Dispersion of Small Amplitude Waves in a Pre-Stressed, Compressible Elastic Plate

Abstract

The dispersion of harmonic waves, propagating along a principal direction in a pre-stressed, compressible elastic plate, is investigated in respect of the most general isotropic strain-energy function. Different cases, dependent on the choice of material parameters and pre-stress, are analysed. A complete long and short wave asymptotic analysis is carried out, with the approximations obtained giving phase speed (and frequency) as explicit functions of wave and mode number. Various wave fronts, both associated with the short wave limit of harmonics and arising through the combination of harmonics in a narrow wave speed region, are discussed. It is mentioned that the case of high compressibility is of particular interest. In contrast with the classical (un-strained) case, the longitudinal body wave speed may be less than the corresponding shear wave speed. In consequence, the short wave limit of all harmonics may be the appropriate longitudinal wave speed; contrasting with the classical case for which this limit is necessarily associated with a shear wave front. A further possible short wave limit is also shown to exist for which the associated wave normal has a component in the direction normal to the plate. Particularly novel numerical results are presented when the longitudinal and shear wave speeds are equal. The analysis is illustrated by numerical calculations for various strain-energy functions.