## 3 Simulation Verification of the Rock-Breaking Process by Disc Cutters  
To validate the accuracy of the calibrated discrete element numerical model in simulating the rock-breaking process by disc cutters, a simulation (Figure 5) was conducted and compared against finite element simulations and the experimental data from the literature. In the model, the disc cutter's rotational speed was set to 2 r/min, corresponding to an angular velocity of 3.7 rad/s. Penetration depths, defined as the cutting depth of the disc cutter, were set at 2, 2.5, 3, 3.5, and 4 mm. As the disc cutter rolled from the right to the left of the model, it fragmented the underlying rock, and the rolling force and rock-breaking specific energy were monitored throughout the process. \begin{table}
\begin{tabular}{c c} \hline \hline
**Rock Type** & **Granite** \\ \hline Friction coefficient & 0.577 \\ Contact modulus (GPa) & 8.7 \\ Normal-to-shear stiffness ratio & 2.5 \\ Minimum particle radius (m) & 0.004 \\ Maximum radius/minimum radius & 1.5 \\ Normal band strength (MPa) & 31.7 \\ Shear bond strength (MPa) & 31.7 \\ Bond friction angle (\({}^{\circ}\)) & 60 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Microscopic parameters of the discrete element model. Figure 4: Three-dimensional rock-breaking model for disc cutters. To objectively evaluate the rock-breaking efficiency of the disc cutter, two indicators were used: the cutter rolling force and rock-breaking specific energy. During the rock-breaking process, the three-dimensional forces acting on the disc cutter contribute to rock fragmentation. However, the cutter primarily relies on the rolling force to overcome rock resistance, as illustrated in Figure 6. The lateral force is relatively small compared to the rolling force, while the normal force has been extensively studied by other researchers [22; 23]. Therefore, this study focused on the variation in rolling force under different tunneling parameters. While the rolling force is critical, it alone is insufficient to fully evaluate the efficiency and energy consumption of the rock-breaking process. To address this, a new indicator, rock-breaking specific energy (denoted as \(A\)), was introduced. This combines the rolling force and the rock fragmentation volume and is defined as:  
\[A=\frac{W}{V} \tag{4}\]  
where \(W\) is the work performed by the rolling force to break the rock, and \(V\) is the volume of the resulting rock fragments. Figure 5: Simulation of the rock-breaking process by disc cutters. Figure 6: Force analysis of the rock-breaking process using disc cutters. Figures 7 and 8 compare the simulation results with the experimental data and finite element simulations for rolling force and rock-breaking specific energy, respectively. The experimental and finite element data were sourced from reference [20], see Table 3. The rolling force increased as the penetration depth increased, while the specify energy decreased as the penetration depth increased. From the perspective of the rolling force, the experimental results are the smallest, with a minimum value of 27.3, a maximum value of 36.9, and an average value of 32.2. The finite element simulation results are the largest, with a minimum value of 30.1, a maximum value of 43.9, and an average value of 36.4. The discrete element simulation results fall in between, with a minimum value of 27.8, a maximum value of 37.8, and an average value of 32.9. In terms of rock fragmentation energy, the experimental results are the smallest, with a minimum value of 124.8, a maximum value of 260.7, and an average value of 179.8. The discrete element simulation results are the largest, with a minimum value of 178, a maximum value of 250.8, and an average value of 213.8. The finite element simulation results are intermediate, with a minimum value of 154.2, a maximum value of 298.2, and an average value of 195.8. Figure 8: Comparison of simulated and experimental results for rock breaking specific energy. Figure 7: Comparison of simulated and experimental results for rolling force.