The initiation of a longitudinal shear crack (mode III fracture) in homogeneous structures and weld joints of elastoplastic materials with the limiting strain is considered. To obtain critical fracture parameters, a coupled (sufficient) fracture criterion in an elastoplastic material is used. Simple relations for critical shear stresses, lengths of pre-fracture zones and critical stress intensity factor (SIF) for mode III fracture were obtained. There is a limiting transition from a sufficient fracture criterion to the necessary criterion, when the length of the prefracture zone tends to zero. Critical stresses obtained according to necessary and sufficient criteria differ significantly. In the framework of the proposed model, the critical SIF obtained by a sufficient criterion is a variable, it depends on the grain diameter, the parameters of the material strain diagram and the crack length. The rule of selection of the parameters of the proposed fracture model is indicated by two laboratory experiments: by approximating the material strain diagram with a simple shear and critical SIF. The obtained critical parameters are conservative estimates of critical stresses. These estimates should be used in the calculation of steel structures operating at temperatures below the cold brittleness.