Application of Flow Theory Relations for Solving Problems of Steady-State Crack Growth

To represent local displacement fields in the problem of the steady-state growth of a crack, which contains a plate of incompressible material, the strain intensity formula is used in the form of a polynomial of the second degree. The case of plane deformation for an elastoplastic material is considered. The solution is obtained by the method of asymptotic expansions. Numerical analysis is carried out for the first term of the expansion. The aim of the work is the process of obtaining analytical solutions to applied problems of the theory of plasticity: finding the components of stress and strain tensors. The paper considers a variant of the method of asymptotic expansions and its application for the problem of the distribution of the stress-strain state in an elastoplastic specimen with a crack. The method of asymptotic expansions has some advantages over the numerical approach in studying the stress-strain state in the vicinity of a crack. It allows to establish exact quantitative relationships between the radial component, the angle, and the components of the stress and strain tensor. Another advantage of this method is the possibility of compiling the mechanical characteristics of an object at the design stage. A system of differential equations has been developed that contains V-0 and its derivatives up to the third order. An example of stress distribution in the vicinity of a crack tip in a steel sample, obtained in a computer system by a numerical method, is given. The deformation diagram has been constructed for the material steel 40. The research results can be used to construct stress and strain fields in the vicinity of a crack, as well as to predict the further direction of crack development.