A biharmonic polynomial airy stress function for the square-end adhesive layer and sandwich core

Abstract

Accurate analysis of the stress, strain, and deformation fields in the square ended adhesive layer or square ended sandwich structure core requires a full elasticity solution. To this end a polynomial Airy stress function that satisfies all displacement boundary conditions, all stress free boundary conditions, and point equilibrium is proposed. The remaining requirement of a full elasticity solution, strain compatibility, is satisfied through a Galerkin approximation of the biharmonic equation on the Polynomial Airy stress function. To explore the utility of this Biharmonic Polynomial solution, the stress fields are explored for the case of idealized simple shear. The peel, shear, and longitudinal normal stress fields are found to be finite, differentiable, and convergent throughout the domain including the sharp corners of the elastic medium. The resulting stress components are compared to the Goland-Reissner Model, the Closed Form Higher Order (CFHO) model, and the Decoupled Biharmonic (DB) model. It is found that previous models significantly underestimate the peak peel stress, and with it the peak von Mises stress and the peak Principal stress values that drive the failure envelope for ductile and brittle materials, respectively.