In order to analyze the optimal dosage of capsule pre-embedded in concrete materials, this article simplified the irregular cracks generated in concrete materials into linear regular hexagonal cracks in two-dimensional plane and planar cracks in three-dimensional space. Then, the probability model describing the interaction between the cracks and capsules was established using the probability theory of integral geometry. The theoretical solutions for the required capsule dosage for the two- and three-dimensional cases were derived. Finally, the calculated results for the theoretical model were compared with those provided by the computer simulation software. The results indicated that the model is suitable for analyzing and optimizing the capsule dosage required to repair cracks in concrete materials. It can provide a theoretical basis for the design of self-healing materials.

中文翻译：

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确定用于修复混凝土裂缝的胶囊剂量的概率分析模型

为了分析混凝土材料中预埋胶囊的最佳用量，本文将混凝土材料中产生的不规则裂缝简化为二维平面的线性规则六边形裂缝和三维空间的平面裂缝。然后，利用积分几何的概率论建立了描述裂纹与胶囊间相互作用的概率模型。推导出了二维和三维情况下所需胶囊剂量的理论解决方案。最后，将理论模型的计算结果与计算机仿真软件提供的结果进行比较。结果表明，该模型适用于分析和优化修复混凝土材料裂缝所需的胶囊剂量。