The Wedge Under a Concentrated Couple: A Paradox in the Two-Dimensional Theory of Elasticity

Abstract

Abstract The classical two-dimensional solution for the stress distribution in an elastic wedge which is subjected to a concentrated couple at the vertex, breaks down when the opening angle 2α of the wedge satisfies the equation tan 2α = 2α, i.e., when 2α is approximately 257 deg. As the foregoing critical opening angle 2α* is approached, all of the nonvanishing components of stress become infinite throughout the field, while the solution displays no obvious pathological characteristics for other values of the wedge angle. It is the purpose of the present paper to account for this peculiar singular behavior and to show that the solution under examination has physical significance only for wedge angles below the critical angle; for opening angles in the range 2α* ≤ 2α < 2π, the notion of a “concentrated couple” at the vertex of an elastic wedge, is found to be inherently deficient in meaning. The present investigation, as a by-product, augments the supply of counterexamples to the traditional version of Saint Venant’s principle.