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1 Introduction 1 Invitation 1 Introduction 4 Overview 8 Omissions 11 Related Reviews and Books 15 2 Review of Continuum Mechanics 17 Vectors and Matrices 17 Deformation 20 Deformation Gradient 22 Rotation and Stretch 26 Kinematic Compatibility 27 3 Continuum Theory of Crystalline Solids 29 Bravais Lattice 29 Deformation of Lattices and Symmetry 32 Lattice-Continuum Link: The Cauchy-Bor Hypothesis 34 Energy Density in Crystalline Solids 36 Symmetry of Deformable Lattices: Ericksen-Pitteri Neighborhood 38 Multi-lattice 43 4 Martensitic Phase Transformation 46 Martensitic Phase Transformation; Bain or Transformation Matrix 46 Energy Density 58 Material Symmetry: Variants of Martensite 60 Frame-indifference: Energy Wells 62 Summary of the Energy Density 64 Multiple Transformations 65 5 Twinning in Martensite 66 Deformation Involving One Variant 66 Deformation Involving Two Variants 67 Interpretation as a Twin 68 Solution of the Twinning Equation and Classification of Twins 69 Cubic to Tetragonal Transformation 74 Cubic to Orthorhombic Transformation 75 Cubic to Monoclinic-I Transformation 78 Parallel Twins 80 Zig-zag Twins 81 Crossing Twins 83 6 Origin of Microstructure 86 Simplified Example in One Dimension 87 Simplified Example in Two Dimensions 89 Example in Three Dimensions: Fine Twins 93 Weakly Converging Sequences and Fine Microstructure 96 Relaxed Energy Density 98 Length-scale of the Microstructure 102 7 Special Microstructures 105 Austenite-Martensite Interface 106 Cubic to Tetragonal Transformation 114 Cubic to Orthorhombic Transformation 117 Cubic to Monoclinic-I Transformation 117 Remarks 118 Twins within Twins 121 Wedge-like Microstructure 124 Cubic to Tetragonal Transformation 127 Cubic to Orthorhombic Transformation 128 8 Analysis of Microstructure 131 Average Compatibility Conditions or the Minors Relations 131 Gradient Young Measure 141 9 The Shape-Memory Effect 143 Self-accommodation 144 Cubic Austenite 146 Tetragonal Austenite 148 Self-accommodating Plate Group 150 Recoverable Strains under Load Control 151 Uniaxial Loading 151 Biaxial Loading 153 Recoverable Strains under Displacement Control 154 10 Thin Films 159 A Theory of Thin Films 161 Deformations with One Variant or Phase 165 Deformations with Two Variants or Two Phases: Interface Conditions 166 Martensite-Martensite Interface 169 Austenite-Martensite Interface 170 Tents and Tunnels 171 11 Geometrically Linear Theory 178 Linearized Kinematics 178 Geometrically Linear Theory for Martensitic Phase Transformations 181 Some Microstructures and Comparisons 185 Twinning 185 Austenite-Martensite Interface 189 Wedge-like Microstructure 192 Zig-zag Twins 192 Recoverable Strains under Displacement Control 194 Self-accommodation 200 Recoverable Strains under Load Control 201 Accuracy of the Geometrically Linear Theory 202 12 Piecewise Linear Elasticity 204 Piecewise Quadratic Energy 204 Stress Due to Incompatible Phase Arrangement 209 Relaxed Energy Density 213 A Modified Variational Principle 219 Connection to Homogenization Theory and Optimal Design 222 13 Polycrystals 224 Examples 227 Bi-crystal 227 Triple Junction 234 Polycrystals 238 Recoverable Strains under Displacement Control 240 Recoverable Strains under Load Control 243 The Consequence of Symmetry 247 The Consequences of Texture 249 Energy Minimization and Recoverable Strain 261 Appendix 267 Bibliography 272 Index 287Library of Congress Subject Headings for this publication: Martensite, Martensite Mathematical models, Microstructure, Microstructure Mathematical models