Theory of Dislocations Peter M

Cambridge University Press 978-0-521-86436-7 — Theory of Dislocations Peter M. Anderson , John P. Hirth , Jens Lothe Frontmatter More Information

Theory of Dislocations

Third Edition

Theory of Dislocations offers coverage of the fundamentals of line defects called dislocations, with applications to speciic metallic and ionic crystals. Step-by- step developments offer an in- depth understanding of the topic and a solid theoretical foundation from which to develop modeling and computational approaches to discrete dislocation plasticity. Important experimental observations related to the effects of crystal structure, temperature, nucleation mechanisms, and speciic systems are discussed. This new edition incorporates margin notes and extensive chapter summaries that highlight key concepts and developments. New topics include a tensorial description of dislocation content, disconnection properties, curvature induced by dislocations, lattice Green's functions, conserved integrals, multipole representations of dislocations, stability of junctions, and discrete dislocation dynamics simulations. Each of the twenty-three chapters begins with a striking image and narrative from an international researcher that highlights a key advance or breakthrough. The text is suitable for graduate and advanced undergraduate study and includes problems at the end of each chapter.

Peter M. Anderson received his ScB degree in Engineering from Brown University in 1981 and his ScM and PhD degrees in Applied Sciences from Harvard University in 1982 and 1986, respectively. Following a two-year postdoctoral fellowship at Cambridge University (UK), he joined The Ohio State University where he is currently Professor and Chair in the Department of Materials Science and Engineering. He has authored/ coauthored over one hundred articles on mechanical behavior of bulk and thin ilm materials, including a chapter on crystal plasticity in “Fundamentals of Metal Forming” and a set of over three hundred PowerPoint lecture slides that serve as an instructors’ resource for the introductory textbook, Materials Science and Engineering: An Introduction. He has held visiting positions at Brown University, National Institute of Standards and Technology, Ruhr-Universität Bochum, and Los Alamos National Laboratory, where he was Bernd T. Matthias Scholar. He is recipient of an Ofice of Naval Research Young Investigator Award, three- time recipient of the Boyer Award for Teaching Innovation, and a recipient of the Lumley Research Award.

Professor John Price Hirth received his undergraduate degree from The Ohio State University in 1953 and his PhD from Carnegie Mellon University (then Carnegie Institute of Technology) in 1957. He taught at the latter institution for a short period after a postdoctor- ate year at Bristol University. He was then a professor at The Ohio State University through 1988 and at Washington State University 1988–1999, when he retired to Arizona, remain- ing afiliated with Los Alamos National Laboratory. His research has been in the areas of dislocation theory, phase transformations, and mechanical properties. In addition to many scientiic articles, he has been the coauthor of two books on these subjects. He has received a number of awards recognizing his work and is a member of the National Academy of Engineering (since 1974) and National Academy of Science (since 1994).

Jens Lothe (deceased) was born in Oslo 1931 and studied physics at UiO (University of Oslo), where he received the cand. real. degree in 1956 and the dr. philos. degree in 1968. Positions at UiO included research assistant (1956– 1959); lecturer (1959– 1963); associate professor (dosent) (1963– 1972); and professor (1972– 1992). Positions while on leave from UiO included Norwegian Research Council Student at University of Bristol (1957–1958), Assistant Professor at Carnegie Institute of Technology (now Carnegie- Mellon University) (1960–1962), and Battelle Visiting Professor at The Ohio State University (1965–1966). Professor Lothe’s research interests included nucleation theory, dislocation theory, and elastic wave theory for anisotropic media. Lothe was author or coauthor of approximately 120 papers. He was coauthor (with J. Hirth) of previous editions of Theory of Dislocations (irst edition, 1968; second edition, 1982; reprint edition, 1992). He coedited/authored (with V. L. Indenbom) Elastic Strain Fields and Dislocation Mobility, volume 31 of the series Modern Problems in Condensed Matter Sciences (1992). Lothe was a member of the Norwegian Academy of Science and Letters (since 1973).

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-0-521-86436-7 — Theory of Dislocations Peter M. Anderson , John P. Hirth , Jens Lothe Frontmatter More Information

About the Book Cover

The book cover image is the product of a competition among students and faculty in the Department of Materials Science and Engineering at The Ohio State University (OSU). The objective was to create the illusion that the book itself is a dislocated crystal. The quality and number of entries received was enhanced by the adoption in 2012 of a three-semester undergraduate sequence in modeling and simulation and a graduate core course in computational materials science, taught by a cadre of faculty: Peter Anderson, Maryam Ghazisaeidi, Stephen Niezgoda, Alison Polasik, Wolfgang Windl, and Ji-Cheng Zhao. The winning image, created by third- year graduate student Daniel Buey, provides a perspective view along the core of an edge dislocation in a simple cubic crystal. The view along the spine gives the illusion that the book is only one unit cell thick. CrystalMaker software was used to create an undistorted simple cubic structure. A dislocation was then inserted by displacing the atoms according to the Volterra solution for an edge dislocation (see Eqs. 3.47, 3.48).

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THEORY OF DISLOCATIONS

Third Edition

PETER MARTIN ANDERSON The Ohio State University

JOHN PRICE HIRTH Washington State University

JENS LOTHE† Universitetet i Oslo

One Liberty Plaza, 20th Floor, New York NY 10006, USA

Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence.

www.cambridge.org Information on this title: www.cambridge.org/ 9780521864367 © Peter Anderson, John Hirth, and Jens Lothe 2017 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2017 Printed in the United States of America by Sheridan Books, Inc. A catalogue record for this publication is available from the British Library.

Library of Congress Cataloging- in- Publication Data Names: Anderson, Peter M. (Peter Martin) | Hirth, John Price, 1930– | Lothe, Jens, 1931–2016 Title: Theory of dislocations / Peter Anderson, Ohio State University, John Hirth, Washington State University, Jens Lothe, Universitetet i Oslo. Description: [2017 edition]. | Cambridge : Cambridge University Press, 2017. | Originally published: New York : McGraw-Hill, 1967. Editions published: New York : Wiley, 1982, and Malabar, FL : Krieger, 1992. | Includes bibliographical references and index. Identiiers: LCCN 2016024204 | ISBN 9780521864367 (hardback : alk. paper) Subjects: LCSH: Dislocations in crystals. | Crystals. Classiication: LCC QD921. H56 2017 | DDC 548/.842–dc23 LC record available at https://lccn.loc.gov/2016024204

ISBN 978- 0- 521- 86436- 7 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third- party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.

Contents

Preface page xix

PART I ISOTROPIC CONTINUA 1

1 INTRODUCTORY MATERIAL ...... 3 1.0 Key Developments 3 1.1 Introduction 4 1.2 Physical Basis for Dislocations 4 1.2a Early Work 4 1.2b Theoretical Shear Strength of a Perfect Crystal 5 1.2c Observations of Dislocations 7 1.3 Some Elementary Geometric Properties of Dislocations 9 1.3a Displacement Associated with a Dislocation 9 1.3b The Burgers Vector 11 1.3c Continuity of a Dislocation 14 1.3d Equivalent Burgers Circuits 15 1.3e Dislocation Glide Planes 16 1.4 A Tensorial Description of Dislocation Density 17 1.5 Curvature Induced by Dislocations 18 1.6 The Standard Model 19 Problems 19 Bibliography 20

2 ELASTICITY ...... 23 2.0 Key Developments 23 2.1 Introduction 24 2.2 Classical Linear Elasticity 24 2.2a Fundamental Equations 24 2.2b Contracted Notation 29 2.3 Transformation of Tensorial Components 31 2.4 Isotropic Elastic Solids 34 2.5 Plane Strain and Anti- Plane Strain 36 2.6 The Displacement Field of a Point Force 38 2.6a Electrostatic Analog 38 2.6b Green’s Function for an Ininite, Isotropic, Elastic Solid 39

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2.6c Point Source of Expansion in a Sphere 40 2.6d Green’s Functions for Discrete Lattices 42 2.7 Interaction Energy between Internal and External Sources of Stress 46 2.7a Theorem Concerning Cross Terms in Interaction Energy 46 2.7b Interaction Energy between a Dilatation Source and Contents External Stress 48 2.7c Interaction Energy between a Dislocation and External Stress 49 2.8 Qualitative Principles in Linear Elasticity 50 2.8a St. Venant’s Principle 50 2.8b Principle of Superposition 50 2.8c The Standard Model 51 Problems 51 Bibliography 52

3 THEORY OF STRAIGHT DISLOCATIONS ...... 55 3.0 Key Developments 55 3.1 Introduction 56 3.2 Screw Dislocations 56 3.2a Stress and Displacement Fields in an Ininite Medium 56 3.2b Screw Dislocation in a Cylinder 57 3.2c Strain Energy 59 3.2d Thermodynamic Forces 60 3.2e Force on a Screw Dislocation 60 3.2f Absence of Self Stress Forces 62 3.3 Image Forces on Screw Dislocations 64 3.3a Screw Dislocation Parallel to a Free Surface 64 3.3b Screw Dislocation in a Circular Cylinder 64 3.4 Edge Dislocations 66 3.4a Stress and Displacement Fields in an Ininite Medium 66 3.4b Edge Dislocation in a Cylinder 69 3.4c Strain Energy 71 3.4d Force on an Edge Dislocation 73 3.5 Image Forces on Edge Dislocations 76 3.5a Edge Dislocation Parallel to a Free Surface 76 3.5b Edge Dislocation Normal to a Free Surface 78 3.6 The Mixed Character Straight Dislocation 79 3.6a Strain Energy 79 3.6b Force on a Mixed Dislocation 80 3.6c Image Force for a Mixed Dislocation Parallel to a Free Surface 81 3.6d Stress Field for a Mixed Dislocation Inclined to a Free Surface 82 3.7 Straight Dislocations in Anisotropic Media 84 3.8 Conserved Integrals 85 3.8a Derivation 85 3.8b J Integral and the Peach- Koehler Force 86 3.8c M Integral and the Prelogarithmic Energy Factor 88 Problems 89 Bibliography 90

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4 THEORY OF CURVED DISLOCATIONS ...... 93 4.0 Key Developments 93 4.1 Introduction 93 4.2 Conservative and Nonconservative Motion 93 4.3 Displacements from Curved Dislocations 94 4.3a Application of Green’s Function Method 94 4.3b The Burgers Displacement Equation 97 Contents 4.4 Self Stress of a Curved Dislocation 98 4.5 Interaction Energy between Loops 100 4.6 Force on a Segment Produced by a Stress 102 4.7 Self Energy of a Dislocation Loop 103 Problems 104 Bibliography 104

5 APPLICATIONS TO DISLOCATION INTERACTIONS ...... 107 5.0 Key Developments 107 5.1 Introduction 108 5.2 Energy and Force: Parallel and Perpendicular Segments 108 5.2a Neglect of End Effects 108 5.2b Parallel and Perpendicular Screw Segments 109 5.2c Parallel Ininite Dislocations 110 5.3 Energy: Coaxial Circular Loops 112 5.3a Case 1: Equal Radii Prismatic Loops 112 5.3b Case 2: Unequal Radii Prismatic Loops 114 5.3c Case 3: Unequal Radii Shear Loops 115 5.4 Force: Nonparallel Ininite Dislocations 115 5.5 Force: Finite and Ininitesimal Segments 118 5.5a Concept of a Dislocation Force Function 118 5.5b Dislocation Force Function: Nonparallel Segments 119 5.5c Dislocation Force Function: Parallel Segments 122 5.5d Self Force: Segments on the Same Dislocation 123 5.6 Stress: A Single Segment 124 5.6a Concept of a Dislocation Segment Stress Function 124 5.6b Dislocation Segment Stress Function: Indicial Form 124 5.6c Dislocation Segment Stress Function: Dyadic Form 126 5.6d Stress Field: Angular Segments 126 5.6e Stress Field: Ininitesimal Loop 128 5.7 Loop Stress & Strain Fields: Line Integrals 129 5.7a The Lothe- Brown Formula 129 5.7b Stress and Force: Segments 132 5.8 Displacement Field: A Single Segment 133 5.8a Concept of a Dislocation Displacement Function 133 5.8b Dislocation Displacement Function: Dyadic Form 133 5.9 General Image Forces 135 5.9a Simple Image Construction: Validity and Limitations 136 5.9b Mixed Dislocation Inclined to a Free Surface 136 5.9c Internal Surface: Parallel Screw Dislocation 137 5.9d Internal Surface: Other Geometries 138 5.9e Alternative Image Constructions 139 5.10 Multipole Expansions for Interaction Energy and Stress 139 Problems 142 Bibliography 143

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6 APPLICATIONS TO SELF ENERGIES...... 145 6.0 Key Developments 145 6.1 Introduction 146 6.2 Energy: Piecewise Straight Coniguration 146 6.3 Interaction Energies: The Jøssang Equations 150 6.3a Non- Coplanar Case 150 6.3b Coplanar Case z = 0 153 Contents 6.3c Parallel Dislocations 153 6.4 Energy: Dislocation Loops 154 6.4a Circular Loop 154 6.4b Hexagonal Loop 155 6.4c Small Bow Out 156 6.4d Large Bow Out 158 6.5 Concept of Line Tension 158 6.5a Orientation-Dependent Line Tension 161 6.5b Energy Flow along Dislocations 163 6.5c Dislocations in Complex Conigurations 165 Problems 165 Bibliography 165

7 DISLOCATIONS AT HIGH VELOCITIES ...... 167 7.0 Key Developments 167 7.1 Introduction 167 7.2 Moving Screw Dislocation 168 7.2a Displacement and Stress Fields 168 7.2b Energy 170 7.2c Effect of a Free Surface 171 7.3 Moving Edge Dislocation 172 7.3a Displacement and Stress Fields 172 7.3b Surface Effects at the Rayleigh Velocity 174 7.4 Acceleration and Radiation 175 7.5 Dislocation Momentum and Radiation Forces 177 7.5a Dislocation Momentum and Effective Mass 177 7.5b Phonon Radiation and Momentum Transfer 180 7.6 Supersonic Dislocations 181 7.7 Dislocation Mobility 183 Problems 186 Bibliography 187

PART II EFFECTS OF CRYSTAL STRUCTURE 189

8 THE INFLUENCE OF LATTICE PERIODICITY...... 191 8.0 Key Developments 191 8.1 Introduction 192 8.2 The Peierls- Nabarro Dislocation Model 193 8.2a Edge Dislocation 193 8.2b Screw Dislocation 197 8.2c Elastic Energy of the Peierls Dislocation 199 8.3 Atomic Calculations 200 8.3a Methods 200 8.3b Core Coniguration 202 8.3c Core Energy 203

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8.4 The Peierls Energy 208 8.4a Review of Original Treatment 208 8.4b Modiications of the Peierls Model 209 8.4c Atomic Calculations and Empirical Representation 211 8.5 Elastic Energies of Kinks 212 8.5a Sharp Kink 215 8.5b Oblique Kink 218 Contents 8.6 Kink Energy and Local Coniguration 220 8.6a Constant Line Tension 220 8.6b Variable Line Energy 221 8.7 Periodicity of the Kink Energy 223 8.8 Line Tension in the Kink Model 223 8.9 Jogs 226 8.9a Screw- Jog Coniguration 226 8.9b Edge- Jog Coniguration 226 8.9c Jog Pair Energy 227 8.9d Single Jog Energy 227 8.9e Line Tension 228 8.9f Jog- Kink Interaction 228 Problems 228 Bibliography 229

9 SLIP SYSTEMS OF PERFECT DISLOCATIONS ...... 231 9.0 Key Developments 231 9.1 Introduction 232 9.2 Perfect Dislocations 232 9.2a Crystallographic Notation 232 9.2b Burgers Vector 233 9.2c Frank Energy Criterion 233 9.3 Slip Systems in Crystals 234 9.3a FCC Slip Systems 235 9.3b BCC Slip Systems 236 9.3c HCP Slip Systems 238 9.3d Diamond Cubic Slip Systems 240 9.3e NaCl Type Slip Systems 240 9.3f General Slip Systems 242 9.4 Resolved Shear Stress 242 9.4a Tensile Tests 242 9.4b Torsion and Simple Shear 243 9.5 Independent Slip Systems 245 9.5a Slip Systems for General Deformation 245 9.5b FCC Metals 246 9.5c BCC Metals 248 9.5d HCP Metals 248 9.5e NaCl Type Crystals 249 9.6 Effects of Free Surfaces and Interfaces 250 9.6a Single Crystals and Bicrystals 250 9.6b Unusual Surface Slip 251 Problems 252 Bibliography 252

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10 PARTIAL DISLOCATIONS IN FCC METALS ...... 255 10.0 Key Developments 255 10.1 Introduction 256 10.2 Stacking Faults 256 10.2a Types of Stacking Faults 256 10.2b Theoretical Stacking Fault Energy 259 10.3 Partial Dislocations 261 Contents 10.3a Shockley Partials 261 10.3b Thompson Tetrahedron 264 10.3c Stair Rod Partials 265 10.3d Frank Partials 267 10.4 Arrays of Partial Dislocations 268 10.4a Extended Superjogs 268 10.4b Extended Unit Jogs or Jog Lines 270 10.4c Stacking Fault Tetrahedra 271 10.4d Extended Nodes 274 10.4e Extended Dipoles 277 10.5 Extrinsic Fault Conigurations 278 10.6 Forces on Partial Dislocations 280 10.6a Dynamic Effects 280 10.6b Static Effects 281 10.7 Peierls Barrier for Partial Dislocations 281 Problems 282 Bibliography 283

11 PARTIAL DISLOCATIONS IN OTHER STRUCTURES...... 285 11.0 Key Developments 285 11.1 Introduction 286 11.2 HCP Lattice 286 11.2a Stacking Faults 286 11.2b Vector Notation 287 11.2c Glissile Shockley Partials 288 11.2d Frank Sessile Partials 289 11.2e Other Partials 289 11.2f Zonal Dislocations 290 11.2g Extended Jogs 291 11.3 BCC Lattice 292 11.3a Stacking Faults 292 11.3b Partial Dislocations 293 11.3c Fractional Screw Dislocations 294 11.3d Vector Notation 296 11.4 Diamond Cubic Lattice 296 11.4a Stacking Faults 296 11.4b Perfect Dislocations 297 11.4c Vector Notation 298 11.4d Glide Set of Dislocations 298 11.4e Shufle Set of Dislocations 299 11.4f Jogs 300 11.4g Glide vs. Shufle Nature of Slip 301 11.5 Layer Structures 301 11.6 Superdislocations in Ordered Structures 301

Contents xi

11.7 β- Brass Superlattice 302 11.7a Structure 302 11.7b Shear Antiphase Boundary Energy 303 11.7c General Antiphase Boundary 304 11.7d Dislocations 305 306 11.8 Cu3Au Superlattice 11.8a Superlattice Structure 306 11.8b Shear Antiphase Boundary Energy 306 Contents 11.8c Dislocations 307 11.9 Synchroshear 308 Problems 309 Bibliography 310

12 DISLOCATIONS IN IONIC CRYSTALS ...... 313 12.0 Key Developments 313 12.1 Introduction 314 12.2 Effective Charge 314 12.3 Edge Dislocations 316 12.3a Charge at the Point of Emergence 316 12.3b Charge at Jogs 317 12.3c Charge at Kinks 318 12.3d Charge along a Core Arising from Vacancies 319 12.4 Screw Dislocations 320 12.4a Charge at the Point of Emergence 320 12.4b Charge at Jogs and Kinks 321 12.4c Moving Jogged Screw 322 12.4d Charged Screw Dislocations 323 12.5 Curved, Mixed Dislocations 323 12.6 Intersection Jogs and Kinks 324 12.6a Vacancy and Interstitial Formation 324 12.6b Charges on Intersection Kinks and Jogs 325 12.6c Contributions to the Gyulai- Hartly Effect 325 12.7 Inluence of Charge on Deformation 326 12.8 Polyvalent Ionic Crystals 326 Problems 327 Bibliography 327

13 DISLOCATIONS IN ANISOTROPIC ELASTIC MEDIA ...... 331 13.0 Key Developments 331 13.1 Introduction 332 13.2 Average Elastic Constants 333 13.2a Voigt Average 334 13.2b Reuss Average 334 13.2c Comparison of Average Elastic Constants 336 13.2d Cubic Crystals 338 13.2e Transformation of Orthogonal Axes 338 13.2f Hexagonal Crystals 341 13.3 Straight Dislocations in Anisotropic Media 342 13.1a Sextic Theory 342 13.3b Displacement Field 342 13.3c Stress Field and Energy 347 13.3d Summary of Method 348

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13.4 Simple Solutions to Anisotropic Equations 349 13.4a Separation into Screw and Edge Components 349 13.4b Pure Edge Dislocation 351 13.4c Pure Screw Dislocation 355 13.4d Third Order Solutions 356 13.5 Hexagonal Crystals 356 13.5a Basal Plane Dislocations 356 Contents 13.5b Basal Plane Isotropy 357 13.5c Nonbasal Dislocations 359 13.6 Cubic Crystals 359 13.6a FCC 359 13.6b BCC 362 13.6c NaCl Type 364 13.7 Advanced Straight Dislocation Formalism 364 13.7a Stroh Theory 365 13.7b Integral Method 369 13.7c Applications 372 13.8 Dislocation Stability 373 13.9 Dislocation Segments 375 Problems 375 Bibliography 376

PART III INTERACTIONS WITH POINT DEFECTS 377

14 EQUILIBRIUM DEFECT CONCENTRATIONS ...... 379 14.0 Key Developments 379 14.1 Introduction 380 14.2 Thermal Kinks 380 14.2a Free Energy of Formation 380 14.2b Equilibrium Concentration 382 14.2c Skew Dislocation 383 14.2d String Model 383 14.2e High Kink Densities 385 14.3 Thermal Jogs 385 14.4 Intrinsic Point Defects in Equilibrium with Dislocations 386 14.4a Interstitial Equilibria 387 14.4b Vacancy Equilibria 389 14.4c Combined Interstitial- Vacancy Equilibria 391 14.4d Higher Order Approximations 391 14.5 Solute- Atom Equilibria 393 14.5a Isotropic Size Effect 393 14.5b Concentrated Solutions 396 14.5c Core Segregation 398 14.5d Other Elastic Interaction Effects 399 14.5e Suzuki Segregation 401 14.6 Fluctuations and Pinning in Segregated Regions 402 14.6a Nearly Saturated Cores 402 14.6b Almost Clean Dislocations 403 14.7 Charged Edge Dislocation 404 14.7a Charged Tube Analogy 404 14.7b Charged Edge Dislocation in a NaCl Crystal 406

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Problems 409 Bibliography 411

15 DIFFUSIVE GLIDE AND CLIMB PROCESSES ...... 413 15.0 Key Developments 413 15.1 Introduction 414 15.2 Glide over the Peierls Barrier 415 15.2a Kink Mobility 415 Contents 15.2b Single Kink Drift 417 15.2c Kink Glide at Small Stress 418 15.2d Kink Pair Nucleation at Large Stress 419 15.2e Modiications of the Kink Pair Theory 421 15.2f Dislocation Velocity 423 15.3 Applications to Internal Friction 424 15.3a Granato- Lücke Theory 425 15.3b Bordoni Peak 428 15.3c Skew Dislocation Line 430 15.4 Diffusion- Controlled Climb 431 15.4a Climb Force: Straight Dislocation 431 15.4b Climb Rate 434 15.4c Interaction between Climbing Dislocations 436 15.4d Climb Force: Curved Dislocations 437 15.4e Climb in a Moving Reference System 439 15.5 Climb of Jogged Dislocations 441 15.5a Single Jog Motion 441 15.5b Climb of Skew Dislocations 443 15.5c Climb by Local Core Diffusion 446 15.5d Jog Pair Nucleation 446 15.5e Quasi- Equilibrium Assumption 447 15.6 Climb of Extended Dislocations 448 15.6a Jog Mobility 448 15.6b Jog Nucleation 450 15.6c Radiation Creep 451 Problems 452 Bibliography 453

16 GLIDE OF JOGGED DISLOCATIONS ...... 455 16.0 Key Developments 455 16.1 Introduction 455 16.2 Screws with Vacancy- Producing Jogs 456 16.2a Diffusion- Controlled Glide 456 16.2b Thermally Activated Glide 457 16.2c Point Forces on Dislocations 459 16.2d Breakaway from Pinning Sites 460 16.3 Screws with Interstitial-Producing Jogs 461 16.4 Screws with Jogs of Both Signs 461 16.5 Glide with Extended Jogs 463 16.5a Glissile Extended Jogs 464 16.5b Sessile Edge Jogs 464 16.5c Sessile Screw Jogs 464 16.5d Tetrahedron Formation 466

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16.5e Motion of Dislocations with Extended Jogs 467 Problems 468

17 DISLOCATION MOTION IN VACANCY SUPERSATURATIONS . . . 471 17.0 Key Developments 471 17.1 Introduction 472 17.2 Climb Forces 472 Contents 17.3 Nucleation of Vacancy Aggregates 473 17.3a Nucleation Theory 474 17.3b Comparison with Experiment 475 17.4 Climb of Straight Dislocations 476 17.5 Bardeen- Herring Sources 477 17.5a Double- Ended Source 477 17.5b Spiral Source 479 17.6 Dislocation Helices 480 17.6a Mechanisms of Formation 481 17.6b Uniform Cylindrical Helices 482 17.6c Fluctuations in Helix Geometry 484 17.6d Tight Winding Case 485 17.6e Other Effects 486 17.7 Stacking- Fault Tetrahedra 486 Problems 487 Bibliography 488

18 EFFECTS OF SOLUTE ATOMS ON DISLOCATION MOTION ...... 491 18.0 Key Developments 491 18.1 Introduction 492 18.2 Diffusion Associated with a Moving Potential Well 493 18.2a Diffusion Solutions 494 18.2b Square Well Potential 497 18.2c Forces in Moving Potential Fields 498 18.3 Drag of a Cottrell Atmosphere by a Dislocation 501 18.3a Interaction Forces 501 18.3b Diffusion Solution 502 18.3c Forces in the Range of the Dislocation Potential 505 18.3d High Velocity Creep 507 18.3e Limitations and Extensions of the Model 510 18.4 Drag of a Snoek Atmosphere by a Dislocation 511 18.5 Dislocation Core Effects 515 18.5a Large Core Drag 515 18.5b Small Core Drag 517 18.5c Breakaway from Core Atmospheres 517 18.5d Breakaway from a Cottrell Atmosphere 520 18.6 Application to Deformation Behavior 521 18.7 Other Solute Effects 522 18.7a Low Temperature Hardening 522 18.7b Ordered Crystals 525 18.7c Suzuki Locking 527 18.8 Ionic Crystals 529 Problems 530 Bibliography 531

Contents xv

PART IV GROUPS OF DISLOCATIONS 533

19 GRAIN BOUNDARIES AND INTERFACES...... 535 19.0 Key Developments 535 19.1 Introduction 536 19.2 Dislocation Models of Grain Boundaries 539 19.2a Simple Boundaries 539 19.2b Net Dislocation Density in an Arbitrary Small- Angle Contents Boundary 543 19.2c Arbitrary Large- Angle Boundary 544 19.2d Disconnections 547 19.2e Dislocation Spacing in the Boundary 547 19.3 Boundaries under Restricted Conditions 549 19.3a One Set of Dislocations 550 19.3b Two Sets of Dislocations 550 19.3c Three Sets of Dislocations 554 19.3d Added Constraints 556 19.4 Local Interactions to Form Stable Boundaries 557 19.4a General Considerations 557 19.4b Grain Boundaries in FCC Crystals 559 19.5 Stress States Near Grain Boundaries 561 19.6 Secondary Grain Boundary Dislocations 564 19.7 Grain Boundary Energies 567 19.7a Perfect Lattice Dislocation Boundaries 567 19.7b Coincidence Lattice Orientations 570 19.8 Other Grain Boundary Effects 571 19.8a Grain Boundary Mobility 571 19.8b Boundary Resistance to Dislocation Motion 572 19.8c Modiications from Anisotropic Elasticity 572 Problems 572 Bibliography 573

20 DISLOCATION SOURCES ...... 575 20.0 Key Developments 575 20.1 Introduction 576 20.2 Frank- Read Sources 576 20.3 Double Cross Slip Sources 579 20.4 Vacancy Disc Sources 580 20.5 Glide Loop Nucleation 580 20.6 Grain Boundary Sources 582 20.7 Kinematic Sources 582 Problems 584

21 DISLOCATION PILEUPS AND CRACKS...... 587 21.0 Key Developments 587 21.1 Introduction 588 21.2 Superdislocation Concept 589 21.3 Stressed Double Pileup 590 21.3a Dislocation Distribution 590 21.3b Force on the Leading Dislocation 592 21.4 Unstressed Single- Sign Pileup 594

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21.4a Dislocation Distribution 594 21.4b Force on the Leading Dislocation 594 21.5 Stressed Single-Sign Pileup 595 21.5a Dislocation Distribution 595 21.5b Back Stress 596 21.5c The Stress Field Ahead of the Pileup 597 21.5d Screw Dislocation Pileup 599 Contents 21.5e Kinematic and Inverse Arrays 600 21.6 Discontinuous Tilt Boundary 600 21.7 Tensile and Shear Cracks 601 21.7a Brittle Cracks and Fracture Mechanics 601 21.7b Ductile Cracks and Crack- Dislocation Interaction 603 21.8 Applications to Macroscopic Flow Phenomena 605 21.8a Blunting of Pileups 605 21.8b Fracture and Yielding Mechanisms at Pileups 605 21.8c Hall- Petch Relation 606 Problems 607 Bibliography 608

22 DISLOCATION INTERSECTIONS AND BARRIERS...... 611 22.0 Key Developments 611 22.1 Introduction 612 22.2 Intersection of Perfect Dislocations 614 22.2a Perpendicular Screw Dislocations 614 22.2b Perpendicular Screw and Edge Dislocations 614 22.2c Energy Fluctuation during a Dislocation Intersection Process 615 22.3 Intersection Jogs 616 22.4 Extended Dislocation Barriers and Intersection Jogs 617 22.4a Barriers with Low Fault Energy 617 22.4b Intersection Jogs 621 22.4c Barriers with High Fault Energy 621 22.5 Cross Slip 622 22.6 Stability of Junctions 625 22.7 Discrete Dislocation Dynamics Simulations 626 22.7a Fundamentals 627 22.7b Applications 630 Problems 631 Bibliography 632

23 DEFORMATION TWINNING...... 635 23.0 Key Developments 635 23.1 Introduction 636 23.2 Crystallography of Twinning 636 23.3 Pole Mechanisms for Twinning 640 23.3a BCC Crystals: The Cottrell- Bilby Pole Mechanism 640 23.3b HCP Crystals 643 23.3c FCC Crystals 644 23.4 Other Mechanisms for Twin Sources 645 23.4a Nucleation 645 23.4b Breakaway of Partial Dislocations 646 23.4c Cross Slip at an Obstacle 647 23.4d Relection Mechanism 647

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23.5 Twin Boundary Phenomena 648 23.5a Emissary Dislocations 648 23.5b Accommodation Bands 649 23.5c Interaction with Glide Dislocations 649 23.6 Twinning in Complex Systems 650 Problems 651 Bibliography 651 Contents

Appendix A1 Elastic Constants 653 A1.1 Elastic Constants, Compliances, and Averages for Cubic Crystals 653 A1.2 Stress- Strain Relations for Isotropic, Linear- Elastic Materials 655 A1.3 Relations between Isotropic Elastic Constants E, K, µ, ν, and λ 655 Appendix A2 Surface Energies 656 Appendix A3 Symbols and Variables 658 References 661 Index 691

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Preface

This book is based on lecture notes that the authors developed for courses on the theory of dislocations. It is intended to be primarily a comprehensive text in the ield of dislocations. For this reason, an exhaustive literature survey is not attempted, although key references are cited throughout the book. This edition incorporates several advances in the theory of dislocations in a format that highlights key developments. New topics include a tensorial description of dislocation content, disconnection properties, curvature induced by dislocations, lattice Green’s functions, conserved integrals, multipole representations of dislocations, stability of junctions, and discrete dislocation dynamics simulations. Expanded and revised topics include classical linear elasticity, the interaction between internal and external stress, solutions for dislocations near surfaces and interfaces, atomic calculations of dislocation cores, dislocation models of grain boundaries, and tensile and shear cracks. Throughout the text, we strive to provide a comprehensive treatment of the theory of dislocations, with detailed elementary portions that can be used for undergraduate instruction, and advanced treatments that are useful in graduate instruction and for researchers in the ield. The basic order of topics from the previous edition is pre- served, with Part I covering dislocations in isotropic continua, Part II discussing the effects of crystal structure, Part III dealing with the effects of point defects at inite temperatures, and Part IV treating groups of dislocations. Aspects of the theory that are well founded are discussed in detail. However, the treatment of some subjects, such as work hardening and discrete dislocation dynamics in Part IV, is still moot. In such cases, we briely outline current theories, point out their shortcomings, and suggest approaches to a general solution to the problem in question. Throughout the book, we assume a background in mathematics through dif- ferential equations. For more advanced mathematical topics, we outline the deri- vations in suficient detail for the student to derive them either directly or with the aid of a reference book, such as I. S. Sokolnikoff and R. M. Redheffer, Mathematics of Physics and Modern Engineering, McGraw- Hill, New York, 1966. A consequence of the variety of topics treated is that the same symbol is sometimes used for different quantities. For example, v denotes velocity, and volume. These are summarized in Appendix A3. The context usually makes the deinitions clear and the use of unfamiliar symbols is avoided. Several new features are intended to make the information more accessible. They include a detailed summary of key developments at the beginning of each chapter; side notes that call out key results, deinitions, principles, and other

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information; margin annotations that conveniently identify each chapter; and an expanded subject index to eficiently ind important results. We are indebted to many individuals and organizations. These include the twenty- three authors who supplied the igures that introduce each chapter and highlight key advances. They are acknowledged in the supporting text for each igure. Several colleagues and students contributed to discussions and comments that shaped and improved the text. They include Robert Ballufi, David Barnett, Preface John Carpenter, Xiang Chen, William Clark, Suliman Dregia, Hamish Fraser, Maryam Ghasizaeidi, Michael Gram, Peter Hazzledine, Richard Hoagland, Lin Li, Qizhen Li, Sivom Manchiraju, Michael Mills, Amit Misra, Terrence Mitchell, William Nix, Harshad Paranjape, Robert Pond, James Rice, Yao Shen, Ryan Smith, Frans Spaepen, Helena van Swygenhoven, Robb Thomson, Jian Wang, Yunzhi Wang, and Hussein Zbib. We are also indebted to several individuals who assisted with countless hours of typing, editing, equation entry, image processing, permis- sions rights, and discussions about grammar and writing style: Nora Anderson, Kathleen Babusci, Richard Blocher, Rachel Fishbein, and James Snyder. Others are acknowledged in earlier editions. We also are pleased to acknowledge the U.S. Department of Energy- Basic Energy Sciences, National Science Foundation, Air Force Ofice of Scientiic Research, Ofice of Naval Research, Los Alamos National Laboratory, and the Physics of Geological Processes Group at the University of Oslo. The research activities supported by these institutions fostered many interactions, discussions, and concepts that appear in the text.