One-dimensional shock and acceleration waves in deformable dielectric materials with memory

Abstract

In this paper we examine the behavior of one-dimensional shock and acceleration waves in deformable dielectric materials with memory. In a sense, this paper may be regarded as a generalization of the earlier studies by CI-IEN & GURTIN I-1 l and COLEMAN & GURTIN [21 on the behavior of shock and acceleration waves in deformable materials with memory. In some respects, our analysis concerning shock waves parallels the recent work of DUNWOOOY [-33 on the behavior of shock waves in heat-conducting deformable materials with memory. After dispensing with the preliminaries, we first derive the differential equation which the amplitude of the shock must obey. We find that the evolutionary behavior of the amplitude of the shock depends on the relative magnitudes of the jump in deformation gradient and the quantity 2 which we call the critical jump in deformation gradient, and that 2 depends on the shock amplitude, the histories of the mechanical and electrical conditions of the material immediately ahead of the shock, and the stress-deformation and stress-electrical displacement relaxation properties of the material. In addition to examining the general behavior of a compressive shock, we study the properties of the electric field during shock transition and derive certain of its properties in terms of the material properties. We then derive the differential equation which the amplitude of the acceleration wave must satisfy. As expected, we find that the evolutionary behavior of the wave is determined by the histories of the mechanical and electrical conditions immediately ahead of the wave and the stress-deformation and stress-electrical displacement relaxation properties of the material. We pay particular attention to the study of waves propagating into materials which have always been in states of homogeneous deformation for all past times. Our motivation for this study is that such a state of deformation characterizes the conditions of two well-defined experimental situations. The first is the voltage mode experiment ~, and the second is the current mode experiment 2. For the voltage mode experiment, we find that the evolutionary behavior of the amplitudes of both the shock and acceleration waves depends, in particular, only on the stress-