The wedge paradox and a correspondence principle

The wedge paradox for stress-free boundaries is the wedge loaded by a concentrated moment at the vertex (Carothers problem). The paradox consists of the fact the for wedges smaller than a half-space the solution exists, while for wedges bigger than a half-space as shown by Sternberg-Koiter the solution does not exist. We examine the possiblity of a paradox for clamped-clamped boundary conditions. We obtain the solution for a clamped-clamped wedge loaded by a concentrated couple at an interior point and show that for wedges smaller than a half-space the limit is zero as the moment approaches the vertex (i.e. the clamped wedge absorbs the concentrated couple), for a half-space it is a constant, and for angles bigger than a half-space the limit does not exist (paradox). Moreover, from the solution for the clamped wedge, taking the limit as Poisson's ratio tends to one, the solution for the stress-free wedge is retrieved (Carothers) according to the correspondence between cavities and rigid inclusions established by Dundurs (1989) and Markenscoff (1993).