Numerical and analytical investigation of shock wave processes in elastoplastic media

The Wilkins model for an elastoplastic medium is considered. A theoretical analysis of discontinuous solutions under the assumption of one-dimensional uniaxial strain is performed. In this approximation, the material equations for the deviator stress tensor components are integrated exactly, and only the conservative part of the governing equations remains, which makes it possible to derive a class of exact analytical solutions for the model. To solve the full nonconservative system of equations (without assuming the uniaxial strain), a Godunov-type numerical method is developed, which uses an approximate Riemann solver based on integrating the system of equations along a path in the phase space. A special choice of path is proposed that reduces the two-wave HLL approximation to the solution of a linear equations. Numerical and exact analytical solutions are compared for a number of problems with various regimes of shockwave processes.