The existence of steady shock waves in nonlinear materials with memory

Abstract

In recent years there has been extensive work on wave propagation in nonlinear materials with fading memory. Shock waves in such materials have been studied by COLEMAN, GURTIN, & HERRERA [1], and COLEMAN & GURTIN [2], while acceleration waves and higher-order discontinuities have been investigated by COLEMAN t~ GURTIN [1, 3-7], VARLEY [8], COLEMAN, GREENBERG, d~ GURTIN [9], and WANG • BOWEN [10]. PIPKIN [11], using a special constitutive equation, has obtained exact solutions of the one-dimensional steady flow equations; these solutions possess both acceleration waves and shock waves. Here I show that for a large class of nonlinear viscoelastic materials it is possible to find one-dimensional steady solutions exhibiting shock waves; thus, this investigation may be regarded as a generalization of PIPKIN'S results. Although this article deals only with compressive motions*, it is possible, by an obvious modification of the underlying hypotheses**, to derive corresponding results for expansive motions***. In section 2 the relevant balance laws are stated, and a representation theorem for steady motions is established. The theorem asserts that if the body occupies the entire real line while in a fixed reference configuration ~ with constant density, then every steady motion (X, t ) -~x(X, t), with material points labeled by their positions X in ~, is of the form