Synopsis ........................................................ 1
I Balance Equations
1 Elements of Tensor Algebra and Analysis ...................... 9
1.1 Vectors and Second-Order Tensors ........................ 9
1.2 Symmetric Tensors ...................................... 14
1.3 Skew and Orthogonal Tensors ............................ 19
1.4 Invertible Tensors ..................................... 22
1.5 Bravais Lattices ....................................... 24
1.6 Higher-Order Tensors ................................... 28
2 Geometry and Kinematics of Continuous Bodies ................ 29
2.1 Processes with Singular Surfaces ....................... 29
2.2 Motion and Deformation ................................. 33
2.3 Compatibility of Deformations at the Interface ......... 38
2.4 Rank 1 Connections ..................................... 47
2.5 Twins .................................................. 51
2.6 Appendix: Piecewise Smooth Objects ..................... 56
3 Balance Equations ........................................... 61
3.1 Extensive Quantities: Fluxes ........................... 61
3.2 Extensive Quantities: Densities and Transport
Theorems ............................................... 65
3.3 Extensive Quantities: Balance Equations ................ 67
3.4 Mass ................................................... 70
3.5 Linear and Angular Momenta ............................. 72
3.6 Energy ................................................. 74
3.7 Entropy ................................................ 76
3.8 Appendix: The Gauss-Green Theorem ...................... 79
II Foundations
4 Material Bodies ............................................. 89
4.1 State Space ............................................ 89
4.2 Local State Functions; Material Bodies ................. 91
5 The First Law of Thermodynamics ............................. 95
5.1 Work and Heat .......................................... 95
5.2 Joule's Relation ....................................... 96
5.3 Energy. The Equation of Balance of Energy .............. 98
6 The Principle of Material Frame Indifference ............... 101
6.1 Formulation ........................................... 101
6.2 The Transformation Law for Work; Mass ................. 104
6.3 Cauchy's Equations of Motion; Internal Energy ......... 107
7 The Second Law of Thermodynamics ........................... 109
7.1 Empirical Temperature. The Heating Measure ............ 109
7.2 Statements of the Second Law .......................... 115
7.3 Ideal Systems ......................................... 116
7.4 The Collection of Bodies .............................. 121
7.5 The Absolute Temperature Scale. The Clausius
Inequality ............................................ 124
7.6 The Entropy. The Clausius-Duhem Inequality ............ 127
7.7 Notes and Complements ................................. 132
III Constitutive Theory
8 Isotropic Functions ........................................ 137
8.1 Isotropic Tensor-Valued Functions ..................... 137
8.2 Isotropic Scalar-Valued Functions ..................... 142
8.3 Objective Functions ................................... 143
8.4 Objective-Isotropic Tensor-Valued Functions ........... 144
8.5 Objective-Isotropic Scalar-Valued Functions ........... 147
9 Constitutive Equations ..................................... 151
9.1 Response Functions .................................... 151
9.2 Consequences of the Clausius-Duhem Inequality ......... 153
9.3 Frame Indifference .................................... 155
9.4 The Symmetry Group .................................... 157
9.5 Supply-Free Processes ................................. 161
10 The Equilibrium Response ................................... 167
10.1 The Legendre Transformation ........................... 167
10.2 Changes of Thermal Variables .......................... 170
10.3 The Eshelby Tensor. The Spatial Description ........... 172
1Q.4 The Generalized Stress and Strain Measures ............ 173
10.5 Isothermal Elastic Constants .......................... 174
10.6 The Thermal Coefficient of Stress ..................... 178
10.7 Adiabatic Elastic Constants ........................... 179
10.8 Specific and Latent Heats; Calorimetry ................ 180
10.9 Approximate Equilibrium Response ...................... 182
11 The Equilibrium Response of Isotropic Bodies ............... 185
11.1 Response Functions for Isotropic Solids ............... 185
11.2 Isotropic States ...................................... 188
11.3 Free Energies of Isotropic Solids ..................... 192
11.4 Response Functions of Fluids .......................... 193
12 The Dynamic Response ....................................... 197
12.1 Linearization, Kinetic Coefficients ................... 197
12.2 Linear Irreversible Thermodynamics. Onsager's
Relations ............................................. 199
12.3 Dissipation Potential ................................. 201
12.4 Relaxation Models. The Extended Linear Irreversible
Thermodynamics ........................................ 202
IV Thermodynamic Equilibrium
13 The Environment ............................................ 209
13.1 States and Processes .................................. 209
13.2 Heating Environments .................................. 210
13.3 Loading Environments .................................. 213
13.4 The Total Canonical Free Energy ....................... 220
13.5 Homogeneous Null Lagrangians .......................... 221
13.6 General Null Lagrangians .............................. 224
13.7 The Form of the Potential Energy ...................... 226
14 Equilibrium States ......................................... 229
14.1 Equilibrium States and Dissipation of Energy .......... 229
14.2 Equilibrium States for Given Environments ............. 230
14.3 Integral Functionals .................................. 233
14.4 Variational Conditions for Thermodynamic Equilibrium .. 236
14.5 Spatial Description. Standard, Inner, and Outer
Variations ............................................ 238
15 Extremum Principles ........................................ 243
15.1 Liapunov Functions and Stability ...................... 243
15.2 The Extremum Principles ............................... 248
15.3 Relationships Among the Principles .................... 250
15.4 Extremum Principles and Variations .................... 251
16 Convexity .................................................. 255
16.1 Convex Sets ........................................... 255
16.2 Convex Functions ...................................... 256
16.3 The Lower Convex Hull ................................. 260
16.4 The Fenchel Transformation ............................ 262
17 Constitutive Inequalities .................................. 267
17.1 Quasiconvexity ........................................ 267
17.2 Quasiconvexity at the Boundary ........................ 272
17.3 Rank 1 Convexity and the Legendre-Hadamard Condition .. 274
17.4 Maxwell's Relation .................................... 279
17.5 Convexity and Polyconvexity ........................... 284
17.6 The Exchange of the Actual and Reference
Configurations ........................................ 288
17.7 Constitutive Inequalities for Fluids .................. 289
17.8 Quasiconvexity and Crystals ........................... 292
18 Convexity Conditions for Isotropic Functions ............... 295
18.1 Symmetric Convex Functions and Sets ................... 295
18.2 Isotropic Convex Functions and Sets ................... 298
18.3 Objective-Isotropic Convex Functions .................. 301
18.4 Invertibility of the Stress Relation .................. 304
18.5 Isotropic Polyconvex Functions ........................ 307
18.6 The Second Differential of the Stored Energy .......... 307
19 Thermostatics of Fluids .................................... 311
19.1 Preview: The Energy Function .......................... 311
19.2 Rest States and Total Quantities ...................... 313
19.3 Extremum Principles for Fluids ........................ 315
19.4 The Equivalence and Consequences of the Extremum
Principles ............................................ 316
19.5 Strict Extremum Principles. The Phase Rule ............ 321
19.6 The Gibbs Function .................................... 323
19.7 Strong Minima and Dynamical Stability of Equilibrium
States ................................................ 326
19.8 The Equilibrium of Fluids Under the Body Forces ....... 327
20 A Local Approach to the Equilibrium of Solids .............. 333
20.1 The Linearized Equations .............................. 333
20.2 Sobolev Spaces ........................................ 338
20.3 The Second Variations and Extrema ..................... 340
20.4 Positivity of the Second Variation (Necessary
Conditions) ........................................... 343
20.5 Positivity of the Second Variation (Sufficient
Conditions) ........................................... 350
20.6 The Second Variation for Stressed Isotropic States .... 351
20.7 Stability and Bifurcation for a Column ................ 360
20.8 Existence in Linearized Elasticity .................... 363
20.9 Existence Via the Implicit Function Theorem ........... 365
21 Direct Methods in Equilibrium Theory ....................... 369
21.1 Weak Convergence and Young Measures ................... 370
21.2 Deformations from Sobolev Spaces ...................... 375
21.3 Weak Convergence of Determinant and Cofactor .......... 379
21.4 States of Rubber-Like Bodies .......................... 381
21.5 Existence of Solutions to Extremum Problems for
Rubber-Like Bodies .................................... 384
21.6 Minimum Energy in Crystals and Young Measure
Minimizers ............................................ 388
V Dynamics
22 Dynamical Thermoelastic and Adiabatic Theories ............. 399
22.1 Equations of Dynamic Thermoelasticity ................. 400
22.2 Extra Conditions for Evolving Phase Boundaries ........ 402
22.3 Adiabatic and Isentropic Dynamics; Shock Waves ........ 405
22.4 Equations in the Form of a First-Order System ......... 409
23 Waves in the Referential Description ....................... 411
23.1 The Characteristic Equation ........................... 411
23.2 Characteristic Fields. Genuine Nonlinearity ........... 414
23.3 Plane, Surface, and Acceleration Waves ................ 415
23.4 The Characteristic Equation and Material Symmetry ..... 421
23.5 Centered Waves ........................................ 424
23.6 Discontinuities ....................................... 426
23.7 The Shock Set ......................................... 429
23.8 The Shock Admissibility Criteria ...................... 434
23.9 The Riemann Problem ................................... 440
24 Adiabatic Fluid Dynamics ................................... 443
24.1 The Equations of Fluid Dynamics ....................... 443
24.2 Shock Waves in Fluids ................................. 445
24.3 Hugoniot's Adiabat .................................... 447
24.4 The Equivalence of the Admissibility Criteria ......... 452
24.5 Shock Layers in Fluids ................................ 453
25 Dissipation of Energy in Solids ............................ 461
25.1 Review of Basic Equations ............................. 461
25.2 Liapunov Functions .................................... 465
25.3 Uniqueness ............................................ 467
25.4 The Existence of the Linear Time Evolution ............ 468
25.5 Asymptotic Stability .................................. 473
25.6 The Linearization About Nonequilibrium States ......... 474
25.1 References ............................................ 479
Subject Index ................................................. 501
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