Modelling wave dynamics of compressible elastic materials. (English) Zbl 1155.74020

Summary: We address an Eulerian conservative hyperbolic model of isotropic elastic materials subjected to finite deformation. It was developed by S. K. Godunov and E. I. Romenskii, Elements of continuum mechanics and conservation laws. Translation from the 1998 Russian original. New York, NY: Kluwer Academic/Plenum Publishers. viii (2003; Zbl 1031.74004)] and G. H. Miller and P. Colella [J. Comput. Phys. 167, No. 1, 131–176 (2001; Zbl 0997.74078)]. 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.


74J40 Shocks and related discontinuities in solid mechanics
74B20 Nonlinear elasticity
74S10 Finite volume methods applied to problems in solid mechanics
76L05 Shock waves and blast waves in fluid mechanics


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