A second-order Godunov method for the conservation laws of nonlinear elastodynamics

Abstract This paper treats the system of conservation laws of nonlinear elastodynamics in the form recently reviewed by B. J. Plohr and D. H. Sharp (Adv. Appl. Math.9, 481–499 (1989)). Following X. Garaizar (Acta Appl. Math., to appear), we consider hyperelastic materials whose specific internal energy function is the sum of an isotropic part and a small anisotropic one. For this system of equations, we present a second order accurate Godunov-type finite difference scheme. Our scheme is based in particular on an approximation of the solution of a Riemann problem obtained by using only shock waves. Several computational results are presented.

[1]  R. Menikoff,et al.  The Riemann problem for fluid flow of real materials , 1989 .

[2]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[3]  David H. Sharp,et al.  A conservative Eulerian formulation of the equations for elastic flow , 1988 .

[4]  X. Garaizar,et al.  Solution of a Riemann problem for elasticity , 1991 .

[5]  M. Wilkins.  Calculation of Elastic-Plastic Flow , 1963 .

[6]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[7]  Bradley J. Plohr,et al.  Shockless acceleration of thin plates modeled by a tracked random choice method , 1988 .

[8]  D. Steinberg,et al.  A constitutive model for metals applicable at high-strain rate , 1980 .

[9]  Heinrich Freistühler,et al.  A standard model of generic rotational degeneracy , 1989 .

[10]  Alexandre J. Chorin,et al.  Random choice solution of hyperbolic systems , 1976 .

[11]  P. Lax.  Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves , 1987 .

[12]  A. Hanyga,et al.  Mathematical Theory of Non-Linear Elasticity , 1985 .

[13]  X. Garaizar,et al.  The small anisotropy formulation of elastic deformation , 1989 .

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

[15]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[16]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

[17]  Gerhard Rayna,et al.  Reduce - software for algebraic computation , 1987, Symbolic computation: artificial intelligence.