On a Lamb-type problem for a bi-axially pre-stressed incompressible elastic plate

Abstract

The far-field response of a bi-axially pre-stressed incompressible elastic plate, subjected to an instantaneous edge impulse loading, is studied using a refined long-wave low-frequency theory. The second-order correction introduced by the refined theory is demonstrated to smooth the discontinuity associated with one of the wave-fronts predicted by the leading order hyperbolic theory. The character of the so-called quasi-front is shown to depend greatly on both the material parameters and the pre-stress and may be either classical receding or advancing. Additionally, and in contrast to the analogous problem in linear isotropic elasticity, in a pre-stressed plate the dilatational quasi-front may propagate slower than the shear wave-front. This situation is demonstrated to lead to the formation of a head-wave quasi-front.