On the Generalized Plane Stress Problem, the Gregory Decomposition and the Filon Mean Method

Abstract

In the present paper a stress general solution is obtained for the generalized plane stress problem with planar body forces, and it is demonstrated that only body force of biharmonic type ensures the compatibility with the generalized plane stress assumptions (σ ₃₃=0). Inspired by the Filon perspective of average values, two more generalized plane stress problems with weak assumptions on the out-of-plane stress averages ($\bar{\sigma}_{33}=0$ or $\nabla^{2} \bar{\sigma}_{33}=0$) are studied, and the averages of the corresponding stress fields are expressed by the Airy stress functions. The authors also provide an alternative proof of the Gregory decomposition theory.