Harmonic wave propagation along a non-principal direction in a pre-stressed elastic plate

Abstract

In this paper harmonic wave propagation in a pre-stressed, incompressible elastic plate is investigated. Specifically the problem in which one principal direction of the primary deformation is normal to the plate and the direction of propagation is at an angle θ to one of the in-plane principal direction is considered. The dispersion relation is derived in respect of a general strain energy function and numerical solutions presented for a Mooney–Rivlin material. The dispersion curves are shown to be more complex in nature than those associated with propagation along an in-plane principal direction, as the solution arising from the horizontally polarised shear wave does not uncouple. An asymptotic analysis, for high and low wave number, is carried out with the high wave number expansions providing a good approximation to the numerical solution over a large wave number region. The paper also includes an investigation of the effects of changes in the normal Cauchy stress and direction of propagation on the existence of surface waves.