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.

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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}}