The propagation of a crack located on the interface line of media in a composite piece-homogeneous material with a structure is considered. Under the action of the tangential (shear) stress applied at infinity, the third mode of destruction is realized. The process of destruction of such a composite is described with the help of the modified Leonov-Panasyuk-Dugdale model, which uses an additional parameter-the width of the zone of plasticity (the width of the pre-destruction zone) of the weakest material. A model of an ideal elastoplastic material with a limiting elongation is chosen as the model of a deformable solid. The investigated class of materials include, for example, low-alloy steels used in structures operating at temperatures below the cold-bridging threshold. The case is analyzed in detail when the elastic characteristics of the materials coincide, and the strength characteristics differ substantially. Formulas are obtained in which the final dimensions of the samples are taken into account, for the critical breaking load and the critical length of the pre-destruction zone. A description is given of the construction of a diagram of quasi-brittle and quasi-viscous fracture under longitudinal shear.