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.