Airy Stress Function-Based Elastoplastic Analysis of Plates with Holes Using the Nonconforming Morley Finite Element Method

Abstract

This paper presents a stress-function-based finite element formulation for two-dimensional elastoplastic plane-stress problems with holes. The Airy stress function is used as the primary global unknown, and the resulting non-homogeneous biharmonic equation is discretized with the nonconforming Morley finite element. The stresses are obtained from the second derivatives of the discrete Airy function. Consequently, the differential equilibrium equations are satisfied inside each element, while interelement coupling is enforced through the Morley weak formulation. Plastic strains enter the governing equation through an element-wise weak-form contribution, which avoids the direct evaluation of their second derivatives. The local material response is described by associated von Mises plasticity with Kuhn–Tucker conditions and is integrated using cutting-plane and return-mapping algorithms. The formulation is applied to a perforated plate subjected to in-plane tension and compared with a classical displacement-based elastoplastic finite element model. The results show good agreement in the elastic stress concentration, yielding load, stress redistribution, and plastic-zone development. For the present benchmark, the Airy–Morley formulation is computationally competitive and provides a stress-function-based alternative for a restricted class of two-dimensional plane-stress problems.