**Finite-volume methods for non-linear elasticity
in heterogeneous media
**

by R. J. LeVeque
*Int. J. Numer. Meth. Fluids* 40(2002), pp. 93-104.

**Abstract.**
An approximate Riemann solver is developed for the equations of
non-linear elasticity in a heterogeneous medium, where each grid cell
has an associated density and stress-strain relation. The non-linear
flux function is spatially varying and a wave decomposition
of the flux difference across a cell interface is used to
approximate the wave structure of the Riemann solution. This solver
is used in conjunction with a high-resolution finite-volume method using
the CLAWPACK software. As a test problem, elastic waves in a periodic
layered medium are studied. Dispersive effects from the heterogeneity,
combined with the non-linearity, lead to solitary wave solutions that
are well captured by the numerical method.

**Errata:**

- Case I is described incorrectly at the bottom of page 98. It should
be

$\rho_A = \rho_B = 2, \sigma_A(\epsilon) = \sigma_B(\epsilon) = 2\epsilon$

so that the sound speed is still 1 but $\sigma = 2\epsilon$ everywhere as seen in Figure 1. Note that this is analogous to Case III.

**Reprint:**

**bibtex entry:**

@article{rjl:elastic2002, Author = {R. J. LeVeque}, Journal = {Int. J. Numer. Meth. Fluids}, Pages = {93--104}, Title = {Finite volume methods for nonlinear elasticity in heterogeneous media}, Volume = {40}, DOI = {10.1002/fld.309}, Year = {2002}}