Approximate phase speed of Lamb waves in a composite plate reinforced with strong fibres

Abstract

In the study of wave propagation through plates and laminates, there are no complete three-dimensional solutions especially if the plate or plies are made of an anisotropic material. In such cases, laminate theories are approximate and often give poor results in comparison with exact solutions. When laminae or plate is highly anisotropic, the small parameter, which is appropriate is usually the modulus ratio, that is, the ratio of some elastic moduli. In the study of wave propagation, in some of the ranges of a wave number, a geometrical parameter, non-dimensional wave number, may be dominant in dispersion relations. The interaction between these two parameters, the modulus ratio and the geometric parameter, is a complicated one and it is not obvious which parameter will dominate in the given circumstances. The influence of these two parameters on dynamical behaviour of the plate, especially in the dispersion relation is examined in the present paper. To consider the singular perturbation problem, where at least one boundary layer arises, it is usual to apply the method of asymptotic expansions. In that case, a matched asymptotic expansion is suitable for the consideration. To satisfy well the corresponding equations, it was necessary to develop outer and inner expansions which are valid in the so-called outer and inner regions, respectively. Then, matching conditions have been derived, which lead to establishing an asymptotic expansion which is valid uniformly in the whole domain.