# Computational Modelling for the Effects of Capsular Clustering on Fracture of Encapsulation-Based Self-Healing Concrete Using XFEM and Cohesive Surface Technique

## Abstract

**:**

## Featured Application

**Simple and reliable fracture modelling of capsular clustering in encapsulation-based self-healing concrete.**

## Abstract

## 1. Introduction

## 2. Clustering Pattern/Degree of Clustering

## 3. Modelling Framework

#### 3.1. The eXtended Finite Element Method (XFEM)

_{I}(x) are shape functions, u

_{I}is a nodal displacement vector, H(x) are jump functions to represent the cracks, a

_{I}and b

_{I}

^{α}are nodal vectors of the enriched degree of freedom, and F

_{α}(x) are crack-tip functions.

#### 3.2. Cohesive Surface Technique (CS)

#### 3.3. Traction-Separation Law

_{n}, t

_{s}, and t

_{t}, which represent the normal and shear tractions in two directions, respectively. The corresponding separations are denoted by δ

_{n}, δ

_{s}, and δ

_{t}. The elastic behavior can then be written as:

_{nn}, K

_{ss}, and K

_{tt}, the so-called penalty stiffness, whose value is calculated as a function of the two adjacent material stiffnesses [22]. It has been proved that its value has no influence on the overall sample stiffness [23] and their values have been taken at 1 × 10

^{6}MPa/mm in this study. Furthermore, the normal and shear penalty stiffnesses are assumed to be decoupled, meaning that a pure normal opening of the interface does not produce shear forces, and vice versa [3]. It is assumed that the initial response is linear [17]. However, damage can be represented by various damage evolution laws, such as a linear and nonlinear traction-separation response, once a damage initiation criterion has been met. In this study, the bilinear traction-separation law will be used, see Figure 4.

#### 3.3.1. Damage Initiation (Separation Initiation)

_{maxps}is the calculated maximum principal stress and σ* is the maximum strength of the material. n, s, t: normal, shear, and tangential components of the interfacial tractions, respectively. (*) represents the maximum interfacial tractions. A purely compressive displacement (contact penetration) or a purely compressive stress state does not initiate damage, as indicated by the Macaulay bracket ⟨⟩.

#### 3.3.2. Damage Evolution (Separation Propagation)

_{n}, G

_{S}, and G

_{t}are the energy release rates calculated from the traction and normal, shear, and tangential displacements during interface opening, respectively, and the power α is a cohesive property parameter that describes the interaction between modes. The critical interface toughness for each direction is represented by the properties ${G}_{n}^{*},{G}_{s}^{*},\mathrm{and}{G}_{t}^{*}$. In this paper, it is assumed that the critical fracture toughness is the same in all directions. In addition, taking into account the effect of this parameter on the response, a value α = 1 has been used [24].

#### 3.4. Implementation in Abaqus

- -
- Time increment required is less than the minimum specified.

^{−9}and the minimum time increment is taken to be 1 × 10

^{−15}.

- -
- Too many attempts were made for this increment.

_{A}, which is the maximum number of cutbacks allowed for an increment within the Step module > Other > General Solution Controls > Edit > Step-1, has to be changed to a higher number to allow many numerical attempts, in this thesis it is taken to be 30.

## 4. Computational Simulations

_{f}). For the cohesive surface representing the interaction between the microcapsule and the concrete matrix, the normal and shear fracture properties are assumed to be equal due to the lack of experimental data available in the literature.

#### 4.1. Study the Mesh Size

#### 4.2. Parametric Studies

## 5. Results and Discussion

#### 5.1. Effects of the Capsular Clustering on the Load-Carrying Capacity

#### 5.2. Effects of the Microcapsule Size on the Load-Carrying Capacity

#### 5.3. Effects of the Capsular Clustering and the Microcapsule Size on the Crack Pattern

## 6. Conclusions

- The microcapsule circumferential contact length (Lcs) between the microcapsule shell and the concrete matrix has a significant role in governing the load-carrying capacity of the specimen. The load-carrying capacity of the sample with the microcapsule core/shell ratio 1:1 and Lcs 100% is reduced by 4.6% in the case of Lcs 25%. While the load-carrying capacity of the sample with the microcapsule core/shell ratio 15:1 and Lcs 100% is reduced by 1.1% in the case of Lcs 25%. This shows the higher the Lcs (i.e., the lower the degree of microcapsule clustering), the higher the load-carrying capacity, and vice versa.
- The Lcs also affects the crack pattern; however, it is not recommended that the microcapsule core/shell ratio be 1:1, which has the largest shell thickness but the smallest volume of the healing agent. Consequently, the debonding of the microcapsule from the concrete will happen regardless of the ratio of Lcs. Otherwise, when the Lcs is lower than 25% of the microcapsule circumference, it will result in a greater possibility for the debonding of the microcapsule from the concrete.
- The microcapsule size has an impact on the load-carrying capacity of SHC specimens. The load-carrying capacity for the sample with Lcs 100% and the microcapsule core/shell ratio 1:1 is reduced by 3.8% in the case of the microcapsule core/shell ratio 15:1. While the load-carrying capacity for the sample with Lcs 25% and the microcapsule core/shell ratio 1:1 is reduced by 0.28% in the case of the microcapsule core/shell ratio of 15:1. It is obvious that the lower the core/shell ratio (larger shell thickness) of the microcapsule, the higher the load-carrying capacity of the specimen, and vice versa.
- The microcapsule core/shell ratio has also a significant effect on the crack pattern as it determines whether the microcapsule will fracture, or debond from, the concrete. The greater the core/shell ratio (smaller shell thickness), the greater the likelihood of microcapsules being fractured.
- It can be concluded from studying the effects of the microcapsule circumferential contact length and the microcapsule core/shell ratio on the load-carrying capacity that the microcapsule clustering has a higher effect than the microcapsule size on the load-carrying capacity of the SHC samples.
- It is important not only to take care of microcapsules size, but also the clustering effects during the mixing process of the SHC, to minimize its effects on the contact surface between the microcapsule and the concrete to ensure the breakage of the microcapsules and the subsequent release of the healing agent.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The geometrical representation of the critical capsular clustering patterns ranging from (

**a**) non-clustered to (

**g**) fully clustered.

**Figure 6.**Three meshes with different discretization: (

**a**) Coarse mesh (5219 elements); (

**b**) Medium mesh (16,435 elements); (

**c**) Fine mesh (32,217 elements).

**Figure 8.**Specimens with different capsular circumferential contact length ratios Lcs ratios: (

**a**) Lcs 100% (Lcs = 6.28 mm); (

**b**) Lcs 75% (Lcs = 4.71 mm); (

**c**) Lcs 50% (Lcs = 3.14 mm); (

**d**) Lcs 25% (Lcs = 1.57 mm).

**Figure 9.**Microcapsule samples with different core shell thickness ratio: (

**a**) Ratio 1:1, (

**b**) Ratio 5:1, (

**c**) Ratio 10:1, and (

**d**) Ratio 15:1.

**Figure 10.**Load displacement curves for microcapsule core/shell ratio 1:1 with different Lcs values.

**Figure 11.**Load displacement curves for microcapsule core/shell ratio 5:1 with different Lcs values.

**Figure 12.**Load displacement curves for microcapsule core/shell ratio 10:1 with different Lcs values.

**Figure 13.**Load displacement curves for microcapsule core/shell ratio 15:1 with different Lcs values.

**Figure 14.**Effects of the Lcs ratios to the maximum load with different microcapsule core/shell ratios.

**Figure 15.**Crack pattern of microcapsule ratio 1:1 with different Lcs ratios: (

**a**) Lcs 100%, (

**b**) Lcs 75%, (

**c**) Lcs 50%, and (

**d**) Lcs 25%.

**Figure 16.**Crack pattern of microcapsule ratio 5:1 with different Lcs ratios: (

**a**) Lcs 100%, (

**b**) Lcs 75%, (

**c**) Lcs 50%, and (

**d**) Lcs 25%.

**Figure 17.**Crack pattern of microcapsule ratio 10:1 with different Lcs ratios: (

**a**) Lcs 100%, (

**b**) Lcs 75%, (

**c**) Lcs 50%, and (

**d**) Lcs 25%.

**Figure 18.**Crack pattern of microcapsule ratio 15:1 with different Lcs ratios: (

**a**) Lcs 100%, (

**b**) Lcs 75%, (

**c**) Lcs 50%, and (

**d**) Lcs 25%.

Material | E (MPa) | ʋ | σ* (MPa) | G_{f}(N/mm) |
---|---|---|---|---|

Concrete | 25,000 | 0.2 | 6 | 0.06 |

Capsule | 3600 | 0.3 | 10 | 0.1 |

Interface | - | - | 6 | 0.06 |

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**MDPI and ACS Style**

Hanna, J.
Computational Modelling for the Effects of Capsular Clustering on Fracture of Encapsulation-Based Self-Healing Concrete Using XFEM and Cohesive Surface Technique. *Appl. Sci.* **2022**, *12*, 5112.
https://doi.org/10.3390/app12105112

**AMA Style**

Hanna J.
Computational Modelling for the Effects of Capsular Clustering on Fracture of Encapsulation-Based Self-Healing Concrete Using XFEM and Cohesive Surface Technique. *Applied Sciences*. 2022; 12(10):5112.
https://doi.org/10.3390/app12105112

**Chicago/Turabian Style**

Hanna, John.
2022. "Computational Modelling for the Effects of Capsular Clustering on Fracture of Encapsulation-Based Self-Healing Concrete Using XFEM and Cohesive Surface Technique" *Applied Sciences* 12, no. 10: 5112.
https://doi.org/10.3390/app12105112