The unconventional approach to the plane elasticity problem for a crack near the curved surface is presented. The solution of the problem is considered in the form of the sum of solutions of two auxiliary problems. The first one describes the plane with a crack, whose surfaces are loaded by some unknown self-balanced force p(x). The second problem is dealing with the semi-infinite region with the boundary conditions equal to the difference of boundary conditions of the problem to be sought and the solution of the first problem on the region border. The unknown function p(x) is supposed to be approximated with the sufficient level of accuracy by N order polynomial with complex coefficients. This paper is aimed to determine the critical loads causing the spontaneous growth of cracks. The angles of propagation of the stationary cracks located in the region with a ledge or a cut are found. The influence of length of a crack on the bearing ability of an elastic body with the curved surface is investigated. The effect of a form of the concentrator and orientation of a crack to the fracture load subject to the different combinations of forces acting both on a surface of a crack and at infinity is analysed. The results of this research can be applied for calculation of the durability of thin-walled vessels of pressure, e.g., chemical reactors, in order to ensure their ecological safety.
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