On Maxwell’s Stress Functions for Solving three Dimensional Elasticity Problems in the Theory of Elasticity

The governing equations of three dimensional elasticity problems include the six Beltrami–Michell stress compatibility equations, the three differential equations of equilibrium, and the six material constitutive relations; and these are usually solved subject to boundary conditions. The system of fifteen differential equations is usually difficult to solve, and simplified methods are usually used to achieve a solution. Stress-based formulation methods, and displacement-based formulation methods are two common simplified methods for solving elasticity problems. This work adopts a stress-based formulation for a three dimensional elasticity problem. In this work, the Maxwell’s stress functions for solving three dimensional problems of elasticity theory are derived from fundamental principles. It is shown that the three Maxwell stress functions identically satisfy all the three differential equations of static equilibrium when body force components are ignored. It is further shown that the three Maxwell stress functions are solutions to the six Beltrami-Michell stress compatibility equations if the Maxwell stress functions are potential functions. It is also shown that the Airy’s stress functions for two dimensional elasticity problems are special cases of the Maxwell stress functions.