×

zbMATH — the first resource for mathematics

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.

MSC:
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
Software:
HLLE
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] B. Despres, A geometrical approach to non-conservative shocks and elastoplastic shocks (preprint of HYKE)(2005).
[2] Davis, S.F., Simplified second-order Godunov-type methods, SIAM J. sci. stat. comput., 9, 445-473, (1988) · Zbl 0645.65050
[3] Friedrichs, K.O.; Lax, P.D., Systems of conservation laws with a convex extension, Proc. nat. acad. sci. USA, 68, 1686-1688, (1971) · Zbl 0229.35061
[4] P. Germain, Cours de Mécanique des Milieux Continus, Masson, Paris, 1973. · Zbl 0254.73001
[5] S.K. Godunov, Elements of Continuum Mechanics, Nauka, Moscow, 1978 (in Russian).
[6] Godunov, S.K.; Romenskii, E.I., Elements of continuum mechanics and conservation laws, (2003), Kluwer Academic Plenum Publishers New York · Zbl 1031.74004
[7] Gouin, H.; Debieve, J.F., Variational principle involving the stress tensor in elastodynamics, Int. J. eng. sci., 24, 7, 1057-1066, (1986) · Zbl 0587.73024
[8] Harten, A.; Lax, P.D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM rev., 25, 35-61, (1983) · Zbl 0565.65051
[9] Kulikovskii, A.G.; Pogorelov, N.V.; Semenov, A.Y., Mathematical aspects of numerical solution of hyperbolic systems, (2001), Chapman & Hall/CRC · Zbl 0965.35001
[10] Miller, G.H.; Colella, P., A high-order Eulerian Godunov method for elastic – plastic flow in solids, J. comput. phys., 167, 131-176, (2001) · Zbl 0997.74078
[11] Miller, G.H.; Colella, P., A conservative three-dimensional Eulerian method for coupled fluid – solid shock capturing, J. comput. phys., 183, 26-82, (2002) · Zbl 1057.76558
[12] Petitpas, F.; Franquet, E.; Saurel, R.; Le Metayer, O., A relaxation-projection method for compressible flows. part II: artificial heat exchanges for multiphase shocks, J. comput. phys., 225, 2214-2248, (2007) · Zbl 1183.76831
[13] Plohr, J.N.; Plohr, B.J., Linearized analysis of richtmyer – meshkov flow for elastic materials, Jfm, 537, 55-89, (2005) · Zbl 1074.74037
[14] J. Serrin, Mathematical principles of classical fluid mechanics, encyclopedia of physics, VIII/1, Springer-Verlag, 1959, pp. 125-263.
[15] Saurel, R.; Franquet, E.; Daniel, E.; Le Metayer, O., A relaxation-projection method for compressible flows. part I: the numerical equation of state for the Euler equations, J. comput. phys., 223, 822-845, (2007) · Zbl 1183.76840
[16] V.A. Titarev, E.I. Romenskii, E.F. Toro, MUSTA-type upwind fluxes for nonlinear elasticity. Int. J. Numer. Meth. Eng., accepted for publication. · Zbl 1159.74046
[17] Toro, E.F.; Spruce, M.; Speares, W., Restoration of the contact surface in the HLL-Riemann solver, Shock waves, 4, 25-34, (1994) · Zbl 0811.76053
[18] E.F. Toro, Exact and approximate Riemann Solvers for the artificial compressibility equations. Technical Report 97-02, Department of Mathematics and Physics, Manchester Metropolitan University, UK, 1997.
[19] Trangenstein, J.A.; Colella, P., A higher-order Godunov method for modelling finite deformation in elastic – plastic solids, Commun. pure appl. math., XLIV, 41-100, (1991) · Zbl 0714.73027
[20] Van Leer, B., Towards the ultimate conservative difference scheme IV. A new approach to numerical convection, J. comput. phys., 23, 276-299, (1991) · Zbl 0339.76056
[21] Wilkins, M.L., Computer simulation of dynamic phenomena, (1999), Springer · Zbl 0926.76001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.