A propagation of an interface crack in a piecewise-homogenous composite square plate is considered. Under the action of tangential (shear) stress, applied at the edges of the plate, the mode II fracture occurs. The case when elastic properties of materials coincide and strength properties are substantially different is analyzed in detail. The spreading of plastic zones in square plates made of bimetal is simulated numerically for quasi-static loading. In the numerical model, the Lagrangian formulation of the solid mechanics is used, as the most preferable formulation, concerning the modeling of finite strain elasto-plasticity of solids. It is found that the results of the numerical simulation are in a good agreement with the predictions by the analytical model for fracture under transverse shear in structured materials.