Geometrical probability of a capsule hitting irregular crack networks: application to capsule-based self-healing materials. (English) Zbl 1481.74667

Summary: Cracks are vitally detrimental to the load-bearing capacity of materials and further to the durability and service-life of various structures. Autonomous self-healing via embedded capsules or hollow fibres with healing agent has recently become more popular to repair the damage of structural materials. Figuring out the exact amount of capsules required to repair the cracks is indispensable to develop and design the capsule-based self-healing materials. In this paper, by means of surveying the irregular crack patterns appear on the surface of composite materials the approximated cracks models are induced in the self-healing materials and the probability model of capsules which are randomly dispersed intersecting with the crack networks is developed from the viewpoint of geometry probability. Further, the proposed probabilities for different crack patterns are applied to gain the theoretical solutions on the exact dosage of capsules containing healing agent that are used to obtain a specific self-healing efficiency in capsule-based self-healing materials (cementitious materials, for instance). Finally, the accuracies of these probabilities values and theoretical solutions are verified via computer simulation where the event of capsules intersection with different crack networks are reproduced.


74R10 Brittle fracture
74A40 Random materials and composite materials
60D05 Geometric probability and stochastic geometry
Full Text: DOI


[1] White, S. R.; Sottos, N. R.; Geubelle, P. H.; Moore, J. S.; Kessler, M. R.; Sriram, S. R.; Brown, E. N.; Viswanathan, S., Autonomic healing of polymer composites, Nature, 409, 794-797 (2001)
[2] Zhang, W.; Zheng, Q. F.; Ashour, A.; Han, B. G., Self-healing cement concrete composites for resilient infrastructures: A review, Compos. Part B: Eng., 189, Article 107892 pp. (2020)
[3] Ghosh, S. K., Self-Healing Materials: Fundamentals, Design Strategies, and Applications (2009), Wiley-VCH: Wiley-VCH Weinheim, Germany
[4] Pang, J. W.C.; Bond, I. P., A hollow fibre reinforced polymer composite encompassing self-healing and enhanced damage visibility, Compos. Sci. Technol., 65, 1791-1799 (2005)
[5] De Nardi, C.; Gardner, D.; Jefferson, A. D., Development of 3D printed networks in self-healing concrete, Materials, 13, 1328 (2020)
[6] van Tittelboom, K.; de Belie, N.; van Loo, D.; Jacobs, P., Self-healing efficiency of cementitious materials containing tubular capsules filled with healing agent, Cem. Concr. Compos., 33, 497-505 (2011)
[7] Jonkers, H. M.; Schlangen, E., Self-healing of cracked concrete: A bacterial approach, (Carpenteri, A.; Gambarova, P. G.; Ferro, G.; Plizzari, G. A., FRACOS6: Fracture Mechanics of Concrete and Concrete Structures (2007), Taylor & Francis/Balkema: Taylor & Francis/Balkema Catania, Italy), 1821-1826
[8] Jonkers, M.; Thijssen, A.; Muyzer, G.; Copuroglu, O.; Schlangen, E., Application of bacteria as self-healing agent for the development of sustainable concrete, Ecol. Eng., 36, 230-235 (2010)
[9] Osada, T.; Wataru, N.; Takahaski, K.; Ando, K., Self-crack-healing behavior in ceramic matrix composites, Advances in Ceramic Matrix Composites, 515-544 (2014), Elsevier
[10] Song, H.; Wang, Z. J.; He, X. D.; Duan, J., Self-healing of damage inside metals triggered by electropulsing stimuli, Sci. Rep., 7, 7097 (2017)
[11] Thomas, S.; Surendran, A., Self-Healing Polymer-Based Systems (2020), Elsevier: Elsevier Amsterdam
[12] Zhang, M. Q.; Rong, M. Z., Self-Healing Polymers and Polymer Composites (2011), John Wiley & Sons: John Wiley & Sons New Jersey
[13] Zhu, D. Y.; Rong, M. Z.; Zhang, M. Q., Self-healing polymeric materials based on microencapsulated healing agents: From design to preparation, Prog. Polym. Sci., 49-50, 175-220 (2015)
[14] Wang, X. F.; Yang, Z. H.; Fang, C.; Han, N. X.; Z, G. M.; Tang, J. N.; Xing, F., Evaluation of the mechanical performance recovery of self-healing cementitious materials – its methods and future development: A review, Constr. Build. Mater., 212, 400-421 (2019)
[15] Lv, Z.; Chen, H. S., Autonomous healing of cracks in cementitious materials - a short review, J. Chin. Chem. Soc., 42, 156-168 (2014)
[16] Pang, J. W.C.; Bond, I. P., Bleeding composites - damage detection and self-repair using a biomimetic approach, Compos. Part A - Appl S, 36, 183-188 (2005)
[17] Trask, R. S.; Williams, G. J.; Bond, I. P., Bioinspired self-healing of advanced composite structures using hollow glass fibres, J. R. Soc. Interface, 4, 363-371 (2007)
[18] Ahmed, A.; Sanada, K.; Fanni, M.; El-Moneim, A., A practical methodology for modeling and verification of self-healing microcapsules-based composites elasticity, Compos. Struct., 184, 1092-1098 (2018)
[19] Lv, Z.; Chen, H. S., A probabilistic method for determining the volume fraction of pre-embedded capsules in self-healing materials, Smart Mater. Struct., 23, Article 115009 pp. (2014)
[20] Mookhoek, S. D.; Fisher, H. R.; van der Zwaag, S., A numerical study into the effects of elongated capsules on the healing efficiency of liquid based systems, Comp. Mater. Sci., 47, 506-511 (2009)
[21] Rule, J. D.; Sottos, N. R.; White, S. R., Effect of microcapsule size on the performance of self-healing polymers, Polymer, 48, 3520-3529 (2007)
[22] Lv, Z.; Li, S. P.; Chen, H. S., Analytical model for effects of capsule shape on the healing efficiency in self-healing materials, PLoS One, 12, Article e0187299 pp. (2017)
[23] Lv, Z.; Chen, H. S., Analytical models for determining the dosage of capsules embedded in self-healing materials, Comp. Mater. Sci., 68, 81-89 (2013)
[24] Lv, Z.; Chen, H. S.; Yuan, H. F., Quantitative solution on dosage of repair-agent for healing of 3D simplified cracks in materials: short capsule model, Mater. Struct., 44, 987-995 (2011)
[25] Lv, Z.; Chen, H. S.; Yuan, H. F., Quantitative solution on dosage of repair agent for healing of cracks in materials: short capsule model vs. two-dimensional crack pattern, Sci. Eng. Compos. Mater., 18, 13-19 (2011)
[26] Katoueizadeh, E.; Zebarjad, S. M.; Janghorban, K., A practical analytic model for predicting the performance of an encapsulated polymer composite, Appl Math Model, 78, 418-432 (2020) · Zbl 07193087
[27] Zemskov, S. V.; Jonkers, H. M.; Vermolen, F. J., Two analytical models for the probability characteristics of a crack hitting encapsulated particles: application to self-healing materials, Comp. Mater. Sci., 50, 3323-3333 (2011)
[28] Zemskov, S. V.; Jonkers, H. M.; Vermolen, F. J., An analytical model for the probability characteristics of a crack hitting an encapsulated self-healing agent in concrete, (Gerdt, V. P., Computer Algebra in Scientific Computing - Lecture Notes in Computer Science (2010), Springer-Verlag: Springer-Verlag Berlin), 280-292 · Zbl 1290.74012
[29] Fang, X. R.; Pan, Z. C.; Chen, A. R., Analytical models to estimate efficiency of capsule-based self-healing cementitious materials considering effect of capsule shell thickness, Constr. Build. Mater., 274, Article 121999 pp. (2021)
[30] Kanellopoulos, A.; Giannaros, P.; Al-Tabbaa, A., The effect of varying volume fraction of microcapsules on fresh, mechanical and self-healing properties of mortars, Constr. Build. Mater., 122, 577-593 (2016)
[31] Quayum, M. S.; Zhuang, X.; Rabczuk, T., Computational model generation and RVE design of self-healing concrete, Front. Struct. Civ. Eng., 9, 383-396 (2015)
[32] Davies, R.; Jefferson, A. D., Micromechanical modelling of self-healing cementitious materials, Int. J. Solids Struct., 113-114, 180-191 (2017)
[33] Lv, Z.; Chen, H. S.; Yuan, H. F., Quantitative solution on dosage of repair agent for healing of cracks in materials: short capsule model vs. two-dimensional crack pattern, Sci. Eng. Compos. Mater., 18, 13-19 (2011)
[34] Lv, Z.; Chen, H. S.; Yuan, H. F., Analytical solution on dosage of self-healing agents in cementitious materials: long capsule model, J. Intell. Mater. Syst. Struct., 25, 47-57 (2014)
[35] Yuan, H. F.; Chen, H. S., Quantitative solution of size and dosage of capsules for self-healing of cracks in cementitious composites, Comput. Concr., 11, 223-236 (2013)
[36] Lin, J. J.; Chen, H. S.; Lv, Z.; Wang, Y. J., Analytical solution on dosage of self-healing capsules in materials with two-dimensional Vmulti-shaped crack patterns, Sci. Eng. Compos. Mater., 25, 1229-1239 (2018)
[37] Zemskov, S. V.; Jonkers, H. M.; Vermolen, F. J., A mathematical model for bacterial self-healing of cracks in concrete, J. Intell. Mater. Syst. Struct., 25, 4-12 (2012)
[38] Gardner, D.; Jefferson, A. D.; Hoffman, A.; Lark, R., Simulation of the capillary flow of an autonomic healing agent in discrete cracks in cementitious materials, Cem. Concr. Res., 58, 35-44 (2014)
[39] Solomon, H., Geometric Probability, Society for Industrial and Applied Mathematics (1978), Philadelphia · Zbl 0382.60016
[40] Mobasher, B.; Peled, A.; Pahilajani, J., Distributed cracking and stiffness degradation in fabric cement composites, Mater. Struct., 39, 317-331 (2006)
[41] Bisschop, J., Drying Shrinkage Microcracking in Cement-Based Materials (2002), Delft University of Technology: Delft University of Technology Delft
[42] Mendenhall, W.; Beaver, R. J.; Beaver, B. M., Introduction to Probability and Statistics (2006), Duxbury: Duxbury Belmont
[43] Jefferson, T.; Javierre, E.; Freeman, B.; Zaoui, A.; Koenders, E.; Ferrara, L., Research progress on numerical models for self-healing cementitious materials, Adv. Mater. Interfaces, 5, Article 1701378 pp. (2018)
[44] Zhu, H. H.; Zhou, S.; Yan, Z. G.; Ju, W.; Chen, Q., A 3D analytical model for the probabilistic characteristics of self-healing model for concrete using spherical microcapsule, Comput. Concr., 15, 37-54 (2015)
[45] Mookhoek, S. D., Novel Routes to Liquid-Based Self-Healing Polymer System (2010), Delft University of Technology: Delft University of Technology Delft
[46] Pelletier, M. M., Self-Healing Concrete (2010), University of Rhode Island: University of Rhode Island Kingston
[47] Huang, H. L.; Ye, G., Application of sodium silicate solution as self-healing agent in cementitious materials, (Leung, C.; Wan, K. T., International RILEM Conference on Advances in Construction Materials Through Science and Engineering. International RILEM Conference on Advances in Construction Materials Through Science and Engineering, Hong Kong, China (2011), RILEM Publications SARL), 530-536
[48] Stroeven, P.; He, H.; Guo, Z. Q.; Stroeven, M., Particle packing in a model concrete at different levels of the microstructure: Evidence of an intrinsic patchy nature, Mater. Charact., 60, 1088-1092 (2009)
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