The stress relaxation during the high-rate straining of materials is followed by the energy exchange between macroscopic and mesoscopic scale levels. A correct description of the shock wave propagation in relaxing medium involving the experimentally measured mesoscopic characteristics such as the mass velocity dispersion and the wave amplitude loss on account of irreversible mesostructure formation had been developed on the basis of a new self-consistent non-local hydrodynamic approach. In scope of this theory the problem of the non-steady shock wave propagation in semi-space had been formulated as a nonlinear operator set with the branching solutions for the mesostructure parameters determining sizes and types (rotational or translational) of mesostructure. These mesostructures are responsible for the dynamic behavior of solids as new carriers of deformation, During the stationary wave propagation the macro-meso-energy exchange inside the wave front is reversible. The structure transition to irreversible mesostructures, when medium properties behind the front are considerably changed, can result in a failure of materials.