# Investigation of the Dynamic Pure-Mode-II Fracture Initiation and Propagation of Rock during Four-Point Bending Test Using Hybrid Finite–Discrete Element Method

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## Abstract

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## 1. Introduction

## 2. Hybrid Finite–Discrete Element Method

## 3. FDEM Modeling of Dynamic Shear Fracture Process

#### 3.1. Under the Loading Rate of 1 m/s

#### 3.2. Under the Loading Rate of 5 m/s

#### 3.3. Under the Loading Rate of 10 m/s and 50 m/s

## 4. Discussion

#### 4.1. Effect of Loading Rate on the Rock Behaviors

#### 4.2. Comparison of the FDEM-Modeled Fractures with Those Reported in the Literature

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Standard geomaterial stress–strain curve under loading (after Liu et al. (2013) [2]).

**Figure 2.**Conceptual FDEM and schematics under different stress conditions (after An et al. (2021) [47]). (

**a**) Node distribution; (

**b**) no stress; (

**c**) under tension conditions; (

**d**) under shear conditions; (

**e**) under mixed tension and shear conditions.

**Figure 3.**Bonding stress and opening/sliding displacement correlation under various stress conditions (after An et al. (2021) [47]): (

**a**) under tension conditions; (

**b**) under shear conditions; (

**c**) under both tension and shear conditions.

**Figure 5.**Geometrical and numerical models for four-point bending test. (

**a**) Geometrical model; (

**b**) numerical model.

**Figure 6.**Stress spread during the 4 PB test under quasi-static loading modeling by FDEM. (

**A**) 0 mm (0 ms), (

**B**) 0.0025 mm (0.0025 ms), (

**C**) 0.0075 mm (0.0075 ms), (

**D**) 0.015 mm (0.015I), (

**E**) 0.05 mm (0.05 ms), (

**F**) 0.075 mm (0.075 ms), (

**G**) 0.08 mm (0.08 ms), (

**H**) 0.1 mm (0.1 ms), (

**I**) 0.14 mm (0.14 ms), (

**J**) 0.38 mm (0.38 ms).

**Figure 7.**Crack initiation and progression during the 4 PB test under a constant 1 m/s loading rate. (

**A**) 0 mm (0 ms), (

**B**) 0.0075 mm (0.0075 ms), (

**C**) 0.08 mm (0.08 ms), (

**D**) 0.11 mm (0.11 ms), (

**E**) 0.14 mm (0.14 ms), (

**F**) 0.38 mm (0.38 ms).

**Figure 8.**Force−loading−related curves for the 4 PB test under a loading rate of 1 m/s. (

**a**) Force−loading displacement curve; (

**b**) force−loading CMOD curve; (

**c**) force−loading CMSD curve. (

**A**−

**D**) loading force rose; (

**D**) loading force reached the maximum; (

**D**−

**E**) loading force declined; (

**F**) the beam lost its load-carrying ability.

**Figure 9.**Crack beginning and progression during the 4 PB test under a constant 5 m/s loading rate. (

**A**) 0 mm (0 ms), (

**B**) 0.075 mm (0.0015 ms), (

**C**) 0.125 mm (0.025 ms), (

**D**) 0.1875 mm (0.0375 ms), (

**E**) 0.325 mm (0.065 ms), (

**F**) 0. 9375 mm (0.1875 ms).

**Figure 10.**Force−loading−related curves for the 4 PB test under a loading rates of 5 m/s. (

**a**) Force−loading displacement; (

**b**) force−loading CMOD curve; (

**c**) force−loading CMSD curve. (

**A**−

**B**) loading force increased; (

**B**) loading force reached maximum; (

**B**−

**E**) loading force declined; (

**F**) the beam lost its load−carrying ability.

**Figure 11.**Crack initiation and progression during the 4 PB test under a constant 10 m/s loading rate. (

**A**) 0 mm (0 ms), (

**B**) 0.075 mm (0.0075 ms), (

**C**) 0.15 mm (0.015 ms), (

**D**) 0.2 mm (0.02 ms), (

**E**) 0.5 mm (0.05 ms), (

**F**) 2.4 mm (0.24 ms).

**Figure 12.**Crack initiation and progression during the 4 PB test under a constant 50 m/s loading rate. (

**A**) 0 mm (0 ms), (

**B**) 0.25 mm, (0.005 ms), (

**C**) 0.75 mm (0.015 mI (

**D**) 1 mm (0.02 ms), (

**E**) 2 mm (0.04 ms), (

**F**) 5 mm (0.10 ms).

**Figure 13.**Crack initiation and progression during the BTS test under different loading rates. (

**a**) 0.1 m/s; (

**b**) 1 m/s; (

**c**) 5 m/s; (

**d**) 10 m/s.

**Figure 15.**Fracture progression process in the four-point bending test modeled using R-T

^{2D}[6] (

**A**) no crack; (

**B**) crack initiated in front of the notch tip; (

**C**) crack initiation angle is about 60$\xb0$; (

**D-1**–

**D-5**) fracture propagated; (

**E**) crack reached the edge of the specimen; (

**F**) specimen was split into two halves.

Symbol | Parameter | Value | Unit |
---|---|---|---|

$E$ | Young’s modulus | 60 | $\mathrm{GPa}$ |

$\nu $ | Poisson’s ratio | 0.26 | $\mathrm{N}/\mathrm{A}$ |

$\rho $ | Density | 2600 | ${\mathrm{Kgm}}^{-3}$ |

${\sigma}_{t}$ | Tensile strength | 20 | $\mathrm{MPa}$ |

${\sigma}_{c}$ | Compressive strength | 200 | $\mathrm{MPa}$ |

$\varnothing $ | Internal friction angle | 30 | $\xb0$(Degree) |

$u$ | Surface friction coefficient | 0.1 | $-$ |

${G}_{fI}$ | Mode-I fracture energy release | 50 | ${\mathrm{Nm}}^{-1}$ |

${G}_{fI}$ | Mode-II fracture energy release | 250 | ${\mathrm{Nm}}^{-1}$ |

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**MDPI and ACS Style**

Song, Y.; Fan, Y.; An, H.; Liu, H.; Wu, S.
Investigation of the Dynamic Pure-Mode-II Fracture Initiation and Propagation of Rock during Four-Point Bending Test Using Hybrid Finite–Discrete Element Method. *Sustainability* **2022**, *14*, 10200.
https://doi.org/10.3390/su141610200

**AMA Style**

Song Y, Fan Y, An H, Liu H, Wu S.
Investigation of the Dynamic Pure-Mode-II Fracture Initiation and Propagation of Rock during Four-Point Bending Test Using Hybrid Finite–Discrete Element Method. *Sustainability*. 2022; 14(16):10200.
https://doi.org/10.3390/su141610200

**Chicago/Turabian Style**

Song, Yushan, Yuqing Fan, Huaming An, Hongyuan Liu, and Shunchuan Wu.
2022. "Investigation of the Dynamic Pure-Mode-II Fracture Initiation and Propagation of Rock during Four-Point Bending Test Using Hybrid Finite–Discrete Element Method" *Sustainability* 14, no. 16: 10200.
https://doi.org/10.3390/su141610200