A low-frequency model for dynamic motion in pre-stressed incompressible elastic structures

An asymptotic one-dimensional theory, with minimal essential parameters, is constructed to help elucidate (two-dimensional) low-frequency dynamic motion in a pre-stressed incompressible elastic plate. In contrast with the classical theory, the long-wave limit of the fundamental mode of antisymmetric motion is non-zero. The occurrence of an associated quasi-front therefore offers considerable deviation from the classical case. Moreover, the presence of pre-stress makes the plate stiffer and thus may preclude bending, in the classical sense. Discontinuities on the associated leading-order wavefronts are smoothed by deriving higher-order theories. Both quasi-fronts are shown to be either receding or advancing, but of differing type. The problems of surface and edge loading are considered and in the latter case a specific problem is formulated and solved to illustrate the theory. In the case of antisymmetric motion, and an appropriate form of pre-stress, it is shown that the leading-order governing equation for the mid-surface deflection is essentially that of waves propagating along an infinite string, a higher-order equation for which is derived.