INTRODUCTION TO ELASTICITY THEORY FOR CRYSTAL DEFECTS. Second Edition

INTRODUCTION TO ELASTICITY THEORY FOR CRYSTAL DEFECTS Second Edition

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INTRODUCTION TO ELASTICITY THEORY FOR CRYSTAL DEFECTS Second Edition Robert W Balluffi Massachusetts Institute of Technology, USA World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI TOKYO

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Balluffi, R. W., author. Title: Introduction to elasticity theory for crystal defects / Robert W. Balluffi, Massachusetts Institute of Technology. Description: 2nd edition. Singapore ; Hackensack, NJ : World Scientific, [2016] 2016 Includes bibliographical references and index. Identifiers: LCCN 2015046960 ISBN 9789814749718 (hardcover ; alk. paper) ISBN 9814749710 (hardcover ; alk. paper) ISBN 9789814749725 (pbk. ; alk. paper) ISBN 9814749729 (pbk. ; alk. paper) Subjects: LCSH: Crystallography, Mathematical. Elasticity. Crystals--Defects. Elastic analysis (Engineering) Classification: LCC QD399.B35 2016 DDC 548/.7--dc23 LC record available at http://lccn.loc.gov/2015046960 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 2017 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore

Preface to First Edition A unified introduction to the theory of anisotropic elasticity for static defects in crystals is presented. The term defects is interpreted broadly to include defects of zero, one, two and three dimensionality: included are point defects (vacancies, self-interstitials, solute atoms, and small clusters of these species) line defects (dislocations) planar defects (homophase and heterophase interfaces) volume defects (inclusions and inhomogeneities) The book is an outgrowth of a graduate course on Defects in Crystals offered by the author for many years at the Massachusetts Institute of Technology, and its purpose is to provide an introduction to current methods of solving defect elasticity problems through the use of anisotropic linear elasticity theory. Emphasis is put on methods rather than a wide range of applications and results. The theory generally allows multiple approaches to given problems, and a particular effort is made to formulate and compare alternative treatments. Anisotropic linear elasticity is employed throughout. This is now practicable because of significant advances in the theory of anisotropic elasticity for crystal defects that have been made over the last thirty five years or so, including the development of anisotropic Green s functions for unit point forces in infinite spaces, half-spaces and joined dissimilar half-spaces. The use of anisotropic theory (rather than the simpler isotropic theory) is important, since even though the results obtained by employing the two approaches often agree to within 25% or so, there are many phenomena v

vi Introduction to Elasticity Theory for Crystal Defects 2nd Ed that depend entirely on elastic anisotropy. Unfortunately, however, the results obtained with the anisotropic theory are usually in the form of lengthy integrals that can be evaluated only by numerical methods and so lack transparency. To assist with this difficulty isotropic elasticity is employed in parallel treatments of many problems where sufficiently simple conditions are assumed so that tractable analytic solutions can be obtained that are more transparent physically. Treatments in the book where isotropic elasticity is employed are clearly distinguished to avoid confusion. The results for the various defects are developed in a sequence of increasing complexity starting with their behavior in isolation in infinite homogeneous regions, where their elastic fields are derived along with, in many cases, corresponding elastic strain energies and induced volume changes. The treatment then progresses to interactions between the defects and imposed applied and internal stresses as well as the image stresses which arise when the defects are in finite homogeneous regions in the vicinity of interfaces. Finally, elastic interactions between the defects themselves are considered in terms of interaction energies and corresponding forces. Due to the breadth of the subject and the impossibility of including all important topics in detail, a selection is made of representative material. This should provide the reader with the background to master omitted topics. The book is designed to be self-sufficient. Included is a preliminary chapter on the basic theory of linear elasticity that includes essentially all the elements of anisotropic and isotropic theory necessary to master the material that follows. A number of appendices is included containing other essentials. A particular effort has been made to write the book in a pedagogical manner useful for graduate students and workers in the field of materials science and engineering. Essentially all results are fully derived, as many intermediate steps as practicable are written out in full, and the use of the phrase it can be shown is avoided. Numerous exercises with solutions are provided, which in many cases expand the scope of the subject matter. Requirements for use of the book are an undergraduate materials science familiarity with the structural aspects of the various defects and knowledge of linear algebra, vector calculus, and differential equations. To avoid long unwieldy expressions, the repeated index summation convention is employed. Consistent sign conventions are used, and introductory lists of the common symbols employed throughout the text are provided. To keep

Preface to First Edition vii the notation as simple as possible, additional symbols are employed locally in various sections of the book and are identified in brief lists in the relevant chapters for the convenience of the reader.

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Preface to Second Edition A new chapter, Defect self-interactions and self-forces, has been added which introduces the important topic of the self-forces experienced by defects, such as dislocations and inclusions, which are extended in at least one dimension and whose self-energies therefore depend upon their shapes. A considerable number of worked exercises has been added which expand the scope of the text and furnish further insights. Numerous sections of the text have been rewritten, and/or expanded, to provide additional aspects and clarity. Again, as in the first edition, a particular effort has been made to include, and compare, different approaches to the solution of various problems. Finally, typographical errors, that mysteriously escaped detection in proofing the first edition, have been corrected. ix

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Acknowledgements I am again particularly indebted to Professor David M. Barnett for permission to include his previously unpublished derivations of the anisotropic Green s functions for unit point forces in infinite spaces, half-spaces and joined elastically dissimilar half-spaces and for providing other valuable assistance. Professor Adrian Sutton again offered encouragement and advice. Professor John Hirth assisted with several questions. I am grateful to the Dept. of Materials Science and Engineering, Cornell University, for hospitality and support during the writing of this book. xi

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Frequently Used Symbols Roman a: A scalar quantities (light face) a :A complex conjugate of a or A ā:ā Fourier transform of a or A a: A vectors (bold face) a i :A i components of a or A â unit vector a = a magnitude of a a : A second rank tensors (bold face, underlined) a ij :A ij components of a : A a : A fourth rank tensors (bold face and double underlined) a ijkl :A ijkl components of fourth rank tensor A [a]: [A] matrices a i :A i elements of [a] or [A] if 1 3 or3 1 matrix a ij :A ij elements of [a] or [A] if 3 3 matrix a ijkl :A ijkl elements of [a] or [A] if 9 9 matrix (ab) jk notation used for element of matrix representing Christoffel tensor: defined by (ab) jk a i C ijkl b l (employs curved brackets rather than the square brackets used for matrices elsewhere throughout book) (ab) matrix representing Christoffel tensor [A] 1 inverse of [A] [A] T transpose of [A] b Burgers vector of dislocation C :C ijkl elastic stiffness tensor xiii

xiv Introduction to Elasticity Theory for Crystal Defects 2nd Ed e ijk alternator symbol: e ijk e i (e j e k ) ê i base unit vector of Cartesian, right-handed, orthogonal coordinate system e dilatation: (sum of the normal elastic strain components: e=ε mm ) E modulus of elasticity (or Young s modulus) E total elasto-mechanical energy, i.e., elastic strain energy plus potential energy of applied forces F force f force per unit length F force per unit area f force density H(x) Heaviside step function: H(x) = 0 when x < 0; H(x) = 1 when x > 0 K bulk elastic modulus ˆl : ˆli unit directional vector: component of l (direction cosine) N number n number per unit volume (density) ˆn unit vector normal to surface (taken to be positive for a closed surface when pointing outwards) P hydrostatic pressure (positive when compressive) k Fourier transform vector r radius r,θ,z cylindrical coordinates (see Fig. A.1a) r,θ,φ spherical coordinates (see Fig. A.1b) R radius of curvature: distance between source point at x and field point at x s arc length along line: distance S E ijkl Eshelby tensor S region of surface S surface area Ŝ surface of unit sphere S :S ijkl elastic compliance tensor sgn(x) sgn(x) = 1 if x > 0: sgn(x) = 1 ifx< 0 ˆt unit vector tangent to dislocation

Frequently Used Symbols xv T traction vector u elastic displacement u T displacement associated with transformation strain u tot total displacement (u tot = u + u T ) V region of volume V volume W:w:W elastic strain energy: elastic strain energy per unit volume: strain energy per unit length W work x 1, x 2, x 3 Cartesian coordinates x :x i : x field vector in Cartesian coordinates: component of x: magnitude of x, i.e., x = x =(x 2 1 +x 2 2 +x 2 3) 1/2 x :x i :x source vector in Cartesian coordinates Greek δ ij Kronecker delta operator (δ ij =1wheni=j:δ ij =0when i j) δ(x x ) Dirac delta function ε: ε ij elastic strain tensor: component of ε ε T ij transformation strain ε T ij transformation strain of equivalent homogeneous inclusion ε tot ij total strain (ε tot ij = ε ij + ε T ij ) θ sum of the normal stress components (θ σ mm ) r,θ,z cylindrical coordinates (see Fig. A.1(a)) r,θ,φ spherical coordinates (see Fig. A.1(b)) λ Lame elastic constant µ Lame elastic constant (elastic shear modulus) ν Poisson s ratio σ : σ ij stress tensor: component of σ Φ potential energy of forces applied to body φ Newtonian potential ψ biharmonic potential Ω atomic volume

xvi Introduction to Elasticity Theory for Crystal Defects 2nd Ed The ± Symbol ± a special symbol used throughout this book, and its relevant literature, which is employed in sums involving Stroh vectors possessing a summation index, such as α, that ranges from 1 to 6. It has the properties that ± =+when α =1, 2, 3and± = when α =4, 5, 6. Hence, for example, 6 ±A sα A kα =A s1 A k1 +A s2 A k2 +A s3 A k3 A s4 A k4 α=1 A s5 A k5 A s6 A k6 Its properties are therefore seen to be the same as those of the sign of the imaginary part of p α,which,byconvention (see Eq. (3.34), is positive when α =1, 2, 3 and negative when α =4, 5, 6. Superscripts D DIS IM INC INH LF M defect dislocation image inclusion inhomogeneity line force matrix

Contents Preface to First Edition Preface to Second Edition Acknowledgements Frequently Used Symbols Roman................................ Greek................................. The ± Symbol............................ Superscripts... v ix xi xiii xiii xv xvi xvi 1. Introduction 1 1.1 Contents of Book....................... 1 1.2 Sources............................ 2 1.3 Symbols and Conventions.................. 3 1.4 On the Applicability of Linear Elasticity.......... 4 2. Basic Elements of Linear Elasticity 7 2.1 Introduction.......................... 7 2.2 Elastic Displacement and Strain Tensor.......... 8 2.2.1 Straining versus rigid body rotation........ 8 2.2.2 Relationships for strain components........ 12 2.3 Traction Vector, Stress Tensor and Body Forces...... 20 2.3.1 Traction vector and components of stress..... 20 2.3.2 Body forces...................... 22 xvii

xviii Introduction to Elasticity Theory for Crystal Defects 2nd Ed 2.3.3 Relationships for stress components andbodyforces... 23 2.4 Linear Coupling of Stress and Strain............ 26 2.4.1 Stress as a function of strain............ 26 2.4.2 Strain as a function of stress............ 31 2.4.3 Corresponding elastic fields............ 33 2.4.4 Stress-strain relationships and elastic constants for isotropic system................. 35 2.5 Elastic Strain Energy..................... 39 2.5.1 General relationships................ 40 2.5.2 Strain energy in isotropic systems......... 41 2.6 St.-Venant s Principle.................... 41 Exercises............................... 42 3. Methods 57 3.1 Introduction.......................... 57 3.2 Basic Field Equation for the Displacement......... 58 3.3 Fourier Transform Method.................. 59 3.4 Green s Function Method.................. 60 3.5 Sextic and Integral Formalisms for Two-Dimensional Problems........................... 63 3.5.1 Sextic formalism................... 64 3.5.2 Integral formalism.................. 80 3.6 Elasticity Theory for Systems Containing Transformation Strains.................... 84 3.6.1 Transformation strain formalism.......... 85 3.6.2 Fourier transform solutions............. 88 3.6.3 Green s function solutions.............. 89 3.7 Stress Function Method for Isotropic Systems....... 90 3.8 Defects in Regions Bounded by Interfaces Method of Image Stresses....................... 91 Exercises............................... 94 4. Green s Functions for Unit Point Force 101 4.1 Introduction.......................... 101 4.2 Green s Functions for Unit Point Force........... 102 4.2.1 In infinite homogeneous region........... 103 4.2.2 In half-space with planar free surface....... 110

Contents xix 4.2.3 In half-space joined to an elastically dissimilar half-space along planar interface.......... 114 4.3 Green s Functions for Unit Point Force in Isotropic System...................... 118 4.3.1 In half-space joined to elastically dissimilar half-space along planar interface.......... 119 4.3.2 In infinite homogeneous region........... 127 4.3.3 In half-space with planar free surface....... 128 Exercises............................... 130 5. Interactions between Defects and Stress 139 5.1 Introduction.......................... 139 5.2 Interaction Energies between a Defect Source of Stress and Various Stresses in Finite Homogeneous Body.... 140 5.2.1 Interaction energy with imposed internal stress.................... 142 5.2.2 Interaction energy with applied stress....... 145 5.2.3 Interaction energy with defect image stress.... 148 5.2.4 Summary....................... 150 5.3 Forces on A Defect Source of Stress in Finite Homogeneous Body...................... 151 5.3.1 General formulation................. 151 5.3.2 Force obtained from change of the system total energy..................... 152 5.3.3 Force obtained from change of the interaction energy......................... 163 5.3.4 Summary....................... 163 5.4 Interaction Energy and Force Between an Inhomogeneity and Imposed Stress...................... 164 Exercises............................... 165 6. Inclusions in Infinite Homogeneous Regions 173 6.1 Introduction.......................... 173 6.2 Characterization of Inclusions................ 173 6.3 Coherent Inclusions...................... 174 6.3.1 Elastic field of homogeneous inclusion by Fourier transformmethod... 175

xx Introduction to Elasticity Theory for Crystal Defects 2nd Ed 6.3.2 Elastic field of inhomogeneous ellipsoidal inclusion....................... 185 6.3.3 Strain energy..................... 188 6.4 Coherent Inclusions in Isotropic Systems.......... 189 6.4.1 Elastic field of homogeneous inclusion byfouriertransformmethod... 189 6.4.2 Elastic field of homogeneous inclusion by Green s function method............ 190 6.4.3 Elastic field of inhomogeneous ellipsoidal inclusion with uniform ε T ij... 203 6.4.4 Strain energy..................... 206 6.4.5 Furtherresults... 210 6.5 Coherent Incoherent Transitions in Isotropic Systems............................ 212 6.5.1 General formulation................. 212 6.5.2 Inhomogeneous sphere................ 214 6.5.3 Inhomogeneous thin-disk.............. 216 6.5.4 Inhomogeneous needle................ 216 Exercises............................... 217 7. Interactions Between Inclusions and Imposed Stress 229 7.1 Introduction.......................... 229 7.2 Interactions between Inclusions and Imposed Stress inisotropicsystems... 229 7.2.1 Homogeneous inclusion............... 229 7.2.2 Inhomogeneous ellipsoidal inclusion........ 232 Exercises............................... 239 8. Homogeneous Inclusions in Finite and Semi-infinite Regions: Image Effects 249 8.1 Introduction.......................... 249 8.2 Homogeneous Inclusions Far From Interfaces in Large Finite Bodies in Isotropic Systems........ 250 8.2.1 Image stress..................... 250 8.2.2 Volume change of body due to inclusion effect of image stress.................... 251 8.3 Homogeneous Inclusion Near Interface in Large Semi-infinite Region..................... 253 8.3.1 Elastic field...................... 253

Contents xxi 8.4 Homogeneous Spherical Inclusion Near Surface of Half-space in Isotropic System.............. 254 8.4.1 Elastic field...................... 254 8.4.2 Force imposed by image stress........... 257 8.5 Strain Energy of Inclusion in Finite Region........ 258 Exercises............................... 258 9. Inhomogeneities 265 9.1 Introduction.......................... 265 9.2 Interaction Between a Uniform Ellipsoidal Inhomogeneity and Imposed Stress............. 266 9.2.1 Elastic field in body containing inhomogeneity and imposed stress.................. 266 9.2.2 Interaction energy between inhomogeneity and imposed stress.................. 268 9.2.3 Some results for isotropic systems......... 271 9.3 Interaction Between an Elastically Non-uniform Inhomogeneity and a Non-uniform Imposed Stress.... 275 Exercises............................... 277 10. Point Defects in Infinite Homogeneous Regions 283 10.1 Introduction.......................... 283 10.2 Symmetry of Point Defects................. 284 10.3 Force Multipole Model.................... 286 10.3.1 Basic model..................... 286 10.3.2 Force multipoles................... 289 10.3.3 Elastic fields of multipoles in isotropic systems........................ 292 10.3.4 Elastic fields of multipoles in general anisotropicsystems... 295 10.3.5 The force dipole moment approximation...... 295 10.4 Small Inclusion Model for Point Defect........... 298 Exercises............................... 299 11. Interactions between Point Defects and Stress: Point Defects in Finite Regions 303 11.1 Introduction.......................... 303

xxii Introduction to Elasticity Theory for Crystal Defects 2nd Ed 11.2 Interaction Between a Point Defect (Multipole) and Stress........................... 304 11.3 Volume Change of Finite Body Due to Single Point Defect.......................... 305 11.4 Statistically Uniform Distributions of Point Defects.... 307 11.4.1 Defect-induced stress and volume change of finite body..................... 307 11.4.2 The (p) tensor... 311 11.4.3 Defect-induced changes in x-ray lattice parameter...................... 312 Exercises............................... 315 12. Dislocations in Infinite Homogeneous Regions 321 12.1 Introduction.......................... 321 12.2 GeometricalFeatures... 321 12.3 Infinitely Long Straight Dislocations and Lines of Force...................... 325 12.3.1 Elastic fields..................... 326 12.3.2 Strainenergies... 332 12.4 Infinitely Long Straight Dislocations inisotropicsystems... 334 12.4.1 Elastic fields..................... 334 12.4.2 Strainenergies... 339 12.5 Smoothly Curved Dislocation Loops............ 340 12.5.1 Elastic fields..................... 340 12.5.2 Strainenergies... 362 12.6 Smoothly Curved Dislocation Loops inisotropicsystems... 363 12.6.1 Elastic fields..................... 363 12.6.2 Strainenergies... 371 12.7 Segmented Dislocation Structures.............. 371 12.7.1 Elastic fields..................... 372 12.7.2 Strainenergies... 381 12.8 Segmented Dislocation Structures in Isotropic Systems............................ 382 12.8.1 Elastic fields..................... 382 12.8.2 Strainenergies... 393 Exercises............................... 398

Contents xxiii 13. Interactions between Dislocations and Stress: Image Effects 413 13.1 Introduction.......................... 413 13.2 Interaction of Dislocation with Imposed Internal or Applied Stress: The Peach Koehler Force Equation......... 413 13.3 Interaction of Dislocation with its Image Stress...... 416 13.3.1 General formulation................. 416 13.3.2 Straight dislocations parallel to free surfaces... 419 13.3.3 Straight dislocation parallel to planar interface between elastically dissimilar half-spaces..... 429 13.3.4 Straight dislocation impinging on planar free surface of half-space................. 438 13.3.5 Dislocation loop near planar free surface of half-space..................... 449 13.3.6 Dislocation loop near planar interface between elastically dissimilar half-spaces.......... 451 Exercises............................... 452 14. Interfaces 459 14.1 Introduction.......................... 459 14.2 Geometrical Features of Interfaces Degrees offreedom... 460 14.3 Iso-elasticInterfaces... 461 14.3.1 Geometrical features................. 461 14.3.2 The Frank Bilby equation............. 465 14.3.3 Elastic fields of interfaces consisting of arrays of parallel dislocations................ 474 14.3.4 Elastic fields of arrays of parallel dislocations in isotropic systems................. 477 14.3.5 Interfacial strain energies in isotropic systems... 479 14.4 Hetero-ElasticInterfaces... 481 14.4.1 Geometrical features................. 481 14.4.2 Elastic fields..................... 482 Exercises............................... 499 15. Interactions between Interfaces and Stress 507 15.1 Introduction.......................... 507 15.2 The Energy-Momentum Tensor Force............ 508

xxiv Introduction to Elasticity Theory for Crystal Defects 2nd Ed 15.3 The Interfacial Dislocation Force.............. 511 15.3.1 Small-angle symmetric tilt interfaces........ 511 15.3.2 Small-angle asymmetric tilt interfaces....... 512 15.3.3 Large-angle homophase interfaces......... 514 15.3.4 Heterophase interfaces................ 515 Exercises............................... 516 16. Interactions between Defects 521 16.1 Introduction.......................... 521 16.2 Point Defect Point Defect Interactions........... 521 16.2.1 General formulation................. 521 16.2.2 Between two point defects in isotropic system.. 523 16.3 Dislocation Dislocation Interactions............ 524 16.3.1 Interaction energies................. 524 16.3.2 Interaction energies in isotropic systems...... 529 16.3.3 Interaction forces.................. 533 16.3.4 Interaction forces in isotropic systems....... 535 16.4 Inclusion Inclusion Interactions............... 536 16.4.1 Between two homogeneous inclusions....... 536 16.4.2 Between two inhomogeneous inclusions...... 539 16.5 Point Defect Dislocation Interactions............ 540 16.5.1 General formulation................. 540 16.5.2 Between point defect and screw dislocation in isotropic system.................. 541 16.6 Point Defect Inclusion Interactions............. 543 16.6.1 General formulation................. 543 16.6.2 Between point defect and spherical inhomogeneous inclusion with ε T ij = εt δ ij in isotropic system... 544 16.7 Dislocation Inclusion Interactions.............. 544 16.7.1 General formulation................. 544 16.7.2 Between dislocation and spherical inhomogeneous inclusion with ε T ij = εt δ ij in isotropic system... 545 Exercises............................... 545 17. Defect Self-interactions and Self-forces 557 17.1 Introduction.......................... 557 17.2 Self-force Experienced by a Smoothly Curved Dislocation...................... 557 17.2.1 Circular planar loop................. 557

Contents xxv 17.2.2 General smoothly curved planar loop....... 568 17.2.3 Some results for isotropic systems......... 569 17.3 Dislocation Line Tension................... 571 17.4 Self-force Experienced by Straight Dislocation Segment..................... 573 17.5 Self-force Experienced by Inclusion............. 576 Exercises............................... 579 Appendix A. Relationships Involving the Operator 589 A.1 Cylindrical Orthogonal Curvilinear Coordinates...... 589 A.2 Spherical Orthogonal Curvilinear Coordinates....... 590 Appendix B Integral Relationships 591 B.1 Divergence (Gauss s) Theorem............... 591 B.2 Stokes Theorem....................... 591 B.3 Another form of Stokes Theorem.............. 592 Appendix C The Tensor Product of Two Vectors 595 Appendix D Properties of the Delta Function 597 Appendix E The Alternator Operator 599 Appendix F Fourier Transforms 601 Appendix G Equations from the Theory of Isotropic Elasticity 603 G.1 Cylindrical Orthogonal Curvilinear Coordinates...... 603 G.2 Spherical Orthogonal Curvilinear Coordinates....... 605 Appendix H Components of the Eshelby Tensor in Isotropic System 607 Appendix I Airy Stress Functions for Plane Strain 609 Appendix J Deviatoric Stress and Strain in Isotropic System 611 References 613 Index 621