Three-dimensional fracture instability of a displacement-weakening planar interface under locally peaked nonuniform loading

•Crack instability on a 3D planar, linear displacement-weakening interface is studied.•The interface is under a locally peaked stress varying quasi-statically in time.•Energy approach with a Rayleigh–Ritz approximation is used for an elliptical crack.•Critical crack size is independent of the curvature of loading stress distribution.•Critical lengths at instability are of the same order for all 2D and 3D cases. We consider stability of fracture on a three-dimensional planar interface subjected to a loading stress that is locally peaked spatially, the level of which increases quasi-statically in time. Similar to the earlier study on the two-dimensional case (Uenishi and Rice, 2003; Rice and Uenishi, 2010), as the loading stress increases, a crack, or a region of displacement discontinuity (opening gap in tension or slip for shear fracture), develops on the interface where the stress is presumed to decrease according to a displacement-weakening constitutive relation. Upon reaching the instability point at which no further quasi-static solution for the extension of the crack on the interface exists, dynamic fracture follows. For the investigation of this instability point, we employ a dimensional analysis as well as an energy approach that gives a Rayleigh–Ritz approximation for the dependence of crack size and maximum displacement discontinuity on the level and quadratic shape of the loading stress distribution. We show that, if the linear displacement-weakening law is applied and the crack may be assumed of an elliptical form, the critical crack size at instability is independent of the curvature of the loading stress distribution and it is of the same order for all two- and three-dimensional cases.