this observation one only needs to recall that very small amounts of volume change can have a highly disproportionate effect on the time dependence of the material, as delineated in Sects. 3.2.8, 3.2.9, and 3.2.10. This is well illustrated by the best known and large effect which a change in temperature has on the relaxation times, where dilatational strains are indeed very small compared to typical shear deformations; responses under pressure and with solvent swelling underscore this observation. The recent publication history for time-dependent material behavior exhibits an increasing number of papers dealing with nonlinear polymer behavior, indicating that efforts are underway to address this lack of understanding in the engineering profession. At the same time it is also becoming clear that the intrinsic time-dependent behavior of polymers is closely

Part A 3.5

tion induces shear forces applied to an Arcan specimen so that there is a pure shear zone in the central portion of the specimen. Other orientations allow the nonlinearly viscoelastic shear behavior to be characterized under loading conditions combining tension/shear, compression/shear, and pure shear. The data processing is illustrated using the results obtained by Knauss and Zhu [3.126,127] as an example. Figure 3.36 shows isochronal creep shear and normal strains at 80 ◦ C using an Arcan specimen under a nominal (maximum) shear stress of 19.3 MPa. At each fixed time, line segments connect points to form an isochronal strain contour. The innermost contour corresponds to a creep time of 10 s, and the outermost contour is the results from 0.8 × 105 s. For comparison purposes we also

90

Part A

Solid Mechanics Topics

connected to the molecular processes that are well represented by the linearly viscoelastic characterization of these solids. It is thus not unreasonable, in retrospect, to have devoted a chapter mostly to describing linearly

viscoelastic solids with the expectation that this knowledge provides a necessary if not sufficient background for dealing with future issues that need to be resolved in the laboratory.

3.6 Recognizing Viscoelastic Solutions if the Elastic Solution is Known

Part A 3.6

Experimental work deals with a variety of situations/test configurations for which boundary value histories are not readily confined to a limited number of cases. For example, when deformations are to be measured by photoelasticity with the help of a viscoelastic, photosensitive material in a two-dimensional domain it may be important to know the local stress state very well, if knowing the stress state in a test configuration is an important prerequisite for resolving an engineering analysis problem. There exists a large class of elastic boundary value problems for which the distribution of stresses (or strains) turns out to be independent of the material properties. In such cases the effect of loading and material properties is expressed through a material-dependent factor and a load factor, both multiplying a function(s) that depends only on the spatial coordinates governing the distribution of stress or strain. It then follows that for the corresponding viscoelastic solution the distribution of stresses is also independent of the material properties and that the time dependence is formulated as the convolution of a material-dependent multiplicative function with the time-dependent load factored out from the spatial distribution function(s). In the experimental context beams and plates fall into this category, though even the simple plate configuration can involve Poisson’s ratio in its deformation field. More important is the class of simply connected two-dimensional domains for which the in-plane stress distribution is independent of the material properties. Consider first two-dimensional, quasistatic problems with (only) traction boundary conditions prescribed on a simply connected domain. For such problems the stress distribution of an elastic solid throughout the interior is independent of the material properties. The same situation prevails for multiply connected domains, provided the traction on each perforation is self-equilibrating. If the latter condition is not satisfied, then a history-dependent Poisson function enters the stress field description so that the stress distribution is, at best, only approximately independent of, or insensitive to, the material behavior. This topic is discussed in a slightly more detailed manner in [3.13].

3.6.1 Further Reading For general background it appears useful to identify the dominant publications to bolster one’s understanding of the theory of viscoelasticity. To that end we summarize here first, without comment or reference number, a list of publications in book or paper form, only one of which appears as an explicit reference in the text, namely the book by J.D. Ferry as part of the text development [3.45] with respect to the special topic of thermorheologically simple solids. 1. T. Alfrey: Mechanical Behavior of High Polymers (Interscience, New York, 1948) 2. B. Gross: Mathematical Structure of the Theories of Viscoelasticity (Herrmann, Paris, 1953) (re-issued 1968) 3. A.V. Tobolsky: Properties and Structure of Polymers (Wiley, New York, 1960) 4. M.E. Gurtin, E. Sternberg: On the linear theory of viscoelasticity, Arch. Rat. Mech. Anal. 11, 291–356 (1962) 5. F. Bueche: Physical Properties of Polymers (Interscience, New York, 1962) 6. M.L. Williams: The structural analysis of viscoelastic materials, AIAA J. 2, 785–809 (1964) 7. Flügge (Ed.): Encyclopedia of Physics VIa/3, M.J. Leitman, G.M.C. Fischer: The linear theory of viscoelasticity (Springer, Berlin, Heidelberg, 1973) 8. J.J. Aklonis, W.J. MacKnight: Introduction to Polymer Viscoelasticity (Wiley, New York 1983) 9. N.W. Tschoegl: The Phenomenological Theory of Linear Viscoelastic Behavior, an Introduction (Springer, Berlin, 1989) 10. A. Drozdov: Viscoelastic Structures, Mechanics of Growth and Aging (Academic, New York, 1998) 11. D.R. Bland: The Theory of Linear Viscoelasticity, Int. Ser. Mon. Pure Appl. Math. 10 (Pergamon, New York, 1960) 12. R.M. Christensen: Theory of Viscoelasticity: An Introduction (Academic, New York, 1971); see also R.M. Christensen: Theory of Viscoelasticity An Introduction, 2nd ed. (Dover, New York, 1982)

Mechanics of Polymers: Viscoelasticity

17. C.W. Folkes: Two systems for automatic reduction of time-dependent photomechanics data, Exp. Mech. 10, 64–71 (1970) 18. P.S. Theocaris: Phenomenological analysis of mechanical and optical behaviour of rheo-optically simple materials. In: Photoelastic Effect and its applications, ed. by J. Kestens (Springer, Berlin New York, 1975) pp. 146–152 19. B.D. Coleman, E.H. Dill: Photoviscoelasticity: Theory and practice. In: The Photoelastic Effect and its Applications, ed. by J. Kestens, (Springer, Berlin New York, 1975) pp. 455–505 20. M.A. Narbut: On the correspondence between dynamic stress states in an elastic body and in its photoviscoelastic model, Vestn. Leningr. Univ. Ser. Mat. Mekh. Astron. (USSR) 1, 116–122 (1978) 21. R.J. Arenz, U. Soltész: Time-dependent optical characterization in the photoviscoelastic study of stress-waver propagation, Exp. Mech. 21, 227–233 (1981) 22. K.S. Kim, K.L. Dickerson, W.G. Knauss (Eds): Viscoelastic effect on dynamic crack propagation in Homalit 100. In: Workshop on Dynamic Fracture (California Institute of Technology, Pasadena, 1983) pp. 205–233 23. H. Weber: Ein nichtlineares Stoffgesetz für die ebene photoviskoelastische Spannungsanalyse, Rheol. Acta. 22, 114–122 (1983) 24. Y. Miyano, S. Nakamura, S. Sugimon, T.C. Woo: A simplified optical method for measuring residual stress by rapid cooling in a thermosetting resin strip, Exp. Mech. 26, 185–192 (1986) 25. A. Bakic: Practical execution of photoviscoelastic experiments, Oesterreichische Ingenieur- und Architekten-Zeitschrift, 131, 260–263 (1986) 26. K.S. Kim, K.L. Dickerson, W.G. Knauss: Viscoelastic behavior of opto-mechanical properties and its application to viscoelastic fracture studies, Int. J. Fract. 12, 265–283 (1987) 27. T. Kunio, Y. Miyano, S. Sugimori: Fundamentals of photoviscoelastic technique for analysis of time and temperature dependent stress and strain. In: Applied Stress Analysis, ed. by T.H. Hyde, E. Ollerton (Elsevier Applied Sciences, London, 1990) pp. 588– 597 28. K.-H. Laermann, C.Yuhai: On the measurement of the material response of linear photoviscoelastic polymers, Measurement, 279–286 (1993)

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References on Photoviscoelasticity 1. R.D. Mindlin: A mathematical theory of photoviscoelasticity, J. Appl. Phys. 29, 206–210 (1949) 2. R.S. Stein, S. Onogi, D.A. Keedy: The dynamic birefringence of high polymers, J. Polym. Sci. 57, 801–821 (1962) 3. C.W. Ferguson: Analysis of stress-wave propagation by photoviscoelastic techniques, J. Soc. Motion Pict. Telev. Eng. 73, 782–787 (1964) 4. P.S. Theocaris, D. Mylonas: Viscoelastic effects in birefringent coating, J. Appl. Mech. 29, 601–607 (1962) 5. M.L. Williams, R.J. Arenz: The engineering analysis of linear photoviscoelastic materials, Exp. Mech. 4, 249–262 (1964) 6. E.H. Dill: On phenomenological rheo-optical constitutive relations, J. Polym. Sci. Part C 5, 67–74 (1964) 7. C.L. Amba-Rao: Stress-strain-time-birefringence relations in photoelastic plastics with creep, J. Polym. Sci. Pt. C 5, 75–86 (1964) 8. B.E. Read: Dynamic birefringence of amorphous polymers, J. Polym. Sci. Pt. C 5, 87–100 (1964) 9. R.D. Andrews, T.J. Hammack: Temperature dependence of orientation birefringence of polymers in the glassy and rubbery states, J. Polym. Sci. Pt. C 5, 101–112 (1964) 10. R. Yamada, C. Hayashi, S. Onogi, M. Horio: Dynamic birefringence of several high polymers, J. Polym. Sci. Pt. C 5, 123–127 (1964) 11. K. Sasguri, R.S. Stain: Dynamic birefringence of polyolefins, J. Polym. Sci. Pt. C 5, 139–152 (1964) 12. D.G. Legrand, W.R. Haaf: Rheo-optical properties of polymers, J. Polym. Sci. Pt. C 5, 153–161 (1964) 13. I.M. Daniel: Experimental methods for dynamic stress analysis in viscoelastic materials, J. Appl. Mech. 32, 598–606 (1965) 14. I.M. Daniel: Quasistatic properties of a photoviscoelastic material, Exp. Mech. 5, 83–89 (1965) 15. A.J. Arenz, C.W. Ferguson, M.L. Williams: The mechanical and optical characterization of a Solithane 113 composition, Exp. Mech. 7, 183– 188 (1967) 16. H.F. Brinson: Mechanical, optical viscoelastic characterization of Hysol 4290: Time and temperature behavior of Hysol 4290 as obtained from creep tests in conjunction with the time-temperature superposition principle, Exp. Mech. 8, 561–566 (1968)

3.6 Recognizing Viscoelastic Solutions if the Elastic Solution is Known

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29. S. Yoneyama, J. Gotoh, M. Takashi: Experimental analysis of rolling contact stresses in a viscoelastic strip, Exp. Mech. 40, 203–210 (2000) 30. A.I. Shyu, C.T. Isayev, T.I. Li: Photoviscoelastic behavior of amorphous polymers during transition

from the glassy to rubbery state, J. Polym. Sci. Pt. B Polym. Phys. 39, 2252–2262 (2001) 31. Y.-H. Zhao, J. Huang: Photoviscoelastic stress analysis of a plate with a central hole, Exp. Mech. 41, 312–18 (2001)

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W.G. Knauss: The mechanics of polymer fracture, Appl. Mech. Rev. 26, 1–17 (1973) C. Singer, E.J. Holmgard, A.R. Hall (Eds.): A History of Technology (Oxford University Press, New York 1954) J.M. Kelly: Strain rate sensitivity and yield point behavior in mild steel, Int. J. Solids Struct. 3, 521–532 (1967) H.H. Johnson, P.C. Paris: Subcritical flaw growth, Eng. Fract. Mech. 1, 3–45 (1968) I. Finnie: Stress analysis for creep and creeprupture. In: Appllied Mechanics Surveys, ed. by H.N. Abramson (Spartan Macmillan, New York 1966) pp. 373–383 F. Garofalo: Fundamentals of Creep and Creep Rupture in Metals (Macmillan, New York 1965) N.J. Grant, A.W. Mullendore (Eds.): Deformation and Fracture at Elevated Temperatures (MIT Press, Cambridge 1965) F.A. McClintock, A.S. Argon (Eds.): Mechanical Behavior of Materials (Addison-Wesley, Reading 1966) J.B. Conway: Numerical Methods for Creep and Rupture Analyses (Gordon-Breach, New York 1967) J.B. Conway, P.N. Flagella: Creep Rupture Data for the Refractory Metals at High Temperatures (Gordon-Breach, New York 1971) M. Tao: High Temperature Deformation of Vitreloy Bulk Metallic Glasses and Their Composite. Ph.D. Thesis (California Institute of Technology, Pasadena 2006) J. Lu, G. Ravichandran, W.L. Johnson: Deformation behavior of the Zr41.2 Ti13.8 Cu12.5 Ni10 Be22.5 bulk metallic glass over a wide range of strain-rates and temperatures, Acta Mater. 51, 3429–3443 (2003) W.G. Knauss: Viscoelasticity and the timedependent fracture of polymers. In: Comprehensive Structural Integrity, Vol. 2, ed. by I. Milne, R.O. Ritchie, B. Karihaloo (Elsevier, Amsterdam 2003) W. Flügge: Viscoelasticity (Springer, Berlin 1975) H. Lu, X. Zhang, W.G. Knauss: Uniaxial, shear and Poisson relaxation and their conversion to bulk relaxation, Polym. Eng. Sci. 37, 1053–1064 (1997) N.W. Tschoegl, W.G. Knauss, I. Emri: Poisson’s ratio in linear viscoelasticity, a critical review, Mech. Time-Depend. Mater. 6, 3–51 (2002) I.L. Hopkins, R.W. Hamming: On creep and relaxation, J. Appl. Phys. 28, 906–909 (1957)

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R.A. Schapery: Approximate methods of transform inversion for viscoelastic stress analysis, Proc. 4th US Natl. Congr. Appl. Mech. (1962) pp. 1075–1085 J.F. Clauser, W.G. Knauss: On the numerical determination of relaxation and retardation spectra for linearly viscoelastic materials, Trans. Soc. Rheol. 12, 143–153 (1968) G.W. Hedstrom, L. Thigpen, B.P. Bonner, P.H. Worley: Regularization and inverse problems in viscoelasticity, J. Appl. Mech. 51, 121–12 (1984) I. Emri, N.W. Tschoegl: Generating line spectra from experimental responses. Part I: Relaxation modulus and creep compliance, Rheol. Acta. 32, 311–321 (1993) I. Emri, N.W. Tschoegl: Generating line spectra from experimental responses. Part IV: Application to experimental data, Rheol. Acta. 33, 60–70 (1994) I. Emri, N.W. Tschoegl: An iterative computer algorithm for generating line spectra from linear viscoelastic response functions, Int. J. Polym. Mater. 40, 55–79 (1998) N.W. Tschoegl, I. Emri: Generating line spectra from experimental responses. Part II. Storage and loss functions, Rheol. Acta. 32, 322–327 (1993) N.W. Tschoegl, I. Emri: Generating line spectra from experimental responses. Part III. Interconversion between relaxation and retardation behavior, Int. J. Polym. Mater. 18, 117–127 (1992) I. Emri, N.W. Tschoegl: Generating line spectra from experimental responses. Part V. Time-dependent viscosity, Rheol. Acta. 36, 303–306 (1997) I. Emri, B.S. von Bernstorff, R. Cvelbar, A. Nikonov: Re-examination of the approximate methods for interconversion between frequency- and timedependent material functions, J. Non-Newton. Fluid Mech. 129, 75–84 (2005) A. Nikonov, A.R. Davies, I. Emri: The determination of creep and relaxation functions from a single experiment, J. Rheol. 49, 1193–1211 (2005) M.A. Branch, T.F. Coleman, Y. Li: A subspace interior, and conjugate gradient method for large-scale bound-constrained minimization problems, Siam J. Sci. Comput. 21, 1–23 (1999) F. Kohlrausch: Experimental-Untersuchungen über die elastische Nachwirkung bei der Torsion, Ausdehnung und Biegung, Pogg. Ann. Phys. 8, 337 (1876)

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J. Bischoff, E. Catsiff, A.V. Tobolsky: Elastoviscous properties of amorphous polymers in the transition region 1, J. Am. Chem. Soc. 74, 3378–3381 (1952) F. Schwarzl, A.J. Staverman: Time-temperature dependence of linearly viscoelastic behavior, J. Appl. Phys. 23, 838–843 (1952) M.L. Williams, R.F. Landel, J.D. Ferry: Mechanical properties of substances of high molecular weight. 19. The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids, J. Am. Chem. Soc. 77, 3701–3707 (1955) D.J. Plazek: Temperature dependence of the viscoelastic behavior of polystyrene, J. Phys. Chem. US 69, 3480–3487 (1965) D.J. Plazek: The temperature dependence of the viscoelastic behavior of poly(vinyl acetate), Polym. J. 12, 43–53 (1980) R.A. Fava (Ed.): Methods of experimental physics 16C. In: Viscoelastic and Steady-State Rheological Response, ed. by D.J. Plazek (Academic, New York 1979) L.W. Morland, E.H. Lee: Stress analysis for linear viscoelastic materials with temperature variation, Trans. Soc. Rheol. 4, 233–263 (1960) L.J. Heymans: An Engineering Analysis of Polymer Film Adhesion to Rigid Substrates. Ph.D. Thesis (California Institute of Technology, Pasadena 1983) G.U. Losi, W.G. Knauss: Free volume theory and nonlinear viscoelasticity, Polym. Eng. Sci. 32, 542–557 (1992) I. Emri, T. Prodan: A Measuring system for bulk and shear characterization of polymers, Exp. Mech. 46(6), 429–439 (2006) N.W. Tschoegl, W.G. Knauss, I. Emri: The effect of temperature and pressure on the mechanical properties of thermo- and/or piezorheologically simple polymeric materials in thermodynamic equilibrium – a critical review, Mech. Time-Depend. Mater. 6, 53–99 (2002) R.W. Fillers, N.W. Tschoegl: The effect of pressure on the mechanical properties of polymers, Trans. Soc. Rheol. 21, 51–100 (1977) W.K. Moonan, N.W. Tschoegl: The effect of pressure on the mechanical properties of polymers. 2. Expansively and compressibility measurements, Macromolecules 16, 55–59 (1983) W.K. Moonan, N.W. Tschoegl: The effect of pressure on the mechanical properties of polymers. 3. Substitution of the glassy parameters for those of the occupied volume, Int. Polym. Mater. 10, 199–211 (1984) W.K. Moonan, N.W. Tschoegl: The effect of pressure on the mechanical properties of polymers. 4. Measurements in torsion, J. Polym. Sci. Polym. Phys. 23, 623–651 (1985)

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G. Williams, D.C. Watts: Non-symmetrical dielectric behavior arising from a simple empirical decay function, Trans. Faraday Soc. 66, 80–85 (1970) S. Lee, W.G. Knauss: A note on the determination of relaxation and creep data from ramp tests, Mech. Time-Depend. Mater. 4, 1–7 (2000) A. Flory, G.B. McKenna: Finite step rate corrections in stress relaxation experiments: A comparison of two methods, Mech. Time-Depend. Mater. 8, 17–37 (2004) J.F. Tormey, S.C. Britton: Effect of cyclic loading on solid propellant grain structures, AIAA J. 1, 1763–1770 (1963) R.A. Schapery: Thermomechanical behavior of viscoelastic media with variable properties subjected to cyclic loading, J. Appl. Mech. 32, 611–619 (1965) L.R.G. Treloar: Physics of high-molecular materials, Nature 181, 1633–1634 (1958) M.-F. Vallat, D.J. Plazek, B. Bushan: Effects of thermal treatment of biaxially oriented poly(ethylene terephthalate), J. Polym. Sci. Polym. Phys. 24, 1303– 1320 (1986) L.E. Struik: Physical Aging in Amorphous Polymers and Other Materials (Elsevier Scientific, Amsterdam 1978) G.B. McKenna, A.J. Kovacs: Physical aging of poly(methyl methacrylate) in the nonlinear range – torque and normal force measurements, Polym. Eng. Sci. 24, 1138–1141 (1984) A. Lee, G.B. McKenna: Effect of crosslink density on physical aging of epoxy networks, Polymer 29, 1812– 1817 (1988) C. G’Sell, G.B. McKenna: Influence of physical aging on the yield response of model dgeba poly(propylene oxide) epoxy glasses, Polymer 33, 2103–2113 (1992) M.L. Cerrada, G.B. McKenna: Isothermal, isochronal and isostructural results, Macromolecules 33, 3065– 3076 (2000) L.C. Brinson, T.S. Gates: Effects of physical aging on long-term creep of polymers and polymer matrix composites, Int. J. Solids Struct. 32, 827–846 (1995) L.M. Nicholson, K.S. Whitley, T.S. Gates: The combined influence of molecular weight and temperature on the physical aging and creep compliance of a glassy thermoplastic polyimide, Mech. TimeDepend. Mat. 5, 199–227 (2001) J.D. Ferry: Viscoelastic Properties of Polymers, 3rd edn. (Wiley, New York 1980) J.R. Mcloughlin, A.V. Tobolsky: The viscoelastic behavior of polymethyl methacrylate, J. Colloid Sci. 7, 555–568 (1952) H. Leaderman, R.G. Smith, R.W. Jones: Rheology of polyisobutylene. 2. Low molecular weight polymers, J. Polym. Sci. 14, 47–80 (1954)

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W.G. Knauss, V.H. Kenner: On the hygrothermomechanical characterization of polyvinyl acetate, J. Appl. Phys. 51, 5131–5136 (1980) I. Emri, V. Pavˇsek: On the influence of moisture on the mechanical properties of polymers, Mater. Forum 16, 123–131 (1992) D.J. Plazek: Magnetic bearing torsional creep apparatus, J. Polym. Sci. Polym. Chem. 6, 621–633 (1968) D.J. Plazek, M.N. Vrancken, J.W. Berge: A torsion pendulum for dynamic and creep measurements of soft viscoelastic materials, Trans. Soc. Rheol. 2, 39–51 (1958) G.C. Berry, J.O. Park, D.W. Meitz, M.H. Birnboim, D.J. Plazek: A rotational rheometer for rheological studies with prescribed strain or stress history, J. Polym. Sci. Polym. Phys. Ed. 27, 273–296 (1989) R.S. Duran, G.B. McKenna: A torsional dilatometer for volume change measurements on deformed glasses: Instrument description and measurements on equilibrated glasses, J. Rheol. 34, 813–839 (1990) J.E. McKinney, S. Edelman, R.S. Marvin: Apparatus for the direct determination of the dynamic bulk modulus, J. Appl. Phys. 27, 425–430 (1956) J.E. McKinney, H.V. Belcher: Dynamic compressibility of poly(vinyl acetate) and its relation to free volume, J. Res. Nat. Bur. Stand. Phys. Chem. 67A, 43–53 (1963) T.H. Deng, W.G. Knauss: The temperature and frequency dependence of the bulk compliance of Poly(vinyl acetate). A re-examination, Mech. TimeDepend. Mater. 1, 33–49 (1997) S. Sane, W.G. Knauss: The time-dependent bulk response of poly (methyl methacrylate), Mech. TimeDepend. Mater. 5, 293–324 (2001) Z. Ma, K. Ravi-Chandar: Confined compression– a stable homogeneous deformation for multiaxial constitutive characterization, Exp. Mech. 40, 38–45 (2000) K. Ravi-Chandar, Z. Ma: Inelastic deformation in polymers under multiaxial compression, Mech. Time-Depend. Mater. 4, 333–357 (2000) D. Qvale, K. Ravi-Chandar: Viscoelastic characterization of polymers under multiaxial compression, Mech. Time-Depend. Mater. 8, 193–214 (2004) S.J. Park, K.M. Liechti, S. Roy: Simplified bulk experiments and hygrothermal nonlinear viscoelasticty, Mech. Time-Depend. Mater. 8, 303–344 (2004) A. Kralj, T. Prodan, I. Emri: An apparatus for measuring the effect of pressure on the timedependent properties of polymers, J. Rheol. 45, 929–943 (2001) J.B. Pethica, R. Hutchings, W.C. Oliver: Hardness measurement at penetration depths as small as 20 nm, Philos. Mag. 48, 593–606 (1983) W.C. Oliver, R. Hutchings, J.B. Pethica: Measurement of hardness at indentation depths as low as 20 nanometers. In: Microindentation Techniques in Materials Science and Engineering ASTM STP 889, ed.

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3.111 R.A. Schapery: A theory of non-linear thermoviscoelasticity based on irreversible thermodynamics, Proc. 5th Natl. Cong. Appl. Mech. (1966) pp. 511–530 3.112 R.A. Schapery: On the characterization of nonlinear viscoelastic materials, Polym. Eng. Sci. 9, 295–310 (1969) 3.113 H. Lu, W.G. Knauss: The role of dilatation in the nonlinearly viscoelastic behavior of pmma under multiaxial stress states, Mech. Time-Depend. Mater. 2, 307–334 (1999) 3.114 H. Lu, G. Vendroux, W.G. Knauss: Surface deformation measurements of cylindrical specimens by digital image correlation, Exp. Mech. 37, 433–439 (1997) 3.115 W.H. Peters, W.F. Ranson: Digital imaging techniques in experimental stress analysis, Opt. Eng. 21, 427–432 (1982) 3.116 M.A. Sutton, W.J. Wolters, W.H. Peters, W.F. Ranson, S.R. McNeil: Determination of displacements using an improved digital image correlation method, Im. Vis. Comput. 1, 133–139 (1983) 3.117 H. Lu: Nonlinear Thermo-Mechanical Behavior of Polymers under Multiaxial Loading. Ph.D. Thesis (California Institute of Technology, Pasadena 1997) 3.118 D.R. Lide: CRC Handbook of Chemistry and Physics (CRC Press, Boca Raton 1995) 3.119 G.B. McKenna: On the physics required for prediction of long term performance of polymers and their composites, J. Res. NIST 99(2), 169–189 (1994) 3.120 P.A. O’Connell, G.B. McKenna: Large deformation response of polycarbonate: Time-temperature, time-aging time, and time-strain superposition, Polym. Eng. Sci. 37, 1485–1495 (1997) 3.121 J.L. Sullivan, E.J. Blais, D. Houston: Physical aging in the creep-behavior of thermosetting and thermoplastic composites, Compos. Sci. Technol. 47, 389–403 (1993) 3.122 I.M. Hodge: Physical aging in polymer glasses, Science 267, 1945–1947 (1995) 3.123 N. Goldenberg, M. Arcan, E. Nicolau: On the most suitable specimen shape for testing sheer strength of plastics, ASTM STP 247, 115–121 (1958) 3.124 M. Arcan, Z. Hashin, A. Voloshin: A method to produce uniform plane-stress states with applications to fiber-reinforced materials, Exp. Mech. 18, 141–146 (1978) 3.125 M. Arcan: Discussion of the iosipescu shear test as applied to composite materials, Exp. Mech. 24, 66– 67 (1984) 3.126 W.G. Knauss, W. Zhu: Nonlinearly viscoelastic behavior of polycarbonate. I. Response under pure shear, Mech. Time-Depend. Mater. 6, 231–269 (2002) 3.127 W.G. Knauss, W. Zhu: Nonlinearly viscoelastic behavior of polycarbonate. II. The role of volumetric strain, Mech. Time-Depend. Mater. 6, 301–322 (2002)

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J.L. Loubet, B.N. Lucas, W.C. Oliver: Some measurements of viscoelastic properties with the help of nanoindentation, International workshop on Instrumental Indentation, ed. by D.T. Smith (NIST Special Publication, San Diego 1995) pp. 31–34 B.N. Lucas, W.C. Oliver, J.E. Swindeman: The dynamic of frequency-specific, depth-sensing indentation testing, Fundamentals of Nanoindentation and Nanotribology, Vol. 522, ed. by N.R. Moody (Mater. Res. Soc., Warrendale 1998) pp. 3–14, MRS Meeting, San Francisco 1998 R.J. Arenz, M.L. Williams (Eds.): A photoelastic technique for ground shock investigation. In: Ballistic and Space Technology (Academic, New York 1960) Y. Miyano, T. Tamura, T. Kunio: The mechanical and optical characterization of Polyurethane with application to photoviscoelastic analysis, Bull. JSME 12, 26–31 (1969) A.B.J. Clark, R.J. Sanford: A comparison of static and dynamic properties of photoelastic materialsm, Exp. Mech. 3, 148–151 (1963) O.A. Hasan, M.C. Boyce: A constitutive model for the nonlinear viscoelastic-viscoplastic behavior of glassy polymers, Polym. Eng. Sci. 35, 331–334 (1995) J.S. Bergström, M.C. Boyce: Constitutive modeling of the large strain time-dependent behavior of elastomers, J. Mech. Phys. Solids 46, 931–954 (1998) P.D. Ewing, S. Turner, J.G. Williams: Combined tension-torsion studies on polymers: apparatus and preliminary results for Polyethylene, J. Strain Anal. Eng. 7, 9–22 (1972) C. Bauwens-Crowet: The compression yield behavior of polymethal methacrylate over a wide range of temperatures and strain rates, J. Mater. Sci. 8, 968– 979 (1973) L.C. Caraprllucci, A.F. Yee: The biaxial deformation and yield behavior of bisphenol-A polycarbonate: Effect of anisotropy, Polym. Eng. Sci. 26, 920–930 (1986) W.G. Knauss, I. Emri: Non-linear viscoelasticity based on free volume consideration, Comput. Struct. 13, 123–128 (1981) W.G. Knauss, I. Emri: Volume change and the nonlinearly thermo-viscoelastic constitution of polymers, Polym. Eng. Sci. 27, 86–100 (1987) G.U. Losi, W.G. Knauss: Free volume theory and nonlinear thermoviscoelasticity, Polym. Eng. Sci. 32, 542–557 (1992) G.U. Losi, W.G. Knauss: Thermal stresses in nonlinearly viscoelastic solids, J. Appl. Mech. 59, S43–S49 (1992) W.G. Knauss, S. Sundaram: Pressure-sensitive dissipation in elastomers and its implications for the detonation of plastic explosives, J. Appl. Phys. 96, 7254–7266 (2004)

References

97

Composite M 4. Composite Materials

Peter G. Ifju

4.2.3 Shear Testing ............................... 105 4.2.4 Single-Geometry Tests .................. 106 4.3

Micromechanics.................................... 4.3.1 In Situ Strain Measurements .......... 4.3.2 Fiber–Matrix Interface Characterization........................... 4.3.3 Nanoscale Testing ........................ 4.3.4 Self-Healing Composites ...............

107 107 108 109 111

4.4 Interlaminar Testing ............................. 4.4.1 Mode I Fracture............................ 4.4.2 Mode II Fracture........................... 4.4.3 Edge Effects .................................

111 111 113 114

4.5 Textile Composite Materials ................... 4.5.1 Documentation of Surface Strain .... 4.5.2 Strain Gage Selection .................... 4.5.3 Edge Effects in Textile Composites ..

114 114 116 117 117 118 118 119 120 120

4.1

Strain Gage Applications ....................... 98 4.1.1 Transverse Sensitivity Corrections ... 98 4.1.2 Error Due to Gage Misalignment ..... 99 4.1.3 Temperature Compensation ........... 100 4.1.4 Self-Heating Effects ...................... 101 4.1.5 Additional Considerations ............. 101

4.6 Residual Stresses in Composites ............. 4.6.1 Composite Sectioning ................... 4.6.2 Hole-Drilling Methods .................. 4.6.3 Strain Gage Methods .................... 4.6.4 Laminate Warpage Methods .......... 4.6.5 The Cure Reference Method ...........

4.2

Material Property Testing ...................... 102 4.2.1 Tension Testing ............................ 103 4.2.2 Compression Testing ..................... 103

4.7

References .................................................. 121

The advancement of civilization has historically been tied to the materials utilized during the era. For example, we document the technological progression of mankind through the stone age, the bronze age, the iron age, etc. [4.1]. If such a description were to be used for the second half of the 20-th century, it is conceivable that the era could be called the composites age. AdR vanced composites such as carbon fiber and Kevlar reinforced polymers, and ceramic matrix materials represent the pinnacle of structural material forms during this era. These are the material systems of choice when high specific strength and stiffness are required [4.2–4].

A composite is defined as a material composed of two or more constituents whose mechanical properties are distinctly different from each other, and phase separated such that at least one of the constituents forms a continuous interconnected region and one of the constituents acts as the reinforcement (and is typically discontinuous) [4.2–4]. The resulting composite has physical properties that differ from the original constituents. By nature, experimental stress analysis on composites can be considerably difficult, since composites can be highly anisotropic and heterogeneous. Many of the well-established experimental techniques

Future Challenges ................................. 121

Part A 4

The application of selected experimental stress analysis techniques for mechanical testing of composite materials is reviewed. Because of the anisotropic and heterogeneous nature of composites, novel methodologies are often adopted. This chapter reviews many of the more applicable experimental methods in specific research areas, including: composite-specific strain gage applications, material property characterization, micromechanics, interlaminar testing, textile composite testing, and residual stress measurements. It would be impossible to review all test methods associated with composites in this short chapter, but many prevalent ones are covered.

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require special procedures, different data analysis, or may not be applicable. Additionally, for anisotropic materials the principal strain direction does not necessarily coincide with the principal stress direction. Failure mechanisms in composites can be also be significantly more complicated than for traditional isotropic materials. There are three main forms of composites: particulate reinforced, lamellar, and fiber reinforced. Some composite materials combine two or more of these forms; for example, a sandwich structure is a lamellar form that can incorporate fiber reinforced face sheets. In this chapter we will focus on advanced fiber and particulate reinforced composites. The objective of this chapter is to help guide the experimentalist by introducing modern experimental stress analysis techniques that are applicable to composites. The chapter is organized by research topics within the composites field. These include: special consideration when testing with electrical resistance strain gages,

coupon and material property testing, micromechanics studies, interlaminar fracture, textile composites, residual stress, and thermal testing. Other chapters of this manual cover time-dependent testing, high-strain-rate experiments, thermoelastic stress analysis, and fiberoptic strain gages, all of which are also important to the study of composites. In 1989 the Composite Materials Technical Division (CMTD) of the Society for Experimental Mechanics, sponsored publication of the Manual on Experimental Methods for Mechanical Testing of Composites [4.5]. The publication was edited by Pendelton and Tuttle. In 1998, Jenkins and the CMTD revised and updated the manual [4.6]. The most recent version is a 264-page book that covers broad aspects of composite testing and analysis. This chapter is significantly more condensed, and highlights applications as well as updates the most recent manual. Often during the course of the chapter, the author will refer the reader to this manual for more details.

4.1 Strain Gage Applications To this day, strain gages are still the most used experimental method to measure mechanical strain on isotropic as well as composite material applications. The previous chapters on strain gages clearly outline the theory and application of strain gages for general testing. For composite applications there are a number of special considerations [4.6] and procedures that should be followed in order to insure accurate and repeatable measurements. This section expands on the use of strain gages for composite material applications by providing a description of these special considerations and showing examples of how severe errors may occur if precautions are not implemented.

4.1.1 Transverse Sensitivity Corrections To obtain accurate strain measurements, transverse sensitivity corrections must be performed in all but two cases [4.7, 8]. The first case is when the transverse sensitivity coefficient K t is equal to zero. This is very rare: typically the transverse sensitivity coefficient is nonzero. The second is when the axial strain is measured on a material with a Poisson’s ratio equal to that of the strain gage calibration device and the state of stress is identical to the calibration device. For the latter case, in practice, this means that the tensile strain is measured for a uniaxial state of stress on a material with a Pois-

son’s ratio ν0 equal to 0.285 (steel). In general, errors due to transverse sensitivity are greatest when the transverse strain with respect to the gage εt is large compared to the axial strain with respect to the gage εa and the transverse sensitivity coefficient is high. Equation (4.1) describes the percentage error E if no corrections are performed due to transverse sensitivity. E=

K t [(εt /εa ) + ν0 ] × 100 . 1 − ν0 K t

(4.1)

For a simple experiment to determine the Poisson’s ratio of metallic material using strain gages, shown in Fig. 4.1, one would apply two strain gages to a tensile specimen, one aligned in the axial direction with respect to the load and one mounted in the transverse direction with respect to the load. If the specimen were aluminum (with a Poisson’s ratio of 0.33) and the transverse sensitivity correction factor of the gage was 2%, then the error in the axial gage would be very small (E = −0.09%) and the error in the transverse gage would be larger (E = −5.46%). If one were to completely ignore these corrections then one would calculate the Poisson’s ratio to be 0.312, rather than 0.33. Although most strain gages have a transverse sensitivity correction factor less than 2%, one can appreciate that corrections should be made.

Composite Materials

σx

4.1 Strain Gage Applications

99

to correct for transverse errors in the other gage. When orthogonal gages are used, as shown in Fig. 4.2, biaxial strain gage rosette corrections may be applied. The correction equations are given in (4.2), which gives the true strains ε1 and ε2 , obtained from the indicated strains ε1 and ε2 , and the transverse sensitivity correction factors K t1 and K t2 for gages 1 and 2.

σx

ε1 (1 − ν0 K t1 ) − K t1 ε2 (1 − ν0 K t2 ) , 1 − K t1 K t2 ε (1 − ν0 K t2 ) − K t2 ε1 (1 − ν0 K t1 ) ε2 = 2 . (4.2) 1 − K t1 K t2 Note that for each gage in the rosette, there may be a different transverse sensitivity coefficient. Similarly, the equations for a rectangular rosette (as shown in Fig. 4.2) are provided in (4.3). Often K t1 and K t3 are equal to each other. ε1 =

Fiber direction

Isotropic material

Unidirectional composite

Even for a seemingly routine test, if the specimen were unidirectional graphite/epoxy, large errors can arise if corrections are not made. For instance, if the objective is to measure ν21 , consider a specimen loaded in the direction transverse to the fibers (Fig. 4.1). For graphite/epoxy a typical ν21 is around 0.015 and therefore the strain gage used to measure the normal strain transverse to the loading direction experiences εt on the order of 66 times that of εa . For K t = 2% the strain gage measuring the strain transverse to the loading direction would yield a value with −132% error if transverse sensitivity corrections are not applied. Therefore the Poisson ratio ν21 would have an error of the order 132% since the error in the axial gage is very small. In order to perform an accurate correction for transverse sensitivity, at least two gages are required. For instance, in the above example, one gage may be used y

y

2

3

2 45°

x 1 Two gage biaxial rosette

1

x

Three gage rectangular rosette

Fig. 4.2 Biaxial and rectangular rosettes

ε1 (1 − ν0 K t1 ) − K t1 ε3 (1 − ν0 K t3 ) , 1 − K t1 K t3 ε (1 − ν0 K t2 ) ε2 = 2 1− K t2 − K t2 ε1 (1 − ν0 K t1 )(1 − K t3 ) + ε3 (1 − ν0 K t3 )(1 − K t1 ) −1 × (1 − K t1 K t3 )(1 − K t2 ) , ε3 (1 − ν0 K t3 ) − K t3 ε1 (1 − ν0 K t1 ) ε3 = . 1 − K t1 K t3

ε1 =

Part A 4.1

Fig. 4.1 Measurement of the Poisson ratio ν for an isotropic material and ν21 for a unidirectional composite material

(4.3)

4.1.2 Error Due to Gage Misalignment Whenever a practitioner mounts strain gages on a mechanical component or test coupon, gage alignment is an important consideration [4.9]. If the gage is not aligned precisely with respect to the loading direction or geometry of the component, the strain information that is recorded may differ from the value obtained if it were aligned properly. This can lead to errors and/or misinterpretation of the data. For isotropic materials, errors of the order of a couple degrees typically do not lead to significant errors in strain measurement. For instance, for a simple tension test on a steel bar, where both axial and transverse (with respect to the load) gages are utilized, there would be an error in the axial strain of − 0.63% and in the transverse strain of 2.2%, for a misalignment of 4◦ . This may or may not be within the acceptable error for such a measurement. Most able practitioners can routinely align gages to within 2◦ . The errors for such a 2◦ misalignment are − 0.16% and 0.55% for the axial and transverse gages, respectively.

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θ

Percent strain measurement error 80

σx

β = –4°

β

60 β = –2°

40 20 0 β

–20 β = 2°

–40 –60 –80

β = 4°

0

10

20

30

40

50

60

70 80 90 Fiber angle (deg)

Fig. 4.5 Transverse strain errors as a function of fiber an-

σx

gle and gage misalignment error (courtesy M. E. Tuttle)

Part A 4.1

Fig. 4.3 Gage misalignment nomenclature

For unidirectional composites, misalignment can introduce significant errors because of the deleterious interaction between the loading direction and the orthotropy of the material. Take the same unidirectional graphite/epoxy material from the previous section where the moduli are E 1 = 170 GPa, E 2 = 8 GPa, G 12 = 6 GPa, and the Poisson’s ratio is ν12 = 0.32. A case where the gage is intended to line up with the load will be used for an example. For a tension test, where the fiber direction with respect to the loading Percent strain measurement error 20

direction is denoted by θ, and the gage misalignment denoted by β, Figs. 4.3, 4.4, and 4.5 illustrate the relationship between the misalignment angle β and the percentage strain measurement error and the fiber angle θ. Figure 4.4 pertains to errors in the axial direction of the specimen and Fig. 4.5 pertains to the transverse direction. Errors exceeding 15% can occur at fiber angles near 12◦ for the axial gage and greater than 60% near a 12◦ fiber angle for the transverse gage. Because of the exaggerated error when testing composites, special procedures may be warranted in the experimental program to ensure proper gage alignment.

4.1.3 Temperature Compensation

β = –4°

15 β = –2°

10 5 0 –5

β = 2°

–10 –15 –20

β = 4°

0

10

20

30

40

50

60

70 80 90 Fiber angle (deg)

Fig. 4.4 Axial strain errors as a function of fiber angle and

gage misalignment angle (courtesy M. E. Tuttle)

When a strain gage is mounted on a mechanical component or test coupon which is subjected to both mechanical loads and temperature variations, the gage measurements are a combination of strains induced by both effects [4.10, 11]. It is often the objective to separate the effects of the mechanical loads, a process known as temperature compensation. The strain gage output which results from temperature variation can occur for two reasons: the electrical resistance of the gage changes with temperature, and the coefficient of expansion of the gage may be different from that of the underlying material. Temperature compensation is more difficult when dealing with composites than with homogeneous isotropic materials. Some of the options for metallic materials are not applicable to composites. For instance,

Composite Materials

4.1.4 Self-Heating Effects According to Joule’s law, the application of voltage to a strain gage (a resistor) creates a power loss which results in the generation of heat [4.7, 12]. This heat is dissipated in the form of conduction by the substrate material and the surrounding environment. The heat that is generated is primarily related to the excitation voltage of the strain gage circuit, as well as to the resistance of the strain gage. The ability of the substrate material to stabilize or neutralize self-heating is related to its power density, which is a measure of

its ability to act as a heat sink. When self-heating is not neutralized it can result in gradual heating of the substrate, which can lead to creep of the underlying material, as well as signal hysteresis and drift. This is typically not acceptable and therefore steps must be implemented in order to avoid self-heating. In general, one can choose a combination of gage resistance, excitation voltage, and gage size to avoid the problem. Larger gage resistance levels develop less self-heating, larger grid areas allow more efficient dissipation, and lower excitation levels decrease self-heating. However, by decreasing the excitation level, the circuit sensitivity also decreases. Additionally, higher-resistance and larger gages can be more expensive. A balance must be struck in order to optimize the experimental configuration [4.12]. Most composite materials are fabricated with polymer matrices. Since polymers have low power densities (and are thus poor heat sinks), it is very important to select gages with a higher resistance and larger grid area, and/or use lower excitation levels as compared to strain gages used for traditional materials. A range of acceptable power densities for composites is 0.31–1.2 kW/m2 (0.2–0.77 W/in.2 ). As such, a rule-of-thumb recommendation was developed by Slaminko [4.11] for strain gage applications on composites: Size: 3 mm(0.125 in.) or larger; Resistance: 350 Ω or higher; Excitation voltage: 3 V or less. Smaller gages can be used, if necessary (to measure in locations with high strain gradients, for instance), by reducing the excitation levels.

4.1.5 Additional Considerations The process of performing a successful experiment with strain gages starts with the selection of the gage. Vishay [4.13] suggests the following order should be followed when selecting a gage: 1. 2. 3. 4. 5. 6.

gage length gage pattern gage series options resistance self-temperature compensation

Note that the gage size is the first, and arguably the most important, consideration. This is especially

101

Part A 4.1

the self-temperature compensation (S-T-C) numbers provided by the gage manufacturer do not match most composites and therefore the temperature compensation data (polynomial and graphical corrections) cannot be applied. As a result other methods of compensation are required. The so-called dummy compensation method is a robust method for performing compensation on composites. Dummy compensation requires a second gage to be mounted in a half-bridge configuration with the active gage. Three requirements must be fulfilled in order to perform dummy compensation. First, the dummy gage must be from the same gage lot as the gage that measures the combined thermal and mechanical components. Second, the dummy gage must be mounted on an unconstrained piece of the same material, of approximately the same volume and aligned in the same orientation with respect to the orthotropy as the mechanically loaded specimen. Finally, the two specimens (one loaded and the other unconstrained) must be in the same thermal environment and in close proximity. Violation of any of these requirements can result in large errors. The two gages are then connected in a half-bridge configuration where the thermal output from each gage cancel. One can also perform a precalibration test to separate the thermal strain from the mechanical strain. This is performed by mounting a gage on the specimen to be mechanically tested through a range of temperatures. Then, the specimen is subjected to the thermal cycle only and the apparent strain is recorded as a function of temperature. Once this is established, the specimen can be mechanically tested over the same temperature range. The strains due to thermal effects are subtracted from the results of the test with both thermal and mechanical contributions. An accurate temperature measurement is required for this technique as well as postprocessing of the data.

4.1 Strain Gage Applications

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true for composites, since the scale associated with homogeneity is often significantly larger than that for traditional materials. For textile composite forms, which will be discussed later, the optimal strain gage size depends on the textile architecture. In general, smaller gages will produce more experimental scatter than larger gages and therefore should be avoided, unless strains are to be measured in areas of high gradients. Additionally, in some instances the average value for a highly nonuniform strain distribution is desired and thus larger gages, or those that span the entire test section, are more appropriate [4.14]. The selection of the gage pattern is largely dependent on the objectives of the experiment, for instance, whether one two or three axes are of interest, or whether normal or shear strains are required. If multidirectional gages are used there is often a choice of planar or stacked configurations. For experiments where high stress gradients are present and the material is more heterogeneous, stacked configurations are desired. Stacked configurations, however, can be more expensive, and create more reinforcement effects and self-heating problems. The gage series relates to the materials (foil and backing) that are used in the strain gage. The available choices relate to the temperature range, fatigue characteristics, and stability. The most common sensing grid alloys used for composites are constantan and Karma. Both have good sensitivity, stability, and fatigue resistance. Karma is more stable at temperatures over 65 ◦ C (150 ◦ F), but is more difficult to solder to than constantan. Polyimide backing materials are well suited for composites and allow for contourability and maximum

elongation, while fiber glass reinforced phenolics offer better temperature stability. As in noncomposite applications there may be instances where options such as preattached leads are required. As far as the resistance of the gage is concerned, Slaminko recommends that 350 Ω or greater should be used. Finally, since S-T-C is generally not an option for composites, there is no need to consider this number and any S-T-C value is appropriate. There are other considerations that the experimentalist should be aware of when using strain gages on composites. For example, the ideal measurement technique should be nonintrusive. For strain gage applications this means that, by applying the gage to the underlying material, the strain data recorded represents that for the case where there is no gage present. Since the gage and the adhesive used to attach the gage have finite stiffness, there is always a reinforcing effect. For measurement of material systems where the modulus is much higher than the gage material this effect is typically considered insignificant and can be ignored. When taking measurements on relatively compliant materials, however, gage reinforcement can lead to errors. In the case of composites, the stiffness of the material is highly orthotropic in nature, ranging from modulus values similar to that of common strain gages to values that are significantly higher. Additionally, when considering the entire installation, adhesive selection may be an important consideration when taking measurements along compliant axes (such as transverse loading). A detailed description of reinforcement effects is presented by Perry in the manual [4.15, 16] for both isotropic and anisotropic materials.

4.2 Material Property Testing With the rapid development of composite material systems throughout the last three decades, there has been a need to characterize them accurately. The primary loading conditions of tension, compression, and shear applied to more than one loading direction introduced many new mechanical test methods. Because of this rapid progress, standardization has not been entirely achieved for all loading conditions. The American Society for Testing and Materials (ASTM), under the D30 subcommittee, is the primary organization for composite test standards. The organization consists of a voluntary membership and conducts round-robin testing and standards writing. Recent activ-

ity by the US Military Standards (Mil Specs) under the MIL-Handbook-17 (Polymer Matrix Composites) has expanded standards to include fabrication. Five volumes of the handbook, released in 2002, include: Guidelines for Characterization of Structural Materials, Polymer Matrix Composites: Material Properties, Materials Usage, Design, and Analysis, Metal Matrix Composites, and “Ceramic Matrix Composites”. Planning is under way for the release of a volume on Structural Sandwich Composites. Internationally, the British Standards Institute (BSI) in the United Kingdom, the Association Française de National (AFNOR) in France, the Deutsches Institut für Normung (DIN) in Germany,

Composite Materials

4.2.1 Tension Testing Tension testing under ASTM D 3039 [4.19] has become a well-accepted standard for testing of high-modulus and high-strength composites. The standard defines a tabbed, straight, flat specimen of high aspect ratio (Fig. 4.6). The tabs, with a beveled entry into the test section, are added to the specimen to allow for substanASTM D 3039

ASTM D 608

Fig. 4.6 ASTM standard tension test specimen geometries

tial clamping forces at the grip. These tabs are typically made from G-10 glass/epoxy and are adhered using a high-strength epoxy. Untabbed, dog-boned specimens (Fig. 4.6), under ASTM D 608 [4.20] may be use for some composites that have lower tensile strength values (for instance transverse tension testing). However, for composite systems with higher tensile strength, often the enlarged portion shears off parallel to the long dimension, thus reducing the specimen to a flat, straight specimen similar to that used in ASTM D 3039, without tabs. In general, ASTM D 3039 is appropriate for almost any composite form. Both mechanical and hydraulic grips may be used for ASTM D 3039. The clamping pressure on wedgetype mechanical grips is generally proportional to the applied loads. This reduces the potential for crushing the specimen in the tabbed area. Hydraulic grips add more versatility, since the clamping force is independent of the load, although care must be taken to avoid specimen crushing.

4.2.2 Compression Testing Compression testing of composites has received a considerable amount of attention and has proven to be more complicated than tension testing [4.17]. Since the compressive strength (in the fiber direction) is generally lower than the tensile strength, and since there is a desire to use a larger fraction of the ultimate load capacity, there is a need to characterize the compressive strength accurately. Because relatively thin specimens are desired, buckling may occur if the unsupported length is too large. Therefore, in general, most compression tests are designed such that the specimen fails due to true compressive failure of the material and not due to Euler buckling. There are three methods [4.21–23] of introducing compressive loading into the specimen: via shear through end tabs (ASTM D 3410), by direct end loading (ASTM D 695), and by bending of a composite sandwich panel (ASTM D 5476). The latter is rarely used since the specimens are typically large, difficult to manufacture, and expensive. Additionally, there is a concern that prevails among the composites community that the compressive failure strength may be artificially elevated by the presence of the core being bonded along the entire face sheet. Introducing the compressive load via shear through end tabs is similar to the way loads are introduced in tensile tests, except that the loads are of opposite sign and therefore the wedges in the grips are inverted. In ASTM D 3410 two test fixtures are described: the

103

Part A 4.2

Japanese Industrial Standards (JIS) and Ministry of International Trade and Industry (MITI) distributed by the Japanese Standards Association, are the primary standardization organizations. Whether testing composite materials in tension, compression or shear, there are three basic objectives that should be realized. First, since material systems are often expensive (especially when they are initially developed), thin laminates and small specimens are generally desired. Second, the test method should be capable of measuring the elastic properties as well as the failure strength. As such, the state of stress in the test section of the specimen must be pure (without other stresses) and uniform. Third, the test should be devised such that common universal testing machines can be utilized. Often, these considerations rule out the use of complicated geometries and testing in specialized testing machines. The Manual on Experimental Methods of Mechanical Testing of Composites [4.17, 18] has a comprehensive chapter written by Adams on test methods. This section will only briefly describe the most used test methods and summarize the appropriate procedures.

4.2 Material Property Testing

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Solid Mechanics Topics

Fig. 4.7 The Celanese compression fixture (Wyoming Test

Fixtures Inc.)

Part A 4.2

Celanese compression fixture [4.24] and the IITRI compression fixture [4.25]; they are shown in Figs. 4.7 and 4.8. Both of these test fixtures have a specified gage length (unsupported length) of 12.8 mm (0.5 in.). Testing of thin specimens may lead to buckling if the slenderness ratio is too high. The original standard was written for the Celanese fixture in 1975. The fixture incorporates cylindrical wedges and has an alignment sleeve. It has been criticized for being potentially unstable if precautions, outlined in the standard, are not followed. These instabilities arise because the wedge grips travel in a conical track and some degrees of freedom are not well constrained. Additionally, the fixture is limited to specimens with a maximum width of 6.4 mm (0.25 in.). For testing of textile composites and materials with heterogeneity on a larger scale, the limited width can be a problem. The IITRI fixture was developed to eliminate the problem of instability by utilizing a flat wedge grip, thus minimizing degrees of freedom. The fixture also allows for testing of wider and thicker specimens. As a result, the fixture is significantly larger than the Celanese fix-

Fig. 4.8 The IITRI compression fixture (Wyoming Test Fixtures Inc.)

ture and weighs in at almost 46 kg (100 lb). This effects ease of handling and increases the cost (typically there is a proportional relationship between the weight and cost of the fixture). By combining the favorable features of the Celanese and IITRI fixtures, Adams et al. developed the Wyoming-modified Celanese compression fixture [4.26], shown in Fig. 4.9. The fixture can accommodate a specimen of 12.8 mm (0.5 in.) wide and 7.6 mm (0.3 in.) thick, twice the values of the original Celanese fixture. Another modification of the Celanese fixture is the German DIN 65 380 compression fixture [4.27]. It uses flat wedge grips and an alignment sleeve. The second common method used to introduce load into the compression specimen is through the ends of the specimen (ASTM D 695) [4.22]. This method can be used with or without end tabs. End tabs increase the bearing surface and thus prevent bearing failure prior to compression failure inside the test section. The clamped portion of the fixture confines the specimen and also suppresses bearing failure, which usually occurs in the form of brooming. Some fixtures can accommodate dog-bone geometries without tabs, again to increase the bearing surface area. Generally speaking, for specimens with compression failure strengths less than 700 MPa (100 ksi) dog-boning may be adequate. For those above that value tabs are usually required because the bog-boned portion shears, and subsequent bearing failure occurs at the ends. Examples of end-loaded fixtures include the Wyoming end-loaded side-supported fixture [4.28], the modified ASTM D 695 compression test fixture, and the NASA short block compression test fixture. Specimens may

Fig. 4.9 The Wyoming modified compression fixture

(Wyoming Test Fixtures Inc.)

Composite Materials

also be loaded through both end loading and shear. An example of such a fixture is the Wyoming combinedloading compression fixture (ASTM D 6641) [4.29,30]. The fixture incorporates the advantages of both load introduction schemes.

4.2.3 Shear Testing

and colleagues at the University of Wyoming along with the development of dedicated loading fixtures. The Iosipescu specimen, illustrated in Fig. 4.10, is a relatively small, rectangular beam with V-shaped notches to concentrate the shear in the test section (the volume between the notches). The notches serve two purposes. They reduce the cross-sectional area in the test section (thus allowing for failure to be confined) and they produce a more uniform state of shear. The latter is highly dependent on the notch angle and the orthotropy of the material being tested. In general, this test violates one of the prerequisites for material strength measurements, since the state of stress in the test section is neither truly pure nor is it perfectly uniform. There exists a free edge in the test section, where the shear stresses are zero (in this case at the notch roots). The shear stresses then rise from the free edge and form a distribution that varies with fiber orientation and notch angle. This large gradient must be accommodated by normal stress gradients, according to equilibrium, and thus the state of shear is neither pure nor uniform. Although, strictly speaking, flat coupons cannot produce the perfect shear field, the Iosipescu geometry provides a practical approach, since the shear stress and strain distribution is near uniform. Shear strain information is recorded with strain gages located in the test section. Two styles have been proposed, a centrally located small ±45◦ rosette and a ±45◦ rosette that spans the entire test section [4.14]. Displacements (μm) –70

0 y x

19 Experimental ux contours –202

–15 Experimental uy contours

Fig. 4.10 The modified Wyoming Iosipescu shear test fix-

ture (Wyoming Test Fixtures Inc.)

Fig. 4.11 Loading condition and displacement fields for a proposed T-shaped specimen (courtesy M. Grediac)

105

Part A 4.2

Shear testing has also received considerable attention in recent years. Unlike tension and compression testing, it is difficult to create a pure and uniform shear field using a specimen made from a flat coupon. Ideally, thin-walled torsion testing provides the most uniform and pure shear state of shear. However, cylindrical composite specimens are inordinately expensive to produce, torsion test stands are relatively rare, and the obtained shear properties extended to flat laminates are questionable. Therefore, numerous specimens have been proposed for shear testing of flat coupons [4.17]. These include the two- and three-rail shear tests, the off-axis tension test, the ±45 tension test, short beam shear, notched compression, picture frame, and notch tests such as the Arcan and Iosipescu specimens. The latter [4.31–33] (ASTM D 5379) has become the most widely used test method for composites. First developed by Nicolai Iosipescu in the early 1960s for isotropic materials, it was extended for use on composites by Adams

4.2 Material Property Testing

106

Part A

Solid Mechanics Topics

Load cell

0°

[(0/90)3]s r = 2

αA

Surface covered with a grating

Fixed

Moveable

265 εxyexp (i, j)

εxyana (i, j)

εxyexp (i, j) – εxyana (i, j)

Part A 4.2 Fig. 4.12 Open-hole specimen and loading configuration with experimental shear strain distribution, analytic shear strain distribution, and differences after the application of difference minimization using the Levenberg–Marquardt algorithm (courtesy J. Molimard)

The latter was developed to correct for the nonuniform shear stress distributions and reduces the experimental scatter when employed on textile composites or materials with significant heterogeneity. It is suggested that gages be placed on both sides of the specimen to cancel the affects of twist [4.34]. By using two Iosipescu strain gages (one on each side of the specimen) the coefficient of variation in modulus can be reduce to less than 2%. A number of fixtures have been proposed for loading the Iosipescu specimen, the most popular being the modified Wyoming shear test fixture (ASTM D 5379) [4.31], which is mounted in a universal testing machine, loads the specimen through wedge clamps, and incorporates a linear bearing for alignment. Loads are imparted to the specimen on the upper and lower edges. For some material systems, tabs may be required if local crushing of the edges is evidenced.

4.2.4 Single-Geometry Tests In order to determine the multiple elastic constants for orthotropic materials, a number of tests must be per-

formed. This can add considerable time and expense. Several researchers have proposed the use of unique geometries and loading conditions in combination with full-field optical methods to determine multiple elastic constants. These tests are generally for elastic constants and are typically not well suited for strength measurements since failure may be from combined effects. Two recent studies in experimental mechanics will be used as examples of such methods [4.35]. The first, proposed in 1999 by Grediac et al., utilizes a unique T-shaped specimen loaded as shown in Fig. 4.11. The investigators used a phased stepped grid method to determine the displacement fields, as shown in the figure, and then determined the strain fields by differentiation. The principal of virtual work, relying on global equilibrium, was employed. Four virtual fields were chosen to satisfy four elastic constants. In practice the four virtual fields act like filters that enable the extraction of the unknown rigidities (constants). From this, the stiffness components can be identified. The axial elastic constants were determined to within 15% of those values obtained using traditional means. It was determined

Composite Materials

that the shear elastic constant could not be determined with reasonable accuracy, since, as the authors point out, the specimen deformation is generally not governed by shear. An open-hole tension specimen was employed by Molimard et al. [4.36] to measure four orthotropic plate constants from a single geometry. They used phase-shifted moiré interferometry to measure full-field surface displacement and strain. Figure 4.12 shows the geometry and loading fixtures used in their tests. Displacement and strain contours were measured in the area around the open hole. The principal employed was

4.3 Micromechanics

107

to minimize the discrepancy between experimental and theoretical strain results using a Levenberg–Marquardt algorithm. Comparisons between the experimental and analytical shearing strain values are presented in Fig. 4.12. The method takes into consideration the optical system, signal processing, and the mechanical aspects. Cost functions were investigated leading to a simple mathematical form. Two models were used: an analytical model based on the Lekhnitskii approach and the finite element method. The researchers were able to identify the four elastic constants to within 6% of those measured using traditional means.

4.3 Micromechanics

a)

b)

be broken down into critical experiments. For instance, the interface can be probed by dedicated tests such as the single-fiber pull-out test. Other critical experiments have been performed on testing of the constituents individually in bulk form. These include polymer studies, fiber strength tests, and nanofiber characterization. Also, by including microencapsulated adhesives with dimensions on the fiber scale, self-healing composites have been developed. In this section full-field fiberscale testing, interface testing, nanocomposites, and self-healing composites will be addressed.

4.3.1 In Situ Strain Measurements As mentioned above, the length scale associated with micromechanics challenges the experimentalist. There have been numerous studies to create scaled-up model materials (to match the capabilities of various experimental techniques such as moiré methods, photoelasticity, etc.) of an ideal geometry (square and hexagonal array) for validation of analytical and numerical modc)

Fig. 4.13 (a) E-beam moiré grating, (b) fringe pattern prior to interlaminar fracture, (c) fringe pattern after fracture (courtesy J. W. Dally)

Part A 4.3

Experimental micromechanical characterization is extremely challenging because of the length scales involved. For example, the diameter of a carbon fiber is of the order 8 μm, boron fibers are of the order 200 μm, and carbon nanotubes are measured in nanometers. As such, there have only been a few notable examples of full-field experimental documentation of the strain field on the fiber scale. Any experimental technique that is suitable for such a task must have extreme spatial resolution, high sensitivity, and be used in conjunction with high magnification capabilities. Traditional experimental techniques, such as moiré interferometry, cannot be used without enhancement. Additionally, most of the interesting micromechanical phenomena occur in the interior of the ply and not on the surface, therefore lack of visual access (whether through traditional optical microscopy or scanning electron microscopy) prohibits the investigation of such problems. As a result, modeling efforts are far ahead of the experimental efforts and precious little experimental evidence supports these models. Nevertheless, aspects of micromechanics can

108

Part A

Solid Mechanics Topics

Fiber pull-out

Fiber push-in

Fiber push-out

Fig. 4.14 Schematic illustrations of the single-fiber pull-out, pushin, and push-out tests

els. This methodology can be useful in understanding the interaction between the fiber and matrix, but often the essence of the micromechanical phenomenon is lost. Here, only implementation of full-field techniques on real composites will be presented.

Part A 4.3

Striker bar

Incident bar

Transmitter bar

Strain gage

Strain gage

3.2 mm Incident bar

Transmitter bar

Punch Specimen

4.3.2 Fiber–Matrix Interface Characterization

Support

Force (N) 0

–1000

5m/s

–2000

7 m/s

–3000 15 m/s

–4000

0

The first example is a novel method developed by Dally and Read called electron beam moiré [4.37– 40]. The method involves writing very high-frequency (10 000 lines/mm) gratings on sectioned composites using e-beam polymethyl methacrylate (PMMA) resists and performing a moiré test in the scanning electron microscope (SEM) using the scanning lines as the reference grating. While details of the technique are given in the reference, the example presented on glass fiber reinforced epoxy will be illustrated here. The interface between the 0◦ and 90◦ plies on the edge surface of a [02 /902 ]2s laminate was interrogated for a specimen loaded in tension using a small testing machine located within the SEM specimen chamber. The crossed-line grating on the surface of a small region of the specimen is shown in Fig. 4.13, along with moiré fringe patterns at two load levels, one prior to and one after a delamination crack. Xing et al. [4.41] also extended the used of e-beam moiré, by measuring the interfacial residual stresses in SiC/Ti-15-3 silicon fiber reinforced ceramic composites. Additional in situ high-spatial-resolution studies using novel experimental techniques include the use of micro-Raman spectroscopy [4.42–44] for the study of fiber breakage in compression specimens and the interaction between neighboring fibers.

10

20

30

40

50

60 70 Time (μs)

Fig. 4.15 Effect of sliding speed on the push-out force for a rough

aluminum fiber (12 mm long) in an epoxy matrix (courtesy J. Lambrose)

Fiber–matrix interface properties can dominate the mechanical behavior of a composite. The properties of the interface can influence delamination, microcracking, kink band formation, residual stresses, and numerous other failure modes. Therefore, the characterization of the interfacial strength is of importance when developing micromechanical models to predict the strength of composites. The interface, defined as the boundary between the fiber and matrix, actually has a finite volume. This is because the fiber typically has a sizing to control adhesion of the matrix and the matrix morphology varies near the fiber. Hence, the term interphase has been adopted to describe this finite volume. The interphase volume is on a length scale even smaller than the fiber scale, and therefore introduces serious challenges to the experimentalist. However, the importance of this volume has spurred numerous researcher efforts. A number of tests that have been proposed to measure the adhesive strength of the fiber–matrix interphase. These tests include the single-fiber pull-out, push-in, and push-out tests shown schematically in Fig. 4.14.

Composite Materials

4.3 Micromechanics

1 μm

Stress σ (×106 Pa) 30 25 20 15 10 Dow DER732+DER331 Nanocomposite 5wt%

5 0

0

1

2

3

4 Strain ε (%)

Fig. 4.16 TEM micrograph of clay platelet reinforced polymer and the tensile stress–strain response of unreinforced and 5% clay reinforced polymer (courtesy I. M. Daniel)

They are common in that a traction is placed on a single fiber to initiate and propagate bond failure between the fiber and matrix either as an adhesive or cohesive failure. An example of such a test is presented in this section. In 2002 Zhouhua et al. [4.45] conducted experiments to determine the dynamic fiber debonding and frictional push-out in a model composite system. The investigators utilized a modified split Hopkinson pressure bar with a tapered punch to apply a compressive load on a single fiber (modeled using aluminum and steel) embedded in an epoxy matrix. In the push-out experiment, the fiber model is pushed from the surface

4.3.3 Nanoscale Testing The quest to achieve improved mechanical properties has led to the development of composites that incorporate nanoparticle, nanofiber, or nanotube reinforcement. These nanocomposites are a class of composite material where one of the constituents has dimensions in the range of 1–100 nm. Over the past decade, the development of nanoreinforcements, processing them into composites, and assessing their mechanics has rapidly evolved. In this section we will briefly describe two studies: the incorporation of nanoclay particles in polymers and a study of the mechanical characterization of multiwalled carbon nanotubes. Some polymer–inorganic nanocomposites have demonstrated pronounced improvements in stiffness, strength, and thermal properties over unreinforced polymers without sacrificing density, toughness, and processibility. The advantages of nanocomposites over traditional composites come from the high surface-areato-volume ratio of the reinforcement, and thus have a direct effect on the interfacial mechanics. Measuring the stress and strain distribution on the nanoscale

Part A 4.3

and punches through a hole in a support. The test allows real-time measurement of the relative fiber–matrix displacement and push-out force. Figure 4.15 shows a schematic of the setup and an example force–time plot to demonstrate the effect of sliding speed (directly related to the loading rate). The technique documented the effects of loading rate, material mismatch, fiber length, and surface roughness on the push-out event. It was observed that surface roughness played an important role in the dynamic interfacial strength, and that the maximum push-out force increased with loading rate. The plot in the figure shows a markedly different behavior between slow and fast sliding speeds, with a transition occurring around 10 m/s. At slow speeds the transmitted force reaches a maximum in two steps, indicating that the load levels required for initiating and propagating the debonding crack differ. A finite element model was constructed to extract the interface strength and toughness values. On the scale of real composites (versus model materials) the single-fiber push-in test is more applicable since the only specimen preparation required is cross-cutting and polishing the specimen. Micro- and nanoindenters have been utilized for applying loads over the small area of the fiber cross-section. These devices have load and position readout and are well suited for static loading cases.

109

110

Part A

Solid Mechanics Topics

a)

increased the tensile modulus and strength by as much 50%, as can be seen in the figure. When adding additional reinforcement, the modulus increased, but the strength significantly degraded. These findings were thought to be attributed to increased voids, inclusions, and agglomeration of the nanoparticles during processing. Recent research by Zhu and Espinosa [4.47] has lead to the development of very small tensile testing machines that are capable of testing single multiwalled carbon nanotubes. The system utilizes a micro-

b)

5 μm

c)

d)

Microcapsule Catalyst Crack

Part A 4.3

Fig. 4.17 The procedure to attach a nanowire to the cross-heads of the MEMS-based testing machine involves welding the wire using electron-beam-induced deposition (courtesy Y. Zhu)

is a current challenge to the experimentalist. Many of the experimental mechanics contributions to this field pertain to using traditional test methods applied to the macroscopic behavior of nanocomposites. Examples include using tensile tests, fracture tests, and shear tests to validate the inclusion of nanoparticles in matrix materials. Daniels et al. [4.46] characterized clay nanoparticle platelet reinforced epoxy via tension tests to determine how process parameters (such as the weight percentage of the reinforcement) affect the stress–strain response. Figure 4.16 shows a tunneling electron microscope micrograph of the exfoliated Cloisite 30B clay reinforced Dow DER331/DER732 epoxy system. It was found that moderate nanoclay concentrations on the order of 5% a)

Healing agent

Polymerized healing agent

20 μm

b)

500 nm

500 nm

Fig. 4.18 Tension tests of a multiwalled carbon nanotube with detailed insets to shown the wall thickness and morphology (courtesy Y. Zhu)

Fig. 4.19 Self-healing concept for thermosetting matrix with SEM image of broken microcapsule (courtesy E. N. Brown)

Composite Materials

4.3.4 Self-Healing Composites Researchers at the University of Illinois have developed a self-healing polymer that they have incorporated into the fabrication of fiber reinforced composite materials [4.48, 49]. The concept is to heal a crack in polymers using a microencapsulated healing agent that

is released upon the intrusion of a crack. The healing agent is then hardened by an embedded catalyst. The dicyclopentadiene (DCPD) healing agent, stabilized by 100–200 ppm p-tert-butylcatechol encapsulated by a poly-ureaformaldehyde shell, catalyzes when exposed to bis(tricyclohexylphosphine)benzylidine ruthenium (IV) dichloride (otherwise called Grubbs’ catalyst), which is imbedded in the composite matrix. Mean capsule diameters are of the order 166 μm. Figure 4.19 shows a schematic of the fracture/healing process, where a crack penetrates through the bulk polymer and the shells of the microcapsules. The healing agent then wicks into the crack and comes into contact with the embedded catalyst, leading to polymerization of the healing agent and subsequent bridging of the crack. The researchers also conducted extensive fracture studies on unreinforced (no fibers) EPON 282 epoxy tapered double-cantilever beam specimens, and found that, once healed, the polymer containing the healing agent exhibited 90% of its virgin fracture toughness [4.50,51]. Studies were then performed by combining the self-healing matrix material with a plain-weave graphite fiber fabric. Fracture tests were performed using a width-tapered double-cantilever beam specimen. It was found that as much as 45% of the virgin fracture toughness was recovered by autonomic healing, and as much as 80% was recovered by increasing the temperature to 80 ◦ C during recovery.

4.4 Interlaminar Testing Since most composite materials are layered with only the matrix material binding each layer to its neighbor, weak planes exist. As a result, delamination between layers is a common failure mode. Delamination may initiate from a number of different loading conditions, including impact (high and low velocity), tension, compression, and flexure. The resulting interlaminar normal and shear stresses can then lead to interlaminar fracture and possible subsequent catastrophic failure of a structural component. Often, interlaminar fracture initiates at a free edge where the interlaminar stresses may be much higher than the nominal applied stresses. Figure 4.20 provides dye penetrant enhanced radiography images showing delamination (dark areas) initiated at the free edge, at a stress concentration and at an impact site. This section will cover the experimental characterization of

mode I (opening mode), mode II (shearing mode), and mixed-mode interlaminar fracture, as well as a study of stresses due to free edge effects.

4.4.1 Mode I Fracture Mode I interlaminar fracture toughness is traditionally measured using a double-cantilever beam (DCB), as shown in Fig. 4.21. The specimen is fabricated such that a starter or initial crack is produced using a Teflon film sandwiched between the layers where fracture is intended to propagate (typically the midplane). When loaded, the crack propagates in the plane of delamination. Mode I fracture toughness can be determined in two ways from the critical energy release rate. First, the load at which the delamination starts to propagate can

111

Part A 4.4

electromechanical system (MEMS) for in situ electron microscopy. Loads in the nanonewton and displacements in the subnanometer range can be resolved with the device. The system was used to measure the stress–strain response of free-standing polysilicon films, metallic nanowires, and carbon nanotubes. Figure 4.17 shows electron microscopy images of a nanowire bridging the two cross-heads of the loading device. Small welds secure the fiber in place. Figure 4.18 shows a tunneling electron microscope (TEM) image of a multiwalled carbon nanotube (with an outer diameter of 130 nm and inner diameter of 99 nm) being loaded. Upon rupture, crystallization is clearly seen in the inset of the lower plot in Fig. 4.18. The failure stress was measured at 15.84 GPa with a failure strain of 1.56%. These correspond to a modulus of more than 1000 GPa. These strength and stiffness values are of the order a fivefold improvement over the best traditional carbon fiber (10 μm diameter), and confirm the extraordinary potential that exists from the processing and mechanics of these materials.

4.4 Interlaminar Testing

112

Part A

Solid Mechanics Topics

be used to compute G IC , the critical energy release rate: F 2 a2 (4.4) G IC = C , bE I where FC is the load at which the crack propagates, a is the current crack length, b is the beams width, and E I is the equivalent flexural rigidity of the upper and lower arms of the specimen. In the second method, the specimen is unloaded after the crack propagates a distance Δa. The area under the load–deflection diagram, bounded by the loading portion, load drop due to crack extension, and unloading portion, represents the work done, ΔU, for crack propagation and the critical energy release rate, G IC , is computed as ΔU (4.5) . bΔa One of the challenges associated with this methodology is the accurate determination of the crack position. Optical microscopes, digital image correlation, and moiré interferometry have been utilized for this purpose. For visual inspection, the edge of the specimen is painted white so that the crack is more pronounced. Perry et al. [4.52] used full-field moiré interferometry to measure the displacement fields for the calculation of the J-integral, in order to determine the strain energy release rate on the DCB specimen. They also developed a method to determine the energy release rate for arbitrary geometries (not DCB) and used the DCB test to validate the methodology. An example

Initial crack

F

Δa a

F Load F

G IC =

Load point displacement

Part A 4.4

Fig. 4.21 The double-cantilever beam specimen, loading, and typical load–displacement diagram

of the phase wrapped moiré displacement pattern for the DCB test is shown in Fig. 4.22. Numerous other researchers have implemented a modified version of the DCB method. Some of these modifications include tapering the specimen in either the vertical direction by adding material to the top and bottom surfaces or tapering the specimen in the widthwise direction [4.53,54]. These modifications transition the crack growth from unstable to stable, and crack propagation occurs at a near-constant load level. For through-thickness stitched composites, it was found that the force required to open the crack breaks the arms of the specimen. The failure was governed by compres-

2 mm

Cracks emanating from the free-edge during loading

Cracks emanating from a stress riser

Damage from impact

Fig. 4.20 Delamination and cracking due to cyclic loading and im-

pact

Fig. 4.22 Moiré interferometry, wrapped, fringe pattern, at the crack tip, for the vertical displacements on a DCB specimen (courtesy K. E. Perry)

Composite Materials

113

in a sliding mode horizontally as the beam bends. The critical energy release rate can be determined using the same analyses as those used for the DCB test. For the instantaneous mode II critical energy release rate

F F/2

F/2

4.4 Interlaminar Testing

Load F

3 FC2 a2 (4.6) . 64 bE I Additionally, by determining the area in the load– displacement diagram (Fig. 4.23), bounded by the loading portion, load drop due to crack extension, and unloading portion, the critical energy release rate is determined in the following equation. G IIC =

ΔU (4.7) . bΔa Typically, the location of the crack front is difficult to determine since the crack does not open, but rather slides. Full-field techniques can be of value for locating the crack tip. Perry et al. used moiré interferometry for both crack front location as well as displacement mapG IIC =

Load point displacement

Fig. 4.23 The end-notched flexure specimen, loading, and typical load–displacement diagram

Ny –5 y

x

5

4.4.2 Mode II Fracture Mode II fracture toughness can be determined using the end-notched flexure (ENF) specimen [4.52]. The specimen is fabricated in the same manner as that used in the double-cantilever beam test, with a Teflon starter crack. It is loaded, however, in a three-point bend configuration, as shown in Fig. 4.23. The crack propagates

V field Ny fringes

0

10

Graphite/ epoxy

15

(O2 /±452 /902)n

20

f = 2400 lines/mm

γxy /|εyav|

γxy x 106

4 8000 2 0

0 –2

–8000 –4

2 mm

–16000

–6

0

8

16

24

32

40

Ply thickness = 0.19 mm (= 0.0075 in.) 48

Ply no.

Fig. 4.24 Horizontal displacement field, at the crack tip,

for an ENF specimen (courtesy K. E. Perry)

Fig. 4.25 Moiré interferometry fringe pattern on a portion of the edge of a quasi-isotropic laminate. Strong interlaminar shear strain are present between +45◦ and −45◦ plies (courtesy D. Post)

Part A 4.4

sive instability. Additional horizontal forces were added by Chen et al. [4.55, 56] to suppress this instability. Under the basic assumptions of superposition of axial load and bending moments in beams, the compressive stresses were reduced and more bending moment could be applied. Crack propagation was realized for stitched materials and traditional DCB analysis showed that the fracture toughness increased by more than 60-fold.

114

Part A

Solid Mechanics Topics

ping to calculate the energy release rate. Figure 4.24 shows the wrapped fringe pattern used in the analysis.

4.4.3 Edge Effects

Part A 4.5

Often, delamination initiates at the free edge of a composite laminate because of highly localized stresses between layers of dissimilar orientation. This free edge effect forms a stress gradient that dissipates to the nominal stress level of the interior within a distance equivalent to a few ply thicknesses. These stress distributions can be highly complex and three dimensional, involving all three normal and all three shear stresses, and are dependent on the loading condition, the stacking sequence, and material properties. Some are restricted to the interior and cannot be measured on any of the free surfaces, making a comprehensive experimental characterization difficult. Additionally, if residual stresses are ignored in the overall analysis, the absolute stresses cannot be determined. In order to study the free edge affect, very high spatial resolution is required, since the strain gradients are high and the geometric features are small. Post et al. [4.57,58] conducted a series of tests using moiré interferometry to study the interlaminar strains

on both cross-ply and quasi-isotropic thick laminates. They performed both in-plane and interlaminar compression tests and documented the strain distributions on the ply scale. As an example of the experiments performed, a (02 / ± 452 /902 )n , graphite–epoxy laminate subjected to in-plane compressive loading will be presented. Figure 4.25 shows the specimen geometry, the moiré interferometry fringe pattern of the displacement in the vertical direction (loading direction), and the shear strain distribution along a line in the thickness direction on the free surface. The fringe pattern shows significant waviness, where there would only be horizontal, straight, evenly spaced, parallel fringes for a homogeneous isotropic specimen (for instance, a monolithic metallic specimen). By analyzing the horizontal fringe gradient ∂v/∂x the shear strains were determined. Strong gradients were present between the +45◦ and −45◦ plies. The shear strains were of the order of five or five times the applied normal strain. Additionally, from the u-field patterns, it was found that tensile strains in the transverse direction (Poisson effect) varied significantly with peak tensile strains of the order of the applied axial compressive strain.

4.5 Textile Composite Materials Textile composites are a form of composite that utilizes reinforcement in the form of a fabric. The fabric may be woven, braided, knit or stitched and then combined with matrix materials, via resin transfer techniques, hand lay-up, or prepreg methods. Textiles have distinct advantages over traditional laminates made from unidirectional layers, including better contourability over complex three-dimensional (3-D) tools, enhanced interlaminar strength, cost savings through near-net-shape production, and better damage tolerance. Disadvantages include lower fiber volume fraction, the existence of resin-rich volumes between yarns, and degraded inplane properties (as a result of the nonstraight path that the yarns must accommodate). Even so, they are extensively used in applications from the sporting goods industry to advanced aerospace vehicles. Textile composites pose additional challenges to the experimentalist since they have an additional level of heterogeneity. Not only is there heterogeneity on the fiber scale, but also on the scale of the textile architecture and the laminate scale (if multiple layers of cloth are used). For many textile forms a yarn may contain

12 000 fibers, or more, and the repeating unit in the textile architecture may have linear dimensions of the order 1 cm or more. This can pose significant challenges to the experimentalist, even for seemingly routine tests to determine the elastic properties.

4.5.1 Documentation of Surface Strain It has been assumed that the textile architecture induces repeating spatial variation of strain on the surface coincident with the architecture itself. In order to document the strain distribution associated with the architecture, high spatial and measurement resolution are required. Recent studies using moiré interferometry, digital phase-shifting shearography, and Michelson interferometry have documented the strain field on the surface of composites. These results can be used to guide instrumentation practices for strain gages as well as to validate modeling efforts. Lee et al. [4.59] used digital phase-shifting grating shearography to characterize experimentally plainweave carbon–epoxy composites under tensile load.

Composite Materials

Measuring area

8 N/mm 14.8 N/mm 22.7 N/mm 28 N/mm

4500 8 N/mm

4000

x1

Two peaks on the fill arm

4

6

115

Fig. 4.26 Tensile strain contour maps for four load levels. Plots along two lines show the variation of strain and correspondence to the textile architecture (courtesy J. Molimard)

Tensile strain (με) 5000

4.5 Textile Composite Materials

3500 3000 2500 2000 1500 1000 500

14.8 N/mm

0

0

2

8

10

12

14

16

Distance along x1-line (mm) Tensile strain (με) 5000

Measuring area

4000 22.7 N/mm

3500

Part A 4.5

8 N/mm 14.8 N/mm 22.7 N/mm 28 N/mm

x2

4500

Two volleys on the warp yarn

3000 2500 2000 1500 1000 500

28 N/mm

0

0

2

4

6

8

10

12

14

16

Distance along x2-line (mm) –353

174

701

1228

1756

2283

Tensile strain (με)

The experimental technique is similar to moiré interferometry and can be used in conjunction with Michelson interferometry to determine the three in-plane strain components (εx , ε y , γxy ) on a full-field basis, as well as the out-of-plane displacements and surface slopes. They found that the strain varied in a cyclic manner and followed the weave architecture. Figure 4.26 shows contour plots as well as line plots (at two different locations with respect to the architecture) of the tensile strain for four load levels. The maximum strains were of the order of three times greater than the minimum strain values. In the transverse direction they found that the normal strain is predominantly compressive (Poisson effect), but locally there are regions of tensile strain. These effects are attributed to yarn crimp and bending as a function of axial load.

2810

3338

Δ = 264

Shrotriya et al. [4.60] used moiré interferometry to measure the local time–temperature-dependent deformation of a woven composite used for multilayer circuit boards. They studied both the deformation fields in the plane as well as over the cross-section through a temperature range of 27–70 ◦ C. Their measurements revealed the influence of the fabric architecture in the deformation field. They noted that the variation in strain was greater when the composite was loaded in the fill direction (versus the warp direction) due to higher crimp angles (the angle that defines the undulation of a yarn as is passes over and under the transverse yarn). They also found that the total deformation increased with temperature and time (reflecting what was previously measured using strain gages) but the shape and distribution remained almost identical for all the loading cases and

116

Part A

Solid Mechanics Topics

A

Stacked nesting case B 1.2 mm

Split – span nesting case

C

y (a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

(i)

Diagonal nesting case

x Prepreg tows

Fig. 4.27a–i Moiré fringe patterns in the fill direction at 27 ◦ C for: (a) initial u-field, (b) u-field at 0.1 min, (c) u-field at 1 min, (d) ufield at 10 min, (e) composite microstructure, (f) initial v-field, (g) vfield at 0.1 min, (h) v-field at 1 min, (i) v-field at 10 min (courtesy

Prepreg tape

Part A 4.5

P. Shrotriya).

sample configurations. Figure 4.27 represents a sample of the fringe patterns taken on the edge of the composite for various times after the load was applied. In the figure, both the horizontal and vertical displacement fields are presented, as well as a schematic of the fabric architecture.

Fig. 4.29 Idealized textile composites can be used to val-

4.5.2 Strain Gage Selection

idate models. Three nesting configurations were tested by the investigators (courtesy D. O. Adams)

Because of the pronounced strain variation on the surface of textile composites, strain gage size (length and Coefficient of variation of strain readings (%) 7 6 Gage data Moire data

5 4 3 2 1 0

0

2

4

6

8 10 12 14 Gage length (mm)

Fig. 4.28 Coefficient of variation (CV) results using differently sized strain gages and CV results from the moiré pattern analysis (courtesy S. Burr)

Tensioned

width) selection is critical in order to avoid significant measurement variation. Studies performed by Burr and Ifju [4.61, 62] showed that, if the strain gage length is sufficiently smaller than the repeating unit cell size of the textile architecture, larger coefficients of variation in material property measurements are observed for simple tension testing. After testing with a variety of gage sizes, it was also observed that, as the gage length increased, the coefficient of variation decreased. Moiré interferometry fringe patterns on the surface of graphite–epoxy triaxial braided specimens were used to determine the appropriate gage length for a variety of braided materials (reinforcement in three directions) with unit cell sizes ranging from 2.5 mm to 22.4 mm. A representative gage area was superimposed over the fringe pattern and a procedure to measure the average strain in the gage area was developed. The representative area was then moved throughout the pattern and the variation of average values was used to determine the relationship between gage area and unit cell size. Figure 4.28 shows a fringe pattern with a represen-

Composite Materials

tative gage area superimposed along with a plot of the gage length versus coefficient of variation for both strain gage experiments and moiré interferometry patterns. Good correlation was found between the moiré prediction of the gage variation and the actual gage variation itself. It was suggested that, prior to performing multiple strain gage experiments, where specimen fabrication and instrumentation costs are prohibitive, a simple moiré experiment can be used to guide gage selection, with the reduction in measurement variability as the payoff.

4.5.3 Edge Effects in Textile Composites

woven composites, where the free edge is typically oriented with the yarn directions, these shearing stresses are less pronounced. It has been shown, however, that small differences in the in-plane fiber angle can lead to significant interlaminar shear on the free edge, where shear would not be expected to exist. For instance, in a weave with fibers running 0/90, there should not be strong shear stresses on the free edge. However, if the fiber angle deviates even a small amount from 90◦ or 0◦ then shear strains will be present, as evidenced by Hale et al. [4.63]. Modeling of the stress and strain fields in textile composites assumes idealized geometries. For multilayered textile composites, the nesting of the layers can influence these fields. Adams and colleagues [4.64] have developed a methodology to produce idealized textile composites with near-perfect nesting, so that experiments performed using full-field optical methods can be directly related to the model. The technique utilizes a method of curing the composite on the actual loom and controlling the tension and precise positions of individual warp and fill yarns. The result is almost perfect alignment of each layer such that layer-to-layer nesting can be controlled. Figure 4.29 shows some of the fabric nesting configurations that were studied.

4.6 Residual Stresses in Composites The act of combining two constituents in the fabrication process of composites typically induces residual stresses in the material.These stresses can arise from two separate sources, the chemical shrinkage of the matrix material (if the matrix undergoes a chemical reaction such as polymerization) as well as stresses due to differences in the coefficient of thermal expansion (CTE) between the fiber and matrix. The stresses can exist on both the fiber scale as well as the ply scale of the laminate. Additionally, for textile composites, these stresses can exist on the fiber architecture scale. The residual stresses due to chemical shrinkage typically occur during cure, but may also evolve with time as additional curing and physical aging transpires. The stresses that arise from CTE mismatch occur when the material is subjected to temperatures that differ from that of the cure temperature, and as such are strongly dependent on temperature. Many composite material systems, especially the high-tech graphite–epoxy systems used for aerospace applications, are cured in a high-pressure high-temperature apparatus called an

autoclave. This allows the materials to be operated at higher temperatures and typically leads to higher fiber volume fraction and lower void fraction. However, when the composite is utilized at a temperature much lower than the cure temperature, large residual stresses can arise. These stresses superimpose with the applied loads and may cause premature failure of the component. In some cases residual stresses may be high enough to lead to failure, even before loads are applied. For laminated forms, residual stresses on the ply scale exist because of the orthotropic nature of the thermal expansion coefficients in the principal material directions. A typical value for the CTE in the fiber direction α1 is very near zero if not slightly negative, while that in the transverse direction α2 is quite high and dominated by the CTE of the matrix material. For instance, a typical value for α1 is − 0.018 × 10−6 /◦ C for graphite–polymer and 6.3 × 10−6 /◦ C for glass– polymer unidirectional material. Typically, α2 is around 24 × 10−6 /◦ C for both graphite–polymer and glass– polymer materials. This value is nearly the same as

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As seen in laminate forms, edge effects can produce significant shear and normal stress components that may be many times greater than in the interior of the laminate. In textile composites a similar phenomenon transpires, except it occurs on the yarn scale, and can be highly localized. If a free edge cuts through a yarn that is neither parallel nor perpendicular to the free surface, large shearing strains may be present in the interface between the yarn and its nearest neighbor. In the case of braided materials, the yarns are typically inclined to the free edge, leading to large shear stresses and strains. For

4.6 Residual Stresses in Composites

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Part A 4.6

that of a typical aluminum. A unidirectional composite is rarely used on mechanical components since the transverse properties are so poor. By laminating the composite, desired mechanical properties can be achieved, however the lamination process leads to residual stresses as neighboring plies constrain each other. A recent example of the seriousness of these stresses is documented by the failure of the NASA X-33 liquid hydrogen fuel tanks [4.65]. A full-size prototype tank was fabricated with honeycomb sandwich core and quasi-isotropic, graphite–epoxy face sheets. Upon filling the tank with liquid hydrogen at cryogenic temperatures, the face sheets failed (cracked) because of residual stress induced by a CTE mismatch on the ply scale superimposed with chemical shrinkage effects on the ply scale. Hydrogen leaked into the core and expanded upon emptying the tank, leading to face sheet separation. The resulting failure led to the cancellation of the entire X-33 program. Given the seriousness of these stresses, there has been a concerted effort to develop experimental techniques to quantify them. This section will document five such methods: the composite sectioning method, the hole-drilling method, the strain gage method, the nonsymmetric lay-up method, and the cure reference method. Generally speaking, the methods can be diHorizontal displacement field (O2/904)13s AS/3501-6 Sectioning here

Mid-plane

vided into two groups, destructive (require damaging the specimen to extract the stress information) and nondestructive. The first two methods are considered destructive, while the following three are nondestructive.

4.6.1 Composite Sectioning Whenever a multidirectional laminate is cut, a free edge is created and stresses are liberated. Stresses at the free edge must satisfy equilibrium and thus the normal stress perpendicular to, and the shear stress components parallel to, the newly developed surface must be zero. Stresses still exist in the interior of the composite, and thus there is a gradient established by forming the free edge. One can measure the strain relief as a result of the creation of the free edge and then relate this to the stresses that existed in the laminate before the cut. There is an underlying assumption that the cutting method itself does not create a deformation field (i. e., no plastic deformation). For most fiber reinforced polymer materials, a well-lubricated slow-speed diamond-impregnated cut-off wheel has been shown to produce negligible local deformation. This can be verified by cutting a unidirectional composite, where-ply level residual stresses do not exist. A number of researchers have utilized this method to back-out the residual stress in laminates [4.66–69]. Joh et al. [4.66] used a combination of moiré interferometry and layer separation to determine residual stresses in a thick (02 /904 )13s , AS/3501-6, cross-ply composite. A grating was attached to the composites before it was sectioned. Cutting was performed to separate plies, and subsequently, moiré experiments were conducted to reveal the displacement field. Figure 4.30 shows the moiré interferometry fringe pattern after ply separation. The displacements were used to determine the strain and subsequent stresses through laminate theory. The method was shown to be capable of determining the residual stresses as a function of ply position and thickness. Tensile stresses of the order the transverse strength of a single ply were reported. Gascoigne [4.67] used a similar procedure to investigate a (020 /9020 /020 )T cross-ply that was composed of thick layers. Similar findings were reported by other investigators [4.69].

4.6.2 Hole-Drilling Methods Fig. 4.30 Moiré interferometry fringe pattern of the hori-

zontal displacements after ply separation (courtesy D. Joh)

The hole-drilling method, originally developed for anisotropic materials by Mathar [4.70], has been ex-

Composite Materials

y x

drilled to two depths (h in the figure). The investigators were able to determine the residual stresses in each ply of a (02 /902 )2s laminate. Later, the method was simplified by Sicot et al. [4.74] by the use of the residual stress rosette in combination with incremental hole drilling. Instead of the displacement information used in Wu’s work, Sicot utilized strain information at the location of the residual stress gages.

4.6.3 Strain Gage Methods Strain gages may be mounted on a cured composite in order to evaluate the thermal coefficient of expansion. This method requires extensive calibration of the strain gage and typically dummy gages mounted on a material that has coefficients of thermal expansion with well-known values. To measure the strain induced during cure however, strain gages must be mounted on the composite and co-cured with it. These gages may be mounted either on top of or embedded in the interior, between plies. The gages must be calibrated through the temperature range and they must be of a gage series that is appropriate for the temperature range. In order to determine the coefficient of expansion through a large temperature range a detailed proce-

υ-field h = 375 μm

u-field h = 112 μm

Finite element model utilizing symmetry about the mid-plane and hole υ-field h = 112 μm

υ-field h = 375 μm

Residual stress (MPa) 80 60 40 20 0 –20 –40 –60

In y-direction

In x-direction

Fig. 4.31 Moiré fringe patterns for

0

112.5

225

337.5

450

562.5

675

119

787.5

900

Thickness direction (μm)

two drill depths, the finite element model used for calibrating coefficients, and the residual stresses as a function of thickness (courtesy Z. Wu)

Part A 4.6

tended for use on orthotropic materials by Schajer et al. [4.71]. This method is similar to the composite sectioning techniques, except instead of a straight cut the free edge is created by a drill-type end mill. The method involves drilling a flat-bottom hole progressively through a laminate (layer by layer), and analyzing the deformation relieved by the process. A special strain gage rosette [4.72] is used to measure the strain field near the hole created by the stress relief. For complicated multidirectional laminates, where the stresses in each individual ply can be different, the simple orthotropic solution cannot be applied. Wu et al. [4.73] introduced the experimental technique of incremental hole drilling, with a 2 mm-diameter drill bit, in combination with the optical method of full-field moiré interferometry. The drilling increments were computer controlled and were coincident with the interfaces between plies. An expression describing the relationship between displacements on the surface, stresses in each layer, in-plane direction cosines, and a set of coefficients was employed. A rigorous calibration of the constants was performed using the full-field moiré displacement fields, and a 3-D finite element model of the 16-ply composite. Figure 4.31 shows two sets of moiré displacement patterns around the hole after the mill had

4.6 Residual Stresses in Composites

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dure was reported by Scalea in [4.75]. He derived analytical expressions for the CTEs of orthotropic lamina, accounting for transverse sensitivity and gage misalignment effects. Experiments were performed on glass–epoxy specimens and Invar as a reference material. He developed rules for temperature dwell time in order to ensure thermal equilibrium.

4.6.4 Laminate Warpage Methods

Part A 4.6

In 1989 [4.76] a method was introduced for the measurement of residual stresses in laminated material systems by fabrication of a nonsymmetric lay-up, more specifically a (0n /90n )T configuration. Numerous researchers have since utilized the method. After cure, the curvature induced from residual stress can be measured for a narrow strip of the laminate and the residual stresses calculated using laminate theory. The curvature can be measured with a variety of methods, including projection moiré, shadow moiré, digital image correlation or simply using a dial gage and assumptions of the shape. The resulting residual stresses are a combination of those induced both thermally and chemically. By heating the specimen to the cure temperature, the thermal contribution can be separated from the contribution due to chemical shrinkage. Figure 4.32 illustrates

the specimen geometry and the deformed shape. In the lower image, a shadow moiré fringe pattern of the curved specimen is presented.

4.6.5 The Cure Reference Method The cure reference method [4.77, 78] introduced in 1999 utilizes moiré interferometry and the application of a diffraction grating on the surface of a composite during the autoclave curing process, as shown in Fig. 4.33. This grating forms a datum from which subsequent thermal stresses can be referenced. Additionally, the method is capable of measuring the combination of the thermally induced and chemically induced components of strain. Unlike many of the former methods, it is capable of determining the residual stresses on any laminate stacking sequence (not just cross-plies), alPorous release film Breather ply Bleeder ply Vacuum bag Vacuum line

Autoclave tool

Non-porous release film Grating Pre-preg

During cure at elevated temperature

Pre-preg 3501-6 epoxy layer Evaporated aluminum 3501-6 epoxy layer

0° 90°

Astrosital autoclave tool (O2/452)2s AS4/3501-6

After cure at room temperature

u displacement field y, υ Shadow moire contour plot documenting the curvature

Fig. 4.32 Warping of a nonsymmetric laminate can be

used to determine residual stresses

x, u

υ displacement field

Fringe patterns were taken over a 1 in. diameter area contered on the face of the composite panel

Fig. 4.33 CRM grating replication and fringe patterns

Composite Materials

the coefficient of thermal expansion α2 , the modulus E 2 , and the shear modulus G 12 on temperature (these three properties are highly dependent on temperature while the others are nearly independent of temperature). These temperature-dependent material properties were then utilized in the analysis. It was shown that the laminate configuration that was presumed to be within a safe operating condition at cryogenic condition, with a predicted factor of safety of 1.3, actually had a safety factor of 0.8, once the temperature-dependent properties and chemical shrinkage residual stresses were incorporated into the analysis. Unfortunately, this analysis was performed after the failure of the fuel tank and the cancellation of the entire X-33 program. In a parallel optimization study, an alternative laminate sequence was developed in order to carry the applied loads, and at the same time, resist residual stress failure. The resulting angle-ply laminate, (±25)n , was tested using the cure reference method in parallel to the X-33 laminate. It was found that it retained a safety factor of 1.8 even when the chemical shrinkage term and the temperature-dependent material properties were used in the analysis.

4.7 Future Challenges Even though many experimental stress analysis techniques have been presented in this chapter, the fact is that modeling efforts are significantly more prevalent. This is especially true on the nanoscale, where molecular and multiscale modeling is proceeding at a strong rate, and the experimental tools to validate these models lag. As such, many of the models remain experimentally unsubstantiated, with little or no validation. Currently, there is a dearth of experimental methods that probe the nanoscale to investigate the complex interaction between composite phases. The traditional continuum breaks down and interatomic mechanics rules the behavior. On the nanoscale, full-field stress analysis techniques are nearly absent and thus this area of research is primed for major contributions. Experimental methods to measure strain must be developed in the atomic force microscope, scanning electron microscope, and tunneling electron microscope.

As new material systems are invented there will be a continuous need to characterize them. Nondestructive evaluation methods discussed in other chapters will play a pivotal role in the adoption of composite materials and their widespread utilization. On the macroscale, with the confidence bought by experimental and modeling efforts and the ability to produce and inspect composite structures, there will be an increase in the number of structures that will be built out of composites. This is especially true in our civil infrastructure, such as bridges and buildings. Continued advances in flight vehicles, lightweight ground vehicles, and sporting goods will require composites because of their advantageous properties. The future of composites is bright and the need to experimentally characterize them will only increase. Tomorrow’s composites may look very different than today’s but the challenges will be similar.

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though standard Kirchoff assumptions are made in the analysis (therefore variations due to through-thickness cure gradients cannot be measured). If the specimen is brought to the cure temperature and the strain is monitored, the thermal and chemical contributions can be separated. By applying the cure reference method to a unidirectional material, the free residual strain, defined by the thermal expansion and chemical shrinkage terms combined, can be measured. Then, by applying the cure reference method to a laminate (in the same autoclave cycle) the stresses can be calculated on the ply scale from the laminate strain information and the free residual strains, within the context of laminate theory. Experiments on the X-33 laminate were conducted at the University of Florida using a combination of the cure reference method and strain gages. The strain on the surface of multidirectional and unidirectional composites was measured through the temperature range from cure to liquid nitrogen. It was determined that approximately 20% of the strain at cryogenic temperatures originates from chemical shrinkage and the remainder from a thermal expansion mismatch. A series of experiments were performed to determine the dependence of

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4.77

4.78

– Carbon/Epoxy [02 /902 ]2s, Proc. SEM Spring Conference (1998) O. Sicot, X.L. Gong, A. Cherout, J. Lu: Determination of residual stress in composite laminates using the incremental hole-drilling method, J. Compos. Mat. 37(9), 831–844 (2003) F.L. Scalea: Measurement of thermal expansion coefficients of composites using strain gages, Exp. Mech. 38(4), 233–241 (1998) I.M. Daniel, T. Wang: Determination of chemical cure shrinkage in composite laminates, J. Comp. Technol. Res. 12(3), 172–176 (1989) P. G. Ifju, B. C. Kilday, X. Niu, S. C. Liu: A novel means to determine residual stress in laminated composites, J. Compos. Mat. 33(16), 1511–1524 (1999) P.G. Ifju, X. Niu, B.C. Kilday, S.-C. Liu, S.M. Ettinger: Residual strain measurement in composites using the cure-referencing method, J. Exp. Mech. 40(1), 22–30 (2000)

Part A 4

125

Fracture Mec 5. Fracture Mechanics

Krishnaswamy Ravi-Chandar

In this chapter, the basic principles of linearly elastic fracture mechanics, elastic plastic fracture mechanics, and dynamic fracture mechanics are first summarized at a level where meaningful applications can be considered. Experimental methods that facilitate characterization of material properties with respect to fracture and analysis of crack tip stress and deformation fields are also summarized. Full-field optical methods and pointwise measurement methods are discussed; many other experimental methods are applicable, but the selection presented here should provide sufficient background to enable implementation of other experimental methods to fracture problems.

5.2

Fracture Mechanics Based on Energy Balance ................................ Linearly Elastic Fracture Mechanics......... 5.2.1 Asymptotic Analysis of the Elastic Crack Tip Stress Field.. 5.2.2 Irwin’s Plastic Correction ............... 5.2.3 Relationship Between Stress Analysis and Energy Balance – The J-Integral .............................. 5.2.4 Fracture Criterion in LEFM ..............

126 128 128 129

Elastic–Plastic Fracture Mechanics.......... 5.3.1 Dugdale–Barenblatt Model............ 5.3.2 Elastic–Plastic Crack Tip Fields ....... 5.3.3 Fracture Criterion in EPFM ............. 5.3.4 General Cohesive Zone Models ....... 5.3.5 Damage Models ...........................

132 132 134 135 136 136

5.4 Dynamic Fracture Mechanics .................. 5.4.1 Dynamic Crack Initiation Toughness ................................... 5.4.2 Dynamic Crack Growth Toughness... 5.4.3 Dynamic Crack Arrest Toughness .....

137 138 139 139

5.5 Subcritical Crack Growth........................ 140 5.6 Experimental Methods .......................... 5.6.1 Photoelasticity ............................. 5.6.2 Interferometry ............................. 5.6.3 Lateral Shearing Interferometry ..... 5.6.4 Strain Gages ................................ 5.6.5 Method of Caustics ....................... 5.6.6 Measurement of Crack Opening Displacement ...... 5.6.7 Measurement of Crack Position and Speed...................................

140 141 143 147 151 152 153 155

130 131

References .................................................. 156

Assessment of the integrity and reliability of structures requires a detailed analysis of the stresses and deformation that they experience under various loading conditions. Many early designs were based on the limits placed by the strength of the materials of construction; in this approach the structure never attains loading that will cause strength-based failure during its entire lifetime. However, many spectacular failures of engineered structures occurred, not by exceeding strength limitations, but due to inherent flaws in the material and/or the structure, or due to flaws that grew to critical dimensions during operation. For example, during World

War II, Liberty ships simply broke into two pieces while sitting still in port as a result of cracking. The first commercial jet aircraft, the de Havilland Comet, failed in flight due to fractures emanating from window corners that grew slowly under repeated loading, eventually becoming critical. Such events provided the impetus for analysis that takes into account material and structural defects and spawned the development of the philosophy of flaw-tolerant design – such that the structure will operate safely, even in the presence of certain anticipated modes of failure. Such a damage- or flawtolerant approach to structural integrity and reliability

Part A 5

5.1

5.3

126

Part A

Solid Mechanics Topics

Part A 5.1

has become possible through a combination of material characterization, flaw detection and fracture mechanics analysis. The success of the flaw-tolerant design approach may be seen in the mid-flight tearing of fuselage panels of the Aloha Airlines Boeing 737 in 1988. While a significant portion of the fuselage was blown off due to linking of multiple cracks, the deflection of crack growth through bulkhead tear-straps limited crack growth, retaining the structural integrity of the airplane, which was able to land safely. The theory of fracture mechanics and the experimental methods used in fracture characterization are described in this chapter. The fundamental ideas of the theory of rupture were formulated elegantly by Griffith [5.1]; the crux of this theory is captured in Sect. 5.1. Griffith’s analysis brought a key element to the theory: that the maximum load-carrying capacity of a structural element depends not only on the strength of the material, but also on the size of flaws that may exist in the structural element – the larger the flaw, the smaller the load capacity or strength. Formalization of this into the practical theory of fracture mechanics – now called linear elastic fracture mechanics (LEFM) – was spearheaded through the efforts of Irwin [5.2, 3] who introduced the idea of the stress intensity factor that combined the stress and flaw issues into one parameter, and by many others;

LEFM is discussed in Sect. 5.2. The limitations of elastic analysis were recognized by many investigators (beginning with Griffith) and a theory of fracture that accounts for elastic–plastic material behavior, the theory of elastic–plastic fracture mechanics, described in Sect. 5.3, was developed. For fast-running cracks, the effect of inertia becomes important; the fundamentals of dynamic fracture theory and practical implementation aspects are discussed in Sect. 5.4. The experience from the Comet airplane incident indicated that, even if the structure was not fracture-critical under monotonic loading, crack extension could occur under repeated loading through a process of fatigue. Other processes such as stress corrosion and creep also engender timedependent crack growth under subcritical conditions. Subcritical crack growth is handled phenomenologically by correlating a loading parameter to crack speed (crack extension per cycle or per unit time). A short description of subcritical crack growth is provided in Sect. 5.5. Experimental investigations and specific experimental methods of analysis have played a crucial role in the development of fracture mechanics. Optical methods that provide full-field information over a region near the crack tip and pointwise measurement of some component of stress or deformation are both in common use; these methods are discussed in detail in Sect. 5.6.

5.1 Fracture Mechanics Based on Energy Balance Consider the equilibrium of a linearly elastic medium containing a crack; a special geometry, called the double-cantilever beam (DCB) is illustrated as an example in Fig. 5.1. This specimen contains a crack of length a and the other relevant geometrical quantities are shown in the figure; the specimen width is taken to be unity, indicating that all loads are considered to be per unit thickness. The total energy of the system E can be partitioned into two components: the potential energy Π and the energy associated with the crack Us . Thus, E = Π + Us .

to be the fracture energy per unit area of crack surface and not specifically attributed to the surface energy. We consider a quasistatic problem and hence the kinetic energy has been neglected. The energy of the system P, ΔT CM P, Δ

(5.1)

The potential energy is the sum of the work done by the applied external tractions WE and the stored energy in the system U. Us was defined originally by Griffith [5.1] as the surface energy of the crack. Orowan [5.4] generalized this to include plastic dissipation during creation of the surface and thus Us can be considered generally

2h

a

Fig. 5.1 Geometry of the double-cantilever beam fracture

specimen

Fracture Mechanics

must be an extremum when the system is in equilibrium; thus dE/ dA = 0, where A represents the crack surface area, which for the unit thickness considered is equal to the crack length a. We define the potential energy release rate per unit crack surface area G and the fracture resistance per unit surface area R as dUs dΠ (5.2) , R≡ . da da The equilibrium condition can be written as an expression that the potential energy released by the body in extending the crack is equal to the fracture resistance: G(a) ≡ −

G(ac ) = R ,

(5.3)

Δ (5.4) , P where P is the load per unit thickness and Δ is the displacement of the load point. Let the specimen be attached to a pinned support at the bottom and attached to a load through a spring at the top. The total displacement ΔT (considered to be fixed) is ΔT = C(a) + CM P , (5.5) C(a) =

where CM is the compliance of the loading machine, with CM = 0 indicating a fixed-grip loading and CM → ∞ implying a dead-weight loading on the specimen. The potential energy in the system – the specimen and the loading machine – at fixed total displacement is Π=

Δ2 (ΔT − Δ)2 + . 2CM 2C(a)

(5.6)

127

The potential energy release rate can then be calculated either at fixed load or fixed grip conditions: 1 2 d C(a) (5.7) P . 2 da Thus, calculation of the compliance of a cracked body will yield the potential energy release rate directly. For the specific example of the double-cantilever beam, elementary beam theory can be used to estimate the 3 compliance of the beam segments: C(a) = E8ah 3 . Substituting in (5.7) and equating to the fracture resistance, we find that the equilibrium crack length can be written either in terms of the beam deflection or the applied load: 3 1/2 3Eh 3 Δ2 1/4 Eh R = . (5.8) ac = 16R 12P 2 G=

Obreimoff [5.5] used this idea and successfully measured the fracture resistance of mica subjected to cleavage between its layers. It is to be noted that the compliance of the loading machine does not enter into the determination of the equilibrium crack length, but will influence the stability of crack growth. The most remarkable aspect of the formulation of the fracture problem in terms of the energy approach is that the issue of analysis of the stresses and deformations in the vicinity of the crack is completely circumvented. In the case of the DCB specimen, as illustrated above, and in a few other simple cases, the compliance may be estimated rather easily and hence the energy approach is easily implemented. In other cases, the compliance can be obtained in the process of experimentation or through a numerical simulation of the structural problem and hence used in the analysis of fracture. The main obstacle to applying this approach in general is the determination of the fracture resistance. If the fracture energy per unit area is to be regarded as a material property, determined in analog laboratory tests and used in applications, the conditions under which crack growth occurs must be similar in the laboratory tests and field applications. This requires consideration of how the deformation and stresses evolve in the vicinity of the crack tip region. Analysis of the stress and deformation in a solid containing a crack is considered next, within the context of the theory of linear elasticity.

Part A 5.1

where ac is the equilibrium crack length. If the fracture resistance is taken to be a material parameter, then the problem of determining the equilibrium crack length is reduced to the problem of determining the potential energy release rate. This is the essence of fracture mechanics as outlined by Griffith [5.1] and augmented by Orowan [5.4]. Calculation of the potential energy release rate can be accomplished experimentally, analytically or numerically by solving the appropriate boundary value problems. A simple analytical method is described here, with application to the double-cantilever beam as an illustrative example. In general, the compliance of any specimen can be written as

5.1 Fracture Mechanics Based on Energy Balance

128

Part A

Solid Mechanics Topics

5.2 Linearly Elastic Fracture Mechanics The Griffith theory of fracture discussed above was developed further through analysis of the details of the stress and deformations in an elastic body containing a crack. We summarize these analyses under the assumption that the material may be modeled as a linearly elastic medium up to the point of failure. Criteria for failure under these conditions are also discussed.

5.2.1 Asymptotic Analysis of the Elastic Crack Tip Stress Field

Part A 5.2

The nature of the stress field near the crack tip in a linearly elastic solid was established through the efforts of Inglis [5.6], Mushkhelisvili [5.7], Williams [5.8], Irwin [5.2], and many other investigators. In general, consideration of three different loading symmetries is sufficient to decompose any arbitrary loading with respect to a crack; as illustrated in Fig. 5.2; these are typically called mode I or opening mode, mode II or inplane shearing mode, and mode III or antiplane shearing mode. For a homogeneous, isotropic, and linearly elastic solid, the structure of the solutions to the equations of equilibrium subjected to traction-free crack surfaces can easily be determined for each of the three modes of loading; this stress field in the vicinity of the crack for modes I and II may be written in the following form: KI I (θ, 1) + σ0x δ1α δ1β Fαβ σαβ (r, θ) = √ 2πr ∞ n I + (θ, n) An r n/2 Fαβ 2 n=3

K II II +√ Fαβ (θ, 1) 2πr ∞ n II + (θ, n) Bnr n/2 Fαβ 2 n=3 ⎧ ⎨0 plane stress , σ33 (r, θ) = ⎩νσ (r, θ) plane strain

(5.9)

where the θ dependence is given by Kobayashi [5.9] I (θ, 1) = cos 12 θ 1 − sin 12 θ sin 32 θ , F11

II F22 (θ, 1) = sin 12 θ cos 12 θ cos 32 θ ,

The corresponding displacement field can be expressed as σ0x 2 KI r I G θ, 1 + rG I θ, 2 u α (r, θ) = μ 2π α μ 1+ν α ∞ An n/2 I r G α θ, n + 2μ n=3 K II r II + G α θ, 1 μ 2π ∞ Bn n/2 II + r G α θ, n , 2μ ⎧ n=3 ⎨− νxE3 σαα r, θ plane stress , (5.10) u 3 r, θ = ⎩0 plane strain where

αα

I F12 (θ, 1) = sin 12 θ cos 12 θ cos 32 θ , I F22 (θ, 1) = cos 12 θ 1 + sin 12 θ sin 32 θ , II F11 (θ, 1) = − sin 12 θ 1 + cos 12 θ cos 32 θ , II F12 (θ, 1) = cos 12 θ 1 − sin 12 θ sin 32 θ ,

I (θ, n) = 2 + (−1)n + 12 n cos 12 n − 1 θ F11 − 12 n − 1 cos 12 n − 3 θ ,

I F12 (θ, n) = − (−1)n + 12 n sin 12 n − 1 θ + 12 n − 1 sin 12 n − 3 θ ,

n I F22 (θ, n) = 2 − −1 − 12 n cos 12 n − 1 θ + 12 n − 1 cos 12 n − 3 θ , for n ≥ 2 ,

n II F11 (θ, n) = − 2 + −1 + 12 n sin 12 n − 1 θ + 12 n − 1 sin 12 n − 3 θ ,

n II F12 (θ, n) = −1 + 12 n cos 12 n − 1 θ + 12 n − 1 cos 12 n − 3 θ ,

n II F22 (θ, n) = − 2 − −1 − 12 n sin 12 n − 1 θ − 12 n − 1 cos 12 n − 3 θ . for n ≥ 2 .

G I1 (θ, 1) = cos 12 θ 1 − 2ν + sin2 G I2 (θ, 1) = sin 12 θ 2 − 3ν − cos2 2 1 G II 1 (θ, 1) = sin 2 θ 2 − 2ν + cos 2 1 G II 2 (θ, 1) = cos 2 θ 1 − 2ν + sin

1 2θ , 1 2θ , 1 2θ , 1 2θ ,

G I1 (θ, 2) = cos θ , G I2 (θ, 2) = −ν sin θ ,

n n n G I1 (θ, n) = (3 − 4ν) cos θ − cos −1 θ 2 2 2 n n n + (−1) cos θ , + 2 2

Fracture Mechanics

n n n G I2 (θ, n) = (3 − 4ν) sin θ + sin −1 θ 2 2 2 n n + (−1)n sin θ , − 2 2 for n ≥ 3 , n n n G II (θ, n) = −(3 − 4ν) sin θ + sin − 1 θ 1 2 2 2 n n + (−1)n sin θ , − 2 2 n n n II −1 θ G 2 (θ, n) = −(3 − 4ν) cos θ + cos 2 2 2 n n + (−1)n cos θ , − 2 2 for n ≥ 3 .

σ22 σ12 σ11

y r θ x

Fig. 5.3 Crack tip coordinate system and notation for

stress components

The stress intensity factors are a function of the applied load, the geometry of the specimen or structure, and the length of the crack, and must be determined by solving the complete boundary value problem in linear elasticity that includes the full geometric description of the cracked structure and the applied load; the stress intensity factor has dimensions √ of (FL3/2 ) and in SI units is typically indicated as MPa m. The displacement field in (5.10), on the other hand, is bounded and u α → 0 as r → 0. While it may be objectionable to consider extending linear elastic calculations to infinite stresses and strains, there are two simple arguments to reconcile this difficulty. First, even though the stresses are unbounded, the energy in a small volume near the crack tip is always bounded because the displacements tend to zero. Secondly, inelastic deformations occur in the regions of high stress and hence, these expressions are taken to be applicable for small values of r relative to structural dimensions, but outside of a distance rp where inelastic, nonlinear, and fracture processes dominate the deformation of the material; an estimate of rp is obtained in the next section. For distances that are very large in comparison to rp , terms involving higher powers of r may become important. The nonsingular term is labeled σ0x in keeping with the literature in experimental mechanics, but it is also called the T-stress in the general fracture literature. This term in (5.9) is independent of r and arises only in the σ11 component of stress. These terms may not necessarily be important in characterizing fracture itself – although there is some debate in recent literature that suggests otherwise – but must be taken into account while evaluating experimental data collected from fullfield optical techniques discussed later in this chapter.

5.2.2 Irwin’s Plastic Correction Fig. 5.2 Three loading symmetries in a cracked specimen

129

Irwin [5.3] suggested that the effect of the plastic zone near the crack tip can be approximated by a very simple

Part A 5.2

The corresponding expressions of the stress and deformation fields for mode III can also be written; we do not address mode III problems in this chapter and hence these fields are not provided here. The corresponding expressions for plane strain may be obtained by replacing ν by ν/(1 + ν). The standard notation of linear elastic theory is used; the Cartesian components of stress are written in terms of the polar coordinates centered at the crack tip (see Fig. 5.3 for the coordinate system and notation). δαβ is the Kronecker delta and E is the modulus of elasticity. Standard index notation is used; Greek subscripts take the range 1, 2 and summation with respect to repeated subscripts is implied. These fields are seen to be separable in r and θ; the θ dependence for the two different modes are given above for completeness. From the first term in (5.9), the stress components are seen to be square-root singular as r → 0 and hence (5.9) cannot be valid very close to the crack tip. The strength of the singularity is determined by the two scalar quantities K I and K II , called the stress intensity factors in modes I and II, respectively.

5.2 Linearly Elastic Fracture Mechanics

130

Part A

Solid Mechanics Topics

such as B, the specimen thickness a, the crack length, and any other length defining the geometry of the specimen. This gives a qualitative meaning to the concept of small-scale yielding, and a more precise restriction must be obtained from experimental characterization of materials.

σ22

5.2.3 Relationship Between Stress Analysis and Energy Balance – The J-Integral

ry

rp

x

Fig. 5.4 Irwin model of the crack tip plastic zone

model; assume that the stress in the plastic zone near the crack tip must be limited to some multiple of the yield stress: σ22 (r < rp , 0) = βσY , where rp is the extent of the plastic zone. However, in order to maintain equilibrium, the above assumption requires that the stress estimated from (5.9) for some distance r < ry must be redistributed to regions r > ry , as indicated in Fig. 5.4. This is expressed by the following equation

Part A 5.2

ry σ22 (r, 0) dr = rp βσY , 0

1 KI ry = 2π βσY

2 . (5.11)

Evaluating this equation yields rp = 2ry . In this estimate, the square-root singular field is considered to have its origin shifted to the point r = ry , effectively considering the crack to be of length aeff = a + ry . Hence the stress intensity factors need to be evaluated using this effective crack length. If we assume a Tresca yield criterion, for thin plates with σ33 = 0, it is easy to show that β = 1. However, for conditions of plane strain expected expects to prevail near the crack tip, with σ33 > 0, one √ β to be larger; Irwin suggested a value of β = 3, resulting in an estimate of plastic zone size 1 KI 2 . (5.12) rp = 3π σY This simple estimate of the zone of yielding does remarkably well in capturing the extent of yielding near the crack tip. More importantly, this analysis indicates clearly that the characteristic scale in the frac2 length ture problem is obtained as σKYI . In applying the ideas of LEFM, it is now clear that one must ensure that (K I /σY )2 is much smaller than other relevant lengths

The energy release rate described in Sect. 5.1 and the stress field discussed above must, of course, be related in some manner. This relationship can be obtained in a formal way by considering the change in potential energy, using ideas of conservation integrals. Eshelby [5.10] introduced these in a general sense for defects; Cherepanov [5.11] and Rice [5.12] defined a special case – the J-integral – for cracks. Knowles and Sternberg [5.13] developed these as general conservation laws for elastostatic problems, but Rice’s paper was the key to recognizing the importance to fracture mechanics. The J-integral is defined as

∂u α (5.13) Un 1 − σαβ n β ds , J= ∂x1 Γ

where U =

ε

σαβ dεαβ is the mechanical work in de-

0

forming to the current strain level, and n α are the components of the unit outward normal to the contour of integration Γ shown in Fig. 5.5; this integral is independent of the path of integration. By considering a contour along the boundaries of the specimen, it can be shown that dΠ (5.14) =G. J =− da Furthermore, by considering a contour to lie in the region close to the crack tip such that the asymptotic field in (5.9) and (5.10) is valid, it can be established that 1 − ν2 2 1 − ν2 2 1 + ν 2 (5.15) KI + K II + K III . J= E E E x2

x1 s Γ

Fig. 5.5 Contour path for the J-integral

n

Fracture Mechanics

For plane stress, the factor (1 − ν2 ) in the first two terms of (5.15) is replaced with unity. This equivalence between the potential energy release rate and the stress intensity factor forms the basis of the local approach to fracture.

5.2.4 Fracture Criterion in LEFM Equations (5.3) and (5.14) provide the basis for formulating the fracture criterion in terms of the energy release rate or the J-integral; for the case of small-scale yielding (5.15) indicates that the fracture criterion may be posed in terms of the stress intensity factor as well. The crack is now considered to be fracture critical under mode I loading when KI = KC ,

(5.16)

KC

the specimen thickness B obey the following inequality K IC 2 a, B ≥ 2.5 . (5.17) σY The characteristic length scale can be seen to arise from Irwin’s scaling analysis in Sect. 5.2.2; the numerical factor was determined through numerous tests in different materials and loading geometries. Standard techniques for characterization of the fracture toughness are provided by the American Society for Testing and Materials, the British Standards Institution, the Japanese Society of Mechanical Engineers, and the International Organization for Standardization (ISO). In particular, conditions for preparation and testing of specimens that produce a valid measurement of the plane-strain fracture toughness that obeys the restriction in (5.17) are discussed in these standards. The typical range of values of the fracture toughness for different materials is provided in Table 5.1. This methodology of assessing fracture with very limited inelastic deformation is called linearly elastic fracture mechanics (LEFM). As an illustration of LEFM, consider a simple example: a large panel with a crack of length 2a subjected to uniform stress σ; the stress√intensity factor for this configuration is simply K I = σ πa. Applying the fracture criterion in√(5.16) results in the critical condition, expressed as σ πa = K IC . This condition can be used in one of three ways in fracture-critical structures. First, at the design stage, one sets the crack length to be at the limit that is detectable by nondestructive inspection techniques. Then, for a desired design load, a material with the appropriate fracture toughness can be selected or alternatively, for a given material, the maximum permissible stress can be determined. Second, for a given structural application with fixed fracture toughness and stress, the critical crack length can be calculated and Table 5.1 Typical values of the fracture toughness for various materials

KIC

2.5 Normalized specimen thickness B

σY KIC

2

Fig. 5.6 Dependence of the fracture toughness on specimen thickness

Material

K IC , MPa

Ductile metals: Cu, Ni, Ag A533 Steel Mild steels High-strength steels Al alloys Ceramics: Al2 O3 , Si3 N4 Polyethylene Polycarbonate Silica glass

100– 350 200 140 50– 150 20– 45 3–6 2 1 – 2.5 0.7

√ m

131

Part A 5.2

where K C is the critical stress intensity factor, considered to be a material property; note that it must be evaluated with special care to ensure that conditions of small-scale yielding are assured. One expects that the plastic work in a thinner specimen is larger than in that in a thicker specimen; this is reflected in the typical variation of K C with specimen thickness as shown in Fig. 5.6. The measured value of K C is high for small specimen thicknesses and reaches a nearly constant lower plateau at large specimen thicknesses; it is this value of the critical stress intensity factor, labeled K IC , that a new material property called the plane strain fracture toughness that is taken to be a new material. The main condition that arises here ensures that the radius rp of the inelastic region near the crack tip is small enough; thus, in tests performed to evaluate the fracture toughness, one must have the crack length a and

5.2 Linearly Elastic Fracture Mechanics

132

Part A

Solid Mechanics Topics

used in inspections to determine how close the structure is to fracture criticality. Considering that cracks grow during subcritical loading (Sect. 5.5), one may also determine how long the crack will grow in the time interval between inspections. Finally, for a given crack length and material, one can impose limits on loading such that critical conditions are not reached during operation. The fracture criterion described above is valid only for the mode I loading symmetry. Under a combination of modes I and II, the crack typically kinks from its initial direction and finds a new direction; hence the fracture criterion should not only determine the load at which the crack will initiate under mixed-mode loading, but also determine the direction of the new crack. Applying the energy balance idea to this problem, the fracture criterion may be stated as follows: The crack will follow the path along which the potential energy release rate is maximized. While this is the most appropriate form of the fracture criterion, it is difficult to apply since the energy release rate has to be calculated for different paths and then maximized. A number of alternative criteria exist: the maximum tangential

stress criterion, the principle of local symmetry, the strain energy density criterion, etc., but their predictions are not clearly distinguishable in experimental observations. The maximum tangential stress criterion, which provides the same estimate for the crack direction as the energy-based criterion, can be stated as follows: The crack will extend in the direction γ such that ∂σθθ /∂θ is maximized. Using the stress field in (5.9), ⎫ ⎧ ⎨ K I − K I2 + 8K II2 ⎬ . γ = 2 arctan (5.18) ⎭ ⎩ 4K II The energy release rate for extension in that direction G(γ ) must be equal to the fracture resistance R at the onset of crack initiation. The fracture criterion appropriate for mode III loading is significantly more complicated. A straightforward extension of the maximum tangential stress criterion discussed in this paragraph for use in combined mode I and mode II can be shown to breakdown simply from symmetry considerations. In fact, the crack front breaks up into a fragmented structure and appropriate criteria for characterizing this are still not well developed.

Part A 5.3

5.3 Elastic–Plastic Fracture Mechanics In cases where the extent of the plastic zone becomes large compared to the specimen thickness and crack length, the LEFM methodology of Sect. 5.2 breaks down and one is forced to contend with the inelastic processes that occur within r < rp . This is the subject of elastic–plastic fracture mechanics (EPFM). First, an elegant and simple way of incorporating the inelastic effects – through the Dugdale–Barenblatt model – is introduced; this is followed by an evaluation of the stress and deformation field using the deformation theory of plasticity. Then we return to an extension of the Dugdale–Barenblatt model in the form of generalized cohesive zone models for inelastic fracture. Finally, a generalized damage model based on void nucleation and growth is described.

the crack tip, interatomic (cohesive) forces that attract the two separating crack surfaces cannot be ignored in the analysis. Dugdale considered a larger-scale picture where a linear region in the prolongation of the crack tip would limit the development of stress to some multiple of the yield stress (similar to the observation of Irwin discussed in Sect. 5.2.2). Both these suggestions amount

δt

α

5.3.1 Dugdale–Barenblatt Model The idea of eliminating the source of the singularities in the elastic analyses was proposed independently by Barenblatt [5.14] and Dugdale [5.15] and has been generalized into what are now commonly called cohesive zone models. Barenblatt suggested that, close to

δt

Fig. 5.7 Schematic diagram of the Dugdale–Barenblatt model of fracture

Fracture Mechanics

Problem a

factors:

K Ia = σ ∞ π(a + rp ) , a + rp a b −1 (5.19) . cos K I = −2βσY π a + rp Note that the negative sign in problem b is a mere formality arising from the fact that the loads pinch the crack tip closed. The condition that the resulting problem does not contain singular stresses is enforced by requiring the sum of the stress intensity factor from the two component problems to be zero; this results in the following estimate for the extent of the inelastic zone: ∞ rp πσ (5.20) −1 . = sec a 2βσY The crack-tip opening displacement (CTOD) is defined as the separation between the top and bottom surfaces of the physical crack tip located at x1 = a; this can also be calculated as the superposition of the elastic solutions to the two problems; for plane stress, this results in δt = u 2 (x1 = a, x2 = 0+ ) − u 2 (a, 0− ) ∞ 8βσY a πσ = (5.21) . ln sec πE 2βσY A key feature of the DB condition is that the crack opening profile is not parabolic, as indicated by the singular solution, but a smooth cusp-like closing from x1 = a to x1 = a + rp ; this is indicated in Fig. 5.9. Crack opening displacement – πE δ 2βσY a 10

5

σ∞ Problem b

133

0

βσ Y –5 2(a+rp) 2(a+rp) –10 –2

–1

0

1 Position –

2 x1 a

Fig. 5.8 Dugdale–Barenblatt model of fracture for a center- Fig. 5.9 Crack opening profile in the Dugdale–Barenblatt cracked specimen model of fracture

Part A 5.3

to a model for the fracture process and provide a way to regularize the stress near the crack tip. Since then generalizations of the cohesive zone ideas to craze failure in polymers (Knauss [5.16] and Schapery [5.17]), fracture in weakening solids such as concrete by Hillerborg et al. [5.18], and for ductile failure in metallic materials [5.19] have been developed. A schematic diagram of the Dugdale–Barenblatt (DB) model is shown in Fig. 5.7. Assuming that the extent of the inelastic zone ahead of the crack tip rp is large in comparison to the specimen thickness B, it can be modeled as a line segment on x2 = 0, along which the normal stress σ22 can be taken to be limited by the yield stress as in the case of the Irwin model: σ22 (r < rp , 0) = βσY . However, the extent of the inelastic zone is obtained by ensuring that the singularity indicated by the elastic analysis is removed, rather than from the local equilibrium analysis of Irwin; this is called the DB condition. This requires the solution of the appropriate boundary value problem for the cracked elastic solid; the procedure is illustrated here for the case of a large plate with a central crack of length 2a loaded in uniform remote tension as shown in Fig. 5.8. Consider that the material in the DB zone is removed and replaced with its equivalent effect, i. e., a constant stress of magnitude βσY over the length rp . (Indeed, there is no reason to limit this to constant stress; the generalization of nonlinear cohesive zone models simply relies on introducing a nonlinear traction separation law over the line of the inelastic zone.) This is represented as the superposition of two elastic problems, problem a, with uniform far-field stress σ ∞ of a crack of length 2(a + rp ), and problem b, with uniform closing stress of magnitude βσY only over the extended crack surfaces rp near either tip. The solutions to these problems result in the following stress intensity

5.3 Elastic–Plastic Fracture Mechanics

134

Part A

Solid Mechanics Topics

While the introduction of the DB model has removed the singularity and provided a way to estimate the extent of the inelastic zone, it has also eliminated the possibility of imposing a fracture criterion based on the stress intensity factor. One has to resort to the energy or J-based criterion for fracture; the latter can be accomplished by evaluating the J-integral for a contour indicated in Fig. 5.7 that closely follows the DB zone; for a < x1 < a + rp , we have n 1 = 0, σ11 = 0, σ12 = 0, σ22 = βσY ; thus, a+α

∂u 2 ∂u 2 J =− − (−βσY ) dx1 ∂x1 (x2 ,0+ ) ∂x1 (x1 ,0− ) a

= βσY δt .

(5.22)

Part A 5.3

The equivalence indicated in (5.22) between the Jintegral and the CTOD holds in general for all elastic–plastic crack tip models, but the particular form above is valid for all crack geometries based on the DB model. Now, the fracture criterion may be applied by setting J = Jc at the onset of crack initiation, or equivalently in terms of the CTOD. For the particular case of the center-cracked geometry, substituting from (5.21), setting J = Jc at the failure stress σ ∞ = σf , and rearranging, we get an estimate for the failure stress under the DB model as: 2 πE σf J = sec−1 exp (5.23) . c βσY π 8(βσY )2 a A plot of the failure stress as a function of the crack length is shown in Fig. 5.10. For comparison, the estiNormalized failure stress – 2

mate based on small-scale yielding is also shown in the figure. It is clear that, for short cracks, the failure stress is large enough that large-scale yielding becomes important and LEFM does not provide a correct estimate of the fracture strength.

5.3.2 Elastic–Plastic Crack Tip Fields In order to account in greater detail for the development of plastic deformation in the vicinity of the crack tip, Hutchinson [5.20], and Rice and Rosengren [5.21] performed an asymptotic analysis of the fields in a nonlinear elastic solid characterized by a power-law constitutive model. The stress–strain relationship is given as n−1 sαβ εαβ 3 σe = α , (5.24) εY 2 σY σY where α is a material constant, n ≤ 1 is the strainhardening exponent, σY is the yield stress in uniaxial tension and εY = σY /E is the corresponding yield strain, sαβ = σαβ − σγγ δαβ /3 are the components of the stress deviatoric, and σe = 3sαβ sαβ /2 is the effective stress. The field near the crack tip was determined to be J 1 1/(n+1) σ˜ αβ (θ, n) , σαβ (r, θ) = σY ασY εY In r εαβ (r, θ) = αεY

ε˜ αβ (θ, n) , (5.26)

σf βσY

J 1 ασY εY In r

(5.25)

n/(n+1)

1 J u α (r, θ) = αεYr ασY εY In r

n/(n+1) u˜ α (θ, n) , (5.27)

1.5

1

0.5

0

0

1

2

3

Normalized crack length –

4 8(βσY)2 a πEJc

Fig. 5.10 Comparison of the failure stress calculated from

LEFM (dashed line) and the DB model (solid line)

where J is the value of the J-integral, In is a dimensionless quantity that depends only on the strain hardening exponent (see, for example, Kanninen and Popelar [5.22] for calculations of I (n)), and σ˜ αβ (θ, n), ε˜ αβ (θ, n), and u˜ αβ (θ, n) express the angular variation of the stress, strain, and displacement variation near the crack tip, determined numerically; the description of the field given above is called the HRR field, in honor of the original authors, Hutchinson, Rice and Rosengren. It is clear that the stress and strain components given in (5.25) and (5.26) are singular as r → 0, but with the order of singularity different from the corresponding linear elastic problem and different for the stress and strain components. The strength of the singularity

Fracture Mechanics

is determined by the value of the J-integral. Equations (5.25)–(5.27) cannot be valid very close to the crack tip and a small region in the vicinity of the crack tip – called the fracture process zone, rfpz – must be excluded from consideration. The fracture process zone is taken to be the region in which the actual process of fracture – void nucleation, growth, and coalescence – occurs and results in the final failure of the material. Furthermore, it is clear from (5.27) that the CTOD, defined arbitrarily as the opening at the intercept of two symmetric 45◦ lines from the crack tip and the crack profile, should be proportional to the J-integral: δt = d(n, εY )J/σY , where d(n, εY ) can be calculated in terms of the HRR field; see Kanninen and Popelar [5.22] for a plot of this variation. Thus, in EPFM, J plays multiple roles – determining the amplitude of the plastic stress and deformation field, determining the CTOD, and determining the potential energy release rate. It is possible to incorporate terms that are higher order in r but evaluation of such terms is much more difficult in the elastic–plastic problem in comparison to the elastic problem. Attempts have been made to incorporate the next-order term (the nonsingular term) in fracture analysis [5.23, 24], but these are not addressed here.

Unlike the case of LEFM, there is no simple way in which to estimate the size of the process zone rfpz and hence of the region of dominance of the HRR field expressions given above. Some attempts using moiré interferometry are discussed in Sect. 5.6.2. However, it is clear from discussions of the plastic zone in Sect. 5.2.2 and the CTOD that the only characteristic length scale relevant to this problem can be written as J/σY . Thus, the remaining uncracked segment b and the thickness of specimen B must obey the following inequality in order to justify the use of the HRR field b, B > κ

J . σY

(5.28)

Estimates of how large this should be vary significantly depending on the geometry of the specimen; estimates based on finite element analysis show that κ varies over a range from about 20–300 depending on the specimen geometry. Under such conditions, the fracture criterion may be posed as the equivalent of the energy criterion discussed above. Hence, we have J(σ , a) = JC .

(5.29)

135

Here it is necessary to calculate J(σ , a), a somewhat more challenging task than the corresponding calculation of the stress intensity factor for a problem in LEFM. Under laboratory testing conditions, since the J-integral is the change in potential energy of the system, one can use (5.14) and the experimentally measured load versus load-point displacement data to evaluate J. Typically, this results in a simple expression of the form J=

ηA , bB

(5.30)

where A is the area under the load–displacement curve, b is the remaining uncracked ligament, B is the specimen width, and η is a factor that depends on the specimen geometry. Therefore, monitoring the load–displacement curve is adequate in the evaluation of the J-integral. The value of the J-integral at onset of crack extension should result in an experimental measurement of the fracture toughness under elastic–plastic conditions, not restricted to small-scale yielding. In applying this approach of EPFM to other structural configurations, one encounters two difficulties. First, whereas many different handbooks tabulate solutions of stress intensity factors for different geometric and loading conditions, few such solutions are available for elastic plastic problems. The Ductile Fracture Handbook [5.25] provides a tabulation of estimates of the J-integral for a few cases associated with circular cylindrical pipes and pressure vessels. However, with modern computational tools, this is hardly a limitation and one should, in principle, be able to calculate the J-integral for arbitrary loading and geometric conditions. The second – and more challenging – difficulty arises from the fact that the measured crack resistance in many common structural materials increases significantly with crack extension and that such an increase depends critically on the constraint to plastic deformation provided in the particular geometric condition of the test. This brings into question the transferability of laboratory test results to full-scale structural integrity assessment. Therefore the methodology presented in this section remains limited in its use to a few configurations of cylindrical pipes and pressure vessels where there is significant experience in its use. Recent advances in elastic–plastic fracture methodology have focused more on generalized cohesive zone models and damage models as discussed in the next two sections. Such generalized cohesive zone models are

Part A 5.3

5.3.3 Fracture Criterion in EPFM

5.3 Elastic–Plastic Fracture Mechanics

136

Part A

Solid Mechanics Topics

now embedded into commercial finite element software packages.

5.3.4 General Cohesive Zone Models

Part A 5.3

Cohesive zone models for nonlinear fracture problems are simply a straightforward extension of the Dugdale– Barenblatt model. In these models, the inelastic region (including the plastic zone and the fracture process zone) is assumed to be in the form of a line ahead of the crack tip, as in the DB model. However a relationship between the traction across this line T and the separation δ between the two surfaces T = F(δ) is postulated. Such a relationship may be obtained directly from experimental measurements (as in the case of some polymeric materials) or more typically from a micromechanical model of the void nucleation, growth, and coalescence that are the underlying fracture processes. A typical traction-separation law for cohesive zone models is shown in Fig. 5.11. Two crucial features can be identified: the peak stress that corresponds to the maximum stress that can be sustained by the material in the fracture process zone and the maximum displacement that corresponds to the critical CTOD. The area under the traction separation curve is the fracture energy JC . Details of such generalized cohesive models may be found in a number of papers by Knauss [5.16], Schapery [5.17], Hilleborg et al. [5.18], Kramer [5.26], Normalized traction T 1.5

1

0.5

0

0

0.2

0.4

0.6 0.8 1 Normalized separation δ

Fig. 5.11 A typical traction–separation law used in cohesive zone models is shown. Traction is normalized by the peak cohesive traction and the separation is normalized by the critical CTOD; the area under the curve is the fracture energy

Carpinteri [5.27], Needleman [5.28], Xu and Needleman [5.29], Yang and Ravi-Chandar [5.30], Ortiz and Pandolfi [5.31], and many other references. Incorporation of such a generalized cohesive zone model into finite element computational strategies allows for estimation of crack growth to be performed automatically, without imposing additional criteria for fracture. Irreversible models of the cohesive zone have also been proposed based on the maximum attained crack surface separation [5.30–32]. In particular, persistence of previous damage in the cohesive zone, and contact and friction conditions upon unloading, are important aspects to include in a proper description of the cohesive zone model. Numerous investigators have shown that crack growth simulations based on cohesive zone models are somewhat insensitive to the detailed form of the traction-separation curve and that it is much more important to capture the correct fracture energy. However, care should be exercised in using this idea since there is an inherent dependence on the mesh size and geometry dependence of the results obtained (Falk et al. [5.33]).

5.3.5 Damage Models In cases where the cohesive zone model may not be appropriate because of the diffuse nature of the inelastic zone, one may resort to continuum damage models for modeling fracture. The Gurson–Tvergaard–Needleman model is perhaps the most commonly used model for simulations of fracture [5.34]. In this model, the yield function describing the plastic constitutive model is represented as σ2 3q2 σH −1 g(σe , σH , f ) = e2 + 2 f ∗ q1 cosh 2σ σ 2 (5.31) − q1 f ∗ , where σe is the effective stress, σH = σii/3 is the mean stress, and σ is the flow stress of the material at the current state, q1 and q2 are fitting parameters used to calibrate to model predictions of periodic arrays of spherical and cylindrical voids, f is the void volume fraction, and f ∗ is a bilinear function of the void volume fractions that accounts for rapid void coalescence at failure and is given as ⎧ ⎨f f < fc , (5.32) f∗ = ⎩ f c + (1/q1 − f c ) ( f − f c ) f ≥ f c ( ff − fc )

where f f represents the void volume fraction corresponding to failure; the value of f ∗ at zero stress f c

Fracture Mechanics

is the critical void volume fraction where rapid coalescence occurs. The above description of the yield function needs to be augmented with evolution equations for both the flow stress σ and the void volume fraction f . The instantaneous rate of growth of the void fraction depends both on nucleation of new voids and growth of preexisting voids. Thus, f˙ = f˙nucleation + f˙growth .

(5.33)

Chu and Needleman [5.35] assumed that void nucleation will obey a Gaussian distribution; then straincontrolled nucleation can be expressed as 1 ε¯ p − εn ˙ p fn ε¯ (5.34) f˙nucleation = √ exp − 2 sn sn 2π where sn is the standard deviation, εn is the nucleation threshold strain, f n is a constant, and ε¯ p is the average plastic strain. The latter is related to the plastic strain through a work equivalence expressed as (1 − f )¯εp σ = σ : ∂Φ ∂σ . The growth rate is obtained by enforcing that the plastic volumetric strain rate in the matrix (outside of the voids) is zero; thus, p f˙growth = (1 − f )˙εkk .

(5.35)

where m is the strain rate hardening exponent, and g(¯ε, T ) is the temperature-dependent effective stress versus effective strain response calibrated through a uniaxial tensile test at a strain rate ε˙¯ 0 and temperature T .

Setting f = 0 results in the above formulation reducing to a standard von Mises yield surface for an isotropic material. Equations (5.31)–(5.35) augment the standard plasticity equations in the sense that void nucleation and growth are now incorporated into the model. If the above model is evaluated under monotonically increasing uniaxial strain on a cell of unit height, initially a linear elastic behavior is obtained; this is followed by nonlinearity associated with plastic deformation and eventually a maximum stress associated with the onset of decohesion or failure of the unit cell. Beyond this level of displacement, the cell is unstable to load control, but deforms in a stable manner under displacement control. Thus, the damage zone model mimics the response that one assumes in the generalized cohesive zone models discussed above. Two main issues need to be addressed with respect to this type of damage model. First, there are numerous constants embedded into the theory. In addition to the usual elastic constants E and ν, one has the plastic constitutive description g(¯ε, T ), the strain-rate hardening parameter m, the four parameters that define the yield function q1 , q2 , f c , f f , and the three parameters that define the nucleation criterion f n , sn , εn . These need to be determined through calibration experiments for each particular material. Second, the introduction of a cell is a micromechanical construct: it introduces an artificial length scale and geometric effect that must be considered carefully. For example, does calibrating the model parameters under uniaxial straining provide sufficient characterization to handle biaxial or triaxial loading conditions? Active research continues in this area to explore these questions further and to validate the approach of the damagemodel-based fracture analysis.

5.4 Dynamic Fracture Mechanics In many applications, it is necessary to consider cracks that grow rapidly at speeds that are a significant fraction of the stress wave speeds in the material. In such cases, material inertia effects cannot be ignored in considerations of crack growth. Within the setting of small-scale yielding, analysis of the stress and displacement field near the tip of a crack moving at a speed v can be accomplished easily within linear elastodynamic theory [5.36, 37]; the crack tip stress and displacement fields for in-plane loading conditions may be written in

terms of polar coordinates moving with the crack tip as KI K II II I f αβ f αβ (θ, v) (θ, v) + √ σαβ (r, θ) = √ 2πr 2πr + σox αd2 − αs2 δα1 δβ1 + · · · , ⎧ ⎨0 plane stress σ33 (r, θ) = , (5.37) ⎩νσ (r, θ) plane strain αα

137

Part A 5.4

Power-law hardening of the plastic material is represented as m σ ˙ε¯ = ε˙¯ 0 , (5.36) g(¯ε, T )

5.4 Dynamic Fracture Mechanics

138

Part A

Solid Mechanics Topics

√ KI r I u α (r, θ) = √ gα,β (θ, v) 2π √ K II r II + √ gαβ (θ, v) + · · · , ⎧ 2π ⎨− νx3 σ (r, θ) plane stress E αα . (5.38) u 3 (r, θ) = ⎩0 plane strain I (θ, v) and f II (θ, v) are The functions f αβ αβ I f 11 (θ, v) =

1 R(v)

G(v) =

cos 12 θd 1 + αs2 1 + 2αd2 − αs2 1/2 γd

− 4αd αs

1

1 2 θs

cos , 1/2 γs 2 cos 12 θd 1 I (θ, v) = − 1 + αs2 f 22 1/2 R(v) γd 1 + 4αd αs 1/2 cos 12 θs , γs 1 + αs2 sin 12 θd sin 12 θs 2α d I − , f 12 (θ, v) = 1/2 1/2 R(v) γ γs d

Part A 5.4

II f 11 (θ, v) = −

2αs R(v)

(5.39)

sin 12 θd 1 + 2αd2 − αs2 1/2 γd

1 − 1 + αs2 1/2 sin 12 θs , γs 2 2αs 1 + αs sin 12 θd 1 II 1 (θ, v) = − sin θ f 22 2 s , 1/2 1/2 R(v) γd γs cos 12 θd 1 II f 12 (θ, v) = 4αd αs 1/2 R(v) γd cos 12 θs − 1 + αs2 (5.40) , 1/2 γs where Cd and Cs are the dilatational and distortional wave speeds, respectively, and γd = 1 − (v sin θ/Cd )2 and (5.41) γs = 1 − (v sin θ/Cs )2 , tan θd = αd tan θ , tan θs = αs tan θ , 2 R(v) = 4αd αs − 1 + αs2 .

gularity dictated by the dynamic stress intensity factor for the appropriate mode of loading. In analogy with the quasistatic crack problems, it is possible to define a dynamic energy release rate; it is the energy released into the crack tip process zone per unit crack extension. Introducing the elastodynamic singular stress field, G can be related to the dynamic stress intensity factor

(5.42) (5.43)

As in the case of quasistatic LEFM, the crack tip stress field is square-root singular, with the strength of the sin-

1 − ν2

AI (v)K I2 + AII (v)K II2 , E

v2 αd and (1 − ν)Cs2 R(v) v2 αs . AII (v) = (1 − ν)Cs2 R(v)

(5.44)

AI (v) =

(5.45)

E is the modulus of elasticity and ν is the Poisson’s ratio. The fracture criterion in (5.3) can be imposed in this problem as well – the dynamic energy release rate must be equal to the dissipation; thus G(v) = R provides the equation of motion for the crack velocity. The functions AI (v) and AII (v) in (5.45) are singular as v → CR . To satisfy the energy balance equation, the dynamic stress intensity factors K I (t, v) and K II (t, v) must tend to zero as v → CR . This implies that the limiting crack speed in modes I and II is the Rayleigh wave speed. Experimental measurements by many investigators, beginning with the pioneering work of Schardin [5.38], have found that in practice cracks never reach that speed, branching into two or more cracks at speeds of only about one half of the Rayleigh wave speed. For use in applications, the deviations from the energy criterion discussed above has been circumvented by the use of distinct criteria for dynamic crack initiation, dynamic crack growth, and dynamic crack arrest as described in the following sections.

5.4.1 Dynamic Crack Initiation Toughness Since the state of stress near the crack tip is described in terms of the dynamic stress intensity factor K I (t), crack initiation can be identified with the stress intensity factor reaching a critical value; the dynamic crack initiation criterion is postulated as (5.46) K I (tf ) = K Id T, K˙ I . The right-hand side represents the dynamic initiation toughness, with the subscript ‘d’ replacing the subscript ‘C’ used for the critical stress intensity factor for the quasistatic fracture toughness. The dependence

Fracture Mechanics

of the dynamic crack initiation toughness on the temperature and rate of loading (represented by the rate of increase of the dynamic stress intensity factor, K˙ I ) is indicated through the arguments; this dependence must be determined through experiments covering the range of temperatures and rates of loading of interest. The temperature dependence of the dynamic crack initiation toughness arises from the increase in ductility with increasing temperature or from heating associated with inelastic deformation in the near-tip zone or a combination of both. The left-hand side of (5.46) represents the applied stress intensity factor at time tf when crack propagation commences. Standard procedures do not exist for characterization of the dynamic crack initiation toughness at rates on the order of K˙ I ≈ 1 × 104 MPa m3/2 s−1 or larger, but there exist a large database on specific materials; a survey is presented by Ravi-Chandar [5.37].

5.4.2 Dynamic Crack Growth Toughness

5.4 Dynamic Fracture Mechanics

139

An interesting consequence of this difference between crack initiation and growth toughness is that the crack will jump to a large finite speed immediately upon initiation. Second, experimental efforts to determine K ID v, K˙ I , T have met with mixed success, particularly for nominally brittle materials; variations in the measurements with different specimen geometries [5.39, 40] and loading rates [5.41] have not been resolved completely. Also, indications that the measurements are hysteretic – meaning that accelerating and decelerating cracks exhibit different behavior – have been reported [5.42]. For ductile materials, (5.47) appears to be a more reasonable characterization of dynamic fracture; see Rosakis et al. [5.43]. As in the case of dynamic crack initiation toughness, standard procedures have not been developed for determination of the dynamic crack growth toughness, but there is a large literature on methods of characterization and data on specific materials; Ravi-Chandar [5.37] presents a survey of available results on this topic.

5.4.3 Dynamic Crack Arrest Toughness

The upper-case subscript ‘D’ is used to indicate the dynamic crack growth toughness instead of the lowercase ‘d’ that was used to indicate the dynamic initiation toughness. Once again, the right-hand side represents the material property to be characterized through experiments and the left-hand side represents the dynamic stress intensity factor calculated from the solution of the boundary initial value problem in elastodynamics. The dynamic crack growth toughness is commonly referred to as the K I –v relation. Some important limitations must be recognized in using (5.47) for dynamic crack growth problems. First, the crack initiation point is not on the curve characterizing the dynamic crack growth criterion. Thus, K ID v → 0, K˙ I , T = K Id T, K˙ I . This could be possibly due to bluntness of the initial crack, the intrinsic rate dependence of the material, or inertial effects.

In applications, the most conservative design approach would utilize (5.48), thus assuring that the dynamic stress intensity factor for all possible loading conditions never exceeds the crack arrest toughness. Thus a dynamically growing crack is never encountered in the lifetime of the structure and one avoids the complications in computing the dynamic stress intensity factors for growing cracks and in determining the dynamic crack growth criterion. Based on round-robin tests and an accumulation of data a standard test procedure, the ASTM E-1221 standard has been established that describes the determination of crack arrest toughness in ferritic steels. A rapid crack growth-arrest sequence is generated in a compact crack arrest specimen by using a displacement-controlled wedge loading; as the crack grows away from the loading wedge, the stress intensity factor drops quickly and hence results in arrest of the

Crack arrest is not the reversal of initiation and hence the initiation toughness discussed above is not relevant for crack arrest; also, the characterization of the crack growth criterion at very slow speeds is quite difficult. This has led to the postulation of a separate criterion for crack arrest: the dynamic crack arrest toughness is defined as the smallest value of the dynamic stress intensity factor for which a growing crack cannot be maintained; thus the crack arrests when K I (t) < K Ia (T ) .

(5.48)

Part A 5.4

Once dynamic crack growth has been initiated as per the conditions of the dynamic initiation criterion, subsequent growth must be determined though a separate criterion that characterizes the energy rate balance during growth. The dynamic stress field near a growing crack is still characterized by the dynamic stress intensity factor, but now this is a function of loading, time, crack position, and speed and is represented as K I (t, v). The energy rate balance in (5.44) can be expressed as a relation between the instantaneous dynamic stress intensity factor and the toughness: (5.47) K I (t, v) = K ID v, K˙ I , T .

140

Part A

Solid Mechanics Topics

crack. The dynamic run–arrest sequence that this specimen experiences under the wedge load clearly indicates the need for a dynamic analysis of the problem. However, according to the ASTM standard, a static analysis is considered to be appropriate assuming that the values

measured at 2 ms after crack arrest do not differ significantly from the values measured at 100 ms after crack arrest. The stress intensity factor at arrest is taken to be the crack arrest toughness, K Ia . Further details can be found in the ASTM E-1221 standard.

5.5 Subcritical Crack Growth

Part A 5.6

The discussion above has been focused on fracture under monotonically increasing loading that approaches the critical conditions resulting in quasistatic or dynamic crack growth. The primary design philosophy of flaw-tolerant design is to ensure that the structure should never reach the critical condition during its useful life. However, this does not imply that crack growth does not occur during the lifetime of the structure; crack extension occurs under subcritical conditions; if such subcritical extension is due to repeated loading– unloading cycles, the resulting crack extension is called fatigue crack growth. Such extension could also be due to operational conditions under steady loading; temperature-dependent accumulation of plastic deformation, resulting in creep crack growth in metals above 0.5 TM , and stress-assisted corrosion in glass and some metallic materials are some other examples of subcritical crack growth. In all these cases, crack extension occurs very slowly over a number of cycles or over a long time; eventually such cracks reach a critical length, resulting in catastrophic failure at some point. Subcritical crack growth is typically handled in a phenomenological sense; regardless of the underlying cause, material characterization of crack extension can be performed under specified condition and represented as da (5.49) = f (K I , . . .) , dξ where ξ is the number of cycles for fatigue, and time for time-dependent crack growth; thus the crack growth

rate per unit cycle or unit time is some function of the stress intensity factor (or an equivalent parameter), the environment, and perhaps other variables. The functional form of the dependence on the stress intensity factor (or its range in the case of cyclic loading) is obtained from experimental characterization. For fatigue, the dependence is typically written as ΔK I da = C(ΔK I )b , dN

(5.50)

where C and b are empirical constants, determined through calibration experiments for each material; it is well known that these are not true material constants, but influenced significantly by material microstructure, crack length, loading range, etc. If the critical crack length, ac , at final fracture is known, then the number of cycles to failure can be obtained by integrating (5.50)

ac Nc = a0

da . C(ΔK I )b

(5.51)

Details of fatigue crack growth analysis may be found in specialty books devoted to fatigue of materials [5.44, 45]. Standard methods for fatigue crack growth characterization, such as E 647, have also been established by the ASTM and other organizations. A good summary of progress in creep fracture is provided by Riedel [5.46]. Stress-corrosion and chemical effects are discussed in detail by Lawn [5.47].

5.6 Experimental Methods Many different experimental methods have been developed and used in investigations of fracture. They may be broadly classified into two categories: those based on optical methods that provide full-field information over a region near the crack tip and those based on pointwise measurement of some component of stress or

deformation. While full-field techniques are preferred in research investigations where the sophistication of apparatus and analysis are required in order to elucidate fundamental phenomena, in applications to industrial practice, simpler methods such as those based on strain gages or compliance methods are to be preferred. The

Fracture Mechanics

methods of photoelasticity, moiré and classical interferometry, and shearing interferometry, fall into the first category and are discussed first. The digital image correlation (DIC) method is rapidly gaining popularity in light of the availability of commercial hardware and software. This method is described in detail in Chap. 25 and hence is not discussed in the present chapter. The optical method of caustics and strain-gage-based methods provide much more limited data, but are quite useful in the determination of stress intensity factors; these are discussed next. Finally, methods based on measurement and analysis of the compliance of the entire specimen and loading system are also of significant use in fracture mechanics; this is a straightforward application of (5.7) and is not discussed further. Three different volumes devoted to experimental methods in fracture characterization have been published in the last 25 years [5.48–50]; while the present chapter provides a survey of the methods, these original references remain invaluable resources for details on the different methods discussed in this chapter.

5.6 Experimental Methods

141

5.6.1 Photoelasticity The method of photoelasticity has been applied to fracture problems under both static and dynamic loading conditions. In a circular polariscope, illustrated in Fig. 5.12, the intensity of the light beam is: Δs(x1 , x2 ) (5.52) , I (x1 , x2 ) ∝ k2 sin2 2 where k is the amplitude of the electric vector and Δs(x1 , x2 ) is the absolute phase retardation (see Chap. 24 for details of the analysis). Thus, in this optical arrangement, the spatial variation of the light intensity is governed only by the phase retardation introduced by the stressed specimen. Introducing the phase difference from (25.2) of Chap. 25, bright fringes corresponding to maximum light intensity are lines in the x1 –x2 plane along which (σ1 − σ2 ) =

N fσ , with N = 0, ±1, ±2, . . . , (5.53) h Axis of polarization Light source

Fast axis Polarizer π/4 σ1 λ/4

σ2 α

Fast axis

Specimen π/4

λ/4

Polarizer

Axis of polarization

Fig. 5.12 Optical arrangement of a circular polariscope in a dark-field configuration

Part A 5.6

*

142

Part A

Solid Mechanics Topics

Table 5.2 Material stress fringe value f σ for selected polymers (λ = 514 nm) Material

fσ (kN/m)

Source

Homalite 100

22.2

Polycarbonate

6.6

Polymethyl methacrylate

129

Dally and Riley [5.51] Dally and Riley [5.51] Kalthoff [5.52]

Part A 5.6

where N is called the fringe order, f σ is the stress-fringe value, which depends on the material, h is the specimen thickness, and σ1 and σ2 are the principal stress components. Typical values of f σ for materials commonly used in fracture studies are given in Table 5.2. Therefore, placing the specimen between crossed circular polarizers reveals lines of constant intensity that are contours of constant in-plane shear stress; these lines are called isochromatic fringes. Such isochromatic fringe patterns are captured with a camera at sufficient spatial and temporal resolution for quasistatic and dynamic problems and analyzed to determine the stress field. In applications to fracture mechanics, it is assumed that the appropriate asymptotic field is applicable in the vicinity of the crack tip and the parameters of the asymptotic field are then extracted by fitting the observed isochromatic fringes to theoretically predicted patterns in a least-squared error minimization process. Assuming that the fringe pattern formation is governed by the asymptotic stress field near the crack tip and substituting for σ1 and σ2 , the geometry of the a)

b)

fringe pattern can be expressed as N fσ (5.54) = g(r, θ, K I , K II , σox , . . .) . h For given values of the stress intensity factors, fringes can be simulated using (5.54). Examples of such simulated isochromatic fringe patterns corresponding to assumed values of K I , K II , and σox are shown in Fig. 5.13; in this representation, only the singular term and the first higher-order term are indicated, whereas in actual applications higher-order terms in the crack tip stress field are also taken into account in interpreting the experimental fringe pattern. An example of the time evolution of the fringe patterns obtained with a high-speed camera is shown in Fig. 5.14 [5.53]. The stress intensity factor K I can be obtained at each instant in time by using a leastsquares matching of the experimentally measured fringe pattern with simulations based on (5.54). We describe the general method of extracting parameters of the stress field from experimentally observed isochromatics. First, the experimental isochromatic fringe pattern is quantified by a collection of (Ni , ri , θi ), measured at M points. The distance r at which these measurements are taken should be appropriate for application of the two-dimensional asymptotic crack tip stress field; this is typically interpreted to hold for r/h > 0.5, based on the experimental results of Rosakis and RaviChandar [5.54], but Mahajan and Ravi-Chandar [5.55] showed that for fringe data based on photoelasticity, one may be as close as r/h > 0.1 and still obtain a good estimate of the stress intensity factors. The sum of the squared error in (5.54) at all measured points is then given by M Ni f σ e(a1 , a2 , . . . ak ) = h i=1 2 − g(ri , θi , a1 , a2 , . . . ak ) . (5.55)

Fig. 5.13a,b Simulated isochromatic fringe patterns corresponding to a dark-field circular polariscope arrangement (a) K I = 1 MPa m1/2 , K II = 0, and σox = 0 (b) K I = 1 MPa m1/2 , K II = 0, and σox = 50 MPa. f σ = 7 kN/m corresponding to polycarbonate has been assumed. The field of view shown in these figures is 40 mm along one side and v/Cd = 0.1

In (5.55), the stress field parameters (K I , K II , σox , . . .) are represented by the vector a = (a1 , a2 , . . . ak ). The stress field parameters must be obtained by minimizing e with respect to the parameters. Near the minimum, the function e can be expanded as a quadratic form e(a1 , a2 , a2 , . . .) ∼ γ − 2

M k=1

βk a k +

M M

αkl ak al ,

k=1 l=1

(5.56)

Fracture Mechanics

5.6 Experimental Methods

143

Fig. 5.14 High-speed sequence showing isochromatic fringe patterns obtained in an experiment with a quasistatically loaded single-edge-notched specimen. Polycarbonate specimen observed in a dark-field circular polariscope arrangement. Frames are 10 μs apart. The height of the field of view presented in the image is about 75 mm (after Taudou et al. [5.53])

where 1 ∂e = 2 ∂ak

i=1

Ni f σ ∂gi − gi h ∂ak

and

M 1 ∂2e ∂gi ∂gi = 2 ∂ak ∂al ∂ak ∂al i=1 2 Ni f σ ∂ gi − − gi h ∂ak ∂al and gi = g(ri , θi , a1 , a2 , a2 . . .). Since Nihf σ − gi is expected to be small, the second term in αkl is usually neglected and only the first derivative of g needs to be evaluated. The estimate for the increment in the parameters is obtained by setting ∇e = 0. This results in the following equation for the increments in the parameters δak :

αkl =

βl =

M

αkl δak .

(5.57)

k=1

With the update of the parameters a = (a1 , a2 , . . . ak ) obtained from the above, the procedure is repeated until the parameters converge. For a thorough discussion of the procedures for such least-squared error fitting see Press et al. [5.56]. Curve fitting routines based on the above are also implemented in Mathematica, Matlab,

and other commercial software packages. There appear to be multiple minima for the function in (5.56) and hence it is usually good practice to simulate the isochromatic fringe patterns with the converged parameters and perform a visual comparison of the recreated fringes to the experimentally observed pattern. Numerous examples of the application of the least-squares fitting method for extraction of the stress intensity factors in quasistatic and dynamic fracture problems can be found in the literature [5.53, 57–61].

5.6.2 Interferometry Classical optical interferometry has been applied by numerous researchers to investigations of fracture problems; two implementations of classical interferometry are commonly used, one to measure the out-of-plane displacement on the free surface of a cracked planar specimen and the other to measure the crack opening displacement in the vicinity of the crack. Complementing this, moiré interferometry has also been used in investigations of the in-plane deformations near a crack tip. The basic principles and interpretation of results are discussed in this section. Measurement of Out-of-Plane Displacements The basic principle of the method is the interference of coherent light beams in a classical two-beam

Part A 5.6

βk = −

M

144

Part A

Solid Mechanics Topics

Object plane

Specimen

Reference mirror

5 mm

Image plane Imaging optics Beam splitter

Beam-forming optics

Laser

Fig. 5.15 Twyman–Green interferometer for the measurement of

out-of-plane displacements (after Schultheisz et al. [5.62, 63])

Part A 5.6

interferometer; the arrangement of a Twyman–Green interferometer, a variant of the Michelson interferometer, shown in Fig. 5.15 was used by Schultheisz et al. [5.62] to accomplish this interference scheme. Light rays reflected from the deformed surface of the specimen and the reference mirror are brought together to interfere; as a result of path differences introduced by the deformation of the surface, fringe patterns are observed in this arrangement as contours of constant out-of-plane displacement component u 3 . This component, corresponding to the singular term of the asymptotic stress field, is given by νK I I Fαα (θ, 1) , u 3 (r, θ) = − √ E 2πr

Fig. 5.16 Interference fringes obtained in a Twyman– Green interferometer corresponding to contours of constant out-of-plane displacements (after Schultheisz et al. [5.62, 63]). 4340 steel specimen

( f -number) limits the acceptance angle of incoming rays; a high f -number is desirable for maximum acceptance angle. Furthermore, as the fringe density becomes high near the crack tip, the resolution of the imaging and reproduction systems limits the visibility of the fringes. The Twyman–Green interferometer has also (mm) 4

(5.58)

where are known functions of θ given in (5.10). Proper alignment of the optical system is essential in order to record these fringe patterns; typically the experimental arrangement is housed on a floating optical table in order to eliminate random vibrations. Also, the specimen surface must be polished to a high degree of flatness and reflectivity; while this may be easily accomplished in inorganic and organic glasses, it is much more difficult to achieve in metallic specimens. An excellent application of this method to the measurement of the out-of-plane deformation of a 4340 steel specimen is demonstrated by Schultheisz et al. [5.62]; an image of the fringe pattern obtained in a Twyman– Green interferometer is shown in Fig. 5.16. Note that, as one approaches the crack tip, the visibility of the fringes becomes poor. This is due to the fact that the slope of the deformed surface increases as the crack tip is approached and the numerical aperture

2

FαI β (θ)

0

–2

–4 –4

–2

0

2

4 (mm)

Fig. 5.17 Out-of-plane displacement field near a rapidly growing crack in a PMMA specimen. Stress waves radiating from the crack tip damage zone can be seen as perturbations in the fringe pattern, especially behind the crack tip. The crack speed was 0.52 mm/μs (about 0.522 Cs ) (after Pfaff et al. [5.64])

Fracture Mechanics

Crack Opening Interferometry The measurement of crack opening displacements with optical interferometry relies on the transparency of the specimen and generation of specularly reflecting fracture surfaces. The basic principle of the method, called crack opening interferometry (COI), is illustrated in Fig. 5.19. When the light rays reflected from the top

145

u3 (μm) 20 0 –20 –40 –60 –80 –100 Expt 35.0 kN FEM 35.0 kN Expt 52.3 kN FEM 52.3 kN Expt 73.5 kN FEM *73.5 kN

–120 –140 –160 –180

0

0.5

1

1.5

2 r/t

Fig. 5.18 Comparison of measured out-of-plane displacement vari-

ation along θ = 0 with a fine-mesh finite element simulation. Good comparison is obtained at the lower load levels, but the discrepancy increases significantly at higher loads (and hence larger plastic zone size) (after Schultheisz et al. [5.63])

and bottom surfaces of the crack are brought together (rays 1 and 2 in the figure), fringes of equal separation between the surfaces can be observed on the image plane. Along the crack surface, the separation distance is the crack opening displacement (COD) denoted as δ(r); the relationship between the fringe number and the Ray 1 Ray 2

y δ (r)

x

Fig. 5.19 Optical arrangement for crack opening interferometry. Rays 1 and 2 reflect from the top and bottom surfaces of the crack and interfere to form fringes of constant COD

Part A 5.6

been used successfully in dynamic fracture problems where the challenges of alignment are significant; fringe patterns captured in a dynamic experiment is shown in Fig. 5.17 [5.64]. Postprocessing was used to subtract out the initial fringe pattern revealing only the surface deformation due to the loading of the crack. Also, the region near the crack tip that is obscured due to aperture limitations has been removed during image processing. Stress waves radiating from the crack tip damage zone can be seen as perturbations in the fringe pattern, especially behind the crack tip. These are caused by the intermittency in the crack propagation process. The displacements calculated from such fringes are pure measurements in the sense that no assumptions about the deformation or stress state are imposed in obtaining the displacements from the measurements. Fitting the observed fringe patterns ahead of the crack to a prediction based on (5.58) using a least-square error method, the stress intensity factor can be determined, in the same way as discussed in Sect. 5.6.1, for the method of photoelasticity. A major difficulty in using this method arises from the fact that, while the measurements of the displacement components are themselves very accurate, the application of the plane asymptotic fields in the interpretation of the crack tip field parameters introduces significant errors, particularly when one approaches the crack tip. Therefore, such experiments may serve as verification on the limitations of models or the accuracy of more precise elastoplastic numerical simulations. Schultheisz et al. [5.63] compared the measured out-of-plane displacement ahead of the crack to a fine-mesh finite element simulation of the same problem (Fig. 5.18) and suggested that good agreement could be obtained with a proper model of the constitutive response of the material as long as the applied loads were low and three-dimensional deformation of the specimen was modeled appropriately. The last aspect is especially important in using classical interferometry since the technique provides the out-ofplane displacements only on the outer surface of the specimen.

5.6 Experimental Methods

146

Part A

Solid Mechanics Topics

COD can be written as nλ δ(r) = u 2 (r, π) − u 2 (r, −π) = , 2 n = 0, 1, 2, . . . ,

(5.59)

where λ is the wavelength of light used and n is the fringe order; it is assumed that the gap between the crack surfaces is filled with air and that the incidence angle of the light beam is normal to the crack surface. Thus, monitoring the fringe patterns results in a direct measurement of the COD; once again, as in the out-ofplane measurements, this is a pure measurement devoid of assumptions on the nature of the deformation and needs to be interpreted in terms of a theoretical model. The measured COD can be compared either with the elastic stress field in (5.10) or the elastic–plastic field corresponding to one of the plasticity models for extraction of the stress intensity factor or the J-integral. For example, if the crack tip deformation is governed by (5.10), then ⎧ ⎨ 8K I r plane stress E 2π . (5.60) δ(r) = 2 ⎩ 8(1−ν )K I r plane strain E 2π

Part A 5.6 Thickness (μm) In unbroken craze stress ≈ 0 for this configuration (note different scales)

2 1 0

Crack

–1 –2

Craze Fracture surface layer thickness ≈ 0.58 μm –80 –70 –60 –50 –40 –30 –20 –10

0

10

20

30

40

Length (μm)

Fig. 5.20 COI fringes for a crack-craze system in PMMA

are shown at the top. The graph indicates the variation of crack and craze opening displacement calculated from the fringe pattern (after Kambour [5.65])

Comparing the measured COD to the calculation based on the elastic field above, the stress intensity factor can be extracted. An example of the COI fringe pattern in a Dugdale–Barenblatt-type craze zone near a crack tip in a polymethyl methacrylate (PMMA) specimen is shown in Fig. 5.20; the displacement profile extracted from this measurement is also shown in the figure. In this example, one can clearly see from the fringe spacing that the COD changes from concave to convex as the crack tip is approached. This corresponds to a physical crack with a craze that develops in the fracture process zone of the PMMA; note the similarity between the COD measured and the one shown in Fig. 5.9. The DB model is more appropriate for this situation; numerous investigators [5.26, 65, 66] have demonstrated that such a model can be applied to crazes. It is also possible to apply nonlinear cohesive zone models to interpret such measurements. One major advantage of COI is that variations in the crack front (three dimensionality) can be rendered visible directly; however, this method comes with big penalties: the optical arrangement is rather complicated since one has to approach the interior of the specimen and more importantly, since only normal opening is measured, effects of mixedmode loading cannot be assessed with this measurement alone. Numerous investigators have used COI to explore problems of brittle fracture [5.67–69], crazing in polymers [5.26, 65, 66], interfacial cracks [5.70], and elastic–plastic problems [5.71]. Moiré Interferometry The method of moiré interferometry is discussed in detail in Chap. 22; here a brief description of the method and its applications to fracture problems is presented. The basic principle relies on diffraction and interference. Consider a diffraction grating adhered to the specimen surface; if two incident beams strike the specimen symmetrically with respect to the normal to the surface such that the first-order diffraction from both beams is normal to the surface, then the two diffracted beams will interfere and provide uniform illumination. If the specimen deforms nonuniformly, then the two diffracted beam will, in general, contain phase differences and hence provide fringe patterns that reflect the deformation of the surface. The fringes are to be interpreted as contours of equal in-plane displacement components with the direction given by the grating used. Details of different systems for implementation of moiré interferometry are discussed by Post et al. [5.72] and are not repeated here. This method is insensitive to out-of-plane displacements near the crack tip

Fracture Mechanics

18 500 N 10 mm

5.6 Experimental Methods

147

in Sect. 5.3.2. It is striking that, while the component of strain perpendicular to the crack line matches the HRR singularity quite well, the strain parallel to the crack exhibits significant deviation. Such detailed strain measurements, obtained without a priori assumptions regarding the material or the deformation, are quite useful in outlining the limits of validity of analytical estimates. Schultheisz et al. [5.62, 63] also performed moiré interferometric experiments on a 4340 steel specimen; however, they compared the measured displacements to a fully three-dimensional finite element simulation (Fig. 5.23) and demonstrated excellent agreement with the experiments, but only for low loading levels. At higher load levels, they found that there was a significant discrepancy; this was attributed to tunneling of the crack front in the interior of the specimen and the resulting three dimensionality of the displacement field not captured in the numerical simulations.

5.6.3 Lateral Shearing Interferometry

Fig. 5.21 Moiré interferometric fringes corresponding to

displacements u 1 (upper image) and u 2 (lower image) (after Schultheisz et al. [5.62])

and hence provides a nice complement to the out-ofplane measurements through classical interferometry. Applications to quasistatic as well as dynamic loading problems in elastic, elastic–plastic, and damaging materials have been considered [5.62, 63, 72–74]. In the case of elastic problems, the interest in application of moiré interferometry is in identifying the crack tip displacement field, outlining the regions of dominance of the elastic singularity, and eventually in the extraction of the stress intensity factor. The moiré interference fringes from a fracture experiment indicating the u 1 and u 2 components of displacement are shown in Fig. 5.21. Dadkhah and Kobayashi [5.73] examined the displacement fields near a crack in a biaxially loaded Al 2024-T3 alloy specimen. Their result for a biaxiality of two is reproduced in Fig. 5.22; here the strain components evaluated from the moiré interferometric fringes are compared with an estimate based on the HRR singularity discussed

where Δs is the absolute phase angle of the components, which depends on the state of stress of the specimen at the point (x1 , x2 ); we assume that the specimen is optically isotropic. Shearing of images is

Part A 5.6

18 500 N 10 mm

The principle of shearing interferometry is very similar to other two-beam interferometric techniques; the main difference is that, rather than using a reference wavefront to interfere with the wavefront of interest, the wavefront is made to interfere with a copy of itself after introducing a predetermined, selectable, (lateral) shear displacement between the two components. Tippur et al. [5.75] developed a variant based on a diffraction grating that enabled them to examine the surface deformations near stationary and dynamically propagating cracks; they termed their implementation the coherent gradient sensor (CGS). Lee and Krishnaswamy [5.76] later introduced a simpler implementation of the experiment based on a calcite crystal; a variant of this scheme is described here. The analysis of the light propagation through Jones calculus makes the manipulations simple [5.77, 78]; see the chapter on photoelasticity for details of Jones’s calculus. The optical arrangement is shown in Fig. 5.24; a light beam polarized at an angle of π/4 with respect to the global x1 -axis passes through the stressed specimen that acts as a phase retarder. The electric vector of the light emerging from the specimen is given by −iΔs(x 1 ,x 2 ) iωt e , (5.61) E = ke e−iΔs(x1 ,x2 )

148

Part A

Solid Mechanics Topics

Strain εx 0.01

Strain εy 0.02 θ = 0° B=2

0.008 Moire

0.006

Moire HRR – 0.93

0.01

θ = 0° B=2

– 0.5

0.004 HRR 0.002

0

0

10 20 Distance from crack tip r (mm)

0

– 0.923 0

10 20 Distance from crack tip r (mm)

Fig. 5.22 Comparison of the strain determined from moiré interferometry with calculations based on the HRR field; note that, while the normal strain ε22 compares favorably, the crack parallel strain ε11 shows significant deviation (after Dadkhah and Kobayashi [5.73])

Part A 5.6

accomplished by introducing a uniaxial optical crystal (for example, calcite) into the path of the light beam. Uniaxial crystals produce two refracted beams – the ordinary ray with a refractive index n and the extraordinary ray with a refractive index n – for each incident beam, each polarized in mutually orthogonal planes (see Born and Wolf [5.79] for a discussion of uniaxial crystals). Let the uniaxial crystal be aligned such that the ordinary ray is polarized along x1 , the extraordinary ray is polarized along the x2 -axis, and that these rays are sheared by an amount Δx1 along the x1 -axis. At every point in the field, there are now two beams: the ordinary ray that entered the specimen at the point (x1 , x2 ) and the extraordinary ray that entered the specimen at the point (x1 + Δx1 , x2 ). Thus the light beam leaving the crystal may be written as e−iΔs(x1 ,x2 ) iωt . (5.62) E = ke e−iΔs(x1 +Δx1 ,x2 ) These two components are brought together by a polarizer oriented at an angle of π/4 with respect to the global x1 -axis. The output electric vector is given by 1 −iΔs(x 1 ,x 2 ) + e−iΔs(x1 +Δx1 ,x2 ) iωt 2 e E = ke . 1 −iΔs(x 1 ,x 2 ) + e−iΔs(x1 +Δx1 ,x2 ) 2 e (5.63)

The intensity of this light beam is the time average of the electric vector E2 , averaged over a time significantly longer than the period; if the beam displacement Δx1 ,

is small, the intensity may be written as k2 Δx1 ∂Δs . cos2 I (x1 , x2 ) = 2 2 ∂x1

(5.64)

Thus, the light intensity variation observed in this optical arrangement depends on the gradients of the phase difference in the x1 -direction. In applying this method to problems in mechanics, it remains to evaluate the angular phase difference Δs; for an optically opaque specimen, if the surfaces are finished to be specularly reflecting, the phase difference is given by hν Δs = − (σ11 + σ22 ) ≡ hcr (σ11 + σ22 ) , (5.65) 2π/λ E where cr = −hν/E represents the sensitivity of the method. Therefore, bright fringes corresponding to maximum light intensity are lines in the x1 –x2 plane along which mλ ∂(σ11 + σ22 ) = , ∂x1 Δx1 with m = 0, ±1, ±2, . . . ,

hcr

(5.66)

where m is the fringe order. Therefore, fringes observed in the shearing interferometer or the coherent gradient sensor are lines of constant gradient ∂(σ11 + σ22 )/∂x1 . Obviously, by reorienting the ordinary axis of the calcite fringes representing lines crystal with the x2 -direction, of constant ∂ σ11 + σ22 /∂x2 can be obtained. A similar analysis can be performed for transparent materials, with only a slight modification in the constant cr .

Fracture Mechanics

Expt 35.0 kN FEM 35.0 kN Expt 52.3 kN FEM 52.3 kN Expt 73.5 kN FEM *73.5 kN

0

0.5

1

1.5

2

2.5

3

r/t

u2 (μm) 400 Expt 35.0 kN FEM 35.0 kN Expt 52.3 kN FEM 52.3 kN Expt 73.5 kN FEM *73.5 kN

350 300 250 200 150

50 0

0.25

0.5

0.75

1

1.25

1.5

r/t

Fig. 5.23 Comparison of the u 1 and u 2 displacement components determined from moiré interferometry with a fine-mesh finite element analysis. While good agreement is observed at lower load levels, a large departure is observed at the highest load (after Schultheisz et al. [5.62])

The above description of the shearing interferometer is quite general and can be applied to many problems in mechanics. Also,from (5.64), it is clear that the gradient of u 3 x1 , x2 , − h2 may be determined directly, without relating it to the plane-stress calculation of the stress component; therefore, the method can also be used to determine surface profiles of objects. Explicit equations for the light intensity can be obtained through the introduction of the asymptotic crack tip stress field in (5.9). The singular term, the higher-order terms in the steadystate asymptotic expansion, and the dynamic asymptotic field have all been used in interpreting the fringe pattern observed near stationary and dynamically growing cracks. We will describe only the quasistatic applica-

The entire analysis presented above carries over if the image shearing is introduced in the x2 -direction; in this case, the x2 gradient of the field is obtained. Simulated interference fringe patterns corresponding to the x1 gradient for assumed values of the stress intensity factors are shown in Fig. 5.25 for pure mode I and mixed-mode loading. An example of the shearing interference fringe patterns obtained in a shearing interferometer is shown in Fig. 5.26. In comparing the images in these figures, it should be noted that only the singular term in the crack tip asymptotic field has been used in the simulations, whereas the complete field will influence the patterns observed in the experiments. In the discussion above, only the singular term was introduced in the evaluation of the fringe patterns. However, since the asymptotic field is not expected to establish dominance at large distances from the crack tip where the fringe patterns are typically analyzed, higher-order nonsingular terms in the expansion in (5.9) must be introduced as we discussed in the case of photoelasticity. The stress intensity factor, K I , can be obtained at each instant in time by using a leastsquares matching of the experimentally measured fringe pattern with simulations based on (5.67). First, the experimental fringe pattern is quantified by a collection of (m i , ri , θi ), measured at M points. The distance r at which these measurements are taken should be appropriate for the application of the two-dimensional asymptotic crack tip stress field; based on the experiments of Rosakis and Ravi-Chandar [5.54], it should be recognized that the distance r must be larger than 0.5h to be away from the zone of three-dimensional deformations. Then, the sum of the squared error in (5.67) at all measured points is then given by e(A0 , A1 , . . . Ak−1 ) 2 M m i πλ − hcr g(ri , θi , A0 , A1 , . . . Ak−1 ) . = Δx2 i=1

(5.68)

In (5.68), the stress field parameters (K I , K II , . . .) are represented by the vector A = (A0 , A1 , . . . Ak−1 ) and g(r, θ, A0 , A1 , . . . Ak−1 ) is used to represent ∂(σ11 + σ22 )/∂x1 determined from the k-term description of the asymptotic crack tip stress field. Also, since

Part A 5.6

100

0

149

tion. Introducing the singular term from (5.9) into (5.66) results in the equation for bright fringes 3θ 3θ mλ hcr . (5.67) K I cos + K II sin = √ 3/2 2 2 Δx 2πr 1

u1 (μm) 65 60 55 50 45 40 35 30 25 20 15 10 5 0 –5

5.6 Experimental Methods

150

Part A

Solid Mechanics Topics

Axis of polarization

*Light source

π/4 σ1 σ2 α

Polarizer

Specimen

Fast axis Image shearing crystal

π/4

x2 Polarizer

Part A 5.6

x3

x1

Fig. 5.24 Optical arrangement for lateral shearing interferometry

shearing interferometry or the coherent gradient sensing method depends on the gradient of the stress field, the constant nonsingular term σox in the asymptotic expansion does not contribute to fringe formation and cannot be determined in the analysis. The remaining a)

b)

stress field parameters must be obtained by minimizing e with respect to the parameters. The least-squared error method described in connection with photoelastic data analysis in Sect. 5.6.1 can be used here as well to a)

b)

10 mm

Fig. 5.25a,b Simulated shearing interferometric fringe patterns corresponding to the optical arrangement in Fig. 5.24 (a) K I =1 MPa m1/2 , K II =0 (b) K I =1 MPa m1/2 , K II =0.5 The field of view shown in these figures is 40 mm along one side and v/Cd = 0.1. The lack of clarity in the region near the crack tip is a numerical artifact

Fig. 5.26a,b Fringe patterns from the coherent gradient sensing (CGS) arrangement of the lateral shearing interferometer: (a) image shearing parallel to the crack line, (b) image shearing perpendicular to the crack line; the dashed lines are reconstructions based on a threeparameter crack tip field

Fracture Mechanics

evaluate the best fit coefficients A = (A0 , A1 , . . . Ak−1 ). The entire procedure carries over to the dynamic problem if the appropriate asymptotic field in (5.37) is used in (5.66). Tippur et al. [5.80] examined the applicability of the asymptotic stress field by comparing the experimentally observed fringe patterns with the analytically estimated patterns; dashed lines in Fig. 5.26 correspond to theoretical estimates based on a threeparameter fit. They concluded that measurements need to be made at a distance of about one-half of the plate thickness, reinforcing earlier results of Rosakis and Ravi-Chandar [5.54]. A complete discussion of the application of this method to quasistatic and dynamic problems is provided by Rosakis [5.81].

5.6.4 Strain Gages

x2

x'2

x'1 α

θ x1

Fig. 5.27 Location and orientation of strain gage relative to the crack tip

151

type, dimensions, and sensitivity of the strain gage are extremely important and are addressed in the chapters that deal with strain gages and are not addressed here. Dally and Berger [5.84] introduced a very simple idea for the use of strain gages in the evaluation of stress intensity factors. Consider a strain gage mounted at a point (r, θ), with the strain gage itself oriented at an angle α with respect to the global x1 -axis (Fig. 5.27); the strain gage will measure the extensional strain ε11 in the direction x1 . For the case of a stationary crack√under mode I loading, retaining up to terms of order r the strain can be evaluated to be E ε (r, θ) (1 + ν) 11 θ 1 K I (t) 3θ k cos − sin θ sin =√ cos 2α 2 2 2 2πr 3θ 1 sin 2α + A1 (k + cos 2α) + sin θ cos 2 2 √ θ + A2 r cos 12 θ k + sin2 cos 2α 2 1 (5.69) − sin θ sin 2α + . . . , 2 where k = (1 − ν)/(1 + ν). Through a proper choice of θ and α the second and third terms in (5.69) can be made to vanish; this is assured by the conditions cos 2α = −k

and

tan 12 θ = − cot 2α .

(5.70)

For a material with ν = 1/3, we get k = 1/3 and α = θ = π/3. Introducing these values in (5.69), we obtain π 3 Eε11 r, θ = (5.71) = KI . 3 8πr Thus, by placing a strain gage aligned along a line oriented at an angle θ = π/3, the measured strain can be directly related to the stress intensity factor. It should be noted that the distance r at which the strain gage is placed is still open, but this can be farther from the crack tip than the K -dominant region, since (5.71) is based on a three-term representation of the strain field. This arrangement can be used in the evaluation of dynamic initiation toughness as well. Examples of application of this method are demonstrated by Dally and Barker [5.85], who evaluated the dynamic initiation toughness of a Homalite 100 specimen at high loading rates by imposing an explosively driven stress wave on a crack. Owen et al. [5.86] determined both the dynamic initiation toughness and dynamic propa-

Part A 5.6

While optical methods yield measurements of the stress field components over the complete field of observation, they also require elaborate instrumentation. Full-field optical methods dominated fracture research in the early years of the discipline, primarily because issues related to the dominance of the various asymptotic fields could be evaluated with these schemes. On the other hand, multiple strain gages can be used more readily with simpler instrumentation requirements; this method was used very successfully by Kinra and Bowers [5.82], Shukla et al. [5.83], and many other investigators for dynamic problems. Recent progress in this area was summarized by Dally and Berger [5.84], who describe the application of strain gages to quasistatic as well as dynamic fracture problems. Here we provide a brief description as applied to dynamic problems. In general, strain gages may be placed at different positions (r, θ) to measure one or more components of the strain tensor. These measurements can be used to determine the stress intensity factors K I and K II as well as the higher-order terms by fitting the experiment in a leastsquare sense to the asymptotic fields of quasistatic or dynamic problems. Practical considerations on the

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Part A 5.6

gation toughness for an aluminum 2024-T3 alloy; they found the initiation toughness to be independent of the dyn rate of loading up to K˙ I = 105 MPa m1/2 s−1 , but then to increase threefold as the rate of loading indyn creased to K˙ I = 106 Pa m1/2 s−1 . Owen et al. [5.86] also determined the dynamic propagation toughness, but using (5.71), justifying its use by the fact that the crack speeds were quite low, about 4% of the shear wave speed. For dynamically running cracks, two major errors are encountered in using (5.71); the first is due to the inertial distortion of the crack tip strain field, which is ignored in developing (5.71). While this error is likely to be small at low crack speeds, it cannot be ignored when the crack speed is high; this error can be removed completely by using the appropriate asymptotic strain field. The second, and perhaps more important, contribution to the error arises from the fact that the distance r and the orientation θ of the position of the strain gage relative to the moving crack tip are continuously changing as a result of crack growth. This must be taken into account in the analysis. Therefore, the simplification introduced in (5.71) is not appropriate and one must incorporate the additional feature that both r and θ are now functions of time, given by (5.72) r(t) = r02 + v2 t 2 − 2r0 vt cos θ0 , r0 sin θ0 θ(t) = arcsin (5.73) , r(t) a)

Specimen

Real screen

where r0 and θ0 are the distance and orientation at time t = 0. Of course, it has been assumed that the crack extension is straight along the x1 -direction. RaviChandar [5.87] used the above analysis to evaluate the measurements of Kinra and Bowers [5.82]. More recently, Berger et al. [5.88] have used measurements from multiple strain gages to set up an overdetermined system of equations and obtain improved estimates of the dynamic propagating toughness. In summary, strain-gage-based methods are just as powerful as full-field optical techniques for crack tip field characterization while at the same time requiring only minimal investment in measuring equipment. This is particularly useful in the development of standard methods of measuring the initiation, propagation, and arrest toughness.

5.6.5 Method of Caustics The method of caustics discovered by Schardin [5.38] was popular in early dynamic fracture investigations due to its simplicity, although significant inherent limitations in the method have made it a little used technique. It provides a direct measurement of the stress intensity factor and is applicable to static and dynamic problems. Here we describe the essential ingredients of the method for completeness. The principle of formation is illustrated in Fig. 5.28 for transparent and opaque specimens. Consider a parallel beam of light incident along the x3 -direction, normal b)

x2

x2 r0 (z0)

r0 (z0) x3

D

Crack front

S1

Virtual screen

Specimen

x3

D

Crack front

S2 z0

z0

Fig. 5.28a,b Schematic illustration of the principle of formation of the caustic curve: (a) transmission arrangement for transparent specimens, (b) reflection arrangement for opaque specimens

Fracture Mechanics

a)

153

b)

D

Dmin Dmax

Fig. 5.29a,b Simulated bitmap image of caustics. The field of view represents a square, 20 mm along one side. (a) K I = 1 MPa m1/2 , K II = 0; (b) K I = 1 MPa m1/2 , K II = 1 MPa m1/2

determined. For mixed-mode loading conditions, the variation of the stress intensity factor with time may be obtained by measuring two length dimensions: Dmax and Dmin . The ratio (Dmax − Dmin )/Dmax depends only on μ; this dependence is shown in Fig. 5.30a. Therefore, μ can be determined first from the measurements of Dmax and Dmin and then K I and K II are determined from the following: √ 3 2 2π Dmax 2 , (5.75) KI = 2ct hz 0 F(v) g(μ) K II = μK I .

(5.76)

The function g(μ) is shown in Fig. 5.30b. While the method has been used by a number of investigators in evaluating mixed-mode stress intensity factors in quasistatic problems, dynamically growing cracks typically follow a locally mode I path and hence there are very few examples of the evaluation of the mixed-mode stress intensity factors. Examples of the application of the method of caustics to dynamic problems may be found in the papers by Kalthoff [5.89], Ravi-Chandar and Knauss [5.41, 90, 91], and Rosakis et al. [5.43].

5.6.6 Measurement of Crack Opening Displacement While the COI discussed before provides an accurate measurement of the crack opening displacement, it has two major limitations: first, it requires a transparent specimen with a complicated optical arrangement, and second, significant postprocessing of fringe data is needed in order to extract the crack opening. In many practical applications, specimens are opaque and hence one needs other means of measuring the COD. In industrial practice, methods based on replica techniques

Part A 5.6

to the specimen free surface. In a transparent material a light ray passing through a stressed plate is deviated from its path partly due to thickness variation generated by the deformation and partly due to the change in refractive index caused by stress-induced birefringence. In an opaque material, the light ray reflects with a deviation from parallelism dictated by the local slope or equivalently the thickness change. If the plate contains a crack, the rays are deviated from the region around the crack tip and these form a singular curve called a caustic on a reference plane at some distance away from the specimen. The size of the caustic curve can be related to the stress intensity factor by introducing an analysis based on geometrical optics and fracture mechanics. This analysis is described in detail in many references [5.81]; we provide just the final results. As can be seen from the illustration in Fig. 5.28, far away from the crack tip, the light rays pass through the transparent specimen and maintain their parallel propagation; the influence of the stress field on the wavefront is small enough to be neglected. On the other hand, in the region near the crack tip, where the specimen exhibits a concave surface due to the Poisson contraction, the light rays deviate significantly from parallelism. As a result, a dark region called the shadowspot forms on the screen at z 0 , where there are no light rays at all. This shadow region is surrounded by a bright curve, called the caustic curve. The line on the specimen plane whose image is the caustic curve on the specimen is called the initial curve. Light rays from outside the initial curve fall outside the caustic; rays from inside the initial curve fall on or outside the caustic curve and rays from the initial curve fall on the caustic curve. Hence the caustic curve is a bright curve that surrounds the dark region. Figure 5.29 shows simulated caustics corresponding to mode I and mixed-mode loading. For mode I loading, the transverse diameter is related to the stress intensity factor through the following relation: √ 3 2 2π D 2 , (5.74) KI = 2cp hz 0 F(v) 3.17 where h is the plate thickness, z 0 is the distance between the midplane and the screen, and specimen F(v) = 2 1 + αs2 αd2 − αs2 /R(v) is a correction factor for a crack propagating dynamically at a speed v, cp = −ν/E for opaque specimens and cp = c − (n − 1)ν/E for transparent specimens, where c is the direct stress-optic coefficient of the material. Thus, from measurements of the transverse diameter of the caustic curve, the dynamic stress intensity factor can be

5.6 Experimental Methods

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a) (Dmax -Dmin )/Dmax

b) Numerical factor g

0.5

4

0.4

3

0.3 2 0.2 1

0.1 0

0

1

2

3

4 5 6 7 8 ∞ Stress intensity factor ratio µ = KII /KI

0

0

1

2 3 4 5 Stress intensity factor ratio µ = KII /KI

Fig. 5.30 (a) Relationship between the caustic dimensions and the mixed-mode μ (b) g(u) versus μ (after Kalthoff [5.52])

Part A 5.6

or clip gages are commonly used. Replica techniques are quite clumsy; they require that a soft polymer be inserted into the crack under load, cured in-place, and then extracted by breaking the specimen. This replica is then characterized to determine the crack opening displacement. A crucial factor in determining the reliability of this method is the uncertainty associated with the flow of the viscous polymer into the crack opening before curing begins; while the replica technique remains a powerful tool in determining metallographic aspects of fracture, is not a very good quantitative tool for precise COD measurements. Clip gages are more reliably used in such applications; these rely on two flexible beam elements, inserted into the crack along the load line, with strain gages attached on them. Opening of the crack results in variations in the bending strains that are then calibrated in terms of the opening of the crack. Many commercial devices suitable a)

b)

A

c)

α d

Fig. 5.31 (a) Vickers indents on the specimen surface 100 μm apart. (b) Optical arrangement for the interferometric strain/displacement gage. (c) Pattern of light observed, indicating the triangular outline

and the interference fringes

for different environmental conditions (high temperature, etc.) are available and therefore clip gage methods are not reviewed here. An optical technique developed by Sharpe [5.92, 93] called the interferometric strain/displacement gage (ISDG) is quite versatile and can be applied in static and dynamic applications and in different environments. We describe the basic principles and capabilities of this technique. The ISDG takes advantage of two-beam interference just as other classical interferometric techniques discussed in Sect. 5.6.2, with the exception that, instead of attempting full-field measurement, reflections from two points on the specimen are considered. In the simplest implementation of the method, two small microindentations are made on the specimen with a pyramidal Vickers indentor; typically the indents are about 10–20 μm long and are spaced about 100 μm apart (Fig. 5.31a), and the reflections from these two points are made to interfere. The two indents could be placed across the crack line if COD is to be measured at that location; they can also be placed anywhere where the local strain is to be measured. If a light beam impinges on these two indents at normal incidence to the nominal surface, the reflected beams from each face of the pyramidal indent will overlap and generate interference fringes in space; the beam diffraction from the indents ensures that overlap of the two beams occurs. This is a variant of the Fresnel mirror arrangement and can be viewed as equivalent to Young’s two-slit interference scheme [5.79]. The scheme is illustrated in Fig. 5.31b; to an observer at point A, the light beams reflected from the two pyramidal faces of the indent ap-

Fracture Mechanics

pear as two slits separated by a distance d sin α. The fringe formation condition for the Young’s two-slit interference scheme is sd sin α = m Lλ ,

m = 0, ±1, ±2, . . . ,

(5.77)

5.6.7 Measurement of Crack Position and Speed Measurements of crack tip position and speed are often required in order to analyze the fracture problem. Different electrical resistance methods have been used to determine the crack position and speed. The two most popular techniques are the resistive grid and potential drop methods. In the grid method a number of electrical wires are laid across the path of the crack. As the crack propagates, it severs the wires sequentially and provides an electrical signal which can then be used to determine the crack position and speed with time (see, for example, Dulaney and Brace [5.94] and Paxson and Lucas [5.95]). Commercial suppliers of strain gages now provide such grids for crack speed measurements; these grids can be incorporated into standard strain gage bridge circuits to provide the history of wire breakage and hence the crack position as a function of time. While very good estimates of the crack speed can be obtained from such grid techniques, the discrete nature

of the grids dictate that the sampling rate of the crack speed will typically be much lower than that obtained using other methods. In contrast to this discrete measurement, if the change in resistance of a conducting specimen is measured as a function of crack length, then the crack length can be inferred at very high spatial and temporal resolution; this is the basis of the potential drop technique. The method can be used even in nonconducting materials, provided that a thin conducting film is deposited on the surface of the specimen. Carlsson et al. [5.96] demonstrated the application of this method to the measurement of crack speed in PMMA; they used a voltage divider circuit to measure the resistance change. Many other investigators have used this method to determine the speed of running cracks [5.97, 98]. Commercial versions of this technique – such as the KrakGageTM – are now available; it is also quite easily accomplished in the laboratory with thin-film coating methods. ASTM guidelines have also been established for measurement of crack position during fatigue crack growth characterization (ASTM E 647). A review of the method can be found in Wilkowski and Maxey [5.99]. A simple arrangement of the method is shown in Fig. 5.32. The specimen or the conducting coating on the specimen is connected to one arm of a Wheatstone bridge as indicated and balanced initially to give a null output; as the crack length increases, the change in resistance of this segment of the bridge circuit results in an output voltage from the bridge. The relationship between this voltage output and the crack length can be obtained through direct calibration or by calculating the electric field numerically for the particular specimen geometry (see Bonamy and Ravi-Chandar [5.100] for a recent application). Analytical and numerical solutions are available for a number of specimen geometries (see the ASTM E 647 standard for references). Both direct- (DC) and alternating-current (AC) excitation and V0 R1

Ra

Vbat Rb

Fig. 5.32 Potential drop technique for measurement of crack tip position

155

Part A 5.6

where s is the distance between fringes, m is the fringe order, λ is the wavelength of light, and L is the distance between the plane of the indents and the plane of observation. The pyramidal shape of the surface of the indent dictates that the light beam emerging from the surface is triangular as well (Fig. 5.31c). Diffraction effects dictate that the reflected beams will acquire a small divergence angle. It is clear from (5.77) that the distance between the fringes will increase as one moves away from the specimen plane (as L increases). However, in the implementation of ISDG, L is fixed and changes in the spacing between the indents Δd cause changes in the fringe spacing. If these changes are monitored with a photosensor, the movement of the fringes over the sensor yields sinusoidal variations in the intensity at a fixed point. High-resolution data can be captured by measuring the intensity variations with a photosensor array and interpreted in terms of fractional fringe orders. Displacement resolutions on the order of 5 nm can be achieved with this kind of displacement gage. The method is suitable for static and dynamic applications, as well as high-temperature applications. Details of the implementation of the method are described in Sharpe [5.92, 93].

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constant-voltage or constant-current excitation can be used; however, for high-speed cracks, it is preferable to use constant-voltage DC excitation. In fatigue applications, this method is used to determine crack position

to within a resolution of a few micrometers. In dynamic applications, where the crack speed may be in the range of 500–1500 m/s, the speed can be determined to within a few tens of m/s.

References 5.1

5.2

5.3 5.4 5.5 5.6

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5.15 5.16

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5.72 5.73

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5.84

5.85

determine crack profiles, Exp. Mech. 22, 383–391 (1982) K.M. Liechti, Y.-S. Chai: Biaxial loading experiments for determining interfacial fracture toughness, J. Appl. Mech. 58, 680–688 (1991) D. Post, B. Han, P. Ifju: High Sensitivity Moiré (Springer, Berlin, Heidelberg 1994) M.S. Dadkhah, A.S. Kobayashi: HRR field of a moving crack: An experimental analysis, Eng. Fract. Mech. 34, 253–262 (1989) J.S. Epstein, M.S. Dadkhah: Moiré interferometry in fracture research. In: Experimental Techniques in Fracture,, ed. by J.S. Epstein (Wiley VCH, Weinheim 1993) pp. 427–508 H.V. Tippur, S. Krishnaswamy, A.J. Rosakis: A coherent gradient sensor for crack tip deformation measurements: Analysis and experimental results, Int. J. Fract. 48, 193–204 (1990) H. Lee, S. Krishnaswamy: A compact polariscope/shearing interferometer for mapping stress fields in bimaterial systems, Exp. Mech. 36, 404–411 (1996) R.C. Jones: A new calculus for the treatment of optical systems Part I, J. Opt. Soc. Am. 31, 488–493 (1941) R.C. Jones: A new calculus for the treatment of optical systems, Part II, J. Opt. Soc. Am. 31, 493–499 (1941) M. Born, E. Wolf: Principles of Optics, 7th Ed (Cambridge Univ. Press, Cambridge 1999) H.V. Tippur, S. Krishnaswamy, A.J. Rosakis: Optical mapping of crack tip deformations using the method of transmission and reflection coherent gradient sensing: a study of the crack tip K-dominance, Int. J. Fract. 52, 91–117 (1991) A.J. Rosakis: Two optical techniques sensitive to the gradients of optical path difference: The method of caustics and the coherent gradient sensor. In: Experimental Techniques in Fracture, Vol. III, ed. by J.S. Epstein (VCH, Weinheim 1993) pp. 327–425 V.K. Kinra, C.L. Bowers: Brittle fracture of plates in tension. Stress field near the crack, Int. J. Solids Struct. 17, 175 (1981) A. Shukla, R.K. Agarwal, H. Nigam: Dynamic fracture studies on 7075-T6 aluminum and 4340 steel using strain gages and photoelastic coatings, Eng. Fract. Mech. 31, 501–515 (1989) J.W. Dally, J.R. Berger: The role of the electrical resistance strain gage in fracture research. In: Experimental Techniques in Fracture, ed. by J.S. Epstein (Wiley VCH, Weinheim 1993) pp. 1–39 J.W. Dally, D.B. Barker: Dynamic measurements of initiation toughness at high loading rates, Exp. Mech. 28, 298–303 (1988)

5.86

D.M. Owen, S. Zhuang, A.J. Rosakis, G. Ravichandran: Experimental determination of dynamic initiation and propagation fracture toughness in thin aluminum sheets, Int. J. Fract. 90, 153–174 (1998) 5.87 K. Ravi-Chandar: A note on the dynamic stress field near a propagating crack, Int. J. Solids Struct. 19, 839–841 (1983) 5.88 J.R. Berger, J.W. Dally, R.J. Sanford: Determining the dynamic stress intensity factor with strain gages using a crack tip locating algorithm, Eng. Fract. Mech. 36, 145–156 (1990) 5.89 J.F. Kalthoff: On the measurement of dynamic fracture toughness – a review of recent work, Int. J. Fract. 27, 277–298 (1985) 5.90 K. Ravi-Chandar, W.G. Knauss: An experimental investigation into dynamic fracture - I. Crack initiation and crack arrest, Int. J. Fract. 25, 247–262 (1984) 5.91 K. Ravi-Chandar, W.G. Knauss: An experimental investigation into dynamic fracture – III. On steady state crack propagation and branching, Int. J. Fract. 26, 141–154 (1984) 5.92 W.N. Sharpe Jr.: An interferometric strain/displacement measuring system, NASA Tech. Memo. 101638 (1989) 5.93 W.N. Sharpe Jr.: Crack-tip opening displacement measurement techniques. In: Experimental Techniques in Fracture, Vol. III, ed. by J.S. Epstein (VCH, Weinheim 1993) pp. 219–252 5.94 E.N. Dulaney, W.F. Brace: Velocity behavior of a growing crack, J. Appl. Phys. 31, 2233–2236 (1960) 5.95 T.L. Paxson, R.A. Lucas: An investigation of the velocity characteristics of a fixed boundary fracture model. In: Dynamic Crack Propagation, ed. by G.C. Sih (Noordhoff, Leiden 1973) pp. 415–426 5.96 J. Carlsson, L. Dahlberg, F. Nilsson: Experimental studies of the unstable phase of crack propagation in metals and polymers. In: Dynamic Crack Propagation, ed. by G.C. Sih (Noordhoff International, Leyden 1973) pp. 165–181 5.97 B. Stalder, P. Beguelin, H.H. Kausch: A simple velocity gauge for measuring crack growth, Int. J. Fract. 22, R47–R54 (1983) 5.98 J. Fineberg, S.P. Gross, M. Marder, H.L. Swinney: Instability in dynamic fracture, Phys. Rev. Lett. 67, 457–460 (1991) 5.99 G.M. Wilkowski, W.A. Maxey: Review and applications of the electric potential method for measuring crack growth in specimens, flawed pipes and pressure vessels, ASTM STP 791, II266–II294 (1983) 5.100 D. Bonamy, K. Ravi-Chandar: Dynamic crack response to a localized shear pulse perturbation in brittle amorphous materials: On crack surface roughening, Int. J. Fract. 134, 1–22 (2005)

159

Active Materi 6. Active Materials

Guruswami Ravichandran

This chapter provides a brief overview of the mechanics of active materials, particularly those which respond to electro/magnetic/mechanical loading. The relative competition between mechanical and electro/magnetic loading, leading to interesting actuation mechanisms, has been highlighted. Key references provided within this chapter should be referred to for further details on the theoretical development and their application to experiments.

6.1

Background ......................................... 159 6.1.1 Mechanisms of Active Materials........................ 160

6.1.2 Mechanics in the Analysis, Design, and Testing of Active Devices ......... 160 6.2

Piezoelectrics ....................................... 161

6.3 Ferroelectrics ....................................... 6.3.1 Electrostriction............................. 6.3.2 Theory ........................................ 6.3.3 Domain Patterns .......................... 6.3.4 Ceramics .....................................

162 162 163 163 165

6.4 Ferromagnets....................................... 166 6.4.1 Theory ........................................ 166 6.4.2 Magnetostriction .......................... 167 References .................................................. 167

6.1 Background shown in Fig. 6.1 [6.1]. Active materials are widely used as sensors and actuators, including vibration damping, Work per volume (J/m3) 108

Experimental Theoretical

Shape memory alloy

7

10

Solid–liquid

106

Fatigued SMA

Thermopneumatic

Ferroelectric EM ES

Thermal expansion

105

PZT

Electromagnetic (EM)

104 Muscle Electrostatic (ES) EM

103 102 0 10

ES Microbubble

101

102

103

ZnO

104 105 106 107 Cycling frequency (Hz)

Fig. 6.1 Characteristics of common actuator systems (after

Krulevitch et al. [6.1])

Part A 6

Active materials in the context of mechanical applications are those that respond by changing shape to external stimuli such as electro/magnetic-mechanical loading, which is in general reversible in nature. The change in shape results in mechanical sensing and actuation that can be exploited in a variety of ways for practical applications. The mechanics of actuation depends on a wide range of mechanisms, which generally depend on some form of phase transition or motion of phase boundaries under external stimuli. There are also active materials which respond to thermomechanical loading such as shape-memory alloys, which are not discussed here. This chapter is confined to materials which respond to electro/magnetic-mechanical loading such as piezoelectric, ferroelectric, and ferromagnetic solids. A common feature of these materials is that cyclic actuation takes place, often accompanied by hysteresis. The common figures of merit used to characterize actuator performance are the work/unit volume and the cycling frequency. The characteristics of common actuator materials/systems in this parameter space are

160

Part A

Solid Mechanics Topics

micro/nanopositioning, ultrasonics, sonar, fuel injection, robotics, adaptive optics, active deformable structures, and micro-electromechanical systems (MEMS) devices such as micropumps and surgical tools.

a)

ε E

E

6.1.1 Mechanisms of Active Materials The mechanics of actuation for most solid-state active materials depend on one or a combination of the following effects, illustrated in Fig. 6.2a–c.

•

•

•

Piezoelectric effect (Fig. 6.2a) is a linear phenomenon in which the mechanical displacement (strain ε) is proportional to the applied field or voltage (E) and the sign of the displacement depends on the sign of the applied field. Electrostriction (Fig. 6.2b) is generally observed in dielectrics and most prominent in single-crystal ferroelectrics and ferroelectric polymers, where the displacement (strain ε) or actuation is a function of the square of the applied voltage (E) and hence the displacement is independent of the sign of the applied voltage. Magnetostriction (Fig. 6.2c) is similar to electrostriction except that the applied field is magnetic in nature, which is most common in ferromagnetic solids. The displacement (strain ε) in single-crystal magnetostrictive material is proportional to the square of the magnetic field (H).

b)

ε ±E E

c)

ε ±H H

Fig. 6.2 (a) Piezoelectricity described by the converse piezoelectric effect is a linear relationship between strain (ε) and applied electric field (E). (b) Electrostriction is a quadratic relationship between strain (ε) and electric field (E), or more generally, an electric-field-induced deformation that is independent of field polarity. (c) Magnetostriction is a quadratic relationship between strain (ε) and the applied magnetic field (H), or more generally, a magnetic-field-induced deformation that is independent of field polarity

Part A 6.1

trasonics, linear and rotary micropositioning devices, and sonar. Potential applications include microrobotics, active surgical tools, adaptive optics, and miniaturized actuators. The problems associated with active materials are multi-physics in nature and involve solving coupled boundary value problems. 6.1.2 Mechanics in the Analysis, Design, The formulation of boundary value problems in and Testing of Active Devices solid mechanics and the solution techniques have been discussed in Chap. 1 and will not be revisited here The quest for the design and analysis of efficient except to restate some of the governing equations inand compact devices for actuation in the form of volving linearized theory and the appropriate boundary micro/nano-electromechanical systems (MEMS/NEMS) conditions. The active materials of interest may undergo places considerable demands on the choice of mater- large deformations at the microstructural scale due ials, processing, and mechanics of actuation. Most of to various phase transformations (reorientation of unit the current applications of actuators except for a few cells); the macroscopic deformations generally do not specialized applications use piezoelectric materials. exceed a few percentage (ranging from 0.2% for piezoA detailed understanding of the various mechanisms electric, 1–6.5% for single-crystal ferroelectrics, and and mechanics of actuation in active materials will pave ≈ 0.1% for single-crystal magnetostrictive solids). Linthe way for the design of new actuation devices, ad- earized theory of elasticity is used throughout, which vancing further application of this promising class of suffices for most experimental design and applications. materials and other emerging multiferroic materials. Appropriate references for provided for readers who The current applications of these materials include ul- are interested in using rigorous large-strain (finiteThough the electrostrictive and magentostrictive effects are most evident in single-crystal materials, they also play an important role in polycrystalline solids, where these effects are influenced by the texture (orientation) of the various crystals in the solid.

Active Materials

deformation) formulations. For the basic notions of materials science such as the unit cell, crystallography, and texture, the reader is referred to Chap. 2. The parameters of interest in solid mechanics for the active materials (occupying a volume V , with surface denoted by S) include the Cauchy stress tensor (σ), the small-strain tensor ε(εij = 12 (u i, j + u j,i )), and the strain energy density W. The materials are assumed to be linearly elastic solids characterized by the fourth-order elastic moduli tensor C. The Cauchy stress is related to the strain through the elastic moduli σij = Cijkl εkl . The mechanical equilibrium of the stress state is governed by the following field equation σij, j + ρ0 bi = 0 in V .

(6.1)

The boundary conditions are characterized by the prescribed traction (force) vector (t 0 ) on S2 and/or the displacement vector (u0 ) on S1 . The traction on a surface is related to the stress through the Cauchy relation, t i = σij n j , where n is the unit outward normal to the surface. ρ0 is the mass density and b is the body force per unit mass.

6.2 Piezoelectrics

161

The parameters of interest in the electromechanics of solids include the polarization ( p) and the electrostatic potential (φ). The governing equation for the electrostatic potential is expressed by Gauss’s equation ∇ · (−ε0 ∇φ + p) = 0 in V ,

(6.2)

where ε0 is the permittivity. The boundary conditions on the electrostrictive solid are characterized by the conductors in the form of the electric field on the electrodes (∇φ = 0 on C1 ), including the ground ( S ∂φ ∂n dS = 0 and φ = 0 on C2 ). The parameters of interest in the magnetomechanics of solids include the magnetization (m) and the induced magnetic field (H). The governing equations are given by ∇ × H = 0,

∇ · (H + 4πm) = 0 in V ,

(6.3)

An important aspect of electro(magnetic) active materials is that the electro(magnetic) field permeates the space (R3 ) surrounding the body that is polarized (magnetized).

6.2 Piezoelectrics

Di = dijk σ jk ,

(6.4)

where σ is the stress tensor and D is the electric displacement vector, which is related to the polarization p according to Di = pi + ε0 E i ,

(6.5)

where E is the electric field vector [6.3]. For materials with large spontaneous polarizations, such as ferroelectrics, the electric displacement is approximately equal to the polarization (D ≈ p). For actuators, a more common representation of piezoelectricity is the converse piezoelectric effect. This is a linear relationship between strain and electric field, as shown in Fig. 6.1a and in the following equation at constant stress, eij = dijk E k ,

(6.6)

where e is the strain tensor and d is the same as in (6.4). These relationships are often expressed in matrix notation as Di = dij σ j , e j = dij E i , σ j = sij E i ,

(6.7) (6.8) (6.9)

where s is the matrix of piezoelectric stress constants [6.2, 3]. The parameters commonly used to characterize the piezoelectric effect are the constants d3i (in particular, d33 ), which are measures of the coupling between the applied voltage and the resultant strain in the specimen.

Part A 6.2

Piezoelectricity is a property of ferroelectric materials, as well as many non-ferroelectric crystals, such as quartz, whose crystal structure satisfy certain symmetry criteria [6.2]. It also exists in certain ceramic materials that either have a suitable texture or exhibit a net spontaneous polarization. The most common piezoelectric materials which are widely used in applications include lead zirconate titanate (PZT, Pb(Zr,Ti)O3 ) and lead lanthanum zirconate titanate (PLZT, Pb(La,Zr,Ti)O3 ). Many polymers such as polyvinylidene fluoride (PVDF) and its copolymers with trifluoroethylene (TrFE) and tetrafluoroethylene (TFE) also exhibit the piezoelectric effect. The typical strain achievable in the common piezoelectric solids is in the range of 0.1–0.2%. The direct piezoelectric effect is defined as a linear relationship between stress and electric displacement or charge per unit area,

162

Part A

Solid Mechanics Topics

6.3 Ferroelectrics The term ferroelectric relates not to a relationship of the material to the element iron, but simply a similarity of the properties to those of ferromagnets. Ferroelectrics exhibit a spontaneous, reversible electrical polarization and an associated hysteresis behavior between the polarization and electric field [6.4–6]. Much of the terminology associated with ferroelectrics is borrowed from ferromagnets; for instance, the transition temperature below which the material exhibits ferroelectric behavior is referred to as the Curie temperature. The ferroelectric phenomenon was first discovered in Rochelle salt (NaKC4 H4 O6 · 4H2 O). Other common examples of ferroelectric materials include barium titanate (BaTiO3 ), lead titanate (PbTiO3 ), and lithium niobate (LiNbO3 ). Materials of the perovskite structure (ABO3 ) appear to have the largest electrostriction and spontaneous polarization.

6.3.1 Electrostriction Electrostriction, in its most general sense, means simply electric-field-induced deformation. However, the term is most often used to refer to an electric-field-induced deformation that is proportional to the square of the electric field, as illustrated in Fig. 6.1b, εij = Mijkl E k El .

(6.10)

Part A 6.3

This effect does not require a net spontaneous polarization and, in fact, occurs for all dielectric materials [6.5]. The effect is quite pronounced in some ferroelectric ceramics, such as Pb(Mgx Nb1−x )O3 (PMN) and (1 − x)[Pb(Mg1/3 Nb2/3 )O3 ] − xPbTiO3 (PMN-PT), which generate strains much larger than those of piezoelectric PZT. The term electrostriction will be defined in a more general sense as electric-field-induced deformation that is independent of electric field polarity. As mentioned earlier, piezoelectricity exists in polycrystalline ceramics which exhibit a net spontaneous polarization. For a ferroelectric ceramic, while each grain may be microscopically polarized, the overall material will not be, due to the random orientation of the grains [6.2, 7]. For this reason, the ceramic must be poled under a strong electric field, often at elevated temperature, in order to generate the net spontaneous polarization. The ceramic is exposed to a strong electric field, generating an average polarization. The most interesting property of a ferroelectric solid is that it can be depolarized by an electric field and/or stress. It is this property which can be

exploited effectively in achieving large electrostriction [6.6]. Poling involves the reorientation of domains within the grains. In the case of PZT it may also involve polarization rotations due to phase changes. PZT is a solid solution of lead zirconate and lead titanate that is often formulated near the boundary between the rhombohedral and tetragonal phases (the so-called morphotropic phase boundary). For these materials, additional polarization states are available as it can choose between any of the 100 polarized states of the tetragonal phase, the 111 polarized states of the rhombohedral phase, or the 11k polarized states of the monoclinic phase. The final polarization of each grain, however, is constrained by the mechanical and electrical boundary conditions presented by the adjacent grains. A typical polarization–electric field hysteresis curve for a ferroelectric material is shown in Fig. 6.3. The spontaneous polarization, P s , is defined by the extrapolation of the linear region at saturation back to the polarization axis. The remaining polarization when the electric field returns to zero is known as the remnant polarization P r . Finally, the electric field at which the polarization returns to zero is known as the coercive field E c [6.8]. Polarization

Ps Pr

Ec

Electric field

Fig. 6.3 Polarization–electric field hysteresis for ferro-

electric materials. The spontaneous polarization (Ps ) is defined by the line extrapolated from the saturated linear region to the polarization axis. The remnant polarization (Pr ) is the polarization remaining at zero electric field. The coercive field (E c ) is the field required to reduce the polarization to zero

Active Materials

6.3.2 Theory

163

W

Based on the concepts for ferroelectricity postulated by Ginzburg and Landau, Devonshire developed a theory in which he treated strain and polarization as order parameters or field variables, which is collectively known as the Devonshire–Ginzburg–Landau (DGL) model [6.2, 9]. This theory was enormously successful in organizing vast amounts of data and providing the basis for the basic studies of ferroelectricity. The adaptation of this theory following Shu and Bhattacharya [6.10] is the most amenable in the context of mechanics and is described below. Consider a ferroelectric crystal V at a fixed temperature subject to an applied traction t 0 on part of its boundary S2 and an external applied electric field E 0 . The displacement u and polarization p of the ferroelectric are those that minimize the potential energy, 1 ∇ p · A∇ p + W(x, ε, p) − E0 · p dx Φ( p, u) = 2 V ε0 − t0 · u dS + (6.11) |∇φ|2 dx , 2 S2

6.3 Ferroelectrics

R3

W(θ, ε, p) = χij pi p j + ωijk pi p j pk + ξijkl pi p j pk pl + ψijklm pi p j pk pl pm + ζijklmn pi p j pk pl pm pn + Cijkl εij εkl + aijk εij pk + qijkl εij pk pl + · · · , (6.12) where χij is the reciprocal dielectric susceptibility of the unpolarized crystal, Cijkl is the elastic stiffness tensor, aijk is the piezoelectric constant tensor, qijkl is the electrostrictive constant tensor, and the coefficients are functions of temperature [6.8].

Fig. 6.4 The multiwell structure of the energy of a ferro-

electric solid with a tetragonal crystal structure as in the case of common perovskite crystals

The third and fourth terms in (6.11) are the potentials associated with the applied electric field and mechanical load, respectively. The final term is the electrostatic field energy that is generated by the polarization distribution. For any polarization distribution, the electrostatic potential φ is determined by solving Gauss’s equation (6.2) in all space, subject to appropriate boundary conditions, especially those on conductors. Thus, this last term is nonlocal. Ferroelectric crystals can be spontaneously polarized and strained in one of K crystallographically equivalent variants below their Curie temperature. Thus, if ε(i) , p(i) are the spontaneous strain and polarization of the i-th variant (i = 1, . . . , K ), then the stored energy W is minimum (zero without loss of generality) on the K [(ε(i) , p(i) )] and grows away from it as shown Z = ∪i=1 in the bottom right of Fig. 6.4.

6.3.3 Domain Patterns A region of constant polarization is known as a ferroelectric domain. The orientation of polarization and strain in ferroelectric crystals is determined by the possible variants of the underlying crystal structure. For example Fig. 6.5a shows the six possible variants that can form by the phase transformation of a perovskite (ABO3 ) crystal from the high-temperature cubic phase to the tetragonal phase when cooled below the Curie temperature. Domains are separated by 90◦ or 180◦ domain boundaries (the angle denotes the orientation between the polarization vectors in adjacent domains, Fig. 6.5b), which can be nucleated or moved by electric field or stress (the ferroelastic effect). Domain patterns are commonly visualized using polarized light microscopy and are shown in Fig. 6.5c for BaTiO3 . The process of changing the polarization direction of a domain by nucleation and growth or domain wall motion is known as domain switching. Electric field can induce both 90◦ or 180◦ switching, while stress

Part A 6.3

where A is a positive-definite matrix so that the first term above penalizes sharp changes in the polarization and may be regarded as the energetic cost of forming domain walls. The second term W is the stored energy density (the Landau energy density), which depends on the state variables or order parameters, the strain ε, and the polarization p, and also explicitly on the position x in polycrystals and heterogeneous media; W encodes the crystallographic and texture information, and may in principle be obtained from first-principles calculations based on quantum mechanics. It is traditional to take W to be a polynomial but one is not limited to this choice; for example, the energy density function W is assumed to be of the form,

ε, p

164

Part A

Solid Mechanics Topics

can induce only 90◦ switching [6.11]. The domain wall structures in the mechanical and electrical domain have recently been experimentally measured using scanning probe microscopy [6.12], which is typically in the range of tens of nanometers. In light of the multiwell structure of W, minimization of the potential energy in (6.11) leads to domain patterns or regions of almost constant strain and polarization close to the spontaneous values separated by domain walls. The width of the domain walls is proportional to the square root of the smallest eigenvalue of A. If this is small compared to the size of the crystal, as is typical, then the domain wall energy has a negligible effect on the macroscopic behavior and may be dropped [6.10]. This leads to an ill-posed problem as the minimizers may develop oscillations at a very fine scale (Fig. 6.5b), however there has been significant recent progress in studying such problems in recent years, motivated by active materials. a)

The minimizers of (6.11) for zero applied load and field are characterized by strain and polarization fields that take their values in Z. So one expects the solutions to be piecewise constant (domains) separated by jumps (domain walls). However the walls cannot be arbitrary. Instead, an energy-minimizing domain wall between variants i and j, i. e., an interface separating regions of strain and polarization (ε(i) , p(i) ) and (ε( j) , p( j) ) as shown in Fig. 6.5d, must satisfy two compatibility conditions [6.10], 1 ε( j) − ε(i) = (a ⊗ n + n ⊗ a) , 2 (6.13) ( p( j) − p(i) ) · n = 0 , where n is the normal to the interface. The first is the mechanical compatibility condition, which assures the mechanical integrity of the interface, and the second is the electrical compatibility conditions, which assures that the interface is uncharged and thus that energy minimizing. At first glance it appears impossible to solve these equations simultaneously: the first equation has at most two solutions for the vectors a and n, and there is no reason that these values of n should satisfy the second. It turns out however, that if the variants are related by two-fold symmetry, i. e., ε( j) = Rε(k) RT , 180◦

b)

90° boundary

Part A 6.3

180° boundary

c)

d)

n (ε(2), p (2))

(ε(1), p (1)) 100 μm

Fig. 6.5 (a) Variants of cubic-to-tetragonal phase transformation, the arrows indicate the direction of polarization. (b) Schematic of 90◦ and 180◦ domains in a ferroelectric crystal. (c) Polarized-light micrograph of the domain pattern in barium titanate. (d) Schematic of a domain wall in a ferroelectric crystal

ε( j) = Rp(k) ,

(6.14)

for some rotation R, then it is indeed possible to solve the two equations (6.14) simultaneously [6.10]. It follows that the only domain walls in a 001c polarized tetragonal phase are 180◦ and 90◦ domain walls, and that the 90◦ domain walls have a structure similar to that of compound twins with a rational {110}c interface and a rational 110c shear direction. The only possible domain walls in the 110c polarized orthorhombic phase are 180◦ domain walls, 90◦ domain walls having a structure like that of compound twins with a rational {100}c interface, 120◦ domain walls having a structure like that of type I twins with a rational {110}c interface, and 60◦ domain walls having a structure like that of type II twins with an irrational normal. The only possible domain walls in the 111c polarized rhombohedral phase are the 180◦ domain walls, and the 70◦ or 109◦ domain walls with a structure similar to that of compound twins. One can also use these ideas to study more-complex patterns involving multiple layers, layers within layers, and crossing layers. The potential energy (6.11) also allows one to study how applied boundary conditions affect the microstructure. The nonlocal electrostatic term is particularly interesting. For example, an isolated ferroelectric that

Active Materials

is homogeneously polarized generates an electrostatic field around it, and its energetic cost forces the ferroelectric to either become frustrated (form many domains at a small scale) or form closure domains or surface layers. In contrast, ferroelectrics shielded by electrodes can form large domains. Hard (high compliance, i. e., stiff) mechanical loading can force the formation of fine domain patterns [6.10], while soft (low compliance, dead loading being the most ideal example) loading leads to large domains. Therefore any strategy for actuation through domain switching must use electrodes to suitably shield the ferroelectric and soft loading in such a manner to create uniform electric and mechanical fields.

6.3 Ferroelectrics

165

by forming a (compatible) microstructure. However, when each pair of variants satisfy the compatibility conditions (6.13), the results of DeSimone and James [6.15] can be adopted to show that Z S equals the set of all possible averages of the spontaneous polarizations and strains of the variants n λi ε(i) , Z S = (ε, p) : ε∗ = i=1

p∗ =

n

λi (2 f i − 1) p(i) , λi ≥ 0 ,

i=1

n

λi = 1, 0 ≤ f i ≤ 1 .

(6.15)

i=1

6.3.4 Ceramics

Z P ⊇ Z T = ∩ Z S (x) = x∈Ω

{(ε, p)|(R(x) ε RT (x), R(x) p) ∈ Z S (x), ∀x ∈ Ω} . (6.16)

This simple bound is easy to calculate and also a surprisingly good indicator of the actual behavior of the material, which has the following implications. A material that is cubic above the Curie temperature and 001c -polarized tetragonal below has a very small set of spontaneous polarizations and no set of spontaneous strains unless the ceramic has a 001c texture. Indeed, each grain has only three possible spontaneous strains so that it is limited to only two possible deformation modes. Consequently the grains simply constrain

Part A 6.3

A polycrystal is a collection of perfectly bonded single crystals with identical crystallography but different orientations [6.13]. The term ceramic in the case of ferroelectrics refers to a polycrystal with numerous small grains, each of which may have numerous domains. The functional (6.11) (with A = 0) describes all the details of the domain pattern in each grain, and thus is rather difficult to understand. Instead it is advantageous to replace the energy density W in the functional (6.1) with ¯ the effective energy density of the polycrystal. The W, energy density W describes the behavior at the smallest length scale, which has a multiwell structure as dis- ˆ x, ε, p cussed earlier. This leads to domains, and W is the energy density of the grain at x after it has formed a domain pattern with average strain η and average polarization p. Note that this energy is zero on a set Z S , which is larger than the set Z. Z S is the set of all possible average spontaneous or remnant strains and polarizations that a single crystal can have by formˆ and Z S can vary from ing domain patterns. However, W grain to grain. The collective behavior of the polycrys¯ W(ε, ¯ tal is described by the energy density, W. p) is the energy density of a polycrystal with grains and domain patterns when the average strain is ε∗ and the average polarization is p∗ . Notice that it is zero on the set Z P , which is the set of all possible average spontaneous or remnant strains and polarization of the polycrystal. The size of the set Z P is an estimate of the ease with which a ferroelectric polycrystal may be poled, and also the strains that one can expect through domain switching. A rigorous discussion and precise definitions are given by Li and Bhattacharya [6.14]. The set Z S is obtained as the average spontaneous polarizations and strains that a single crystal can obtain

This is the case in materials with a cubic nonpolar high-temperature phase, and tetragonal, rhombohedral or orthorhombic ferroelectric low-temperature phases; explicit formulas are given in [6.14]. This is not the case in a cubic–monoclinic transformation, but one can estimate the set in that case. In a ceramic, the grain x has its own set Z S (x), which is obtained from the reference set by applying the rotation R(x) that describes the orientation of the grain relative to the reference single crystal. The set Z p of the polycrystal may be obtained as the macroscopic averages of the locally varying strain and polarization fields, which take their values in Z S (x) in each grain x. An explicit characterization remains an open problem (and sample dependent). However, one can obtain an insight into the size of the set by the so-called Taylor bound Z T , which assumes that the (mesoscale) strain and polarization are equal in each grain. Z T is simply the intersection of all possible sets Z S (x) corresponding to the different grains as x varies over the entire crystal. It is a conservative estimate of the actual set Z p :

166

Part A

Solid Mechanics Topics

each other. Similar results hold for a material which is cubic above the Curie temperature and 111c -polarized rhombohedral below it unless the ceramic has a 111c texture. These results imply that tetragonal and rhombohedral materials will not display large strain unless they are single crystals or are textured. Furthermore, it shows that it is difficult to pole these materials. This is the situation in BaTiO3 at room temperature, PbTiO3 , or PZT away from the morphotropic phase boundary (MPB). The situation is quite different if the material is either monoclinic or has a coexistence of 001c -polarized

tetragonal and 111c -polarized rhombohedral states below the Curie temperature. In either of these situations, the material has a large set of spontaneous polarizations and at least some set of spontaneous polarizations irrespective of the texture. Thus these materials will always display significant strain and can easily be poled. This is exactly the situation in PZT at the MPB. In particular, this shows that PZT has large piezoelectricity at the MPB because it can easily be poled and because it can have significant extrinsic strains [6.14].

6.4 Ferromagnets

Part A 6.4

Magnetostrictive materials are ferromagnetics that are spontaneously magnetized and can be demagnetized by the application of external magnetic field and/or stress. Ferromagnets exhibit a spontaneous, reversible magnetization and an associated hysteresis behavior between magnetization and magnetic field. All magnetic materials exhibit magnetostriction to some extent and the spontaneous magnetization is a measure of the actuation strain (magnetostriction) that can be obtained. Application of external magnetic fields to materials such as iron, nickel, and cobalt results in a strain on the order of 10−5 –10−4 . Recent advances in materials development have resulted in large magnetostriction in ironand nickel-based alloys on the order of 10−3 –10−2 . The most notable examples of these materials include Tb0.3 Dy0.7 Fe2 (Terfenol-D) and Ni2 MnGa, a ferromagnetic shape-memory alloy [6.15,16]. A well-established approach that has been used to model magnetostrictive materials is the theory of micromagnetics due to Brown [6.17]. A recent approach known as constrained theory of magnetoelasticity proposed by DeSimone and James [6.15, 18] is presented here, which is more suited for applications related to experimental mechanics.

6.4.1 Theory The potential energy of a magnetostrictive solid can be written as [6.15, 18], Φ(ε, M, θ) = Φexch + Φmst + Φext + Φmel ,

(6.17)

where Φexch is the exchange energy, Φmst is the magnetostatic or stray-field energy, Φext is the energy associated with the external magnetomechanical loads consisting of a uniform prestress σ0 applied at the boundary of S, and of a uniform applied magnetic field H0 in V . The expression (6.17) is similar to the ex-

pression for the potential energy of a ferroelectric solid in (6.11). The arguments for the ferroelectric solid for neglecting the first two terms in the exchange energy and the stray-field energy are also applicable to the magnetostrictive solids under appropriate conditions, namely 1. that the specimen is much larger than the domain size so that the exchange energy associated with the domain walls can be neglected, and 2. that the sample is shielded so that energy associated with the stray fields is negligible. Such an approach enables one to explore the implications of the energy minimization of (6.17). However, shielding the sample is a challenge and needs careful attention. The energy associated with the external loading is written (6.18) Φext = − t0 · u dS − H0 · m dV . S2

V

The magnetoelastic energy that accounts for the deviation of the magnetization from the favored crystallographic direction can be written (6.19) Φmel = W(ε, m) dV . V

The magnetoelastic energy density W is a function of the elastic moduli, the magentostrictive constants, and the magnetic susceptibility, and is analogous to (6.12). For a given material, W can be minimized when evaluated on a pair consisting of a magnetization along an easy direction and of the corresponding stress-free strain. In view of the crystallographic symmetry, there will be several, symmetry-related energy-minimizing magnetization (m) and strain (ε) pairs, analogous to the

Active Materials

electrostrictive solids explored in Sect. 6.3.2. Assuming that the corresponding minimum value of Wis zero, one can define the set of energy wells of the material, K [(ε(i) , m(i) )], which is analogous to the energy Z = ∪i=1 wells for ferroelectric solids shown in Fig. 6.5; W increases steeply away from the energy wells. Necessary conditions for compatibility between adjacent domains in terms of jumps in strain and magnetization have been established and have a form similar to (6.13). Much of the discussion concerning domain patterns in ferroelectric solids presented in Sect. 6.3 is applicable to magentostrictive solids, with magnetization in place of electric polarization.

6.4.2 Magnetostriction The application of mechanical loading (stress) can demagnetize ferromagnets by reorienting the magnetization axis, which can be counteracted by an applied magnetic field. This competition between applied magnetic field and dead loading (constant stress) acting on the solid provides a competition, leading to nucleation and propagation of magnetic domains across a specimen that can give rise to large magnetic actuation. For Terfenol-D, the material with the largest known room-temperature magnetostriction, the energy wells comprising Z are eight symmetry-related variants. Experiments on Terfenol-D by Teter et al., illustrate the effect of crystal orientation and applied stress on magnetostriction and are reproduced in Fig. 6.6 [6.19]. Interesting features of the results

References

Magnetostriction (x 10 –3 ) 2.5 [111]

2 [112]

1.5 1 0.5 [110]

0 –2000

–1000

0

1000

2000 Field (Oe)

Fig. 6.6 Magnetostriction versus applied magnetic field for three mutually orthogonal directions in single-crystal Terfenol-D at 20 ◦ C and applied constant stress of 11 MPa. (after 6.19 with permission. Copyright 1990, AIP)

include the steep change in energy away from the minimum (th ereference state at zero strain) and the varying amounts of hysteresis for different orientations. The constrained theory of magnetoelasticity described above can reproduce qualitative features of the experiments based on energy-minimizing domain patterns. An important aspect of modeling magnetostriction hinges on the ability to measure the magnetoelastic energy density, W, and the associated constants.

6.2

6.3 6.4

6.5 6.6

P. Krulevitch, A.P. Lee, P.B. Ramsey, J.C. Trevino, J. Hamilton, M.A. Northrup: Thin film shape memory alloy microactuators, J. MEMS 5, 270–282 (1996) D. Damjanovic: Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics, Rep. Prog. Phys. 61, 1267–1324 (1998) L.L. Hench, J.K. West: Principles of Electronic Ceramics (Wiley, New York 1990) C.Z. Rosen, B.V. Hiremath, R.E. Newnham (Eds): Piezoelectricity. In: Key Papers in Physics (AIP, New York 1992) Y. Xu: Ferroelectric Materials and Their Applications (North-Holland, Amsterdam 1991) K. Bhattacharya, G. Ravichandran: Ferroelectric perovskites for electromechanical actuation, Acta Mater. 51, 5941–5960 (2003)

6.7

6.8 6.9 6.10

6.11

6.12

L.E. Cross: Ferroelectric ceramics: Tailoring properties for specific applications. In: Ferroelectric Ceramics, ed. by N. Setter, E.L. Colla (Monte Verita, Zurich 1993) pp. 1–85 F. Jona, G. Shirane: Ferroelectric Crystals (Pergamon, New York 1962), Reprint, Dover, New York (1993) A.F. Devonshire: Theory of ferroelectrics, Philos. Mag. Suppl. 3, 85–130 (1954) Y.C. Shu, K. Bhattacharya: Domain patterns and macroscopic behavior of ferroelectric materials, Philos. Mag. B 81, 2021–2054 (2001) E. Burcsu, G. Ravichandran, K. Bhattacharya: Large electrostrictive actuation of barium titanate single crystals, J. Mech. Phys. Solids 52, 823–846 (2004) C. Franck, G. Ravichandran, K. Bhattacharya: Characterization of domain walls in BaTiO3 using simultaneous atomic force and piezo response

Part A 6

References 6.1

167

168

Part A

Solid Mechanics Topics

6.13 6.14

6.15

force microscopy, Appl. Phys. Lett. 88, 1–3 (2006), 102907 C. Hartley: Introduction to materials for the experimental mechanist, Chapter 2 J.Y. Li, K. Bhattacharya: Domain patterns, texture and macroscopic electro-mechanical behavior of ferroelectrics. In: Fundamental Physics of Ferroelectrics 2001, ed. by H. Krakauer (AIP, New York 2001) p. 72 A. DeSimone, R.D. James: A constrained theory of magnetoelasticity, J. Mech. Phys. Solids 50, 283–320 (2002)

6.16

6.17 6.18

6.19

G. Engdahl, I.D. Mayergoyz (Eds.): Handbook of Giant Magnetostrictive Materials (Academic, New York 2000) W.F. Brown: Micromagnetics (Wiley, New York 1963) A. DeSimone, R.D. James: A theory of magnetostriction oriented towards applications, J. Appl. Phys. 81, 5706–5708 (1997) J.P. Teter, M. Wun-Fogle, A.E. Clark, K. Mahoney: Anisotropic perpendicular axis magnetostriction in twinned Tbx Dy1−x Fe1.95 , J. Appl. Phys. 67, 5004–5006 (1990)

Part A 6

169

Biological So 7. Biological Soft Tissues

Jay D. Humphrey

A better understanding of many issues of human health, disease, injury, and the treatment thereof necessitates a detailed quantification of how biological cells, tissues, and organs respond to applied loads. Thus, experimental mechanics can, and must, play a fundamental role in cell biology, physiology, pathophysiology, and clinical intervention. The goal of this chapter is to discuss some of the foundations of experimental biomechanics, with particular attention to quantifying the finite-strain behavior of biological soft tissues in terms of nonlinear constitutive relations. Towards this end, we review illustrative elastic, viscoelastic, and poroelastic descriptors of softtissue behavior and the experiments on which they are based. In addition, we review a new class of much needed constitutive relations that will help quantify the growth and remodeling processes within tissues that are fundamental to long-term adaptations and responses to disease, injury, and clinical intervention. We will see that much has

Constitutive Formulations – Overview .... 171

7.2

Traditional Constitutive Relations .......... 7.2.1 Elasticity ..................................... 7.2.2 Viscoelasticity .............................. 7.2.3 Poroelasticity and Mixture Descriptions .............. 7.2.4 Muscle Activation ......................... 7.2.5 Thermomechanics ........................

176 177 177

7.3

Growth and Remodeling – A New Frontier 7.3.1 Early Approaches.......................... 7.3.2 Kinematic Growth ........................ 7.3.3 Constrained Mixture Approach .......

178 178 179 180

7.4

Closure ................................................ 182

7.5

Further Reading ................................... 182

172 172 176

References .................................................. 183 been learned, yet much remains to be discovered about the wonderfully complex biomechanical behavior of soft tissues.

there was a need to await the development of a nonlinear theory of material behavior in order to quantify well that of soft tissues. It is not altogether surprising, therefore, that biomechanics did not truly come into its own until the mid-1960s. It is suggested that five independent developments facilitated this: (i) the post World War II renaissance in nonlinear continuum mechanics [7.2] established a general foundation needed for developing constitutive relations suitable for soft tissues; (ii) the development of computers enabled the precisely controlled experimentation [7.3] that was needed to investigate complex anisotropic behaviors and facilitated the nonlinear regressions [7.4] that were needed to determine best-fit material parameters from data; (iii) related to this technological advance, development of sophisticated numerical methods, particularly the

Part A 7

At least since the time of Galileo Galilei (1564–1642), there has been a general appreciation that mechanics influences the structure and function of biological tissues and organs. For example, Galileo studied the strength of long bones and suggested that they are hollow as this increases their strength-to-weight ratio (i. e., it increases the second moment of area given a fixed amount of tissue, which is fundamental to increasing the bending stiffness). Many other figures in the storied history of mechanics, including G. Borelli, R. Hooke, L. Euler, T. Young, J. Poiseuille, and H. von Helmholtz, contributed much to our growing understanding of biomechanics. Of particular interest herein, M. Wertheim, a very productive experimental mechanicist of the 19-th century, showed that diverse soft tissues do not exhibit the linear stress–strain response that is common to many engineering materials [7.1]. That is,

7.1

170

Part A

Solid Mechanics Topics

Mechanical stimuli ECM

α β

Cell-cell G-proteins

Ion channels

GF receptor Cadherins Cell membrane

Mechanobiologic factors

Cell-matrix Integrins FAC changes

Signaling pathways

CSK changes

Transcription factors Gene expression

Altered mechanical properties

Altered cell function

Fig. 7.1 Schema of cellular stimuli or inputs – chemical, mechanical, and of course, genetic – and possible mechanobio-

logic responses. The cytoskeleton (CSK) maintains the structural integrity of the cell and interacts with the extracellular matrix (ECM) through integrins (transmembrane structural proteins) or clusters of integrins (i. e., focal adhesion complexes or FACs). Of course, cells also interact with other cells via special interconnections (e.g., cadherins) and are stimulated by various molecules, including growth factors (GFs). Although the pathways are not well understood, mechanical stimuli can induce diverse changes in gene expression, which in turn alter cell function and properties as well as help control the structure and properties of the extracellular matrix. It is clear, therefore, that mechanical stimuli can affect many different aspects of cell function and thus that of tissues and organs

Part A 7

finite element method [7.5], enabled the solution of complex boundary value problems associated with soft tissues, including inverse finite element estimations of material parameters [7.6]; and (iv) the space race of the 1960s increased our motivation to study the response of the human body to applied loads, particularly high G-forces associated with lift-off and reentry as well as the microgravity environment in space. Finally, it is not coincidental that the birth of modern biomechanics followed closely (v) the birth of modern biology, often signaled by discoveries of the structure of proteins, by L. Pauling, and the structure of DNA, by J. Watson and F. Crick. These discoveries ushered in the age of molecular and cellular biology [7.7]. Indeed, in contrast to the general appreciation of roles of mechanics in biology in the spirit of Galileo, the last few decades have revealed a fundamentally more important role of mechanics and mechanical fac-

tors. Experiments since the mid-1970s have revealed that cells often alter their basic activities (e.g., their expression of particular genes) in response to even subtle changes in their mechanical environment. For example, in response to increased mechanical stretching, cells can increase their production of structural proteins [7.8]; in response to mechanical injury, cells can increase their production of enzymes that degrade the damaged structural proteins [7.9]; and, in response to increased flow-induced shear stresses, cells can increase their production of molecules that change permeability or cause the lumen to dilate and thereby to restore the shear stress towards its baseline value [7.10]. Cells that are responsive to changes in their mechanical loading are sometimes referred to as mechanocytes, and the study of altered cellular activity in response to altered mechanical loading is called mechanobiology (Fig. 7.1). Consistent with the mechanistic philosophy

Biological Soft Tissues

of R. Descartes, which motivated many of the early studies in biomechanics, it is widely accepted that cells are not only responsive to changes in their mechanical environment, they are also subject to the basic postulates of mechanics (e.g., balance of linear momen-

7.1 Constitutive Formulations – Overview

171

tum). As a result, basic concepts from mechanics (e.g., stress and strain) can be useful in quantifying cellular responses and there is a strong relationship between mechanobiology and biomechanics. Herein, however, we focus primarily on the latter.

7.1 Constitutive Formulations – Overview There are, of course, three general approaches to quantify complex mechanical behaviors via constitutive formulations: (i) theoretically, based on precise information on the microstructure of the material; (ii) experimentally, based directly on data collected from particular classes of experiments; and (iii) via trial and error, by postulating competing relations and selecting preferred ones based on their ability to fit data. Although theoretically derived microstructural relations are preferred in principle, efforts ranging from C. Navier’s attempt to model the behavior of metals to L. Treloar’s attempt to model the behavior of natural rubber reveal that that this is very difficult in practice, particularly for materials with complex microstructures such as soft tissues. Formulating constitutive relations directly from experimental data is thus a practical preference [7.11], but again history reveals that it has been difficult to identify and execute appropriate theoretically based empirical approaches. For this reason, trial-and-error phenomenological formulations, based on lessons learned over years of investigation and often motivated by limited microstructural information, continue to be common in biomechanics just as they are in more traditional areas of applied mechanics. Regardless of approach, there are five basic steps that one must follow in any constitutive formulation:

•

Specifically, the first step is to classify the behavior of the material under conditions of interest, as, for example, if the material exhibits primarily a fluid-like or a solid-like response, if the response is dissipative or not, if it is isotropic or not, if it is isochoric or not, and so forth. Once sufficient observations enable one to classify the behavior, one can then establish an appro-

Fibroblast Endothelial cell Collagen

Smooth muscle Adventitia

Elastin

Media Intima

Basal lamina

b)

Fig. 7.2 (a) Schema of the arterial wall, which consists of three basic layers: the intima, or innermost layer, the media, or middle layer, and the adventitia, or outermost layer. Illustrated too are the three primary cells types (endothelial, smooth muscle, and fibroblasts) and two of the key structural proteins (elastin and collagen). (b) For comparison, see the histological section of an actual artery: the dark inner line shows the internal elastic lamina, which separates the thin intima from the media; the lighter shade in the middle shows the muscle dominated media, and the darker shade in the outer layer shows the collagendominated adventitia

Part A 7.1

• • • •

delineate general characteristics of the material behavior, establish an appropriate theoretical framework, identify specific functional forms of the relations, calculate values of the material parameters, evaluate the predictive capability of the final relations.

a)

172

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Solid Mechanics Topics

priate theoretical framework (e.g., a theory of elasticity within the context of the definition of a simple material by W. Noll), with suitable restrictions on the possible constitutive relations (e.g., as required by the Clausius– Duhem inequality, material frame indifference, and so forth). Once a theory is available, one can then design appropriate experiments to quantify the material behavior in terms of specific functional relationships and best-fit values of the material parameters. Because of the complex behaviors, multiaxial tests are often preferred, including in-plane biaxial stretching of a planar specimen, extension and inflation of a cylindrical specimen, extension and torsion of a cylindrical specimen, or inflation of an axisymmetric membrane, each of which has a tractable solution to the associated finite-strain boundary value problem. The final step, of course, is to ensure that the relation has predictive capability beyond that used in formulating the relation. With regard to classifying the mechanical behavior of biological soft tissues, it is useful to note that they often consist of multiple cell types embedded within an extracellular matrix that consists of diverse proteins as well as proteoglycans (i. e., protein cores with polysaccharide branches) that sequester significant amounts of water. Figure 7.2 shows constituents in the arterial wall, an illustrative soft tissue. The primary structural proteins in arteries, as in most soft tissues, are elastin and different families of collagen. Elastin is perhaps the most elastic protein, capable of extensions of over 100% with little dissipation. Moreover, elastin is one of the most stable proteins in the body, with a normal halflife of decades. Collagen, on the other hand, tends to be very stiff and not very extensible, with the exception that it is often undulated in the physiologic state and thus can undergo large displacements until straightened.

The half-life of collagen varies tremendously depending on the type of tissue, ranging from a few days in the periodontal ligament to years in bone. Note, too, that the half-life as well as overall tissue stiffness is modulated in part by extensive covalent cross-links, which can be enzymatic or nonenzymatic (which occur in diabetes, for example, and contribute to the loss of normal function of various types of tissues, including arteries). In general, however, soft tissues can exhibit a nearly elastic response (e.g., because of elastin) under many conditions, one that is often nonlinear (due to the gradual recruitment of undulated collagen), anisotropic (due to different orientations of the different constituents), and isochoric (due to the high water content) unless water is exuded or imbibed during the deformation. Many soft tissues will also creep or stress-relax under a constant load or a constant extension, respectively. Hence, as with most materials, one must be careful to identify the conditions of interest. In the cardiovascular and pulmonary systems, for example, normal loading is cyclic, and these tissues exhibit a nearly elastic response under cyclic loading. Conversely, implanted prosthetic devices in the cardiovascular system (e.g., a coronary stent) may impose a constant distension, and thus induce significant stress relaxation early on but remodeling over longer periods due to the degradation and synthesis of matrix proteins. Finally, two distinguishing features of soft tissues are that they typically contain contractile cells (e.g., muscle cells, as in the heart, or myofibroblasts, as in wounds to the skin) and that they can repair themselves in response to local damage. That is, whether we remove them from the body for testing or not, we must remember that our ultimate interest is in the mechanical properties of tissues that are living [7.12].

Part A 7.2

7.2 Traditional Constitutive Relations At the beginning of this section, we reemphasize that constitutive relations do not describe materials; rather they describe the behavior of materials under welldefined conditions of interest. A simple case in point is that we do not have a constitutive relation for water; we have different constitutive relations for water in its solid (ice), liquid (water), or gaseous (steam) states, which is to say for different conditions of temperature and pressure. Hence, it is unreasonable to expect that any single constitutive relation can, or even should, describe a particular tissue. In other words, we should expect that

diverse constitutive relations will be equally useful for describing the behavior of individual tissues depending on the conditions of interest. Many arguments in the literature over whether a particular tissue is elastic, viscoelastic, poroelastic, etc. could have been avoided if this simple truth had been embraced.

7.2.1 Elasticity No biological soft tissue exhibits a truly elastic response, but there are many conditions under which the

Biological Soft Tissues

assumption of elasticity is both reasonable and useful. Toward this end, one of the most interesting, and experimentally useful, observations with regard to the behavior of many soft tissues is that they can be preconditioned under cyclic loading. That is, as an excised tissue is cyclically loaded and unloaded, the stress versus stretch curves tend to shift rightward, with decreasing hysteresis, until a near-steady-state response is obtained. Fung [7.12] suggested that this steadystate response could be modeled by separately treating the loading and unloading curves as nearly elastic; he coined the term pseudoelastic to remind us that the response is not truly elastic. In practice, however, except in the case of muscular tissues, the hysteresis is often small and one can often approximate reasonably well the mean response between the loading and the unloading responses using a single elastic descriptor, similar to what is done to describe rubber elasticity. Indeed, although mechanisms underlying the preconditioning of soft tissues are likely very different from those underlying the Mullin effect in rubber elasticity [7.13], in both cases initial cyclic loading produces stress softening and enables one to use the many advances in nonlinear elasticity. This and many other parallels between tissue and rubber elasticity likely result from the long-chain polymeric microstructure of both classes of materials, thus these fields can and should borrow ideas from one another (see discussion in [7.14] Chap. 1). For example, Cauchy membrane stress (g/cm) 120 RV epicardium equibiaxial stretch Circumferential Apex–to–base 60

1

1.16

1.32 Stretch

Fig. 7.3 Representative tension–stretch data taken, fol-

lowing preconditioning, from a primarily collagenous membrane, the epicardium or covering of the heart, tested under in-plane equibiaxial extension. Note the strong nonlinear response, anisotropy, and negligible hysteresis over finite deformations (note: membrane stress is the same as a stress resultant or tension, thus having units of force per length though here shown as a mass per length)

advances in rubber elasticity have taught us much about the importance of universal solutions, common types of material and structural instabilities, useful experimental approaches, and so forth [7.14–16]. Nonetheless, common forms of stress–strain relations in rubber elasticity – for example, neo-Hookean, Mooney–Rivlin, and Ogden – have little utility in soft-tissue biomechanics and at times can be misleading [7.17]. A final comment with regard to preconditioning is that, although we desire to know properties in vivo (literally in the body), it is difficult in practice to perform the requisite measurements without removing the cells, tissues, or organs from the body so that boundary conditions can be known. This process of removing specimens from their native environment necessarily induces a nonphysiological, often poorly controlled strain history. Because the mechanical behavior is history dependent, the experimental procedure of preconditioning provides a common, recent strain history that facilitates comparisons of subsequent responses from specimen to specimen. For this reason, the preconditioning protocol should be designed well and always reported. Whereas a measured linear stress–strain response implies a unique functional relationship, the nonlinear, anisotropic stress–strain responses exhibited by most soft tissues (Fig. 7.3) typically do not suggest a specific functional relationship. In other words, one must decide whether the observed characteristic stiffening over finite strains (often from 5% to as much as 100% strain) is best represented by polynomial, exponential, or more complex stress–strain relations. Based on one-dimensional (1-D) extension tests on a primarily collagenous membrane called the mesentery, which is found in the abdomen, Fung showed in 1967 that it can be useful to plot stiffness, specifically the change of the first Piola–Kirchhoff stress with respect to changes in the deformation gradient, versus stress rather than to plot stress versus stretch as is common [7.12]. Specifically, if P is the 1-D first Piola– Kirchhoff stress and λ is the associated component of the deformation gradient (i. e., a stretch ratio), then seeking a functional form P = P(λ) directly from data is simplified by interpreting dP/ dλ versus P. For the mesentery, Fung found a near-linear relation between stiffness and stress, which in turn suggested directly (i. e., via the solution of the linear first-order ordinary differential equation) an exponential stress–stretch relationship (with P(λ = 1) = 0): α β(λ−1) dP e −1 , = α+βP → P = dλ β

(7.1)

173

Part A 7.2

0

7.2 Traditional Constitutive Relations

174

Part A

Solid Mechanics Topics

where α and β are material parameters, which can be determined via nonlinear regressions of stress versus stretch data or more simply via linear regressions of stiffness versus stress data. Note, too, that one could expand and then linearize the exponential function to relate these parameters to the Young’s modulus of linearized elasticity if so desired. Albeit a very important finding, a 1-D constitutive relation cannot be extended to describe the multiaxial behavior that is common to many soft tissues, ranging from the mesentery to the heart, arteries, skin, cornea, bladder, and so forth. At this point, therefore, Fung made a bold hypothesis. Given that this 1-D first Piola–Kirchhoff versus stretch relationship was exponential, he hypothesized that the behavior of many soft tissues could be described by an exponential relationship between the second Piola–Kirchhoff stress tensor S and the Green strain tensor E, which are conjugate measures appropriate for large-strain elasticity. In particular, he suggested a hyperelastic constitutive relation of the form: ∂W ∂Q whereby S = = c eQ , W = c eQ − 1 ∂E ∂E (7.2)

Part A 7.2

where W is a stored energy function and Q is a function of E. Over many years, Fung and others suggested, based on attempts to fit data, that a convenient form of Q is one that is quadratic in the Green strain, similar to the form of a stored energy function in terms of the infinitesimal strain in linearized elasticity. Fung argued that, because Q is related directly to ln W, material symmetry arguments were the same for this exponential stored energy function as they are for linearized elasticity. Q thus contains two, five, or nine nondimensional material parameters for isotropic, transversely isotropic, or orthotropic material symmetries, respectively, and this form of W recovers that for linearized elasticity as a special case. The addition of a single extra material parameter, c, having units of stress, is a small price to pay in going from linearized to finite elasticity, and this form of W has been used with some success to describe data on the biaxial behavior of skin, lung tissue, arteries, heart tissue, urinary bladder, and various membranes including the pericardium (which covers the heart) and the pleura (which covers the lungs). In some of these cases, this form of W was modified easily for incompressibility (e.g., by introducing a Lagrange multiplier) or for a twodimensional (2-D) problem. For example, a commonly used 2-D form in terms of principal Green strains

is 2 2 + c2 E 22 + 2c3 E 11 E 22 − 1 , W = c exp c1 E 11 (7.3)

where ci are material parameters. It is easy to show that physically reasonable behavior is ensured by c > 0 and ci > 0 and that convexity is ensured by the additional condition c1 c2 > c23 [7.18, 19]. Nonetheless, (7.2) and (7.3) are not without limitations. This 2-D form limits the degree of strain-dependent changes in anisotropy that are allowed, and experience has shown that it is typically difficult to find a unique set of material parameters via nonlinear regressions of data, particularly in three dimensions. Perhaps more problematic, experience has also revealed that convergent solutions can be difficult to achieve in finite element models based on the three-dimensional (3-D) relation except in cases of axisymmetric geometries. The later may be related to the recent finding by Walton and Wilber [7.20] that the 3-D Fung stored energy function is not strongly elliptic. Hence, despite its past success and usage, there is clearly a need to explore other constitutive approaches. Moreover, this is a good reminder that there is a need for a strong theoretical foundation in all constitutive formulations. The Green strain is related directly to the right Cauchy Green tensor C = F F, where F is the deformation gradient tensor, yet most work in finite elasticity has been based on forms of W that depend on C, thus allowing the Cauchy stress t to be determined via (as required by Clausius–Duhem, and consistent with (7.2) for the relation between the second Piola–Kirchhoff stress and Green strain): t=

∂W 2 F F . det F ∂C

(7.4)

For example, Spencer [7.21] suggested that materials consisting of a single family of fibers (i. e., exhibiting a transversely isotropic material symmetry) could be described by a W of the form ˆ II, III, IV, V) , W = W(I,

(7.5)

where I = tr C,

2II = (tr C)2 − tr C2 ,

IV = M · CM,

V = M·C M , 2

III = det C, (7.6)

and M is a unit vector that identifies the direction of the fiber family in a reference configuration. Incompress-

Biological Soft Tissues

ibility, III = 1, reduces the number of invariants by one while introducing an arbitrary Lagrange multiplier p, namely p ˜ II, IV, V) − (III − 1) , (7.7) W = W(I, 2 yet it is still difficult to impossible to rigorously determine specific functional forms directly from data. Indeed, this problem is more acute in cases of twofiber families, including orthotropy when the families are orthogonal. Hence, subclasses of this form of W have been evaluated in biomechanics. For example, Humphrey et al. [7.22] showed, and Sacks and Chuong [7.23] confirmed, that a stored energy funcˆ IV), determined directly from tion of the form W(I, in-plane biaxial tests (with I and IV separately maintained constant) on excised slabs of noncontracting heart muscle to be a polynomial function, described well the available in-plane biaxial stretching data. This 1990 paper [7.22] also illustrated the utility of performing nonlinear regressions of stress–stretch data using data sets that combined results from multiple biaxial tests as well as the importance of respecting Baker–Ericksen-type inequalities [7.2] in the parameter estimations. Holzapfel et al. [7.18] and others have similarly proposed specific forms of W for arteries based on a subclass of the two-fiber family approach of Spencer. Specifically, they propose a form of W that combines that of a neo-Hookean relation with a simple exponential form for two-fiber families, namely ˜ = c(tr C − 3) W c1 exp b1 (M1 · CM1 − 1)2 − 1 + b1 c2 exp b2 (M2 · CM2 − 1)2 − 1 , + b2

(7.8)

175

depend on traditional invariants of C is not optimal, hence alternative invariant sets should be identified and explored [7.18, 24, 25]. Experiments designed based on these invariants remain to be performed, however. Preceding the one- and two-fiber family models were the microstructural models proposed by Lanir [7.26]. Briefly, electron and light microscopy reveal that the elastin and collagen fibers within many soft tissues have complex spatial distributions (notable exceptions being collagen fibers within tendons, which are coaxial, and those within the cornea of the eye, which are arranged in layered orthogonal networks). Moreover, it appears that, despite extensive cross-linking at the molecular level, these networks are often loosely organized. Consequently, Lanir suggested that a stored energy function for a tissue could be derived in terms of strain energies for straightened individual fibers if one accounted for the undulation and distribution of the different types of fibers, or alternatively that one could postulate exponential stored energy functions for the individual fibers (cf. (7.3)) and simply account for distribution functions for each type of fiber. For example, for a soft tissue consisting primarily of elastin and type I collagen (i. e., only two types of constituents), one could consider

Ri (ϕ, θ) wif λif sin ϕ dϕ dθ , (7.9) φi W= i=1,2

φi

where and Ri are, respectively, the volume fraction and distribution function for constituent i (elastin or collagen), and wif is the 1-D stored energy function for a fiber belonging to constituent i. Clearly, the stress could be computed as in (7.1) provided that the fiber stretch can be related to the overall strain, which is easy if one assumes affine deformations. In principle, the distribution function could be determined from histology and the material parameters for the fiber stored energy function could be determined from straightforward 1-D tests, thus eliminating, or at least reducing, the need to find many free parameters via nonlinear regressions in which unique estimates are rare. In practice, however, it has been difficult to identify the distribution functions directly, thus they have often been assumed to be Gaussian or a similarly common distribution function. Although proposed as a microstructural model, the many underlying assumptions render this approach microstructurally motivated at best; that is, there is no actual modeling of the complex interactions (including covalent cross-links, van der Waals forces, etc.) between the many different proteins and proteoglycans that endow the tissue with its bulk properties,

Part A 7.2

where Mi (i = 1, 2) denote the original directions of the two-fiber families. Although neither of these forms is derived directly from precise knowledge of the microstructure nor inferred directly from experimental data, this form of W was motivated by the idea that elastin endows an artery with a nearly linearly elastic (neo-Hookean)-type response whereas the straightening of multiple families of collagen can be modeled by exponential functions in terms of fiber stretches. Thus far, this and similar forms of W have proven useful in large-scale computations and illustrates well the utility of the third approach to modeling noted above – trial and error based on experience with other materials or similar relations. It is important to note, however, that inferring forms of W for incompressible behaviors that

7.2 Traditional Constitutive Relations

176

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Solid Mechanics Topics

which are measurable using standard procedures. Indeed, perhaps one reason that this approach did not gain wider usage is that it failed to predict material behaviors under simple experimental conditions, thus like competing phenomenological relations it had to rely on nonlinear regressions to obtain best-fit values of the associated material parameters. As noted earlier, the extreme complexity of the microstructure of soft tissues renders it difficult to impossible to derive truly microstructural relations. Nevertheless, as we discuss below, microstructurally motivated formulations can be a very useful approach to constitutive modeling provided that the relations are not overinterpreted. Among others, Bischoff et al. [7.27] have revisited microstructurally motivated constitutive models with a goal of melding them with phenomenological models. In summary, although no soft tissue is truly elastic in its behavior, hyperelastic constitutive relations have proven useful in many applications. We have reviewed but a few of the many different functional forms reported in the literature, thus the interested reader is referred to Fung [7.12], Maurel et al. [7.28], and Humphrey [7.19] for additional discussion.

7.2.2 Viscoelasticity Although the response of many soft tissues tends to be relatively insensitive to changes in strain rate over physiologic ranges, soft tissues creep under constant loads and they stress-relax under constant deformations. Among others, Fung [7.12] suggested that single-integral heredity models could be useful in biomechanics just as they are in rubber viscoelasticity [7.29]. For example, we recently showed that sets of strain-dependent stress relaxation responses of a collagenous membrane, before and after thermal damage, can be modeled via [7.30]

Part A 7.2

τ G(τ − s) t(τ) = − p(τ)I + 2F(τ) 0

∂ ∂W × (s) ds F (τ) , ∂s ∂C

(7.10)

where G is a reduced relaxation function that depends on fading time (τ − s) and W was taken to be an exponential function similar to (7.3); all other quantities are the same as before except with explicit dependence on the current time. Various forms of the reduced relaxation function can be used, including a simple form that

we found to be useful in thermal damage G(x) =

1− R +R, 1 + (x/τR )n

(7.11)

where n is a free parameter, τR is a characteristic time of relaxation, and R is the stress remainder, that is, the fraction of the elastic response that is left after a long relaxation (e.g., R = 0 for a viscoelastic fluid). If short-term responses are important, numerous models can be used; for example, the simple viscohyperelastic approach of Beatty and Zhou [7.31] is useful in modeling biomembranes [7.32]. Briefly, the Cauchy stress is assumed to be of the general form t = tˆ(B, D), where B = FF is the left Cauchy–Green tensor, with F the deformation gradient, and D = (L + L )/2 is the ˙ −1 . stretching tensor, with the velocity gradient L = FF Specifically, assuming incompressibility, the Cauchy stress has three contributions: a reaction stress, an elastic part, and a viscous part, namely t = − pI + 2F

˜ ∂W F + 2μD , ∂C

(7.12)

where p is again a Lagrange multiplier that enforces incompressibility, W is the same strain energy function that was used in the elastic-only description noted above, and μ is a viscosity. Hence, this description of short-term nonlinear viscohyperelasticity adds but one additional material parameter to the constitutive equation, and the elastic response can be quantified first via quasistatic tests, thereby reducing the number of parameters in each regression. Some have considered a synthesis of the short- and long-term models. For a discussion of other viscoelastic models in softtissue mechanics, see Provenzano et al. [7.33] or Haslach [7.34] and references therein.

7.2.3 Poroelasticity and Mixture Descriptions Not only do soft tissues consist of considerable water, every cell in these tissues is within ≈ 50 μm of a capillary, which is to say close to flowing blood. Clearly then, it can be advantageous to model tissues as solid–fluid mixtures under many conditions of interest. A basic premise of mixture theory [7.2] is that balance relations hold both for the mixture as a whole and for the individual constituents, with the requirement that summation of the balance relations for the constituents must yield the classical relations. Moreover, it is assumed that the constituent balance relations include additional constitutive relations, particularly those that model the

Biological Soft Tissues

exchanges of mass, momentum, or energy between constituents. The first, and most often used, approach to model soft tissues via mixture theory was proposed by Mow et al. [7.35]. Briefly, their linear biphasic theory treated cartilage as a porous solid (i. e., the composite response due to type II collagen, proteoglycans, etc.), which was assumed to exhibit a linearly elastic isotropic response, with an associated viscous fluid within. They proposed constitutive relations for the solid and fluid stresses of the form t (s) = −φ(s) pI + λs tr(ε)I + 2μs ε , t (f) = −φ(f) pI − 23 μf div v(f) I + 2μf D ,

(7.13)

and, for the momentum exchange between the solid and fluid, − p(f) = p(s) = p∇φ(f) + K (v(f) − v(s) ) ,

(7.14)

177

7.2.4 Muscle Activation Another unique feature of many soft tissues is their ability to contract via actin–myosin interactions within specialized cells called myocytes. Examples include the cardiac muscle of the heart, skeletal muscle of the arms and legs, and smooth muscle, which is found in many tissues including the airways, arteries, and uterus. The most famous equation in muscle mechanics is that postulated in 1938 by A. V. Hill to describe force–velocity relations. This relation, like many subsequent ones, focuses on 1-D behavior along the long axis of the myocyte or muscle; data typically comes from tests on muscle fibers or strips, or in some cases rings taken from arteries or airways. Although much has been learned, much remains to be learned particularly with respect to the multiaxial behavior. The interested reader is referred to Fung [7.12]. Zahalak et al. [7.42], and Rachev and Hayashi [7.43]. In addition, however, note that modeling muscle activity in the heart (i. e., the electromechanics) has advanced significantly and represents a great example of the synthesis of complex theoretical, experimental, and computational methods. Toward this end, the reader is referred to Hunter et al. [7.44, 45].

7.2.5 Thermomechanics The human body regulates its temperature to remain within a narrow range, and for this reason there has been little attention to constitutive relations for thermomechanical behaviors of cells, tissues, and organs. Nevertheless, advances in laser, microwave, highfrequency ultrasound, and related technologies have encouraged the development and use of heating devices to treat diverse diseases and injuries. For example, supraphysiologic temperatures can destroy cells and shrink collagenous tissue, which can be useful in treating cancer and orthopedic injuries, respectively. Laser-based corneal reshaping, or LASIK, is another prime example. Due to space limitations here, the interested reader is referred to Humphrey [7.46], and references therein, for a brief review of the growing field of biothermomechanics and insight into ways in which experimental mechanics and constitutive modeling can contribute. Also see Diller and Ryan [7.47] for information on the associated bioheat transfer. Of particular note, however, it has been shown that increased mechanical loading can delay the rate at which thermal damage accrues, hence there is a strong thermomechanical coupling and a pressing need for more mechanics-based studies – first experimental, then computational.

Part A 7.2

where the superscripts and subscripts ‘s’ and ‘f’ denote solid and fluid constituents, hence v(s) and v(f) are solid and fluid velocities, respectively. Finally, φ(i) are constituent fractions, μs and λs are the classical Lamé constants for the solid, μf is the fluid viscosity, and ε is the linearized strain in the solid. In some cases the fluid viscosity is neglected, thus allowing tissue viscoelasticity to be accounted for solely via the momentum exchange between the solid and diffusing fluid, where K is related to the permeability coefficient. Mow and colleagues have developed this theory over the years to account for additional factors, including the presence of diffusing ions [7.36, 37] (see also [7.38] for a related approach). Because of the complexity of poroelastic and mixture theories, as well as the inherent geometric complexities associated with most real initial–boundary value problems in soft tissues, finite element methods will continue to prove essential; see, for example, Spilker et al. [7.39] and Simon et al. [7.40] for such formulations. In summary, one can now find many different applications of mixtures in the literature on soft tissues (e.g., Reynolds and Humphrey [7.41] address capillary blood flow within a tissue using mixture theory) and, indeed, the past success and future promise of this approach mandates intensified research in this area, research that must not simply be application, but rather should include development and extension of past theories. Moreover, whereas many of the experiments in biomechanics have consisted of unconfined or confined uniaxial compression tests using porous indenters, there is a pressing need for new multiaxial tests.

7.2 Traditional Constitutive Relations

178

Part A

Solid Mechanics Topics

7.3 Growth and Remodeling – A New Frontier As noted above, it has been thought at least since the time of Galileo that mechanical stimuli play essential roles in governing biological structure and function. Nevertheless an important step in our understanding of the biomechanics of tissues began with Wolff’s law for bone remodeling, which was put forth in the late 19-th century. Briefly, it was observed that the fine structure of cancellous (i. e., trabecular) bone within long bones tended to follow lines of maximum tension. That is, it appeared that the stress field dictated, at least in part, the way in which the microstructure of bone was organized. This observation led to the concept of functional adaptation wherein it was thought that bone functionally adapts so as to achieve maximum strength with a minimum of material. For a discussion of bone growth and remodeling, see Fung [7.12], Cowin [7.48], and Carter and Beaupre [7.49]. Although the general concept of functional adaptation appears to hold for most tissues, it is emphasized that bone differs significantly from soft tissues in three important ways. First, bone growth occurs on surfaces, that is, via appositional growth rather than via interstitial growth as in most soft tissues; second, most of the strength of bone derives from an inorganic component, which is not true in soft tissues; third, bone experiences small strains and exhibits a nearly linearly elastic, or poroelastic, behavior, which is very different from the nonlinear behavior exhibited by soft tissues over finite strains. Hence, let us consider methods that have been applied to soft tissues.

7.3.1 Early Approaches

Part A 7.3

Murray [7.50] suggested that biological “organization and adaptation are observed facts, presumably conforming to definite laws because, statistically at least, there is some sort of uniformity or determinism in their appearances. And let us assume that the best quantitative statement embodying the concept of organization is a principle which states that the cost of operation of physiological systems tends to be a minimum. . . ” Murray illustrated his ideas by postulating a cost function for “operating an arterial segment.” He proposed that the radius of a blood vessel results from a compromise between the advantage of increasing the lumen, which reduces the resistance to flow and thereby the workload on the heart, and the disadvantage of increasing overall blood volume, which increases the metabolic demand of maintaining the blood (e.g., red blood cells have a life-

span of a few months in humans, which necessitates a continual production and removal of cells). Murray’s findings suggest that “. . . the flow of blood past any section shall everywhere bear the same relation to the cube of the radius of the vessel at that point.” Recently, it has been shown that Murray’s ideas are consistent with the observation that the lumen of an artery appears to be governed, in part, so as to keep the wall shear stress at a preferred value – for a simple, steady, incompressible, laminar flow of a Newtonian fluid in a circular tube, the wall shear stress is proportional to the volumetric flow rate and inversely proportional to the cube of the radius [7.19]. Clearly, optimization approaches should be given increased attention, particularly with regard to the design of useful biomechanical experiments and the reduction of the associated data. Perhaps best known for inventing the Turing machine for computing, Turing also published a seminal paper on biological growth [7.51]. Briefly, he was interested in mathematically modeling morphogenesis, that is, the development of the form, or shape, of an organism. In his words, he sought to understand the mechanism by which “genes . . . may determine the anatomical structure of the resulting organism.” Turing recognized the importance of both mechanical and chemical stimuli in controlling morphogenesis, but he focused on the chemical aspects, especially the reaction kinetics and diffusion of morphogens, substances such as growth factors that regulate the development of form. For example, he postulated linear reaction– diffusion equations of the form ∂M1 = a (M1 − h) + b (M2 − g) + D1 ∇ 2 M1 , ∂t ∂M2 = c (M1 − h) + d (M2 − g) + D2 ∇ 2 M2 , ∂t

(7.15)

where M1 and M2 are concentrations of two morphogens, a, b, c, and d are reaction rates, and D1 and D2 are diffusivities; h and g are equilibrium values of M1 and M2 , and t represents time during morphogenesis. It was assumed that the local concentration of a particular morphogen tracked the local production or removal of tissue. Numerical examples revealed that solutions to such systems of equations could “develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances.” These solutions were proposed as possible descriptors of the morphogenesis. It was not until the

Biological Soft Tissues

1980s, however, that there was an increased interest in the use of reaction–diffusion models to study biological growth and remodeling, which now includes studies of wound healing, tumor growth, angiogenesis, and tissue engineering in addition to morphogenesis [7.52–56]. As noted by Turing, mechanics clearly plays an important role in such growth and remodeling, thus it is not surprising that there has been a trend to embed the reaction–diffusion framework within tissue mechanics (albeit often within the context of linearized elasticity or viscoelasticity). For example, Barocas and Tranquillo [7.53] suggested that reaction–diffusion models for spatial–temporal information on cells could be combined with a mixture theory representation of a tissue consisting of a fluid constituent and solid network. In this way, they studied mechanically stimulated cell migration, which is thought to be an early step towards mechanically stimulated changes in deposition of structural proteins. See the original paper for details. In summary, there has been significant attention to modeling the production, diffusion, and half-life of a host of molecules (growth factors, cytokines, proteases) and how they affect cell migration, mitosis, apoptosis, and the synthesis and reorientation of extracellular matrix. In some cases the reaction–diffusion models are used in isolation, but there has been a move towards combining such relations with those of mechanics (mass and momentum balance). Yet because of the lack of attention to the finite-strain kinematics and nonlinear material behavior characteristic of soft tissues [7.57], there is a pressing need for increased generalization if this approach is to become truly predictive. Moreover, there is a pressing need for additional biomechanical experiments throughout the evolution of the geometry and properties so that appropriate kinetic equations can be developed.

7.3.2 Kinematic Growth

being conserved, the overall mass density appeared to remain constant, thus focusing attention on changes in volume.) Tozeren and Skalak [7.59] suggested further that finite-strain growth and remodeling in a soft tissue (idealized as fibrous networks) could be described, in part, by considering that “The stress-free lengths of the fibers composing the network are not fixed as in an inert elastic solid, but are assumed to evolve as a result of growth and stress adaptation. Similarly, the topology of the fiber network may also evolve under the application of stress.” One of the remarkable aspects of Skalak’s work is that he postulated that, if differential growth is incompatible, then continuity of material may be restored via residual stresses. Residual stresses in arteries were reported soon thereafter (independently in 1983 by Vaishnav and Fung; see the discussion in [7.19]), and shown to affect dramatically the computed stress field in the arterial wall. The basic ideas of incompatible kinematic growth, residual stress, and evolving material symmetries and stress-free configurations were seminal contributions. Rodriguez et al. [7.60] built upon these ideas and put them into tensorial form – the approach was called “finite volumetric growth”, which is now described in brief. The primary assumption is that one models volumetric growth through a growth tensor Fg , which describes changes between two fictitious stress-free configurations: the original body is imagined to be fictitiously cut into small stress-free pieces, each of which is allowed to grow separately via Fg , with det Fg = 1. Because these growths need not be compatible, internal forces are often needed to assemble the grown pieces, via Fa , into a continuous configuration. This, in general, produces residual stresses, which are now known to exist in many soft tissues besides arteries. The formulation is completed by considering elastic deformations, via Fe , from the intact but residually stressed traction-free configuration to a current configuration that is induced by external mechanical loads. The initial–boundary value problem is solved by introducing a constitutive relation for the stress response to the deformation Fe Fa , which is often assumed to be isochoric and of the Fung type, plus a relation for the evolution of the stress-free configuration via Fg (actually Ug since the rotation Rg is assumed to be I). Thus, growth is assumed to occur in stress-free configurations and typically not to affect material properties. See, too, Lubarda and Hoger [7.61], who consider special cases of transversely isotropic and orthotropic growth. Among others, Taber [7.62] and Rachev et al. [7.63] independently embraced the concept of kinematic

179

Part A 7.3

In a seminal paper, Skalak [7.58] offered an approach very different from the reaction–diffusion approach, one that brought the analysis of biological growth within the purview of large-deformation continuum mechanics. He suggested that “any finite growth or change of form may be regarded as the integrated result of differential growth, i. e. growth of the infinitesimal elements making up the animal and plant.” His primary goal, therefore, was to “form a framework within which growth and deformation may be discussed in regard to the kinematics involved.” (Note: Although it was realized that mass may change over time, rather than

7.3 Growth and Remodeling – A New Frontier

180

Part A

Solid Mechanics Topics

growth and solved initial–boundary value problems relating to cardiac development, arterial remodeling in hypertension and altered flow, and aortic development. For the purposes of discussion, briefly consider the model of aortic growth by Taber [7.62]. The aortic wall was assumed to have material properties (given by Fung’s exponential relation) that remained constant during growth, which in turn was modeled via additional constitutive relations for time rates of change of the growth tensor Fg = diag[λgr , λgθ , 1], namely dλgr 1 ge t¯θθ (s) − t¯θθ , = dt Tr dλgθ 1 ge t¯θθ (s) − t¯θθ = dt Tθ 1 ge + τ¯w (s) − τ¯w eα(R/Ri −1) , Tτ

(7.16)

where Ti are time constants, t¯jj and τ¯w are mean values of wall stress and flow-induced wall shear stress, respectively, the superscript ‘ge’ denotes growthequilibrium, α is a parameter that reflects the intensity of effects, at any undeformed radial location R, of growth factors produced by the cells that line the arterial wall and interact directly with the blood. Clearly, growth (i. e., the time rate of change of the stress-free configuration in multiple directions) continues until the stresses return to their preferred or equilibrium values. Albeit not in the context of vascular mechanics, Klisch et al. [7.64] suggested further that the concept of volumetric growth could be incorporated within the theory of mixtures (with solid constituents k = 1, . . . , n and fluid constituent f ) to describe growth in cartilage. The deformation of constituent k was given by Fk = Fke Fak Fkg , where Fkg was related to a scalar mass growth function m k via t

Part A 7.3

det Fkg (t) = exp

m k dτ,

(7.17)

0

and mass balance requires dk ρ k + ρk div vk = ρk m k , dt dfρf + ρ f div v f = 0 . dt

(7.18)

This theory requires evolution equations for Fkg (or similarly, m k ), which the authors suggested could depend on the stresses, deformations, growth of other constituents, etc., as well as constitutive relations for the mass growth

function. It is clear that such a function could be related to the reaction–diffusion framework of Turing, and thus chemomechanical stimulation of growth. Although it is reasonable, in principle, to consider a full mixture theory given that so many different constituents contribute to the overall growth and structural stability of a tissue or organ, it is very difficult in practice to prescribe appropriate partial traction boundary conditions and very difficult to identify the requisite constitutive relations for momentum exchanges. Indeed, it is not clear that there is a need to model such detail, such as the momentum exchange between different proteins that comprise the extracellular matrix or between the extracellular matrix and a migrating cell, for example, particularly given that such migration involves complex chemical reactions (e.g., degradation of proteins at the leading edge of the cell) not just mechanical interactions. For these and other reasons, let us now consider an alternative mixture theory.

7.3.3 Constrained Mixture Approach Although the theory of kinematic growth yields many reasonable predictions, we have suggested that it models consequences of growth and remodeling (G&R), not the processes by which they occur. G&R necessarily occur in stressed, not fictitious stress-free, configurations, and they occur via the production, removal, and organization of different constituents; moreover, G&R need not restore stresses exactly to homeostatic values. Hence, we introduced a conceptually different approach to model G&R, one that is based on tracking the turnover of individual constituents in stressed configurations [7.65]. Here, we illustrate this approach for 2-D (membrane-like) tissues [7.66]. Briefly, let a soft tissue consist of multiple types of structurally important constituents, each of which must deform with the overall tissue but may have individual material properties and associated individual natural (i. e., stress-free) configurations that may evolve over time. We employ the concept of a constrained mixture wherein constituents deform together in current configurations and tacitly assume that they coexist within neighborhoods over which a local macroscopic homogenization would be meaningful. Specifically, not only may different constituents coexist at a point of interest, the same type of constituent produced at different instants can also coexist. Because of our focus on thin soft tissues consisting primarily of fibrillar collagen, one can consider a constitutive relation for the principal Cauchy stress resultants for the tissue (i. e., constrained mixture) of the

Biological Soft Tissues

form 1 ∂wk T1 (t) = T10 (t) + , λ2 (t) ∂λ1 (t) k

1 ∂wk , T2 (t) = T20 (t) + λ1 (t) ∂λ2 (t)

(7.19)

k

where Ti0 (i = 1, 2) represent contributions by an amorphous matrix (e.g., elastin-dominated or synthetic/reconstituted in a tissue equivalent) that can degrade but cannot be produced, λi are measurable principal stretches that are experienced by the tissue, and wk is a stored energy function for collagen family k, which may be produced or removed over time. Note, too, that 2 2 k (7.20) λ (t) = λ1 cos α0k + λ2 sin α0k are stretches experienced by fibers in collagen family k relative to a common mixture reference configuration, with α0k the angle between fiber family k and the 1 coordinate axis. To account for the deposition of new collagen fibers within stressed configurations, however, we further assume the existence of a preferred (i. e., homeostatic) deposition stretch G kh , whereby the stretch experienced by fiber family k, relative to its unique natural configuration, can be shown to be λkn(τ) (t) = G kh λk (t)/λk (τ) ,

(7.21)

with t the current time and τ the past time at which family k was produced. Finally, to account for continual production and removal, let the constituent stored energies be [7.66] wk (t) =

0

(7.22)

where ρ is the mixture mass density, M k (0) is the 2-D mass density of constituent k at time 0, when G&R commences, Q k (t) ∈ [0, 1] is the fraction of constituent k that was produced before time 0 but survives to the current time t > 0, m k is the current mass density production of constituent k, W k λkn(τ) (t) is the strain energy function for a fiber family relative to its unique natural configuration, and q k is

181

an associated survival function describing that fraction of constituent k that was produced at time τ (after time 0) and survives to the current time t. Hence, consistent with (7.4), the principal Cauchy stress resultants of the constituents that may turnover are k M (0)Q k (t)G kh ∂W k ∂λk (t) 1 T1k (t) = λ2 (t) ρλk (0) ∂λkn(τ) (t) ∂λ1 (t) t

m k (τ)q k (t − τ)G kh ρλk (τ) 0 ∂W k ∂λk (t) × k dτ , ∂λn(τ) (t) ∂λ1 (t) k M (0)Q k (t)G kh ∂W k ∂λk (t) 1 T2k (t) = λ1 (t) ρλk (0) ∂λkn(τ) (t) ∂λ2 (t) +

t

m k (τ)q k (t − τ)G kh ρλk (τ) 0 ∂W k ∂λk (t) × k dτ . ∂λn(τ) (t) ∂λ2 (t) +

(7.23)

As in most other applications of biomechanics, the key challenge therefore is to identify specific functional forms for the requisite constitutive relations, particularly the individual mass density productions, the survival functions, and the strain energy functions for the individual fibers, not to mention relations for muscle contractility and its adaptation. Finally, there is also a need to prescribe the alignment of newly produced fibers, not just their rate of production and removal. These, too, will require contributions from experimental biomechanics. Illustrative simulations are found nonetheless in the original paper [7.66], which show that stable versus unstable growth and remodeling can result, depending on the choice of constitutive relation. Given that the biomechanics of growth and remodeling is still in its infancy, it is not yet clear which approaches will ultimately prove most useful. The interested reader is thus referred to the following as examples of alternate approaches [7.67–71]. Finally, it is important to emphasize that, regardless of the specific theoretical framework, the most pressing need at present is an experimental program wherein the evolving mechanical properties and geometries of cells, tissues, and organs are quantified as a function of time during adaptations (or maladaptations) in response to altered mechanical loading, and that such information must be

Part A 7.3

M k (0) k Q (t)W k λkn(0) (t) ρ t k m (τ) k + q (t − τ)W k λkn(τ) (t) dτ , ρ

7.3 Growth and Remodeling – A New Frontier

182

Part A

Solid Mechanics Topics

correlated with changes in the rates of production and removal of structurally significant constituents, which in turn depend on the rates of production, removal, and diffusion of growth factors, proteases, and related substances. Clearly, biomechanics is not simply mechanics

applied to biology; it is the extension, development, and application of mechanics to problems in biology and medicine, which depends on theoretically motivated experimental studies that seek to identify new classes of constitutive relations.

7.4 Closure In summary, much has been accomplished in our quest to quantify the biomechanical behavior of soft tissues, yet much remains to be learned. Fortunately, continuing technological developments necessary for advancing experimental biomechanics (e.g., optical tweezers, atomic force and multiphoton microscopes, tissue bioreactors) combined with traditional methods of testing (e.g., computer-controlled in-plane biaxial testing of planar specimens, inflation and extension testing of tubular specimens, and inflation testing of membranous specimens; see [7.19] Chap. 5) as well as continuing advances in theoretical and computational mechanics are helping us to probe deeper into the mechanobiology and biomechanics every day. Thus,

both the potential and the promise of engineering contributions have never been greater. It is hoped, therefore, that this chapter provided some background, and especially some motivation, to contribute to this important field. The interested reader is also referred to a number of related books, listed in the Bibliography, and encouraged to consult archival papers that can be found in many journals, including Biomechanics and Modeling in Mechanobiology, the Journal of Biomechanics, and the Journal of Biomechanical Engineering. Indeed, an excellent electronic search engine is NIH PubMed, which can be found via the National Institutes of Health web site (www.nih.gov); it will serve us well as we continue to build on past achievements.

7.5 Further Reading

Part A 7.5

Given the depth and breadth of the knowledge base in biomedical research, no one person can begin to gain all of the needed expertise. Hence, biomechanical research requires teams consisting of experts in mechanics (theoretical, experimental, and computational) as well as biology, physiology, pathology, and clinical practice. Nevertheless, bioengineers must have a basic understanding of the biological concepts. I recommend, therefore, that the serious bioengineer keep nearby books on (i) molecular and cell biology, (ii) histology, and (iii) medical definitions. Below, I list some books that will serve the reader well.

• • • •

H. Abe, K. Hayashi, M. Sato: Data Book on Mechanical Properties of Living Cells, Tissues, and Organs (Springer, New York 1996) Dorland’s Illustrated Medical Dictionary (Saunders, Philadelphia 1988) S.C. Cowin, J.D. Humphrey: Cardiovascular Soft Tissue Mechanics (Kluwer Academic, Dordrecht 2001) S.C. Cowin, S.B. Doty: Tissue Mechanics (Springer, New York 2007)

• • • • • • • • •

D. Fawcett: A Textbook of Histology (Saunders, Philadelphia 1986) Y.C. Fung: Biomechanics: Mechanical Properties of Living Tissues (Springer, New York 1993) F. Guilak, D.L. Butler, S.A. Goldstein, D.J. Mooney: Functional Tissue Engineering (Springer, New York 2003) G.A. Holzapfel, R.W. Ogden: Biomechanics of Soft Tissue in Cardiovascular Systems (Springer, Vienna 2003) G.A. Holzapfel, R.W. Ogden: Mechanics of Biological Tissue (Springer, Berlin, Heidelberg 2006) J.D. Humphrey, S.L. Delange: An Introduction to Biomechanics (Springer, New York 2004) V.C. Mow, R.M. Hochmuth, F. Guilak, R. TransSon-Tay: Cell Mechanics and Cellular Engineering (Springer, New York 1994) W.M. Saltzman: Tissue Engineering (Oxford Univ Press, Oxford 2004) L.A. Taber: Nonlinear Theory of Elasticity: Applications to Biomechanics (World Scientific, Singapore 2004)

Biological Soft Tissues

References

183

References 7.1

7.2

7.3

7.4 7.5 7.6

7.7

7.8

7.9

7.10

7.11

7.12 7.13

7.14

7.16 7.17

7.18

7.19

7.20

7.21 7.22

7.23

7.24

7.25

7.26 7.27

7.28

7.29

7.30

7.31

7.32

7.33

7.34

7.35

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7.48 7.49 7.50

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in compression: Theory and experiments, ASME J. Biomech. Eng. 102, 73–84 (1980) W.M. Lai, V.C. Mow, W. Zhu: Constitutive modeling of articular cartilage and biomacromolecular solutions, J. Biomech. Eng. 115, 474–480 (1993) V.C. Mow, G.A. Ateshian, R.L. Spilker: Biomechanics of diarthroidal joints: A review of twenty years of progress, J. Biomech. Eng. 115, 460–473 (1993) J.M. Huyghe, G.B. Houben, M.R. Drost, C.C. van Donkelaar: An ionised/non-ionised dual porosity model of intervertebral disc tissue experimental quantification of parameters, Biomech. Model. Mechanobiol. 2, 3–20 (2003) R.L. Spilker, J.K. Suh, M.E. Vermilyea, T.A. Maxian: Alternate Hybrid, Mixed, and Penalty Finite Element Formulations for the Biphasic Model of Soft Hydrated Tissues,. In: Biomechanics of Diarthrodial Joints, ed. by V.C. Mow, A. Ratcliffe, S.L.Y. Woo (Springer, New York 1990) pp. 401–436 B.R. Simon, M.V. Kaufman, M.A. McAfee, A.L. Baldwin: Finite element models for arterial wall mechanics, J. Biomech. Eng. 115, 489–496 (1993) R.A. Reynolds, J.D. Humphrey: Steady diffusion within a finitely extended mixture slab, Math. Mech. Solids 3, 147–167 (1998) G.I. Zahalak, B. de Laborderie, J.M. Guccione: The effects of cross-fiber deformation on axial fiber stress in myocardium, J. Biomech. Eng. 121, 376–385 (1999) A. Rachev, K. Hayashi: Theoretical study of the effects of vascular smooth muscle contraction on strain and stress distributions in arteries, Ann. Biomed. Eng. 27, 459–468 (1999) P.J. Hunter, A.D. McCulloch, H.E.D.J. ter Keurs: Modelling the mechanical properties of cardiac muscle, Prog. Biophys. Mol. Biol. 69, 289–331 (1998) P.J. Hunter, P. Kohl, D. Noble: Integrative models of the heart: Achievements and limitations, Philos. Trans. R. Soc. London A359, 1049–1054 (2001) J.D. Humphrey: Continuum thermomechanics and the clinical treatment of disease and injury, Appl. Mech. Rev. 56, 231–260 (2003) K.R. Diller, T.P. Ryan: Heat transfer in living systems: Current opportunities, J. Heat. Transf. 120, 810–829 (1998) S.C. Cowin: Bone stress adaptation models, J. Biomech. Eng. 115, 528–533 (1993) D.R. Carter, G.S. Beaupré: Skeletal Function and Form (Cambridge Univ. Press, Cambridge 2001) C.D. Murray: The physiological principle of minimum work: I. The vascular system and the cost of blood volume, Proc. Natl. Acad. Sci. 12, 207–214 (1926) A.M. Turing: The chemical basis of morphogenesis, Proc. R. Soc. London B 237, 37–72 (1952)

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7.57

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7.60

7.61

7.62

7.63

7.64

7.65

7.66

7.67

R.T. Tranquillo, J.D. Murray: Continuum model of fibroblast-driven wound contraction: Inflammationmediation, J. Theor. Biol. 158, 135–172 (1992) V.H. Barocas, R.T. Tranquillo: An anisotropic biphasic theory of tissue equivalent mechanics: The interplay among cell traction, fibrillar network, fibril alignment, and cell contact guidance, J. Biomech. Eng. 119, 137–145 (1997) L. Olsen, P.K. Maini, J.A. Sherratt, J. Dallon: Mathematical modelling of anisotropy in fibrous connective tissue, Math. Biosci. 158, 145–170 (1999) N. Bellomo, L. Preziosi: Modelling and mathematical problems related to tumor evolution and its interaction with the immune system, Math. Comput. Model. 32, 413–452 (2000) A.F. Jones, H.M. Byrne, J.S. Gibson, J.W. Dodd: A mathematical model of the stress induced during avascular tumour growth, J. Math. Biol. 40, 473–499 (2000) L.A. Taber: Biomechanics of growth, remodeling, and morphogenesis, Appl. Mech. Rev. 48, 487–545 (1995) R. Skalak: Growth as a finite displacement field, Proc. IUTAM Symposium on Finite Elasticity, ed. by D.E. Carlson, R.T. Shield (Martinus Nijhoff, The Hague 1981) pp. 347–355 A. Tozeren, R. Skalak: Interaction of stress and growth in a fibrous tissue, J. Theor. Biol. 130, 337– 350 (1988) E.K. Rodriguez, A. Hoger, A.D. McCulloch: Stressdependent finite growth in soft elastic tissues, J. Biomech. 27, 455–467 (1994) V.A. Lubarda, A. Hoger: On the mechanics of solids with a growing mass, Int. J. Solid Struct. 39, 4627– 4664 (2002) L.A. Taber: A model for aortic growth based on fluid shear and fiber stresses, ASME J. Biomech. Eng. 120, 348–354 (1998) A. Rachev, N. Stergiopulos, J.-J. Meister: A model for geometric and mechanical adaptation of arteries to sustained hypertension, ASME J. Biomech. Eng. 120, 9–17 (1998) S.M. Klisch, T.J. van Dyke, A. Hoger: A theory of volumetric growth for compressible elastic biological materials, Math. Mech. Solid 6, 551–575 (2001) J.D. Humphrey, K.R. Rajagopal: A constrained mixture model for growth and remodeling of soft tissues, Math. Model. Meth. Appl. Sci. 12, 407–430 (2002) S. Baek, K.R. Rajagopal, J.D. Humphrey: A theoretical model of enlarging intracranial fusiform aneurysms, J. Biomech. Eng. 128, 142–149 (2006) R.A. Boerboom, N.J.B. Driessen, C.V.C. Bouten, J.M. Huyghe, F.P.T. Baaijens: Finite element model of mechanically induced collagen fiber synthesis and degradation in the aortic valve, Ann. Biomed. Eng. 31, 1040–1053 (2003)

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Part A 7

187

Electrochemo 8. Electrochemomechanics of Ionic Polymer–Metal Composites

Sia Nemat-Nasser

The ionomeric polymer–metal composites (IPMCs) consist of polyelectrolyte membranes, with metal electrodes plated on both faces and neutralized with an amount of counterions, balancing the charge of anions covalently fixed to the membrane. IPMCs in the solvated state form soft actuators and sensors; they are sometimes referred to as artificial muscles. Here, we examine the nanoscale chemoelectromechanical mechanisms that underpin the macroscale actuation and sensing of IPMCs, as well as some of their electromechanical properties.

8.2.2 Pressure in Clusters ...................... 192 8.2.3 Membrane Stiffness ...................... 192 8.2.4 IPMC Stiffness .............................. 192 8.3 Voltage-Induced Cation Distribution ...... 193 8.3.1 Equilibrium Cation Distribution ...... 194 8.4 Nanomechanics of Actuation ................. 8.4.1 Cluster Pressure Change Due to Cation Migration ................ 8.4.2 Cluster Solvent Uptake Due to Cation Migration ................ 8.4.3 Voltage-Induced Actuation............

195 195 196 197

8.5 Experimental Verification ...................... 197 8.5.1 Evaluation of Basic Physical Properties ........... 197 8.5.2 Experimental Verification .............. 198

8.2 Stiffness Versus Solvation ...................... 191 8.2.1 The Stress Field in the Backbone Polymer .............. 191

8.6 Potential Applications ........................... 199 References .................................................. 199

Polyelectrolytes are polymers that carry covalentlybound positive or negative charges. They occur naturally, such as deoxyribonucleic acid (DNA) and ribonucleic acid (RNA), or they have been manuR factured for various applications, such as Nafion R or Flemion , which consist of three-dimensionally structured backbone perfluorinated copolymers of polytetrafluoroethylene, having regularly spaced long perfluorovinyl ether pendant side-chains that terminate in ionic sulfonate (Nafion) or carboxylate (Flemion) groups. The resulting Nafion or Flemion membranes are permeable to water or other polar solvents and cations, while they are impermeable to anions. The ionomeric polymer–metal composites (IPMCs) consist of polyelectrolyte membranes, about 200 μm thick, with metal electrodes (5–10 μm thick) plated on both faces [8.1] (Fig. 8.1). The polyelectrolyte matrix is neutralized with an amount of counterions, balancing

the charge of anions covalently fixed to the membrane. When an IPMC in the solvated (e.g., hydrated) state is stimulated with a suddenly applied small (1–3 V, depending on the solvent) step potential, both the fixed anions and mobile counterions are subjected to an electric field, with the counterions being able to diffuse toward one of the electrodes. As a result, the composite undergoes an initial fast bending deformation, followed by a slow relaxation, either in the same or in the opposite direction, depending on the composition of the backbone ionomer, and the nature of the counterion and the solvent. The magnitude and speed of the initial fast deflection also depend on the same factors, as well as on the structure of the electrodes, and other conditions (e.g., the time variation of the imposed voltage). IPMCs that are made from Nafion and are neutralized with alkali metals or with alkyl-ammonium cations (except for tetrabutylammonium, TBA+ ), invariably first bend to-

Microstructure and Actuation................. 8.1.1 Composition ................................ 8.1.2 Cluster Size .................................. 8.1.3 Actuation ....................................

Part A 8

188 188 189 191

8.1

188

Part A

Solid Mechanics Topics

a)

b)

Au plating

Fig. 8.1a,b Cross section of (a) a Pt/Au-plated Nafion-117 membrane at electrode region; the length of the bar is 408 nm; and (b) an Auplated Flemion at electrode region; the length of the bar is 1 μm

Au plating

Platinum particles

Au 408 nm

Flemion

wards the anode under a step direct current (DC), and then relax towards the cathode while the applied voltage is being maintained, often moving beyond their starting position. In this case, the motion towards the anode can be eliminated by slowly increasing the applied potential at a suitable rate. For Flemion-based IPMCs, on the other hand, the initial fast bending and the subsequent relaxation are both towards the anode, for all counterions that have been considered. With TBA+ as the counterion, no noticeable relaxation towards the cathode has been recorded for either Nafionor Flemion-based IPMCs. Under an alternating electric

1μm

potential, cantilevered strips of IPMCs perform bending oscillations at the frequency of the applied voltage, usually no more than a few to a few tens of hertz, depending on the solvent. When an IPMC membrane is suddenly bent, a small voltage of the order of millivolts is produced across its faces. Hence, IPMCs of this kind can serve as soft actuators and sensors. They are sometimes referred to as artificial muscles [8.2, 3]. In this chapter, we examine the nanoscale chemoelectromechanical factors that underpin the macroscale actuation and sensing of IPMCs, as well as some of their electromechanical properties.

8.1 Microstructure and Actuation 8.1.1 Composition The Nafion-based IPMC, in the dry state, is about 180 μm thick and the Flemion-based one is about 160 μm thick (see [8.4–7] for further information on IPMC manufacturing). Samples consist of 1. backbone perfluorinated copolymer of polytetrafluoroethylene with perfluorinated vinyl ether sulfonate pendants for Nafion-based and perfluorinated b) Nanosized

a)

interconnected clusters

Cracked metal overlayer

Metal coating

Part A 8.1

Ionic polymer Naflon or Flamion

0.2 mm

Cluster group

Fig. 8.2 (a) A schematic representation of an IPMC and (b) a transmission electron microscopy (TEM) photo of the cluster structure (see Fig. 8.3 for more detail)

propyl ether carboxylate pendants for Flemionbased IPMCs, forming interconnected nanoscale clusters ([8.8]; primary physical data for fluorinated ionomers have been summarized in [8.9]) 2. electrodes, which in Nafion-based IPMCs consist of 3–10 nm-diameter platinum particles, distributed mainly within a 10–20 μm depth of both faces of the membrane, and usually covered with about 1 μmthick gold plating to improve surface conductivity, while in Flemion-based IPMCs, the electrodes are gold, with a dendritic structure, as shown in Fig. 8.1 3. neutralizing cations 4. a solvent Figure 8.2 shows a schematic representation of a typical IPMC, including a photograph of the nanostructure of the ionomer (Nafion, in this case), and a sketch of ionic polymer with metal coating. Figure 8.3 shows some additional details of a Nafion-based IPMC. The dark spots within the inset are the left-over platinum crystals. The ion exchange capacity (IEC) of an ionomer represents the amount of sulfonate (in Nafion) and carboxylate (in Flemion) group in the material, mea-

Electrochemomechanics of Ionic Polymer–Metal Composites

a) Metal overlayer

1μm

8.1 Microstructure and Actuation

b)

189

c)

O O

O

O

O O

Platinum diffusion zone (0 –25 μm)

Fig. 8.4 (a) Chemical structure of 18-Crown-6 (each node is CH2 ); (b) Na+ ion; and (c) K+ ion within a macrocyclic 18-Crown-6

ligand

8.1.2 Cluster Size Left-over platinum particles 20 nm

Fig. 8.3 Near-surface structure of an IPMC; the lower in-

set indicates the size of a typical cluster in Nafion

2 3NkT 3 NEWion Uela = − aI , ρ ∗ NA h 2 ρB + wρs ρ∗ = , 1+w

(8.1)

where N is the number of dipoles inside a typical cluster, k is Boltzmann’s constant, T is the temperature, h 2 is the mean end-to-end chain length, ρ∗ is the effective density of the solvated membrane, and NA is Avogadro’s number (6.023 × 1023 ). The electrostatic en-

Part A 8.1

sured in moles per unit dry polymer mass. The dry bare ionomer equivalent weight (EW) is defined as the weight in grams of dry ionomer per mole of its anion. The ion exchange capacity and the equivalent weight of Nafion are 0.91 meq/g and 1.100 g/mol, and those of Flemion are 1.44 meq/g and 694.4 g/mol, respectively. For neutralizing counterions, we have used Li+ , + Na , K+ , Rb+ , Cs+ , Mg2+ , and Al3+ , as well as alkylammonium cations TMA+ , TEA+ , TPA+ , and TBA+ . The properties of the bare ionomer, as well as those of the corresponding IPMC, change with the cation type for the same membrane and solvent. In addition to water, ethylene glycol, glycerol, and crown ethers have been used as solvents. Ethylene glycol or 1,2ethanediol (C2 H6 O2 ) is an organic polar solvent that can be used over a wide range of temperatures. Glycerol, or 1,2,3-propanetriol (C3 H8 O3 ) is another polar solvent with high viscosity (about 1000 times the viscosity of water). Crown ethers are cyclic oligomers of ethylene glycol that serve as macrocyclic ligands to surround and transport cations (Fig. 8.4). The required crown ether depends on the size of the ion. The 12-Crown-4 (12CR4) matches Li+ , 15-Crown-5 (15CR5) matches K+ , and 18-Crown-6 matched Na+ and K+ . For example, an 18-Crown-6 (18CR6) molecule has a cavity of 2.7 Å and is suitable for potassium ions of 2.66 Å diameter. A schematic configuration of this crown with sodium and potassium ions is shown in Fig. 8.4.

X-ray scanning of the Nafion membranes [8.10] has shown that, in the process of solvent absorption, hydrophilic regions consisting of clusters are formed within the membrane. Hydrophilicity and hydrophobicity are generally terms used for affinity or lack of affinity toward the polar molecule of water. In the present work we use these terms for interaction toward any polar solvent (e.g., ethylene–glycol). Cluster formation is promoted by the aggregation of hydrophilic ionic sulfonate groups located at the terminuses of vinyl ether sulfonate pendants of the polytetrafluoroethylene chain. While these regions are hydrophilic, the membrane backbone is hydrophobic and it is believed that the motion of the solvent takes place among these clusters via the connecting channels. The characteristics of these clusters and channels are important factors in IPMC behavior. The size of the solvated cluster radius aI depends on the cation form, the type of solvent used, and the amount of solvation. The average cluster size can be calculated by minimizing the free energy of the cluster formation with respect to the cluster size. The total energy for cluster formation consists of an elastic Uela , an electrostatic Uele , and a surface Usur component. The elastic energy is given by [8.11]

190

Part A

Solid Mechanics Topics

a13 (nm3) 15

10

5

0

Model x-ray 0

0.5

1

1.5

2

2.5

3 η 105

Fig. 8.5 Cluster size in Nafion membranes with different solvent uptakes

ergy is given by Uele = −g

N 2 m2 , 4πκe aI3

where n is the number of clusters present in the membrane. Minimizing this energy with respect to cluster tot = 0 , gives the optimum cluster size at size, ∂U ∂aI which the free energy of the ionomer is minimum. In this manner Li and Nemat-Nasser [8.11] have obtained ⎡ γ h 2 EWion (w + ΔV ) aI = ⎣ 2RT ρB −2 ⎤1/3 4πρ B ⎦ × 1− 3 ∗ , 3ρ (w + ΔV ) NA Vi ρd , (8.5) EWion where Vi is the volume of a single ion exchange site. Assuming that h 2 = EWion β [8.12], it can be seen that, ΔV =

(8.2)

aI3 =

−2 EW2ion (w + ΔV ) 4πρB 3 η= 1− . ρB 3ρ∗ (w + ΔV )

where g is a geometric factor, m is the dipole moment, and κe is the effective permittivity within the cluster. The surface energy can be expressed as

(8.6)

Usur = 4πaI2 γ ,

(8.3)

where γ is the surface energy density of the cluster. Therefore, the total energy due to the presence of clusters in the ionomer is given by Utot = n(Uele + Uela + Usur ) , a)

γβ η, 2RT

(8.4)

(c) t = 4m 45s

Figure 8.5 shows the variation of the cluster size (aI3 in nm3 ) for different solvent uptakes. Data from Nafion-based IPMC samples with different cations and different solvent uptakes are considered. The model is compared with the experimental results on the cluster size based on x-ray scanning, shown as circles

b)

c) –

(d) t = 6 m 27s –

R

+

t = 3ms t = 210ms

+

t = 5ms

t= 129ms

1cm

(a) t = 0 (b) t = 0.6 s

t = 10ms

t = 54ms t = 36ms

t = 15ms t = 24ms

Fig. 8.6 (a) A Nafion-based sample in the thallium (I) ion form is hydrated and a 1 V DC signal is suddenly applied

Part A 8.1

and maintained during the first 5 min, after which the voltage is removed and the two electrodes are shorted. Initial fast bend toward the bottom ((a) to (b), anode) occurs during the first 0.6 s, followed by a long relaxation upward (towards the cathode (c)) over 4.75 min. Upon shorting, the sample displays a fast bend in the same upward direction (not shown), followed by a slow downward relaxation (to (d)) during the next 1.75 min. (b) A Nafion-based sample in the sodium ion form is solvated with glycerol, and a 2 V DC signal is suddenly applied and maintained. It deforms into a perfect circle, but its qualitative response is the same. (c) A Flemion-based sample in tetrabutylammonium ion form is hydrated, and a 3 V DC signal is suddenly applied and maintained, resulting in continuous bending towards the anode (no back relaxation)

Electrochemomechanics of Ionic Polymer–Metal Composites

in Fig. 8.5 for a Nafion ionomer in various cation forms and with water as the solvent [8.10]. We have set β = 1.547, γ = 0.15, and Vi = 68 × 10−24 cm3 to calculate the cluster size [8.12].

8.1.3 Actuation A Nafion-based IPMC sample in the solvated state performs an oscillatory bending motion when an alternating voltage is imposed across its faces, and it produces a voltage when suddenly bent. When the same strip is subjected to a suddenly imposed and sustained constant voltage (DC) across its faces, an initial fast displacement (towards the anode) is gen-

8.2 Stiffness Versus Solvation

191

erally followed by a slow relaxation in the reverse direction (towards the cathode). If the two faces of the strip are then shorted during this slow relaxation towards the cathode, a sudden fast motion in the same direction (towards the cathode) occurs, followed by a slow relaxation in the opposite direction (towards the anode). Figure 8.6 illustrates these processes for a hydrated Tl+ -form Nafion-based IPMC (left), Na+ -form with glycerol as solvent (middle), and hydrated Flemion-based in TBA+ -form (right), under 1, 2, and 3 V DC, respectively. The magnitudes of the fast motion and the relaxation that follows the fast motion change with the corresponding cation and the solvent.

8.2 Stiffness Versus Solvation To model the actuation of the IPMC samples in terms of the chemoelectromechanical characteristics of the backbone ionomer, the electrodes, the neutralizing cation, the solvent, and the level of solvation, it is first necessary to model the stiffness of the corresponding samples. This is discussed in the present section. A dry sample of a bare polymer or an IPMC placed in a solvent bath absorbs solvent until the resulting pressure within its clusters is balanced by the elastic stresses that are consequently developed within its backbone polymer membrane. From this observation the stiffness of the membrane can be estimated as a function of the solvent uptake for various cations. Consider first the balance of the cluster pressure and the elastic stresses for the bare polymer (no metal plating) and then use the results to calculate the stiffness of the corresponding IPMC by including the effect of the added metal electrodes. The procedure also provides a way of estimating many of the nanostructural parameters that are needed for the modeling of the actuation of the IPMCs.

8.2.1 The Stress Field in the Backbone Polymer

σr (r0 ) = − p0 + K [(r0 /a0 )−3 (w/w0 − 1) + 1]−4/3 , σθ (r0 ) = σϕ (r0 ) = − p0 + K [(r0 /a0 )−3 (w/w0 − 1) + 1]2/3 ,

(8.7)

where r0 measures the initial radial length from the center of the cluster, and w0 is the initial (dry) volume fraction of the voids. Theeffective elastic resistance of the (homogenized solvated) membrane balances the cluster’s pressure pc which is produced by the combined osmotic and electrostatic forces within the cluster.

Part A 8.2

The stresses within the backbone polymer may be estimated by modeling the polymer matrix as an incompressible elastic material [8.13, 14]. Here, it will prove adequate to consider a neo-Hookean model for the matrix material. In this model, the principal stresses σI are related to the principal stretches λI by σI = − p0 + K λ2I ,

where p0 is an undetermined parameter (pressure) to be calculated from the boundary data and K is an effective stiffness, approximately equal to a third of the overall Young’s modulus. The aim is to calculate K and p0 as functions of the solvent uptake w for various ionform membranes. To this end, examine the deformation of a unit cell of the solvated polyelectrolyte (bare membrane) by considering a spherical cavity of initial (i. e., dry state) radius a0 (representing a typical cluster), embedded at the center of a spherical matrix of initial radius R0 , and placed in a homogenized solvated membrane, referred to as the matrix. In micromechanics, this is called the double-inclusion model [8.15]. Assume that the stiffness of both the spherical shell and the homogenized matrix is the same as that of the (as yet unknown) overall effective stiffness of the hydrated membrane. For an isotropic expansion of a typical cluster, the two hoop stretches (and stresses) are equal, and using the incompressibility condition and spherical coordinates, it follows that

192

Part A

Solid Mechanics Topics

8.2.2 Pressure in Clusters For the solvated bare membrane or an IPMC in the M+ -ion form, and in the absence of an applied electric field, the pressure within each cluster pc consists of osmotic Π(M+ ) and electrostatic pDD components. The electrostatic component is produced by the ionic interaction within the cluster. The cation–anion conjugate pairs can be represented as uniformly distributed dipoles on the surface of a spherical cluster, and the resulting dipole–dipole (DD) interaction forces pDD calculated. The osmotic pressure is calculated by examining the difference between the chemical potential of the free (bath) solvent and that of the solvent within a typical cluster of known ion concentration within the membrane. In this manner, one obtains [8.16] pc =

νQ − B K0φ

w RT K0 = , F

±α2

1 −2 Q , 3κe B w2 ρB F , Q− B = EWion +

by the volume of solid), respectively. The membrane Young’s modulus may now be set equal to YB = 3K (w), assuming that both the elastomer and solvent are essentially incompressible under the involved conditions.

8.2.4 IPMC Stiffness To include the effect of the metal plating on the stiffness, note that for the Nafion-based IPMCs the gold plating is about a 1 μm layer on both faces of an IPMC strip, while platinum particles are distributed through the first 10–20 μm surface regions, with diminishing density. Assume a uniaxial stress state and average the axial strain and stress over the strip’s volume to obtain their average values, as ε¯ IPMC = f MH ε¯ B + (1 − f MH )¯εM , σ¯ IPMC = f MH σ¯ B + (1 − f MH )σ¯ M ,

f MH =

fM , 1+w (8.10)

(8.8)

where φ is the practical osmotic coefficient, α is the effective length of the dipole, F is the Faraday constant, ν is the cation–anion valence (ν = 2 for monovalent cations), R = 8.314 J/mol/K is the universal gas constant, T is the cluster temperature, ρB is the density of the bare ionomer, and κe is the effective permittivity.

where the barred quantities are the average values of the axial strain and stress in the IPMC, bare (solvated) membrane, and metal electrodes, respectively (indicated by the corresponding subscripts ‘B’ and ‘M’, respectively); and f M is the volume fraction of the metal plating in a dry sample. Setting σ¯ B = YB ε¯ B , σ¯ M = YM ε¯ M , and σ¯ IPMC = Y¯IPMC ε¯ IPMC , it follows that YM YB , BAB YM + (1 − BAB )YB (1 + w)(1 − fM) ¯ B= , w = w(1 ¯ − fM) , 1 + w(1 − fM) ¯

Y¯IPMC =

8.2.3 Membrane Stiffness The radial stress σr must equal the pressure pc in the cluster at r0 = a0 . In addition, the volume average of the stress tensor, taken over the entire membrane, must vanish in the absence of any externally applied loads. This is a consistency condition that to a degree accounts for the interaction among clusters. These conditions are sufficient to yield the undetermined pressure p0 and the stiffness K in terms of solvation volume fraction w and the initial dry void volume fraction w0 for each ion-form bare membrane, leading to the following final closed-form results: 1+w , w0 In − (w0 /w)4/3 1 + 2An 0 1 + 2A w − , A= −1 , In = w0 n 0 (1 + An 0 )1/3 (1 + A)1/3 K = pc

Part A 8.2

(8.9)

where n 0 and w0 = n 0 /(1 − n 0 ) are the initial (dry) porosity and initial void ratio (volume of voids divided

(8.11)

where AB is the concentration factor, giving the average stress in the bare polymer in terms of the average stress of the IPMC σ¯ IPMC . Here YB = 3K is evaluated from (8.9) at solvation w ¯ when the solvation of the IPMC is w. The latter can be measured directly at various solvation levels. Experimentally, the dry and solvated dimensions (thickness, width, and length) and masses of each bare or plated sample are measured. Knowing the composition of the ionomer, the neutralizing cation, and the composition of the electrodes, all physical quantities in equations (8.8) and (8.11) are then known for each sample, except for the three parameters α, κe , and AB , namely the effective distance between the neutralizing cation and its conjugate covalently fixed anion, the effective electric permittivity of the cluster, and the concentration factor that defines the fraction of axial stress

Electrochemomechanics of Ionic Polymer–Metal Composites

carried by the bare elastomer in the IPMC. The parameters α and κe are functions of the solvation level. They play critical roles in controlling the electrostatic and chemical interaction forces within the clusters, as will be shown later in this chapter in connection with the IPMC’s actuation. Thus, they must be evaluated with care and with due regard for the physics of the process. Estimate of κe and α The solvents are polar molecules, carrying an electrostatic dipole. Water, for example, has a dipole moment of about 1.87 D (Debyes) in the gaseous state and about 2.42 D in bulk at room temperature, the difference being due to the electrostatic pull of other water molecules. Because of this, water forms a primary and a secondary hydration shell around a charged ion. Its dielectric constant as a hydration shell of an ion (about 6) is thus much smaller than that in bulk (about 78). The second term in expression (8.8) for the cluster pressure pc , can change by a factor of 13 for water as the solvent, depending on whether the water molecules are free or constrained to a hydration shell. Similar comments apply to other solvents. For example, for glycerol and ethylene glycol, the room-temperature dielectric constants are about 9 and 8 as solvation shells, and 46 and 41 when in bulk. When the cluster contains both free and ion-bound solvent molecules, its effective electric permittivity can be estimated using a micromechanical model. Let κ1 = ε1 κ0 and κ2 = ε2 κ0 be the electric permittivity of the solvent in a solvation shell and in bulk, respectively, where κ0 = 8.85 × 10−12 F/m is the electric permittivity of free space. Then, using a double-inclusion

8.3 Voltage-Induced Cation Distribution

193

model [8.15] it can be shown that κ2 + 2κ1 + f (κ2 − κ1 ) κ1 , κe = κ2 + 2κ1 − f (κ2 − κ1 ) m w − CN EWion w f = , mw = (8.12) . mw Msolvent ρB ν Here, CN is the static solvation shell (equal to the coordination number), m w is the number of moles of solvent per mole of ion (cation and anion), and ν = 2. When m w is less than CN, then all solvent molecules are part of the solvation shell, for which κe = ε1 κ0 . On the other hand, when there are more solvent molecules, equation (8.12) yields the corresponding value of κe . Thus, κe is calculated in a cluster in terms of the cluster’s volume fraction of solvent, w. Similarly, the dipole arm α in (8.8), which measures the average distance between a conjugate pair of anion– cation, is expected to depend on the effective dielectric constant of the solvation medium. We now calculate the parameter ±α2 , as follows. As a first approximation, we let α2 vary linearly with w for m w ≤ CN, i. e., we set ± α 2 = a1 w + a2 ,

for

m w ≤ CN ,

(8.13)

and estimate the coefficients a1 and a2 from the experimental data. For m w > CN, furthermore, we assume that the distance between the two charges forming a pseudodipole is controlled by the effective electric permittivity of the their environment (e.g., water molecules), and hence is given by 2 κe (a1 w + a2 ) . (8.14) α2 = 10−20 κ1 Note that for m w > CN, we have a1 w + a2 > 0. An illustrative example is given in Sect. 8.5 where measured results are compared with model predictions.

8.3 Voltage-Induced Cation Distribution terized by [8.17–19], Ci Di ∂μi (8.15) + Ci υi , RT ∂x where Di is the diffusivity coefficient, μi is the chemical potential, Ci is the concentration, and υi is the velocity of species i. The chemical potential is given by Ji = −

μi = μ0 + RT ln(γi Ci ) + z i φF ,

(8.16)

where μ0 is the reference chemical potential, γi is the affinity of species i, z i is the species charge, and

Part A 8.3

When an IPMC strip in a solvated state is subjected to an electric field, the initially uniform distribution of its neutralizing cations is disturbed, as cations on the anode side are driven out of the anode boundary clusters while the clusters in the cathode side are supplied with additional cations. This redistribution of cations under an applied potential can be modeled using the coupled electrochemical equations that characterize the net flux of the species, caused by the electrochemical potentials (chemical concentration and electric field gradients). The total flux consists of cation migration and solvent transport. The flux Ji of species i is charac-

194

Part A

Solid Mechanics Topics

φ = φ(x, t) is the electric potential. For an ideal solution where γi = 1, and if there is only one type of cation, the subscript i may be dropped, as will be done in what follows. The variation in the electric potential field in the membrane is governed by the basic Poisson’s electrostatic equations [8.20, 21], ∂(κ E) = z(C − C − )F , ∂x

E=−

∂φ , ∂x

(8.17)

where E and κ are the electric field and the electric permittivity, respectively; C − is the anion concentration in moles per unit solvated volume; and C = C(x, t) is the total ion (cation and anion) concentration. Since the solvent velocity is very small, the last term in (8.15) may be neglected and, in view of (8.17) and continuity, it follows that

∂C(x, t) J(x, t) = −D ∂x F − zC(x, t) E(x, t) , RT ∂C(x, t) ∂J(x, t) =− , ∂t ∂x ∂E(x, t) F (8.18) =z (C(x, t) − C − ) . ∂x κ¯ Here κ¯ is the overall electric permittivity of the solvated IPMC sample that can be estimated from its measured effective capacitance. The above system of equations can be directly solved numerically, or they can be solved analytically using approximations. In the following sections, the analytic solution is considered. First, from (8.18) it follows that

∂ ∂(κ E) −D ∂x ∂t

∂ 2 (κ E) ∂x 2

−

C− F2 κ RT

(κ E)

=0. (8.19)

Part A 8.3

This equation provides a natural length scale and a natural time scale τ that characterize the ion redistribution, 2 κ¯ RT 1/2 , τ= (8.20) , = − 2 D C F where κ¯ is the overall electric permittivity of the IPMC. If Cap is the measured overall capacitance, then we set √ κ¯ = 2HCap. Since is linear in κ¯ and κ¯ is proportional to the capacitance, it follows that is proportional to the square root of the capacitance.

8.3.1 Equilibrium Cation Distribution To calculate the ion redistribution caused by the application of a step voltage across the faces of a hydrated strip of IPMC, we first examine the time-independent equilibrium case with J = 0. In the cation-depleted (anode) boundary layer the charge density is −C − F, whereas in the remaining part of the membrane the charge density is (C + − C − )F. Let the thickness of the cation-depleted zone be denoted by and set Q(x, t) =

C+ − C− , C−

Q 0 (x) = lim Q(x, t) . t→∞

(8.21)

Then, it follows from (8.17) and boundary and continuity conditions that [8.16] the equilibrium distribution is given by ⎧ ⎪ ⎪ ⎨−1 for x ≤ −h + Q 0 (x) ≡

F RT [B0 exp(x/) − B1 exp(−x/)] , ⎪ ⎪ ⎩ for − h + < x < h ,

(8.22)

2φ0 F B1 = K 0 exp(−a ) , = −2 , RT

2 φ0 1 B2 = − K0 +1 +1 , 2 2 B0 = exp(−a) φ0 /2 + B1 exp(−a) + B2 , F K0 = (8.23) , RT where φ0 is the applied potential, a ≡ h/, and a ≡ (h − )/. Since is only 0.5–3 μm, a ≡ h/, a ≡ h / 1, and hence exp(−a) ≈ 0 and exp(−a ) ≈ 0. The constants B0 and B1 are very small, of the order of 10−17 or even smaller, depending on the value of the capacitance. Therefore, the approximation used to arrive at (8.22) does not compromise the accuracy of the results. Remarkably, the estimated length of the anode boundary layer with charge √constant negative density −C − F, i. e., = 2φ0 F/RT − 2 , depends only on the applied potential and the effective capacitance through the characteristic length. We use as our length scale. From (8.23) it can be concluded that, over the most central part of the membrane, there is charge neutrality within all clusters, and the charge imbalance in clusters occurs only over narrow boundary layers, with the anode boundary layer being thicker

Electrochemomechanics of Ionic Polymer–Metal Composites

than the cathode boundary layer. The charge imbalance in the clusters is balanced by the corresponding electrode charges. Therefore, the charge imbalance defined by (8.23) applies to the clusters within each boundary layer and not to the boundary layer itself, since the combined clusters and the charged metal particles within the boundary layers are electrically balanced. We now examine the time variation of the charge distribution that results upon the application of a step voltage and leads to the equilibrium solution given above. To this end, rewrite equation (8.19) as 2 ∂ Q Q ∂Q − 2 =0, (8.24) −D ∂t ∂x 2 and set Q = ψ(x, t) exp(−t/τ) + Q 0 (x) to obtain the following standard diffusion equation for ψ(x, t): ∂ψ ∂2ψ =D 2 , ∂t ∂x

ψ(x, 0) = −Q 0 (x) ,

(8.25)

from which it can easily be concluded that, to a good degree of accuracy, one may use the following simple approximation in place of an exact infinite series

8.4 Nanomechanics of Actuation

195

solution [8.16]: Q(x, t) ≈ g(t)Q 0 (x) , g(t) = 1 − exp(−t/τ) . (8.26) Actually, when the electric potential is suddenly applied and then maintained, say, at a constant level φ0 , the cations of the anode clusters are initially depleted at the same rate as cations being added to the clusters within the cathode boundary layer. Thus, initially, the cation distribution is antisymmetric through the thickness of the membrane, as shown by Nemat-Nasser and Li [8.22], C(x) = C − +

κφ ¯ 0 sinh(x/) . 2F2 sinh(h/)

(8.27)

After a certain time period, say, τ1 , clusters near the anode face are totally depleted of their cations, so only the length of the anode boundary layer can further increase, while the cathode clusters continue to receive additional cations. Recognizing this fact, recently Nemat-Nasser and Zamani [8.23] have used one time scale for the first event and another time scale for the remaining process of cation redistribution. To simulate the actuation, however, these authors continue to use the one-time-scale approximation similar to (8.26).

8.4 Nanomechanics of Actuation changes in the pressure within the clusters in the anode and cathode boundary layers, and the resulting diffusive flow of solvent into or out of the corresponding clusters.

8.4.1 Cluster Pressure Change Due to Cation Migration The elastic pressure on a cluster can be calculated from (8.7) by simply setting r0 = aI , where aI is the initial cluster size just before the potential is applied, and using the spatially and temporally changing volume fraction of solvent and ion concentration w(x, t) and ν(x, t), since the osmotic pressure depends on the cluster’s ion concentration and it is reduced in the anode and is increased in the cathode clusters. The corresponding osmotic pressure can be computed from Π(x, t) =

φQ − C + (x, t) B K0 +1 , ν(x, t) , ν(x, t) = w(x, t) C− (8.28)

C + (x, t)

where is the cation concentration; note from (8.21) that ν(x, t) = Q(x, t) + 2. The pressure pro-

Part A 8.4

The application of an electric potential produces two thin boundary layers, one near the anode and the other near the cathode electrodes, while maintaining overall electric neutrality in the IPMC strip. The cation imbalance within the clusters of each boundary layer changes the osmotic, electrostatic, and elastic forces that tend to expand or contract the corresponding clusters, forcing the solvents out of or into the clusters, and producing the bending motion of the sample. Thus, the volume fraction of the solvent within each boundary layer is controlled by the effective pressure in the corresponding clusters produced by the osmotic, electrostatic, and elastic forces. These forces can even cause the cathode boundary layer to contract during the back relaxation that is observed for Nafion-based IPMCs in various alkali metal forms, expelling the extra solvents onto the IPMC’s surface while cations continue to accumulate within the cathode boundary layer. This, in fact, is what has been observed in open air during the very slow back relaxation of IPMCs that are solvated with ethylene glycol or glycerol [8.24]. To predict this and other actuation responses of IPMCs, it is thus necessary to model the

196

Part A

Solid Mechanics Topics

duced by the electrostatic forces among interacting fixed anions and mobile cations within each cluster varies as the cations move into or out of the cluster in response to the imposed electric field. As the cations of the clusters within the anode boundary layer are depleted, the dipole–dipole interaction forces diminish. We model this in the anode boundary layer by calculating the resulting pressure, pADD , as a function of the cation concentration, as follows: 2 + 1 C (x, t) −2 α(x, t) pADD (x, t) = . Q 3κ(x, t) B w(x, t) C− (8.29)

Parallel with the reduction in the dipole–dipole interaction forces is the development of electrostatic interaction forces among the remaining fixed anions, which introduces an additional pressure, say pAA , 1 2 pAA (x, t) = Q− B 18κ(x, t) R02 C + (x, t) × 1− , (8.30) C− [w(x, t)]4/3 where R0 is the initial (unsolvated, dry) cluster size. This expression is obtained by considering a spherical cluster with uniformly distributed anion charges on its surface. The total pressure within a typical anode boundary layer cluster hence is tA (x, t) = −σr (a0 , t) + Π(x, t) + pAA (x, t) + pADD (x, t) .

(8.31)

Part A 8.4

Consider now the clusters in the cathode boundary layer. In these clusters, in addition to the osmotic pressure we identify two forms of electrostatic interaction forces. One is repulsion due to the cation–anion pseudodipoles already present in the clusters, and the other is due to the extra cations that migrate into the clusters and interact with the existing pseudo-dipoles. The additional stresses produced by this latter effect may tend to expand or contract the clusters, depending on the distribution of cations relative to fixed anions. We again model each effect separately, although in fact they are coupled. The dipole–dipole interaction pressure in the cathode boundary layer clusters is assumed to be reduced as 1 α(x, t) 2 2 Q− pCDD (x, t) = 3κ(x, t) B w(x, t) C− × (8.32) , C + (x, t)

while at the same time new dipole–cation interaction forces are being developed as additional cations enter the clusters and disturb the pseudo-dipole structure in the clusters. The pressure due to these latter forces is represented by pDC (x, t) =

2 2Q − R0 α(x, t) C(x, t) B − 1 . 9κ(x, t) [w(x, t)]5/3 C− (8.33)

However, for sulfonates in a Nafion-based IPMC, we expect extensive restructuring and redistribution of the cations. It appears that this process underpins the observed reverse relaxation of the Nafion-based IPMC strip. Indeed, this redistribution of the cations within the clusters in the cathode boundary layer may quickly diminish the value of pCD to zero or even render it negative. To represent this, we modify (8.33) by a relaxation factor, and write 2Q − R0 αC (x, t) B pDC (x, t) ≈ 9κC (x, t) [wC (x, t)]5/3 + C (x, t) × − 1 g1 (t) , C− g1 (t) = [r0 + (1 − r0 ) exp(−t/τ1 )], 2

r0 < 1 , (8.34)

where τ1 is the relaxation time and r0 is the equilibrium fraction of the dipole–cation interaction forces. The total stress in clusters within the cathode boundary layer is now approximated by tC = −σr (a0 , t) + ΠC (x, t) + pCDD (x, t) + pDC (x, t) . (8.35)

8.4.2 Cluster Solvent Uptake Due to Cation Migration The pressure change in clusters within the anode and cathode boundary layers drives solvents into or out of these clusters. This is a diffusive process that can be modeled using the continuity equation ∂υ x, t w ˙ x, t + (8.36) =0, ∂x 1 + w x, t and a constitutive model to relate the gradient of the solvent velocity, ∂υ(x, t)/∂x, to the cluster pressure. We may use a linear relation and obtain w ˙ i x, t = DBL ti x, t , i = A, C , (8.37) 1 + wi x, t

Electrochemomechanics of Ionic Polymer–Metal Composites

where DBL is the boundary-layer diffusion coefficient, assumed to be constant here.

8.4.3 Voltage-Induced Actuation As is seen from Fig. 8.6, an IPMC strip can undergo large deflections under an applied electric potential. Since the membrane is rather thin (0.2 mm) even if it deforms from a straight configuration into a circle of radius, say 1 cm still the radius-to-thickness ratio would be 50, suggesting that the maximum axial strain in the membrane is very small. It is thus reasonable to use a linear strain distribution through the thickness and use the following classical expression for the maximum strain: εmax ≈ ±

H , Rb

(8.38)

8.5 Experimental Verification

197

where Rb is the radius of the curvature of the membrane and H is half of its thickness. The strain in the membrane is due to the volumetric changes that occur within the boundary layers, and can be estimated from εv (x, t) = ln[1 + w(x, t)] .

(8.39)

We assume that the axial strain is one-third of the volumetric strain, integrate over the thickness, and obtain L L = Rb 2H 3 (3Y IPMC − 2YB ) h × YBL (w(x, t))x ln[1 + w(x, t)] dx , (8.40) −h

where all quantities have been defined before and are measurable.

8.5 Experimental Verification We now examine some of the experimental results that have been used to characterize this material and to verify the model results.

8.5.1 Evaluation of Basic Physical Properties Both the bare ionomer and the corresponding IPMC can undergo large dimensional changes when solvated. The phenomenon is also affected by the nature of the neutralizing cation and the solvent. This, in turn, substantially changes the stiffness of the material. Therefore, techniques have been developed to measure the dimensions and the stiffness of the ma-

terial in various cation forms and at various solvation stages. The reader is referred to Nemat-Nasser and Thomas [8.25, 26], Nemat-Nasser and Wu [8.27], and Nemat-Nasser and Zamani [8.23] for the details of various measurements and extensive experimental results. In what follows, a brief account of some of the essential features is presented. One of the most important quantities that characterize the ionic polymers is the material’s equivalent weight (EW), defined as dry mass in grams of ionic polymer in proton form divided by the moles of sulfonate (or carboxylate) groups in the polymer. It is expressed in grams per mole, and is measured by neutralizing the same sample sequentially with deferent

Table 8.1 Measured stiffness of dry and hydrated Nafion/Flemion ionomers and IPMCs in various cation forms Thickness

⎧ ⎨ Bare

Nafion

⎧ ⎨ Flemion IPMC

⎩

Nafion-based Flemion-based

K+ Cs+ Na+ K+ Cs+ Cs+ Cs+

Stiffness

Thickness

Water-saturated form Density Stiffness

(μm)

(g/cm3 )

(MPa)

(μm)

(g/cm3 )

(MPa)

Hydration volume (%)

182.1 178.2 189.1 149.4 148.4 150.7 156.0 148.7

2.008 2.065 2.156 2.021 2.041 2.186 3.096 3.148

1432.1 1555.9 1472.2 2396.0 2461.2 1799.2 1539.5 2637.3

219.6 207.6 210.5 167.6 163.0 184.2 195.7 184.1

1.633 1.722 1.836 1.757 1.816 1.759 2.500 2.413

80.5 124.4 163.6 168.6 199.5 150.6 140.4 319.0

71.3 50.0 41.4 42.0 34.7 53.7 54.1 58.1

Part A 8.5

⎩

Na+

Dry form Density

198

Part A

Solid Mechanics Topics

Part A 8.5

cations, and measuring the changes in the weight of the sample. This change is directly related to the number of anion sites within the sample and the difference in the atomic weight of the cations. Another important parameter is the solvent uptake w, defined as the volume of the solvent divided by the volume of the dry sample. Finally, it is necessary to measure the weight fraction of the metal in an IPMC sample, which is usually about 40%. The uniaxial stiffness of both the bare ionomer and the corresponding IPMC changes with the solvent uptake. It is also a function of the cation form, the solvent, and the backbone material. Table 8.1 gives some data on bare and metal-plated Nafion and Flemion in the indicated cation forms. Note that the stiffness can vary by a factor of ten for the same sample depending on whether it is dry or fully hydrated. Surface conductivity is an important electrical property governing an IPMC’s actuation behavior. When applying a potential across the sample’s thickness at the grip end, the bending of the cantilever is affected by its surface resistance, which in turn is dependent on the electrode morphology, cation form, and the level of solvation. An applied electric field affects the cation distribution within an IPMC membrane, forcing the cations to migrate towards the cathode. This change in the cation distribution produces two thin layers, one near the anode and the other near the cathode boundaries. In time, and once an equilibrium state is attained, the anode boundary layer is essentially depleted of its cations, while the cathode boundary layer has become cation rich. If the applied constant electric potential is V and the corresponding total charge that is accumulated within the cathode boundary layer once the equilibrium state is attained is Q, then the effective electric capacitance of the IPMC is defined as C = Q/V . From this, one obtains the corresponding areal capacitance, measured in mF/cm2 , by dividing by the area of the sample. Usually, the total equilibrium accumulated charge can be calculated by time integration of the measured net current, and using the actual dimensions of the saturated sample. For alkali-metal cations, one may have capacitance of 1–50 mF/cm2 for an IPMC.

8.5.2 Experimental Verification As an illustration of the model verification, consider first the measured and modeled stiffness of the bare and metal-plated Nafion-based samples (Fig. 8.7). Here, the

Stiffness (MPa) 2500 Experimental_bare Naflon Experimental_IPMC (Sh2) Experimental_IPMC (Sh5) Model_bare Naflon Model_Naflon IPMC

2000 1500 1000 500 0

0

10

20

30

40

50

60

70

80

Hydration (%)

Fig. 8.7 Uniaxial stiffness (Young’s modulus) of bare Nafion-117 (lower data points and the solid curve, model) and an IPMC (upper data points and solid curve) in the Na+ -form versus hydration water

lower data points and the solid curve are for the bare ionomer in Na+ -form, and the upper data points and the solid curve are for the corresponding IPMC. The model results have been obtained for the bare Nafion using (8.9) and for the IPMC from (8.11). In (8.9), φ is taken to be 1 and n 0 to be 1%. The dry density of the bare membrane is measured to be 2.02 g/cm3 , the equivalent weight for Nafion-117 in Na+ -form (with 23 atomic weight for sodium) is calculated to be 1122, the electric permittivity is calculated from (8.12) with CN = 4.5, the temperature is taken to be 300 K (room temperature), and κe is calculated using (8.12), with α2 Tip displacement/gauge length 0.12 Experiment Model

0.08 0.04 0 –0.04 –0.08 –0.12 –0.16 –10

0

10

20

30

40

50

60

Time (s)

Fig. 8.8 Tip displacement of a 15 mm cantilevered strip of a Nafion-based IPMC in Na+ -form, subjected to a 1.5 V step potential for about 32 s, then shorted; the solid curve is the model and the geometric symbols are experimental points

Electrochemomechanics of Ionic Polymer–Metal Composites

being calculated from (8.13) or (8.14) depending on the level of hydration, i. e., the value of w. The only free parameters are then a1 and a2 , which are calculated to be a1 = 1.5234 × 10−20 and a2 = −0.0703 × 10−20 , using two data points. In (8.11), f M is measured to be 0.0625, YM is 75 GPa, and AB (the only free parameter) is set equal to 0.5. To check the model prediction of the observed actuation response of this Nafion-based IPMC, consider a hydrated cantilevered strip subjected to a 1.5 V step potential across its faces that is maintained for 32 s and then shorted. Figure 8.8 shows (geometric symbols) the measured tip displacement of a 15 mm-long cantilever strip that is actuated by applying a 1.5 V step potential across its faces, maintaining the voltage for about 32 s and then removing the voltage while the two faces are shorted. The initial water uptake is wIPMC = 0.46, and the volume fraction of metal plating is 0.0625. Hence, the initial volume fraction of water in the Nafion part of the IPMC is given by wI = wIPMC /(1 − f M ) = 0.49. The formula weight of sodium is 23, and the dry density of the bare membrane is 2.02 g/cm3 . With EWNa+ = 1122 g/mol, the initial value of C − for the

References

199

bare Nafion becomes 1208. The thickness of the hydrated strip is measured to be 2H = 224 μm, and based on inspection of the microstructure of the electrodes, we set 2h ≈ 212 μm. The effective length of the anode boundary layer for φ0 = 1.5 V is L A = 9.78, and in order to simplify the model calculation of the solvent flow into or out of the cathode clusters, an equivalent uniform boundary layer is used near the cathode whose thickness is then estimated to be L C = 2.84, where = 0.862 μm for φ0 = 1.5 V. The electric permittivity and the dipole length are calculated using a1 = 1.5234 × 10−20 and a2 = −0.0703 × 10−20 , which are the same as for the stiffness modeling. The measured capacitance ranges from 10 to 20 mF/cm2 , and we use 15 mF/cm2 . Other actuation model parameters are: DA = 10−2 (when pressure is measured in MPa), τ = 1/4 and τ1 = 4 (both in seconds), and r0 = 0.25. Since wI ≈ (a/R0 )3 , where a is the cluster size at wa−1/3 ter uptake wI , we set R0 ≈ w0 a, and adjust a to fit the experimental data; here, a = 1.65 nm, or an average cluster size of 3.3 nm, prior to the application of the potential. The fraction of cations that are left after shorting is adjusted to r = 0.03.

8.6 Potential Applications tors in space-based applications [8.2, 37], as their lack of multiple moving parts is ideal for any environment where maintenance is difficult. Besides these, IPMCs find applications in other disciplines, including fuel-cell membranes, electrochemical sensing [8.38], and electrosynthesis [8.2, 37, 39], as their lack of multiple moving parts is ideal for any environment where maintenance is difficult. Besides these, IPMCs find applications in other disciplines, including fuelcell membranes, electrochemical sensing [8.38], and electrosynthesis [8.39]. The commercial availability of Nafion has enabled alternate uses for these materials to be found. Growth and maturity in the field of IPMC actuators will require exploration beyond these commercially available polymer membranes, as the applications for which they have been designed are not necessarily optimal for IPMC actuators.

References 8.1

P. Millet: Noble metal-membrane composites for electrochemical applications, J. Chem. Ed. 76(1), 47– 49 (1999)

8.2

Y. Bar-Cohen, T. Xue, M. Shahinpoor, J.O. Simpson, J. Smith: Low-mass muscle actuators using elec-

Part A 8

The ultimate success of IPMC materials depends on their applications. Despite the limited force and frequency that IPMC materials can offer, a number of applications have been proposed which take advantage of IPMCs’ bending response, low voltage/power requirements, small and compact design, lack of moving parts, and relative insensitivity to damage. Osada and coworkers have described a number of potential applications for IPMCs, including catheters [8.28, 29], elliptic friction drive elements [8.30], and ratchet-and-pawl-based motile species [8.31]. IPMCs are being considered for applications to mimic biological muscles; Caldwell has investigated artificial muscle actuators [8.32, 33] and Shahinpoor has suggested applications ranging from peristaltic pumps [8.34] and devices for augmenting human muscles [8.35] to robotic fish [8.36]. Bar-Cohen and others have discussed the use of IPMC actua-

200

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8.3

8.4

8.5

8.6

8.7

8.8

8.9

8.10

8.11

8.12

8.13 8.14 8.15

8.16

8.17

Part A 8

8.18 8.19

troactive polymers (EAP), Proc. SPIE. 3324, 218–223 (1998) Y. Bar-Cohen, S. Leary, M. Shahinpoor, J.O. Harrison, J. Smith: Electro-active polymer (EAP) actuators for planetary applications, SPIE Conference on Electroactive Polymer Actuators, Proc. SPIE. 3669, 57–63 (1999) R. Liu, W.H. Her, P.S. Fedkiw: In situ electrode formation on a Nafion membrane by chemical platinization, J. Electrochem. Soc. 139(1), 15–23 (1992) M. Homma, Y. Nakano: Development of electrodriven polymer gel/platinum composite membranes, Kagaku Kogaku Ronbunshu 25(6), 1010–1014 (1999) T. Rashid, M. Shahinpoor: Force optimization of ionic polymeric platinum composite artificial muscles by means of an orthogonal array manufacturing method, Proc. SPIE 3669, 289–298 (1999) M. Bennett, D.J. Leo: Manufacture and characterization of ionic polymer transducers with non-precious metal electrodes, Smart Mater. Struct. 12(3), 424–436 (2003) C. Heitner-Wirguin: Recent advances in perfluorinated ionomer membranes – structure, properties and applications, J. Membr. Sci. 120(1), 1–33 (1996) R.E. Fernandez: Perfluorinated ionomers. In: Polymer Data Handbook, ed. by J.E. Mark (Oxford Univ. Press, New York 1999) pp. 233–238 T.D. Gierke, C.E. Munn, P.N. Walmsley: The morphology in Nafion perfluorinated membrane products, as determined by wide- and small-angle X-ray studies, J. Polym. Sci. Polym. Phys. Ed. 19, 1687–1704 (1981) J.Y. Li, S. Nemat-Nasser: Micromechanical analysis of ionic clustering in Nafion perfluorinated membrane, Mech. Mater. 32(5), 303–314 (2000) W.A. Forsman: Statistical mechanics of ion-pair association in ionomers, Proc. NATO Adv. Workshop Struct. Properties Ionomers (1986) pp. 39–50 L.R.G. Treolar: Physics of Rubber Elasticity (Oxford Univ. Press, Oxford 1958) R.J. Atkin, N. Fox: An Introduction to the Theory of Elasticity (Longman, London 1980) S. Nemat-Nasser, M. Hori: Micromechanics: Overall Properties of Heterogeneous Materials, 1st edn. (North-Holland, Amsterdam 1993) S. Nemat-Nasser: Micromechanics of actuation of ionic polymer-metal composites, J. Appl. Phys. 92(5), 2899–2915 (2002) J. O’M Bockris, A.K.N. Reddy: Modern Electrochemistry 1: Ionics, Vol. 1 (Plenum, New York 1998) P. Shewmon: Diffusion in Solids, 2nd edn. (The Minerals Metals & Materials Society, Warrendale 1989) M. Eikerling, Y.I. Kharkats, A.A. Kornyshev, Y.M. Volfkovich: Phenomenological theory of electro-osmotic effect and water management in polymer electrolyte proton-conducting membranes, J. Electrochem. Soc. 145(8), 2684–2699 (1998)

8.20 8.21 8.22

8.23

8.24

8.25

8.26

8.27

8.28

8.29

8.30

8.31

8.32

8.33 8.34

8.35

8.36

J.D. Jackson: Classical Electrodynamics (Wiley, New York 1962) W.B. Cheston: Elementary Theory of Electric and Magnetic Fields (Wiley, New York 1964) S. Nemat-Nasser, J.Y. Li: Electromechanical response of ionic polymer-metal composites, J. Appl. Phys. 87(7), 3321–3331 (2000) S. Nemat-Nasser, S. Zamani: Modeling of electrochemo-mechanical response of ionic-polymermetal composites with various solvents, J. Appl. Phys. 38, 203–219 (2006) S. Nemat-Nasser, S. Zamani, Y. Tor: Effect of solvents on the chemical and physical properties of ionic polymer-metal composites, J. Appl. Phys. 99, 104902 (2006) S. Nemat-Nasser, C. Thomas: Ionomeric polymermetal composites. In: Electroactive Polymer (EAP) Actuators as Artificial Muscles, ed. by Y. Bar-Cohen (SPIE, Bellingham 2001) pp. 139–191 S. Nemat-Nasser, C. Thomas: Ionomeric polymermetal composites. In: Electroactive Polymer (EAP) Actuators as Artificial Muscles 2nd edn, ed. by Y. BarCohen (SPIE, Bellingham 2004) pp. 171–230 S. Nemat-Nasser, Y. Wu: Comparative experimental study of Nafion-and-Flemion-based ionic polymermetal composites (IPMC), J. Appl. Phys. 93(9), 5255– 5267 (2003) S. Sewa, K. Onishi, K. Asaka, N. Fujiwara, K. Oguro: Polymer actuator driven by ion current at low voltage, applied to catheter system, Proc. IEEE Ann. Int. Workshop Micro Electro Mech. Syst. 11th (1998) pp. 148–153 K. Oguro, N. Fujiwara, K. Asaka, K. Onishi, S. Sewa: Polymer electrolyte actuator with gold electrodes, Proc. SPIE 3669, 64–71 (1999) S. Tadokoro, T. Murakami, S. Fuji, R. Kanno, M. Hattori, T. Takamori, K. Oguro: An elliptic friction drive element using an ICPF actuator, IEEE Contr. Syst. Mag. 17(3), 60–68 (1997) Y. Osada, H. Okuzaki, H. Hori: A polymer gel with electrically driven motility, Nature 355(6357), 242– 244 (1992) D.G. Caldwell: Pseudomuscular actuator for use in dextrous manipulation, Med. Biol. Eng. Comp. 28(6), 595 (1990) D.G. Caldwell, N. Tsagarakis: Soft grasping using a dextrous hand, Ind. Robot. 27(3), 194–199 (2000) D.J. Segalman, W.R. Witkowski, D.B. Adolf, M. Shahinpoor: Theory and application of electrically controlled polymeric gels, Smart Mater. Struct. 1(1), 95–100 (1992) M. Shahinpoor: Ionic polymeric gels as artificial muscles for robotic and medical applications, Iran. J. Sci. Technol. 20(1), 89–136 (1996) M. Shahinpoor: Conceptual design, kinematics and dynamics of swimming robotic structures using active polymer gels, Act. Mater. Adapt. Struct. Proc. ADPA/AIAA/ASME/SPIE Conf. (1992) pp. 91–95

Electrochemomechanics of Ionic Polymer–Metal Composites

8.37

8.38

Y. Bar-Cohen, S. Leary, M. Shahinpoor, J.O. Harrison, J. Smith: Electro-active polymer (EAP) actuators for planetary applications, SPIE Conf. Electroactive 8.39 Polymer Actuators Proc. SPIE 3669, 57–63 (1999) D.W. DeWulf, A.J. Bard: Application of Nafion/platinum electrodes (solid polymer electrolyte structures) to voltammetric investigations of highly resistive

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solutions, J. Electrochem. Soc. 135(8), 1977–1985 (1988) J.M. Potente: Gas-phase Electrosynthesis by Proton Pumping Through a Metalized Nafion Membrane: Hydrogen Evolution and Oxidation, Reduction of Ethene, and Oxidation of Ethane and Ethene. Ph.D. Thesis (North Carolina State University, Raleigh 1988)

Part A 8

203

Wendy C. Crone

The expanding and developing fields of micro-electromechanical systems (MEMS) and nano-electromechanical (NEMS) are highly interdisciplinary and rely heavily on experimental mechanics for materials selection, process validation, design development, and device characterization. These devices range from mechanical sensors and actuators, to microanalysis and chemical sensors, to micro-optical systems and bioMEMS for microscopic surgery. Their applications span the automotive industry, communications, defense systems, national security, health care, information technology, avionics, and environmental monitoring. This chapter gives a general introduction to the fabrication processes and materials commonly used in MEMS/NEMS, as well as a discussion of the application of experimental mechanics techniques to these devices. Mechanics issues that arise in selected example devices are also presented.

9.1

Background ......................................... 203

9.2

MEMS/NEMS Fabrication ......................... 206

9.3 Common MEMS/NEMS Materials and Their Properties ............................. 206 9.3.1 Silicon-Based Materials ................ 207 9.3.2 Other Hard Materials .................... 208

9.3.3 Metals ........................................ 9.3.4 Polymeric Materials ...................... 9.3.5 Active Materials ........................... 9.3.6 Nanomaterials ............................. 9.3.7 Micromachining ........................... 9.3.8 Hard Fabrication Techniques ......... 9.3.9 Deposition .................................. 9.3.10 Lithography ................................. 9.3.11 Etching .......................................

208 208 209 209 210 211 211 211 212

9.4 Bulk Micromachining versus Surface Micromachining .............. 213 9.5 Wafer Bonding ..................................... 214 9.6 Soft Fabrication Techniques................... 215 9.6.1 Other NEMS Fabrication Strategies .. 215 9.6.2 Packaging ................................... 216 9.7

Experimental Mechanics Applied to MEMS/NEMS .......................... 217

9.8 The Influence of Scale ........................... 9.8.1 Basic Device Characterization Techniques .................................. 9.8.2 Residual Stresses in Films .............. 9.8.3 Wafer Bond Integrity .................... 9.8.4 Adhesion and Friction...................

217 218 219 220 220

9.9 Mechanics Issues in MEMS/NEMS ............. 221 9.9.1 Devices ....................................... 221 9.10 Conclusion ........................................... 224 References .................................................. 225

9.1 Background The acronym MEMS stands for micro-electromechanical system, but MEMS generally refers to microscale devices or miniature embedded systems involving one or more micromachined component that enables higher-level functionality. Similarly NEMS, nanoelectromechanical system, refers to such nanoscale devices or nanodevices. MEMS and NEMS are fab-

ricated microscale and nanoscale devices that are often made in batch processes, usually convert between some physical parameter and a signal, and may be incorporated with integrated circuit technology. The field of MEMS/NEMS encompasses devices created with micromachining technologies originally developed to produce integrated circuits, as well as

Part A 9

A Brief Introd 9. A Brief Introduction to MEMS and NEMS

204

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Part A 9.1

Table 9.1 Sample applications of MEMS/NEMS Sensors Actuators

Passive structures

Accelerometers, biochemical analyzers, environmental assay devices, gyroscopes, medical diagnostic sensors, pressure sensors Data storage, drug-delivery devices, drug synthesis, fluid regulators, ink-jet printing devices, micro fuel cells, micromirror devices, microphones, optoelectric devices, radiofrequency devices, surgical devices Atomizers, fluid spray systems, fuel injection, medical inhalers

non-silicon-based devices created by the same micromachining or other techniques. They can be classified as sensors, actuators, and passive structures (Table 9.1). Sensors and actuators are transducers that convert one physical quantity to another, such as electromagnetic, mechanical, chemical, biological, optical or thermal phenomena. MEMS sensors commonly measure pressure, force, linear acceleration, rate of angular motion, torque, and flow. For instance, to sense pressure an intermediate conversion step, such as mechanical stress, can be used to produce a signal in the form of electrical energy. The sensing or actuation conversion can use a variety of methods. MEMS/NEMS sensing can employ change in electrical resistance, piezoresistive, piezoelectric, change in capacitance, and magnetoresistive methods (Table 9.2). MEMS/NEMS actuators provide the ability to manipulate physical parameters at the micro/nanoscale, and can employ electrostatic, thermal, magnetic, piezoelectric, piezoresistive, and shape-memory transformation methods. Passive MEMS structures such as micronozzles are used in atomizers, medical inhalers, fluid spray systems, fuel injection, and ink-jet printers.

MEMS have a characteristic length scale between 1 mm and 1 μm, whereas NEMS devices have a characteristic length scale below 1 μm (most strictly, 1–100 nm). For instance a digital micromirror device has a characteristic length scale of 14 μm, a quantum dot transistor has components measuring 300 nm, and molecular gears fall into the 10–100 nm range [9.2]. Additionally, although an entire device may be mesoscale, if the functional components fall in the microscale or nanoscale regime it may be referred to as a MEMS or NEMS device, respectively. MEMS/NEMS inherently have a reduced size and weight for the function they carry out, but they can also carry advantages such as low power consumption, improved speed, increased function in one package, and higher precision. There is no distinct MEMS/NEMS market, instead there is a collection of niche markets where MEMS/NEMS become attractive by enabling a new function, bringing the advantage of reduced size, or lowering cost [9.1]. Despite this characteristic, the MEMS industry is already valued in tens of billions of dollars and growing rapidly. The Small Times Tech Business DirectoryTM

Table 9.2 Physical quantities used in MEMS/NEMS sensors and actuators (after Maluf [9.1]) Method

Description

Physical and material parameters

Order of energy density (J/cm3 )

Electrostatic

Attractive force between two components carrying opposite charge Certain materials that change shape under an electric field Thermal expansion or difference in coefficient of thermal expansion Electric current in a component surrounded by a magnetic field gives rise to an electromagnetic force Certain materials that undergo a solid–solid phase transformation producing a large shape change

Electric field, dielectric permittivity

≈ 0.1

Electric field, Young’s modulus, piezoelectric constant Coefficient of expansion, temperature change, Young’s modulus Magnetic field, magnetic permeability

≈ 0.2

Transformation temperature

≈ 10

Piezoelectric Thermal Magnetic

Shape memory

≈5 ≈4

A Brief Introduction to MEMS and NEMS

ples include biofluidic chips for biochemical analyses, biosensors for medical diagnostics, environmental assays for toxin identification, implantable pharmaceutical drug delivery, DNA and genetic code analysis, imaging, and surgery. NEMS is often associated with biotechnology because this size scale allows for interaction with biological systems in a fundamental way. BioNEMS may be used for drug delivery, drug synthesis, genome synthesis, nanosurgery, and artificial organs comprised of nanomaterials. The sensitivity of such bioNEMS devices can be exquisite, selectively binding and detecting a single biomolecule. More complete background information on microfluidic devices can be found in Beeby [9.7], Koch [9.9] and Kovacs [9.10]. Semiconductor NEMS devices can offer microwave resonance frequencies, exceptionally high mechanical quality factors, and extraordinarily small heat capacities [9.11, 12]. Examples of NEMS devices also include transducers, radiating energy devices, nanoscale integrated circuits, and optoelectronic devices [9.13, 14]. NEMS manufacturing is being further enabled by the drive towards nanometer feature sizes in the microelectronics industry. Terascale computational ability will require nanotransistors, nanodiodes, nanoswitces, and nanologic gates [9.15]. NEMS also opens the door for fundamental science at the nanometer scale investigating phonon-mediated mechanical processes [9.16] and quantum behavior of mesoscopic mechanical systems [9.17]. Although there is some discussion as to whether the NEMS definition requires a characteristic length scale below 1000 nm or 100 nm, there is no argument that the field of NEMS is in its infancy. Existing commercial devices are limited at this point, but research on NEMS is extremely active and highly promising. Many challenges remain, including assembly of nanoscale devices and mass production capabilities. In the long term, a number of issues must be addressed in analysis, design, development, and fabrication for high-performance MEMS/NEMS to become ubiquitous. Of most relevance to the focus of this handbook, advanced materials must be well characterized and MEMS/NEMS testing must be further developed. Additionally for commercialization, MEMS/NEMS design must consider issues of market (need for product, size of market), impact (enabling new systems, paradigm shift for the field), competition (other macro and micro/nanoproducts existing), technology (available capability and tools), and manufacturing (manufacturability in volume at low cost) [9.18].

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lists more than 700 manufacturers/fabricators of microsystems and nanotechnologies [9.3]. High-volume production with lucrative sales have been achieved by several companies making devices such as accelerometers for automobiles (Analog Devices, Motorola, Bosch), micromirrors for digital projection displays (Texas Instruments), and pressure sensors for the automotive and medical industries (NovaSensor). Currently, the MEMS markets with the largest commercial value are ink-jet printer heads, optical MEMS (which includes the Digital Micromirror DeviceTM discussed below), and pressure sensors, followed by microfluidics, gyroscopes, and accelerometers [9.4]. The MEMS market was reported to be US $5.1 billion in 2005 and projected to reach US $9.7 billion by 2010 [9.4]. The NEMS industry, while still young, has been growing in value. The market research firm Report Buyer recently released a market report on Nanorobotics and NEMS indicating that the global market for NEMS increased from US $29.5 million to US $34.2 million between 2004 and 2005 and projecting that the market will reach US $830.4 million by 2011 [9.5]. Microfluidic and nanofluidic devices also fall under the umbrella of MEMS/NEMS and are often classified as bioMEMS/bioNEMS devices when involving biological materials. These devices incorporate channels with at least one microscale or nanoscale dimension in which fluid flows. The small scale of these devices allow for smaller sample size, faster reactions, and higher sensitivity. Microfluidic devices commonly use both hard and soft fabrication techniques to produce channels and other fluidic structures [9.6]. The common feature of these devices is that they allow for flow of gas and/or liquid and use components such as pumps, valves, nozzles, and mixers. Commercial and defense applications include automotive controls, pneumatics, environmental testing, and medical devices. The advantages of the microscale in these applications include high spatial resolution, fast time response, small fluid volumes required for analysis, low leakage, low power consumption, low cost, appropriate compatibility of surfaces, and the potential for integrate signal processing [9.7]. At the microscale, pressure drop over a narrow channel is high and fluid flow generated by electric fields can be substantial. The micrometer and nanometer length scales are particularly relevant to biological materials because they are comparable to the size of cells, molecules, diffusions length for molecules, and electrostatic screening lengths of ionic conducting fluids [9.8]. Device exam-

9.1 Background

206

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Part A 9.3

The field of MEMS/NEMS is highly multidisciplinary, often involving expertise from engineering, materials science, physics, chemistry, biology, and medicine. Because of the breadth of the field and the range of activities that fall under the scope of MEMS/NEMS, a comprehensive review is not possible in this chapter. After providing general background, the focus will be on mechanics and specifically experimental mechanics as it is applied to MEMS/NEMS.

Mechanics is critical to the design, fabrication, and performance of MEMS/NEMS. A broad range of experimental tools has been applied to MEMS/NEMS. This chapter will provide an overview of such work. Additional information on the application of mechanics to MEMS/NEMS can be found in the proceedings of the annual symposium held by the MEMS and Nanotechnology Technical Division of the Society for Experimental Mechanics (see, for example, [9.19]).

9.2 MEMS/NEMS Fabrication Traditionally MEMS/NEMS are thought of in the context of microelectronics fabrication techniques which utilize silicon. This approach to MEMS/NEMS brings with it the momentum of the integrated circuits industry and has the advantage of ease of integration with semiconductor devices, but fabrication is expensive in both the infrastructure and equipment required and the time investment needed to create a working prototype. An alternative approach that has seen significant success, especially in its application to microfluidic devices, is the use of soft materials such as polydimethylsiloxane (PDMS). Soft MEMS/NEMS fabrication can often be conducted with bench-top techniques with no need for the clean-room facilities used in microelectronics fabrication. Additionally, polymers offer a range of properties not available in siliconbased materials such as mechanical shock tolerance, biocompatibility, and biodegradability. However, polymers can carry disadvantages for certain applications because of their viscoelastic behavior and low thermal stability. Ultimately a combination of function and

economics decides the medium of choice for device construction. Whether we talk about hard or soft MEMS/NEMS, the basic approach to device construction is similar. Material is deposited onto a substrate, a lithographic step is used to produce a pattern, and material removal is conducted to create a shape. For traditional microelectronics fabrication, the substrate is often silicon, material deposition is achieved by vapor deposition or sputtering, lithography involves patterning of a chemically resistant polymer, and material is removed by a chemical etch. Alternatively, for soft MEMS/NEMS materials, fabrication often utilizes a glass or plastic substrate, material in the form of a monomer is flowed into a region, a lithographic mask allows exposure of a pattern to ultraviolet (UV) radiation triggering polymerization, and the unpolymerized monomer is removed with a flushing solution. For both hard and soft MEMS/NEMS fabrication there are a number of variations on these basic steps which allow for a wide array of structures and devices to be constructed.

9.3 Common MEMS/NEMS Materials and Their Properties Materials used in MEMS/NEMS must simultaneously satisfy a range requirements for chemical, structural, mechanical, and electrical properties. For biomedical and bioassay devices, material biocompatibility and bioresistance must also be considered. Most MEMS/NEMS devices are created on a substrate. Common substrate materials include singlecrystal silicon, single-crystal quartz, fused quartz, gallium arsenide, glass, and plastics. Devices are made with a range of methods by machining into the substrate and/or depositing additional material on top of the

substrate. The additional materials may be structural, sacrificial, or active. Although traditionally MEMS in particular have relied on silicon, the materials used in MEMS/NEMS are becoming more heterogeneous. Selected properties are given in Table 9.3 for comparative purposes, but an extensive list of properties for the wide range of materials used in MEMS/NEMS cannot be included here. It should be noted, however, that the constitutive behavior of materials used in MEMS/NEMS applications can be sensitive to fabrication method, processing parame-

A Brief Introduction to MEMS and NEMS

9.3 Common MEMS/NEMS Materials and Their Properties

Property

Si

SiO2

Si3 N4

Quartz

SiC

Si(111)

Stainless steel

Al

Young’s modulus (GPa) Yield strength (GPa) Poisson’s ratio Density (g/cm3 ) Coefficient of thermal expansion (10−6 /◦ C) Thermal conductivity at 300 K (W/cm · K) Melting temperature (◦ C)

160 7 0.22 2.4 2.6

73 8.4 0.17 2.3 0.55

323 14 0.25 3.1 2.8

107 9 0.16 2.65 0.55

450 21 0.14 3.2 4.2

190 7 0.22 2.3 2.3

200 3 0.3 8 16

70 0.17 0.33 2.7 24

1.57

0.014

0.19

0.0138

5

1.48

0.2

2.37

1415

1700

1800

1610

2830

1414

1500

660

ters, and thermal history due to the relative similarity between characteristic length scales and device dimensions. A good resource compiling characterization data from a number of sources is the material database at http://www.memsnet.org/material/ [9.20]. The following books, used as references for the discussion here, are valuable resources for more extensive information: Senturia [9.18], Maluf [9.1], and Beeby [9.7].

9.3.1 Silicon-Based Materials Silicon, Polysilicon, and Amorphous Silicon Silicon-based materials are the most common materials currently used in MEMS/NEMS commercial production. MEMS/NEMS devices often exploit the mechanical properties of silicon rather than its electrical properties. Silicon can be used in a number of different forms: oriented single-crystal silicon, amorphous silicon, or polycrystal silicon (polysilicon). Single-crystal silicon, which has cubic crystal structure, exhibits anisotropic behavior which is evident in its mechanical properties such as Young’s modulus. A high-purity ingot of single-crystal silicon is grown, sawn to the desired thickness, and polished to create a wafer. Single-crystal silicon used for MEMS/NEMS are usually the standard 100 mm (4 inch diameter, 525 μm thickness) or 150 mm (6 inch diameter, 650 μm thickness) wafers. Although larger 8-inch and 12-inch wafers are available, they are not used as prevalently for MEMS fabrication. The properties of the wafer depend on both the orientation of crystal growth and the dopants added to the silicon (Fig. 9.1). Impurity doping has a significant impact on electrical properties but does not generally impact the mechanical properties if the concentration is approximately < 1020 cm−3 . Silicon is a group IV semiconductor. To create a p-type material, dopants

from group III (such as boron) create mobile charge carriers that behave like positively charged species. To create an n-type material, dopants from group V (such as phosphorous, arsenic, and antimony) are used to create mobile charge carriers that behave like negatively charged electrons. Doping of the entire wafer can be accomplished during crystal growth. Counter-doping can be accomplished by adding dopants of the other type to an already doped substrate using deposition followed by ion implantation and annealing (to promote diffusion and relieve residual stresses). For instance, p-type into n-type creates a pn-junction. Amorphous and polysilicon films are usually deposited with thicknesses of < 5 μm, although it is also possible to create thick polysilicon [9.21]. The residual stress in deposited polysilicon and amorphous silicon thin films can be large, but annealing can be used to provide some relief. Polysilicon has the disadvantage of a somewhat lower strength and lower piezoresistivity than single-crystal silicon. Additionally, Young’s

(100) n-type

Secondary flat

(111) n-type

Primary flat

(100) p-type

Primary flat

Secondary flat

Primary flat

(111) p-type

Primary flat

Secondary flat

Fig. 9.1 Flats on standard commercial silicon wafers used

to identify crystallographic orientation and doping (after Senturia [9.18])

Part A 9.3

Table 9.3 Properties of selected materials (after Maluf [9.1] and Beeby [9.7])

207

208

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Part A 9.3

modulus may vary significantly because the diameter of a single grain may comprise a large fraction of a component’s width [9.22]. Silicon, polysilicon, and amorphous silicon are also piezoresistive, meaning that the resistivity of the material changes with applied stress. The fractional change in resistivity, Δρ/ρ, is linearly dependent on the stress components parallel and perpendicular to the direction of the resistor. The proportionality constants are the piezoresistive coefficients, which are dependent on the crystallographic orientation, and the dopant type/concentration in single-crystal silicon. This property can be used to create a strain gage. Silicon Dioxide The success of silicon is heavily based on its ability to form a stable oxide which can be predictably grown at elevated temperature. Dry oxidation produces a higherquality oxide layer, but wet oxidation (in the presence of water) enhances the diffusion rate and is often used when making thicker oxides. Amorphous silicon dioxide can be used as a mask against etchants. It should be noted that these films can have large residual stresses. Silicon Nitride Silicon nitride can be deposited by chemical vapor deposition (CVD) as an amorphous film which can be used as a mask against etchants. It should be noted that these films can have large residual stresses. Silicon Carbide Silicon carbide is an attractive material because of its high hardness, good thermal properties, and resistance to harsh environments. Additionally, silicon carbide is piezoresistive. Although it can be produced as a bulk polycrystalline material it is generally grown or deposited on a silicon substrate by epitaxial growth (single crystal) or by chemical vapor deposition (polycrystal).

Quartz Single-crystal quartz, which has a hexagonal crystal structure, can be used in natural or synthesized form. Like silicon, it can be etched selectively but the results are less ideal than in silicon because of unwanted facets and poor edge definition. Single-crystal quartz can be used as substrate material in a range of cuts which have different temperature sensitivities for piezoelectric or mechanical properties. Detailed information about quartz cuts can be found in Ikeda [9.25]. Quartz is also piezoelectric, meaning that there is a relationship between strain and voltage in the material. Fused quartz (silica) is a glassy noncrystalline material that is also occasionally used in MEMS/NEMS devices. Glass Glasses such as phosphosilicate and borosilicate (Pyrex) can be used as a substrate or in conjunction with silicon and other materials using wafer bonding (discussed below). Diamond Diamond is also attractive because of its high hardness, high fracture strength, low thermal expansion, low heat capacity, and resistance to harsh environments. Diamond is also piezoresistive and can be doped to produce semiconducting and metal-like behavior [9.26]. Because of its hardness, diamond is particularly attractive for parts exposed to wear. The most promising synthetic forms are amorphous diamond-like carbon, nanocrystalline diamond, and ultra-nanocrystalline diamond films created by pulsed laser deposition or chemical vapor deposition [9.27–31].

9.3.3 Metals

Silicon on Insulator (SOI) Silicon on insulator (SOI) wafers are also used for MEMS sensors and actuators [9.23]. Different SOI materials are distinguished by their properties. Buried oxide layers can be produced either through ion implantation or wafer bonding processes; these techniques are discussed further below [9.24].

Metals are usually deposited as a thin film by sputtering, evaporation or chemical vapor deposition (CVD). Gold, nickel and iron can also be electroplated. Aluminum is the most common metal used in MEMS/NEMS, and is often used for light reflection and electrical conduction. Gold is used for electrochemistry, infrared (IR) light reflection, and electrical conduction. Chromium is often used as an adhesion layer. Alloys of Ni, such as NiTi and PermalloyTM , can be used for actuation and are discussed in more detail below.

9.3.2 Other Hard Materials

9.3.4 Polymeric Materials

Gallium Arsenide Gallium arsenide (GaAs) is a III–V compound semiconductor which is often used to create lasers, optical devices, and high-frequency components.

Photoresists Polymeric photoresist materials are generally used as a spin-cast film as part of a photolithographic process. The film is modified by exposure to radiation

A Brief Introduction to MEMS and NEMS

Polydimethylsiloxane Polydimethylsiloxane (PDMS) is an elastomer used as both a structural component in MEMS devices and a stamping material for creating micro- and nanoscale features on surfaces. PDMS is a common silicone rubber and is used extensively because of its processibility, low curing temperature, stability, tunable modulus, optical transparency, biocompatibility, and adaptability by a range functional groups that can be attached [9.32,33].

9.3.5 Active Materials There are several types of active materials that successfully perform sensing and actuation functions at the microscale. Several examples of active materials are given below. NiTi Near-equiatomic nickel titanium alloy can be deposited as a thin film and used an as active material. This material is of particular interest to MEMS because the actuation work density of NiTi is more than an order of magnitude higher than the work densities of other actuation schemes. These shape-memory alloys (SMAs) undergo a reversible phase transformation that allows the material to display dramatic and recoverable stress- and temperature-induced transformations. The behavior of NiTi SMA is governed by a phase transformation between austenite and martensite crystal structures. Transformation between the austenite (B2) and martensite (B19) phases in NiTi can be produced by temperature cycling between the hightemperature austenite phase and the low-temperature martensite phase (shape-memory effect), or loading and unloading the material to favor either the highstrain martensite phase or the low-strain austenite phase (superelasticity). Thus both stress and temperature produce the transformation between the austenite and martensite phases of the alloy. The transfor-

mation occurs in a temperature window, which can be adjusted from −100 ◦ C to +160 ◦ C by changing the alloy composition and heat treatment processing [9.34]. PermalloyTM PermalloyTM , Nix Fe y , displays magnetoresistance properties and is used for magnetic transducing. Multilayered nanostructures of this alloy give rise to a giant-magnetoresistance (GMR) phenomenon which can be used to detect magnetic fields. It has been widely applied to read the state of magnetic bits in data storage media. Lead Zirconate Titanate (PZT) Lead zirconate titanate (PZT) is a ceramic solid solution of lead zirconate (PbZrO3 ) and lead titanate (PbTiO3 ). PZT is a piezoelectric material that can be deposited in thin film form by sputtering or using a sol–gel process. In addition to natural piezoelectric materials such as quartz, other common synthetic piezoelectric materials include polyvinylidene fluoride (PVDF), zinc oxide, and aluminum nitride. Actuation performed by piezoelectrics has the advantage of being capable of achieving reasonable displacements with fast response, but the material processing is complex. Hydrogels Hydrogels, such as poly(2-hydroxyethyl methacrylate (HEMA)) gel, with volumetric shape-memory capability are now being employed as actuators, fluid pumps, and valves in microfluidic devices. In an aqueous environment, hydrogels will undergo a reversible phase transformation that results in dramatic volumetric swelling and shrinking upon exposure and removal of a stimulus. Hydrogels have been produced that actuate when exposed to such stimuli as pH, salinity, electrical current, temperature, and antigens. Since the rate of swelling and shrinking in a hydrogel is diffusion limited, the temporal response of hydrogel structures can be reduced to minutes or even seconds in microscale devices.

9.3.6 Nanomaterials Nanostructuring of materials can produce unique mechanical, electrical, magnetic, optical, and chemical properties. The materials themselves range from polymers to metals to ceramics, it is their nanostructured nature that gives them exciting new behaviors. Increased hardness with decreasing grain size allows for

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such as visible light, ultraviolet light, x-rays or electrons. Exposure is usually conducted through a mask so that a pattern is created in the photoresist layer and subsequently on the substrate through an etching or deposition process. Resists are either positive or negative depending on whether the radiation exposure weakens or strengthens the polymer. In the developer step, chemicals are used to remove the weaker material, leaving a patterned photoresist layer behind. Important photoresist properties include resolution and sensitivity, particularly as feature sizes decrease.

9.3 Common MEMS/NEMS Materials and Their Properties

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Table 9.4 Micromachining processes and their applications (after Kovacs [9.10]) Process

Example applications

Lithography Thin-film deposition

Photolithography, screen printing, electron-beam lithography, x-ray lithography Chemical vapor deposition (CVD), plasma-enhanced chemical vapor deposition (PECVD), physical vapor deposition (PVD) such as sputtering and evaporation, spin casting, sol–gel deposition Blanket and template-delimited electroplating of metals Electroplating, LIGA, stereolithography, laser-driven CVD, screen printing, microcontact printing, dip-pen lithography Plasma etching, reactive-ion enhanced etching (RIE), deep reactive-ion etching (DRIE), wet chemical etching, electrochemical etching Drilling, milling, electrical discharge machining (EDM), focused ion beam (FIB) milling, diamond turning, sawing Direct silicon-fusion bonding, fusion bonding, anodic bonding, adhesives Wet chemical modification, plasma modification, self assembled monolayer (SAM) deposition, grinding, chemomechanical polishing Thermal annealing, laser annealing

Electroplating Directed deposition Etching Machining Bonding Surface modification Annealing

hard coatings and protective layers, lower percolation threshold impacts conductivity, and narrower bandgap with decreasing grain size enable unique optoelectronics [9.35]. Hundreds of different synthesis routes have been created for manufacturing nanostructured materials. (See, for example, the proceedings of the International Conferences on Nanostructured Materials [9.36].) A few examples of such materials are given below. Carbon Nanotubes and Fullerenes Carbon nanotubes (CNTs) and fullerenes (buckyballs, e.g., C60 ) are self-assembled carbon nanostructures. CNTs are cylindrical graphene structures of single- or multiwall form which are extremely strong and flexible. They possess metallic or semiconducting electronic behavior depending on the details of the structure (chirality). They can be created in an arc plasma furnace, laser ablation, or grown by chemical vapor deposition (CVD) on a substrate using catalyst particles [9.37]. Quantum Dots, Quantum Wires, and Quantum Films Quantum behavior occurs in semiconductor materials (such as GaAs) when electrons are confined to nanoscale dimensions. The confined space forces electrons to have energy states that are clustered around specific peaks, producing fundamentally different electrical and optical properties than would be found in the same material in bulk form. The number of directions free of confinement is used to classify structures, thus two-dimensional (2-D) confinement leads to a quan-

tum film, one-dimensional (1-D) confinement leads to a quantum wire, and zero-dimensional (0-D) confinement leads to a quantum dot. The dimension of the confined direction(s) is so small that the energy states are quantized in that direction [9.37]. Nanowires A variety of methods have been developed for making nanowires of a wide range of metals, ceramics, and polymers. Examples include gold nanowires made by a solution method [9.38], palladium nanowires created by electroplating on a stepped surface [9.39], and zinc oxide nanowires created by a vapor/liquid/solid method [9.40]. In one popular technique, electroplating is conducted inside a nanoporous template of alumina or polycarbonate to direct the growth of nanowires [9.41, 42]. The template can be chemically removed, leaving the nanowires behind. In another application, lithographically patterned metal is used as a catalyst for silicon nanowire growth, creating predefined regions of nanowires on a surface [9.43, 44]. Using various combinations of metal catalysts and gases, a wide range of nanowire compositions can be created from chemical vapor deposition methods.

9.3.7 Micromachining Micromachining is a set of material removal and forming techniques for creating microscale movable features and complex structures, often from silicon. The micromachining processes listed in Table 9.4 can be applied to other materials such as glasses, ce-

A Brief Introduction to MEMS and NEMS

9.3.8 Hard Fabrication Techniques Hard MEMS utilizes enabling technologies for fabrication and design from the microelectronics industry. The MEMS industry has modified advanced techniques, leveraging well beyond the capability to fabricate integrated circuits. Micromachining involves three fundamental processes: deposition, lithography, and etching. Deposition may employ oxidation, chemical vapor deposition, physical vapor deposition, electroplating, diffusion, or ion implantation. Lithography methods include optical and electron-beam techniques. Etching methods include wet and dry chemical etches, which can be either isotropic (uniform etching in all directions, resulting in rounded features) or anisotropic (etching in one preferential direction, resulting in well-defined features).

9.3.9 Deposition Physical Vapor Deposition (PVD) Physical vapor deposition (PVD) includes evaporation and sputtering. The evaporation method is used to deposit metals on a surface from vaporized atoms removed from a target by heating with an electron beam. This technique is performed under high vacuum and produces very directional deposition and can create

shadows. Sputtering of a metallic or nonmetallic material is accomplished by knocking atoms off a target with a plasma of an inert gas such as argon. Sputtering is less directional and allows for higher deposition rates. Chemical Vapor Deposition (CVD) In chemical vapor deposition (CVD), precursor material is introduced into a heated furnace and a chemical reaction takes place on the surface of the wafer. The CVD process is generally performed under low-pressure conditions and is sometimes explicitly referred to as low-pressure CVD (LPCVD). A range of materials can be deposited by CVD, including films of silicon (formed by decomposition of silane (SiH4 )), silicon nitride formed by reacting dichlorosilane (SiH2 Cl2 ) with ammonia (NH3 )), and silicon oxide (formed by silane with an oxidizing species). LPCVD can produce amorphous inorganic dielectric films and polycrystalline polysilicon and metal films. Epitaxy is a CVD process where temperature and growth rate are controlled to achieve ordered crystalline growth in registration with the substrate. PECVD is a plasma-enhanced CVD process. Electroplating A variety of electroplating techniques are used to make micro- and nanoscale components. A mold is created into which metal is plated. Gold, copper, chromium, nickel, and iron are common plating metals. Spin Casting Spin casting is used to create films from a solution. The most common spin-cast material is polymeric photoresist. Sol–Gel Deposition A range of sol–gel processes can be used to make films and particles. The general technique involves a colloidal suspension of solid particles in a fluid that undergo a reaction to generate a gelatinous network. After deposition of the gel, the solvent can be removed to transform the network into a solid phase which is subsequently sintered. Piezoelectric materials such as PZT can be deposited with this method.

9.3.10 Lithography Most of the micromachining techniques discussed below utilize lithography, or pattern transfer, at some point in the manufacturing process. Depending on the resolution required to produce the desired feature sizes and the aspect ratio necessary, lithography is either per-

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ramics, polymers, and metals, but silicon is favored because of its widespread use and the availability of design and processing techniques. Other advantages of silicon include the availability of relatively inexpensive pure single-crystal substrate wafers, its desirable electrical properties, its well-understood mechanical properties, and ease of integration into a circuit for control and signal processing [9.7]. Although often performed in batch processes, micromachining for MEMS application may make large-aspect-ratio features and incorporation of novel or active materials a higher priority than batch manufacturing. This opens the door for a wider range of fabrication techniques such as focused ion-beam milling, laser machining, and electron-beam writing [9.22, 45, 46]. A brief overview of micromachining is provided below. The following books, used as references for the discussion here, are valuable resources for more extensive information [9.1, 2, 10, 18, 22, 33]. Additional information can be found in Taniguchi [9.47] and Evans [9.48] on microfabrication technology, Bustillo [9.49] on surface micromachining, and Gentili [9.50] on nanolithography.

9.3 Common MEMS/NEMS Materials and Their Properties

Solid Mechanics Topics

Part A 9.3

formed with ultraviolet light, an ion beam, x-rays, or an electron beam. X-ray lithography can produce features down to 10 nm and electron beams can be focused down to less than 1 nm [9.50]. Optical lithography allows aspect ratios of up to three whereas x-ray lithography can produce aspect ratios > 100. This large depth of focus, lack of scattering effects, and insensitivity to organic dust make x-ray lithography very attractive for NEMS production. Electron-beam lithograph has the attractive feature that a pattern can be directly written onto a resist, as well as the fact that it produces lower defect densities with a large depth of focus, but the process must be performed in vacuum. In most cases a mask that carries either a positive or negative image of the features to be created must first be produced. Masks are commonly made with a chromium layer on fused silica. Photoresist covering the chromium is exposed with an optical pattern generated from a sequence of small rectangles used to draw out the pattern desired. Other mask production techniques include photographic emulsion on quartz, electron-beam lithography with electron-beam resist, and high-resolution ink-jet printing on acetate or mylar film. Photolithographic fabrication techniques have a long history of use with ceramics, plastics, and glasses. In the case of silicon fabrication, the wafer is coated with a polymeric photoresist layer sensitive to ultraviolet light. Exposure of the photoresist layer is conducted through a mask. Depending on whether a positive or negative photoresist is used, the light either weakens the polymer or strengthens the polymer. In the developer step, chemicals are used to remove the weaker material, leaving a patterned photoresist layer behind. The photoresist acts as a protective layer when etching is conducted. Contact lithography produces a 1:1 ratio of the mask size and feature size. Proximity lithography also gives a 1:1 ratio with slightly lower resolution because a gap is left between the mask and the substrate to minimize damage to the mask. A factor of 5–10 reduction is common for projection step-and-repeat lithography. Because this technique allows for the production of feature sizes smaller then the mask, only a small region is exposed at one time and the mask must be stepped across the substrate.

9.3.11 Etching A number of wet and dry etchants have been developed for silicon. Important properties include orientation

dependence, selectivity, and the geometric details of the etched feature (Fig. 9.2). A common isotropic wet etchant for silicon is HNA (a combination of HF, HNO3 , and CH3 COOH), while anisotropic wet etchants include KOH, which etches {100} planes 100 times faster than {111} planes, tetramethylammonium hydroxide (called TMAH or (CH3 )4 NOH), which etches {100} planes 30–50 times faster than {111} planes but leaves silicon dioxide and silicon nitride unetched, and ethylenediamine pyrochatechol (EDP), which is very hazardous but does not etch most metals. Wet etchants such as HF for silicon dioxide, H3 PO4 for silicon nitride, KCl for gold, and acetone for organic layers, can be performed in batch processes with little cost [9.51]. An important feature of an etchant is its selectivity; for example, the etch rate of an oxide by HF is 100 nm/min compared to 0.04 nm/min for silicon nitride [9.51]. The etching reaction can be either reaction rate controlled or mass transfer limited. Because wet etchants act quickly, making it hard to control depth of the etch, electrochemical etching is sometimes employed using an electric potential to moderate the reaction along with a precision thickness epitaxial layer used for etch stop. The challenge comes with drying after the wet etching process is complete. Capillary forces can easily draw surfaces together, causing damage and stiction. Supercritical drying, where the liquid is converted to a gas, can be used to prevent this. Alternatively, application of a hydrophobic passivation layer such as a fluorocarbon polymer can be used to prevent stiction. Chemically reactive vapors and plasmas are highly effective dry etchants. Xenon difluoride (XeF2 ) is a commercially important highly selective vapor etchant for silicon. Dry etchants such as CHF3 + O2 for silicon dioxide, SF6 for silicon nitride, Cl2 + SiCl4 for

Wet etch

Plasma (dry) etch

Isotropic

Part A

Anisotropic

212

{111}

Fig. 9.2 Trench profiles produced by different etching processes (after Maluf [9.1])

A Brief Introduction to MEMS and NEMS

all parts of the wafer. Ion milling refers to selective sputtering and can be done uniformly over a wafer or with focusing electrodes by focused ion-beam milling (FIB). FIB is also becoming a more important technique for test sample production and the application of gratings used for interferometry [9.52]. In addition to FIB, techniques such as scanning probe microscope (SPM) lithography and molecular-beam epitaxy can also be used to create micro- and nanoscale gratings [9.53]. Beyond the use of etching as part of the initial fabrication of a device, some small adjustments may be required after the device is fabricated due to small variations that occur in processing. Compensation can be performed by trimming resistors and altering mechanical dimensions via techniques such as laser ablation and FIB milling. Calibration can be performed electronically with correction coefficients.

9.4 Bulk Micromachining versus Surface Micromachining The processes for silicon micromachining fall into two general categories: bulk (subtraction of substrate material) and surface (addition of layers to the substrate). Other techniques used on a range of materials include surface micromachining, wafer bonding, thin film screen printing, electroplating, lithography galvanoforming molding (LIGA, from the German Lithografie-Galvanik-Abformung), injection molding,

electric-discharge machining (EDM), and focused ion beam (FIB). Figure 9.3 provides a basic comparison of bulk micromachining, surface micromachining, and LIGA. Bulk Micromachining Removal of significant regions of substrate material in bulk micromachining is accomplished through

a)

b)

c)

Bulk micromachining

Surface micromachining

LIGA Resist structure Base plate

Deposition of sacrificial layer

Deposition of silica layers on Si Membrane face

Metal structure Electroforming

Patterning with mask

Gate plate

Patterning with mask and etching of Si to produce cavity Silicon

Silica

Lithography

Mold insert Mold fabrication

Deposition of microstructure layer Molding mass

Mold filling

Etching of sacrificial layer to produce freestanding structure Silicon

Polysilicon

Sacrificial material

Plastic structure

Unmolding

Fig. 9.3a–c Schematic diagrams depicting the processing steps required for (a) bulk micromachining (b) surface micromachining, and (c) LIGA. All views are shown from the side (after Bhushan [9.2] Chap. 50)

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aluminum, and O2 for organic layers, are used as a plasma [9.51]. The process is conducted in a specially designed system that generates a chemically reactive plasma species of ion neutrals and accelerates them towards a substrate with an electric or magnetic field. Plasma etching is the spontaneous reaction of neutrals with the substrate materials, while reactive-ion etching involves a synergistic role between the ion bombardment and the chemical reaction. Deep reactive-ion etching (DRIE) allows for the creation of high-aspectratio features. DRIE involves periodic deposition of a protective layer to shield the sidewalls either through condensation of reactant gasses produced by cryogenic cooling of the substrate or interim deposition cycles to put down a thin polymer film. Ions can also be used to sputter away material. For example, argon plasma will remove material from

9.4 Bulk Micromachining versus Surface Micromachining

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anisotropic etching of a silicon single-crystal wafer. The fabrication process includes deposition, lithography, and etching. Bulk micromachining is commonly used for high-volume production of accelerometers, pressure sensors, and flow sensors.

are used to produce a free-standing structure. Surface micromachining is attractive for integrating MEMS sensors with electronic circuits, and is commonly used for micromirror arrays, motors, gears, and grippers.

Surface Micromachining Alternating structural and sacrificial thin film layers are built up and patterned in sequence for surface micromachining. The process used by Sandia National Laboratory uses up to five structural polysilicon and five sacrificial silicon dioxide layers, whereas Texas Instrument’s digital micromirror device (discussed below) is made from a stack of structural metallic layers and sacrificial polymer layers [9.1, 54]. Deposition methods include oxidation, chemical vapor deposition (CVD), and sputtering. Annealing must sometimes be used to relax the mechanical stresses that build up in the films. Lithography and etching

LIGA Lithography galvanoforming molding (LIGA, from the German Lithografie-Galvanik-Abformung) is a lithography and electroplating method used to create very high-aspect-ratio structures (aspect ratios of more than 100 are common). The use of x-rays in the lithography process takes advantage of the short wavelength to create a larger depth of focus compared to photolithography [9.14]. Devices can be up to 1 mm in height with another dimension being only a few microns and are commonly made of materials such as metals, ceramics, and polymers. See Guckel [9.55], Becker [9.56], and Bley [9.57] for additional details.

9.5 Wafer Bonding Although microelectronics fabrication processes allow stacking layers of films, structures are relatively two dimensional. Wafer bonding provides an opportunity for a more three-dimensional structure and is commonly used to make pressure sensors, accelerometers, and microfluidic devices (Fig. 9.4). Anodic and direct bonding are the most common techniques, but

2.5μm LPCVD polysilicon mask

Sheat Sample inlet injector holes

Fused silica substrate Electrochemical-discharge machined through-holes

Two substrates thermally bonded together

Fig. 9.4 Schematic diagram depicting a wafer bonding process used to create a microfluidic channel for flow cytometry (after Kovacs [9.10])

bonding can also be achieved by using intermediate layers such as polymers, solders, and thin-film metals. Anodic (electrostatic) bonding can be used to bond silicon to a sodium-containing glass substrate (with a matched coefficient of thermal expansion) using an applied electric field. This is accomplished with the application of a large voltage at elevated temperature to make positive Na+ ions mobile. The positively charged silicon is held to the negatively charged glass by electrostatic attraction. Direct (silicon-fusion) bonding requires two flat, clean surfaces in intimate contact. Direct bonding of a silicon/glass stack can be achieved by applying pressure. Direct wafer bonding allows joining of two silicon surfaces or silicon and silicon dioxide surfaces and is used extensively to create SOI wafers. After treatment of the surfaces to produce hydroxyl (OH) groups, intimate contact allows van der Waals forces to make the initial bond followed by an annealing step to create a chemical reaction at the interface. Grinding and polishing is sometimes needed to thin a bonded wafer. Annealing must be performed afterwards to remove defects incurred during grinding. Alternatively, chemomechanical polishing can be used to combine chemical etching with the mechanical action of polishing.

A Brief Introduction to MEMS and NEMS

9.6 Soft Fabrication Techniques

Self-Assembly Partly because of the high cost of nanolithography and the time-consuming nature of atom-by-atom placement using probe microscopy techniques, self-assembly is an important bottom-up approach to NEMS fabrication [9.59]. To offset the time it takes to build unit by unit to create a useful device, massive parallelism and autonomy is required. The advantage of self-assembly is that it occurs at thermodynamic minima, relying on naturally occurring phenomena that govern at the nanoscale and create highly perfect assemblies [9.58]. The atoms, molecules, collections of molecules, or nanoparticles self-organize into functioning entities using thermodynamic forces and kinetic control [9.60]. Such self-organization at the nanoscale is observed naturally in liquid crystals, colloids, micelles, and self-assembled monolayers [9.61]. Reviews of self-assembly can be found in [9.62–65]. At the nanoparticle level, a variety of methods have been used to promote self-assembly. Three basic requirements must be met: there must be some sort of bonding force present between particles or the particles and a substrate, the bonding must be selective, and the particles must be in random motion to facilitate chance interactions with a relatively high rate of occurrence. Additionally, for the technique to be practical, the particles must be easily synthesized. Selectivity can be facilitated by micromachining the substrate including patterns with geometric designs that allow for only certain orientations of the mating particle. Particularly powerful are self-assembly methods using complementary pairs and molecular building blocks (analogous to DNA replication). Complementary pairs can bind electrostatically or chemically (using functional groups with couple monomers). Molecular Table 9.5 Techniques for creating patterned SAMs [9.58] Method

Scale of features

Microcontact printing Micromachining Microwriting with pen Photolithography/lift-off Photochemical patterning Photo-oxidation Focused ion-beam writing Electron-beam writing Scanning tunneling microscope writing

100 nm – some cm 100 nm – some μm ≈ 10–100 μm > 1 μm > 1 μm > 1 μm ≈ some μm 25–100 nm 15–50 nm

building blocks can use a number of different bonds and linkages (ionic bonds, hydrogen bonds, transition metal complex bonds, amide linkages, and ester linkages) to create building blocks for three-dimensional (3-D) nanostructures and nanocrystals such as quantum dots. Self-assembled monolayers (SAMs) can be produced in patterned form by several techniques that produce features in a range of micro- and nanoscale sizes (Table 9.5). Combined with lithography, defined areas of self-assembly on a surface can be created. Applications of SAMs include fundamental studies of wetting and electrochemistry, control of adhesion, surface passivation (to protect from corrosion, control oxidation, or use as resist), tribology, directed assembly, optical systems, colloid fabrication, and biologically active surfaces for biotechnology [9.58]. Soft Lithography The term soft lithography encompasses a number of techniques that can be used to fabricate microand nanoscale structures using replica molding and self-assembly. These techniques include microcontact printing, replica molding, microtransfer molding, micromolding in capillaries, and solvent-assisted micromolding [9.66]. As an example, microcontact printing uses a selfassembled monolayer as ink in a stamping operation that transfers the SAM to a surface (Fig. 9.5). The stamp is fabricated from of an elastomeric material such as PDMS by casting onto a master with surface features. The master can be produced with a range of photolithographic techniques. The polymeric replica mold is used as a stamp to enable physical pattern transfer. The advantages of microcontact printing are its simplicity, conformal contact with a surface, the reusability of the stamp, and the ability to produce multiple stamps from one master. Although defect density and registration of patterns over large scales can be issues, the flexibility of the stamp can be use to make small features (≈ 100 nm) using compression or pattern transfer onto curved surfaces [9.58]. The aspect ratio of features is a constraint with PDMS however. Ratios between 0.2 and 2 must be used to ensure defect-free stamps and molds [9.67].

9.6.1 Other NEMS Fabrication Strategies Nanoscale structures can be created from both topdown and bottom-up approaches. Because of the push

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9.6 Soft Fabrication Techniques

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to miniaturize commercial electronics, many top-down methods are refinements of micromachining techniques with the goal of achieving manufacturing accuracy on the nanometer scale. Bottom-up methods rely on additive atomic and molecular techniques, such as self-organization, self-assembly, and templating, using building blocks similar and size to those used in nature [9.68]. A brief review of some additional examples is provided below. Nanomachining Scanning probe microscopes (SPMs) are a valuable set of tools for NEMS characterization, but these tools Photoresist

Si Photolithography is used to create a master Photoresist pattern (1–2 μm thickness)

Si

PDMS is poured over master and cured

PDMS Photoresist pattern

Si PDMS is peeled away from the master

PDMS PDMS is exposed to a solution containing HS(CH2)15CH3

PDMS

Alkanethiol

Stamping onto gold substrate transfers thiol to form SAM

SAMs (1–2 nm) Au (5–2000 nm)

Si

Ti (5–10 nm) Metal not protected by SAM is removed by exposure to selective chemical etchant

Si

Fig. 9.5 Schematic diagram depicting the processing steps required for microcontact printing. All views are shown from the side (after Wilbur [9.58])

can also be used for NEMS manufacturing. These microscopes share the common feature that they employ a nanometer-scale probe tip in the proximal vicinity of a surface. They are many times more powerful than scanning electron microscopes because their resolution is not determined by wavelength for the interaction with the surface under investigation. The scanning tunneling microscope (STM) can be used to create a strong electric field in the vicinity of the probe tip to manipulate individual atoms. Atoms can be induced to slide over a surface in order to move them into a desired arrangement by mechanosynthesis [9.69]. Resolution is effectively the size of a single atom but practically the process is exceptionally time consuming and the sample must be held at very low temperature to prevent movement of atoms out of place [9.70]. With slightly less resolution but still less than 100 nm, an STM can also be used to write on a chemically amplified negative electron-beam resist. Nanolithography Surface micromachining can be conducted at the nanoscale using electron-beam lithography to create free-standing or suspended mechanical objects. Although the general approach parallels standard lithography (see above), the small-scale ability of this technique is enabled by the fact that an electron beam with energy in the keV range is not limited by diffraction. The electron beam can be scanned to create a desired pattern in the resist [9.8]. Nanoscale resolution can also be obtained using alternative lithographic techniques such as dip-pen nanolithography (DPN) [9.71]. This technique employs an atomic force microscope (AFM) probe tip to deposit a layer of material onto a surface, much as a pen writes on paper. A pattern can be drawn on a surface using a wide range of inks such as thiols, silanes, metals, sol–gel precursors, and biological macromolecules. Although the DPN process is inherently slower that standard mask lithographic techniques, it can be used for intricate functions such as mask repair and the application of macromolecules in biosensor fabrication, or it can be parallelized to increase speed [9.72]. This and other nanofabrication techniques using AFM to modify and pattern surfaces are reviewed by Tang [9.73].

9.6.2 Packaging Packaging of a MEMS/NEMS device provides a protective housing to prevent mechanical damage, minimize stresses and vibrations, guard against contamination,

A Brief Introduction to MEMS and NEMS

considerations include thickness of the device, wafer dicing (separation of the wafer into separate dice), sequence of final release, cooling of heat-dissipating devices, power dissipation, mechanical stress isolation, thermal expansion matching, minimization of creep, protective coatings to mitigate damaging environmental effects, and media isolation for extreme environments [9.1]. In the die-attach process, each individual die is mounted into a package, by bonding it to a metal, ceramic or plastic platform with a metal alloy solder or an adhesive. For silicon and glass, a thin metal layer must be placed over the surface prior to soldering to allow for wetting. Electrical interconnects can be produced with wire bonding (thermosonic gold bonding with ultrasonic energy and elevated temperature) and flip-chip bonding (using solder bumps between the die and package pads). Fluid interconnects are created by insertion of capillary tubes, mating of self-aligning fluid ports, and laminated layers of plastic [9.1].

9.7 Experimental Mechanics Applied to MEMS/NEMS With a basic understanding of the materials and processes used to make MEMS/NEMS devices, the role of mechanics in materials selection, process validation, design development, and device characterization can now be discussed. The remainder of this chapter will

focus on the forces and phenomena dominant at the micrometer and nanometer scales, basic device characterization techniques, and mechanics issues that arise in MEMS/NEMS devices.

9.8 The Influence of Scale To gain perspective on the micrometer and nanometer size scales, consider that the diameter of human hair is 40–80 μm and a DNA molecule is 2–3 nm wide. The weight of a MEMS structure can be about 1 nN and that of a NEMS components about 10−20 N. Compare this to the mass of a drop of water (10 μN) or an eyelash (100 nN) [9.2]. The minuscule size of forces that influence behavior at these small scales is hard to imagine. For instance, if you take a 10 cm length of your hair and hold it like a cantilever beam, the amount of force placed on the tip of the cantilever to deflect it by 1 cm is on the order of 1 pN. That piece of hair is 40–80 μm in diameter, which is large compared to most MEMS/NEMS components. In dealing with micro- and nanoscale devices, engineering intuition developed through experience with

macroscale behavior is often misleading. It should be noted that many macroscale techniques can be applied at the micro- and nanoscales, but advantages come not from miniaturization but rather working at the relevant size scale using the uniqueness of the scale. The balance of forces at these scales differs dramatically from the macroscale (Table 9.6). Compared to a macroscale counterpart of the same aspect ratio, the structural stiffness of a microscale cantilever increases relative to inertially imposed loads. When the length scale changes by a factor of a thousand, the area decreases by a factor of a million and the volume by a factor of a billion. Surface forces, proportional to area, become a thousand times large than forces that are proportional to volume, thus inertial and electromagnet forces become negligible. At small scales, adhesion, friction, stiction

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protect from harsh environmental conditions, dissipate heat, and shield from electromagnetic interference [9.15]. Packaging is critical because it enables the usefulness, safety, and reliability of the device. Hermetic packaging made of metal, ceramic, glass or silicon is used to prevent the infiltration of moisture, guard against corrosion, and eliminate contamination. The internal cavity is evacuated or filled with an inert gas. For MEMS/NEMS the packaging may also be required to provide access to the environment through electrical and/or fluid interconnects and optically transparent windows. In these cases, the devices are left more vulnerable in order for them to interact with the environment to perform their function. Although there are well-established techniques for packaging of common microelectronics devices, packaging of MEMS/NEMS presents particular challenges and may account for 75–95% of the overall cost of the device [9.1]. Packaging design must be conducted in parallel with design of the MEMS/NEMS component. Design

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Table 9.6 Scaling laws and the relative importance of phenomena as they depend on linear dimension, i (after Madou [9.22]) Importance at small scale Diminished

Increased

Phenomena

Power of linear dimension

Flow Gravity Inertial force Magnetic force Thermal emission Electrostatic force Friction Pressure Piezoelectricity Shape-memory effect Velocity Surface tension Diffusion van der Waal force

l4 l3 l3 l 2 , l 3 or l 4 l 2 or l 4 l2 l2 l2 l2 l2 l l l 1/2 l 1/4

(static friction), surface tension, meniscus forces, and viscous drag often govern. Acceleration of a small object becomes rapid. At the nanoscale, phenomena such as quantum effects, crystalline perfection, statistical time variation of properties, surface interactions, and interface interactions govern behavior and materials properties [9.35]. Additionally, the highly coupled nature of thermal transport properties at the microscale can be either an advantage or disadvantage depending on the device. Enhanced mass transport due to large surface-to-volume ratio can be a significant advantage for applications such as capillary electrophoresis and gas chromatography. However, purging air bubbles in microfluidic systems can be extremely difficult due to capillary forces. The interfacial surface tension force will cause small bubble less than a few millimeters in diameter to adhere to channel surfaces because the mass of liquid in a capillary tube produces an insubstantial inertial force compared to the surface tension [9.10]. Some scaling effects favor particular micro- and nanoscale situations but others do not. For instance, large surface-to-volume ratio in MEMS devices can undermine device performance because of the retarding effects of adhesion and friction. However, electrostatic force is a good example of a phenomena that can have substantial engineering value at small scales. Transla-

tional motion can be achieved in MEMS by electrostatic force because this scales as l 2 as compared to inertial force which scales as l 3 . Microactuation using electrostatic forces between parallel plates is used in comb drives, resonant microstructures, linear motors, rotary motors, and switches. In relation to MEMS testing, gripping of a tension sample can be achieved using an electrostatic force between a sample and the grip [9.74]. It is also important to note that, as the size scale decreases, breakdown in the predictions of continuumbased theories can occur at various length scales. In the case of electrostatics, electrical breakdown in the air gap between parallel plates separated by less than 5 μm does not occur at the predicted voltage [9.75]. In optical devices, nanometer-scale gratings can produce an effective refractive index different from the natural refractive index of the material because the grating features are smaller than the wavelength of light [9.76]. For resonant structures continuum mechanics predictions break down when the structure’s dimensions are on the order of tens of lattice constants in cross section [9.11]. Detailed discussions of issues related to size scale can be found in Madou [9.22] and Trimmer [9.77].

9.8.1 Basic Device Characterization Techniques A range of mechanical properties are needed to facilitate design, predict allowable operating limits, and conduct quality control inspection for MEMS. As with any macroscale device or component, structural integrity is critical to MEMS/NEMS. Concerns include friction/stiction, wear, fracture, excessive deformation, and strength. Properties required for complete understanding of the mechanical performance of MEMS/NEMS materials include elastic modulus, strength, fracture toughness, fatigue strength, hardness, and surface topography. In MEMS devices the minimum feature size is on the order of 1 μm, which is also the natural length scale for microstructure (such as the grain size, dislocation length, or precipitate spacing) in most materials. Because of this, many of the mechanical properties of interest are size dependent, which precipitates the need for new testing methods given that knowledge of material properties is essential for predicting device reliability and performance. A detailed discussion about micro- and nanoscale testing can be found in Part B of this handbook as well as in references such as Sharpe [9.78], Srikar [9.79], Haque and Saif [9.80], Bhushan [9.2], and Yi [9.81]. The following sections

A Brief Introduction to MEMS and NEMS

9.8.2 Residual Stresses in Films Many MEMS/NEMS devices involve thin films of materials. Properties of thin-film material often differ from their bulk counterparts due to the high surfaceto-volume ratio of thin films and the influence of surface properties. Additionally, these films must have good adhesion, low residual stress, low pinhole density, good mechanical strength, and good chemical resistance [9.22]. These properties often depend on deposition and processing details. The stress state of a thin film is a combination of external applied stress, thermal stress, and intrinsic residual stress that may arise due to factors such as doping (in silicon), grain boundaries, voids, gas entrapment, creep, and shrinkage with curing (in polymeric materials). Stresses that develop during deposition of thin-film material can be either tensile or compressive and may give rise to cracking, buckling, blistering, delaminating, and void formation, all of which degrade device performance. Residual stresses can arise because of coefficient of thermal expansion mismatch, lattice mismatch, growth processes, and nonuniform plastic deformation. Residual stresses that do not cause mechanical failure may still significantly affect device performance by causing warping of released structures, changes in resonant frequency of resonant structures, and diminished electrical characteristics. In some instances, however, residual stresses can be used productively, such as in shape setting of shape-memory alloy films or stress-modulated growth and arrangement of quantum dots. There are numerous techniques for measuring residual stresses in thin films. Fundamental techniques rely on the fact that stresses within a film will cause bending in its substrate (tension causing concavity, compression causing convexity). Simple displacement measurements can be conducted on a circular disk or a micromachined beam and stress calculated from the radius of curvature of the bent substrate or the deflection of a cantilever. Strain gages may also be made directly in the film and used to make local measurements. Freestanding portions of the thin film can be created by micromachining so that the films stresses can be explored by applied

pressure, external probe, critical length for buckling, or resonant frequency measurements. For instance, the critical stress to cause buckling in a doubly supported beam can be estimated from: π 2t2 , K L2 where K is a constant determined by the boundary conditions (3 for a doubly supported beam), E is Young’s modulus, t is the beam thickness, and L is the shortest length of beam displaying buckling [9.10]. The stress or strain gradient over a region of a film can be found by measuring deflections in a simple cantilever. The upward or downward deflection along the length of the beam can be measured by optical methods and used to estimate the internal bending moment M from the expression: σCR = E

(1 − ν2 ) δ (x) =K+ Mx , x 2E I where δ(x) is the vertical deflection at a distance x from the support, E is Young’s modulus, ν is Poisson’s ratio, I is the moment of the beam cross section about the axis of bending, and K is a constant determined by the boundary conditions at the support [9.83]. A number of techniques have been developed for determining residual stresses including an American Society for Testing and Materials (ASTM) standard involving optical interferometry [9.84]. The bulge test is a basic technique for measuring residual stress in a freestanding thin film [9.85]. The bulge test structure can be easily created by micromachining with well-defined boundary conditions. The M-test is an on-chip test that uses bending of an integrated free-standing prismatic beam [9.86]. The principle of an electrostatic actuator is used to conduct the test to find the onset of instability in the structure. The wafer curvature test is regularly used for residual stress measurement in nonintegrated film structures, and can be used even when the film thickness is much smaller than the substrate thickness [9.87]. Dynamic testing can be used to measure resonant frequency and extract information about residual stress and modulus. Resonant frequency increases with tension and decreases in compression [9.88, 89]. Air damping can significantly impact theses measurements, however, so they must be conducted in vacuum [9.18]. Other established techniques that can be employed to measure residual stresses in films include passive strain sensors, Raman spectroscopy, and nanoindentation [9.79]. More recently, nanoscale gratings created by focused ion-beam (FIB) milling have been used to

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provide a review of some mechanics issues that arise at the device level. The following sources, used as references for the discussion below, should be consulted for additional background on mechanics, metrology, and MEMS: Trimmer [9.77], Madou [9.22], Bhushan [9.2], and Gorecki [9.82].

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produce moiré interference between the grating on the specimen surface and raster scan lines of a scanning electron microscope (SEM) image [9.90]. This technique can be used to provide details of residual strains in microscale structures as they evolve with etching of the underlying sacrificial layer [9.52]. Digital image correlation (DIC) has also been applied to SEM and atomic force microscopy (AFM) images in combination with FIB. DIC is used to capture deformation fields while nearby FIB milling of the specimens releases residual stresses, allowing very local evaluation [9.91].

9.8.3 Wafer Bond Integrity Wafer bonding is often an essential device fabrication step, particularly for microfluidic devices, microengines, and microscale heat exchangers. Although direct bonding of silicon can achieve strengths comparable to bulk silicon, the process is sensitive to bonding parameters such as temperature and pressure. The appearance of voids and bubbles at the interface is particularly undesirable for both strength and electrical conductivity [9.92]. An important nondestructive technique for assessing the bond quality of bonded silicon wafers is infrared transmission. At IR wavelengths of about 1.1 μm silicon is transparent [9.1]. Quantification of bond strength can be conducted with techniques such as the pressure burst test, tensile/shear test, knife-edge test, or four-point bend-delamination test [9.93]. Although a range of techniques and processes can be employed to bond both similar and dissimilar materials, the stresses and deformation of the wafers that develop are consistent. The residual stress stored in the bonded wafers is important because it may provide the elastic strain energy to drive fracture. Details of the wafer geometry can impact the final shape of the bonded pair and the integrity of the bond interface [9.94].

9.8.4 Adhesion and Friction Adhesion is both essential and problematic for MEMS/NEMS. For multilayered devices, good adhesion between layers is critical for overall performance and reliability, where delamination under repetitive applied mechanical stresses must be avoided. Adhesion between material layers can be enhanced by improved substrate cleanliness, increased substrate roughness, increased nucleation sites during deposition, and addition of a thin adhesion-promoting layer. Standard tests for film adhesion include: the scotch-tape test, abrasion,

scratching, deceleration using ultrasonic and ultracentrifuge techniques, bending, and pulling [9.95]. In situ testing of adhesion can also be conducted by pressurizing the underside of a film until initiation of delamination. This method also allows the determination of the average work of adhesion. Adhesion can be problematic if distinct components or a component and the nearby substrate come into contact, causing the device to fail. For example, although the mass in an accelerometer device is intended to be free standing at all points of operation, adhesion can occur in the fabrication process. Commonly with freestanding portions of MEMS structures, the capillary forces present during the drying of a device after etching to remove sacrificial material are large enough to cause collapse of the structure and failure due to adhesion [9.96]. To avoid this problem, supercritical drying is used. Contacting surfaces that must move relative to one another in MEMS/NEMS are minimized or eliminated altogether, the reason being that friction and adhesion at these scales can overwhelm the other forces at play. Because silicon readily oxidizes to form a hydrophilic surface, it is much more susceptible to adhesion and accumulation of static charge [9.97]. When contacting surfaces are involved, lubricant films and hydrophobic coatings with low surface energy can be applied to minimize wear and stiction (the large lateral force required to initiate relative motion between two surfaces). For instance, Analog Devices uses a nonpolar silicone coating in its accelerometers to resist charge buildup and stiction [9.98]. Processing plays a major role in surface properties such as friction and adhesion. Polishing will dramatically affect roughness, as in the case of polysilicon where roughness can be reduced by an order of magnitude from the as-deposited state [9.2]. The doping process can also lead to higher roughness. Organic monolayer films show promise for lubrication of MEMS to reduce friction and prevent wear. The atomic force microscope and the surface force apparatus used to quantify friction and MEMS test structures such as those developed at Sandia National Laboratory are aiding the development of detailed mechanics models addressing friction [9.99–102]. Flow Visualization Flow in the microscale domain occurs in a range of MEMS devices, particularly in bioMEMS, microchannel networks, ink-jet printer heads, and micropropulsion systems. The different balance of forces at micro-

A Brief Introduction to MEMS and NEMS

flow fields in microfluidic devices, where micron-scale spatial resolution is critical [9.103]. Microparticle image velocimetry (μPIV) has been used to characterize such things as microchannel flow [9.104] and microfabricated ink-jet printer head flow [9.105]. For the high-velocity, small-length-scale flows found in microfluidics, high-speed lasers and cameras are used in conjunction with a microscope to image the particles seeded in the flow. With μPIV techniques, the flow boundary topology can be measured to within tens of nanometers [9.106].

9.9 Mechanics Issues in MEMS/NEMS 9.9.1 Devices A wide range of MEMS/NEMS devices is discussed in the literature, both as research and commercialized devices. These devices are commonly planar in nature and employ structures such as cantilever beams, fixed–fixed beams, and springs that are loaded in bending and torsion. A range of mechanics calculations are needed for device characterization, including the effective stiffness of composite beams, deflection analysis of beams, modal analysis of a resonant structures, buckling analysis of a compressively loaded beams, fracture and adhesion analysis of structures, and contact mechanics calculations for friction and wear of surfaces. A substantial literature is available on the application of mechanics to MEMS/NEMS devices. The selected MEMS/NEMS examples presented below were chosen for their illustrative nature. Digital Micromirror Device Optical MEMS devices range from bar-code readers to fiber-optic telecommunication, and use a range of wideband-gap materials, nonlinear electro-optic polymers, and ceramics [9.107]. (See Walker and Nagel [9.108] for more information on optical MEMS.) A wellestablished commercial example of an optical MEMS device is the Digital Micromirror DeviceTM (DMD) by Texas Instruments used for projection display (Fig. 9.6) [9.109]. These devices have superior resolution, brightness, contrast, and convergence performance compared to conventional cathode ray tube technology [9.2]. The DMD contains a surface micromachined array of half a million to two million independently controlled, reflective, hinged micromirrors that have a mechanical switching time of 15 μs [9.110]. This de-

vice steers a reflected beam of light with each individual mirrored aluminum pixel. Pixel motion is driven by an electrostatic field between the yoke and an underlying electrode. The yoke rotates and comes to rest on mechanical stops and its position is restored upon release by torsional hinge springs [9.111]. Almost all commercial MEMS structures avoid any contact between structural members in the operation of the device, and sliding contact is avoided completely because of stiction, friction, and wear. The DMD is currently the only commercial device where structural components come in and out of contact, with contact occurring between the mirror spring tips and the underlying mechanical stops, which act as landing sites. To prevent adhesion problems in the DMD, a self healing perfluorodecanoic acid coating is used on the structural aluminum components [9.112]. Other challenges for the DMD include creep and fatigue behavior in the hinge, shock and vibration, and sensitivity to debris within the package [9.2]. The primary failure mechanisms are surface contamination and hinge memory due to creep in the metallic alloy resulting in a residual tilt angle [9.1]. Heat transfer, which contributes to the creep problem, is also an issue for micromirrors. When the reflection coefficient is less than 100% some of the optical power is absorbed as heat and can cause changes in the flatness of the mirror, damage to the reflective layer, and alterations in the dynamic behavior of the system [9.113]. Micromirrors for projection display involve rotating structures and members in torsion. Such torsional springs must be well characterized and their mechanics well modeled. For production devices extensive finite element models are developed to optimize performance [9.114]. For initial design calculations however,

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scopic length scales can influence fluid flow to produce counterintuitive behavior in microscopic flows. Additionally, the breakdown in continuum laws for fluid flow begins to occur at the microscale. For instance, the no-slip condition no longer applies and the friction factor starts to decrease with channel reduction. Particle image velocimetry (PIV) is a technique commonly used at macroscopic length scales to measure velocity fields through the use of particles seeded in the fluid. The technique has been adapted to measure

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some closed-form solutions for mechanics analysis can be employed. For instance, an appropriate material can be chosen or the basic dimensional requirements can be found from calculation of the maximum shear stress τmax in a beam of elliptical cross section in torsion (with a and b the semi-axis lengths) using: 2Gαa2 b , for a > b , a 2 + b2 where G is the shear modulus and α in the angular twist [9.107]. Mechanical integrity of the DMD relies on low stresses in the hinge, thus the tilt angle is limited to ±10◦ [9.1]. τmax =

atomic force microscopy [9.117, 118], and magnetic beads [9.119], but these techniques have the disadvantage of requiring external probes, labeling, and/or optical excitation. Alternatively, there are several methods using molecular recognition and the small-scale forces created by events such as DNA hybridization and receptor–ligand binding to produce bending in cantilevers to create sensors with high selectivity and resolution [9.115, 120]. Microcantilever sensors have been used for some time to detect changes in relative humidity, temperature, pressure, flow, viscosity, sound, natural gas, mercury vapor, and ultraviolet and infrared radiation. More re-

Biomolecular Recognition Device Biological molecules can be probed by external methods using techniques such as optical tweezers [9.116], a)

100 μm Oligonucleotide

x

b)

Mirror –10 deg Mirror +10 deg

Hybridization

Δx Hinge Yoke Landing tip

CMOS substrate

Fig. 9.6 (a) SEM image of yoke and hinges of one pixel with mirror removed (b) Schematic of two tilted pixels with mirrors (shown as transparent) (reprinted with permission, Hornbeck [9.111], Bhushan [9.2])

Fig. 9.7 SEM image of a portion of the cantilever sensor

array and schematics illustrating functionalized cantilevers with selective sensing capability (reprinted with permission, Fritz [9.115])

A Brief Introduction to MEMS and NEMS

Δm ≈ 2

Meff Δω ω0

where Meff is the effective vibratory mass of the resonator, and ω0 is the resonance frequency of the device [9.11]. The mass sensitivity of NEMS devices with micromachined cantilevers can be as small as a single small molecule (in the range of a single Dalton). In a device such as that shown in Fig. 9.7, a liquid medium, which contains molecules that dock to a layer of receptor molecules attached to one side of the cantilever, is injected into the device. Sensitizing an array of cantilevers with different receptor allows docking of different substances in the same solution [9.115]. Hybridization can be done with short strands of single-stranded DNA and proteins known a)

to recognize antibodies. When docking occurs, the increase in the molecular packing density leads to surface stress, causing bending (10–20 nm of deflection). This deflection can be measured by a laser beam reflected off of the end of the cantilever [9.115]. Alternatively, simple geometric interference by interdigitated cantilevers that act as diffraction gratings can be used to provide output of a binding event [9.120]. Thermomechanical Data Storage Device Much of the drive to nanometer-scale devices originates in the desire for higher density and faster computational devices. Magnetic data storage has been pushed into the nanoscale regime, but limitations have prompted the development of alternative methods for data storage such as the NEMS device known as the Millipede, developed by IBM. The Millipede, or scanning probe array memory device, is an array of individually addressable scanning probe tips (similar to atomic force microscope probe tips) that makes precisely positioned indentations in a polymer thin film. The Millipede is scanned to address a large area for data storage. The indentations are bits of digital information. A polymer thin film (50 nm thick) of polymethyl methacrylate (PMMA) is used for write, read, erase, and rewrite operations. Each individual bit is a nanoscale feature, which allows the Millipede to extend storage density to the Tbit/in2 range with a bit size of 30–40 nm [9.123]. The device uses multiple cantilever probe tips equipped with integrated heaters which allow for data transfer rates of up to a few Mb/s [9.124].

Multiplex driver

2-D cantilever array chip

b) Highly doped silicon cantilever leg

Nickel bridge

x

Metal 1 (Gold) Metal 2 (Nickel)

z3 Low-doped Schottky diode area y

Polymer storage media on x/y/z scanner

z1 z2

Stress-controlled nitride

Silicon cantilever

Heater platform

Fig. 9.8 (a) Schematic illustration of the millipede device, (b) with a detail of one cantilever cell (reprinted with permission, Despont [9.122])

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cently micromachined cantilevers have been used to interact and probe material at the molecular level. Devices employing these micromachined cantilevers can be dynamic, which are sensitive to mass changes down to 10–21 g (the single molecule level), or static, which are sensitive to surface stress changes in the low mN/m range (changes in Gibbs free energy caused by binding site-analyte interactions) [9.121]. In this case adhesion is required between the device and the material to be detected. In a functionalized cantilever array device produced to measure biomechanical forces created by DNA hybridization or receptor–ligand binding, detection of the mass change is accomplished by measuring a shift in resonant frequency. The responsiveness of the device to a change in mass is given by the expression:

9.9 Mechanics Issues in MEMS/NEMS

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Substrate Polymer Cantilever Write current Pit: 25nm deep 40nm wide (maximum)

Erasure current

Sensing current

Inscribed pins Data stream 0 0 1 1 1 1 1 0 1 0 0 0 1 0 1 0 1 0 0 1 0 1 1 1 0 1 Output signal 0

The Millipede device is a massively parallel structure with a large array of thousands of probe tips (100 cantilevers/mm2 ), each of which is able to address a region of the substrate where it produces indentations for use as data storage bits (Fig. 9.8) [9.125]. As illustrated in Fig. 9.9, the probes writes a bit by heating and a mechanical force applied between a can-

Fig. 9.9 Schematic of the writing, erasing, and reading op-

erations in the Millipede device. Data are mechanically stored in pits on a surface (reprinted with permission, Vettiger [9.127])

tilever tip and a polymer film. Erasure of a bit is also conducted with heating by placing a small pit just adjacent to the bit to be erased or using the spring back of the polymer when a hot tip is inserted into a pit. Reading is also enabled by heat transfer since the sensing relies on a thermomechanical sensor that exploits temperature-dependent resistance [9.126]. The change in temperature of a continuously heated resistor is monitored while the tip is scanned over the film and relies on the change in resistance that occurs when a tip moves into a bit [9.123]. Scanning x, y manipulation is conducted magnetically with the entire array at once. The data storage substrate is suspended above the cantilever array with leaf springs which enables the nanometer-scale scanning tolerances required. The cantilevers are precisely curved using residual stress control of a silicon nitride layer in order to minimize the distance between the heating platform of the cantilever and the polymer film while maximizing the distance between the cantilever array and the film substrate to ensure that only the tips come into contact [9.123]. Fabrication details are given in Despont [9.125]. Thermal expansion is a major hurdle for this device since a shift of ≈ 30 nm can cause misalignment of the data storage substrate and the cantilever array. A 10 nm tip position accuracy of a 3 mm × 3 mm silicon area requires that temperature of the device be controlled to 1 ◦ C using several sensors and heater elements [9.123]. Tip wear due to contact between the tip and the underlying silicon substrate is an issue for device reliability. Additionally, the PMMA is prone to charring at the temperatures necessary for device operation (around 350 ◦ C) [9.128] so new polymeric formulations had to be developed to minimize this problem. The feasibility of using thin-film NiTi shape-memory alloy (SMA) for thermomechanical data storage as an alternative to the polymer thin film has also been shown [9.129].

9.10 Conclusion The sensors, actuators, and passive structures developed as MEMS and NEMS devices require a highly interdisciplinary approach to their analysis, design, de-

velopment, and fabrication. Experimental mechanics plays a critical role in design development, materials selection, prediction of allowable operating lim-

A Brief Introduction to MEMS and NEMS

MEMS/NEMS testing must be further developed. This chapter has provided a brief review of the fabrication processes and materials commonly used and experimental mechanics as it is applied to MEMS and NEMS.

References 9.1 9.2 9.3

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9.10 9.11

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N. Maluf: An Introduction to Microelectromechanical Systems Engineering (Artech House, Boston 2000) B. Bhushan (Ed.): Springer Handbook of Nanotechnology (Springer, Berlin, Heidelberg 2006) Small Tech Business DirectoryTM Guide, Small Times Media, http://www.smalltechdirectory.com/ directory/directory.asp. Accessed July 20, 2007. J.C. Eloy: The MEMS industry: Current and future trends, OnBoard Technol., 22–24 (2006), http://www.onboard-technology.com/ pdf_settembre2006/090604.pdf (Accessed July 20, 2007) NewswireToday: Nanorobotics and NEMS to Reach $830.4 million by 2011, 05/31/2007, http://www.newswiretoday.com/news/18867/ (Accessed July 20, 2007) B. Ziaie, A. Baldi, M. Lei, Y. Gu, R.A. Siegel: Hard and soft micromachining for BioMEMS: review of techniques and examples of applications in microfluidics and drug delivery, Adv. Drug Deliv. Rev. 56, 145–172 (2004) S. Beeby, G. Ensell, M. Kraft, N. White: MEMS Mechanical Sensors (Artech House, Boston 2004) H.G. Craighead: Nanoelectromechanical systems, Science 290, 1532–1536 (2000) M. Koch, A.G.R. Evans, A. Brunnschweiler: Microfluidic Technology and Applications (Research Studies, Baldock 2000) G.T.A. Kovacs: Micromachined Transducers – Sourcebook (McGraw-Hill, New York 1998) K.L. Ekinci: Electromechanical transducers at the nanoscale: Actuation and sensing of motion in nanoelectromechanical systems (NEMS), Small 1(8-9), 786–797 (2005) K.L. Ekinci, M.L. Roukes: Nanoelectromechanical systems, Rev. Sci. Instrum. 76(061101), 1–12 (2005) P. Gammel, G. Fischer, J. Bouchaud: RF MEMS and NEMS technology, devices, and applications, Bell Labs Tech. J. 10(3), 29–59 (2005) W.A. Goddard III, D.W. Brenner, S.E. Lyshevski, G.J. Iafrate (Eds.): Handbook of Nanoscience, Engineering and Technology (CRC, Boca Raton 2002) S.E. Lyshevski: Nano- and Microelectromechanical Systems: Fundamentals of Nano- and Microengineering (CRC, Boca Raton 2000) T.S. Tighe, J.M. Worlock, M.L. Roukes: Direct thermal conductance measurements on suspended

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9.28

monocrystalline nanostructures, Appl. Phys. Lett. 70(20), 2687–2689 (1997) R.G. Knobel, A.N. Cleland: Nanometre-scale displacement sensing using a single electron transistor, Nature 424(6946), 291–293 (2003) S.D. Senturia: Microsystem Design (Kluwer Academic, Boston 2001) SEM: Society for Experimental Mechanics, Bethel, 2007 SEM Annual Conference Proceedings, Springfield (2007). MEMS and Nanotechnology Clearinghouse: Material Index, http://www.memsnet.org/material/ (Accessed July 20, 2007) P. Lange, M. Kirsten, W. Riethmuller, B. Wenk, G. Zwicker, J.R. Morante, F. Ericson, J.A. Schweitz: Thisck polycrystalline silicon for surface micromechanical applications: deposition, structuring and mechanical characterization, 8th International conference on solid-state sensors and actuators (Transducers ’95), Stockholm (1995) pp. 202–205 M. Madou: Fundamentals of Microfabrication (CRC, Boca Raton 1997) B. Diem, M.T. Delaye, F. Michel, S. Renard, G. Delapierre: SOI(SIMOX) as a substrate for surface micromachining of single crystalline silicon sensors and actuators, 7th International Conference on Solid-State Sensors and Actuators (Transducers ’93), Yokohama (1993) pp. 233–236 J.M. Noworolski, E. Klaassen, J. Logan, K. Petersen, N. Maluf: Fabricationof SOI wafers with buried cavities using silicon fusion bonding ans electrochemical etchback, 8-th International conference on solid-state sensors and actuators (Transducers ’95), Stockholm (1995) pp. 71–74 T. Ikeda: Fundamentals of Piezoelectricity (Oxford Univ., New York 1990) P. Gluche, A. Floter, S. Ertl, H.J. Fecht: Commercial applications of diamond-based nano- and microtechnolgy. In: The Nano-Micro Interface, ed. by H.-J. Fecht, M. Werner (Wiley-VCH, Weinheim 2004) pp. 247–262 S. Cho, I. Chasiotis, T.A. Friedmann, J.P. Sullivan: Young’s modulus, Poisson’s ratio and failure properties of tetrahedral amorphous diamond-like carbon for MEMS devices, J. Micromech. Microeng. 15, 728–735 (2005) A.R. Krauss, O. Auciello, D.M. Gruen, A. Jayatissa, A. Sumant, J. Tucek, D.C. Mancini, N. Moldovan,

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its, device characterization, process validation, and quality control inspection. Commercial devices exist, and research in the area of MEMS/NEMS is extremely active, but many challenges remain. Advanced materials must be well characterized and

References

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Part A 9 9.29 9.30

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9.32 9.33 9.34 9.35 9.36

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229

Hybrid Metho 10. Hybrid Methods

Analyses involving real structures and components are, by their very nature, only partially specified. The central role of modern experimental analysis is to help complete, through measurement and testing, the construction of an analytical model for the given problem. This chapter recapitulates recent developments in hybrid methods for achieving this and demonstrates through examples the progress being made.

10.1 Basic Theory of Inverse Methods ............ 10.1.1 Partially Specified Problems and Experimental Mechanics ......... 10.1.2 Origin of Ill-Conditioning in Inverse Problems...................... 10.1.3 Minimizing Principle with Regularization ...................... 10.2 Parameter Identification Problems......... 10.2.1 Sensitivity Response Method (SRM) . 10.2.2 Experimental Data Study I: Measuring Dynamic Properties ....... 10.2.3 Experimental Data Study II: Measuring Effective BCs................. 10.2.4 Synthetic Data Study I: Dynamic Crack Propagation ........................ 10.3 Force Identification Problems ................ 10.3.1 Sensitivity Response Method for Static Problems ....................... 10.3.2 Generalization for Transient Loads .

231 231 232 234 235 235 237 239 240 240

10.3.3 Experimental Data Study I: Double-Exposure Holography ........ 243 10.3.4 Experimental Data Study II: One-Sided Hopkinson Bar ............. 244 10.4 Some Nonlinear Force Identification Problems ............................................. 10.4.1 Nonlinear Data Relations .............. 10.4.2 Nonlinear Structural Dynamics ....... 10.4.3 Nonlinear Space–Time Deconvolution ............................. 10.4.4 Experimental Data Study I: Stress Analysis Around a Hole ........ 10.4.5 Experimental Data Study II: Photoelastic Analysis of Cracks ....... 10.4.6 Synthetic Data Study I: Elastic–Plastic Projectile Impact ..... 10.4.7 Synthetic Data Study II: Multiple Loads on a Truss Structure 10.4.8 Experimental Data Study III: Dynamic Photoelasticity ................ 10.5 Discussion of Parameterizing the Unknowns...................................... 10.5.1 Parameterized Loadings and Subdomains .......................... 10.5.2 Unknowns Parameterized Through a Second Model ............... 10.5.3 Final Remarks ..............................

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Experimental methods do not provide a complete stress analysis solution without additional processing of the data and/or assumptions about the structural system. As eloquently stated by Kobayashi [10.1] in the earlier edition of this Handbook: “One of the frustrations of an experimental stress analyst is the lack of a universal experimental procedure that solves all problems”. Figure 10.1 shows experimental whole-field data for some sample stress analysis problems; these example prob-

lems were chosen because they represent a range of difficulties often encountered when doing experimental stress analysis using whole-field optical methods. The photoelastic data of Fig. 10.1b can directly give the stresses along a free edge; however, because of edge effects, machining effects, and loss of contrast, the quality of photoelastic data is poorest along the edge, precisely where we need good data. Furthermore, a good deal of additional data collection and processing

Part A 10

James F. Doyle

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Initial

p (x)

c)

Fig. 10.1a–c Example whole-field experimental data. (a) Moiré inplane u-displacement fringes for a point-loaded plate (the inset is the initial fringe pattern). (b) Photoelastic stress-difference fringes for a plate with a hole. (c) Double-exposure holographic out-ofplane displacement fringes for a circular plate with uniform pressure

is required if the stresses away from the free edge are of interest (this would be the case in contact and thermal problems). By contrast, the moiré methods give objective displacement information over the whole field but suffer the drawback that the fringe data must be spatially differentiated to give the strains and subsequently the stresses. It is clear from Fig. 10.1a that the fringes are too sparse to allow for differentiation; this is especially true if the stresses at the load application point are of interest. Also, the moiré methods invariably have an initial fringe pattern that must be subtracted from the loaded pattern, which leads to further deterioration of the computed strains. Double-exposure holography directly gives the deformed pattern but is so sensitive that fringe contrast is easily lost (as seen in Fig. 10.1c) and fringe localization can become a problem. The strains in this case are obtained by double spatial differentiation of the measured data on the assumption that the plate is correctly described by classical thin-plate theory – otherwise it is uncertain as to how the strains are to be obtained. Conceivably, we can overcome the limitations of each of these methods in special circumstances; in this work, however, we propose to tackle each difficulty dir-

ectly and in a consistent manner across the different experimental methods and different types of problems. That is, given a limited amount of sometimes sparse (both spatially and temporally) and sometimes (visually) poor data, determine the complete stress and strain state of the structure. The basic approach is to supplement the experimental data with analytical modeling of the problem; this is referred to as hybrid stress analysis. Experimental mechanics has a long tradition of using hybrid methods; early examples in stress analysis were the use of finite differences combined with the equilibrium and/or compatibility equations to separate the stress components in photoelasticity. Kobayashi [10.1, 2] gives an in-depth study of this early literature with an impressive array of applications and results. More recent hybrid literature is represented by Rowlands and coworkers [10.3–5] and their cited bibliography. Applications include strain gages for stress concentrations, stress pattern analysis by thermal emissions (SPATE) for thermoelastic problems, and moiré for crack analysis, to name a few. This chapter builds on this earlier research but tries to extend it beyond stress to more general applications in experimental mechanics. The ideas expressed here are elaborated in [10.6], where the thesis is advanced that the central role of experimental mechanics is to help complete construction of the analytical model using the finite element method (FEM). A single approach, called the sensitivity response method, is used across all problems and this is made feasible because of the coupling with finite element methods. The chapter begins with a general discussion of inverse problems and places the hybrid methods within the context of partially specified problems. It is shown that the two main categories of problems are parameter identification and force identification. The following sections then show solutions of these in a variety of situations involving static/dynamic and linear/nonlinear problems, and using point/whole-field data. The examples discussed are a mixture of those using experimental data to assess the practicality of the methods and those using synthetic data to assess the robustness of the algorithms. The chapter ends with a general discussion of methods of parameterizing the problem and identifying the fundamental set of unknowns.

Hybrid Methods

10.1 Basic Theory of Inverse Methods

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10.1 Basic Theory of Inverse Methods

10.1.1 Partially Specified Problems and Experimental Mechanics To begin making the connection between analysis (or model building) and experiment, consider a simple situation using the finite element method for analysis of a linear static problem; the problem is mathematically represented by [K ]{u} = {P} , where [K ] is the stiffness of the structure, {P} is the vector of applied loads, and {u} is the unknown vector of nodal displacements. (The notation is that of [10.11, 12].) The solution of these problems can be put in the generic form {u} = [K −1 ]{P} or {response} = [system]{input} . We describe forward problems as those where both the system and the input are known and only the response is unknown. A finite element analysis of a problem where the geometry and material properties (the system) are known, where the loads (the input) are known, and where we wish to determine the displacement (the response) is an example of a forward problem. An inverse problem is one where something about the system or the input is unknown but inferred by utilizing some measured response information. A common example is the measurement of load (input) and strain (response) on a uniaxial specimen to infer the Young’s modulus (system). In fact, all experimental problems can be thought of as inverse problems because we use response information to infer something about the system or the input. A word about terminology. Strictly, an inverse problem is one requiring the inverse of the system matrix.

Thus all problems involving simultaneous equations (which means virtually every problem in computational mechanics) are inverse problems. But we do not want to refer to these FEM problems as inverse problems; indeed, we want to refer to them as forward problems. Consequently, we shall call those methods, techniques, algorithms, etc., for the robust inversion of matrices inverse methods and give looser meanings to forward and inverse problems based on their cause and effect relationship. Thus a forward problem (irrespective of whether a matrix inversion is involved or not) is one for which the system is known, the inputs are known, and the responses are the only unknowns. An inverse problem is one where something about the system or the input is unknown and knowledge of something about the response is needed in order to effect a solution. Thus the key ingredient in inverse problems is the use of response data and hence its connection to experimental methods. A fully specified forward problem is one where all the materials, geometries, boundary conditions, loads, and so on are known and the displacements, stresses, and so on are required. A partially specified forward problem is one where some input information is missing. A common practice in finite element analyses is to try to make all problems fully specified by invoking reasonable modeling assumptions. Consider the following set of unknowns and the type of assumptions that could be made:

• • • •

Dimension (e.g., ice on power lines): Assume a standard deposit thickness. Report the results for a range of values about this mean. Boundary condition (e.g., loose bolt connection): Assume an elastic support/attachment. Report the results for a range of support values from very stiff to somewhat flexible. Loading (e.g., hail impacting a windshield): Assume an interaction model (hertzian, plastic). Report the results for a range of model parameters. Model (e.g., sloshing fuel tank): Assume no dynamic effects, just model the mass and stiffness. Since the unknown was not parameterized, its effect cannot be gauged.

Note that in each case because assumptions are used the results must be reported over the possible latitude of the assumptions. This adds considerably to the total cost of the analysis. Additionally, and ultimately more impor-

Part A 10.1

Before developing specific algorithms for solving problems in experimental mechanics, we will first review some of the basic theory of solving inverse problems. Inverse problems are difficult to solve since they are notoriously ill-conditioned (i.e., small changes in data result in large changes of response); a robust solution must incorporate (in addition to the measurements and our knowledge of the system) extra information either about the underlying system, or about the nature of the response functions, or both. A key idea to be introduced is that of regularization. Additional details can be found in [10.6–10].

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Solid Mechanics Topics

Part A 10.1

tantly, the use of assumptions makes the results of the analyses uncertain and even unreliable. It may appear that the most direct way of minimizing uncertainty in our assumptions is simply to measure the unknown. For an unknown dimension this may be straightforward but what about the force due to impact of hail on an aircraft wing? This is not something that can be measured directly. It is our objective here to use indirect measurements in the solution of partially specified problems to infer, rather than assume, quantities that are not directly measurable. The unknowns in a problem fall into two broad categories: system and input. Those associated with the system will be referred to as parameter identification problems, whereas those associated with the input will be referred to as force identification problems. As will be shown, by using the idea of sensitivity responses generated by the analytical model, we can formulate both sets of problems using a common underlying foundation; we will refer to the approach as the sensitivity response method (SRM). Also, quantities of interest in usual hybrid analyses (such as stress and strain distributions) are not considered basic unknowns; once the problem is fully specified these are obtained as a postprocessing operation on the model. What emerges is the central role played by models: the FEM model, the loading model, and so on. Because of the wide range of possible models needed, it would not make sense to embed these models in the inverse methods. We instead will try to develop inverse methods that use FEM and the like as external processes in a distributed computing sense. In this way, the power and versatility of stand-alone commercial packages can be harnessed.

10.1.2 Origin of Ill-Conditioning in Inverse Problems We begin with a simple illustration of the distinction between forward and inverse problems and how a system can become ill-conditioned. Reference [10.10] introduces the scenario of a zookeeper with an unknown number of animals and, based on the number of heads and feet, we try to infer the number of animals of each type. Let us continue with this scenario and suppose the zookeeper has just a number of lions (L) and tigers (T ), then the number of heads (H) and paws (P) of all his animals can be computed as H = L +T H 1 1 L or = . P = 4L + 4T P 4 4 T

This is a well-stated forward set of equations; that is, given the populations of lions and tigers (the input), and the number of heads and paws per each animal (the system), we can compute precisely the total number of heads and paws (the output). Let us invert the problem, that is, given the total number of heads and paws, determine the populations of lions and tigers. The answer is −1 H L 1 1 . = P T 4 4 Unfortunately, the indicated matrix inverse cannot be computed because its determinant is zero. When the determinant of a system matrix is zero, we say that the system is singular. It is important to realize that our data are precise and yet we cannot get a satisfactory solution. Also, it is incorrect to say we cannot get a solution as, in fact, we get an infinity of solutions that satisfy the given data. Thus it is not always true that N equations are sufficient to solve for N unknowns; the equations may be consistent but contain insufficient information to separate the N unknowns. Indeed, we can even overspecify our problem by including the number of eyes as additional data and still not get a satisfactory solution. The origin of the problem is that our choice of features to describe the animals (heads, paws, eyes) does not distinguish between the animals because lions and tigers have the same number of heads and paws. Let us choose the weight as a distinguishing feature and say (on average) that each lion weighs 100 and each tiger weighs 100 + Δ. The heads and weight totals are H 1 1 L = , W 100 100 + Δ T 1 100 + Δ −1 H L . = Δ −100 1 W T As long as (on average) the animals have a different weight (Δ = 0) then we get a unique (single) inverse solution for the two populations. The heads are counted using integer arithmetic and we can assume it is precise, but assume there is some error in the weight measurement, that is, W → W ± δ. Then the estimated populations of lions and tigers are given by 1 100 + Δ −1 H L∗ = , or Δ −100 1 T∗ W ±δ δ δ L∗ = L ∓ , T ∗ = T ± , Δ Δ

Hybrid Methods

a)

I2 I1 y1

S2 d

y2

D

d S1

from our choice of weight as a purported distinguishing feature between the animals. The second thing we notice about the estimated populations is that the errors go in opposite directions. As a consequence, if the estimated populations are substituted back into the forward equations then excellent agreement with the measured data is obtained even though the populations may be grossly in error. The sad conclusion is that excellent agreement between model predictions and data is no guarantee of a good model if we suspect that the model system is ill-conditioned. As a more pertinent example of ill-conditioning, consider the measurement situation shown in Fig. 10.2 where two detectors, I1 and I2 , are used to determine the strength of two sources, S1 and S2 . Suppose the intensity obeys an inverse power law so that it diminishes as 1/r, then the detectors receive the intensities 1/r11 1/r12 S1 I1 = , I2 1/r21 1/r22 S2 rij = D2 + (yi + d j )2 , where the subscripts on r(i, j) are (sensor, source). We wish to investigate the sensitivity of this system to errors. Figure 10.2b shows the reconstructions of the sources (labeled as direct) when there is a fixed 0.001% source of error in the second sensor. The sources have strengths of 1.0 and 2.0, respectively. There is a steady increase in error as the sensors become more remote (D increases for fixed d; a similar effect occurs as d is decreased for fixed D). Least squares is popularly used to normalize a system of equations that is overdetermined (more equations than unknowns) [10.8]; let us use least squares here even though to system is determined. This leads to S1 1/r11 1/r21 1/r11 1/r12 1/r12 1/r22 1/r21 1/r22 S2 I1 1/r11 1/r21 . = 1/r12 1/r22 I2

b) Strength 14 12 10 8 6 4 2 0 –2 –4 –6 –8 –10

Direct Least squares

–12 10

20

30

40

50 60 Distance (D/d)

Fig. 10.2a,b Two detectors measuring emissions from two sources: (a) geometry and (b) reconstructions of source strengths for fixed error in the second sensor

The inverse results are also shown in Fig. 10.2b. The least-squares solution shows very erratic behavior as D/d is increased. It is sometimes mistakenly thought that least squares will cancel errors in the data and thereby give the correct (true or smoothed) answer. Least squares will average errors in the reconstructions of the data but in all likelihood the computed solu-

233

Part A 10.1

where L and T are the exact (true) values, respectively. We notice two things about this result. First, for a given measurement error δ, the difference in weight between the animals acts as an amplifier on the errors in the population estimates; that is, the smaller Δ is, the larger is δ/Δ. When Δ is small (but not zero) we say the system is ill-conditioned – small changes in data (δ) cause large changes in results. If there were no measurement errors, then we would not notice the ill-conditioning of the problem, but the measurement errors do not cause the ill-conditioning – this is a system property arising

10.1 Basic Theory of Inverse Methods

Solid Mechanics Topics

B(u) = {u} [H]{u} with [H] being an [M × M] positive-definite matrix (the unit matrix, say), then minimizing this will lead to a unique solution for {u}. Of course, the solution will not be very accurate since none of the data were used. The key point in the inverse solution is: if we add any multiple γ times B(u) to A(u) then minimizing A(u) + γ B(u) will lead to a unique solution for {u}. Thus, the essential idea in our inverse theory is the objective to minimize A + γ B = {d} − [A]{u} W {d} − [A]{u}

B∝

du dx

2 dx ∝

M−1

({u}m − {u}m+1 )2

m=1

= ([D]{u})2

La

rg e

rm

ro r

where {d} is the data vector and [A] is the system matrix. If [A] has fewer rows (N, data) than columns (M, unknowns) then minimizing A(u), will give the correct number of equations (M); however, it will not give a unique solution for {u} because the system matrix will be rank deficient. Suppose instead we choose a nondegenerate quadratic form B(u), for example,

These equations can be solved by standard techniques such as [LU] decomposition [10.11]. As opposed to the standard least-squares form, these equations are not ill-conditioned because of the role played by γ and the regularization matrix [H] in preventing the system eigenvalues from becoming very small, and, as will be discussed shortly, can be used to add levels of smoothness to the inverse solution. The concept of regularization is a relatively recent idea in the history of mathematics; the first few papers on the subject seem to be those of [10.13, 14], although [10.9] cites earlier literature in Russian. A readable discussion of regularization is given in [10.8] and a more extensive mathematical treatment can be found in [10.7]. This latter reference has an extensive bibliography. The regularization method we will use here is generally called Tikhonov regularization [10.9, 15, 16]. Typically, the functional B involves some measure of smoothness that derives from first or higher derivatives. Consider a distribution represented by u(x) or its discretization {u} which we wish to determine from some measurements. Suppose that our a priori belief is that, locally, u(x) is not too different from a constant, then a reasonable functional to minimize is associated with the first derivative

“Best” solution od

er

Suppose that {u} is an unknown vector that we plan to solve for by some minimizing principle. Suppose, further, that our functional A(u) has the particular form of the χ 2 error function [10.8], A(u) = {d} − [A]{u} {d} − [A]{u} ,

[A] W[A] + γ [H] {u} = [A] W{d} .

at a

10.1.3 Minimizing Principle with Regularization

so-called trade-off curve of Fig. 10.3. The minimization problem is reduced to a linear set of normal equations by the usual method of differentiation to yield

el

rd

Part A 10.1

tion will not be smooth. Closer inspection shows that when one solution is up the other is down, and vice versa. What we are seeing is the manifestation of illconditioning. It should also be emphasized that these erratic values of S1 and S2 reconstruct the measurements, I1 and I2 , very closely (i. e., within the error bounds of the measurements). Ill-conditioning is a property of the system and not the data; errors in the data manifests the illconditioning. As it happens, the condition number of the matrix product [A ][A] (the ratio of its largest eigenvalue to its smallest) is the square of that of the matrix [A] and this is why least-squares solutions are more apt to be ill-conditioned. In this work, we alleviate this problem by introducing regularization.

er

ro r

rg e

Part A

La

234

+ γ u [H]u , where W is a general weighting array, although we will take it as diagonal. We minimize the error functional for various values of 0 < γ < ∞ along the

More regularization

Fig. 10.3 Inverse solutions involve a trade-off between achieving good data agreement and smooth modeling

Hybrid Methods

since it is nonnegative and equal to zero only when u(x) is constant. We can write the second form of B as B = {u} [D] [D]{u} ,

0 −1 · · · ⎡ ⎤ 1 −1 0 · · · ⎢ ⎥ ⎢−1 2 −1 · · ·⎥ [H] = [D] [D] = ⎢ ⎥. ⎣ 0 −1 2 · · ·⎦ ··· −1 · · · The above choice of [D] is only the simplest in an obvious sequence of derivatives. References [10.6, 8] give explicit forms for the matrices [D] and [H] for higherorder smoothing. An apparent weakness of regularization methods is that the solution procedure does not specify the amount of regularization to be used. This is not exactly true, but choosing an appropriate value for γ is an important aspect of solving inverse problems. First, we must understand that there is no correct value; the basis of the 0

choice can range from the purely subjective (the curvefit looks reasonable) to somewhat objective based on the variances of the data. A reasonable initial value of γ to try is

γ = tr [A] [A] /tr[H] , where tr is the trace of the matrix computed as the sum of diagonal components. This choice will tend to make the two parts of the minimization (model error versus data error) have comparable weights; adjustments (in factors of ten) can be made from there. To emphasize this last point, the purpose of regularization is to provide a mechanism for enforcing on the solution a priori expectations of the behavior of the solution; its determination cannot be fully objective in the sense of only using the immediate measured data. We will see in the following examples that other judgments can and should come into play. For example, suppose that we are determining a transient force history: before the event begins, we would expect the reconstructed force to show zero behavior, and if it is an impact problem we would expect the force to be of only one sign. Subjectively judging these characteristics of the reconstructions can be very valuable in assessing the amount of regularization to use.

10.2 Parameter Identification Problems In some ways the parameter identification (ID) problem is simpler than the force identification problem, so we will begin with it. It will be a good vehicle to illustrate the concepts behind the sensitivity response method. We will generalize the programming to the case of many measurements and parameters (and hence the possible need for regularization). Since the parameters can be of very different types (e.g., dimension, modulus, boundary condition), we will also include scaling of the data and parameters. It should be mentioned, however, that there is one category of parameter ID problem that is in a class of its own: the location problem (force location, damage location, and so on). What makes these so difficult is that, in addition to a search over the parameter space, they also involve a global search over the physical space of the structure. These problems are not considered here, but note is made of [10.17], which introduces some innovative ideas using the sensitivity response method.

235

10.2.1 Sensitivity Response Method (SRM) The dynamics of a general nonlinear system are described in detail in [10.12] and given by the relationship ¨ + [C]{u} ˙ = {P} − F(u) , [M]{u}

(10.1)

where [M] is the mass matrix, [C] the damping matrix, {P} the vector of applied loads, and {F} the vector of element nodal forces. The latter is a function of the deformations and would therefore be updated as part of an incremental/iterative-type solution procedure. How this is done is of no immediate concern here since we will treat the FEM as a black box that provides solutions to fully specified forward problems. Consider a situation in which we have measured a set of response histories di (t) for a known load excitation {P(t)}, and there are M unknown parameters a j that need to be identified. Let us have a set of reasonable initial guesses for the unknown parameters, then solve

Part A 10.2

where [D] is the [(M − 1) × M] first difference matrix and [H] is the [M × M] symmetric matrix given by, respectively, ⎡ ⎤ −1 1 0 · · · ⎢ ⎥ [D] = ⎣ 0 −1 1 · · ·⎦ ,

10.2 Parameter Identification Problems

Part A

Solid Mechanics Topics

Part A 10.2

StrlDent: Make guess {a}0 Iterate ∂u Compute: ψ = ∂a Solve: ~ [ΨT W Ψ + γH]{P } T = {Ψ Wd} – {ΨTWu0}

FEM: {a}, {a + Δa}

+ {ΨTW Ψ P 0} Update: a = a + Δa

Script file

SDF

GenMesh

{P}

Script file

{u}

StaDyn/NonStaD

where a¯ j is a normalizing factor, typically based on the search window for the parameter. The small change da j should also be based on the allowable search window; typically it is taken as 5%, but in some circumstances it could be chosen adaptively. Additional discussions of this point are given in [10.6, 18]. The response solution corresponding to the true parameters is different from the guessed parameter solution according to u j − u 0 Δa j {u} = {u}0 + da j j = {u}0 + {ψ} j Δ P˜ j , j

Fig. 10.4 Algorithm schematic of the distributed computing rela-

tionship between the inverse program and the FEM programs for implementation of the sensitivity response method (SRM) for parameter identification

the system [M]{u} ¨ 0 + [C]0 {u} ˙ 0 = {P} − {F}0 , where the subscript 0 means that the forward problem was solved with the initial guesses. In turn, change each of the parameters (one at a time) by the definite amount a j −→ a j + da j and solve ˙ j = {P} − {F} j . ¨ j + [C]0 {u} [M]{u} The sensitivity of the solution to the change of parameters is constructed from u j − u0 a¯ j , {ψ} j ≡ da j a)

Δa j . Δ P˜ j ≡ aj It remains to determine Δa j or the equivalent nondimensional scales Δ P˜ j . Replacing Δ P˜ j with P˜ j − P˜ 0j , and using subscript i to indicate measurements di , we can write the error function as Wi di − Qu0i + Q ψij P˜ 0j A= i j 2 −Q ψij P˜ j , j u0i = u xi , a0j , where Q acts as a selector. This is reduced to standard form by minimizing with respect to P˜ j to give

˜ [QΨ ] W[QΨ ] + γ [H] { P} = [QΨ ] W{d − Qu0 + Qψ P˜ 0 } . b)

b L = 254 mm b = 25.4 mm h = 6.3 mm 40 elements 2024 aluminum

h

L

#1

L

#2

L

P (t)

Acceleration (g)

236

υ1

100 0

–100

2

υ2

P (t) 0

2

4

6

Fig. 10.5a,b An impacted aluminum beam: (a) geometry and properties and (b) recorded accelerations

8 10 Time (ms)

Hybrid Methods

#1

Acceleration (g)

Iterated

100 0 –100

100 0 –100 Nominal

Nominal

1000

Part A 10.2

Acceleration (g)

237

#2

Forward Recorded

Iterated

0

10.2 Parameter Identification Problems

2000

3000

4000

5000 Time (μs)

0

1000

2000

3000

4000

5000 Time (μs)

Fig. 10.6 Comparison of the recorded responses and the forward solution responses

The system of equations is solved iteratively with the new parameters used to compute updated responses and sensitivities. An implementation of this approach is shown in Fig. 10.4: StrIDent is the inverse solver, GenMesh, the FEM modeler program, produces the changed structure data files (SDF) that are then used in the FEM (StaDyn/NonStaD) programs. Note that there is a nice separation of the inverse problem and the FEM analysis. The significant advantage of the algorithm is that it is independent of the underlying problem, be it static/dynamic, linear/nonlinear, and so on.

10.2.2 Experimental Data Study I: Measuring Dynamic Properties This example shows how the dynamic properties of an impacted beam were obtained. Note that wave propagation characteristics are dominated by the ratio √ c0 = E/ρ and therefore wave responses are not the most suitable for obtaining separated values of E and ρ. Since mass density can be easily and accurately determined by weighing, we will illustrate the determination of the Young’s modulus and damping. Fig. 10.8a–c Pressure-loaded circular plate: (a) geometry and gage positions, (b) center mesh, and (c) the elastic

boundary condition, modeled with an overhang of unknown modulus

Pressure gauge

Ring and bolts

Pressurized cylinder

Plate and gages p

ε1

ε2

ε3

ε4

ε5

80

778

292

590

140

140

Fig. 10.7 Photograph of pressurized plate and maximum recorded

strains

a)

b) 5 3

1

2

4 Diameter: 107mm (4,214 in.) Thickness: 3.3 (0.131in.) 2024 T-3 aluminum Gage: EA-13-062AQ-350

c)

p (x,y) = specified

E2 = ?

238

Part A

Solid Mechanics Topics

a) Strain (µε)

b) Strain (µε)

1200

–4 FEM Theory Experiment

1000

1200 FEM Experiment

1000

Part A 10.2

800

1

800

600

3

600

400

400 2

200

5

0

200 0

0

20

40

60 80 Nominal pressure (psi)

0

20

40

60 80 Nominal pressure (psi)

Fig. 10.9a,b Pressurized plate comparisons: (a) plate with fixed boundary and (b) plate with identified flexible boundary

Data frame

P(l ) P(t)

a0

Symmetry

a0: 6.35 mm L = 63.5 mm 2h = 12.7 mm b = 25.4 mm Aluminum

Fig. 10.10 Double cantilevered beam specimen: geometry, properties and mesh

Figure 10.5 shows the geometry and the acceleration data recorded for the impact of the beam. The force transducer described in [10.6] was used to record

the force history and PCB 303 A accelerometers (made by Piezotronics, Inc.) were used for the response. The DASH-18 system recorded the data. The impact was on the wide side of the beam, which increased the amount of damping due to movement through the air. The damping is implemented in the FEM code as [C] = α[M] = (η/ρ)[M], which is a form of proportional damping. The parameter α has units of s−1 and this will be identified along with the modulus. The search window for this type of problem can be set quite narrow since we generally have good nominal values. Consequently, convergence was quite rapid (three or c)

a) u (x, y) δ = 64 nm σ0 = 241MPa

υ (x, y)

δ

b)

σ0

0

5

10

15 Iterations

Fig. 10.11a–c Double cantilever beam synthetic data: (a) moiré u(x, y) and v(x, y) fringes at 10.6 μs, (b) deformed shape (exaggerated ×10), and (c) convergence behavior using a single frame of data with 0.2% noise

Hybrid Methods

a)

10.2 Parameter Identification Problems

239

b)

P21 P19

a (t)

(60×103 in./s) a· (t) P3 P1 0

5

10

15 Time (μs)

0

5

10

15

20

25 30 Time (μs)

Fig. 10.12a,b Reconstruction of the DCB specimen behavior. (a) Cohesive force history. (b) Crack tip position and speed

four iterations) and the values obtained were Nominal:

c0 = 5080.00 m/s (200 000 in./s) ,

Iterated:

α = 0.0 s−1 , c0 = 4996.18 m/s (196 700 in./s) , α = 113.37 s−1 .

Figure 10.6 shows the response reconstructions using the nominal values and the iterated values compared to the measured responses. It is apparent that taking the damping into account can improve the quality of the comparisons considerably. To improve the correspondence over a longer period of time would require iterating on the effective properties of the clamped condition. Physically, the beam was clamped in a steel vise and not fixed as in the model. For wave propagation problems, some energy is transmitted into the vise. Determining effective boundary conditions is discussed next.

10.2.3 Experimental Data Study II: Measuring Effective BCs This second experiment is that of a plate with a uniform pressure. In this case, the pressure loading is actually known with a good deal of confidence but, because the plate is formed as the end of a pressurized cylinder with bolted clamps, the true boundary conditions are not rigidly clamped as was originally intended. The ob-

jective will be to get a more realistic modeling of the boundary condition. A photograph of the plate forming the end of a cylinder is shown in Fig. 10.7. Sixteen bolts along with an aluminum ring are used to clamp the plate to the walls of the cylinder. Note that the cylinder has an increased wall thickness to facilitate this. Figure 10.8 shows the geometry of the plate and the gage locations. Gages 4 and 5 are not exactly on the edge of the plate, but the gage backing was trimmed so that they could be put as close as possible to the edge. Each gage ended up being approximately 1.8 mm (0.07 in.) from the edge; the meshes, such as that in Fig. 10.8b, were adjusted so as to have a node at the gage positions. Figure 10.9a shows the recorded strains. There is very good linearity with very little offset; consequently, the data was used directly as part of the inverse solutions. The strains at the maximum pressure are shown in Fig. 10.7. Figure 10.9a shows that modeling the boundary condition as fixed does not give a good comparison with experiment. Figure 10.8b models the problem as an overhanging plate with specified thickness but unknown modulus; the pressure load will be taken as specified. With E 2 as the single unknown, the converged result gave E 2 = 0.26E 1 and the reconstructions are shown in Fig. 10.9b; the standard deviation in the strain is 18 μe. The comparison between the model and ex-

Part A 10.2

1500m/s

240

Part A

Solid Mechanics Topics

~

10

1 × P1 ~

5

1 × P2 +

Part A 10.3

~

{ψ}1 × P1

+

=

~

{ψ}2 × P2

=

{d}

Fig. 10.13 The data {d} are conceived as synthesized from the superpositions of unit-load solutions {ψ}1 and {ψ}2 modified by scalings P˜1 and P˜2

periment is considerably improved. The results hardly changed when both moduli were assumed simultaneous unknowns.

10.2.4 Synthetic Data Study I: Dynamic Crack Propagation A theme of this work is the central role played by the model in stress analysis. We illustrate here how it can be used in the design of a complicated experiment. That is, it is generally too expensive in both time and money to design experiments by trial and error and a wellconstructed model can be used to narrow the design parameters. To make the scenario explicit, consider a double cantilever beam (DCB) specimen made from two aluminum segments bonded together as shown in Fig. 10.10. The crack (or delamination) is constrained to grow along the interface, and the interface properties of the bond line are the basic unknowns of the experiment. This delamination is modeled as a cohesive layer described by σ (v) = σ0 [v/δ] e(−v/δ) , where v is the separation distance and (σ0 , δ) are the parameters to be identified. Reference [10.19] contains details about this type of crack modeling that is implemented in the nonlinear FEM code NonStaD. To get accurate estimates of the interface properties, it is necessary to record the data as the crack passes

through the frame of view. Typical frames are shown in Fig. 10.11a along with the exaggerated deformed shape in Fig. 10.11b. The applied load is a rectangular pulse of long duration with a rise time of 3 μs and, for identification purposes, is taken as known. This is a problem with a wide range of scales. As seen from Fig. 10.11b, the tip deflection is of order h/10 but h is of order δ × 105 . At rupture, the deflections along the line of symmetry are 1δ ∼ 10δ. To increase the accuracy of the data, the frames came from just the 12 mm × 12 mm window indicated in Fig. 10.10a. The noise added to the synthetic data had a specified standard deviation of 0.2 × δ; with a moiré sensitivity of 100 line/mm, this works out to about 0.2% of a fringe. The convergence behavior is shown in Fig. 10.11c where a single v(x, y) frame of data at 10.6 μs was used; both parameters have converged by the tenth iteration. The results are robust with reasonable initial guesses for the parameters. In the present tests, the allowable search windows spanned 10–200% of the exact values, with the initial guesses being 50%. Keep in mind that, during the iteration process, at each iteration a complete crack propagation problem is solved three times (once for the nominal parameters, and once each for the variation of the parameters); thus the computational cost is quite significant. Once the parameters are determined, reconstructions of any quantity can be obtained, which is the utility of the hybrid methods. Figure 10.12a shows, for example, the interface force histories at every other node. These histories are almost identical to the exact histories, which is not surprising because of the goodness of the parameter estimates. The quality of these reconstructions is greatly superior to those achieved in [10.20], which determined the interface forces directly; this underlines the value of a reparameterization of unknowns of a problem. The position of the crack tip is quite apparent in Fig. 10.12a; using the peak value of the force as the indicator, Fig. 10.12b shows the crack tip position and its speed. The speed is not constant because of the decrease in beam stiffness as the crack propagates.

10.3 Force Identification Problems In the inverse problems of interest here, the applied loads and displacements are unknown but we know some information about the responses (either in the

form of displacements or strains). In particular, assume that we have a vector of measurements d(t) which are related to the structural degree of freedom (DoF) ac-

Hybrid Methods

cording to {d(t)} ⇔ [Q]{u(t)} ,

10.3.1 Sensitivity Response Method for Static Problems Since there is a relation between {u} and {P}, then (for minimization purposes) we can take either as the basic set of unknowns; here it is more convenient to use the loads because for most problems the size of {P} is substantially smaller than the size of {u}. Let the unknown forces be represented as the collection ˜ = [Φ]{ P} ˜ , (10.2) {P} = [{φ}1 {φ}2 · · · {φ} Mp ]{ P} where each {φ}m is a known distribution of forces and P˜m the corresponding unknown scale. As a particular case, {φ}m could be a single unit force, but it can also represent a uniform or a Gaussian pressure distribution; for traction distribution problems it has a triangular distribution [10.20]. We will refer to each {φ}m distribution as a unit or perturbation load; there are Mp such loads. Solve the series of forward FEM problems [K ]{ψ}1 = {φ}1 , [K ]{ψ}2 = {φ}2 , [K ]{ψ} Mp = {φ} Mp .

... ,

The {ψ}m responses will have imbedded in them the effects of the complexity of the structure; consequently, usual complicating factors (for deriving analytical solutions used in hybrid methods) such as geometry, material, and boundary conditions are all included in {ψ}m , which in turn is handled by the finite element method. Furthermore, note that the responses are not necessarily just displacement; they could be strain or principal stress difference as required by the experimental method, and consequently the inverse program does not need to perform differentiation to get strains. In other words, anything particular to the modeling or mechanics of the structure is handled by the FEM program. We can therefore write the actual forward solution as ˜ , {u} = [Ψ ]{ P} [Ψ ] = [{ψ}1 {ψ}2 · · · {ψ} Mp ] , (10.3)

˜ is not known. In this way, the coefficients although { P} ˜ act as scale factors on the solutions in [Ψ ]. of { P} This idea is shown schematically in Fig. 10.13 where the two-load problem is conceived as the scaled superposition of two separate one-load problems. This superposition idea leads to simultaneous equations to ˜ solve for the unknown scales { P}. The measured data at a collection of locations are related to the computed responses as ˜ . {d} ⇐⇒ [Q]{u} = [Q][Ψ ]{ P} If supplementary information is available (e.g., additional data or a relationship among the unknowns), it is incorporated simply by appending to this set of equations before the least-squares procedure is used. The error functional can be written as a function of P˜ only and minimized to get ˜ = [QΨ ] W{d} . [Ψ Q WQΨ + γ H]{ P} (10.4)

This reduced system of equations is of size [Mp × Mp ] and is relatively inexpensive to solve since it is symmetric and small. Once the loads are known, the actual forward solution is then obtained from (10.3), which involves a simple matrix product. The key to understanding our approach to the computer implementation is the series of solutions in the matrix [Ψ ] – it is these that can be performed externally (in a distributed computing sense) by a commercial FEM code. The basic relationship is shown in Fig. 10.14 and is laid out in a similar style to Fig. 10.4. If the number of unknowns are computationally too large, then a parameterized loading scheme can be used, where the loading is assumed to be synthesized from speci-

StrlDent: Form: – – – [A ] = [Q ][Ψ ] –

φm (t)

FEM: GenMesh StaDyn

–

[B ] = [A ]T{d} Solve: – – – – – [A TA + γH ]{P } = {B }

Script file

{ψ}m (t)

Response file

Fig. 10.14 Algorithm schematic of the distributed computing relationship between the inverse program StrIDent and the FEM programs. For static problems, the FEM program is run once for each load position; for dynamic problems, a {ψ}m (t) history at all data locations is computed for each force location

241

Part A 10.3

where [Q] is a selection matrix that plays the role of selecting the subset of responses participating in forming the data. We want to find the forces {P(t)} that make the system best match the measurements. We will introduce the ideas behind the method by first considering static problems, and then generalize them to the dynamic case.

10.3 Force Identification Problems

242

Part A

Solid Mechanics Topics

fied forms [10.21]. The experimental study to follow is a simple example.

10.3.2 Generalization for Transient Loads

of the structure. Because of the linearity of the system, the responses are also similar to each other but shifted an amount mΔT . This basic idea is illustrated in Fig. 10.15. The error functional is

Let the force history be represented as

Part A 10.3

P(t) =

Mp

P˜m φm (t) ,

˜ , {P} = [Φ]{ P}

(10.5)

m=1

where φm (t) are specified functions of time; they are similar to each other but shifted an amount mΔT . We will consider specific forms later. The array [Φ] has as columns each φm (t) discretized in time. The vector {P} has the force history P(t) discretized in time. This force causes the response at a particular point in the model u(x, t) =

Mp

P˜m ψm (x, t) ,

A = {d − Qu} W{d − Qu} ˜ . ˜ W{d − QΨ P} = {d − QΨ P} Force reconstructions show the manifestations of illconditioning as ΔT is changed because, with small ΔT , the data are incapable of distinguishing between neighbor force values Pn + Δ, Pn+1 − Δ. Time regularization is therefore important and connects components within ˜ that is, { P}, ˜ = { P} ˜ [H]{ P} ˜ . ˜ [D] [D]{ P} B = { P} Minimizing with respect to P˜m then gives

˜ = [QΨ ] W{d} . [QΨ ] W[QΨ ] + γ [H] { P}

˜ , {u(x)} = [Ψ ]{ P}

m=1

(10.6)

(10.7)

where ψm (x, t) is the response to force φm (t) at the particular point, and [Ψ ] are all the discretized responses arranged as columns. As in the static case, we refer to φm (t) as unit or perturbation forces and ψm (t) as sensitivity responses. These responses are obtained by the finite element method, and therefore will have imbedded in them the effects of the complexity

The core array is symmetric of size [Mp × Mp ]. There are a variety of functions that could be used for φm (t); the main requirement is that they have compact support in both the time/space and frequency domains. Reference [10.22] used a Gram–Schmidt reduction scheme to generate a series of smooth orthogonal functions from the basic triangular pulse; a set

a) P (t)

b) Actual force

φ (t)

Pertubation force

ψ (t)

Sensitivity response

Time P (t) P3 P2 P1

P4

P5

P6

Measured response

P7 Time

0

200

400

600

800

1000 1200 1400 Time (μs)

Fig. 10.15a,b The sensitivity response concept for transient load identification. (a) A discretized history is viewed as a series of scaled triangular perturbations. (b) How the scaled perturbations superpose to give the actual histories

Hybrid Methods

¯ = {[Φ]1 { P} ˜ 1 , [Φ]2 { P} ˜ 2 , · · · , [Φ] Np { P} ˜ N p } , { P} ˜ n is the history Pn (t) arranged as a vector where each { P} of Mp time components. This collection of force vectors causes the collection of responses ⎫ ⎡ ⎧ ⎤ ⎪ [Ψ ]11 [Ψ ]12 · · · [Ψ ]1Np {u}1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎬ ⎢ ⎨ {u}2 ⎪ ⎢ [Ψ ]21 [Ψ ]22 · · · [Ψ ]2Np ⎥ ⎢ ⎥ .. ⎪ = ⎢ .. .. .. .. ⎥ ⎪ ⎪ ⎪ . ⎣ . ⎦ . . . ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ {u} Nd [Ψ ] Nd 1 [Ψ ] Nd 1 · · · [Ψ ] Nd Np ⎫ ⎧ ˜ 1 ⎪ ⎪ { P} ⎪ ⎪ ⎪ ⎪ ⎪ ˜ ⎪ ⎨ { P}2 ⎬ (10.8) × . ⎪ ⎪ .. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˜ Np ,⎭ { P} where {u}i is a history of a displacement (DoF) at the i measurement location. We also write this as ˜ j or {u} ¯ . ¯ P} [Ψ ]ij { P} {u}i = ¯ = [ψ]{ j

The [Mp × Mp ] array [Ψ ]ij relates the i location response history to the j applied perturbation force. It contains the responses ψm (t) arranged as a vector. Let the Nd sensor histories of data be organized as ¯ = {{d}1 , {d}2 , · · · , {d} Nd } , {d}

243

where {d}n is a vector of Md components (it is simplest to have Md = Mp but this is not required). The measured data DoF are a subset of the computed responses, that is, ¯ u} ¯ i ⇐⇒ [Q]{ {d} ¯ or

¯ ψ]{ ¯ ⇐⇒ [Q ¯ . ¯ P} {d}

(10.9)

Define a data error functional as ¯ ψ¯ P} ¯ ψ¯ P} ¯ ¯ = {d¯ − Q ¯ W{d − Q E( P)

¯ [Ht ]{ P} ¯ + γs { P} ¯ [Hs ]{ P} ¯ , + γt { P} which has incorporated regularization in both time [Ht ] and space [Hs ]. Time regularization connects components within each P˜ j only, while space regularization connects components in P˜ j across near locations. The sparsity of the matrices [Ht ] and [Hs ] are quite different and therefore their effect on the solution is also quite different, even for same order regularizations. Minimiz¯ gives ing with respect to { P}

¯ ψ] ¯ ψ] ¯ ¯ + γt [Ht ] + γs [Hs ] { P} ¯ W[Q [Q ¯ ψ] ¯ . ¯ W{d} = [Q

(10.10)

The core matrix is symmetric, of size [(Mp × Np ) × (Mp × Np )], and in general is fully populated. Only when Np is large (there are a large number of force histories) does the computational cost of solving (10.10) become significant. The computer implementation is shown in Fig. 10.14. The sensitivity response is computed only once for a given force location irrespective of the number of sensors, hence the FEM cost is simply that of a forward solution (times the number of forces).

10.3.3 Experimental Data Study I: Double-Exposure Holography Figure 10.1c shows a double-exposure holographic fringe pattern for the out-of-plane displacement of a pressure-loaded plate. The plate forms one end of a pressurized cylinder and the inner white dots in the figure show the flexing part of the plate. The small change of pressure was applied accurately using a manometer. As sometimes happens with holography, the fringe pattern is not of uniform quality. Furthermore, there is a slight asymmetry in the fringe pattern which could be due to a slight overall rigid-body motion of the cylinder. Figure 10.16b shows the limited number of discretized data points used. Because of the expected symmetries in the results the points were distributed somewhat uniformly so as to provide averaging. Figure 10.16c shows

Part A 10.3

of these is illustrated in Fig. 10.15b. Here we will use the triangular pulse directly; this is equivalent to using a linear interpolation between the discretized points, as shown in Fig. 10.15a. The advantage of the triangle is that the scales P˜m have the direct meaning of being the discretized force history and hence there is no need to actually perform the summation in (10.5). In fact, without losing the advantage of the straight triangular pulse, it is also possible to use a smoothed triangular pulse; this is the scheme implemented in StrIDent. There are a number of features we now wish to add to the above developments, primarily the ability to determine multiple unknown force histories. There are two situations of interest here: the first is that of multiple isolated forces and the second is traction distributions. The main difference is whether the forces are related to each other or not; isolated forces are a finite number of forces which are not related, but traction distributions contain very many forces which are continuously related to their neighbors and therefore space regularization can be used. Let the Np unknown force histories be organized as

10.3 Force Identification Problems

244

Part A

Solid Mechanics Topics

a)

b)

c)

Diameter: 107 mm (4.215 in) Thickness: 3.12 mm (0.123 in) 6061 aluminum

Part A 10.3

α: 17 deg, β: 0 deg λ: 632nm (24.9 μin.)

Fig. 10.16a–c Holographic experiment of a pressure-loaded plate. (a) Cylindrical pressure vessel properties and holographic parameters. (b) Discretized data positions. (c) FEM mesh

a)

b)

Nominal: 7.63 kPa (1.09psi) Computed: 7.77 kPa (1.11psi)

σxx

σxy

Fig. 10.17a,b Reconstructions. (a) Displacement fringe pattern: top recorded, bottom reconstructed. (b) Stress contours

the FEM mesh used. The data were relocated onto the nearest nodes of this mesh by the scheme discussed in [10.6]. The edge of the plate was modeled as being rigidly clamped. The pressure was taken as the only unknown in formulating the inverse problem. Since the number of

Strain gages P (t)

Steel

Rs (t)

Ra (t)

Accelerometers

data far exceed the number of unknowns, regularization is not required. The nominal measured and computed pressures are given in Fig. 10.17. The comparison is very good. Figure 10.17a shows a comparison of the reconstructed fringe pattern (bottom) with that measured (top). Since the pressure comparison is good, it is not surprising that this comparison is also good. A theme running through the hybrid solutions in this work is that, once the unknown forces (or whatever parameterization of the unknowns is used) have been determined, the analytical apparatus is then already available for the complete stress analysis. Figure 10.17b shows, for example, the σxx and σxy stress distributions. What is significant about these reconstructions is that (in comparison to traditional ways of manipulating such holographic data) the process of directly differentiating experimental data is avoided. Furthermore, this would still be achievable even if the plate was more complicated by having, say, reinforcing ribs (in which case the strains are not related to the second derivatives of the displacement). Note also that, because the experimental data were not directly differentiated, only a limited number of data points needed to be recorded.

aluminum

51

Accelerometers #1 #2 L

L = 305 mm D = 102 mm h = 6.3 mm 6061 aluminum

Fig. 10.18 Square plate with a central hole. The Hopkinson bar is

attached to the left side and its two strain gages are sufficient to determine both the input load and the transmitted load

10.3.4 Experimental Data Study II: One-Sided Hopkinson Bar It often occurs that we need to apply a known force history to a structure. In some instances, where the frequency of excitation is low, we can use an instrumented hammer as is common in modal analysis [10.23]. In other instances, we need a high-energy high-frequency input, and common force transducers do not have the required specifications. The one-sided Hopkinson bar

Hybrid Methods

b)

Strain (ue)

0

0.1

Part A 10.3

100

0 –0.1

–100

0

1000

2000

3000

4000 Time (μs)

0

1000

2000

3000

4000 Time (μs)

Fig. 10.19a,b Experimentally recorded data. (a) Strains from the OSHB. (b) Velocities of the plate computed from measured accelerations

force input to the plate. These force reconstructions are shown labeled as P and Rs , respectively, in Fig. 10.20. The impact force, as expected, is essentially that of a single pulse input. The reconstructions of the force at the connection point, on the other hand, are quite complicated, persisting with significant amplitude for the full duration of the recording. Note that the connection force is larger than the impact force because of the high impedance of the plate. The Rs force from the bar was used as input to the forward problem for the plate and the computed

P (t) Accelerometers Hopkinson bar

R s (t), R 2a (t)

#2 1

P/P0

(OSHB) was designed for this purpose. More details and examples can be found in [10.24, 25]. Figure 10.18 shows a schematic of a problem used to test the ability of accelerometers to reconstruct forces on a twodimensional plate. This force will be compared to that determined by the strain gage responses from the onesided Hopkinson bar. Figure 10.18 shows the specimen dimensions and placement of the accelerometers. The specimen was machined from a 6.3 mm (1/4 in.)-thick sheet of aluminum to be a 305 mm (12 in.) square with a 76 mm (4 in.)-diameter hole at its center. A small hole was drilled and tapped at the midpoint of the left edge for the connection of the OSHB. Accelerometers were placed on the same edge as the force input and on the opposite edge, at 25.4 mm (2 in.) off the force input line. The data were collected at a 4 μs rate over 4000 μs. The force input to the OSHB was from the impact of a steel ball attached to a pendulum. The acceleration data were converted to velocities and detrended, the strain gage data were modified to account for the slight nonlinearity of the Wheatstone bridge (since the resistance change of semiconductor gages is relatively large). These data are shown in Fig. 10.19. The inverse problem was divided into two separate problems corresponding to the bar and the plate. The data from the two strain gages on the bar were used simultaneously to reconstruct the impact force and the

245

Measured Forward

Velocity (m/s)

a)

10.3 Force Identification Problems

R s (t), R 1a (t)

0 #1

0

1000

2000

3000

4000 Time (μs)

Fig. 10.20 Force reconstructions for the plate with a hole. The reference load is P¯0 = 2224 N (500 lb)

246

Part A

Solid Mechanics Topics

Part A 10.4

velocities are compared to the measured in Fig. 10.19. The comparison is quite good considering the extended time period of the comparison. The data from the accelerometers were then used as separate single sensor inputs to reconstruct the force at the connection point. These force reconstructions are shown labeled as R1a and R2a in Fig. 10.20. The reconstructed forces at the connection point agree quite closely both in character and magnitude. This is significant because the force transmitted across the boundary

is quite complex and has been reconstructed from different sensor types placed on different structural types. Reference [10.6] discusses a collection of possible variations of the one-sided Hopkinson bar. The basic idea expressed is that, because a general finite element modeling underlies the force identification method, and because we can determine multiple unknown forces simultaneously, we are given great flexibility in the design of the rod and its attachment to the structure.

10.4 Some Nonlinear Force Identification Problems Nonlinearities can arise in a number of forms: material nonlinearity such as plasticity and rubber elasticity, geometric nonlinearity as occurs with large displacements and rotations, and nonlinearities associated with the data relation. We discuss examples of each.

10.4.1 Nonlinear Data Relations In photoelasticity there is a nonlinear relationship between observed fringes and applied loads. To see this, recall that fringes are related to stresses by [10.26]

fσ N h

2 2 = (σxx − σ yy )2 + 4σxy .

From this, while it is true that a doubling of the load doubles the stress and therefore doubles the fringe orders, this is not true for nonproportional loading. That is, the fringe pattern caused by two loads acting simultaneously is not equal to the sum of the fringe patterns of the separate loads even though the stresses would be equal. Since our inverse method is based on superposition, we must modify it to account for this nonlinear relation. Based on this discussion, it may seem that photoelasticity is not as attractive as other methods. This is perhaps true, but the main reason it is included here is as an example of data with a nonlinear relation to mechanical variables. It may well be that sensors of the future will also have a nonlinear relationship and the algorithms developed here will be available to take advantage of them. For example, there is currently a good deal of activity in developing pressure-sensitive paints, fluorescent paints, and the like; see [10.27, 28] for a discussion of their properties and some of the literature. These inexpensive methods, most likely, will have

a nonlinear relation between the data and the mechanical variables. We need to generate a linear relation between the fringe data and the applied loads. As usual for nonlinear problems, we begin with a linearization about a known state. That is, suppose we have a reasonable estimate of the stresses as [σxx , σ yy , σxy ]0 ; then the true stresses are only a small increment away. A Taylor series expansion about this known (guessed) state provides

2

fσ 2 N ≈ (σxx − σ yy )2 + 4σxy 0 h + 2 (σxx − σ yy )0

× (Δσxx − Δσ yy ) + (4σxy )0 Δσxy . Let the stress increments (Δσxx , Δσ yy , Δσxy ) be computed from the scaled unit loads so that for all components of force we have {ψxx , ψ yy , ψxy } j Δ P˜ j . {Δσxx , Δσ yy , Δσxy } = j

The stress sensitivities {ψ} j must be computed for each component of force. Noting that Δ P˜ = P˜ − P˜0 , we write the approximate fringe relation as

2 fσ G¯ j P˜o j + G¯ j P˜o j (10.11) N ≈ σ02 − h j

j

with

2 σ02 ≡ (σxx − σ yy )2 + 4σxy , 0 G¯ j ≡ [2(σxx − σ yy )0 (ψxx − ψ yy ) j + 8(σxy )0 ψxy j ] . Equation (10.11) is our linear relation between fringes and applied loads. Let the data be expressed in the form

Hybrid Methods

N¯ ≡ N f σ /h so that the minimization principle leads to

[QG] W[QG] + γ [H] P˜ = [QG] W{ N¯ 2 − Qσ02 + QG P˜0 } , (10.12)

10.4.2 Nonlinear Structural Dynamics The dynamics of a general nonlinear system are described by (10.1) and rewritten here as ˜ − {F} . [M]{u} ¨ + [C]{u} ˙ = {P} − {F} = [Φ]{ P} There are two sets of loads: {P}, which is the set of unknown applied loads to be identified, and the element nodal forces {F}, which depend on the deformations and are also unknown. Both {P} and {F} must be identified as part of the solution. We begin the solution by approximating the element nodal forces by a Taylor series expansion about a known (guessed) configuration ! ∂F {Δu} = {F}0 + [K T ]0 {u − u0 } {F} ≈ {F}0 + ∂u = {F}0 − [K T ]0 {u}0 + [K T ]0 {u} , where the subscript 0 means known state and {F0 } − [K T ]0 {u}0 is a residual force vector due to the mismatch between the correctly reconstructed nodal forces {F} and its linear approximation [K T ]0 {u}0 ; at convergence, it is not zero. The square matrix [K T ]0 is the tangent stiffness matrix. We have thus represented the nonlinear problem as a combination of known forces {F0 } − [K T ]0 {u}0 (known from the previous iteration)

StrlDent: ~

Make guess: {P }

[Φ]

Script file

[Ψ]

StaDyn

{P}

Script file

247

Iterate Update: G, σ0 Solve: ~ [GT WG + γH]{P }

[σxx, σyy, σxy ]

–2

= {GTWN } – {GTWσ02} ~ + {GTWP0}

StaDyn FEM:

Fig. 10.21 Algorithm schematic of the distributed computing relationship between the inverse program and the FEM programs when nonlinear updating is needed

and a force linearly dependent on the current displacements, [K T ]0 {u}. The governing equation is rearranged in the form [M]{u} ¨ + [C]{u} ˙ + [K T ]0 {u} ˜ − {F}0 + [K T ]0 {u}0 . = [Φ]{ P} This is a linear force identification problem with un˜ and known {F}0 and [K T ]0 {u}0 . Solve the known { P} three linear dynamics problems ¨ + [C][ψ] ˙ + [K T ]0 [Ψ ] = [Φ] , [M][ψ] [M]{ψ¨ F } + [C]{ψ˙ F } + [K T ]0 {ψF } = −{F}0 , [M]{ψ¨ u } + [C]{ψ˙ u } + [K T ]0 {ψu } = [K T ]0 {u}0 . Note that the left-hand operator is the same in each case. The total response can be represented as ˜ + {ψF } + {ψu } , {u} = [Ψ ]{ P} and the data relation becomes [Q]{u} − {d} = 0

⇒

˜ = {d} − {QψF } − {Qψu } = {d ∗ } . [QΨ ]{ P} The minimum principle leads to

˜ [QΨ ] W[QΨ ] + γ [H] { P} = [QΨ ] {d − QψF − Qψu } ,

(10.13)

which is identical in form to the linear dynamic case except for the carryover terms {ψF } and {ψu }. The core array is symmetric, of size [Mp × Mp ]; note that it does not change during the iterations and hence the decomposition need be done only once. The right-hand side

Part A 10.4

where, as before, [Q] plays the role of a selector. This ˜ to is solved repeatedly using the newly computed { P} ˜ 0. replace { P} In terms of the computer implementation shown in Fig. 10.21, the sensitivity solution {ψ} is determined only once for each load component, but the updating of the stress field {σ0 } requires an FEM call on each iteration. Both the left- and right-hand side of the system of equations must be reformed (and hence decomposed) on each iteration. In all iterative schemes, it is important to have a method of obtaining good initial guesses. There are no hard and fast rules for the present type of problem; however, the rate of convergence depends on the number of unknown forces, this suggests using minimal forces initially to obtain an idea of the unknowns to use as starter values. Furthermore, the effect of regularization is sensitive to the number of unknowns and therefore it is wise to obtain starter values without regularization.

10.4 Some Nonlinear Force Identification Problems

248

Part A

Solid Mechanics Topics

StrlDent: Make guess: P Form: [ΨTΨ + γH ] Iterate

Part A 10.4

[Φ]

Script file

[Ψ]

StaDyn

{P}

Script file

[F ], [u]

NonStaD

[F ], [u]

Script file

FEM:

Update: P, u, F Solve: ~ [ΨTΨ + γH ] {P } = {ΨTd} +{ΨTψu} +{ΨTψF}

{ψu}, {ψF}

StaDyn

Fig. 10.22 A possible algorithm schematic for the distributed computing relationship between the nonlinear dynamic inverse program and the FEM programs

10

∑ ⇒ {d}

9

3

8

Δt

7

Data frames

6

∑ ⇒ {d}

5

2

4

Whole-field {ψ} snapshots Perturbation φm forces

3 2 1 φm 0

∑ ⇒ {d} 1

1

2

3

4

5

6

7

8

9

10

∑ ⇒ P (t)

Fig. 10.23 Relationship between frames due to the perturbation

forces and the measured frames

a)

b)

P

Fig. 10.24a,b Photoelastic fringe pattern due to distributed load: (a) experimentally recorded and (b) reconstructed

changes during the iterations due to the updating associated with {ψF } and {ψu }.

The basic algorithm is: starting with an initial guess for the unknown force histor