International Conference «Mathematical and Informational Technologies, MIT-2013»
(X Conference «Computational and Informational Technologies for Science,
Engineering and Education»)

Vrnjacka Banja, Serbia, September, 5–8, 2013

Budva, Montenegro, September, 9-14, 2013

Sadovskii V.M.   Sadovskaya O.V.   Varygina M.P.  

Thermodynamically consistent system of equations of the dynamics of an elastic medium with moment properties

Reporter: Varygina M.P.

One of the well known models of the theory of elasticity under small strains is the model of the Cosserat continuum, which is intended to describe the mechanical behavior of deformable materials with microstructure (soils, rocks, granular, porous, microfractured and micropolar media). In contrast to the classical theory of elasticity based on the concept of a medium as a continuum of material points, in this theory a medium is a continuum of material particles – solids of small volume, possessing the rotational degrees of freedom. Our goal is to reduce the equations, modeling the finite thermoelastic deformation of a spatial medium with independent rotations of particles, to the thermodynamically consistent by Godunov system of conservation laws. This procedure is an essential step in the study of mathematical model, since the formulation in the form of a thermodynamically consistent system allows writing it as a symmetric hyperbolic system, providing a simple proof of the uniqueness and continuous dependence of the solution of the Cauchy problem and boundary value problems with dissipative boundary conditions on the initial data. On the basis of this system the concept of a generalized solution can be formulated, which allows one to analyze the case of strong discontinuities of velocities and stresses (contact discontinuities, shock waves). It also makes possible the use of well-developed efficient numerical methods to analysis of the model. The general equations of gas dynamics and magneto-hydrodynamics, the equations of linear and nonlinear elasticity and some other special models of mechanics are reduced to the form of thermodynamically consistent system of conservation laws. The question of reducing the nonlinear system of equations of the Cosserat elasticity theory to this form is not entirely solved, because in addition to the formal introduction of generating potentials it is also required that the generating potential under the time derivative must be a strongly convex function with respect to the parameters of strain state.

This work was supported by the Russian Foundation for Basic Research (grant no. 11–01–00053) and the Complex Fundamental Research Program no. 18 “Algorithms and Software for Computational Systems of Superhigh Productivity” of the Presidium of RAS.

Abstracts file: Садовский, Садовская, Варыгина.doc

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