Large amplitude waves in bounded media II. The deformation of an impulsively loaded slab: the first reflexion

Abstract

This paper continues the investigation of large amplitude waves in bounded media started by Cekirge & Varley (1973). It describes the early stages of the deformation produced in an elastic slab contained between two parallel, plane interfaces when the normal traction at one of them changes discontinuously. During the subsequent deformation energy is radiated across these interfaces to adjacent elastic materials. Typically, the disturbance in the slab could be caused by the arrival of a constant strength shock wave travelling through an adjacent material or when the slab, which forms the front part of a composite material, impacts some other elastic material. It is assumed that the dynamic response of the slab can be approximated by that of one of the model materials introduced in the first part of this study. It is shown that this is possible for a whole host of materials. These include polycrystalline solids, metals when subjected to high pressure, water, explosive products, gases, yarns as well as elastic-plastic, rigid-plastic and rigid-elastic materials. The results reported are obtained by showing that for these model materials a simple, but exact, representation can be found that describes the interaction of a centred wave with any wave travelling in the opposite direction. The arbitrary functions occurring in this representation are then found for the special case when this opposite travelling wave is the wave reflected from an interface with some other elastic material during the arrival of the centred wave. The limiting cases of a perfectly free interface, a perfectly rigid interface, and an interface with a Hookean material are analysed in great detail. Although the terminology used in this paper is that associated with nonlinear elastodynamics, the results are directly applicable to any system whose response is described by the nonlinear wave equation. For example, the slab could represent a layer of nonlinear dielectric embedded in some other nonlinear dielectric and the disturbance could be generated by the arrival of an electromagnetic shock. Alternatively, the slab could represent sea water which is bounded by air from above and by rock from below while the disturbance is produced by a sudden motion of the water/rock interface.