We perform calculation of the initial prefracture zone at the tip of a mode I crack that reaches a nonsmooth interface of two dissimilar elastic media at its corner point by the Wiener–Hopf method. The zone is modeled by a line of normal displacement fracture on a crack continuation. Expressions for the length of the prefracture zone and the potential energy accumulated in it are obtained. Their numerical values are compared with the corresponding values for the prefracture zone in a bonding material on the interface of the media, on the basis of which we draw a conclusion on the possible direction of development of the zone. The problem of an interface crack that lies on the interface of two dissimilar media has been extensively studied for a long time [14–17]. At the same time, the class of problems of a crack reaching the interface of a medium was given much less attention, though this problem is no less important for the mechanics of composite materials and adhesive and welded joints than the problem of an interface crack. The existing solutions were predominantly obtained for a crack that reaches a plane interface [1, 8, 18], whereas the problems of a crack with a tip at a corner point of the nonsmooth boundary of a joint, which is a stress concentrator with a power singularity, is of obvious interest [4]. Under loading, the stress concentration at the tip of a crack, at the corner point of the interface of a medium, leads to the formation of a prefracture zone in the vicinity of the crack. This zone affects substantially the stress-strain state in the region most dangerous from the viewpoint of further development of the crack and explains the necessity to investigate it in detail. In [3, 5–7], the initial stages of development of the prefracture zones localized on the interface of two media in the binding material were studied within the framework of models with lines of discontinuity of displacements. In particular, in [6], a calculation of the prefracture zone at the tip of a mode I crack that, under the conditions of plane deformation, reaches the nonsmooth interface of two dissimilar elastic homogeneous isotropic media at its corner point was performed under the assumption that the material that binds the parts of the body into a single whole is less crack-resistant than the material of the bound parts. However, if the condition of the smaller crack resistance of the binding material is discarded, another mechanism of formation of the prefracture zone is possible, namely, under a tensile normal stress, on a continuation of the crack, a prefracture zone can form in this direction. In what follows, we investigate the initial stage of the nucleation and development of such a prefracture zone.