A Mathematical Model for Bacterial Self-healing of Cracks in Concrete

Self-healing materials are smart materials with the capability to repair itself autonomously after being damaged. Since cracks commonly occur in structures consisting of materials like concrete, polymers, etc., a self-healing mechanism is crucial for the durability of the material. In this research, we present a two-dimensional mathematical model of the self-healing process in concrete. This model can help to estimate the influence of various parameters involved on the rate and quality of the crack healing and to find conditions such that healing proceeds in an optimal way, with a minimal influence on other material properties, such as strength, ductility etc. Bacteria-based self-healing concrete is produced by adding a two-component biochemical self-healing agent to the concrete mixture. The first component, spores of bacteria of a special kind (Bacillus cohnii, Bacillus pseudofirmus), is incorporated in the concrete matrix at the stage of preparation of the cement paste by mixing the spore suspension with the paste make up water. The second component, the biochemical healing agent, is immobilized in expanded clay capsules prior to addition to the concrete mixture. The homogeneously distributed clay particles will, after setting of the concrete mixture, release the encapsulated healing agent when hit and opened by a crack forming in the concrete matrix. Water entering the crack subsequently disperses the chemicals over the crack surface and activates the bacterial spores which germinate to become metabolically active vegetative cells that are able to convert the organic bio-mineral precursor compound into insoluble inorganic calciumcarbonate-based minerals. Copious production of bacteria-mediated minerals results in sealing off the crack, hence the permeability of concrete and leakage are reduced. The capsule content is a solid mixture of three components which dissolve in water that fills the crack. About 97% of the mass of the capsule content is calcium lactate (Ca(C3H5O3)2) which is converted by bacteria on the crack surface into calcium carbonate (limestone) (CaCO3). Two other components, monosodium glutamate (C5H8NNaO4) and monopotassium phosphate (KH2PO4), are, respectively, sources of nitrogen and phosphorus which are necessary nutrients for bacteria in order to be able to multiply and to carry out the conversion process. Since clay capsules with the healing agent have a lower strength than the surrounding concrete matrix, they influence the process of crack propagation. As the crack approaches the capsule close enough, it may change its direction to intersect the weakest material parts, and, therefore, the crack will pass through the capsule. Hence, the capsule “attracts” the crack, and in the most typical case the crack passes through the centre area of the capsule. Serguey V. Zemskov, Henk M. Jonkers and Fred J. Vermolen Because of this observation, the following geometry of a computational domain for the mathematical model was chosen in order to build a two-dimensional mathematical model of the self-healing process (Fig. 1 (a) and (b)).