Perturbation Solution of Plane Stress Problems in Anisotropic Elasticity

Abstract

The problem on the approximate determination of the mean stress distribution across a thin crystal plate by the method of perturbation is considered. The plate is free from surface tractions and body forces, but is stressed under the action of prescribed self-equilibrating edge forces. The anisotropic elastic compliance matrix $S_{IJ} $ is written in the form $S_{IJ} = S_{IJ}^0 + \epsilon S_{IJ}^ * $, where $S_{IJ}^0 $, is the isotropic elastic compliance matrix nearest to $S_{IJ} $. A power series expansion in $ \epsilon $ is then assumed for the Airy stress function from which the stress components may be derived. The problem of the zeroth order in $ \epsilon $ is that of an isotropic plate. The perturbed problems satisfy nonhomogeneous differential equations and vanishing stress boundary values. The 7090/94 Ibsys Formac programming is used to handle some of the tedious algebraic and analytic routines involved in developing the perturbation solutions. Application of the perturbation method is carried out explicitly for a circular plate.