A Normally Loaded Half-plane with an Edge Crack

The study of brittle or fatigue fracture that involves contact along the crack faces is difficult for several reasons. On the one hand, material data pertaining to the growth of cracks in a compressive environment are scant, and, on the other, frictional stresses may be transmitted across the contacting crack faces. The analysis of cracks under these conditions is complicated. Nevertheless, solutions to some idealized problems involving a layer-substrate geometry[l-41 or a subsurface crack parallel to the free surface[S] have been presented. In the present study, we examine the case of a crack perpendicular to the surface of the half-plane. This geometry is motivated by the following application. When a component suffers impact loading, little wear occurs other than by some surface degradation in the form of surface cracks[6]. These may form anywhere within the contact zone. However, it is found that if the load is imposed by a punch of high rigidity, or if there is a small additional shear present[7], the crack tends to grow in a plane delimiting the extent of loading. In the present calculation, only cracks at this location will be considered, since this reduces the number of independent variables, and the severest state of orthogonal shear is found here. Figure I shows the four possible cases that might obtain. If there is a stick zone, it is found that this will always be attached to the upper end a of the crack. Note that cases IIa and IIb are similar, since if the crack is locked at the upper end, it is immaterial whether the incision extends to the free surface or stops short. It also transpires that the normal traction N(x) transmitted across the crack is always negative, i.e. that the crack is always closed, and hence the slip zones may be modeled in terms of continuous distributions of glide dislocations alone.