Preface page ................................................. xiii
Part I Introduction
1 Introduction ................................................. 3
1.1 What this monograph is about ............................ 3
1.2 Some experiments ........................................ 7
1.3 Continuum mechanics ..................................... 9
1.4 Quasilinear systems .................................... 10
1.5 Outline of monograph ................................... 11
Part II Purely Mechanical Theory
2 Two-Well Potentials, Governing Equations and Energetics ..... 19
2.1 Introduction ........................................... 19
2.2 Two-phase nonlinearly elastic materials ................ 20
2.3 Field equations and jump conditions .................... 25
2.4 Energetics of motion, driving force and dissipation
inequality ............................................. 27
3 Equilibrium Phase Mixtures and Quasistatic Processes ........ 32
3.1 Introduction ........................................... 32
3.2 Equilibrium states ..................................... 33
3.3 Variational theory of equilibrium mixtures of phases ... 37
3.4 Quasistatic processes .................................. 42
3.5 Nucleation and kinetics ................................ 44
3.6 Constant elongation rate processes ..................... 47
3.7 Hysteresis ............................................. 53
4 Impact-Induced Transitions in Two-Phase Elastic Materials ... 59
4.1 Introduction ........................................... 59
4.2 The impact problem for trilinear two-phase materials ... 61
4.2.1 The constitutive law ............................ 61
4.2.2 The impact problem .............................. 64
4.3 Scale-invariant solutions of the impact problem ........ 66
4.3.1 Solutions without a phase transition ............ 66
4.3.2 Solutions with a phase transition: The two-
wave case ....................................... 67
4.3.3 Solutions with a phase transition: The one-
wave case ....................................... 68
4.3.4 The totality of solutions ....................... 69
4.4 Nucleation and kinetics ................................ 71
4.5 Comparison with experiment ............................. 74
4.6 Other types of kinetic relations ....................... 77
4.7 Related work ........................................... 77
Part III Thermomechanical Theory
5 Multiple-Well Free Energy Potentials ........................ 85
5.1 Introduction ........................................... 85
5.2 Helmholtz free energy potential ........................ 86
5.3 Potential energy function and the effect of stress ..... 88
5.4 Example 1: The van der Waals Fluid ..................... 90
5.5 Example 2: Two-phase martensitic material with cubic
and tetragonal phases .................................. 95
6 The Continuum Theory of Driving Force ...................... 105
6.1 Introduction .......................................... 105
6.2 Balance laws, field equations and jump conditions ..... 106
6.2.1 Balances of momentum and energy in integral
form ........................................... 106
6.2.2 Localization of the balance laws ............... 106
6.3 The second law of thermodynamics and the driving
force ................................................. 108
6.3.1 Entropy production rate ........................ 108
6.3.2 Driving force and the second law ............... 110
6.3.3 Driving force in the case of mechanical
equilibrium .................................... 111
7 Thermoelastic Materials .................................... 113
7.1 Introduction .......................................... 113
7.2 The thermoelastic constitutive law .................... 113
7.2.1 Relations among stress, deformation gradient,
temperature and specific entropy ............... 113
7.2.2 The heat conduction law ........................ 116
7.2.3 The partial differential equations of
nonlinear thermoelasticity ..................... 116
7.2.4 Thermomechanical equilibrium ................... 117
7.3 Stability of a thermoelastic material ................. 118
7.4 A one-dimensional special case: uniaxial strain ....... 120
8 Kinetics and Nucleation .................................... 124
8.1 Introduction .......................................... 124
8.2 Nonequilibrium processes, thermodynamic fluxes
and forces, kinetic relation .......................... 124
8.3 Phenomenological examples of kinetic relations ........ 127
8.4 Micromechanically based examples of kinetic
relations ............................................. 128
8.4.1 Viscosity-strain gradient model ................ 130
8.4.2 Thermal activation model ....................... 131
8.4.3 Propagation through a row of imperfections ..... 133
8.4.4 Kinetics from atomistic considerations ......... 134
8.4.5 Frenkel-Kontorowa model ........................ 136
8.5 Nucleation ............................................ 139
Part IV One-Dimensional Thermoelastic Theory and Problems
9 Models for Two-Phase Thermoelastic Materials in One
Dimension .................................................. 149
9.1 Preliminaries ......................................... 149
9.2 Materials of Mie-Grüneisen type ....................... 151
9.3 Two-phase Mie-Grüneisen materials ..................... 153
9.3.1 The trilinear material ......................... 153
9.3.2 Stability of phases of the trilinear
material ....................................... 156
9.3.3 Other two-phase materials of Mie-Griineisen
type ........................................... 159
10 Quasistatic Hysteresis in Two-Phase Thermoelastic
Tensile Bars ............................................... 163
10.1 Preliminaries ......................................... 163
10.2 Thermomechanical equilibrium states for a two-phase
material .............................................. 164
10.3 Quasistatic processes ................................. 166
10.4 Trilinear thermoelastic material ...................... 167
10.5 Stress cycles at constant temperature ................. 169
10.6 Temperature cycles at constant stress ................. 173
10.7 The shape-memory cycle ................................ 175
10.8 The experiments of Shaw and Kyriakides ................ 176
10.9 Slow thermomechanical processes ....................... 178
11 Dynamics of Phase Transitions in Uniaxially Strained
Thermoelastic Solids ....................................... 181
11.1 Introduction .......................................... 181
11.2 Uniaxial strain in adiabatic thermoelasticity ......... 182
11.2.1 Field equations, jump conditions and
driving force .................................. 182
11.2.2 The trilinear Mie-Grüneisen thermoelastic
material ....................................... 183
11.3 The impact problem .................................... 185
11.3.1 Formulation: Scale-invariant solutions ......... 185
11.3.2 Solutions with no phase transition ............. 186
11.3.3 Solutions with a phase transition .............. 188
Part V Higher Dimensional Problems
12 Statics: Geometric Compatibility ........................... 197
12.1 Preliminaries ......................................... 197
12.2 Examples .............................................. 200
13 Dynamics: Impact-Induced Transition in a CuAlNi Single
Crystal .................................................... 209
13.1 Introduction .......................................... 209
13.2 Preliminaries ......................................... 210
13.3 Impact without phase transformation ................... 212
13.4 Impact with phase transformation ...................... 214
13.5 Application to austenite-β1 martensite
transformation in CuAlNi .............................. 217
13.5.1 Experimental data .............................. 217
13.5.2 Phase boundary speed ........................... 218
13.5.3 Driving force .................................. 218
13.5.4 Kinetic law .................................... 219
14 Quasistatics: Kinetics of Martensitic Twinning ............. 221
14.1 Introduction .......................................... 221
14.2 The material and loading device ....................... 222
14.3 Observations .......................................... 223
14.4 The model ............................................. 225
14.5 The energy of the system .............................. 226
14.5.1 Elastic energy of the specimen ................. 226
14.5.2 Loading device energy .......................... 227
14.5.3 Summary ........................................ 228
14.6 The effect of the transition layers: Further
observations .......................................... 229
14.7 The effect of the transition layers: Further
modeling .............................................. 230
14.8 Kinetics .............................................. 231
Author Index .................................................. 235
Subject Index ................................................. 238
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