Influence of heterogeneity due to toughness variations on weakest-link modeling for brittle failure

Abstract

The effect of heterogeneous microstructures on the macroscopic probability of failure is studied by use of weakest-link modeling. Heterogeneity is here associated with a local variation of toughness, where a size scale characteristic of this variation defines a length parameter. The ratio between this length parameter and the size of the active fracture process zone, defined as the heterogeneity ratio, is key to evaluating the impact of a heterogeneous microstructure. Two extremes are identified; small-scale heterogeneity (SSH) and large-scale heterogeneity (LSH). For these cases, it is possible to formulate analytical expressions based on the weakest-link concept, and references are made to existing models in the literature. Typically, heterogeneity along the crack front, where gradients of the mechanical fields are small, falls under the category of SSH. On the other hand, the effect of heterogeneity in a plane perpendicular to the crack front depends strongly on the heterogeneity ratio. Cases that can neither be identified with SSH nor LSH must be addressed with care. How this can be done is discussed, and examples are given for four different microstructure configurations of interest. The investigation is carried out by numerical analysis of a modified boundary layer model. The cumulative probability of failure by cleavage fracture is evaluated in a post-processing step, where two different statistical models are examined; the Beremin model and the Kroon–Faleskog model. Both models render the same conclusion about the alteration of the overall failure probability distributions caused by heterogeneity.