Two analytical models for the probability characteristics of a crack hitting encapsulated particles: Application to self-healing materials

Two two-dimensional analytical models are considered for the computation of the probability that a crack hits an encapsulated particle. Most computations are performed in closed algebraic form which allows to perform the last calculation steps numerically with a higher accuracy. The functions built in this paper allow to estimate combinations of crack lengths, capsule size, and mean intercapsule distance in order to analyse the efficiency of a self-healing material. The study is performed in the framework of the investigation of the potential of bacteria to act as a catalyst of the self-healing process in concrete, i.e. their ability to repair occurring cracks autonomously. The self-healing technology that is dealt with in this paper is based on use of spherical clay capsules, containing the healing agent (calcium lactate) and nutrients for bacteria, embedded in the concrete structure. The concepts developed are generic and can be applied to other cases in which encapsulated particles are possibly hit by cracks.