Solution of a Riemann problem for elasticity

Abstract

This paper describes a numerical algorithm for the Riemann solution for nonlinear elasticity. We assume that the material is hyperelastic, which means that the stress-strain relations are given by the specific internal energy. Our results become more explicit under further assumptions: that the material is isotropic and that the Riemann problem is uniaxial. We assume that any umbilical points lie outside the region of physical relevance. Our main conclusion is that the Riemann solution can be obtained by the iterative solution of functional equations (Godunov iterations) each defined in one-or two-dimensional spaces.

MUSTA‐type upwind fluxes for non‐linear elasticity

“…The procedure is not general and is applicable only in two special cases: an isolated contact discontinuity with possible changes in all components of the vector of primitive variables W (test problem (24)) and the purely one-dimensional shock-tube problem, in which no disturbances in the transverse direction exist (test problems with data (21)-(23), (25)). For more details on the construction of the exact Riemann problem solutions see [9,11].…”

“…A simple way of removing the dependence of the truncation error on the reciprocal of the Courant number in the FORCE flux while retaining its simplicity is to use locally defined mesh parameters x g , t g in (14); the time step t g is estimated from the data U L , U R in (9); the resulting flux is upwind [14]. A further improvement is the generalized FORCE (GFORCE) flux [12], which is given by a convex average of (12) and (13), again with the local selection of 903 the time step:…”

“…Developing exact or approximate Riemann solvers for non-linear elasticity is a difficult task due to the complexity of the system and its associated equation of state. The conservative form of the equations was used in [9] for the theoretical study of the Riemann problem solution. A second-order Godunov-type method based on an approximate Riemann solver is developed in [10].…”

The Quasilinear Wave Equation for Antiplane Shearing of Nonlinearly Elastic Bodies

“…The eigenvalues λ 1 , λ 2 of A are 12) and the corresponding right and left eigenvectors of A are 15) system (3.6) is genuinely nonlinear in the sense of Lax where f uu = 0. The flux function f , which is odd by definition, could be super-or sublinear for u > 0.…”

“…We also carry out a careful validation of our methods, described below. For analyses of the Riemann problem see [15,22,30,35].…”