Analytical models for determining the dosage of capsules embedded in self-healing materials

Crack-repairing technology via embedded capsules containing repair-agent is becoming a promising approach to sustain the performance of structure materials. From the viewpoint of geometric probability and the application of concept of integral geometry, this contribution aims to develop and determine the theoretical solution on dosage of capsules required to repair the cracks. Based on a general fact that the capsules are randomly dispersed in the matrix and the discrete cracks occur independently in matrix, we present the probability of capsules hit by the cracks to develop the theoretical solution on dosage of capsules for self-healing technology via embedded capsules in two- and three-dimensional self-healing system. Then, according to the targeted healing level, a philosophy proposed to fulfill the healing expectations from the probabilistic (risk-based) healing approach, the volume fraction of capsules required in the matrix is determined. At the same time, under the assumption that the healing capacity of a capsule is sufficient to repair the crack meeting the capsule, it shows that the volume fraction of capsules required is opposite to the size of cracks as the cracks grow. Also the influence of the elongated capsules fabricated with different aspect ratios on the hitting probability is discussed and it shows that for discrete cracks mode the hitting probability of elongated capsules with aspect ratio is not always larger than that of spherical capsules. The hitting probability referred the size, shape and amount of capsules and the geometric characterizations of cracks would be helpful for designing self-healing materials with pre-embedded capsules. Finally, computer simulation is employed to verify the reliability of these theoretical models.