Micrornechanics of defects in solids

Transcription

1 Micrornechanics of defects in solids

2 Monographs and textbooks on mechanics of solids and fluids editor-in-chief: G.AE. Oravas Mechanics of elastic and inelastic solids editor: S. Nemat-Nasser Books published under this series are: 1. G.M.L. Gladwell Contact problems in the c1assical theory of elasticity 2. G. Wempner Mechanics of solids with applications to thin bodies 3. T. Mura Micromechanics of defects in solids

3 Micromechanics of defects in solids Toshio Mura Department 01 Civil Engineering and Materials Research Center, Northwestern University, Evanston, I/I., U.S.A MARTINUS NIJHOFF PUBLISHERS THE HAGUE/BOSTON/LONDON

4 Distributors: for the United States and Canada Kluwer Boston, Inc. 190 Old Derby Street Hingharn, MA USA for all other count ries Kluwer Academic Publishers Group Distribution Center P.O. Box AH Dordrecht The Netherlands Library of Congress Cataloging in Publication Data Mura, Toshio, Micromechanics of defects in solids. (Monographs and textbooks on mechanics of solids and fluids) Includes bibliographical references and indexes. 1. Micromechanics. 2. Solids. 3. Crystals- Defects: 11. Senes. QC176.8.M5M87 4. Strains and stresses. I. Title 'O'i AACR2 ISBN / ISBN (ebook) Copyright /982 hy Martillus Nijhoff Puhlishers, The Hague Softcover reprint of the hardcover 1 st edition 1982 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording, or otherwise, without the prior written permiuion of the publisher. Martinus NiJhoff Publishers, P.O. Box 566, 2501 CN The Hague, The Netherlands.

5 Preface This book sterns from a course on Micromechanics that I started about fifteen years ago at Northwestern University. At that time, micromechanics was a rather unfamiliar subject. Although I repeated the course every year, I was ne ver convinced that my notes have quite developed into a final manuscript because new topics emerged constantly requiring revisions, and additions. I finally came to realize that if this is continued, then I will never complete the book to my total satisfaction. Meanwhile, T. Mori and I had coauthored a book in Japanese, entitled Micromechanics, published by Baifu-kan, Tokyo, in It received an extremely favorable response from students and researchers in Japan. This encouraged me to go ahead and publish my course notes in their latest version, as this book, which contains further development of the subject and is more comprehensive than the one published in Japanese. Micromechanics encompasses mechanics related to microstructures of materials. The method employed is a continuum theory of elasticity yet its applications cover a broad area relating to the mechanical behavior of materials: plasticity, fracture and fatigue, constitutive equations, composite materials, polycrystals, etc. These subjects are treated in this book by means of a powerful and unified method which is called the 'eigenstrain method.' In particular, problems relating to inclusions and dislocations are most effectively analyzed by this method, and therefore, special emphasis is placed on these topics. When this book is used as a text for a graduate course, Sections 3, 11, and 22 should be emphasized. Eigenstrain is a generic name given by the author to such nonelastic strains as thermal expansion, phase transformation, and misfit strains. J.D. Eshelby, who is a pioneer in this area, refers to eigenstrains as stress-free transformation strains in his celebrated papers (1957, 1959). v

6 Preface The term eigenstrain should not be confused with the term 'eigenvalue' which occurs in mathematical physics, and relates to an entirely different concept. No particular background is required of readers of this book because necessary mathematics and physics are explained in the text and Appendix. Although I have tried to be fair in citing Iiterature, I have to apologize if so me papers do not receive proper credit or are not cited. The sections and subsections marked with an asterisk (*), can be skipped in the first reading, since the subjects discussed there are peripheral to the main theme. I wish to express my thanks to all the people who have helped me during the course of the preparation of the manuscript: my previous graduate students, Zissis A. Moschovidis, Minoru Taya, Carl R. Vilmann, and Ronald B. Castles, as weil as my friends R. Furuhashi, N. Kinoshita, T. Morita, M. Inokuti, and T. Mori. Mori receives my special thanks for having advised me on the subject matter, for discussing with me whole chapters, and for helping me to write Chapter 7. The manuscript reached the final form in his hands. I also wish to thank S. Nemat-Nasser who has read through the manuscript and has given valuable comments. I give my thanks to Vera Fisher for her skillful typing and her great patience with me, to the secretaries whom I involved in various aspects of the work over the years: Erika Ivansons, Miriam LittelI, Masa Sumikura, and Carolyn Andrews, and to my family for their patience and understanding. Finally, I acknowledge the National Science Foundation and the U.S. Army Research Office for their support of my research in the area of micromechanics. November 13, 1980 T.M. VI

7 Contents Preface Chapter 1. General theory of eigenstrains 1. Definition of eigenstrains 2. Fundamental equations of elasticity Hooke's law Equilibrium eonditions Compatibility eonditions 3. General expressions of elastie fields for given eigenstrain distributions Periodie solutions Method of Fourier series and Fourier integrals Method of Green's funetions Isotropie materials Cubie erystals Hexagonal erystals (transversely isotropie) 4. Exereises of general formulae A straight serew disloeation A straight edge disloeation Periodie distribution of euboidal precipitates 5. Statie Green's funetions Isotropie materials * Anisotropie materials *Transversely isotropie materials *Kröner's formula *Derivatives of Green's funetions *Two-dimensional Green's function 6. Inclusions and inhomogeneities Inclusions Inhomogeneities Effeet of isotropie elastie moduli on stress v VII

8 Contents 7. Dislocations Volterra and Mura formulas *The Indenborn and Orlov formula *Disclinations 8. Dynamic solutions U niformly moving edge dislocation Uniformly moving screw dislocation *9. Dynamic Green's functions *Isotropic materials Steady-state 10. Incompatibility *Riemann-Christoffel curvature tensor Chapter 2. Isotropie inclusions 11. Eshelby's solution Interior points Sphere Elliptic cylinder Penny-shape Flat ellipsoid Oblate spheroid Prolate spheroid Exterior points Thermal expansion with central symmetry * 12. Ellipsoidal inclusions with polynomial eigenstrains *The I-integrals *Sphere *Elliptic cylinder *Oblate spheroid *Prolate spheroid *Elliptical plate *The Ferrers and Dyson formula 13. Energies of inclusions Elastic strain energy Interaction energy Strain energy due to a spherical inclusion Elliptic cylinder Penny-shaped ftat ellipsoid Spheroid * 14. Cuboidal inclusions 15. Inclusions in a half space VIII

9 Contents Green's funetions Ellipsoidal inclusion with a uniform dilatational eigenstrain *Cuboidal inclusion with uniform eigens trains *Periodie distribution of eigenstrains Joined half-spaees Chapter 3. Anisotropie inclusions 16. Elastie field of an ellipsoidal inclusion 17. Formulae for interior points Uniform eigens trains Spheroid Cylinder (elliptie inclusion) Flat ellipsoid Eigenstrains with polynomial variation Eigenstrains with a periodie form * 18. Formulae for exterior points Examples 19. Ellipsoidal inclusions with polynomial eigenstrains in anisotropie media Special eases *20. Harmonie eigenstrains 21. Periodie distribution of spherieal inclusions Chapter 4. Ellipsoidal inhomogeneities 22. Equivalent inclusion method Isotropie materials Sphere Penny shape Rod Anisotropie inhomogeneities in isotropie matriees Stress field for exterior points 23. Numerieal ealculations Two ellipsoidal inhomogeneities *24. Impotent eigenstrains 25. Energies of inhomogeneities Elastie strain energy Interaction energy Colonnetti's theorem Uniform plastie deformation in a matrix Energy balance IX

10 Contents 26. Precipitates and martensites Isotropie precipitates Anistropic precipitates Incoherent precipitates Martensitic transformation Stress orienting precipitation Chapter 5. Cracks Critical stresses of cracks in isotropie media 204 Penny-shaped cracks 204 Slit-like cracks 206 Flat ellipsoidal cracks 207 Crack opening displacement Critical stresses of cracks in anisotropie media 210 Uniform applied stress 210 Non-uniform applied stress 215 *TI integrals for a penny-shaped crack 216 *TI integrals for cubic crystals 217 *TI integrals for transversely isotropie materials Stress intensity factor for a ftat ellipsoidal crack 220 Uniform applied stresses 224 Non-uniform applied stresses Stress intensity factor for a slit-like crack 229 Uniform applied stresses 230 Non-uniform applied stresses 232 Isotropie materials Stress concentration factors 234 Simple tension 235 Pure shear Dugdale-Barenblatt cracks 238 BCS model 245 Penny-shlwed crack 248 *33. Stress intensity factor for an arbitrarily shaped plane crack 253 *Numerical examples Crack growth 262 Energy release rate 262 The J-integral 265 Fatigue 268 Dynamie crack growth 272 Chapter 6. Dislocations Displacement fields 277 x

11 Contents Parallel dislocations 278 A straight dislocation Stress fields 280 Dislocation segments 280 Willis' formula 284 The Asaro et al. formula 285 Dislocation loops Dislocation density tensor 289 Surface dislocation density 292 Impotent distribution of dislocations Dislocation ftux tensor 295 *Line integral expression of displacement and plastic distortion fields Energies and forces 301 Dynamic consideration Plasticity 308 Mathematical theory of plasticity 308 Dislocation theory 310 Plane strain problems 312 Beams and cylinders Dislocation model for fatigue crack initiation 325 Chapter 7. Material properties and related topics Macroscopic average 334 Average of internal stresses 334 Macroscopic strains 335 Tanaka-Mori's theorem 336 Image stress 338 Random distribution of inclusions-mori and Tanaka's ~~ ~ 43. Work-hardening of dispersion hardened alloys 342 Work-hardening in simple shear 343 Dislocations around an inclusion 347 Uniformity of plastic deformation Diffusional relaxation of internal and external stresses 350 Relaxation of the internal stress in a plastically deformed dispersion strengthened alloy 351 Diffusional relaxation process, climb rate of an Orowan loop 353 Recovery creep of a dispersion strengthened alloy 356 Interfacial diffusional relaxation 358 XI

12 Contents 45. Average elastic moduli of composite materials 364 The V oigt approximation 364 The Reuss approximation 367 Hill's theory 368 Eshelby's method 370 Self-consistent method 372 U pp er and lower bounds 375 Other related works Plastic behavior of polycrystalline metals and eomposites 380 Taylor's analysis 380 Self-eonsistent method 384 Embedded weakened zone Viseoelastieity of eomposite materials 389 Homogeneous inclusions 389 Inhomogeneous inclusions 391 Waves in an infinite medium Elastie wave scattering Interaction between disloeations and inclusions 398 Inclusions and dislocations 398 Cracks in two-phase materials Eigenstrains in lattiee theory 410 A uniformly moving serew dislocation 413 Appendix Einstein summation eonvention 417 Kroneeker delta 417 Permutation tensor 417 Appendix The elastie moduli for isotropie materials 419 Appendix Fourier series and integrals 423 Dirae's delta function and Heaviside's unit function 424 Laplace transformation 425 Appendix Dislocations pile-up 427 References 429 Author index 481 Subject index 489 XII