On one version of the Godunov method for calculating elastoplastic deformations of a medium

Abstract

Godunov's hybrid method suitable for numerical calculation of elastoplastic deformation of a solid body within the framework of the classical Prandtl-Reis model with the non-barotropic state equation is described. Mises’ fluidity condition is used as a criterion for the transition from elastic to plastic state. A characteristic analysis of the model equations was carried out and their hyperbolicity was shown. It is noted that, if one takes the Maxwell-Cattaneo law instead of the Fourier law, then the Godunov hybrid method can be applied to calculate the deformation of a thermally conductive elastoplastic medium, since in this case the medium model is of a hyperbolic type. The algorithm for solving the systems in which there are equations that do not lead to divergence form is described in detail; Godunov's original method serves to integrate systems of equations represented in divergence form. When calculating stream variables on the faces of adjacent cells, a linearized Riemannian solver is used, the algorithm of which includes the right eigenvectors of the model equations. In the proposed approach, the equations written in divergence form look like finite-volume formulas, and others that do not lead to divergence form look like finite-difference relations. To illustrate the capabilities of the Godunov hybrid method, several one- and two-dimensional problems were solved, in particular, the problem of hitting an aluminum sample against a rigid barrier. It is shown that, depending on the rate of interaction, either single-wave or two-wave reflections described in the literature can be implemented with an elastic precursor.