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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.