Prandtl-Meyer solutions of viscoplasticity equations with negative strain rate sensitivity

We present a further study of a viscoplasticity model with nonmonotonic strain rate dependence ensuring the complete integrability of the two-dimensional equilibrium and consistency equations. The considered nonlinear equations change their type from hyperbolic to elliptic at a certain critical value of the strain rate intensity; the type change is accompanied by the formation of an interphase in the solid. This model is of interest for describing spatial autowave processes in active continua, and the integrability of equations allows one to construct efficient methods for the numerical solution of boundary value problems and ensures the existence of closed-form solutions. The present paper shows that the considered material function satisfies a criterion for the separation of the system of these equations into two noninteracting subsystems. We derive kinematic equations on the characteristics. We obtain and analyze centered self-similar solutions (Prandtl-Meyer solutions) in the domain of hyperbolicity of the equations, which describe flows in convergent and divergent channels.