We examined a two-dimensional mathematical model for the problem of elasticity theory on welded dissimilar elastic half-planes containing rectilinear cracks under the action of mechanical efforts on the shores of a crack. As a consequence, the intensity of stresses in the vicinity of tops of the cracks increases, which significantly affects strength of the body. This may lead to the growth of a crack and to the local destruction of a structure. Such a model represents to some extent a mechanism of destruction of the elements of engineering structures with cracks when the water, contained in them, freezes to ice. It creates normal pressure on the shores of the cracks. Based on the application of the apparatus of singular integral equations (SIE), the problem is reduced to the system of SIE of the first kind on the contours of cracks. We obtained numerical solutions to the corresponding integral equation in particular cases of two welded dissimilar half-planes with one randomly-oriented crack, as well as a two-link irregular crack, which crosses the line of junction when the crack’s shores are exposed to uniformly distributed normal pressure. By employing these solutions, we determined stress intensity coefficients (SIC) at the tops of the crack, which are subsequently used to determine critical values of the normal pressure on the shores of the crack. We built graphic dependences of SIC, which characterize distribution of the intensity of stresses at the tops of a crack, on the angle of crack inclination and elastic characteristics of half-planes. This makes it possible to analyze the intensity of stresses in the vicinity of a crack’s tops depending on the geometrical and mechanical factors, as well as to determine the limit of permissible values of normal pressure on the shores of the crack at which the growth of the crack starts, as well as the local destruction of the body. It is shown that the proper selection of elastic characteristics of the components of welded dissimilar half-planes can help achieve an improvement in the strength of the body in terms of the mechanics of destruction by reducing SIC at the crack’s tops