Continuum Mechanics

This book offers a broad overview of the potential of continuum mechanics to describe a wide range of macroscopic phenomena in real-world problems. Building on the fundamentals presented in the authors’ previous book Continuum Mechanics using Mathematica, this new work explores interesting models of continuum mechanics having applications in a wide variety of fields.Specific topics covered include:* various aspects of nonlinear elasticity, including equilibrium equations and their variational formulation, nonlinear constitutive equations, existence and uniqueness theorems of Van Buren and Stoppelli, and Signorini's method with some extensions to live loads and acceleration waves* continua with directors* a model of a continuum with a nonmaterial moving interface* mixture theory (The Gibbs Rule in a binary mixture)* interaction between electric or magnetic fields with matter* micromagnetism* continua in special relativity and relativistic interactions between matter and electromagnetic fieldsMathematica® notebooks accompanying the text are available for download at http://www/birkhauser.com/978-0-8176-4869-5. Aimed at advanced graduate students, applied mathematicians, mathematical physicists, and engineers, the work will be an excellent self-study reference or supplementary textbook in graduate-level courses focusing on advanced topics and research trends in continuum mechanics.

Preface	p. ix
Nonlinear Elasticity	p. 1
Preliminary Considerations	p. 1
The Equilibrium Problem	p. 2
Remarks About Equilibrium Boundary Problems	p. 4
Variational Formulation of Equilibrium	p. 7
Isotropic Elastic Materials	p. 11
Homogeneous Deformations	p. 12
Homothetic Deformation	p. 13
Simple Extension of a Rectangular Block	p. 16
Simple Shear of a Rectangular Block	p. 18
Universal Static Solutions	p. 21
Constitutive Equations in Nonlinear Elasticity	p. 24
Treolar's Experiments	p. 25
Rivlin and Saunders' Experiment	p. 26
Nondimensional Analysis of Equilibrium	p. 28
Signorini's Perturbation Method for Mixed Problems	p. 29
Signorini's Method for Traction Problems	p. 31
Loads with an Equilibrium Axis	p. 34
Second-Order Hyperelasticity	p. 36
A Simple Application of Signorini's Method	p. 38
Van Buren's Theorem	p. 40
An Extension of Signorini's Method to Live Loads	p. 45
Second-Order Singular Surfaces	p. 47
Singular Waves in Nonlinear Elasticity	p. 51
Principal Waves in Isotropic Compressible Elastic Materials	p. 53
A Perturbation Method for Waves in Compressible Media	p. 56
A Perturbation Method for Analyzing Ordinary Waves in Incompressible Media	p. 60
Micropolar Elasticity	p. 67
Preliminary Considerations	p. 67
Kinematics of a Micropolar Continuum	p. 68
Mechanical Balance Equations	p. 73
Energy and Entropy	p. 76
Elastic Micropolar Systems	p. 78
The Objectivity Principle	p. 81
Some Remarks on Boundary Value Problems	p. 86
Asymmetric Elasticity	p. 87
Continuous System with a Nonmaterial Interface	p. 91
Introduction	p. 91
Velocity of a Moving Surface	p. 92
Velocity of a Moving Curve	p. 94
Thomas' Derivative and Other Formulae	p. 95
Differentiation Formulae	p. 96
Balance Laws	p. 101
Entropy Inequality and Gibbs Potential	p. 106
Other Balance Equations	p. 109
Integral Form of Maxwell's Equations	p. 111
Phase Equilibrium	p. 113
Boundary Value Problems in Phase Equilibrium	p. 113
Some Phenomenological Results of Changes in State	p. 114
Equilibrium of Fluid Phases with a Planar Interface	p. 117
Equilibrium of Fluid Phases with a Spherical Interface	p. 119
Variational Formulation of Phase Equilibrium	p. 122
Phase Equilibrium in Crystals	p. 125
Wulff's Construction	p. 130
Stationary and Time-Dependent Phase Changes	p. 133
The Problem of Continuous Casting	p. 133
On the Evolution of the Solid-Liquid Phase Change	p. 138
On the Evolution of the Liquid-Vapor Phase Change	p. 142
The Case of a Perfect Gas	p. 146
An Introduction to Mixture Theory	p. 149
Balance Laws	p. 150
Classical Mixtures	p. 155
Nonclassical Mixtures	p. 159
Balance Equations of Binary Fluid Mixtures	p. 161
Constitutive Equations	p. 163
Phase Equilibrium and Gibbs' Principle	p. 167
Evaporation of a Fluid into a Gas	p. 168
Electromagnetism in Matter	p. 171
Integral Balance Laws	p. 171
Electromagnetic Fields in Rigid Bodies at Rest	p. 174
Constitutive Equations for Isotropic Rigid Bodies	p. 178
Approximate Constitutive Equations for Isotropic Bodies	p. 180
Maxwell's Equations and the Principle of Relativity	p. 181
Quasi-electrostatic and Quasi-magnetostatic Approximations	p. 185
Balance Equations for Quasi-electrostatics	p. 189
Isotropic and Anisotropic Constitutive Equations	p. 192
Polarization Fields and the Equations of Quasi-electrostatics	p. 194
More General Constitutive Equations	p. 197
Lagrangian Formulation of Quasi-electrostatics	p. 198
Variational Formulation for Equilibrium in Quasi-electrostatics	p. 201
Introduction to Magnetofluid Dynamics	p. 205
An Evolution Equation for the Magnetic Field	p. 205
Balance Equations in Magnetofluid Dynamics	p. 207
Equivalent Form of the Balance Equations	p. 208
Constitutive Equations	p. 211
Ordinary Waves in Magnetofluid Dynamics	p. 212
Alfven's Theorems	p. 216
Laminar Motion Between Two Parallel Plates	p. 217
Law of Isorotation	p. 222
Continua with an Interface and Micromagnetism	p. 225
Ferromagnetism and Micromagnetism	p. 225
A Ferromagnetic Crystal as a Continuum with an Interface	p. 227
Variations in Surfaces of Discontinuity	p. 228
Variational Formulation of Weiss Domains	p. 229
Weiss Domain Structure	p. 231
Weiss Domains in the Absence of a Magnetic Field	p. 234
Weiss Domains in Uniaxial Crystals	p. 236
A Variational Principle for Elastic Ferromagnetic Crystals	p. 239
Weiss Domains in Elastic Uniaxial Crystals	p. 241
A Possible Weiss Domain Distribution in Elastic Uniaxial Crystals	p. 243
A More General Variational Principle	p. 244
Weiss Domain Branching	p. 250
Weiss Domains in an Applied Magnetic Field	p. 252
Relativistic Continuous Systems	p. 257
Lorentz Transformations	p. 257
The Principle of Relativity	p. 261
Minkowski Spacetime	p. 264
Physical Meaning of Minkowski Spacetime	p. 268
Four-Dimensional Equation of Motion	p. 270
Integral Balance Laws	p. 272
The Momentum-Energy Tensor	p. 274
Fermi and Fermi-Walker Transport	p. 277
The Space Projector	p. 281
Intrinsic Deformation Gradient	p. 283
Relativistic Dissipation Inequality	p. 286
Thermoelastic Materials in Relativity	p. 289
About the Physical Meanings of Relative Quantities	p. 293
Maxwell's Equation in Matter	p. 295
Minkowski's Description	p. 297
Ampere's Model	p. 298
Brief Introduction to Weak Solutions	p. 301
Weak Derivative and Sobolev Spaces	p. 301
A Weak Solution of a PDE	p. 305
The Lax-Milgram Theorem	p. 307
Elements of Surface Geometry	p. 309
Regular Surfaces	p. 309
The Second Fundamental Form	p. 311
Surface Gradient and the Gauss Theorem	p. 316
First-Order PDE	p. 319
Monge's Cone	p. 319
Characteristic Strips	p. 321
Cauchy's Problem	p. 324
The Tensor Character of Some Physical Quantities	p. 327
References	p. 331
Index	p. 345
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