7 - Two-Dimensional Formulation

Abstract

This chapter examines the formulation of two-dimensional problems in elasticity. The initial formulation presented in this study results in a boundary-value problem cast within a two-dimensional domain in the x, y-plane using Cartesian coordinates. This work is then reformulated in polar coordinates to allow for the development of important solutions in that coordinate system. Because all real elastic structures are three-dimensional, the theories described here are approximate models. The nature and accuracy of the approximation depend on problem and loading geometry. Although four different formulations are developed, the two basic theories of plane strain and plane stress represent the fundamental plane problem in elasticity. These two theories apply to different types of two-dimensional bodies; however, their formulations yield very similar field equations. The study shows that these two theories can be reduced to one governing equation in terms of a single unknown stress function. This reduction then allows for the generation of many solutions to the problems of engineering interest.