The 3D problem of out-of-plane perturbation of a semi-infinite crack in an infinite body revisited

Abstract

Bueckner–Rice’s theory, in its original form (Rice, 1985; Bueckner, 1987; Rice, 1989), provided the general first-order expression of the variation of the displacement field arising from a small, but otherwise arbitrary tangential perturbation of the front of a crack in a 3D elastic body. This theory was recently extended (Leblond and Lebihain, 2023) to completely arbitrary geometric perturbations of the crack front and surface, including a normal component to the surface. The aim of this paper is to illustrate how the extended theory permits to treat elasticity problems of out-of-plane perturbations of planar cracks, and potentially of normal perturbations of cracks with arbitrary warped surface, in a more direct and simpler way than was previously possible. The principle consists of deriving the first-order expression of the variation of the stress intensity factors along the crack front from some detailed asymptotic study of the variation of the displacement near this front. This method parallels, for normal perturbations of the crack surface, that proposed and applied by Rice (1985); Gao and Rice (1986, 1987a,b); Gao (1988) to the calculation, in a number of crack configurations of practical interest, of the variation of the stress intensity factors resulting from tangential perturbations of the front. It is illustrated here in the simplest case of out-of-plane perturbation of a semi-infinite crack in an infinite 3D body. The results obtained confirm and complete, with a reduced technical effort, those previously derived by Movchan et al. (1998) using a “direct” approach implying a full solution of the complex 3D elasticity problem. Two new applications to problems of crack propagation in mixed-mode are presented as illustrations. The fundamental simplicity of the method, which circumvents the search for a general method of solution of the perturbed elasticity problem by reducing the treatment to finding the limits of some integrals, should permit to envisage next more complex cracked geometries, resembling more those encountered in actual experiments of crack propagation, and previously out of reach of theoretical analyses.