One-dimensional shock waves in simple materials with memory

Asymptotic evolution laws for plane dilatational shock waves travelling in simple materials with memory are derived in this paper by using two approximation methods. The first method is a combination of singular surface theory and perturbation methods. A system of two coupled first-order ordinary differential equations is derived for the shock amplitude and the amplitude of the accompanying second-order discontinuity. The shock amplitude is assumed to be small, but the accompanying second-order discontinuity may be taken either to be finite or to be small with the shock amplitude. The first case corresponds to the situation in which the duration time of the applied load is small compared with the viscous relaxation time and we show that the evolutionary behaviour of the two discontinuities is strongly affected by material nonlinearity. The second case, however, corresponds to the situation in which the duration time is comparable with the viscous relaxation time and we are able to show that the evolutionary behaviour is as predicted by the linear theory of viscoelasticity. In both cases the corresponding elastic results are obtained on allowing the viscous relaxation time to tend to infinity. The second approximation method is the shock-fitting method applied to a modulated simple wave theory, which is itself an approximation based on a small-amplitude finite-rate assumption equivalent to the first case discussed above. The two approximation methods are shown to yield the same evolution laws within their common range of validity.