Elements of Continuum Mechanics and Conservation Laws

This work 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 book 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. It 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.