Branching of plastic strips near the tip of a notch under longitudinal shear

We solve the problem of a crack of longitudinal shear with a plastic zone, formed by an arbitrary number of strips that go out from the crack tip. Tangential stresses along the strips are equal to the yield limit of the material. The obtained solution shows that stresses exceed the yield limit outside the strips, and this excess decreases with an increase in the number of strips and tends to zero as their number tends to infinity. The lengths of the strips of the zone under study asymptotically approach the lengths of similar stripe, on which the conditions of plasticity are satisfied. We show that, for localization of deformations in a single strip near the tip of a V-shaped notch, it is sufficient that the yield limit of the material along the bisecting plane of the notch be smaller than the yield limit in other directions.