Asymptotic analysis of out-of-plane strain and displacement fields at angular corners

Abstract

This paper presents three-dimensional analytical solutions for displacement and strain fields near angular corners in plates subjected to in-plane loading. The approach is based on the first-order plate theory, which represents an elementary extension of the classical plane theory of linear elasticity. Utilising the Kontorovich–Lebedev transform, the asymptotic behaviour of the out-of-plane displacement field near the apex is investigated for both Mode I and Mode II loadings. The analytical solutions obtained in the present work correctly predict several three-dimensional effects, such as the order of stress singularity and the intensity of the coupled (local) out-of-plane singular mode under remote in-plane shear loading as well as the scale effect associated with the plate thickness. The developed solutions are more general than many previous analytical results obtained for crack and notch geometries and, essentially, generalise the classical William's solution for angular corners. The analytical predictions compare favourably with three-dimensional finite element modelling and experimental results.