Modelling wave dynamics of compressible elastic materials

An Eulerian conservative hyperbolic model of isotropic elastic materials subjected to finite deformation is addressed. It was developed by Godunov [S.K. Godunov, Elements of continuum mechanics, Nauka, Moscow, 1978 (in Russian) and G.H. Miller, P. Colella, A high-order Eulerian Godunov method for elastic–plastic flow in solids, J. Comput. Phys. 167 (2001) 131–176]. Some modifications are made concerning a more suitable form of governing equations. They form a set of evolution equations for a local cobasis which is naturally related to the Almansi deformation tensor. Another novelty is that the equation of state is given in terms of invariants of the Almansi tensor in a form which separates hydrodynamic and shear effects. This model is compared with another hyperbolic non-conservative model which is widely used in engineering sciences. For this model we develop a Riemann solver and determine some reference solutions which are compared with the conservative model. The numerical results for different tests show good agreement of both models for waves of very small and very large amplitude. However, for waves of intermediate amplitude important discrepancies between results are clearly visible.