A new hydrodynamic theory based on non-equilibrium statistical mechanics is developed to describe the structure formation in dynamically deformed materials. Self-consistent non-local formulation of the boundary-value problem for a high-strain-rate process is reduced to a nonlinear operator set similar to some resonance problems. The branching of solutions to the problem determines both scales and types of the formed internal structure. A penetration problem for a long flat rigid plate into a viscous elastic medium is considered accounting for the dynamic structure formation following the high-rate straining in the framework of the nonlocal self-consistent approach. The obtained approximate analytical solution has shown to describe three regimes: initial, transient and quasi-stationary. It has been demonstrated that the mesoscopic structure formation had been initiated by relative accelerations in a medium localized near the plate surface moving at high velocity. The mesoscopic structures formed during the initial stage of penetration can affect the steady-state stage. It is very important that the proposed self-consistent theory allows taking into account the feed-back influence of the mesoscopic effects on macroscopic movement of the plate.