The wedge subjected to tractions: a paradox resolved

Abstract

The classical two-dimensional solution provided by Lévy for the stress distribution in an elastic wedge, loaded by a uniform pressure on one face, becomes infinite when the opening angle 2α of the wedge satisfies the equation tan 2α_✱ = 2α_✱. Such pathological behavior prompted the investigation in this paper of the stresses and displacements that are induced by tractions of O(r ^−ω) as r→0. The key point is to choose an Airy stress function which generates stresses capable of accommodating unrestricted loading. Fortunately conditions can be derived which pre-determine the form of the necessary Airy stress function. The results show that inhomogeneous boundary conditions can induce stresses of O(r ^−ω), O(r ^−ω ln r), or O(r ^−ω ln² r) as r→0, depending on which conditions are satisfied. The stress function used by Williams is sufficient only if the induced stress and displacement behavior is of the power type. The wedge loaded by uniform antisymmetric shear tractions is shown in this paper to exhibit stresses of O(ln r) as r→0 for the half-plane or crack geometry. At the critical opening angle 2α_✱, uniform antisymmetric normal and symmetric shear tractions also induce the above type of stress singularity. No anticipating such stresses, Lévy used an insufficiently general Airy stress function that led to the observed pathological behavior at 2α_✱.