Evaluation of the basic formulations for the cumulative probability of brittle fracture with two different spatial distributions of microcracks

The random distribution of microcracks in terms of their size, shape, orientation and spatial location has direct impact on the cumulative probability of brittle fracture induced failure, with the effect of spatial distribution being rarely explored. Recently, two weakest link theory‐based formulations for the cumulative probability of brittle fracture induced failure have been proposed for the spatial distribution of microcracks obeying the Poisson postulates and the uniform distribution, respectively. This work compares these two new formulations with the currently commonly adopted one built on the Poisson postulates under both the uniform and the non‐uniform uniaxial loading conditions. It is concluded that under general loading conditions involving non‐uniform stress states, the existing formulation is equivalent to or closely approximate to neither of the two new formulations thus should be discarded, because of its inaccurate derivation. The new formulations are featured with unique symmetry or self‐similarity in their expressions. Their capability in revealing the size effect or the scaling law of failure is highlighted and validated by a set of published uniaxial and biaxial flexural strength data of brittle material.