Stress functions for torsion-free axisymmetric state of stress

Abstract

The solution of problems of the theory of elasticity in terms of stresses requires the solution of equations of equilibrium. In two and three-dimensional application, the components of stress are frequently expressed in terms of partial derivatives of stress function, the correct expressions being chosen so that the conditions of static equilibrium become a consequence of partial derivatives independent of the order of differentiation. In this manner the solution of an equilibrium equation can be obtained without any difficulty. In this paper a systematic process is devised to derive the stress functions for the problems of torsion-free axisymmetric state of stresses. The applied method is based on the theory of total differentials. The stress boundary conditions are also formulated in terms of stress functions. The relations between the strain compatibility conditions and the stress functions are discussed. Different forms of solutions of equilibrium equations in terms of stress functions are also analyzed.