The Wiener-Hopf method is used to find a solution to a symmetric problem that gives a criterion for the onset of growth of long shear cracks emanating from a corner point on the boundary of a hole and filled with a plastic material Keywords: shear crack, elastic body, corner point, hole, plastic material Establishing criteria for the growth of shear cracks based on the use of modern approaches of fracture mechanics [10][11][12][13][14] is of great importance for rock mechanics, tectonics, and composite mechanics.In this paper, we use the Wiener-Hopf technique to find the exact solution of a symmetric equilibrium problem for an elastic plane with two rectilinear shear cracks emanating from a corner point and filled with a plastic material. These cracks may also be considered as tectonic fractures with low-strength filler. This solution will further be used to study the growth of long shear cracks from a corner point on the boundary of a hole under plane-strain conditions. We will establish the dependence of the failure stress on the initial crack length, the mode II fracture toughness of the material, and the yield stress of the plastic filler.

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The Wiener-Hopf method is used to find the exact solution to the static symmetric plane problem of elasticity for a homogeneous isotropic plate with a finite-length crack emerging from the point of intersection of two semi-infinite straight slip (dislocation) lines. An expression for the crack-tip stress intensity factor is derived. Crack initiation is described by the Cottrell mechanism. The equilibrium of the crack is analyzed for stability