Wei Yang . W.B. Lee

Mesoplasticity and its Applications

With 197 Figures

Springer-Verlag Berlin Heidelberg GmbH

Wei Yang Departrnent of Engineering Mechanies Tsinghua University Beijing 100084 China

W.B. Lee Department of Manufacturing Engineering Hong Kong Polytechnic, Hung Horn Kowloon, Hong Kong

Series Editors

Prof. Bernhard Ilschner Polytechnique Federale de Lausanne Laboratoire de Metallurgie Mecanique MX-D Ecublens Ecole CH-iOI5 Lausanne/Switzerland

Prof. Kenneth C. Russe) Departrnent of Materials Science and Engineering and Departrnent of Nuc1ear Engineering Roorn 8-411 Massachusetts Institute of Technology Carnbridge, MA 02139IUSA

ISBN 978-3-642-50042-8 ISBN 978-3-642-50040-4 (eBook) DOI 10.1007/978-3-642-50040-4

Library of Congress Cataloging-in-Publication Dara Yang, Wei, 1954-Mesoplasticity and its applications I Wei Yang, W.B. Lee. p. cm. Includes indexes.

New York Berlin Hddelberg : acid-free paper) : $ 169.00 1. Plasticity. I. Lee, W.B. 11. Title.

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© Sprißger-Verlag Berliß Heidelberg 1993 Originally published by Springer-Verlag Berlin Heidelberg New York in 1993.

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Editor's Preface

This monograph written by two Chinese scientists of the younger generation opens a window into the world of thoughts on Mechanical Metallurgy in this fascinating area of our world, characterized by age old cultural heritage as weil as by its dynamic evolution into the future. Based on notions and names all so familiar to the western scientist, and regarding the subject from the point of view of the theoretical mechanical engineer (Yang) as weil as that of the materials and manufacturing engineer (Lee), the authors present a synthesis of both approaches and endeavour to guide the reader from basic theory to engineering applications. Between structural defects in the micrometer scale and the meter-measures of engineering components, the term of mesoplasticity is meant to place the reader right in the center: This is certainly achallenging enterprise, and the editor expresses his sincere wishes as to enrichment and stimulation which will emanate from this interesting book and its creative perspective.

March, 1993. Prof.B. Ilschner

Preface

In the past two decades, enormous advances in materials and manufacturing tech­nology have been achieved, which upgrade the material design, processing and precision manufacturing as quantitative and concise scientific disciplines. Rapid improvements on mechanics understanding have been instrumental in the above­mentioned development. A topic of great interest and importance in plasticity re­search has been the design and processing of materials themselves on the mesoscale to achieve the desired macroscopic properties. In recent years, the studies on the plastic behaviour of various materials and their constitutive representations have been the focus point in the field of plasticity. Several excellent survey articles fo­cused on the subsets of these knowledge data appeared. However, near the end of 1989, we still can not find a book which overviews the whole subject from a mesoplasticity viewpoint.

As a mechanist and a material/manufacturing scientist, the two authors of the present book happened to meet together in Beijing and discuss their common interests in the interdisciplinary field where their expertness overlaps. After several rounds of communication back and forth, an outline of the book was set. Through a Hong Kong Polytechnic Special Fund for Academic Exchange with China, WY was able to visit Hong Kong in early 1990 for two months, and in that period of time a substantial part of the book came out as the product of our collaboration. Later on, our writing project was encouraged by Prof. B.Ilschner of Ecole Polytechnic Federale de Lausanne, the Editor-in-Chief of Material Science and Engineering Series, and the Engineering Editor of the Springer-Verlag Press. We cannot thank them more for their sincere supports to a book on a relatively new field by two relatively young authors. To meet their expectations, WBL travelled back to Beijing in the autumn of 1991 to finalize the manuscript. We tried our best to write out whatever we know about mesoplasticity, and needless to mention that we ourselves learned a lot during the preparation of the manuscript.

The intention of this book is to treat the subject of plasticity as weil as its engineering applications through a somewhat unconventional angle, in which both mathematical rigor and engineering appeal could be maintained. In the book, the scope of macroplasticity is firstly reviewed and a systematic framework of meso­plasticity is laid down from a combined approach of solid mechanics and mate­rials science, encompassing various physical deformation mechanisms. Besides a systematic discussion on the constitutive formulation derived from crystalline and

viii Preface

geological materials, other interesting aspects of mesoplasticity , like the hardening mechanism and the meso-damage theory are also discussed in details. The appli­cations of mesoplasticity in materials science and manufacturing engineering are examined from the author's own work as weIl as from other researchers' findings. This book will be suitable for research scientists and engineers, and postgraduate and senior undergraduate students working in the field of materials science, solid mechanics and physical theory of plasticity. We would be overjoyed if the readers could find this book interesting and beneficial to their own studies and research es.

The authors are deeply indebted to many organizations and individuals who made this book possible. We would like to acknowledge the supports from the Research Sub-Committee of the Hong Kong Polytechnic, Fok Ying-Tung Education Foundation, and the State Education Commission of China for the encouragement in mesoplasticity research in the recent years. We are also very grateful to Professor K.c. Hwang of Tsinghua University for the stimulating discussions and sincere advices during the preparation of the manuscript. Special thanks are due to Dr. BJ.Duggan of the University of Hong Kong for introducing texture work to one of the authors, and Dr. K.C.Chan for his help in computer programming and the preparation of some of the drawings used in the book.

Our wives - Cheng Li and Louisa Lee - are the greatest support behind the work and without their patience and endurance, this book would not have been materialized.

January, [992. Yang Wei Tsinghua University, Beijing, China.

W.B.Lee Hong Kong Polytechnic Hong Kong.

Contents

1 Introduction

1.1 What is Mesoplastieity? 1.2 Seales for Plastieity Investigation 1.3 Methods for Mesoplastieity Investigation 1.4 Layout of the Book 1.5 Referenees

Part I Fundamentals of Mesoplasticity

2 Scope and Limitations of Macroplasticity

2.1 Maeroplastie Observations 2.1.1 Introduetion 2.1.2 Uniaxial Stress-strain Curve

2.2 One-Dimensional Elastie-Plastie Formulation 2.3 Three-Dimensional Formulation for Infinitesimal Deformation

2.3.1 Tensor Notations ...... . 2.3.2 Additive Strain Deeomposition . . 2.3.3 Prineiple of Maximum Plastic Work 2.3.4 Metal Plastieity ....... .

2.4 Three-Dimensional Formulation for Finite Deformation 2.4.1 Elastie-Plastie Deeomposition .... 2.4.2 Seleetion of Stress and Strain Measures 2.4.3 Objeetive Corotational Rate . . . . 2.4.4 Elastie-Plastie Constitutive Equations

2.5 h Flow Theory 2.5.1 Isotropie Hardening 2.5.2 Mixed Hardening 2.5.3 Exampies

2.6 h Deformation Theory 2.6.1 Formulation of h Deformation Theory 2.6.2 Comparison to h Flow Theory . . . 2.6.3 Reassessment of h Deformation Theory

2 3 5 6

9

II

11 11 12 15 16 16 17 19 21 23 23 25 27 29 30 30 33 36 41 41 42 43

x Contents

2.7 1z Corner Theory 45 2.7.1 Essential Features of Corner Theory 45 2.7.2 Nonlinear Rate Form 46 2.7.3 Cone Transition Function 47 2.7.4 Constitutive Laws 48

2.8 Theory of Internal State Variables 49 2.8.1 Internal Variables and Free Energy 50 2.8.2 Thermodynamic Restrictions 52 2.8.3 Flow Potential and Normality 53

2.9 Non-Associated Flow Rules 56 2.9.1 Pressure Sensitive Yielding 56 2.9.2 Plastic Dilatancy 57

2.10 Limitations of Macroplasticity 59 2.11 References 62

3 Introduction to Materials Structures 64

3.1 Introduction 64 3.2 The Crystalline State of Matter 65

3.2.1 Crystalline versus Amorphous State 65 3.2.2 Basic Crystallographic Geometry 66 3.2.3 Miller Indices and the Reciprocal Lattice 71 3.2.4 Crystal Structures 74

3.3 Defects in Crystalline Materials 77 3.3.1 Point, Line and Planar Defects 77 3.3.2 Grain and Twin Boundaries 82 3.3.3 Coherency of Interphase Boundaries 86

3.4 Quantitative Description of Microstructures 88 3.4.1 Basic Topological Features 89 3.4.2 Quantitative Metallography 91

3.5 Crystallographic Textures 94 3.5.1 Introduction 94 3.5.2 Representation of Grain Orientation 95 3.5.3 Pole Figures and Inverse Pole Figures 97 3.5.4 Crystallite Orientation Distribution Function 101

3.6 Evolution of Deformed Microstructures 104 3.7 References 109

4 Deformation Mechanisms I : Dislocations 111

4.1 Basic Mechanisms for Plastic Deformations 11 I 4.2 Introduction to Dislocations 112

4.2.1 Origin of Dislocation Theory 112 4.2.2 Dislocation Kinematics 115

4.3 Elasticity Theory of Dislocation Fields 121 4.3.1 Simple Dislocation Fields 121

4.3.2 Green Function for Linear Elasticity . 4.3.3 General Solution for Dislocation Loops 4.3.4 Dislocation Energy . . . . . . 4.3.5 Forces on Dislocations ....

4.4 Dislocation Induced Plastic Deformation 4.4.l Orowan Equation 4.4.2 Dislocation Dynamics

4.5 Crystallography of Dislocations 4.5.1 Lattice Burgers Vectors 4.5.2 Stacking Faults 4.5.3 Partial Dislocations and Dislocation Reactions

Appendix 4A Line integral expression of Green function Appendix 4B Decomposition of dislocation energy Appendix 4C Brown formula of dislocation field 4.6 References ............ .

5 Deformation Mechanisms 11 : Miscellaneous

5.1 Deformation Twinning . . . 5.l.l Twinning versus Slip . 5.1.2 Kinematics of Twinning 5.1.3 Directionality of Twinning

5.2 Shear Banding . . . . . . . 5.2.1 Structure of Shear Bands 5.2.2 Crystallographic Model of Shear Band Formation 5.2.3 Symmetry Requirement in Shear Band Formation 5.2.4 Shear Banding and Bifurcation . . .

5.3 Grain Boundary and Grain Boundary Sliding 5.3.1 Kinematics of Grain Boundary 5.3.2 Dislocation Model of Grain Boundary 5.3.3 Grain Boundary Energy . . 5.3.4 Grain Boundary Sliding . .

5.4 Transformation Induced Plasticity 5.4.1 Phase Transformation . . . 5.4.2 Mechanics of Martensitic Transformation 5.4.3 Micromechanics Modelling 5.4.4 Transformation Toughening

5.5 References ...... .

6 Strengthening Mechanisms

6.1 Strain Hardening . . . . 6.1.1 Taylor' s Theory 6.1.2 Lomer-Cottrell Lock 6.1.3 Forest Dislocations 6.1.4 Frank-Read Source .

Contents xi

125 128 132 135 138 138 140 142 142 144 145 148 149 152 154

155

155 155 156 158 159 159 161 165 168 170 170 172 173 176 180 180 181 185 187 189

191

191 191 192 194 195

xii Contents

6.1.5 Stress-Strain Curve of a Single Crystal 6.1.6 Seeger Theory

6.2 Solution Hardening . . . . . . . . . . 6.2.1 Point Defects . . . . . . . . . . 6.2.2 Displacement Field Induced by a Force Dipole 6.2.3 Interactions between Point Defects and Dislocations 6.2.4 Cottrell Atmosphere ........ . 6.2.5 Solution Hardening and Yield Point Drop 6.2.6 Creep Controlled by Point-Defect Diffusion

6.3 Precipitation and Dispersion Hardening 6.3.1 Second Phase Partic1es 6.3.2 Long Range Stress 6.3.3 Orowan Stress 6.3.4 Ashby Theory . . .

6.4 Grain Boundary Hardening 6.4.1 Dislocation Pileups at Grain Boundary 6.4.2 Hall-Petch Relation

6.5 References

7 Plasticity for Crystalline and Geological Materials

7.1 Single Crystals . . . . . . . . 7.1.1 Single Crystal Deformation 7.1.2 Elastic Distortion of Lattices 7.1.3 Schmid Yield Criterion 7.1.4 Hardening Laws for Rate Independent and Rate Dependent Crys-

196 198 201 201 202 203 205 205 206 207 207 208 210 211 213 213 214 216

217

217 217 220 222

tals .. . . . . . . . . . . . . . 224 7.1.5 Constitutive Relations of Single Crystals 7.1.6 Latent Hardening Measurement . . . . 7.1.7 Uniaxial Tension of a Single Crystal Bar

7.2 Polycrystals ....... . 7.2.1 Selection of Slip Systems 7.2.2 Sachs Theory . . . . . 7.2.3 Dual Principles of Plastic Work 7.2.4 Taylor Theory . . . . . . . 7.2.5 Transformation Strain Problem of Eshelby 7.2.6 Self Consistent Theory .. . . . . . . 7.2.7 Examples ............ . 7.2.8 Miscellaneous results of Crystalline Plasticity

7.3 Macroscopic Responses 7.3.1 Macroscopic Averaging . 7.3.2 Plastic Strain Rate Tensor 7.3.3 Geometry of n Surfaces 7.3.4 Loading Surfaces of LiF Single Crystals

7.4 Plasticity for Geological Materials . . . . .

225 226 229 231 231 232 233 234 236 238 241 243 246 246 246 247 248 251

7.4.1 Slip Induced Dilatancy ..... . 7.4.2 Matrix Elasticity . . . . . . . . . 7.4.3 Slip Friction and Pressure Sensitivity 7.4.4 Elastic-Plastic Constitutive Equations 7.4.5 Planar Double Slip Model . . . . . 7.4.6 Normality, Vertex Modulus and Coaxiality

7.5 References .............. .

8 Meso-Damage Theory

8.1 Introduction to Meso-Damage 8.1.1 What is meso-Damage ? 8.1.2 Meso-Damage Structures 8.1.3 Approach to Meso-Damage Study

8.2 Meso-Damage by Void Evolution 8.2.1 Void Nucleation, Growth and Coalescence 8.2.2 Rate Independent Ductile Damage Formulation 8.2.3 Gurson's Model . . . . . . . . . . . . . 8.2.4 Improvements and Applications of Gurson's Model 8.2.5 Rate Dependent Creep Damage . . . 8.2.6 Grain Boundary Cavitation and Sliding 8.2.7 Hutchinson-Tvergaard Model 8.2.8 Damage Diffusion, Concentration and Localization

8.3 Meso-Damage by Progressive Cracking 8.3.1 Crack Damaging Configuration . 8.3.2 Single Crack Solution .... 8.3.3 Randomly Aligned Crack Arrays 8.3.4 Self Consistent Scheme . . . 8.3.5 Fabric Tensor . . . . . . . . 8.3.60rderly Aligned Crack Arrays 8.3.7 Some Related Topics of Microcrack Damage

8.4 References ............... .

Part 11 Engineering Applications

9 Modelling of Sheet Metal Textures

9.1 Development of Rolling Textures in FCC Metals 9.2 Prediction of Lattice Rotation . . . . . . . . 9.3 Effect of Strain Path on Slip Rotation .... 9.4 Generation of New Orientations in Recrystallization 9.5 Lattice Curvature and Recrystallization . . . . . 9.6 Recrystallization Textures of Stretched Sheet Metals Appendix 9 Pascal program for the calculation of crystal rotation in

FCC metals 9.7 References

Contents xiii

251 252 252 253 254 255 256

257

257 257 259 260 262 262 262 263 269 274 278 279 281 283 283 284 287 288 290 291 294 295

297

299

299 300 304 308 310 317

320 331

xiv

10 Prediction of Formability 333

10.1 Structure and Formability 333 10.2 CalcuIation of PIastic Strain Ratio 337 10.3 Formation of Ears in Deep Drawing 343 10.4 Prediction of the Size of the M-K Groove in BiaxiaI Stretching 350 10.5 References ................ 359

11 Grain Boundary Engineering and Related Topics

11.1 Introduction to Grain Boundary Engineering 1 i.2 Micrograin SuperpIasticity ...... . 11.3 Design of Turbine BIade Materials . . . . 11.4 Ultra-Precision machining of Crystalline Materials 11.5 References .............. .

Subject Index Author Index

361

361 364 371 377 389

391 395