On the existence of surface waves and the propagation of plate waves in pre-stressed fibre-reinforced composites

Abstract

A theoretical investigation of the effect of pre-stress on wave propagation in a fibre-reinforced plate is carried out using a continuum material model. In the case of both extensional and flexural waves the high wave number phase speed limit of the fundamental mode of the dispersion relation is found to be the corresponding surface wave speed. Whenever such a wave exists it is shown to be unique. The existence of such waves is dependent on the underlying primary deformation; necessary and sufficient conditions for existence are established. For the harmonics, numerical calculations indicate the possibility of two distinct classes of material response. One of these is similar to classical linear isotropic theory. The other is quite different, with a shear wave front arising from the cumulative effect of the harmonics, the contribution of successive harmonics arising in adjacent wave number regimes, and flexural and extensional modes intersecting in both the moderate and high wave number regimes. This shear wave front is further illuminated through examination of the associated group velocity curves, in which corresponding distinct flat maxima are observed. High and low wave number representations of the dispersion relations are derived for each harmonic, giving phase speed as a function of wave number, pre-stress and the arbitrary tension and provide excellent agreement with some illustrative numerical solutions. The paper is concluded with a brief discussion of stability and bifurcation.