# 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

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**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