Crack nucleation in an isotropic medium under non-uniform stress field

Abstract

We give a mathematical description of a calculation model for cracking in an isotropic medium under influence of non-uniform stress field. When the isotropic medium is loading by traction load in the material of medium was appear a prefracture zone which is modeled as a zone of weakened interparticle bonds of the material. A model of the pre-fracture zone with bonds between the faces is used. Cracking is assumed as the transition from the pre-fracture zone to zone of the broken bonds between the surfaces of the isotropic medium material. The interaction between the faces of prefracture zone is modeled by bonds between the faces of prefracture zone with given deformation law. Size of the prefracture zone is unknown in advance and determined in the process of problem solving. The equilibrium problem of the prefracture zone (zone of weakened interparticle bonds of material) in an isotropic medium under the influence of non-uniform stress field is reduced to solving a system of two integro-differential equations. Then the integral equations are reduced to a system of nonlinear algebraic equations which is solved by method of successive approximations. Directly from the solution of algebraic systems the tractions in the bonds and disclosure of prefracture zone faces are determined. Criterion of the crack initiation is formulated. The tractions in the bonds between the prefracture zone faces, the size of the pre-fracture zone and the limit external load, at which in the medium a crack is occurrence, are found. Analysis of limit-equilibrium state of the isotropic medium, at which a crack is occurrence, is reduced to the parametric studies of obtained algebraic systems and the criterion of crack appearance with the various laws of bonds deformation, elastic constants of the material and geometric characteristics of the medium.