Mixed boundary value problems for a penny-shaped cut

The axially symmetric problem for a penny-shaped cut is solved in the case the displacements are prescribed on its upper surface and stresses on its lower surface. The solution is achieved by using integral transforms along with certain representations that reduce the general problem to the solution of uncoupled Hilbert problems. For polynomial loadings the Hilbert problems can be solved exactly by methods given by Muskhelishvili. As in the related plane problem two singularities at the edge of the disc are noted in the solution.