Potentials and Stress functions

Abstract

Similar to other fields theories such as fluid mechanics , heat conduction and electromagnetic , the study and applications of elasticity theory requires knowledge of several areas of applied mathematics .The theory is formulated in terms of a variety of variables including scalar , vector and tensor fields ,through the use of particular principles from continuum mechanics , the theory is developed as a system of partial differential field equations , that are to be solved in a region of space coinciding with the body under study .we begin development of the basic field equations of elasticity theory by first investigating the kinematics of material deformation. As result of applied loadings , elastic solids will change the shape or deform , and these deformations can be quantified by knowing the displacement of material points in the body . we introduce the basic deformations and displacement and strain , and establish relations between these two field quantities . We establish the definition and properties of traction vector and stress tensor and develop the equilibrium equation, which becomes another set of field equation necessary in the over all formulation of elasticity theory . several different partial techniques have been developed in order to solve problems with in both displacement and stress formulations .Finally we used the Airy and other common stress functions for the solution of particular elasticity problems