Thermodynamic consistency and mathematical well-posedness in the theory of elastoplastic, granular, and porous materials

Mathematical models of the dynamics of elastoplastic, granular, and porous media are reduced to variational inequalities for hyperbolic differential operators that are thermodynamically consistent in the sense of Godunov. On this basis, the concept of weak solutions with dissipative shock waves is introduced and a priori estimates of smooth solutions in characteristic conoids of operators are constructed, which suggest the well-posedness of the formulation of the Cauchy problem and boundary value problems with dissipative boundary conditions. Additionally, efficient shock-capturing methods adapted to solution discontinuities are designed.