Analytical Model for the Probability Characteristics of a Crack Penetrating Capsules in Capsule-Based Self-Healing Cementitious Materials
Autonomous crack healing using pre-embedded capsules containing healing agent is becoming a promising approach to restore the strength of damaged structures. In addition to the material properties, the size and volume fraction of capsules influence crack healing in the matrix. Understanding the crack and capsule interaction is critical in the development and design of structures made of capsule-based self-healing materials. Continuing our previous study, in this contribution a more practical rupturing mode of capsules characterizing the rupturing manner of capsules fractured by cracks in cementitious materials is presented, i.e., penetrating mode. With the underlying assumption that a crack penetrating capsules undoubtedly leads to crack healing, geometrical probability theory is employed to develop the quantitative relationship between crack size and capsule size, capsule concentration in capsule-based self-healing virtual cementitious material. Moreover, an analytical expression of probability of a crack penetrating with randomly dispersed capsules is developed in two-dimensional material matrix setup. The influences of the induced rupturing modes of capsules embedded on the self-healing efficiency are analyzed. Much attention is paid to compare the penetrating probability and the hitting probability, in order to assist the designer to make a choice of the optimal rupturing modes of capsules embedded. The accuracy of results of the theoretical model is also compared with Monte-Carlo numerical analysis of crack interacting with capsules. It shows that the developed probability characteristics of a crack interaction with capsules for different rupturing modes is helpful to provide guidelines for designer working with capsule-based self-healing cementitious materials.
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