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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.