Romano / Marasco Continuum Mechanics

Advanced Topics and Research Trends
1. Auflage 2010
ISBN: 978-0-8176-4870-1
Verlag: Birkhäuser Boston
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 348 Seiten

Reihe: Modeling and Simulation in Science, Engineering and Technology

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, with an emphasis on exploring the flexibility of their applications in a wide variety of fields.

Autoren/Hrsg.

Romano, Antonio

Marasco, Addolorata

Inhaltsverzeichnis

1;Contents;6
2;Preface;10
3;Chapter 1 Nonlinear Elasticity;13
3.1;1.1 Preliminary Considerations;13
3.2;1.2 The Equilibrium Problem;14
3.3;1.3 Remarks About Equilibrium Boundary Problems;16
3.4;1.4 Variational Formulation of Equilibrium;19
3.5;1.5 Isotropic Elastic Materials;23
3.6;1.6 Homogeneous Deformations;24
3.7;1.7 Homothetic Deformation;25
3.8;1.8 Simple Extension of a Rectangular Block;28
3.9;1.9 Simple Shear of a Rectangular Block;30
3.10;1.10 Universal Static Solutions;33
3.11;1.11 Constitutive Equations in Nonlinear Elasticity;36
3.12;1.12 Treolar’s Experiments;37
3.13;1.13 Rivlin and Saunders’ Experiment;38
3.14;1.14 Nondimensional Analysis of Equilibrium;40
3.15;1.15 Signorini’s Perturbation Method for Mixed Problems;41
3.16;1.16 Signorini’s Method for Traction Problems;43
3.17;1.17 Loads with an Equilibrium Axis;46
3.18;1.18 Second-Order Hyperelasticity;48
3.19;1.19 A Simple Application of Signorini’s Method;50
3.20;1.20 Van Buren’s Theorem;52
3.21;1.21 An Extension of Signorini’s Method to Live Loads;57
3.22;1.22 Second-Order Singular Surfaces;59
3.23;1.23 Singular Waves in Nonlinear Elastic;63
3.24;1.24 PrincipalWaves in Isotropic Compressible ElasticMaterials;65
3.25;1.25 A Perturbation Method for Waves in CompressibleMedia;68
3.26;1.26 A Perturbation Method for Analyzing OrdinaryWaves in Incompressible Media;72
4;Chapter 2 Micropolar Elasticity;79
4.1;2.1 Preliminary Considerations;79
4.2;2.2 Kinematics of a Micropolar Continuum;80
4.3;2.3 Mechanical Balance Equations;85
4.4;2.4 Energy and Entropy;88
4.5;2.5 Elastic Micropolar Systems;90
4.6;2.6 The Objectivity Principle;93
4.7;2.7 Some Remarks on Boundary Value Problems;98
4.8;2.8 Asymmetric Elasticity;99
5;Chapter 3 Continuous System with aNonmaterial Interface;103
5.1;3.1 Introduction;103
5.2;3.2 Velocity of a Moving Surface;104
5.3;3.3 Velocity of a Moving Curve;106
5.4;3.4 Thomas’ Derivative and Other Formulae;107
5.5;3.5 Differentiation Formulae;108
5.6;3.6 Balance Laws;113
5.7;3.7 Entropy Inequality and Gibbs Potential;118
5.8;3.8 Other Balance Equations;121
5.9;3.9 Integral Form of Maxwell’s Equations;123
6;Chapter 4 Phase Equilibrium;124
6.1;4.1 Boundary Value Problems in Phase Equilibrium;124
6.2;4.2 Some Phenomenological Results of Changes in State;125
6.3;4.3 Equilibrium of Fluid Phases with a Planar Interface;128
6.4;4.4 Equilibrium of Fluid Phases with a Spherical Interface;130
6.5;4.5 Variational Formulation of Phase Equilibrium;133
6.6;4.6 Phase Equilibrium in Crystals;136
6.7;4.7 Wulff’s Construction;141
7;Chapter 5 Stationary and Time-Dependent Phase Changes;144
7.1;5.1 The Problem of Continuous Casting;144
7.2;5.2 On the Evolution of the Solid–Liquid Phase Change;149
7.3;5.3 On the Evolution of the Liquid–Vapor Phase Change;153
7.4;5.4 The Case of a Perfect Gas;157
8;Chapter 6 An Introduction to Mixture Theory;160
8.1;6.1 Balance Laws;161
8.2;6.2 Classical Mixtures;166
8.3;6.3 Nonclassical Mixtures;170
8.4;6.4 Balance Equations of Binary Fluid Mixtures;172
8.5;6.5 Constitutive Equations;174
8.6;6.6 Phase Equilibrium and Gibbs’ Principle;178
8.7;6.7 Evaporation of a Fluid into a Gas;179
9;Chapter 7 Electromagnetism in Matter;182
9.1;7.1 Integral Balance Laws;182
9.2;7.2 Electromagnetic Fields in Rigid Bodies at Rest;185
9.3;7.3 Constitutive Equations for Isotropic Rigid Bodies;189
9.4;7.4 Approximate Constitutive Equations for IsotropicBodies;191
9.5;7.5 Maxwell’s Equations and the Principle ofRelativity;192
9.6;7.6 Quasi-electrostatic and Quasi-magnetostaticApproximations;196
9.7;7.7 Balance Equations for Quasi-electrostatics;200
9.8;7.8 Isotropic and Anisotropic Constitutive Equations;203
9.9;7.9 Polarization Fields and the Equations of Quasielectrostatics;205
9.10;7.10 More General Constitutive Equations;208
9.11;7.11 Lagrangian Formulation of Quasi-electrostatics;209
9.12;7.12 Variational Formulation for Equilibrium in Quasielectrostatics;212
10;Chapter 8 Introduction to MagnetofluidDynamics;216
10.1;8.1 An Evolution Equation for the Magnetic Field;216
10.2;8.2 Balance Equations in Magnetofluid Dynamics;218
10.3;8.3 Equivalent Form of the Balance Equations;219
10.4;8.4 Constitutive Equations;222
10.5;8.5 Ordinary Waves in Magnetofluid Dynamics;223
10.6;8.6 Alfven’s Theorems;227
10.7;8.7 Laminar Motion Between Two Parallel Plates;228
10.8;8.8 Law of Isorotation;233
11;Chapter 9 Continua with an Interface andMicromagnetism;236
11.1;9.1 Ferromagnetism and Micromagnetism;236
11.2;9.2 A Ferromagnetic Crystal as a Continuum with anInterface;238
11.3;9.3 Variations in Surfaces of Discontinuity;239
11.4;9.4 Variational Formulation of Weiss Domains;240
11.5;9.5 Weiss Domain Structure;242
11.6;9.6 Weiss Domains in the Absence of a Magnetic Field;245
11.7;9.7 Weiss Domains in Uniaxial Crystals;247
11.8;9.8 A Variational Principle for Elastic FerromagneticCrystals;250
11.9;9.9 Weiss Domains in Elastic Uniaxial Crystals;252
11.10;9.10 A Possible Weiss Domain Distribution in ElasticUniaxial Crystals;254
11.11;9.11 A More General Variational Principle;255
11.12;9.12 Weiss Domain Branching;261
11.13;9.13 Weiss Domains in an Applied Magnetic Field;263
12;Chapter 10 Relativistic Continuous Systems;267
12.1;10.1 Lorentz Transformations;267
12.2;10.2 The Principle of Relativity;271
12.3;10.3 Minkowski Spacetime;274
12.4;10.4 Physical Meaning of Minkowski Spacetime;278
12.5;10.5 Four-Dimensional Equation of Motion;280
12.6;10.6 Integral Balance Laws;282
12.7;10.7 The Momentum–Energy Tensor;284
12.8;10.8 Fermi and Fermi–Walker Transport;287
12.9;10.9 The Space Projector;291
12.10;10.10 Intrinsic Deformation Gradient;293
12.11;10.11 Relativistic Dissipation Inequality;296
12.12;10.12 Thermoelastic Materials in Relativity;299
12.13;10.13 About the Physical Meanings of Relative Quantities;303
12.14;10.14 Maxwell’s Equation in Matter;305
12.15;10.15 Minkowski’s Description;307
12.16;10.16 Ampere’s Model;308
13;Appendix A Brief Introduction to WeakSolutions;310
13.1;A.1 Weak Derivative and Sobolev Spaces;310
13.2;A.2 A Weak Solution of a PDE;314
13.3;A.3 The Lax–Milgram Theorem;316
14;Appendix B Elements of Surface Geometry;318
14.1;B.1 Regular Surfaces;318
14.2;B.2 The Second Fundamental Form;320
14.3;B.3 Surface Gradient and the Gauss Theorem;325
15;Appendix C First-Order PDE;328
15.1;C.1 Monge’s Cone;328
15.2;C.2 Characteristic Strips;330
15.3;C.3 Cauchy’s Problem;333
16;Appendix D The Tensor Character of SomePhysical Quantities;335
17;References;338
18;Index;351