A geometrically nonlinear analysis of coplanar crack propagation in some heterogeneous medium

In a recent paper (Leblond et al., 2012), we established, using some results of Rice (1989), the second-order expression of the variation of the mode I stress intensity factor resulting from some small, but otherwise arbitrary coplanar perturbation of the front of a semi-infinite tensile crack in an infinite body. The aim of the present work is to apply the expression found to a geometrically nonlinear analysis of quasistatic, coplanar crack propagation in some heterogeneous medium. In a first step, we recall Leblond et al. (2012)’s formula, extending it to the case where the unperturbed stress intensity factor, for the straight configuration of the front, depends on the position of this front; in addition to being intrinsically interesting, such an extension is necessary in order to avoid meaningless divergent integrals in what follows. In a second step, assuming the local energy-release-rate to be equal everywhere on the crack front to its critical value, we derive an expression of the shape of this front accurate to second order in the fluctuations of toughness of the material. In a third step, as an application, we present a second-order calculation of the equilibrium shape of the crack front, when it penetrates a single infinitely elongated obstacle or a periodic distribution of such obstacles. Special attention is paid to the case, of particular physical interest, where the derivative of the unperturbed stress intensity factor with respect to the position of the crack front can be neglected.