Two Tangential Forces and a Penny-Shaped Crack: A Complete Solution

The following problem is considered: A penny-shaped crack is located in the plane z = 0 of a transversely isotropic elastic space and interacts with two equal tangential forces, acting in the same direction, which are located arbitrarily, but symmetrically with respect to the plane of the crack. An exact closed-form solution is obtained here. Some of the stresses and displacements in the whole space are expressed in terms of elementary functions. Two methods of solution are considered. The first method considers superposition of an uncracked space subjected to two forces and a space weakened by a crack and subjected to a specially defined loading. The second method reduces the problem to that of a half-space, one force is acting inside this half-space, and its surface is loaded by a pressure, providing zero displacements outside the crack. It is shown that both methods give the same results.