Cracking is one of the critical causes of material failure, and early damage management by implanting microcapsules in the matrix is practical. Existing studies have focused on the improvement of self-healing systems and preparation technology of microcapsules', but have paid less attention to the issue of the probability of triggering the self-healing mechanism in microencapsulated self-healing composites (MSC), which in turn is a critical factor in improving the self-healing capability of the MSC. To improve the design capability of the MSC, a mathematical model affecting the intersection of a random single crack with microcapsules in a chaotic state is constructed based on the geometric probability theory. Further, the effects of a single factor on the intersection probability of a crack with microcapsules are studied. The results show that (1) the intersection mathematical probability of a randomly expanding single crack with a microcapsule is simultaneously affected by material shape, crack width, microcapsule volume fraction, and microcapsule diameter. (2) To improve the self-healing ability of the MSC, pay more attention to the mathematical expectation value of the cracks intersecting the microcapsules. (3) Under the conditions of ensuring the mechanical properties of the MSC and microcapsules, increasing the volume fraction and diameter of microcapsules can improve the self-healing efficiency of the MSC.
