# Research on Shear Behavior and Crack Evolution of Symmetrical Discontinuous Rock Joints Based on FEM-CZM

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## Abstract

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## 1. Introduction

## 2. FEM-CZM Simulation of Rock Shear

#### 2.1. Initial Linear Elastic Traction-Displacement

#### 2.2. Linear Damage Stage of Cohesive Element

#### 2.3. Cohesive Element Inserting Process

## 3. Model Establishment

#### 3.1. Parameters Determination

#### 3.2. Model Establishment

## 4. Simulation Results and Analysis

#### 4.1. Influence of Joint Distribution on Shear Resistance

#### 4.2. Effect of Joint Persistence on Shear Resistance

#### 4.3. Crack Evolution Analysis

## 5. Conclusions

- (1)
- The shearing process can be divided into four stages: elastic stage, strengthening stage, plastic stage, and residual stress stage. In the stress-strain curve, the residual stress stage of type-II has a higher slope, which shows that the type-II has more brittleness and the type-I has higher plasticity. At the same time, with the decrease of joint persistence, the specimen turns more brittle, and the joint shear is closer to the direct shear test of intact rock.
- (2)
- Under the same conditions, the specimen in the type-I is more likely to produce an unbalanced moment around the centroid of the specimen, which causes the opposite ends of the specimen to produce vertical displacements in opposite directions. Due to the strengthening effect of the rock bridge, the vertical displacement on the side away from the loading site is smaller, and as a result, the overall displacement field of the specimen is not simply center-symmetric. At the same time, the distribution of rock bridges in type-II is more dispersed, providing more reinforcement for the joints, and the specimens are subjected to lower dilatancy.
- (3)
- The crack propagation process can be divided into three stages: crack initiation stage, crack evolution stage, and final failure stage. Under the load, the joint tips are prone to stress concentration, where the cohesive elements accumulate more fracture energy and reach the damage evolution stage faster. Initial cracks always start from the joint ends. Affected by the unbalanced moment, more tensile cracks are generated in type-I, while the type-II and type-III penetrated cracks are all shear cracks, and the cracks mainly propagate along the rock bridge.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Separation-Traction (S-T) model: (

**a**) mixed-mode cohesive traction response, and (

**b**) constitutive Model of S-T model.

**Figure 2.**Insertion process of cohesive elements: (

**a**) continuous solid elements, (

**b**) split solid elements, (

**c**) re-number nodes, and (

**d**) insert a cohesive element.

**Figure 3.**The uniaxial compression test: (

**a**) uniaxial compression experiment, and (

**b**) numerical model and failure pattern.

**Figure 5.**Numerical model of the Brazilian test: (

**a**) numerical simulation model, (

**b**) failure pattern, and (

**c**) experimental results.

**Figure 7.**Symmetrical calculation model used in this paper. $\sigma $: Normal stress, $\tau $: Shear stress, ${L}_{j}$: Joint length, ${L}_{r}$: Rock bridge length, $H$: Model width, $L$: Model length, ${e}_{j}$: Joint aperture.

**Figure 8.**Symmetrical three types of joint distribution: (

**a**) Type-I: both sided, (

**b**) Type-II: scattered, and (

**c**) Type-III: central.

**Figure 11.**The vertical displacement field of specimens: (

**a**) initial displacement field and (

**b**) displacement field at failure stage.

**Figure 12.**Stress-strain curve at normal stress of 1.0 MPa: (

**a**) Type-I, (

**b**) Type-II, (

**c**) Type-III, and (

**d**) peak stress.

**Figure 13.**Relationship of the peak shear strength of the specimens with joints’ persistence and normal stress.

**Figure 14.**Crack evolution process: (

**a**) crack initiation stage, (

**b**) crack evolution stage, (

**c**) final failure stage, and (

**d**) failure form of cohesive elements.

**Figure 15.**Stress concentration: (

**a**) stress concentration at joint ends, and (

**b**) accumulation of fracture energy.

Materials | Parameters | Value |
---|---|---|

Solid element | Density (kg·m^{−3}) | 2.5 × 10 ^{3} |

Young’s modulus (GPa) | 15 | |

Poisson’s ratio | 0.3 | |

Cohesive element | Initial tensile stiffness (GPa·m^{−1}) | 15 |

Initial shear stiffness (GPa·m^{−1}) | 5.28 | |

Normal traction force (MPa) | 6 | |

Tangential traction force (MPa) | 22 | |

Model-I fracture energy (N·mm^{−1}) | 6 × 10^{−2} | |

Model-II fracture energy (N·mm^{−1}) | 1.65 × 10^{−1} | |

Loading plate | Density (kg·m^{−3}) | 7.8 × 10 ^{3} |

Young’s modulus (GPa) | 2.1 × 10 ^{2} | |

Poisson’s ratio | 0.3 |

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**MDPI and ACS Style**

Wu, X.; Wang, G.; Li, G.; Han, W.; Sun, S.; Zhang, S.; Bi, W.
Research on Shear Behavior and Crack Evolution of Symmetrical Discontinuous Rock Joints Based on FEM-CZM. *Symmetry* **2020**, *12*, 1314.
https://doi.org/10.3390/sym12081314

**AMA Style**

Wu X, Wang G, Li G, Han W, Sun S, Zhang S, Bi W.
Research on Shear Behavior and Crack Evolution of Symmetrical Discontinuous Rock Joints Based on FEM-CZM. *Symmetry*. 2020; 12(8):1314.
https://doi.org/10.3390/sym12081314

**Chicago/Turabian Style**

Wu, Xianlong, Gang Wang, Genxiao Li, Wei Han, Shangqu Sun, Shubo Zhang, and Wangliang Bi.
2020. "Research on Shear Behavior and Crack Evolution of Symmetrical Discontinuous Rock Joints Based on FEM-CZM" *Symmetry* 12, no. 8: 1314.
https://doi.org/10.3390/sym12081314