A three-dimensional analysis of anisotropic inhomogeneous and laminated plates

Abstract

An asymptotic theory for bending and stretching of anisotropic inhomogeneous and laminated plates is developed based on the three-dimensional elasticity without a priori assumptions. The inhomogeneities are considered to vary through the plate thickness, and laminated plates belong to an important class of this inhomogeneous plate. Through appropriate nondimensionalization of the basic equations and expansion of the displacements and stresses in powers of a small parameter, we obtain sets of differential equations of various orders, that can be integrated successively to determine the three-dimensional solutions for the anisotropic inhomogeneous plate under lateral tractions and edge loads. We show that the governing equations for the asymptotic solutions are precisely those in the classical laminated plate theory (CLT) with nonhomogeneous terms. As a result, all the displacement and stress components can be determined in a systematic way, using the same solution method as that for the CLT solution. While the solution is no more difficult than the CLT, the asymptotic solution converges rapidly and gives accurate results. The basic theory and the solution approach are illustrated by considering a problem of symmetric laminated plates under the action of edge loads, and Pagano's problem of a bi-directional laminated plate under lateral transverse loads. Elasticity solutions for the problems are obtained in a simple manner, without treating the individual layers and considering the interfacial continuity conditions in particular.