Self-consistent scheme for toughness homogenization

Considering a semi-infinite planar crack propagating along a plane where the local toughness is a random field, the addressed problem is to compute the effective (or homogeneous and macroscopic) toughness. After a brief introduction to the two regimes — strong and weak pinning — that are expected depending on the system size, a self-consistent homogenization scheme is introduced. It is shown that this scheme allows one to predict not only the mean value but also the standard deviation and even the complete probability distribution function of the toughness. A discussion about the quality of this prediction as compared with direct numerical simulations is proposed.