The Linear Elastic Wedge Under a Tip Couple at the Critical Angle. Where Is the Paradox?

Abstract

AbstractThe classical problem of a two-dimensional infinite linear elastic wedge, with opening angle of $2\alpha $ 2 α , subjected to the action of a concentrated couple at its tip, was considered by Carothers. The solution, which presents a quadratic singularity in the stress field, suffers a spurious behavior at the critical angle $\alpha = \alpha ^{*} \simeq \, 0.715 \,\pi $ α = α ∗ ≃ 0.715 π , when the stress grows unboundedly within the body. This inconsistency is usually referred to as the “wedge paradox”. Here, relying on rather intuitive arguments, we present a representation for the stress field in the wedge which captures other states characterized by a quadratic singularity. This is first obtained by considering an auxiliary problem, in which the wedge is ideally split into three wedges under tip couples, whose values are determined by compatibility conditions in terms of stress and displacement on the common lines. Remarkably, we find that the state of stress varies continuously on $\alpha $ α ; at $\alpha =\alpha ^{*}$ α = α ∗ the stress does not vanish, although the action at the wedge tip has zero resultant and zero moment resultant. An alternative derivation of the elastic solution in the wedge, which relies upon the notion of nuclei of strain, suggests that, at the critical angle, the applied action is that of dipoles without moment at the tip point. We conclude that it is the form of Carothers’ solution that cannot account for all states of stress characterized by a quadratic singularity at the vertex. The paradox naturally disappears in the proposed representation.