Bridging micro to macroscale fracture properties in highly heterogeneous brittle solids: weak pinning versus fingering

The effect of strong toughness heterogeneities on the macroscopic failure properties of brittle solids is investigated in the context of planar crack propagation. The basic mechanism at play is that the crack is locally slowed down or even trapped when encountering tougher material. The induced front deformation results in a selection of local toughness values that reflect at larger scale on the material resistance. To unravel this complexity and bridge micro to macroscale in failure of strongly heterogeneous media, we propose a homogenization procedure based on the introduction of two complementary macroscopic properties: An apparent toughness defined from the loading required to make the crack propagate and an effective fracture energy defined from the rate of energy released by unit area of crack advance. The relationship between these homogenized properties and the features of the local toughness map is computed using an iterative perturbation method. This approach is applied to a circular crack pinned by a periodic array of obstacles invariant in the radial direction, which gives rise to two distinct propagation regimes: A weak pinning regime where the crack maintains a stationary shape after reaching an equilibrium position and a fingering regime characterized by the continuous growth of localized regions of the fronts while the other parts remain trapped. Our approach successfully bridges micro to macroscopic failure properties in both cases and illustrates how small scale heterogeneities can drastically affect the overall failure response of brittle solids. On a broader perspective, we believe that our approach can be used as a powerful tool for the rational design of heterogeneous brittle solids and interfaces with tailored failure properties.