We study the equations describing the dynamics of radial motions for isotropic elastic materials; these form a system of non-homogeneous conservation laws. We construct a variational approximation scheme that decreases the total mechanical energy and at the same time leads to physically realizable motions that avoid interpenetration of matter.

In addition, we consider a variational scheme developed by S. Demoulini, D. Stuart and A. Tzavaras that approximates the equations of three-dimensional elastodynamics with polyconvex stored energy. We establish the convergence of the time-continuous interpolates constructed in the scheme to a solution of polyconvex elastodynamics before shock formation. The proof is based on a relative entropy estimation for the time-discrete approximants in an environment of L^p-theory bounds, and provides an error estimate for the approximation before the formation of shocks.