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More About This Textbook
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.
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From the reviews:"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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Table of Contents
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.