Silhavy M. The mechanics and thermodynamics of continuous media (Berlin; Heidelberg, 1997). - ОГЛАВЛЕНИЕ / CONTENTS

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ОбложкаŠilhavý M. The mechanics and thermodynamics of continuous media. - Berlin; Heidelberg: Springer, 1997. - XIV,504 p. - (Texts and monographs in physics). - Bibliogr.: p.479-500. – Sub. ind.: p.501-504. - ISBN 978-3-642-08204-7; ISSN 0172-5998
 

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Оглавление / Contents
 
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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