The initiation of a longitudinal shear crack (mode III fracture) in elastoplastic materials with the limiting strain is considered. The crack propagation criterion is formulated using a modified Leonov–Panasyuk–Dugdale model using an additional parameter—the width of the plasticity zone (the width of the pre-fracture zone). Conditions of a small-scale yielding in the presence of a stress field singularity in the vicinity of the crack tip are considered. A two-parameter criterion for quasi-brittle fracture of an elastoplastic material is formulated for mode III cracks. The proposed fracture criterion includes the deformation-based criterion, which is formulated at the crack tip, as well as the force-based criterion, formulated at the tip of a model crack. The lengths of the initial and model cracks differ by the length of the pre-fracture zone. The application of the proposed strength criterion to the determination of fracture loads for bodies containing longitudinal shear cracks is demonstrated. The diagrams of quasi-brittle fracture are constructed for a strip of finite width with an edge crack in case of the out-of-plane deformation. It is proposed to determine the model parameters using an idealized simple shear diagram and the critical stress intensity factor. For quasi-ductile and ductile types of fracture, the limiting loads are determined numerically. The propagation of plastic zones in the vicinity of the crack tip under quasistatic loading is successively described by the nonlinear finite element method. It is shown that the shapes of the numerically simulated plastic zones differ significantly from the known classical concepts.