Inclined circular punch with one or two ends in smooth contact with a cracked half plane

An inclined circular punch with one or two ends in smooth contact (incomplete contact) with a half plane with a vertical edge crack is considered. The solution is analytically derived, and the conditions of smooth contact with the edge-cracked half plane are expressed as the restraining conditions. The inclined angle is related to the restraining condition at the smooth end of the punch. Because the restraining conditions belong to nonlinear equations, the problem cannot be simply solved using the method of superposition. Different inclined angles are assumed to iterate the corresponding stress intensity factors of the crack, resultant moment on the contact region and length of the contact region in different cases. The relations among the contact length, resultant moment about the origin, stress intensity factor and inclined angle of the punch are shown in the figures.