# Global Sensitivity Analysis for the Polymeric Microcapsules in Self-Healing Cementitious Composites

^{*}

## Abstract

**:**

## 1. Introduction

## 2. A Micromechanical Model for the Microcapsule-Contained Cementitious Composite

#### 2.1. Multilevel Homogenization Scheme for Predicting the Effective Properties

#### 2.2. The First-Level Homogenization

_{r}, can be expressed as [41]

_{eq}, E

_{1}and E

_{2}denote Young’s moduli of the equivalent particle, the shell and the healing agent, respectively. Further, v

_{eq}, v

_{1}and v

_{2}mean the Poisson’s ratios of the equivalent particle, the shell and the healing agent, respectively.

_{eq}can be obtained by [42]

_{1}and f

_{2}are the volume fraction of the shell and healing agents, respectively.

#### 2.3. The Second-Level Homogenization

^{(r)}and

**C**

^{(r)}are the volume fraction and the elastic stiffness tensor of r-th phase, respectively. 0-th represents the intrinsic concrete.

**I**denotes the fourth-order identity tensor. ${T}_{MT}^{(r)}$ and ${R}^{(r)}$ are two tensors relative to interfacial properties, and can be calculated by [27]

**S**

^{MD}is the modified Eshelby tensor and can be obtained by the Direct Computation method [27]

## 3. Global Sensitivity Analysis Method

_{i}is great. The main process of EFAST is summarized as follows:

_{1}, x

_{2},…, x

_{n}), with parameters in the domain of unit hypercube

_{i}is a search-curve. There are many forms of x

_{i}. Here, we take the transformation proposed by Saltelli et al. [43]

_{i}is a set of different, linearly independent of integer frequencies associated with each factor x

_{i}. s varies in (−π/2, π/2). By using Fourier transform, the first-order sensitivity index ${\widehat{S}}_{i}$ can be obtained [43]

## 4. Results and Discussion

_{1}, v

_{1}and k) have a medium influence on the outputs. They make up approximately 20% altogether. The interfacial sliding property ɑ only takes up about 0.42% of the influence, which can be neglected. The result was acceptable since the interfacial sliding parameter has little influence on the bulk modulus as illustrated in previous studies [27].

_{1}> v

_{1}> k, while PRCC values yielded a slightly different order k > v

_{1}> E

_{1}. The other rankings were the same. These results support that the results of EFAST are correct, and PRCC provides a similar identification of sensitive parameters.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Baron, R.I.; Bercea, M.; Avadanei, M.; Lisa, G.; Biliuta, G.; Coseri, S. Green route for the fabrication of self-healable hydrogels based on tricarboxy cellulose and poly(vinyl alcohol). Int. J. Biol. Macromol.
**2019**, 123, 744–751. [Google Scholar] [CrossRef] - Bercea, M.; Biliuta, G.; Avadanei, M.; Baron, R.I.; Butnaru, M.; Coseri, S. Self-healing hydrogels of oxidized pullulan and poly(vinyl alcohol). Carbohyd. Polym.
**2019**, 206, 210–219. [Google Scholar] [CrossRef] - Morariu, S.; Bercea, M.; Gradinaru, L.M.; Rosca, I.; Avadanei, M. Versatile poly (vinyl alcohol)/clay physical hydrogels with tailorable structure as potential candidates for wound healing applications. Mat. Sci. Eng. C
**2020**, 109, 110395. [Google Scholar] [CrossRef] - Nita, L.E.; Chiriac, A.; Rusu, A.G.; Bercea, M.; Ghilan, A.; Dumitriu, R.; Mititelu-Tartau, L. New self-healing hydrogels based on reversible physical interactions and their potential applications. Eur. Polym. J.
**2019**, 118, 176–185. [Google Scholar] [CrossRef] - Rusu, A.; Nita, L.E.; Bercea, M.; Tudorachi, N.; Diaconu, A.; Pamfil, D.; Rusu, D.; Ivan, F.E.; Chiriac, A. Interpenetrated polymer network with modified chitosan in composition and self-healing properties. Int. J. Biol. Macromol.
**2019**, 132, 374–384. [Google Scholar] [CrossRef] [PubMed] - White, S.R.; Sottos, N.; Geubelle, P.; Moore, J.; Kessler, M.R.; Sriram, S.; Brown, E.; Viswanathan, S. Autonomic healing of polymer composites. Nature
**2001**, 409, 794–797. [Google Scholar] [CrossRef] [PubMed] - Shahabudin, N.; Yahya, R.; Gan, S.N. Microcapsules filled with a palm oil-based alkyd as healing agent for epoxy matrix. Polymers
**2016**, 8, 125. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lee, J.; Park, S.J.; Park, C.S.; Kwon, O.S.; Chung, S.Y.; Shim, J.; Lee, C.S.; Bae, J. Effect of a surfactant in microcapsule synthesis on self-healing behavior of capsule embedded polymeric films. Polymers
**2018**, 10, 675. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Yang, H.J.; Mo, Q.F.; Li, W.Z.; Gu, F.M. Preparation and properties of self-healing and self-lubricating epoxy coatings with polyurethane microcapsules containing bifunctional linseed oil. Polymers
**2019**, 11, 1578. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zheng, N.; Liu, J.; Li, W. TO/TMMP-TMTGE double-healing composite containing a transesterification reversible matrix and tung oil-loaded microcapsules for active self-healing. Polymers
**2019**, 11, 1127. [Google Scholar] [CrossRef] [Green Version] - Kim, S.; Kim, B.H.; Oh, M.; Park, D.H.; Lee, S. Repeatable crack self-healing by photochemical [2+2] cycloaddition of TCE-co-DCE monomers enclosed in homopolymer microcapsules. Polymers
**2019**, 11, 104. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Jeoung, H.J.; Kim, K.W.; Chang, Y.J.; Jung, Y.C.; Ku, H.; Oh, K.W.; Choi, H.M.; Chung, J.W. Self-healing EPDM rubbers with highly stable and mechanically-enhanced urea-formaldehyde (UF) microcapsules prepared by multi-step in situ polymerization. Polymers
**2020**, 12, 1918. [Google Scholar] [CrossRef] [PubMed] - Yang, Z.X.; Hollar, J.; He, X.D.; Shi, X.M. A self-healing cementitious composite using oil core/silica gel shell microcapsules. Cem. Concr. Comp.
**2011**, 33, 506–512. [Google Scholar] [CrossRef] - Zhuang, X.; Zhou, S. The prediction of self-healing capacity of bacteria-based concrete using machine learning approaches. CMC-Comput. Mater. Con.
**2019**, 1, 57–77. [Google Scholar] [CrossRef] [Green Version] - Zhuang, X.; Zhou, S. Molecular dynamics study of an amorphous polyethylene/silica interface with shear tests. Materials
**2018**, 11, 929. [Google Scholar] [CrossRef] [Green Version] - Zhou, S.; Vu-Bac, N.; Arash, B.; Zhu, H.; Zhuang, X. Interface characterization between polyethylene/ silica in engineered cementitious composites by molecular dynamics simulation. Molecules
**2019**, 24, 1497. [Google Scholar] [CrossRef] [Green Version] - Zhou, S.; Xie, L.; Jia, Y.; Wang, C. Review of cementitious composites containing polyethylene fibers as repairing materials. Polymers
**2020**, 12, 2624. [Google Scholar] [CrossRef] - Zhou, S.; Zhuang, X. Characterization of loading rate effects on the interactions between crack growth and inclusions in cementitious material. CMC-Comput. Mater. Contin.
**2018**, 57, 417–466. [Google Scholar] [CrossRef] - Zhou, S.; Zhuang, X. Micromechanical study of loading rate effects between a hole and a crack. Undergr. Space
**2019**, 4, 22–30. [Google Scholar] [CrossRef] - Zhou, S.; Zhu, H.; Ju, J.; Yan, Z.; Chen, Q. Modeling microcapsule-enabled self-healing cementitious composite materials using discrete element method. Int. J. Damage Mech.
**2017**, 26, 340–357. [Google Scholar] [CrossRef] - Zhou, S.; Zhu, H.; Yan, Z.; Ju, J.; Zhang, L. A micromechanical study of the breakage mechanism of microcapsules in concrete using PFC2D. Constr. Build. Mater.
**2016**, 115, 452–463. [Google Scholar] [CrossRef] - Zhu, H.; Zhou, S.; Yan, Z.; Ju, J.W.; Chen, Q. A two-dimensional micromechanical damage-healing model on microcrack-induced damage for microcapsule-enabled self-healing cementitious composites under compressive loading. Int. J. Damage Mech.
**2016**, 25, 727–749. [Google Scholar] [CrossRef] - Zhu, H.; Zhou, S.; Yan, Z.; Ju, J.W.; Chen, Q. A two-dimensional micromechanical damage-healing model for microcapsule-enabled self-healing cementitious composites under tensile loading. Int. J. Damage Mech.
**2015**, 24, 95–115. [Google Scholar] [CrossRef] - Zhu, H.; Zhou, S.; Yan, Z.; Ju, J.W.; Chen, Q. A 3D analytical model for the probabilistic characteristics of self-healing model for concrete using spherical microcapsule. Comput. Concr.
**2015**, 15, 37–54. [Google Scholar] [CrossRef] - Yuan, K.Y.; Ju, J.W.; Yuan, W.; Yang, J.M. Numerical predictions of mechanical behavior of innovative pothole patching materials featuring high toughness, low-viscosity nano-molecular resins. Acta Mech.
**2014**, 225, 1141–1151. [Google Scholar] [CrossRef] - Zhu, H.; Chen, Q.; Yan, Z.; Ju, J.W.; Zhou, S. Micromechanical models for saturated concrete repaired by the electrochemical deposition method. Mater. Struct.
**2014**, 47, 1067–1082. [Google Scholar] [CrossRef] - Yanase, K.; Ju, J.W. Effective elastic moduli of spherical particle reinforced composites containing imperfect interfaces. Int. J. Damage Mech.
**2012**, 21, 97–127. [Google Scholar] [CrossRef] - Li, C.; Wang, J.; Dou, F. An estimation approach for the effective elastic modulus of lightweight bulk filling material with compressible inclusions and imperfect interfaces. Materials
**2020**, 13, 3563. [Google Scholar] [CrossRef] - Chen, Q.; Zhu, H.H.; Ju, J.W.; Yan, Z.G.; Jiang, Z.W.; Chen, B.; Wang, Y.Q.; Fan, Z.H. Stochastic micromechanical predictions for the probabilistic behavior of saturated concrete repaired by the electrochemical deposition method. Int. J. Damage Mech.
**2020**, 29, 435–453. [Google Scholar] [CrossRef] - Chen, X.; Li, R.; Sun, L.Z. Dynamic magneto-viscoelastic model for magnetorheological nanocomposites with imperfect interface. Int. J. Damage Mech.
**2019**, 28, 1248–1260. [Google Scholar] [CrossRef] - Zhang, Y.; Ju, J.W.; Zhu, H.H.; Yan, Z.G. A novel multi-scale model for predicting the thermal damage of hybrid fiber-reinforced concrete. Int. J. Damage Mech.
**2020**, 29, 19–44. [Google Scholar] [CrossRef] - Chen, Q.; Zhu, H.H.; Ju, J.W.; Li, H.X.; Jiang, Z.W.; Yan, Z.G. Stochastic micromechanics-based investigations for the damage healing of unsaturated concrete using electrochemical deposition method. Int. J. Damage Mech.
**2020**, 29, 1361–1378. [Google Scholar] [CrossRef] - Shodja, H.M.; Hashemian, B. Variational bounds and overall shear modulus of nano-composites with interfacial damage in anti-plane couple stress elasticity. Int. J. Damage Mech.
**2020**, 29, 246–271. [Google Scholar] [CrossRef] - Hong, S.X.; Qin, S.F.; Dong, B.Q.; Xing, F. Corrosion features of the reinforcing bar in concrete with intelligent OH- regulation of microcapsules. Materials
**2019**, 12, 3966. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zuo, J.D.; Li, H.B.; Zhan, J.; Dong, B.Q.; Wang, L.; Chen, D.Z. Preparation and properties of cement mortar/metal hydroxide microcapsules composites. Cem. Concr. Comp.
**2020**, 105, 103438. [Google Scholar] [CrossRef] - Confalonieri, R. Monte Carlo based sensitivity analysis of two crop simulators and considerations on model balance. Eur. J. Agron.
**2010**, 33, 89–93. [Google Scholar] [CrossRef] - Marino, S.; Hogue, I.B.; Ray, C.J.; Kirschner, D.E. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol.
**2008**, 254, 178–196. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Donckels, B.M.R.; Kroll, S.; Van Dorpe, M.; Weemaes, M. Global sensitivity analysis of an in-sewer process model for the study of sulfide-induced corrosion of concrete. Water Sci. Technol.
**2014**, 69, 647–655. [Google Scholar] [CrossRef] [Green Version] - Zhao, J.; Tiede, C. Using a variance-based sensitivity analysis for analyzing the relation between measurements and unknown parameters of a physical model. Nonlinear Proc. Geoph.
**2011**, 18, 269–276. [Google Scholar] [CrossRef] [Green Version] - Saltelli, A.; Marivoet, J. Non-parametric statistics in sensitivity analysis for model output: A comparison of selected techniques. Reliab. Eng. Syst. Safe.
**1990**, 28, 229–253. [Google Scholar] [CrossRef] - Timoshenko, S.P.; Goodier, J.N.; Abramson, H.N. Theory of elasticity. J. Appl. Mech.
**1970**, 37, 888. [Google Scholar] [CrossRef] - Ahmed, S.; Jones, F. A review of particulate reinforcement theories for polymer composites. J. Mater. Sci.
**1990**, 25, 4933–4942. [Google Scholar] [CrossRef] - Saltelli, A.; Tarantola, S.; Chan, K.S. A quantitative model-independent method for global sensitivity analysis of model output. Technometrics
**1999**, 41, 39–56. [Google Scholar] [CrossRef] - Saltelli, A. Sensitivity analysis for importance assessment. Risk Anal.
**2002**, 22, 579–590. [Google Scholar] [CrossRef] [PubMed] - Shin, M.J.; Guillaume, J.H.; Croke, B.F.; Jakeman, A.J. Addressing ten questions about conceptual rainfall–runoff models with global sensitivity analyses in R. J. Hydrol.
**2013**, 503, 135–152. [Google Scholar] [CrossRef]

**Figure 2.**The homogenization process: (

**a**) the first level: the homogenization of the shell and healing agents inside, and (

**b**) the second level: the homogenization of the intrinsic concrete, the equivalent inclusion and interfaces.

**Figure 5.**First-order sensitivity indices (FSIs) computed by the Extended Fourier Amplitude Sensitivity Test (EFAST) for the bulk modulus in the C30 concrete.

**Figure 6.**FSIs computed by the EFAST sensitivity analysis for the shear modulus in the C30 concrete.

**Figure 7.**The Partial Rank Correlation Coefficient (PRCC) sensitivity analysis for the (

**a**) bulk modulus and (

**b**) shear modulus in the C30 concrete.

**Figure 8.**FSIs computed by the EFAST sensitivity analysis for the bulk modulus of the (

**a**) C30 concrete, (

**b**) the C40 concrete, and (

**c**) the C50 concrete.

**Figure 9.**FSI values using an (

**a**) original and (

**b**) adjusted range of the volume fraction of microcapsules for the C30 concretes with the bulk modulus objective function.

Parameter | Description | Unit | Scope |
---|---|---|---|

E_{1} | Elastic modulus of the shell | GPa | (1, 10) |

v_{1} | Poisson’s ratio of the shell | - | (0.001, 0.499) |

k | The core-shell ratio | - | (0.1, 0.9) |

ɑ | The interfacial sliding compliance | 1/MPa | (0.001, 0.01) |

β | The interfacial separation compliance | 1/MPa | (0.001, 0.01) |

f | The volume fraction of microcapsules | - | (1%, 10%) |

Type | Elastic Modulus | Poisson’s Ratio |
---|---|---|

C30 | 30 GPa | 0.2 |

C40 | 32.5 GPa | 0.2 |

C50 | 34.5 GPa | 0.2 |

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**MDPI and ACS Style**

Zhou, S.; Jia, Y.; Wang, C.
Global Sensitivity Analysis for the Polymeric Microcapsules in Self-Healing Cementitious Composites. *Polymers* **2020**, *12*, 2990.
https://doi.org/10.3390/polym12122990

**AMA Style**

Zhou S, Jia Y, Wang C.
Global Sensitivity Analysis for the Polymeric Microcapsules in Self-Healing Cementitious Composites. *Polymers*. 2020; 12(12):2990.
https://doi.org/10.3390/polym12122990

**Chicago/Turabian Style**

Zhou, Shuai, Yue Jia, and Chong Wang.
2020. "Global Sensitivity Analysis for the Polymeric Microcapsules in Self-Healing Cementitious Composites" *Polymers* 12, no. 12: 2990.
https://doi.org/10.3390/polym12122990