# The State of the Art and New Insight into Combined Finite–Discrete Element Modelling of the Entire Rock Slope Failure Process

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## Abstract

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## 1. Introduction

## 2. Combined Finite–Discrete Element Method

## 3. Calibration of Hybrid Finite–Discrete Element Method for Modelling the Rock Slope Failure Process

## 4. Application of the Hybrid Finite–Discrete Element Method in Modelling the Entire Rock Failure Process

## 5. Discussion

#### 5.1. Discussion on the Numerical Modelling Entire Slope Failure Process

#### 5.2. Discussion on the Advantages and Limitations of the FDEM in Rock Fracture Modelling

## 6. New Insight into GPGPU-Parallelized FDEM Modelling of Rock Slope Failure Process

**M**

^{scale}is the scaled lumped mass,

**f**

_{tot}is the nodal out-of-balance force,

**v**is the nodal velocity, ||

**f**

_{tot}|| is the absolute value of each component of

**f**

_{tot}, sgn(∙) is the sign function automatically determined by the sign of (∙), and α is the local damping coefficient. Please refer to Fukuda et al. (2019) [53] for the implementation of the local damping method in detail. This process is completely different from that implemented in reference [19].

## 7. Conclusions

- The FDEM can effectively model the entire rock slope failure process from the fracture initiation, wedge sliding, and fragmentation due to the transition from the continuum to discontinuum technique implemented in the FDEM;
- The continuum or the discontinuum methods have their limitations in naturally modelling the entire rock failure process. However, the distinctive character, i.e., the transition from continuum to discontinuum, means that the FDEM can effectively model the natural failure process for the rock slopes, even without any failure modes implemented in the proposed method;
- The GPGUP-parallelized FDEM with the implementation of SRM is a promising technique in the back analysis and prediction of the slope failure process, as it combined the advantages of the continuum methods and discontinuum methods, and it can naturally model the transition for rock from continuum to discontinuum through fracture and fragmentations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Examples of rock slopes (from reference [1]). (

**a**) Rock slope in Hong Kong. (

**b**) Palabora open-pit mine.

**Figure 5.**Contact force due to penetration [26].

**Figure 7.**Toppling failure process of the rock slope modelling using FDEM (from reference [46]). (

**a**) Initial state; (

**b**) sliding along the discontinuity surface; (

**c**) interaction between rock blocks; (

**d**) piling to the toe of the slope.

**Figure 8.**FDEM modelling of stress distribution and displacement destitution of the equilibrium state of a benchmark (from reference [19]). (

**a**) Stress distribution; (

**b**) displacement distribution.

**Figure 9.**Plane view of the Alpetto Mine in Northern Italy (from reference [21]).

**Figure 10.**FDEM modelling of rock slope sliding process of the Alpetto high rock cut slope (from reference [21]). (

**a**) Section A; (

**b**) Section B.

**Figure 11.**FDEM modelling of the entire rock slope failure process [19].

**Figure 12.**Comparison of the slope failure process modelled by: (

**a**) the UDEC; (

**b**) the FDEM (from reference [46]).

**Figure 13.**Geometrical model of high rock slope (from reference [19]).

**Figure 14.**GPGUP-parallelized Y-HFDEM IDE modelling of rock slope failure process. (

**a**) Stress equilibrium state; (

**b**) 3 s; (

**c**) 5 s; (

**d**) 10 s; (

**e**) 25 s.

Numerical Code | Modelled Results | Reference |
---|---|---|

Y-Slope | Y-Slope considers the tensile and shear failure. The failure is caused by gravity. By decreasing the strength parameters, the cracks initiate from the toe of the slope and propagate further into the slope. Finally, the cracks form a discontinuity surface. The crack initiation, propagation, colliding, fragmentation, and piling are modelled. | [19] |

FDEM realized using ABAQUS/Explicit | The FDEM framework is implemented in the ABAQUS/Explicit. The cohesive zone model (CZM) is employed to model the fracture occurring along the bulk elements boundary. The gravity increase method is implanted in the ABAQUS/Explicit-based FDEM program to model the slope failure process. The failure processes of the laboratory-scale slope with various joint inclination surfaces are modelled. | [20] |

Y-Geo based on Munjiza’s Y-code | Y-Geo is used to model the evaluation of a rock slide that occurred in Italy in 1997. The modelled results in terms of the runout profiles and evaluation of the slopes agree well with the site observation. | [21] |

ELFEN | A modified Mohr–Coulomb elastoplastic model is implemented in ELFEN to model the material softening, and deal with both the tension and shear states. Then, the ELFEN was employed to model the 1991 Randa rockslide. Due to strength degradation, the rock mass breaks into blocks and are modelled using ELFEN. | [23] |

Y-2D with Y-GUI | The failure process of the rock avalanche is modelled. The weak interface in the slope was firstly produced, then, the rock avalanche was initiated. A large volume of rock mass started to move, which further fragmented. During the process, the blocks were progressively broken into smaller fragments. | [22] |

Properties | Values |
---|---|

Young’s modulus (GPa) | 10 |

Poisson’s ratio | 0.25 |

Density (kg/m^{3}) | 2000 |

Friction angle (degrees) | 32 |

Cohesion (MPa) | 1 |

Tensile strength (MPa) | 0.5 |

Properties | Values |
---|---|

Mode-I fracture energy (J/m^{2}) | 4 |

Mode-II fracture energy (J/m^{2}) | 12 |

Normal contact penalty (GPa) | 100 |

Tangent contact penalty (GPa) | 100 |

Artificial stiffness penalty (GPa) | 1000 |

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**MDPI and ACS Style**

An, H.; Fan, Y.; Liu, H.; Cheng, Y.; Song, Y.
The State of the Art and New Insight into Combined Finite–Discrete Element Modelling of the Entire Rock Slope Failure Process. *Sustainability* **2022**, *14*, 4896.
https://doi.org/10.3390/su14094896

**AMA Style**

An H, Fan Y, Liu H, Cheng Y, Song Y.
The State of the Art and New Insight into Combined Finite–Discrete Element Modelling of the Entire Rock Slope Failure Process. *Sustainability*. 2022; 14(9):4896.
https://doi.org/10.3390/su14094896

**Chicago/Turabian Style**

An, Huaming, Yuqing Fan, Hongyuan Liu, Yinyao Cheng, and Yushan Song.
2022. "The State of the Art and New Insight into Combined Finite–Discrete Element Modelling of the Entire Rock Slope Failure Process" *Sustainability* 14, no. 9: 4896.
https://doi.org/10.3390/su14094896