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This article presents a numerical method for determining the dosage of pre-embedded capsules in self-healing materials with complex crack patterns. The crack distribution on the surface of materials is simplified into a two-dimensional (2D) multi-shaped geometrical structure composed of triangles, rhombuses, and hexagons with specified area fractions, and further decomposed into three separate mono-shaped crack systems. Then, the dosage of capsules required to heal the cracks in each mono-shaped crack system is computed. According to the area fraction of each mono-shaped polygon in the whole system, the integrated models of crack-hitting probability by the capsules and the capsule dosage for the multi-shaped crack system are derived. The analytical results reveal that the dosage of capsules significantly depends on the spatial distribution of the cracks and the ratio of the capsule length to the crack size. For a certain fixed crack pattern, the size and dosage of capsules will strongly affect crack healing efficiency.

Analytical solution on dosage of self-healing capsules in materials with two-dimensional multi-shaped crack patterns