Continuum models for phase transitions and twinning in crystals / Mario Pitteri, G. Zanzotto.

Format:

Book

Author/Creator:

Pitteri, Mario.

Contributor:

Zanzotto, Giovanni.

Series:

Applied mathematics ; 19.

Applied mathematics ; 19

Language:

English

Subjects (All):

Twinning (Crystallography).

Continuum mechanics.

Phase transformations (Statistical physics).

Physical Description:

385 pages : illustrations ; 25 cm.

Place of Publication:

Boca Raton, Fla. : Chapman & Hall/CRC, [2003]

Contents:

1.2 Some experimental observations 28

2.2 Some notions of elementary group theory 42

2.2.2 Conjugacy 42

2.2.3 Group actions and symmetry 43

2.3 Linear and orthogonal transformations 45

2.3.1 Tensors with period two 48

2.3.2 Simple shears 48

2.3.3 Finite groups of tensors or matrices 50

2.4 Affine transformations 52

2.5 Continuum mechanics 53

2.5.1 Deformation 54

2.5.2 Thermodynamic potentials and their invariance 56

2.5.3 Stability of equilibrium 58

3 Simple lattices 61

3.1 Definitions and global symmetry 62

3.2 Geometric symmetry and crystal systems 66

3.2.1 Crystallographic point groups and holohedries 66

3.2.2 Crystal classes and crystal systems 68

3.2.3 Laue groups 70

3.3 Arithmetic symmetry and Bravais lattice types 72

3.3.1 Lattice groups 72

3.3.2 Conjugacy in O (crystal systems) and in GL(3, Z) (Bravais lattice types) 73

3.3.3 Centerings 76

3.4 The fourteen Bravais lattices 78

3.5 Fixed sets of lattice groups 87

3.6 Symmetry-preserving stretches for simple lattices 91

3.6.1 Commutation relations 91

3.6.2 Structure of the fixed sets 94

3.6.3 The Bain stretch in the centered cubic lattices 96

3.7 Lattice subspaces, packings and indices 96

3.7.1 Lattice rows and lattice planes 96

3.7.2 Close-packed structures 97

3.7.3 Miller indices and crystallographic equivalence 98

3.7.4 Miller-Bravais indices for hexagonal lattices 101

3.8 Lattice groups and fixed sets for planar lattices 102

4 Weak-transformation neighborhoods and variants 107

4.1 Reconciliatio of global and local symmetries 108

4.2 Symmetry-breaking stretches for simple lattices 111

4.3 Small deformations and weak phase transformations 113

4.3.1 Small symmetry-preserving stretches 114

4.3.2 Small symmetry-breaking stretches 115

4.4 Constructing the small symmetry-breaking stretches 118

4.5 Variant structures (local orbits) in the wt-nbhds 120

4.5.3 Variants and cosets 123

4.5.4 Variant structures and conjugacy classes 124

5 Explicit variant structures 127

5.1 Variant structures in cubic wt-nbhds 128

5.1.1 Tetragonal conjugacy class and variant structure 130

5.1.2 Rhombohedral conjugacy class and variant structure 131

5.1.3 Orthorhombic conjugacy classes and variant structures 132

5.1.3.1 Orthorhombic 'cubic edges' variants 133

5.1.3.2 Orthorhombic 'mixed axes' variants 134

5.1.4 Monoclinic conjugacy classes 135

5.1.4.1 Monoclinic 'cubic edges' variants 135

5.1.4.2 Monoclinic 'face-diagonals' variants 135

5.1.5 Triclinic conjugacy class and variant structure 138

5.2 Variant structures in hexagonal wt-nbhds 138

5.2.1 Orthorhombic conjugacy class and variant structure 141

5.2.2 Monoclinic conjugacy classes and variant structures 142

5.2.2.1 Monoclinic 'basal diagonals' variants 142

5.2.2.2 Monoclinic 'basal side-axes' variants 142

5.2.2.3 Monoclinic 'optic axis' variants 146

5.2.3 Triclinic conjugacy class and variant structure 146

5.3 Kinematics of weak phase transformations 146

5.4 Irreducible invariant subspaces for the holohedries 149

5.4.1 General properties 150

5.4.2 Reduced actions and reduced symmetry groups on the i.i. subspaces 153

5.4.3 Decompositions of Sym under the action of the holohedries 154

5.4.3.1 Triclinic decompositions 155

5.4.3.2 Monoclinic decompositions 155

5.4.3.3 Orthorhombic decompositions 156

5.4.3.4 Rhombohedral decompositions 157

5.4.3.5 Tetragonal decompositions 160

5.4.3.6 Hexagonal decompositions 162

5.4.3.7 Cubic decompositions 163

6 Energetics 165

6.1 Invariance of simple-lattice energies 167

6.2 The Cauchy-Born hypothesis 169

6.2.1 The Born rule 170

6.2.2 Failures of the Born rule 171

6.3 Thermoelastic constitutive equations for crystals 172

6.3.1 Invariance of the response functions of elastic crystals 173

6.4 Energy minimizers and their general properties 175

6.4.1 Multiplicity of the symmetry-related minimizers 175

6.4.2 Multiphase crystals: minimizers that are not symmetry-related 176

6.4.3 Lack of convexity and symmetry-induced instabilities 178

6.5 Constitutive functions for weak phase transitions 179

6.5.1 Weak and symmetry-breaking transformations 180

6.5.2 Domain restrictions for the constitutive functions 181

6.5.3 Energy wells in the wt-nbhds 182

6.6 In the vicinity of an energy well 186

6.6.1 Thermal expansion and compressibility of a crystal 187

6.6.2 The elasticity tensor 189

6.6.3 Temperature-dependence of the elastic moduli 193

6.7 Anisotropic elasticity 194

7 Bifurcation patterns 199

7.1.1 The Landau theory 200

7.2 Isolated critical points and bifurcation points 204

7.2.1 Neighborhoods of bifurcation points 206

7.2.2 Genericity 207

7.3 Reduced bifurcation problems; order parameters 208

7.4 Analysis of the reduced bifurcation problems 211

7.4.7 Comparison with the kinematic transitions of [section]5.3 223

7.5 Behavior of the moduli along the transitions 225

7.6 Examples of energy functions for simple lattices 226

7.6.1 A schematic 1-dimensional example 227

7.6.2 Energies for cubic-to-tetragonal and for tetragonal-to-monoclinic transitions 228

7.6.3 Orientation relationships and lattice correspondence 231

7.7 Relation with the Landau theory 234

8 Mechanical twinning 239

8.1 Coherence and rank-1 connections 240

8.2 The twinning equation 244

8.3 Solutions of the twinning equation 248

8.3.1 Different descriptions of the same twin and cosets 249

8.3.2 Crystallographically equivalent twins 250

8.3.3 Reciprocal twins 251

8.3.4 Generic twins 252

8.3.5 Type-1 and Type-2 (conventional) twins 252

8.3.6 Compound twins 254

8.3.7 Conventional twins and rationality conditions 256

8.4 Short remarks 257

8.4.1 Experimental data 257

8.4.2 Mechanical twinning and the Born rule 257

8.4.3 Growth twins 259

9 Transformation twins 263

9.1 General properties 263

9.1.1 Procedure to determine the transformation twins 268

9.2 Rk-1 connections in a cubic wt-nbhd 270

9.2.1 Tetragonal variant structure 270

9.2.2 Rhombohedral variant structure 271

9.2.3 Orthorhombic variant structures 271

9.2.3.1 Orthorhombic 'cubic edges' wells 271

9.2.3.2 Orthorhombic 'mixed axes' wells 272

9.2.4 Monoclinic variant structures 273

9.2.4.1 Monoclinic 'cubic edges' wells 273

9.2.4.2 Monoclinic 'face-diagonals' wells 278

9.2.5 Triclinic variant structure 280

9.3 Rk-1 connections in a hexagonal wt-nbhd 281

9.3.1 Orthorhombic variant structure 281

9.3.2 Monoclinic variant structures 281

9.3.2.1 Monoclinic 'basal diagonals' wells 281

9.3.2.2 Monoclinic 'basal side-axes' wells 282

9.3.2.3 Monoclinic 'optic axis' wells 283

9.3.3 Triclinic variant structure 284

9.4 The Mallard law 285

10 Microstructures 287

10.1 Piecewise homogeneous equilibria 287

10.2 Generalized solutions 289

10.2.1 The minors relations 291

10.2.2 The N-well problem 292

10.3 Examples of microstructures that are not laminates 295

10.4 Habit planes in martensite 297

10.4.1 Geometrically nonlinear theory 297

10.4.2 Self-accommodation in shape memory alloys 299

10.4.3 Wedges and other microstructures 300

11 Kinematics of multilattices 301

11.1 Crystals as multilattices 302

11.1.1 Descriptors and configuration spaces for deformable multilattices 304

11.1.2 Essential descriptions of multilattices 305

11.2 The global symmetry of multilattices 306

11.2.1 Indeterminateness of the descriptors (P[subscript 0,...], P[subscript n-1], e[subscript a]) 306

11.2.2 Indeterminateness of the descriptors (P[subscript 0], [varepsilon subscript [sigma]) 309

11.2.3 Nonessential descriptors of multilattices 311

11.3 The affine symmetry of multilattices 314

11.3.1 Space groups; crystal class and crystal system of a multilattice 315

11.4 The arithmetic symmetry

of multilattices 318

11.4.1 Lattice groups of multilattices 318

11.4.2 Fixed sets of lattice groups 320

11.4.3 Relation between the arithmetic and the space-group symmetries 321

11.5.1 Three-dimensional 2-lattices and hexagonal close-packed structures 324

11.5.2 The structure of quartz as a 3-lattice 327

11.6 Weak-transformation neighborhoods 333

11.7 The energy of a multilattice and its invariance 335

11.7.1 Minimizing out the internal variables of complex crystals 336

11.7.2 Local invariance of multilattice energies; the example of quartz 339

11.8 Twinning in multilattices 341

11.8.1 A proposal for a class of twins 343

11.8.3 A model for stress relaxation 348.

Notes:

Includes bibliographical references (pages 351-370) and indexes.

ISBN:

0849303273

OCLC:

49901846