Elements of Continuum Mechanics and Conservation Laws

Abstract

I: Elementary Properties of Deformations and Stresses. Deformations and Strain. General Properties of Stresses. Stressed State at a Point. Isotropic Elastic Medium, Equation of State, and the Murnaghan Formulas. Viscous Stresses in an Isotropic Medium. Strain Tensors and Distortion Tensor. Murnaghan Formulas in the Tensor Form and Small Deformations. Moving Medium. Murnaghan Formulas. II: Effective Elastic Deformation. Relaxation of Stresses. Equations for the Metric Effective Elastic Deformation Tensor. Compatibility Conditions. Description of Relaxation of Shear Stresses by Equations for Effective Distortion. Defects of Crystal Lattice. Review of Notions. Dislocations. III: Differential Equations of Dynamical Processes. Conservation Laws. Consequences of Conservation Laws. Deformation of a Layer and a Bar. Equations for One-Dimensional Nonstationary Processes and Structure of Stationary Waves in a Maxwell Medium. IV: Well Posedness of Differential Equations and Thermodynamics. Well Posedness Conditions for Equations of Elasticity. Equations of Linear Elasticity as Symmetric Hyperbolic Equations. Symmetrization of Equations of Gas Dynamics. Influence of Small Viscosity on the Behavior of Solutions. Well Posedness Stability and Postulates of Phenomenological Thermodynamics. V: Multi-Dimensional Thermodynamically Compatible Conservation Laws. Gas Dynamics Equations and Magnetic Hydrodynamics Equations. Equations of Elasticity. Structure of Multi Dimensional Equations. Symmetric Hyperbolicity in the Three-Dimensional Case. Appendix: Structure of Thermodynamically Compatible Systems S.K. Godunov. Mathematical Aspects. The Simplest Galilei-Invariant Thermodynamically Compatible Systems. Methods of Constructing Equations. Some Facts of the Theory of Representations of Orthogonal Transformations. The Clebsch-Gordan Coefficients. Orthogonal Invariants. Literature. Subject Index.