A Relaxation Theory with Polyconvex Entropy Function Converging to Elastodynamics

The equations of polyconvex elastodynamics can be embedded to an augmented symmetric hyperbolic system. This property provides a stability framework between solutions of the viscosity approximation of polyconvex elastodynamics and smooth solutions of polyconvex elastodynamics. We devise here a model of stress relaxation motivated by the format of the enlargement process which formally approximates the equations of polyconvex elastodynamics. The model is endowed with an entropy function which is not convex but rather of polyconvex type. Using the relative entropy we prove a stability estimate and convergence of the stress relaxation model to polyconvex elastodynamics in the smooth regime.