Short wave motion in a pre-stressed incompressible elastic plate

Abstract

An asymptotically consistent theory for short wave motion in a pre‐stressed elastic layer is presented. Its derivation is motivated by, and fully consistent with, the appropriate dispersion relation. The two‐dimensional scalar governing equation contains a minimal number of essential parameters and is reduced to a one‐dimensional equation by expanding the transverse displacement as a trigonometric series in terms of the thickness variable, resulting in a telegraphy‐type equation. In contrast to the classical linear isotropic theory, two distinct types of material response are possible. Although one type is a natural extension of the classical theory, the other is quite different and not a possible feature of the analogous classical problem. The latter is characterized by abnormal dispersion curve behaviour, loss of hyperbolicity of the asymptotic governing equation and degeneration of the boundary layer near the shear wave front. To illustrate the theory, the problem of appropriate edge loading of a semi‐infinite plate is considered.