# Hybrid Finite-Discrete Element Modelling of Various Rock Fracture Modes during Three Conventional Bending Tests

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## Abstract

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## 1. Introduction

## 2. HFDEM for Modelling Dynamic Rock Fracture

#### 2.1. Transition from Continuum to Discontinuum

#### 2.2. The Straite Rate Effect

## 3. Calibration of HFDEM

#### 3.1. Numerical Modelling UCS Test78

^{16}Mpa, compressive strength 100 × 10

^{16}Mpa, Young’s modulus 200 Gpa, surface friction coefficient 0.1, and Mode-I and Mode-II fracture energy release 3 × 10

^{12}${\mathrm{Nm}}^{-1}$. During the test, the loading rates of 1 m/s is applied on the two loading plates on the vertical direction while the plates are fixed in the horizontal direction. The loading rate, i.e., 1 m/s, is much higher than those in the laboratory test, which is about 0.05 mm/s. The reason to use a high loading rate is to significantly decrease computational time, as the increase in the loading rate can dramatically decrease the computational time. The processor used for the simulation is inter(R) Core (TM) i7-4500U with CPU from 1.80 GHz to 2.40 GHz, while the installed memory (RAM) is 16.0 GB. The system installed in the computer is Windows 8.1 with system type 64-bit Operating System. The current loading rate, i.e., 1 m/s, can reduce at least one order of magnitude in terms of the calculation time compared with loading rate for static tests in laboratory, i.e., around 0.05 m/s.

#### 3.2. Numerical Modelling Rock Fracture during BTS Tests

## 4. HFDEM Modelling Three Conventional Bending Tests

#### 4.1. 3PB Test

#### 4.2. Four-Point Bending Test (Pure Mode-II Fracture)

#### 4.3. Asymmetrical Three-Point Bending Test (Mixed-Mode I–II Fracture)

## 5. Discussion

#### 5.1. Effect of the Strain Rate Rock Strength

#### 5.2. Loaidng Rate Influence on Rock Toughness

#### 5.3. Effect of the Mesh Orientation

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Typical brittle material stress–strain curve. $\sigma $, ${\sigma}_{p}$, $\mathsf{\epsilon}$, ${\epsilon}_{p}$ represent the stress loading, peak stress, strain, and peak strain, respectively.

**Figure 2.**Hybrid Finite–discrete fracture model (red line represents edges of joint elements while the black line represents the edge of the finite elements): (

**a**) Without stress; (

**b**) Under tension condition; (

**c**) Under shear condition; (

**d**) Under both tension and shear condition.

**Figure 3.**Bonding stress and opening or sliding displacement: (

**a**) Mode-I; (

**b**) Mode-II; (

**c**) Mixed mode I–II.

**Figure 4.**Geometrical and numerical model of uniaxial compression test: (

**a**) Geometrical model; (

**b**) Numerical model.

**Figure 8.**HFDEM modelling of Brazilian tensile strength tests: (

**a**) Minor principal stress; (

**b**) fracture propagation; (

**A**) d = 1$\text{}\mathsf{\mu}$m; (

**B**) d = $20\text{}\mathsf{\mu}$m; (

**C**) d = $33.5\text{}\mathsf{\mu}$m; (

**D**) d = $39\text{}\mathsf{\mu}$m; (

**E**) d = $41\text{}\mathsf{\mu}$m; (

**F**) d = $58\text{}\mathsf{\mu}$m; (

**G**) d = $166\text{}\mathsf{\mu}$m.

**Figure 9.**HFDEM obtained curves during BTS test: (

**a**) force from the top plate; (

**b**) force from the bottom plate.

**Figure 10.**Comparison of the HFDEM obtained results with analytical solutions in terms of stress distribution.

**Figure 12.**Geometrical mode for 3PB and 4PB tests: (

**a**) 3PB test; (

**b**) 4PB test; (

**c**) Asymmetrical A3PB test.

**Figure 14.**HFDEM modelling rock failure processes in 3PB test. (

**A**) Initial state; (

**B**) Crack initation; (

**C**) Crack propagation; (

**D**) Crack continual propagation; (

**E**) Crack completion.

**Figure 15.**Curves for force-loading with displacement, CMOD, and Time during 3PB test: (

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

**b**) Force-loading CMOD curve; (

**c**) Force-loading CMOD curve.

**Figure 17.**Force loading related curves for 4PB test: (

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

**b**) Force-loading CMOD curve; (

**c**) Force-loading CMSD curve.

**Figure 22.**The modelled result and the experimental results [33] comparison t in terms of relationship between the fracture toughness (mode-I) and loading rate.

Symbols | Properties | Values | Units |
---|---|---|---|

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

$\nu $ | Poisson’s ration | 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 coefficient | 30 | °C |

$u$ | Surface friction coefficient | 0.1 | $\mathrm{N}/\mathrm{A}$ |

${G}_{fI}$ | 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**

An, H.; Wu, S.; Liu, H.; Wang, X.
Hybrid Finite-Discrete Element Modelling of Various Rock Fracture Modes during Three Conventional Bending Tests. *Sustainability* **2022**, *14*, 592.
https://doi.org/10.3390/su14020592

**AMA Style**

An H, Wu S, Liu H, Wang X.
Hybrid Finite-Discrete Element Modelling of Various Rock Fracture Modes during Three Conventional Bending Tests. *Sustainability*. 2022; 14(2):592.
https://doi.org/10.3390/su14020592

**Chicago/Turabian Style**

An, Huaming, Shunchuan Wu, Hongyuan Liu, and Xuguang Wang.
2022. "Hybrid Finite-Discrete Element Modelling of Various Rock Fracture Modes during Three Conventional Bending Tests" *Sustainability* 14, no. 2: 592.
https://doi.org/10.3390/su14020592