STRESS-STRAIN STATE OF A PRESTRESSED HALFPLANE WITH CRACK IN MODE I NEAR THE SURFACE

Abstract

An analytic-numerical method of investigation of the plane fracture mechanics problem for a semi-bounded body containing a near-surface crack of mode I parallel to the boundary surface, taking into account the action of initial (residual) stresses directed along the crack, is proposed. The method is based on the relations of three-dimensional linearized mechanics of deformable bodies. Using representations of general solutions of linearized equilibrium equations in terms of potential harmonic functions and applying the Fourier integral transformation, the formulated boundary value problem is reduced first to dual integral equations and then to a system of inhomogeneous Fredholm integral equations of the second kind. From the analysis of the asymptotic stress distribution in the vicinity of the crack it is concluded that the order of singularity in the stress distribution near the crack tips in the considered problem coincides with the order of singularity obtained in the plane problem for a halfplane with a mode I crack in the absence of initial stresses, and analytical expressions for the stress intensity factors have been obtained. For a highly elastic (hyperelastic) body, the material of which is described by the Treloar elastic potential (a body of neo-Hookian type), the dependences of the stress intensity coefficients on the initial (residual) stresses are calculated and the effect of the interaction between the crack and the boundary surface of the body on them has been estimated. A resonant increase in the values of stress intensity coefficients when the initial compressive stresses reach certain critical values, which for a given material correspond to a local loss of stability of the equilibrium state in the vicinity of the crack, has also been found.