Asymptotic Analysis for Plane Stress Problems

In this classroom note, the old and well-known plane-stress elastic class of problems is revisited, using an analysis technique which is different than that commonly found in the literature, and with a pedagogical benefit. An asymptotic analysis is applied to problems of thin linear elastic plates, made of a homogeneous and rather general anisotropic material, under the plane stress assumption. It is assumed that there are no body forces, that the boundary conditions are uniform over the thickness, and that the material (hence also the solution) is symmetric about the middle plane. The small parameter in this analysis is ϵ =t/D where t is the (uniform) thickness of the plate and D is a measure of its overall size. The goal of this analysis is to show how the three-dimensional (3D) problem of this type is reduced asymptotically to a sequence of essentially two-dimensional (2D) problems for a small ϵ . As expected, the leading problem in this sequence is shown to be the classical plane-stress problem. The solutions of the higher-order problems are corrections to the plane-stress solution. The analysis also shows that all six 3D compatibility equations are satisfied as ϵ goes to zero, and that the error incurred by the plane stress assumption is O(ϵ ^2) . For the special case of an isotropic in-the-plane material, the second-order solution is shown to be the exact solution of the 3D problem, up to an O(ϵ ^2) error in the close vicinity of the edge (which agrees with a well-known result for an isotropic material).