Fracture probability of a randomly oriented microcrack under multi-axial loading for the normal tensile stress criterion

Microcracks distribute randomly in a solid with respect to their size, shape, spatial location and orientation. This makes it attractive from the angle of micromechanics to evaluate the fracture probability of a randomly oriented microcrack under arbitrary loading in order to improve the physical and mechanistic understanding of statistical approaches to brittle fracture. The fracture probability is largely affected by the specific microscopic fracture criteria adopted. This work presents a closed-form solution to the fracture probability of a randomly oriented microcrack subjected to arbitrary multi-axial loading under the normal tensile stress criterion. The set theory is adopted to derive this solution, with the expectation that it can be also used for other more complex fracture criteria.