<p>We consider two problems of wedging by a finite wedge of constant width: along a semi-infinite crack on the boundary of an elastic half plane and along the crack located on the boundary of a thin rigid semi-infinite inclusion in the elastic plane. We take into account both the presence of contact of the crack faces near the crack tip and the action of friction forces in the contact zones. The analytic solutions of the problems are obtained by the Wiener–Hopf method and a simple relationship between them is established. For each problem, we find the size of the contact zone near the crack tip, determine the distributions of contact stresses and stresses acting on the crack extension, and find the stress intensity factors.</p>