Mechanical Behaviour of Materials

SOLID MECHANICS AND ITS APPLICATIONSVolume 180

Series Editors: G.M.L. GLADWELLDepartment of Civil EngineeringUniversity of WaterlooWaterloo, Ontario, Canada N2L 3GI

Aims and Scope of the Series

The fundamental questions arising in mechanics are: Why?, How?, and Howmuch? The aim of this series is to provide lucid accounts written by authoritativeresearchers giving vision and insight in answering these questions on the subject ofmechanics as it relates to solids.

The scope of the series covers the entire spectrum of solid mechanics. Thus itincludes the foundation of mechanics; variational formulations; computationalmechanics; statics, kinematics and dynamics of rigid and elastic bodies: vibrationsof solids and structures; dynamical systems and chaos; the theories of elasticity,plasticity and viscoelasticity; composite materials; rods, beams, shells andmembranes; structural control and stability; soils, rocks and geomechanics; fracture;tribology; experimental mechanics; biomechanics and machine design.

The median level of presentation is the first year graduate student. Some textsare monographs defining the current state of the field; others are accessible to finalyear undergraduates; but essentially the emphasis is on readability and clarity.

For further volumes:http://www.springer.com/series/6557

Dominique Francois • Andre Pineau • Andre Zaoui

Mechanical Behaviourof Materials

Volume 1: Micro- and MacroscopicConstitutive Behaviour

123

Dominique FrancoisEcole Centrale de Parisrue des Envierges 1375020 ParisFrance

Andre ZaouiAcademie des Sciences23 Quai de Conti75006 ParisFranceAcademy of EngineeringAvenue F.D. Roosevelt75008 ParisFrance

Andre PineauEcole des Mines de ParisParisTechCentre des Materiaux UMR CNRSB. P. 8791003 Evry CedexAcademy of EngineeringAvenue F.D. Roosevelt75008 ParisFrance

This is the second updated edition of the earlier book entitled Mechanical Behaviour ofMaterials, Volume I, published by Kluwer Academic Publishers in 1998.

ISSN 0925-0042ISBN 978-94-007-2545-4 e-ISBN 978-94-007-2546-1DOI 10.1007/978-94-007-2546-1Springer Dordrecht Heidelberg London New York

Library of Congress Control Number: 2011944979

© Springer Science+Business Media B.V. 2012No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or byany means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without writtenpermission from the Publisher, with the exception of any material supplied specifically for the purposeof being entered and executed on a computer system, for exclusive use by the purchaser of the work.

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Springer is part of Springer Science+Business Media (www.springer.com)

Foreword

Man discovered a long time ago by a succession of trials and errors the way toproduce steel and to increase its hardness by quenching. The empirical recipeswere more or less kept secret within craft guilds. It is not until 1772 that Rene-Antoine Ferchault de Reaumur1 shed some light on the difference in carboncontent of cast iron and steel. He thus opened the way, and Alexandre-TheophileVandermonde, Gaspard Monge, Claude Louis Berthollet2 after him, to a truescientific understanding of metallurgy, enabling to master transformations and theeffects of metal treatments. Scientific researches following, among which shouldbe emphasised the pioneering works of Henry le Chatelier, of Floris Osmond,of Georges Charpy and of Leon Guillet, allowed understanding the chemistry ofalloys, the links between microstructure and mechanical behaviour and improvingindustrial processes. The science of metals thus founded opened the way to the widersubject of materials science. In the same way the practical problems of constructionled, earlier than metallurgy as shown by the interest of Leonardo da Vinci andGalileo Galilei for the strength of materials, scientists like Robert Hooke, JosephLouis Lagrange, Leonhard Euler, Augustin Louis Cauchy to build solid mechanicsas a branch of applied mathematics. For a long time the constitutive equationsneeded in structural mechanics remained crude idealisations of actual behaviour.The pioneers in this field could correspond equally well with their peers aboutmetallurgy (or alchemy) as about mechanics (or astrology). Later, scientists havebecome more and more specialised, and there is yet not enough overlap betweenmaterials science and solid mechanics.

As technical equipment of ever-greater sophistication has become available, therisk of catastrophes, of a scale that can affect the environment and kill many people,

1Rene-Antoine Ferchault de Reaumur (1772) L’art de convertir le fer forge en acier et l’artd’adoucir le fer fondu; ou De faire des Ouvrages de fer fondu aussi finis que de fer forge, MichelBrunet Paris.2Vandermonde A T, Monge G and Berthollet C (1790) Avis aux ouvriers en fer sur la fabricationde l’acier. Comite de salut public.

v

vi Foreword

has increased; and safety has become a major concern. As we write this foreword,hour after hour we learn that the consequences of the earthquake and tsunami inJapan keep being more and more severe. Economic considerations press for longerlifetimes and smaller safety factors; these generate strong incentives to use morerealistic constitutive equations and better failure criteria in the calculations, and thecomputer now makes this possible. Materials design has become much more of apractical possibility, and materials can be produced with better and more reliableproperties.

All this shows that establishing relations, as quantitative as possible, between themicrostructure of materials and their macroscopic properties is nowadays essential.Thanks to fruitful cooperation between materials scientists and solid-mechanicsspecialists, recent research has led to promising achievements in this direction;but the number of training programs covering both fields, which we considered toremain low when writing the first edition of this book, tends even to decrease. Itwas the awareness of the need for advanced courses here that led us, some 26 yearsago, to create in France what was called a Diplome d’Etudes Approfondies (DEA) –Advanced Studies Diploma – with the title “Mecanique et Materiaux” – “Mechanicsand Materials”. The notes provided for the courses were the root of two bookswritten in French3 concerning mechanical properties of materials. The need wasprobably greater in France than in English-speaking countries, where the famousbook of McClintock and Argon, Mechanical Properties of Materials, was alreadymuch in use. This, however, was published in 1966 and so did not deal with recentdevelopments. This gave us the incentive to embark on writing these books, eventhough we felt that it was hard to match McClintock and Argon.4 The D.E.A.“Mecanique et materiaux” trained some 500 students. It is estimated that about300 of those then prepared a PhD. They have pursued careers in University andIndustry, contributing to continued technical progresses. Unfortunately, for someof those highly political reasons, the D.E.A. was discontinued. As a consequence,textbooks about mechanics and materials are probably more than ever needed.

In the mean time, Kluwer asked us the permission to translate the books inEnglish. Dr. Jack Howlett was appointed for this thankless work. It gave birth totwo new books: “Mechanical behaviour of materials” in the Solid mechanics andits applications series.5 Apparently, they met a certain audience, so that Springer,continuing the series asked us to prepare a new edition. When we agreed to proceed,we underestimated the amount of revision required. Now at last, we are glad to finishVolume I.

3Francois D, Pineau A and Zaoui A (1991, 1992) Comportement mecanique des materiaux.I Elasticite et plasticite. II Viscoplasticite, endommagement, mecanique de la rupture, mecaniquedu contact. Hermes Paris.4Argon AS and Mc Clintock FA (1966) Mechanical behaviour of materials. Addison Wesley,Reading.5Francois D, Pineau A and Zaoui A (1998) Mechanical behaviour of materials. I Elasticity andplasticity. II Viscoelasticity, Damage, Fracture Mechanics, Contact Mechanics. Kluwer AcademicPub., Dordrecht.

Foreword vii

The organisation of this volume follows the main classes of mechanicalbehaviour: elasticity, elastoplasticity, elastoviscoplasticity and viscoelasticity.Throughout we attempt to describe the physical processes that are responsiblefor the kinds of behaviour studied, the way in which the constitutive equationscan represent the behaviour and how they relate to the microstructures. Revisingthe book, we improved much the existing material, in particular in modifyingthe organisation, and we added new up to date content. Understanding the subjectmatter requires a good knowledge of solid mechanics and materials science; we givethe main elements of these fields in a set of annexes at the end of the first volumes.We thought interesting for the readers to give as footnotes some information aboutthe many scientists whose names are attached to theories and formulae and whosememories must be celebrated. Wikipedia proved extremely helpful in doing so.

We are now to undertake the revision of the second volume, which will bedevoted to fracture mechanics and damage as well as elements of contact mechanics,friction and wear. We have realised that exercises to illustrate the various chapters,and case studies also, would occupy too much space to be included in each bookand thus this will need a third volume. Now, each volume will be self-sustained.

Whilst the present book, as well as the following ones, is addressed primarily tograduate students, part of it could possibly be used in undergraduate courses; andwe hope that practising engineers and scientists will find the information it conveysuseful. We hope also that English-speaking readers will be interested in the aspectsof French culture, and more particularly of the French school of micromechanics ofmaterials, which our treatment will undoubtedly display.

The authors are very grateful to all their colleagues, in particular those whoparticipated in the DEA “Mecanique et Materiaux”, for their contributions andencouragements. We wish to thank all those people who have provided photographsto illustrate the book. We also thank Professor Gladwell of the University ofWaterloo, Canada, for including it in the series of which he is responsible. In thecourse of the original translation the frequent questions and suggestions of Dr. JackHowlett helped to improve many paragraphs significantly. We found cooperationwith him very stimulating and we thank him for his excellent work. Nathalie Jacobs,of Springer, followed our work and kept helping and encouraging us in answeringour many questions. Thanks to her are extended to all the people who took greatcare in editing a book of the best possible quality.

We are particularly indebted to Professor Georges Cailletaud (Ecole nationalesuperieure des mines de Paris) who contributed greatly to Chap. 4 about vis-coplasticity, and to Professor Jacques Verdu (Arts et Metiers ParisTech) who notonly wrote the part devoted to polymers in Chap. 5 about viscoelasticity but alsocontributed to Annex 1. Joelle Pineau and Odile Adam have been extremely helpfuland they deserve our warmest thanks.

Acknowledgements

Illustrations in this book are for the most part originals or adapted from varioussources. Many figures were provided by courtesy of authors. Let them all bethanked.

Permissions for reproduction were solicited for the reproduction of originalfigures and photographs. Would publishers and authors who would not have beenidentified signal it to Springer so acknowledgments could be given in future editions.

The authors would like to acknowledge the following publishers for theirpermission to use a number of figures included in the text:

- MANEY Publishing

Pope SS, Ezz DP (1984) Critical resolved shear stress for Ni3Al. Int Mater Rev29:136–167 – (Figure 13) for Fig. 3.58

Gourgues-Lorenzon AF (2007) Application of electron backscatter diffraction to thestudy of phase transformations. Int Mater Rev 52:65–128 – (Figures 1 & 13) forFig. 1.21a, b

Flower HM, Gregson PJ (1987) Solid state phase transformations in aluminiumalloys containing lithium. Mater Sci Technol 3:81–90 – (Figures 3 & 8) for Figs.1.43 and 1.45

Gregson PJ, Dinsdale K, Harris SJ, Noble B (1987) Evolution of microstructure inAl-Li-Zn-Mg-Cu alloys. Mater Sci Technol 3:7–13 – (Figure 4a) for Fig. 1.44

- SPRINGER Netherlands

Cozar R, Pineau A (1973) Morphology of ”0 and ”00 precipitates and thermalstability of Inconel 718 type alloys. Met Trans 4:47–59 – (Figure 4) forFig. 1.27a

- TAYLOR & FRANCIS

Foreman AJE, Makin MJ (1966) Dislocation movement through random arrays ofobstacles. Philos Mag 14:911–924 – (Figures 1, 3, & 5) for Figs. 3.95a, b

ix

x Acknowledgements

Campbell JD, Ferguson WG (1970) The temperature and strain rate dependence ofthe shear strength of mild steel. Philos Mag 21:63–82 – (Figures 5 & 11a) forFig. 4.24

- ELSEVIER

Miller MK (2001) Contributions of atom probe tomography to the understanding ofnickel based superalloys. Micron 32:757–764 – (Figure 11) for Fig. 1.27b

Clavel M, Pineau A (1982) Fatigue behaviour of two nickel base alloys II. MaterSci Eng 55:173–180 – (Figures 4 & 5a) for Figs. 1.54 and 1.55

- IOP Publishing

Verdier M, Fivel M, Groma I (1998) Mesoscopic scale simulation of dislocationdynamics in fcc metals. Model Simul Mater Sci Eng 6:755–770 – (Figure 13) forFig. 3.63

- TRANS TECH Publications

Polak J, Man J, Obrtlik K (2005) Atomic force microscopy study of the early fatiguedamage. Mater Sci Forum 482:45–50. doi: 10.4028/www.scientific.net/MSF.482.45 – (Figures 1 & 3) for Figs. 1.25a, b

- WILEY & Sons

Willems B, Nistor LC et al (2005) Strain mapping around dislocations in diamondand cBN. Phys Status Solidi A 202:2224–2228. doi: 10.1002/pssa.200561923 –(Figure 3) for Fig. 3.41

Motoyashiki Y, Bruckner-Foit A, Sugeta A (2007) Investigation of small crackbehaviour under cyclic loading. Fatigue Fract Eng Mater Struct 30:556–564 –(Figure 8) for Fig. 1.17

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Main Classes of Materials from the Point of View

of Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 What Is a Material? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.2 Industrial Importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 Importance of Mechanical Properties . . . . . . . . . . . . . . . . . . . 71.1.4 Bond Types for the Main Classes of Materials . . . . . . . . . 91.1.5 Main Types of Mechanical Behaviour . . . . . . . . . . . . . . . . . . 101.1.6 Modes of Failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.1.7 Mechanical Properties as Criteria for

Choice of Materials. From Properties to Performances 131.1.8 As a Conclusion: The Structure of the Books . . . . . . . . . . 15

1.2 Microstructures of Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.2.1 The Importance of Microstructural Observations . . . . . . 161.2.2 Scales of Observation. Available Means of

Observation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.2.3 Examples of Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.3 Characterisation of Mechanical Properties . . . . . . . . . . . . . . . . . . . . . . . . . 531.3.1 Aim of Mechanical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531.3.2 Servo-Controlled Testing Machines.. . . . . . . . . . . . . . . . . . . . 531.3.3 Tensile Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541.3.4 Compression Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601.3.5 Other Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601.3.6 Measurements of Elastic Characteristics. . . . . . . . . . . . . . . . 651.3.7 Tendencies in the Evolution of Methods for

Studying Mechanical Properties of Materials . . . . . . . . . . 671.4 Constitutive Equations: General Considerations . . . . . . . . . . . . . . . . . . . 67

1.4.1 Modelling.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671.4.2 Constitutive Equations: Main Classes. . . . . . . . . . . . . . . . . . . 681.4.3 General Formulation of the Constitutive Equations . . . . 701.4.4 Anisotropy and Heterogeneity. . . . . . . . . . . . . . . . . . . . . . . . . . . 72

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80xi

xii Contents

2 Elastic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.1 Elastic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

2.1.1 Strain Energy .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.1.2 Elastic Potential and the Complementary Energy . . . . . . 852.1.3 Thermodynamic Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 862.1.4 Isothermal and Adiabatic Elastic Compliances .. . . . . . . . 88

2.2 Two Major Classes of Elastic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.2.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.2.2 Rubber Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 912.2.3 Cohesion Energy, Elastic Constants. . . . . . . . . . . . . . . . . . . . . 97

2.3 Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 992.3.1 Elastic Moduli and Compliances . . . . . . . . . . . . . . . . . . . . . . . . 992.3.2 Anisotropy.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1032.3.3 Stability of the Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1092.3.4 Field Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102.3.5 Example: Propagation of Plane Sine Waves . . . . . . . . . . . . 110

2.4 Variational Methods: Introduction to the Finite Elements Method 1122.4.1 Extremal Theorems in Linear Elasticity . . . . . . . . . . . . . . . . 1132.4.2 Principle of the Finite Element Method.. . . . . . . . . . . . . . . . 115

2.5 Heterogeneous Materials: Estimates and Boundsin Linear Elasticity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1172.5.1 Effective Moduli and Compliances . . . . . . . . . . . . . . . . . . . . . 1182.5.2 The Case of Initial Deformations . . . . . . . . . . . . . . . . . . . . . . . 1192.5.3 Bounds for the Overall Moduli and Compliances .. . . . . 122

2.6 Elasticity of Heterogeneous Materials: Estimatesfor the Overall Moduli and Compliances . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

2.7 The Inclusion Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272.7.1 Inclusion with Uniform Stress-Free Strain

in a Load-Free Infinite Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 1272.7.2 The Case of Isotropic Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 1302.7.3 Other Problems Concerning Ellipsoidal Inclusions . . . . 1322.7.4 Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

2.8 Improved Bounds and Estimates for ElasticInhomogeneous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1372.8.1 Methodology for Getting Sharper Bounds (Outline) . . . 1372.8.2 Improved Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

2.9 Systematic Theory of the Elasticity of Random Media (Outline) . 1462.9.1 General Equation for Heterogeneous Elastic Media . . . 1462.9.2 Properties of the Modified Green’s Operator . . . . . . . . . . . 1482.9.3 Equation for the Effective Moduli. . . . . . . . . . . . . . . . . . . . . . . 1502.9.4 Bounds and Estimates for the Effective Moduli . . . . . . . . 151

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

Contents xiii

3 Elastoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553.2 General: Phenomenological Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

3.2.1 One-Dimensional Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1573.2.2 Three-Dimensional Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 161

3.3 Physical Mechanisms of Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.3.1 The Problem .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.3.2 Deformation of a Single Crystal.

Discrepancy Between Experiment and Theory .. . . . . . . . 1663.3.3 Dislocations: Definition, Geometrical Properties. . . . . . . 1733.3.4 Dislocations in Face Centred Cubic (FCC) Crystals . . . 1833.3.5 Dislocations in Other Crystalline Structures. . . . . . . . . . . . 1873.3.6 Force on Dislocations and Deformation

Produced by Their Displacement. . . . . . . . . . . . . . . . . . . . . . . . 1923.3.7 Stress, Strain Fields and Deformation

Energy Associated with a Dislocation . . . . . . . . . . . . . . . . . . 1963.3.8 Interaction of Dislocations with Other

Dislocations and Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2103.3.9 Twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

3.4 Hardening Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2213.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2213.4.2 Obstacles of Crystallographic Nature .. . . . . . . . . . . . . . . . . . 2253.4.3 Interaction of Dislocations with Foreign Atoms. . . . . . . . 2493.4.4 Volumetric Shear Deformation:

Twinning, Martensitic Transformation,Transformation Plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

3.4.5 An Application of Hardening:Strengthening of Steels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

3.4.6 Fibres Reinforcement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2983.5 Macroscopic Formulation of Plastic Behaviour . . . . . . . . . . . . . . . . . . . . 304

3.5.1 From Microscopic to Macroscopic Plasticity . . . . . . . . . . . 3043.5.2 Criteria, Work-Hardening and Plastic Flow . . . . . . . . . . . . 3143.5.3 Introduction to the Plastic Design of Structures . . . . . . . . 3363.5.4 Introduction to the Plasticity of

Heterogeneous Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

4 Elastoviscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3634.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3634.2 Typical Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364

4.2.1 One-Dimensional Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3654.2.2 Multiaxial Loading.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3754.2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378

xiv Contents

4.3 The Physical Mechanisms Responsible for Viscoplasticity . . . . . . . 3794.3.1 Low-Temperature Activation of Plastic and

Viscoplastic Deformation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3794.3.2 Physical Models of High-Temperature

Viscoplasticity in Metals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3964.3.3 Creep in Ceramic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4174.3.4 Creep in Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

4.4 Mechanical Models of Macroscopic Viscoplasticity . . . . . . . . . . . . . . . 4224.4.1 Viscoplastic Potential for a Single Crystal . . . . . . . . . . . . . . 4224.4.2 Viscoplastic Potential for a Polycrystal . . . . . . . . . . . . . . . . . 4254.4.3 Time-Independent Plasticity and

Viscoplasticity Compared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4254.4.4 Specific Constitutive Equations . . . . . . . . . . . . . . . . . . . . . . . . . 4264.4.5 Simultaneous Treatment of Plasticity and

Viscoplasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4324.5 Methods for Reinforcing Against Creep. . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

4.5.1 Reinforcement by Reducing Diffusion.. . . . . . . . . . . . . . . . . 4344.5.2 Creep of Solid Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4354.5.3 Creep of Alloys Reinforced by Particles . . . . . . . . . . . . . . . . 437

4.6 Exercise: Activation Energy Needed for Dislocationsto By-Pass Ordered Precipitates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4384.6.1 Shearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4404.6.2 Orowan By-Passing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443

5 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4455.1 Phenomenological Analysis of 1-D Mechanical Responses . . . . . . . 446

5.1.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4465.1.2 Linear Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4495.1.3 Non-ageing Linear Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . 452

5.2 Microstructural Aspects and Physical Mechanisms.. . . . . . . . . . . . . . . 4655.2.1 Viscoelasticity of Polymers and Related Phenomena . . 4655.2.2 Internal Friction of Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483

5.3 Viscoelastic Structures and Heterogeneous Materials . . . . . . . . . . . . . 4945.3.1 Local Viscoelastic Constitutive Equations .. . . . . . . . . . . . . 4945.3.2 The Correspondence Theorem: Principle

of Structural Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4985.3.3 Homogenisation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504

Appendix A Annex 1: Atomic and Molecular Structures. . . . . . . . . . . . . . . . . . . . 507A1.1 Types of Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507A1.2 Crystalline Solids – Elements of Crystallography . . . . . . . . . . . . . . . . . 508

A1.2.1 Symmetry Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508A1.2.2 Crystallographic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

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A1.2.3 Ordered Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510A1.2.4 Miller Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 510A1.2.5 Reciprocal Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514A1.2.6 Stereographic Projection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515A1.2.7 Twinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517A1.2.8 X-Ray Diffraction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

A1.3 Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524A1.3.1 Chemical Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524A1.3.2 Structural Arrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526A1.3.3 Main Polymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

A1.4 Amorphous Materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536A1.4.1 Glasses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536A1.4.2 Amorphous Metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

A1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538A1.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542

Appendix B Annex 2: Phase Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543A2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543A2.2 Equilibrium Diagrams .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544

A2.2.1 The Nature of Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544A2.2.2 Thermodynamics of Equilibrium.. . . . . . . . . . . . . . . . . . . . . . . 545A2.2.3 Multi-phase Equilibria – Equilibrium Phase Diagrams 546

A2.3 Kinetics – Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554A2.3.1 Basic Diffusion Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555A2.3.2 The Diffusion Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557

A2.4 Nucleation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559A2.4.1 Free Energy Associated with Variations

in the Configuration .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560A2.4.2 Heterogeneous Nucleation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561

A2.5 Thermally-Activated Growth.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562A2.5.1 Growth Governed by a Reaction at the Interface .. . . . . . 563A2.5.2 Growth Governed by Diffusion: Zener’s Theory .. . . . . . 563A2.5.3 Coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563

A2.6 Phenomenological Theories of Kinetics and Phase Changes . . . . . . 566A2.6.1 Isothermal Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566A2.6.2 Non-isothermal Transformations .. . . . . . . . . . . . . . . . . . . . . . . 566

A2.7 Solidification .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569A2.7.1 Nucleation in the Solid Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 569A2.7.2 Growth of the Solid Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569A2.7.3 Morphology of the Solid Phase . . . . . . . . . . . . . . . . . . . . . . . . . 571A2.7.4 Solidification of Eutectics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572A2.7.5 Structure of Solidified Material . . . . . . . . . . . . . . . . . . . . . . . . . 574

A2.8 Precipitation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576A2.8.1 The Two Types of Precipitation . . . . . . . . . . . . . . . . . . . . . . . . . 577A2.8.2 Coherency Between the Precipitates and the Matrix . . . 577

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A2.9 Martensitic Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579A2.9.1 General Features of Martensitic

Transformations in Steels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579A2.9.2 Critical Points of the Transformation, and a

Note on Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582A2.10 Further Reading .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584A2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585

A2.11.1 Equilibrium Diagram, Purification by Zone Melting .. . 585A2.11.2 Steel Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585A2.11.3 Martensitic Transformation.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586A2.11.4 Steel Microstructures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587A2.11.5 Surface Treatment: Cementation . . . . . . . . . . . . . . . . . . . . . . . . 587A2.11.6 Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589A2.11.7 Hardening by Precipitation and

Coalescence of the Precipitates in a FerriticStainless Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

A2.11.8 Precipitation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591A2.11.9 Effect of Applied Stress on the Morphology

of Precipitates in Nickel-Based Alloys . . . . . . . . . . . . . . . . . 593

Appendix C Annex 3: Continuum Mechanics: Basic Conceptsand Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595A3.1 Deformations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596

A3.1.1 Strain Tensor for a Solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596A3.1.2 Strain Tensor for a Variety (Curve or Surface) . . . . . . . . . 597A3.1.3 Eulerian Tensors for Virtual Strain and

Strain Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598A3.1.4 Compatibility Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598

A3.2 Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598A3.2.1 Definitions and Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598A3.2.2 Field Equations.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599

A3.3 Problems in Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600A3.3.1 Navier Equations for Linear Homogeneous

Isotropic Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601A3.3.2 Beltrami Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

Appendix D Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

Appendix E Physical Constants; Conversion Factors . . . . . . . . . . . . . . . . . . . . . . . . 611

Appendix F Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613F.1 Rectangular Cartesian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613F.2 Cylindrical Polars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614F.3 Spherical Polars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616

Contents xvii

Appendix G Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 619G.1 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626G.2 Vector and Tensor Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635