# Numerical Analysis of Microcrack Propagation Characteristics and Influencing Factors of Serrated Structural Plane

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## Abstract

**:**

## 1. Introduction

## 2. Establishment of the Numerical Calculation Model

## 3. Crack Evolution Rule in the Shear Process of the Structural Plane

#### 3.1. The Crack Development Rule

#### 3.2. The Evolution of Microcracks and Contact Force of Structure Plane

## 4. Effect of Micro-Parameters on Shear Mechanical Behavior

#### 4.1. Parallel Bond Stiffness Ratio ${k}_{n}/{k}_{s}$

#### 4.2. Particle Contact Stiffness Ratio ${k}_{n}{}^{\ast}/{k}_{s}{}^{\ast}$

#### 4.3. Particle Contact Modulus $E$

#### 4.4. Parallel Bond Modulus ${E}^{\ast}$

#### 4.5. Effects of Micro-Parameters on Shear Strength of the Structural Plane

## 5. Quantitative Analysis of Micro-Parameters Based on Machine Learning

^{2}) as the criterion, the algorithm with the best fitting effect was selected to calculate the weight share of each parameter and analyze the degrees of influence of micro-parameters to provide a theoretical reference for related studies.

#### 5.1. Extreme Gradient Boosting Decision Tree (XGBoost)

#### 5.2. Particle Swarm Optimization (PSO) and Whale Optimization Algorithm (WOA)

#### 5.3. Model Validation and Analysis

^{2}of 0.759, and WOA-XGBoost has the highest prediction accuracy and the best agreement between the predicted data and test data with an R

^{2}of 0.902. PSO-XGBoost had a prediction accuracy between those of XGBoost and WOA-XGBoost with an R

^{2}of 0.847. WOA-XGBoost achieved a very good fit between the predicted data and the test data for shear strength. Therefore, the WOA-XGBoost model was selected as the prediction model in this study, and the weight share of each micro-parameter was calculated based on the weight calculation module of the algorithm, and the results are shown in Figure 14.

## 6. Conclusions

- (1)
- The relationships among the number of microcracks, the normal stress and the shear stress during the shearing process of the serrated structural plane were analyzed. The results show that the greater the normal stress, the higher the number of microcracks. Before the shear stress reached its peak, the number of cracks increased slowly; after the shear stress reached its peak, the degrees of crack sprouting and expansion increased, the number of cracks kept increasing and the damage accumulated. Eventually, when the shear stress reached the residual shear strength, the number of microcracks tended to stabilize.
- (2)
- During the shearing process, the contact force distribution on the serrated surface has similarity with the evolutionary law of microcracks. In the initial stage of shearing, the contact force chain is mainly concentrated at the root of the sawtooth. As the shear progresses, the contact force chain becomes more and more concentrated, and the force on the root of the sawtooth becomes larger and larger. When the failure finally occurred, a large number of cracks appeared at the root of the sawtooth.
- (3)
- The four meso-parameters (particle contact modulus $E$, particle contact stiffness ratio ${k}_{n}{}^{\ast}/{k}_{s}{}^{\ast}$, parallel bond modulus ${E}^{\ast}$, and parallel bond stiffness ratio ${k}_{n}/{k}_{s}$) have certain influences on the shear stress–shear displacement curve, shear strength and peak shear displacement. The particle contact stiffness ratio ${k}_{n}{}^{\ast}/{k}_{s}{}^{\ast}$ and parallel bond stiffness ratio ${k}_{n}/{k}_{s}$ are negatively correlated with the shear strength; and the particle contact modulus $E$ and parallel bond modulus ${E}^{\ast}$ are positively correlated with the shear strength. The peak shear displacement gradually decreases as the particle contact modulus $E$ and parallel bond modulus ${E}^{\ast}$ increase. The parallel bond stiffness ratio ${k}_{n}/{k}_{s}$ was negatively correlated with the peak shear displacement to some extent. The effect of parallel bond stiffness ratio ${k}_{n}/{k}_{s}$ on the peak shear displacement was not significant.
- (4)
- To further quantitatively study the relationships between micro-parameters and shear strength, XGBoost, WOA-XGBoost, and PSO-XGBoost algorithms were introduced to construct the quantitative prediction model. It was found that WOA-XGBoost had the highest prediction accuracy and the best agreement between the predicted data and test data with an R
^{2}of 0.902. Using this model to calculate the weight shares of micro-parameters, we realized the weight share of ${k}_{n}{}^{\ast}/{k}_{s}{}^{\ast}$ should be the highest, 0.812, followed by ${E}^{\ast}$ at 0.106, and the weight shares of $E$ and ${k}_{n}/{k}_{s}$ were the lowest, adding up to less than 0.1. This shows that ${k}_{n}{}^{\ast}/{k}_{s}{}^{\ast}$ is the factor that has the greatest influence on shear strength; ${E}^{\ast}$ has a small influence; $E$ and ${k}_{n}/{k}_{s}$ have almost no influence.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The numerical mode of PFC2D in a serrated structural plane (1–8 are the numbers of the walls respectively).

**Figure 3.**Numerical simulation results and laboratory testing results [39] with different normal stress of serrated structural plane. (

**a**) Normal stress = 0.4 MPa. (

**b**) Normal stress = 0.8 MPa. (

**c**) Normal stress = 1.2 MPa. (

**d**) Normal stress = 1.6 MPa. (

**e**) Normal stress = 2.0 MPa.

**Figure 5.**Distribution of cracks expansion and contact force with different shearing displacement. (

**a**) Shear displacement = 1 mm. (

**b**) Shear displacement = 2 mm. (

**c**) Shear displacement = 3 mm. (

**d**) Shear displacement = 4 mm. (

**e**) Shear displacement = 5 mm. (

**f**) Shear displacement = 10 mm.

**Figure 6.**The shear stress and number of cracks in the serrated structural plane. (a, b, c, d, e, and f correspond to the monitoring points where the shear displacements are 1 mm, 2 mm, 3 mm, 4 mm, 5 mm and 10 mm, respectively).

**Figure 7.**Shear stress–shear displacement curves of corresponding different parallel bond stiffness ratios.

**Figure 8.**The corresponding shear stress–shear displacement relationships of different particle contact stiffness ratios.

**Figure 9.**The shear stress–shear displacement curve under different values of particle contact modulus.

**Figure 10.**The corresponding shear stress–shear displacement curves with different parallel bond moduli.

**Figure 13.**The comparison between the test data and predicted data being used by different models: (

**a**) XGBoost, (

**b**) WOA-XGBoost, (

**c**) PSO-XGBoost.

Min Particle Diameter (mm) | Particle Radius Ratio | Contact Modulus (GPa) | Contact Stiffness Ratio | Bond Modulus (GPa) | Bond Stiffness Ratio | Parallel Bond Normal Strength (MPa) | Parallel Bond Cohesion (MPa) | Internal Friction Angle of Parallel Bonding |
---|---|---|---|---|---|---|---|---|

0.4 | 1.6 | 4 | 2.1 | 0.08 | 2.2 | 7.2 | 23.56 | 53.3 |

Test Condition | Simulation Condition | |
---|---|---|

Cohesion (MPa) | 2.466 | 1.713 |

Internal friction angle (°) | 55.55 | 64.21 |

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**MDPI and ACS Style**

Zhang, X.; Lin, H.; Qin, J.; Cao, R.; Ma, S.; Hu, H.
Numerical Analysis of Microcrack Propagation Characteristics and Influencing Factors of Serrated Structural Plane. *Materials* **2022**, *15*, 5287.
https://doi.org/10.3390/ma15155287

**AMA Style**

Zhang X, Lin H, Qin J, Cao R, Ma S, Hu H.
Numerical Analysis of Microcrack Propagation Characteristics and Influencing Factors of Serrated Structural Plane. *Materials*. 2022; 15(15):5287.
https://doi.org/10.3390/ma15155287

**Chicago/Turabian Style**

Zhang, Xing, Hang Lin, Jianxin Qin, Rihong Cao, Shaowei Ma, and Huihua Hu.
2022. "Numerical Analysis of Microcrack Propagation Characteristics and Influencing Factors of Serrated Structural Plane" *Materials* 15, no. 15: 5287.
https://doi.org/10.3390/ma15155287