Geometrical probability of a capsule hitting irregular crack networks: Application to capsule-based self-healing materials

•Several regular and irregular crack patterns were induced based on the practical cracking appearances of composites•The probabilities of a capsule intersection with irregular crack patterns were determined by an elementary probability method•As an example, the amount of capsules required to be embedded in the matrix was developed from the viewpoint of probability Cracks are vitally detrimental to the load-bearing capacity of materials and further to the durability and service-life of various structures. Autonomous self-healing via embedded capsules or hollow fibres with healing agent has recently become more popular to repair the damage of structural materials. Figuring out the exact amount of capsules required to repair the cracks is indispensable to develop and design the capsule-based self-healing materials. In this paper, by means of surveying the irregular crack patterns appear on the surface of composite materials the approximated cracks models are induced in the self-healing materials and the probability model of capsules which are randomly dispersed intersecting with the crack networks is developed from the viewpoint of geometry probability. Further, the proposed probabilities for different crack patterns are applied to gain the theoretical solutions on the exact dosage of capsules containing healing agent that are used to obtain a specific self-healing efficiency in capsule-based self-healing materials (cementitious materials, for instance). Finally, the accuracies of these probabilities values and theoretical solutions are verified via computer simulation where the event of capsules intersection with different crack networks are reproduced.