# Comparison of DEM Models with Different Elemental Dimensions for TBM Disc Cutter Rock Fragmentation

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## Abstract

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## 1. Introduction

^{2D}to simulate the process of TBM indentation. Choi et al. [20] analyzed the cutting power of a disc cutter in a jointed rock mass. Moon et al. [21] used DEM to study the optimal rock-cutting conditions of a hard rock TBM, based on a two-dimensional model. Gong et al. [22] used chip thickness and chipping area to determine the efficiency of TBM excavation. These DEM applications mainly used a two-dimensional model to reduce the calculation time. Although a three-dimensional model is more realistic, it is not often used with a DEM because of the large amount of calculation required. A few researchers have employed three-dimensional DEM models to offer a more realistic simulation of the cutting process. Choi et al. [23] used the contact bond model to simulate the rock in PFC

^{3D}and analyzed the rock-cutting behavior, but this model is not suitable for hard rock because of its contact model. Bahr et al. [24] discussed some of the challenges facing the three-dimensional modeling of rock cutting. Wu et al. [25] established PFC

^{3D}models to analyze the relationship between the mean and peak force during cutting with a disc cutter. To analyze the differences between the two- and three-dimensional models, the current study focuses on comparing the impact of elemental dimensions, to provide a more realistic simulation of rock-cutting, as represented by comparing the results with measured normal forces and specific energy. Validation of the modeling and related parameters, including the optimal size of particles in PFC, allows for the development of the method of determining the optimal conditions for a TBM. The remainder of this paper is organized as follows. Section 2 presents the simulation models and methods, including the contact model and the optimal condition of the simulation models. The three-dimensional simulation results and specific energy of different disc-cutter spacings and penetrations are presented in Section 3. Section 4 compares the cutting force of the two- and three-dimensional models in PFC. Section 5 concludes this article.

## 2. Numerical Modeling

#### 2.1. Disc Cutter Geometry

^{3D}software as two walls. For the two-dimensional model, the disc cutter was simplified, and the tip width of the disc cutter was set to 13 mm to match the available data for the LCM tests. Figure 2 shows the two-dimensional disc cutter model.

#### 2.2. Modeling Rock Specimens

#### 2.2.1. Flat-Jointed Contact Model

#### 2.2.2. Mesoscopic Parameter Calibration

^{3D}were determined and are summarized in Table 1. Therefore, the macroscopic parameters of CRG selected for this study were the UCS, BTS, and EM.

- (1)
- For the two and three-dimensional models, the minimum and maximum particle radii were 1.5 and 2 mm, respectively. The particles were evenly distributed by radius.
- (2)
- The radius multiplier of the FJCM is 1.0.
- (3)
- The particle density is equal to the rock density (i.e., 2650 kg/m
^{3}).

#### 2.3. Development of Numerical Models of Rock-Cutting

^{3D}was used to construct three-dimensional models of the disc cutters and the rock specimen. The rock specimen had dimensions of 300 mm × 200 mm × 100 mm and consisted of 241,135 particles. Because the disc-cutter wear was outside the scope of this study, we used a wall model to simulate the disc cutters. We used the principle of relative motion to simulate the process of the disc-cutting of rock. The geometries and boundary conditions used in the numerical model are shown in Figure 8. The rock specimen model was fixed in all directions, and the disc cutter model moved in the horizontal direction (y-direction) with a constant velocity for linear cutting. Simultaneously, the cutter was fixed to maintain a given penetration depth in the vertical direction (z-direction) and was rotated at a constant angular velocity (x-direction). To match the LCM cutting conditions, two disc cutters that were cutting in sequence were modeled. For comparison with the LCM test results, six sets of numerical simulations with disc-cutter spacings of 62.5 and 75 mm and penetrations of 3.2, 4.4, and 6.4 mm were modeled.

#### 2.4. Determining the Optimal Conditions for Disc-Cutter Rock Fragmentation

_{total}is the total energy consumption during the cutting process, V

_{cut}is the cutting volume, MRF is the mean rolling force, l

_{y}is the cutting distance, MNF is the mean normal force, and p is the penetration. In this study, we calculated the cutting volume from the number of eliminated particles after cutting. If the contact number of a particle was less than 4, then the particle was assumed to be cut off and was deleted.

## 3. Simulation Results of Three-Dimensional Models

#### 3.1. Effect of Cutting Velocities on the Simulation Results

^{3D}, so we could record the wall forces in different directions. The two disc cutters are parallel to the stratification in the numerical simulation tests. Figure 10, Figure 11, Figure 12 and Figure 13 show the normal and rolling forces for different spacings. Table 3 presents the mean normal force and rolling force under various conditions.

#### 3.2. Comparison of the Simulated Forces with the LCM Results

#### 3.3. Disc Cutters’ Performance Comparison

^{3D}simulation than when measured in the LCM tests. This agrees with the results of Choi and Lee [23].

^{3D}is the difference between the two calculations of excavated volume. On the one hand, in LCM testing, SE is calculated based on spacing and penetration, thereby assuming full clearance between the lines and nominal excavation volume, which is true as the cuts proceed and the rock is removed. On the other hand, the rock’s calculated excavate volume in the simulation is based on the volume estimated to have been removed from the main sample. This warrants a closer look at the way that the excavated volume is estimated in the simulation; it could be that the values that are calculated are slightly larger than those of the nominal excavation volume used in the calculation of the SE in LCM tests.

## 4. Discussion

## 5. Conclusions

^{3D}and PFC

^{2D}were used to simulate rock fragmentation when using disc cutters in different configurations. FJCM was used to simulate the rock specimen and offer more realistic fracturing behavior in the context of rock excavation, where the contact parameters could be better controlled to show the behavior of rock, as measured by typical rock mechanics testing. The cutting force calculated under various conditions could be used to calculate the SE for various disc-cutter spacings and penetrations. The numerical simulation results were verified in two stages, one by comparing the calculated results with the results of rock mechanics testing, thus determining the mesoscopic parameters for the PFC models. This level of verification was used to compare the results of the numerical analysis with full-scale rock-cutting tests using LCM data for a selected granite (CRG) at the CSM rock excavation laboratory. The recorded force in the LCM tests was slightly lower than the forces calculated in the numerical simulations. The rolling coefficients of RC and SE showed the same trends as the LCM tests, which validates the numerical simulation results.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Three-dimensional disc cutter model (unit: mm): (

**a**) geometric model of a disc cutter; (

**b**) the CCS profile of the cutter used in this research (unit: mm).

**Figure 4.**Stress–strain curves and three-dimensional simulation model of UCS and BTS: (

**a**) UCS; (

**b**) BTS.

**Figure 5.**Stress–strain curves and two-dimensional simulation model of UCS and BTS: (

**a**) UCS; (

**b**) BTS.

**Figure 6.**Development of crushed zones and cracks when using TBM disc cutters. (

**a**) single disc cutter, (

**b**) double disc cutters.

**Figure 8.**Geometries and boundary conditions used in the numerical model (unit: mm): (

**a**) side view; (

**b**) front view.

**Figure 9.**Calculation times and normal forces at various cutting speeds in the three-dimensional numerical simulation.

**Figure 10.**Normal force with a disc-cutter spacing of 62.5 mm: (

**a**) penetration of 3.2 mm; (

**b**) penetration of 4.4 mm; (

**c**) penetration of 6.4 mm.

**Figure 11.**Rolling force with a disc-cutter spacing of 62.5 mm: (

**a**) penetration of 3.2mm; (

**b**) penetration of 4.4 mm; (

**c**) penetration of 6.4 mm.

**Figure 12.**Normal force with a disc-cutter spacing of 75 mm: (

**a**) penetration of 3.2 mm; (

**b**) penetration of 4.4 mm; (

**c**) penetration of 6.4 mm.

**Figure 13.**Rolling force with a disc-cutter spacing of 75 mm: (

**a**) penetration of 3.2 mm; (

**b**) penetration of 4.4 mm; (

**c**) penetration of 6.4 mm.

**Figure 19.**Calculation time and normal force for the two-dimensional numerical simulation at various penetration speeds.

**Figure 20.**Variations of normal force as a function of penetration, at a spacing of 62.5 mm: (

**a**) penetration of 3.2 mm; (

**b**) penetration of 4.4 mm; (

**c**) penetration of 6.4 mm.

**Figure 21.**Development of the crushed zones and cracks after cutting with two disc cutters at a spacing of 62.5mm: (

**a**) penetration of 3.2mm; (

**b**) penetration of 4.4mm; (

**c**) penetration of 6.4mm.

**Table 1.**Summary of the results of the rock mechanics testing and numerical simulation by PDF, including the calibration error rate of the macro parameters.

Marco Test Parameters | UCS), (MPa) | BTS, Mpa | Elastic Modulus (EM), (Gpa) |
---|---|---|---|

Average Value of 3 tests | 178.5 | 8.9 | 21.8 |

Simulation Results in PFC^{3D} | 179.1 | 8.8 | 21.4 |

Error Rate in PFC^{3D} | 0.34% | 1.12% | 1.83% |

Simulation Results in PFC^{2D} | 177.8 | 9.0 | 22.5 |

Error Rate in PFC^{2D} | 0.39% | 1.12% | 3.21% |

Marco Parameters | Uniaxial Compressive Strength (UCS) | Brazilian Tensile Strength (BTS) | Elastic Modulus (EM) | |||||
---|---|---|---|---|---|---|---|---|

Mesoscopic Parameters | Bond Gap (fj_gap0) | Deformability Effective Modulus (fj_emod) | Effective Normal-to-Shear Stiffness Ratio(fj_krat) | Number of Elements in Radial Direction (fj_nr) | Number of Elements in Circumferential Direction (fj_nal) | Tensile Strength (fj_ten) | Cohesion (fj_coh) | Friction Angle (fj_fa) |

PFC^{3D} | 0.0 | 11 × 10^{9} | 1.5 | 2 | 4 | 4.4 × 10^{6} | 38 × 10^{6} | 30 |

PFC^{2D} | 0.0 | 27 × 10^{9} | 1.5 | 4 | \ | 13 × 10^{6} | 150 × 10^{6} | 30 |

Spacing (mm) | Penetration (mm) | PFC^{3D} | LCM Tests | Error Rate | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Mean Normal Force (kN) | Mean Rolling Force (kN) | Specific Energy (MJ/m³) | Mean Normal Force (kN) | Mean Rolling Force (kN) | Specific Energy (MJ/m³) | Mean Normal Force | Mean Rolling Force | Specific Energy | ||

62.5 | 3.2 | 129.14 | 6.10 | 22.93 | 128 | 7 | 37.80 | 0.89% | 12.86% | 39.34% |

4.4 | 199.69 | 7.95 | 35.92 | 139 | 11 | 39.96 | 43.66% | 27.73% | 10.11% | |

6.4 | 196.08 | 13.17 | 38.43 | 155 | 18 | 45.72 | 26.50% | 26.83% | 15.94% | |

75 | 3.2 | 130.04 | 7.48 | 27.53 | 186 | 12 | 52.56 | 30.09% | 37.67% | 47.62% |

4.4 | 205.61 | 8.79 | 34.48 | 144 | 12 | 37.44 | 42.78% | 26.75% | 7.91% | |

6.4 | 196.33 | 15.84 | 42.06 | 186 | 25 | 53.28 | 5.55% | 36.64% | 21.06% |

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**MDPI and ACS Style**

Xu, C.; Zhu, Y.; Song, D.; Guo, X.; Liu, X.; Wang, E.; Lu, R.
Comparison of DEM Models with Different Elemental Dimensions for TBM Disc Cutter Rock Fragmentation. *Sustainability* **2022**, *14*, 12909.
https://doi.org/10.3390/su141912909

**AMA Style**

Xu C, Zhu Y, Song D, Guo X, Liu X, Wang E, Lu R.
Comparison of DEM Models with Different Elemental Dimensions for TBM Disc Cutter Rock Fragmentation. *Sustainability*. 2022; 14(19):12909.
https://doi.org/10.3390/su141912909

**Chicago/Turabian Style**

Xu, Chen, Yujie Zhu, Danqing Song, Xiaogang Guo, Xiaoli Liu, Enzhi Wang, and Runhu Lu.
2022. "Comparison of DEM Models with Different Elemental Dimensions for TBM Disc Cutter Rock Fragmentation" *Sustainability* 14, no. 19: 12909.
https://doi.org/10.3390/su141912909