Continuum Mechanics (eBook)

Contents 6
Preface 10
Chapter 1 Nonlinear Elasticity 13
1.1 Preliminary Considerations 13
1.2 The Equilibrium Problem 14
1.3 Remarks About Equilibrium Boundary Problems 16
1.4 Variational Formulation of Equilibrium 19
1.5 Isotropic Elastic Materials 23
1.6 Homogeneous Deformations 24
1.7 Homothetic Deformation 25
1.8 Simple Extension of a Rectangular Block 28
1.9 Simple Shear of a Rectangular Block 30
1.10 Universal Static Solutions 33
1.11 Constitutive Equations in Nonlinear Elasticity 36
1.12 Treolar’s Experiments 37
1.13 Rivlin and Saunders’ Experiment 38
1.14 Nondimensional Analysis of Equilibrium 40
1.15 Signorini’s Perturbation Method for Mixed Problems 41
1.16 Signorini’s Method for Traction Problems 43
1.17 Loads with an Equilibrium Axis 46
1.18 Second-Order Hyperelasticity 48
1.19 A Simple Application of Signorini’s Method 50
1.20 Van Buren’s Theorem 52
1.21 An Extension of Signorini’s Method to Live Loads 57
1.22 Second-Order Singular Surfaces 59
1.23 Singular Waves in Nonlinear Elastic 63
1.24 PrincipalWaves in Isotropic Compressible ElasticMaterials 65
1.25 A Perturbation Method for Waves in CompressibleMedia 68
1.26 A Perturbation Method for Analyzing OrdinaryWaves in Incompressible Media 72
Chapter 2 Micropolar Elasticity 79
2.1 Preliminary Considerations 79
2.2 Kinematics of a Micropolar Continuum 80
2.3 Mechanical Balance Equations 85
2.4 Energy and Entropy 88
2.5 Elastic Micropolar Systems 90
2.6 The Objectivity Principle 93
2.7 Some Remarks on Boundary Value Problems 98
2.8 Asymmetric Elasticity 99
Chapter 3 Continuous System with aNonmaterial Interface 103
3.1 Introduction 103
3.2 Velocity of a Moving Surface 104
3.3 Velocity of a Moving Curve 106
3.4 Thomas’ Derivative and Other Formulae 107
3.5 Differentiation Formulae 108
3.6 Balance Laws 113
3.7 Entropy Inequality and Gibbs Potential 118
3.8 Other Balance Equations 121
3.9 Integral Form of Maxwell’s Equations 123
Chapter 4 Phase Equilibrium 124
4.1 Boundary Value Problems in Phase Equilibrium 124
4.2 Some Phenomenological Results of Changes in State 125
4.3 Equilibrium of Fluid Phases with a Planar Interface 128
4.4 Equilibrium of Fluid Phases with a Spherical Interface 130
4.5 Variational Formulation of Phase Equilibrium 133
4.6 Phase Equilibrium in Crystals 136
4.7 Wulff’s Construction 141
Chapter 5 Stationary and Time-Dependent Phase Changes 144
5.1 The Problem of Continuous Casting 144
5.2 On the Evolution of the Solid–Liquid Phase Change 149
5.3 On the Evolution of the Liquid–Vapor Phase Change 153
5.4 The Case of a Perfect Gas 157
Chapter 6 An Introduction to Mixture Theory 160
6.1 Balance Laws 161
6.2 Classical Mixtures 166
6.3 Nonclassical Mixtures 170
6.4 Balance Equations of Binary Fluid Mixtures 172
6.5 Constitutive Equations 174
6.6 Phase Equilibrium and Gibbs’ Principle 178
6.7 Evaporation of a Fluid into a Gas 179
Chapter 7 Electromagnetism in Matter 182
7.1 Integral Balance Laws 182
7.2 Electromagnetic Fields in Rigid Bodies at Rest 185
7.3 Constitutive Equations for Isotropic Rigid Bodies 189
7.4 Approximate Constitutive Equations for IsotropicBodies 191
7.5 Maxwell’s Equations and the Principle ofRelativity 192
7.6 Quasi-electrostatic and Quasi-magnetostaticApproximations 196
7.7 Balance Equations for Quasi-electrostatics 200
7.8 Isotropic and Anisotropic Constitutive Equations 203
7.9 Polarization Fields and the Equations of Quasielectrostatics 205
7.10 More General Constitutive Equations 208
7.11 Lagrangian Formulation of Quasi-electrostatics 209
7.12 Variational Formulation for Equilibrium in Quasielectrostatics 212
Chapter 8 Introduction to MagnetofluidDynamics 216
8.1 An Evolution Equation for the Magnetic Field 216
8.2 Balance Equations in Magnetofluid Dynamics 218
8.3 Equivalent Form of the Balance Equations 219
8.4 Constitutive Equations 222
8.5 Ordinary Waves in Magnetofluid Dynamics 223
8.6 Alfven’s Theorems 227
8.7 Laminar Motion Between Two Parallel Plates 228
8.8 Law of Isorotation 233
Chapter 9 Continua with an Interface andMicromagnetism 236
9.1 Ferromagnetism and Micromagnetism 236
9.2 A Ferromagnetic Crystal as a Continuum with anInterface 238
9.3 Variations in Surfaces of Discontinuity 239
9.4 Variational Formulation of Weiss Domains 240
9.5 Weiss Domain Structure 242
9.6 Weiss Domains in the Absence of a Magnetic Field 245
9.7 Weiss Domains in Uniaxial Crystals 247
9.8 A Variational Principle for Elastic FerromagneticCrystals 250
9.9 Weiss Domains in Elastic Uniaxial Crystals 252
9.10 A Possible Weiss Domain Distribution in ElasticUniaxial Crystals 254
9.11 A More General Variational Principle 255
9.12 Weiss Domain Branching 261
9.13 Weiss Domains in an Applied Magnetic Field 263
Chapter 10 Relativistic Continuous Systems 267
10.1 Lorentz Transformations 267
10.2 The Principle of Relativity 271
10.3 Minkowski Spacetime 274
10.4 Physical Meaning of Minkowski Spacetime 278
10.5 Four-Dimensional Equation of Motion 280
10.6 Integral Balance Laws 282
10.7 The Momentum–Energy Tensor 284
10.8 Fermi and Fermi–Walker Transport 287
10.9 The Space Projector 291
10.10 Intrinsic Deformation Gradient 293
10.11 Relativistic Dissipation Inequality 296
10.12 Thermoelastic Materials in Relativity 299
10.13 About the Physical Meanings of Relative Quantities 303
10.14 Maxwell’s Equation in Matter 305
10.15 Minkowski’s Description 307
10.16 Ampere’s Model 308
Appendix A Brief Introduction to WeakSolutions 310
A.1 Weak Derivative and Sobolev Spaces 310
A.2 A Weak Solution of a PDE 314
A.3 The Lax–Milgram Theorem 316
Appendix B Elements of Surface Geometry 318
B.1 Regular Surfaces 318
B.2 The Second Fundamental Form 320
B.3 Surface Gradient and the Gauss Theorem 325
Appendix C First-Order PDE 328
C.1 Monge’s Cone 328
C.2 Characteristic Strips 330
C.3 Cauchy’s Problem 333
Appendix D The Tensor Character of SomePhysical Quantities 335
References 338
Index 351