Self-healing materials can typically be divided into two types: intrinsic healing that harnesses dynamic bones to autonomously repair fractures, and extrinsic healing that uses the externally added components to enable the bonding of fractured interfaces. Although the theoretical modeling of intrinsic self-healing materials has been recently studied by Wang et al., the fundamental understanding and theoretical modeling of the extrinsic self-healing materials remain elusive. Without a deep understanding of the extrinsic healing mechanics, the design of extrinsic-healing materials and corresponding applications are still at the trial-and-error stage. Here, taking bacterial-precipitation-enabled healing as an example, we construct a modeling framework to explain the bacteria-assisted extrinsic healing mechanics. A model for the growth of crystal pillars is developed to explain the bacteria-assisted growth of the calcium carbonate (CaCO₃) crystal forest within the fracture interface, and a cohesive zone model is built to explain the interfacial bonding. Our modeling framework can explain the evolution of the interfacial healing strength over the healing time and reveal the effects of interface distance and concentrations of bacteria and calcium ions on the healing performance. The modeling results are consistent with the bacteria-assisted healing experiments of ceramics and cement.

自修复材料通常可以分为两种类型：利用动态骨骼自动修复骨折的内在修复，以及使用外部添加的组件实现骨折界面粘合的外在修复。尽管Wang等人最近研究了固有的自我修复材料的理论模型，但是对外部自我修复材料的基本理解和理论模型仍然难以捉摸。在没有对外在愈合机制的深入了解的情况下，外在愈合材料的设计和相应的应用仍处于试错阶段。在这里，以细菌沉淀促成的愈合为例，我们构建了一个建模框架来解释细菌辅助的外在愈合机制。₃）裂缝界面内的晶体林，并建立了一个内聚区模型来解释界面结合。我们的建模框架可以解释随着愈合时间界面愈合强度的变化，并揭示界面距离以及细菌和钙离子浓度对愈合性能的影响。模拟结果与陶瓷和水泥的细菌辅助愈合实验一致。