Abstract: The equations of elastodynamics are a paradigm of a system of conservation laws where the lack of uniform convexity of the stored energy function poses challenges in the mathematical theory.
First, we show how the existence of certain additional nonlinear transport constraints reinforces the efficacy of the entropy as a stabilizing factor and recovers the strength associated wth uniformly convex entropies in hyperbolic systems. It turns out that elastodynamics with polyconvex stored energy can be embedded into a larger symmetric hyperbolic system and visualized as constrained evolution. This leads to a variational approximation scheme and an existence theory for measure valued solutions satisfying certain kinematic constraints in the weak sense. It provides a framework, in conjunction with the relative entropy method, to establish convergence of viscocity (or relaxation) approximations to smooth solutions of polyconvex elastodynamics in several space dimensions. In addition, to show that when a smooth solution is present it is unique within the class of measure valued solutions.
Then we focus on the system of radial elastodynamics for isotropic elastic materials. We present the format of the enlarged system in this special case with the objective of assigning a mechanical interpretation in the nonlinear transport constraints. It turns out that one can construct via variational approximation solutions that obey the impenetrability of matter constraint. Finally, we recall a non-uniqueness result associated to cavitating solutions due to S. Spector and K. Pericak-Spector, and will present a perspective for that construction. |