Chapter 13 - Displacement potentials and stress functions: Applications to three-dimensional problems

Abstract

We now wish to investigate the method of potentials to generate solutions to elasticity problems. Several different potential techniques have been developed in order to solve problems within both displacement and stress formulations. Methods related to the displacement formulation include the scalar and vector potentials from the Helmholtz decomposition, Galerkin vector, and Papkovich–Neuber functions. These schemes provide general solution forms for Navier's equations. Potentials used in the stress formulation are those related to the Maxwell and Morera stress functions, and these lead to Airy and other common stress functions that we have already used for the solution of particular elasticity problems. As observed previously, these stress functions normally satisfy the equilibrium equations identically and when combined with the compatibility relations they yield a simpler and more tractable system of equations. For many cases these potential approaches are useful to solve particular three-dimensional elasticity problems, and we will investigate several such solutions.