Consider a semi-infinite layer of an incompressible elastic material subjected to a finite static homogeneous pre-deformation. This paper describes the long-wave low-frequency limiting behaviour of extensional edge waves that may propagate along the free edge of such a layer. The analysis is done by constructing an asymptotic plate theory that describes the extensional motion of thin pre-stressed incompressible plates. The secular equation for edge waves propagating along one of the principal axes of the pre-stress is studied for bi-axial deformations and several specific material models. We show that certain combinations of material models and configurations of pre-stresses do not support the propagation of edge waves, which mirrors the situation with surface waves in pre-stressed media. More unusually, we present a seemingly first explicit example of the non-unique edge wave solutions existing in thin pre-stressed plates.
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