# Probability Characteristics of a Crack Hitting Spherical Healing Agent Particles: Application to a Self-Healing Cementitious System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Problem Statement

_{skew}) and the angles (θ

_{skew}), where θ

_{skew}corresponds to the deviation the propagating crack makes relative to its initial angle θ

_{c}from the y-axis. Moreover, the aggregates are not specifically considered. The presence of fine aggregates is assumed to have a minor effect on the capsules’ randomized distribution and the initiation and propagation of surface cracks. As for assessing the effectiveness of the self-healing system, the following measurements are compiled: (1) capsule hit probability, which is the likelihood of a single randomly oriented crack to intersect at least one capsule; (2) depth of first capsule hit, which is a measure of the unhealed crack depth or, in other words, the depth at which a crack will first intersect a capsule and initiate healing at that location; and (3) the fill ratio, which is the ratio between the total volume of the encapsulated healing agent in hit capsules to the total volume of the crack.

#### 2.2. Design of Experiment

_{f}, capsule diameter d, crack width L

_{w}and depth L

_{D}, and length L

_{L}. Accordingly, 43 combinations of factors are considered, in addition to 16 replicates at the center point to allow for a more uniform estimate of the prediction variance over the entire design space. Table 1 provides the levels selected for each factor. The range of crack dimensions reflects typical crack opening and depth of early age microcracks caused by drying [24] or thermal shrinkage [25]. For each combination, 500 numerical simulations are performed wherein the random variables δ

_{i}(perturbation of capsule position), n

_{skew}, θ

_{skew}, and θ

_{c}are varied within their corresponding range, given in Table 2 and based on a uniformly distributed random distribution. Typical capsule properties, also given in Table 2 [10,12,26], are adopted in this numerical experiment. The measured responses per simulation are as follows: (1) the number of capsules hit by the crack; (2) the depth of the first capsule hit by the crack; and (3) the total volume of capsules hit by the crack.

#### 2.3. Geometric Model

#### 2.3.1. Capsule Distribution

_{x}or d

_{z}) and vertical (d

_{y}) directions, as illustrated in Figure 2d. Perturbation of each capsule is assumed to be independent of other capsules, with the position being determined as follows:

#### 2.3.2. Crack Generation

_{c}between 0° (parallel to y-axis) and 45° from the vertical, with an angle θ

_{skew}between zigzag crack segments and the overall crack propagation direction, as illustrated in Figure 1. For ease of computation, crack width is assumed to be constant, i.e., not tapered along the depth in the x–y plane direction and along the length in the z-direction.

#### 2.3.3. Agglomeration

_{agg}is the total number fraction of capsules to be agglomerated and m

_{f}is the original mass fraction of capsules with respect to cement. Constants A, B, and C are assumed to be A = 1, B = 20, and C = 0.1 to produce an agglomeration curve that reflects capsule clustering trend observed in experimental studies [14]. Total agglomeration is limited to a maximum of 80%, i.e., at least 20% mass fraction of capsules are not in agglomerated clusters.

_{i}is the number of agglomerates of size i, and n

_{0}is the original number of non-agglomerated capsules. κ is a constant determined by rearranging Equation (3) via

#### 2.4. Statistical Model

#### 2.4.1. Capsule Hit Probability ${P}_{h}$

#### 2.4.2. First Hit Depth ${h}_{0}$

#### 2.4.3. Crack Fill Ratio ${R}_{f}$

_{0}is the total volume of healing agent released into the crack.

#### 2.5. Regression Analysis

^{2}value as well as the relative error distribution.

^{2}of the estimators.

#### 2.5.1. Hit Probability ${P}_{h}$

_{1}x

_{4}as a potential candidate, a final non-linear expression is selected based on simplicity and distribution of errors:

^{2}values and the maximum p-value of coefficients are 0.94 and 10

^{−5}, respectively, for model-P

_{h}. Table 3 summarizes the regression results of the model.

_{h}in terms of fit with numerical simulation results, residual error, and relative error distributions. By examining the results, the following conclusions are deduced: the regression model predictions have a maximum residual error less than ±0.100 except for two data points that correspond to extreme variables value (i.e., level ±2.378), and the model R

^{2}value is 0.95.

_{h}results when values of x

_{i}(i = 1 to 4) are out of the typical range specified by levels [−1, +1] in Table 1. Figure 8 presents the model relative errors at extreme values. The results reveal a higher residual error for mass fraction (${x}_{1}$) and crack depth (${x}_{4}$) when their values are outside [−1, +1] levels, while variations in capsule diameter (${x}_{2}$) and crack width (${x}_{3}$) result in a residual error typical of the model estimate. Accordingly, model-P

_{h}should be limited to $0.02<{x}_{1}<0.08$ and $20\mathrm{mm}{x}_{4}100\mathrm{mm}$ for mass fraction and early age crack depths, respectively.

#### 2.5.2. First Hit Depth ${h}_{0-95}$

^{2}values and the maximum p-value of coefficients are 0.91 and 0.01, respectively, for model-H

_{0}. The majority of the residual errors are less than 0.01 for points within levels [−1, +1], as shown in Figure 9a,c. A larger residual error and relative error are observed for extreme values of x

_{2}and x

_{4}at level ±2.378, notably for small diameters (${x}_{2}=0.124\mathrm{mm}$) and short cracks (${x}_{4}=2.432\mathrm{mm}$), as illustrated in Figure 9b–d. Table 4 summarizes the regression results of model-H

_{0}.

_{0}for hit probability presented in Equation (14), model-H

_{0}for hit depth ${h}_{0-95}$ has a higher residual error $\Delta {h}_{0-95}$ when estimating the results at extreme input values (level ±2.378). The sensitivity of $\Delta {h}_{0-95}$ to variables ${x}_{1}$ to ${x}_{4}$ is illustrated in Figure 10. It is evident that $\Delta {h}_{0-95}$ for higher mass fractions or small size capsules significantly exceeds the typical $\Delta {h}_{0-95}$ range of level [−1, +1] points. Accordingly, it is not recommended to use model-H

_{0}represented by Equation (15) when mass fraction ${x}_{1}>8\%$ and capsule diameter ${x}_{2}>0.3\mathrm{mm}$.

#### 2.5.3. Crack Fill Ratio ${R}_{\mathrm{f}-95}$

^{2}= 0.93 is obtained and given by the following:

_{f}is presented in Table 5. Figure 11 presents the distribution of errors associated with the model-R

_{f}regression model. It should be noted that the distribution of errors in Figure 11 does not show two extreme points at level ±2.378 for high mass fractions and small size capsules.

_{f}. The residual error $\Delta {R}_{f-95}$ is not as sensitive to larger variations in other variables; however, caution should be used in situations of smaller crack depths. $\Delta {R}_{f-95}$ for very large or very narrow crack widths significantly exceeds the typical $\Delta {R}_{f-95}$ range of level [−1, +1] points. As such, it is not recommended to use model-R

_{f}for very short and long cracks lying outside the noted size range of ($0.1<{x}_{4}<0.5$). When adjusting a single variable while holding other variables constant, the value of ${R}_{f-95}$ tends to increase with the mass fraction and diameter of capsules (Figure 13a,b). As one may expect, the healing effectiveness decreases as the crack becomes wider (Figure 13c). The influence of crack depth on ${R}_{f-95}$ can be considered negligible, as shown in Figure 13d.

- To achieve the desired ${R}_{f-95}$ with a given capsule size ${x}_{2}$, coefficient values for ${x}_{1}$ and ${x}_{3}$ show that the mass fraction of capsules must be increased to heal wider cracks. This is in agreement with the experimental observations [32].
- For a desired ${R}_{f-95}$ and fixed capsule mass fraction m
_{f}, coefficient values for ${x}_{2}$ and ${x}_{3}$ show that large size capsules must be used to heal wider cracks.

## 3. Regression Models’ Evaluation

_{D}= 30–70 mm), capsule diameter ($d=0.4-1.0\mathrm{mm}$), and mass fraction of capsules (m

_{f}= 1–10%); meanwhile, the probability functions developed by Zemskov et al. [19] were for capsules of size $d=2.0\mathrm{mm}$ and crack depths L

_{D}= 0.05–8.5 mm. The results from Lv and Chen [17] correspond $d=2.0\mathrm{mm}$ and L

_{D}= 0.05–8.5 mm. Given the difference in the range of variables and the assumptions about the cracks, it is not reasonable to compare the results directly. As such, a qualitative/semi-quantitative evaluation is carried out to examine the general trends as well as the similarities and differences in the results.

_{f}< 5% (or m

_{f}< 7%), such as in Figure 15c, ${P}_{h}$ increases as v

_{f}increases. However, for the case of v

_{f}> 15% in Figure 15d, ${P}_{h}$ tends to decrease slightly as v

_{f}increases, likely owing to the effects of capsule agglomeration, which become more pronounced at a high mass fraction according to Equation (2).

_{f}< 7%, the ${P}_{h}=F\left({L}_{D},{m}_{f}\right)$ contours in Figure 16a,b have the same trend of variation as that in Figure 16c; in particular, the value of ${P}_{h}$ increases with L

_{D}and m

_{f}. Figure 16a,b show that, at any mass fraction in the range 5% < m

_{f}< 7%, a long crack will hit at least one capsule. For lower mass fractions of capsules, the maximum value of ${P}_{h}$ may only be reached at certain capsule diameters.

_{f}> 7%, which show a deviation from the results of Zemskov et al. [19]. Agglomeration causes capsule clustering and reduces the number density of capsules in the mix, which in turn reduces the probability of a crack hitting a capsule. The agglomeration effect becomes more pronounced with a continual increase in m

_{f}. This trend of ${P}_{h}$ variation is clearly demonstrated in Figure 16a,b.

## 4. Conclusions

- The proposed framework has captured the observations previously reported in the literature, including the effect of capsule size and dosage and crack opening on hit probability, filling ratio, and hit depth.
- The 95% confidence level adopted in this study is recommended for the design of a self-healing system as it reduces the uncertainties in the design and significantly increases the efficacy of a healing system.
- Agglomeration with an increasing dosage of capsules reduces hit probability while increasing the crack fill volume. Further addition of capsules past the noted threshold of 7% mass fraction yields adverse effects on hit probability. This shows that agglomeration effects are an important factor that must be considered.
- Crack tortuosity increases the potential intersection region and results in a higher number of capsules intersected.
- Irregular cracks have a larger crack volume compared with a straight crack of the same depth, resulting in an overall increase in fill ratio.
- Higher crack tortuosity slightly increases the uncertainty in the expected fill ratio.

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Three-dimensional (3D) visualization of cross section planes with respect to crack placement. (

**a**) View of crack plane, (

**b**) View of y–z plane with projection of crack; (

**c**) View of x–y plane; (

**d**) Schematic diagram of structured random distribution on x–y plane.

**Figure 5.**Characteristics of a crack related to the calculation of crack fill ratio. (

**a**) View of crack plane; (

**b**) View of y–z plane with projection of crack; (

**c**) Definition of ${A}_{crack}^{x-y}$ and ${A}_{crack}^{y-z}$.

**Figure 6.**Frequency distribution (x-axis, number of occurrence) of fill ratio (y-axis, %) for different combinations of variable levels.

**Figure 7.**Comparison of regression models for P

_{h}. (

**a**) Regression and Monte-Carlo simulation results; (

**b**) residual error; (

**c**) relative error; (

**d**) relative error against residual error; and (

**e**) relative error distribution for P

_{h}.

**Figure 8.**Influence of individual factors on residual errors of model-P

_{h}. The results of extreme cases at Level ±2.378 are connected by a red line.

**Figure 9.**Comparison of different regression models for: (

**a**) regression and Monte-Carlo simulation results; (

**b**) regression and Monte-Carlo simulation results for level ±2.378 simulations; (

**c**) residual error; (

**d**) relative error; and (

**e**) residual error distribution.

**Figure 10.**Influence of individual factors on residual errors of model-H

_{0}. The results of extreme cases at Level ±2.378 are connected by a red line.

**Figure 11.**(

**a**) Regression and Monte-Carlo simulation results with all data points displayed; (

**b**) regression and Monte-Carlo simulation results with extreme mass fraction and diameter removed from the plot; and (

**c**) residual error distribution for model-R

_{f}.

**Figure 12.**Influence of individual factors on residual errors for ${R}_{f-95}$. The results of extreme cases at Level ±2.378 are connected by a red line.

**Figure 14.**Comparison of the simulation results obtained from numerical modelling to those from theoretical data (Zhang and Qian [21]).

**Figure 16.**Contour plot of hitting probability ${P}_{h}$ as a function of L

_{D}and m

_{f}for (

**a**) $d=0.2\mathrm{mm},{L}_{w}=0.3\mathrm{mm}$ and (

**b**) $d=0.6\mathrm{mm},{L}_{w}=0.3\mathrm{mm}$; (

**c**) Hitting probability as a function of normalized crack depth and volume fraction of capsules, for a cube of side length equivalent to 1cm containing 27 capsules, with V

_{capsule}/V

_{cube}varying from 0.1–0.3 [19].

**Figure 17.**Effect of capsule agglomeration and crack geometry on the variation in fill ratio. (

**a**) Comparison of variances between numerical study and simulation; (

**b**) Uncertainty in data with different crack geometry; (

**c**) Effect of crack geometry on average fill ratio.

Level | |||||
---|---|---|---|---|---|

Variable | −2.378 | −1 | 0 | 1 | 2.378 |

Capsule Properties | |||||

Mass fraction, m_{f} (%) | 0.243% | 3% | 5% | 7% | 9.757% |

Diameter, d (mm) | 0.024 | 0.3 | 0.5 | 0.7 | 0.976 |

Crack Properties | |||||

Crack width, L_{W} (mm) | 0.062 | 0.2 | 0.3 | 0.4 | 0.538 |

Crack depth, L_{D} (mm) | 2.432 | 30 | 50 | 70 | 97.568 |

Crack length, L_{L} (mm) | 6.216 | 20 | 30 | 40 | 53.784 |

Variable | Value |
---|---|

Material Properties | |

Cement density | ρ_{cement} = 3150 kg/m^{3} |

Water-to-cement ratio | 0.5 |

Capsule core material density (DCPD) | ρ_{core} = 980 kg/m^{3} [27] |

Shell material density (urea-formaldehyde) | ρ_{shell} = 1170 kg/m^{3} [28] |

Shell thickness | t_{shell} = 1 µm |

Domain Properties | |

Width of sample area | L_{x} = 150 mm |

Depth of sample area | L_{y} = 100 mm |

Length of sample area | L_{z} = 150 mm |

Perturbation of capsule position | ${\delta}_{i}=\left[-{d}_{i}/2,{d}_{i}/2\right],i=x,y$ |

Crack Properties | |

Angle from vertical (y-axis) | θ_{c} range = [−π/4, +π/4] |

Skewness (angle of zigzag segments) | θ_{skew} range = [0, π/4] |

Number of segments of zigzag | n_{skew} range = [0, 10] |

Variables | Coefficient | Value | Standard Error | t-Ratio | p-Value |
---|---|---|---|---|---|

- | a_{0} | −0.3632 | 0.0626 | −5.8006 | 0 |

$100{x}_{1}$ | a_{1} | 0.1966 | 0.0155 | 12.6209 | 0 |

${x}_{2}$ | a_{2} | −0.2362 | 0.0368 | −6.4090 | 0 |

${x}_{3}$ | a_{3} | 0.3714 | 0.0737 | 5.0382 | 10^{−5} |

${x}_{4}$ | a_{4} | 0.0179 | 0.0015 | 11.4808 | 0 |

${x}_{1}^{2}$ | b_{11} | −0.0138 | 0.0015 | −9.1103 | 0 |

${x}_{4}^{2}$ | b_{44} | −0.0001 | 0.0000 | −7.3440 | 0 |

Variables | Coefficient | Value | Standard Error | t-Ratio | p-Value |
---|---|---|---|---|---|

${x}_{4}$ | a_{4} | 0.0274 | 0.0014 | 18.8052 | 0 |

${x}_{1}{x}_{2}$ | b_{12} | 0.0844 | 0.0208 | 4.0569 | 0.00017 |

${x}_{1}{x}_{4}$ | b_{14} | −0.0018 | 0.0002 | −7.2618 | 0 |

${x}_{2}^{2}$ | b_{22} | −0.4373 | 0.1475 | −2.9636 | 0.00458 |

${x}_{2}{x}_{4}$ | b_{24} | 0.0070 | 0.0026 | 2.6739 | 0.01 |

${x}_{3}{x}_{4}$ | b_{34} | −0.0106 | 0.0024 | −4.4041 | 0.00005 |

${x}_{4}^{2}$ | b_{44} | −0.0001 | 0.0001 | −3.3545 | 0.00149 |

Variables | Coefficient | Value | Standard Error | t-Ratio | p-Value |
---|---|---|---|---|---|

- | a_{0} | 0.01661 | 3.944 × 10^{−3} | 4.2124 | 0.00010 |

x_{2} | a_{2} | 0.00829 | 8.098 × 10^{−4} | 10.2392 | 0.00000 |

x_{3} | a_{3} | −0.00774 | 1.633 × 10^{−3} | −4.7371 | 0.00002 |

x_{1}x_{3} | b_{13} | −0.00774 | 1.633 × 10^{−3} | −4.7371 | 0.00002 |

x_{2}x_{2} | b_{22} | 0.01566 | 5.432 × 10^{−3} | 2.8822 | 0.00573 |

x_{2}x_{3} | b_{23} | −0.10013 | 2.000 × 10^{−2} | −5.0072 | 0.00001 |

x_{3}x_{3} | b_{33} | 0.26245 | 3.353 × 10^{−2} | 7.8267 | 0.00000 |

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## Share and Cite

**MDPI and ACS Style**

Guo, S.; Chidiac, S.E.
Probability Characteristics of a Crack Hitting Spherical Healing Agent Particles: Application to a Self-Healing Cementitious System. *Materials* **2022**, *15*, 7355.
https://doi.org/10.3390/ma15207355

**AMA Style**

Guo S, Chidiac SE.
Probability Characteristics of a Crack Hitting Spherical Healing Agent Particles: Application to a Self-Healing Cementitious System. *Materials*. 2022; 15(20):7355.
https://doi.org/10.3390/ma15207355

**Chicago/Turabian Style**

Guo, Shannon, and Samir E. Chidiac.
2022. "Probability Characteristics of a Crack Hitting Spherical Healing Agent Particles: Application to a Self-Healing Cementitious System" *Materials* 15, no. 20: 7355.
https://doi.org/10.3390/ma15207355