The distribution characteristics of microcapsules incorporated into matrix can influence the quantity of microcapsules required for embedding in the cementitious matrix, thereby determining the self-healing efficiency. Self-healing systems are essentially passive systems that autonomously initiate the healing process when cracks occur. The efficacy of such systems necessitates both adequate distribution and sufficient bonding of the capsules with the matrix. The recent preparation of microcapsules in self-healing composites presents significant challenges in achieving optimal self-healing effects. One of the key obstacles is the aggregation and uneven distribution of capsules within the matrix, which can significantly impair the healing efficiency of microcapsule self-healing cementitious materials. This nonuniform distribution diminishes the effectiveness of the capsules and adversely affects properties like early mechanical strength. Furthermore, the uneven distribution of capsules in the matrix will lead to the reduction of the number of capsules encountered by cracks. It is therefore essential to ensure the uniform distribution of capsules within the matrix in order to maximise the probability of microcapsule rupture and self-healing capacity.

In order to avoid agglomeration of microcapsules, a uniformed distribution model for microcapsules inspired by the structural form of a honeybee's hive is designed in self-healing cementitious materials and some embedded microcapsules are dispersed within the honeycomb. This honeycomb model ensures the uniform distribution of microcapsules inside the matrix and prevent the aggregation of microcapsules, thereby achieving a design strategy for the uniform distribution of microcapsules in self-healing cementitious materials.

The probabilistic characteristics of a planar crack that appears inside a composite matrix and intersects with embedded microcapsules is investigated from the viewpoint of geometrical probability. The probability of a randomly occurring planar crack in the matrix when it encounters a spherical or spherocylindrical microcapsule is derived, and an intersection probability formula is obtained. Furthermore, the influences of capsule dimensions and quantities on the intersection probability is examined in capsule-based self-healing composites. In addition, comparative analysis is conducted on microencapsulated composite matrix models with varying shapes. Finally, the results of the theoretical model are validated using Monte Carlo simulation.

The honeycomb-like composite structure is a three-dimensional structure composed of several small cubes (cells), with a certain number of spherical or spherocylindrical microcapsules randomly dispersed within the small cubes, ensuring the uniform distribution of the capsules and avoiding the aggregation of microcapsules. When a random planar crack appears within the composite, the crack will meet some microcapsules and causes the ones to rupture releasing a healing agent to heal the crack, which is necessary for self-healing to work. Three-dimensional probabilistic analytical models are developed for the spherical microcapsules and spherocylindrical microcapsules in order to represent the intersection of a planar crack with randomly dispersed microcapsules, respectively.

The findings indicate that uniformly dispersed microcapsules can increase the probability of capsule rupture and improve the self-healing efficiency. The developed three-dimensional geometric probability model of a planar crack intersecting with randomly distributed microcapsules in the matrix cells is close to a binomial probability distribution. The use of spherocylindrical microcapsules can increase the healing potential of the same volume fraction of spherical microcapsules if the microcapsules are correctly oriented. The theoretical probability values are verified by comparison with the results obtained independently by computer programming simulations.

The proposed theoretical formulas quantitatively express the effect of the size and dosage of different-shaped microcapsules within the matrix on the probability of a planar crack hitting microcapsules and to predict the number of microcapsules that are most likely to intersect. If the volume fraction (or size) of microcapsules is given and a planar crack requires a fixed number of microcapsules to heal, the maximum intersection probability can be calculated from the theoretical equation to determine the optimum size (or volume fraction) of microcapsules to be mixed within the model. The results of this research will help to provide a theoretical basis for the quantitative design of self-healing materials based on microcapsules of different shapes. Furthermore, employing 3D printing technology could be a feasible approach to realize the design strategy for the uniform distribution of microcapsules within self-healing cementitious materials.