Selected topics in the history of the two-dimensional biharmonic problem

The classical two-dimensional solution for the stress distribution in an elastic wedge loaded by a uniform pressure on one side of the wedge becomes infinite when the wedge angle 2α satisfies the equation tan 235-1. This paradox was resolved recently by Dempsey who obtained a solution which is bounded at 235-2. However, for α not equal but very close to 235-3, the classical solution can still be very large as α approaches 235-4. In this paper we re-examine the paradox. We obtain a solution which remains bounded as α approaches 235-5 and reproduces Dempsey's solution in the limit 235-6. At 235-7 the stress distribution contains a (ln r) term for general loadings. The (ln r) term disappears under a special loading and the stresses are bounded for all r. Moreover, the solution is not unique. In other words, for the wedge angle 235-8 under a special loading, infinitely many solutions exist for which the stresses are bounded for all r. We also obtain solutions which are bounded and approach Dempsey's solutions when 2α=π and 2π. Again, under a special loading infinitely many solutions exist for which the stresses are bounded for all r. Care has been exercised in this paper to present the solutions in a form in which the terms r-λ and ln r are replaced by R-gl and ln R where R=r/r0is the dimensionless radial distance and r0 is an arbitrary constant having the dimension of length.

The wedge subjected to tractions: a paradox re-examined

Multiple root solutions, wedge paradoxes and singular stress states that are not variable-separable

The classical solution for an isotropic elastic wedge loaded by uniform tractions on the sides of the wedge becomes infinite everywhere in the wedge when the wedge angle 2α equals π, 2π or 2α* where tan 2α* = 2α*. When the wedge is loaded by a concentrated couple at the wedge apex the solution also becomes infinite at 2α = 2α*. A similar situation occurs when the wedge is anisotropic except that 2α* is governed by a different equation and depends on material properties. Solutions which do not become infinite everywhere in the wedge are available for isotropic elastic wedges. In this paper we present solutions for the anisotropic elastic wedge at critical wedge angles. The main feature of the solutions obtained here is that they are in a real form even though Stroh's complex formalism is employed.

T. C. T. Ting

Papers

A Paradox on the Elastic Wedge Subjected to a Concentrated Couple and on the Jeffery‐Hamel Viscous Flow Problem