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.

[1]  Mark L. Wilkins,et al.  Computer Simulation of Dynamic Phenomena , 1999 .

[2]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .

[3]  Henri Gouin,et al.  Variational Principle Involving the Stress Tensor in Elastodynamics , 2008 .

[4]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[5]  S. Godunov,et al.  Elements of Continuum Mechanics and Conservation Laws , 2003, Springer US.

[6]  Phillip Colella,et al.  A high-order Eulerian Godunov method for elastic-plastic flow in solids , 2001 .

[7]  Bruno Despres,et al.  A Geometrical Approach to Nonconservative Shocks and Elastoplastic Shocks , 2007 .

[8]  Kurt Friedrichs,et al.  [71-1] Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci. USA, 68 (1971), 1686–1688 , 1986 .

[9]  P. Lax,et al.  Systems of conservation equations with a convex extension. , 1971, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Bradley J. Plohr,et al.  Linearized analysis of Richtmyer–Meshkov flow for elastic materials , 2005, Journal of Fluid Mechanics.

[11]  S. F. Davis.  Simplified second-order Godunov-type methods , 1988 .

[12]  A. Kulikovskii,et al.  Mathematical Aspects of Numerical Solution of Hyperbolic Systems , 1998, physics/9807053.

[13]  Richard Saurel,et al.  A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations , 2007, J. Comput. Phys..

[14]  Phillip Colella,et al.  A higher-order Godunov method for modeling finite deformation in elastic-plastic solids , 1991 .

[15]  E. Toro,et al.  Restoration of the contact surface in the HLL-Riemann solver , 1994 .

[16]  Mondher Neifar.  Cours de Mécanique des Milieux Continus , 2009 .

[17]  Richard Saurel,et al.  A relaxation-projection method for compressible flows. Part II: Artificial heat exchanges for multiphase shocks , 2007, J. Comput. Phys..

[18]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection , 1977 .

[19]  S. K. Godunov,et al.  Elements of continuum mechanics , 1978 .

[20]  James Serrin,et al.  Mathematical Principles of Classical Fluid Mechanics , 1959 .

[21]  Eleuterio F. Toro,et al.  MUSTA‐type upwind fluxes for non‐linear elasticity , 2008 .

[22]  P. Colella,et al.  A conservative three-dimensional Eulerian method for coupled solid-fluid shock capturing , 2002 .