Elements of Continuum Mechanics and Conservation Laws / Edition 1

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Overview

Elements of Continuum Mechanics and Conservation Laws presents a systematization of different models in mathematical physics, a study of the structure of conservation laws, thermodynamical identities, and connection with criteria for well-posedness of the corresponding mathematical problems. The theory presented in this book stems from research carried out by the authors concerning the formulations of differential equations describing explosive deformations of metals. In such processes, elasticity equations are used in some zones, whereas hydrodynamics equations are stated in other zones. Plastic deformations appear in transition zones, which leads to residual stresses. The suggested model contains some relaxation terms which simulate these plastic deformations. Certain laws of thermodynamics are used in order to describe and study differential equations simulating the physical processes. This leads to the special formulation of differential equations using generalized thermodynamical potentials. The structure of conservation laws and new ideas and methods of constructing mathematical models are presented. The final chapter: Structure of Thermodynamically Compatible Systems reflects Godunov's latest research. It presents an approach to the formalization of equations of continuum mechanics, in particular, relationships between the structure of thermodynamical conservation laws and representations of the rotation group. The material covered in this chapter was written during the preparation of the English edition and intensively discussed with specialists in different countries and presented at lectures given by Godunov in 2002. This book describes the theory developed by Godunov together with his former student Evgenii Romenskii which presents a systematization of different models of elastic media and related classification of hyperbolic equations.

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Editorial Reviews

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"The authors present a systematic investigation of a variety of models in mathematical physics. … The theory presented here stems from some very beautiful results obtained by the authors concerning the formulation of differential equations describing explosive deformations of metals." (Konstantina Trivisa, Mathematical Reviews, 2005 e)

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Product Details

  • ISBN-13: 9780306477355
  • Publisher: Springer US
  • Publication date: 5/31/2003
  • Edition description: 2003
  • Edition number: 1
  • Pages: 266
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.69 (d)

Table of Contents

Chapter I. Elementary Properties of Deformations and Stresses 1
1. Deformations and Strain Rate 1
2. General Properties of Stresses 9
3. Stressed State at a Point 22
4. Isotropic Elastic Medium, Equation of State, and the Murnaghan Formulas 29
5. Viscous Stresses in an Isotropic Medium 42
6. Strain Tensors and Distortion Tensor 52
7. Murnaghan Formulas in the Tensor Form and Small Deformations 57
8. Moving Medium. Murnaghan Formulas 63
Chapter II. Effective Elastic Deformation 71
9. Relaxation of Stresses 71
10. Equations for the Metric Effective Elastic Deformation Tensor 82
11. Compatibility Conditions 88
12. Description of Relaxation of Shear Stresses by Equations for Effective Distortion 103
13. Defects of Crystal Lattice. Review of Notions (Dislocations) 112
Chapter III. Differential Equations of Dynamical Processes 121
14. Conservation Laws 121
15. Consequences of Conservation Laws 126
16. Deformation of a Layer and a Bar 130
17. Equations for One-Dimensional Nonstationary Processes and Structure of Stationary Waves in a Maxwell Medium 139
Chapter IV. Well-Posedness of Differential Equations and Thermodynamics 153
18. Well-Posedness Conditions for Equations of Elasticity 153
19. Equations of Linear Elasticity as Symmetric Hyperbolic Equations 161
20. Symmetrization of Equations of Gas Dynamics 167
21. Influence of Small Viscosity on the Behavior of Solutions 176
22. Well-Posedness, Stability, and Postulates of Phenomenological Thermodynamics 187
Chapter V. Multi-Dimensional Thermodynamically Compatible Conservation Laws 203
23. Gas Dynamics Equations and Magnetic Hydrodynamics Equations 203
24. Equations of Elasticity 206
25. Structure of Multi-Dimensional Equations 211
26. Symmetric Hyperbolicity in the Three-Dimensional Case 215
Appendix Structure of Thermodynamically Compatible Systems 217
1. Mathematical Aspects 217
2. The Simplest Galilei-Invariant Thermodynamically Compatible Systems 219
3. Methods of Constructing Equations 227
4. Some Facts of the Theory of Representations of Orthogonal Transformations of Three-Dimensional Space 238
5. The Clebsch-Gordan Coefficients 245
6. Orthogonal Invariants 251
Literature 255
Subject Index 257
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