The mechanics and thermodynamics of continuous media

Miroslav Silhavy3540583785, 9783540583783

This book presents the non-linear theories of continuum thermomechanics. The author emphasizes issues that are foundational in nature and seeks results common to materials of arbitrary symmetry. The central part deals with thermoelastic bodies with heat conduction and viscosity, including the inviscid or ideal dissipationless bodies. A surprising variety of phenomena can be modeled within this framework. The main ideas can be transferred into more complicated theories of materials with memory or microstructure. A large portion is an extension of Gibbs’ ideas to bodies of general symmetry by the methods of the calculus of variations. The interplay between the convexity properties of the stored energy functions, the resulting equations, and the physics is also discussed.

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Table of contents :
Front cover……Page 1
Series……Page 2
Title page……Page 3
Date-line……Page 4
Dedication……Page 5
Preface……Page 7
Contents……Page 9
Synopsis……Page 15
I Balance Equations……Page 21
1.1 Vectors and Second-Order Tensors……Page 23
1.2 Symmetric Tensors……Page 28
1.3 Skew and Orthogonal Tensors……Page 33
1.4 Invertible Tensors……Page 36
1.5 Bravais Lattices……Page 38
1.6 Higher-Order Tensors……Page 42
2.1 Processes with Singular Surfaces……Page 43
2.2 Motion and Deformation……Page 47
2.3 Compatibility of Deformations at the Interface……Page 52
2.4 Rank 1 Connections……Page 61
2.5 Twins……Page 65
2.6 Appendix: Piecewise Smooth Objects……Page 70
3.1 Extensive Quantities: Fluxes……Page 75
3.2 Extensive Quantities: Densities and Transport Theorems……Page 79
3.3 Extensive Quantities: Balance Equations……Page 81
3.4 Mass……Page 84
3.5 Linear and Angular Momenta……Page 86
3.6 Energy……Page 88
3.7 Entropy……Page 90
3.8 Appendix: The Gauss-Green Theorem……Page 93
II Foundations……Page 101
4.1 State Space……Page 103
4.2 Local State Functions; Material Bodies……Page 105
5.1 Work and Heat……Page 109
5.2 Joule’s Relation……Page 110
5.3 Energy. The Equation of Balance of Energy……Page 112
6.1 Formulation……Page 115
6.2 The Transformation Law for Work; Mass……Page 118
6.3 Cauchy’s Equations of Motion; Internal Energy……Page 121
7.1 Empirical Temperature. The Heating Measure……Page 123
7.2 Statements of the Second Law……Page 129
7.3 Ideal Systems……Page 130
7.4 The Collection of Bodies……Page 135
7.5 The Absolute Temperature Scale. The Clausius Inequality……Page 138
7.6 The Entropy. The Clausius-Duhem Inequality……Page 141
7.7 Notes and Complements……Page 146
III Constitutive Theory……Page 149
8.1 Isotropic Tensor-Valued Functions……Page 151
8.2 Isotropic Scalar-Valued Functions……Page 156
8.3 Objective Functions……Page 157
8.4 Objective-Isotropic Tensor-Valued Functions……Page 158
8.5 Objective-Isotropic Scalar-Valued Functions……Page 161
9.1 Response Functions……Page 165
9.2 Consequences of the Clausius-Duhem Inequality……Page 167
9.3 Frame Indifference……Page 169
9.4 The Symmetry Group……Page 171
9.5 Supply-Free Processes……Page 175
10.1 The Legendre Transformation……Page 181
10.2 Changes of Thermal Variables……Page 184
10.3 The Eshelby Tensor. The Spatial Description……Page 186
10.4 The Generalized Stress and Strain Measures……Page 187
10.5 Isothermal Elastic Constants……Page 188
10.6 The Thermal Coefficient of Stress……Page 192
10.7 Adiabatic Elastic Constants……Page 193
10.8 Specific and Latent Heats; Calorimetry……Page 194
10.9 Approximate Equilibrium Response……Page 196
11.1 Response Functions for Isotropic Solids……Page 199
11.2 Isotropic States……Page 202
11.3 Free Energies of Isotropic Solids……Page 206
11.4 Response Functions of Fluids……Page 207
12.1 Linearization, Kinetic Coefficients……Page 211
12.2 Linear Irreversible Thermodynamics. Onsager’s Relations……Page 213
12.3 Dissipation Potential……Page 215
12.4 Relaxation Models. The Extended Linear Irreversible Thermodynamics……Page 216
IV Thermodynamic Equilibrium……Page 221
13.1 States and Processes……Page 223
13.2 Heating Environments……Page 224
13.3 Loading Environments……Page 227
13.4 The Total Canonical Free Energy……Page 234
13.5 Homogeneous Null Lagrangians……Page 235
13.6 General Null Lagrangians……Page 238
13.7 The Form of the Potential Energy……Page 240
14.1 Equilibrium States and Dissipation of Energy……Page 243
14.2 Equilibrium States for Given Environments……Page 244
14.3 Integral Functionals……Page 247
14.4 Variational Conditions for Thermodynamic Equilibrium……Page 250
14.5 Spatial Description. Standard, Inner, and Outer Variations……Page 252
15.1 Liapunov Functions and Stability……Page 257
15.2 The Extremum Principles……Page 262
15.3 Relationships Among the Principles……Page 264
15.4 Extremum Principles and Variations……Page 265
16.1 Convex Sets……Page 269
16.2 Convex Functions……Page 270
16.3 The Lower Convex Hull……Page 274
16.4 The Fenchel Transformation……Page 276
17.1 Quasiconvexity……Page 281
17.2 Quasiconvexity at the Boundary……Page 286
17.3 Rank 1 Convexity and the Legendre-Hadamard Condition……Page 288
17.4 Maxwell’s Relation……Page 293
17.5 Convexity and Polyconvexity……Page 298
17.6 The Exchange of the Actual and Reference Configurations……Page 302
17.7 Constitutive Inequalities for Fluids……Page 303
17.8 Quasiconvexity and Crystals……Page 306
18.1 Symmetric Convex Functions and Sets……Page 309
18.2 Isotropic Convex Functions and Sets……Page 312
18.3 Objective-Isotropic Convex Functions……Page 315
18.4 Invertibility of the Stress Relation……Page 318
18.6 The Second Differential of the Stored Energy……Page 321
19.1 Preview: The Energy Function……Page 325
19.2 Rest States and Total Quantities……Page 327
19.3 Extremum Principles for Fluids……Page 329
19.4 The Equivalence and Consequences of the Extremum Principles……Page 330
19.5 Strict Extremum Principles. The Phase Rule……Page 335
19.6 The Gibbs Function……Page 337
19.7 Strong Minima and Dynamical Stability of Equilibrium States……Page 340
19.8 The Equilibrium of Fluids Under the Body Forces……Page 341
20.1 The Linearized Equations……Page 347
20.2 Sobolev Spaces……Page 352
20.3 The Second Variations and Extrema……Page 354
20.4 Positivity of the Second Variation (Necessary Conditions)……Page 357
20.5 Positivity of the Second Variation (Sufficient Conditions)……Page 364
20.6 The Second Variation for Stressed Isotropic States……Page 365
20.7 Stability and Bifurcation for a Column……Page 374
20.8 Existence in Linearized Elasticity……Page 377
20.9 Existence Via the Implicit Function Theorem……Page 379
21 Direct Methods in Equilibrium Theory……Page 383
21.1 Weak Convergence and Young Measures……Page 384
21.2 Deformations from Sobolev Spaces……Page 389
21.3 Weak Convergence of Determinant and Cofactor……Page 393
21.4 States of Rubber-Like Bodies……Page 395
21.5 Existence of Solutions to Extremum Problems for Rubber-Like Bodies……Page 398
21.6 Minimum Energy in Crystals and Young Measure Minimizers……Page 402
V Dynamics……Page 411
22 Dynamical Thermoelastic and Adiabatic Theories……Page 413
22.1 Equations of Dynamic Thermoelasticity……Page 414
22.2 Extra Conditions for Evolving Phase Boundaries……Page 416
22.3 Adiabatic and Isentropic Dynamics; Shock Waves……Page 419
22.4 Equations in the Form of a First-Order System……Page 423
23.1 The Characteristic Equation……Page 425
23.2 Characteristic Fields. Genuine Nonlinearity……Page 428
23.3 Plane, Surface, and Acceleration Waves……Page 429
23.4 The Characteristic Equation and Material Symmetry……Page 435
23.5 Centered Waves……Page 438
23.6 Discontinuities……Page 440
23.7 The Shock Set……Page 443
23.8 The Shock Admissibility Criteria……Page 448
23.9 The Riemann Problem……Page 454
24.1 The Equations of Fluid Dynamics……Page 457
24.2 Shock Waves in Fluids……Page 459
24.3 Hugoniot’s Adiabat……Page 461
24.4 The Equivalence of the Admissibility Criteria……Page 466
24.5 Shock Layers in Fluids……Page 467
25.1 Review of Basic Equations……Page 475
25.2 Liapunov Functions……Page 479
25.3 Uniqueness……Page 481
25.4 The Existence of the Linear Time Evolution……Page 482
25.5 Asymptotic Stability……Page 487
25.6 The Linearization About Nonequilibrium States……Page 488
References……Page 493
Subject Index……Page 515
Back cover……Page 519