Chapter 7 - Two-Dimensional Formulation

Abstract

Because of the complexity of the elasticity field equations, analytical closed-form solutions to fully three-dimensional problems are very difficult to accomplish. Thus, most solutions are developed for reduced problems that typically include axisymmetry or two-dimensionality to simplify particular aspects of the formulation and solution. We now wish to examine in detail the formulation of two-dimensional problems in elasticity. Our initial formulation will result in a boundary-value problem cast within a two-dimensional domain in the x , y -plane using Cartesian coordinates. This work will then be reformulated in polar coordinates to allow for the development of important solutions in that coordinate system. Since all real elastic structures are three-dimensional, the theories set forth here will be approximate models. The nature and accuracy of the approximation will depend on problem and loading geometry. Although four different formulations will be developed, the two basic theories of plane strain and plane stress represent the fundamental plane problem in elasticity. While these two theories apply to significantly different types of two-dimensional bodies, their formulations yield very similar field equations. It will be shown that these two theories can be reduced to one governing equation in terms of a single unknown stress function. This reduction then allows many solutions to be generated to problems of engineering interest.