The Mechanics and Thermodynamics of Continuous Media (Theoretical and Mathematical Physics)

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. 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 This book presents the nonlinear theories of continuum thermomechanics. Through­ out 1 emphasize issues that are foundational in nature, and seek results common to materials of arbitrary symmetry. The central part of the book deals with thermoelastic bodies with heat conduction and viscosity, including the inviscid or ideal dissipation­ less bodies. A surprising variety of phenomena can be modeled within this frame­ work. Moreover, the main ideas can be transferred into more complicated theories. At present, the major challenge to the non linear thermoelasticity is posed by phase transformations with changes in symmetry. 1. W. Gibbs'immensely inftuen­ tiaI treatise On the equilibrium of heterogeneous substances has provided a highly successful theory of phase transitions in ftuids. Gibbs brought the view that the ther­ modynamics is not only the theory of heat, but also a theory of equilibrium, with the of the book is an extension of main tool the minimum principles. A large portion Gibbs'ideas to bodies of general symmetry by the methods of the calculus of varia­ tions. The interplay between the convexity properties of the stored energy functions, the resulting equations, and the physics of the phenomena is a leading theme. This textbook presents the theories of nonlinear continuum thermomechanics. It emphasizes issues that are foundational in nature and seeks results common materials of arbitrary symmetry. The central part deals with thermoelastic bodies, with heat conduction and viscosity From the reviews: "The book is excellent, and covers a very broad area (usually treated as separate topics) from a unified perspective. [...] It will be very useful for both mathematicians and physicists." EMS Newsletter