Recently there has been interest in studying a new class of elasticmaterials, which is described by implicit constitutive relations. Under somebasic assumption for elasticity constants, the system of governing equations ofmotion for this elastic material is strictly hyperbolic but without theconvexity property. In this paper, all wave patterns for the nonclassicnonlinearly elastic materials under Riemann data are established completely byseparating the phase plane into twelve disjoint regions and by using anonnegative dissipation rate assumption and the maximally dissipative kineticsat any stress discontinuity. Depending on the initial data, a variety of wavepatterns can arise, and in particular there exist composite waves composed of ararefaction wave and a shock wave. The solutions for a physically realizablecase are presented in detail, which may be used to test whether the materialbelongs to the class of classical elastic bodies or the one wherein the stretchis expressed as a function of the stress.
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