## 4 Simulation Experiments and Results  
### Effect of Back-Rake Angle  
This paper uses simulation to analyze the fidelity of the cutting teeth of PDC drills under different cutting structure parameters and to obtain the corresponding laws. The depth of cut is 2 mm, the side-rake angle is 0\({}^{\circ}\), and the back rake is 5.0\({}^{\circ}\), 10.0\({}^{\circ}\), 15.0\({}^{\circ}\), 20.0\({}^{\circ}\), 25.0\({}^{\circ}\), and 30.0\({}^{\circ}\), respectively. The simulated cutting forces with time at a back-rake angle of 15\({}^{\circ}\) are shown in Figure 7:  
As can be observed from the graph, the cutting forces exhibit significant fluctuations over time. This is attributed to the rock reaching its yield limit, leading to fracture damage. Specific units fail to absorb plastic deformation energy, resulting in a rapid decrease in cutting force. However, to proceed with the cutting process for the subsequent division, the cutting force experiences a rapid increase once again. Throughout the cutting process, while maintaining a constant depth of cut, the tangential and axial forces fluctuate within a specific range. Nonetheless, there is an underlying stable average value. By utilizing this average value as the representative tangential and axial forces, the average cutting force at a back-rake angle of 15\({}^{\circ}\) is determined to be 822.5 N. In the ABAQUS 2023 post-processing, the X-directional support force CFN2 at the RP point, which is attached to the rigid body, is extracted [29; 30]. This force represents the tangential force exerted by the cutting tooth on \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Density/ kg m\({}^{-3}\)** & **Elasticity Modulus/Gpa** & **Poisson Ratio** & **Shear Strength/MPa** & **Cohesion/MPa** & **Internal Friction Angle/\({}^{C}\)** & **Flow Stress Ratio** \\ \hline Cutter & 3510 & 890 & 0.077 & & & & \\ Rock & 2250 & 27.1 & 0.29 & 17.3 & 27.2 & 46.934 & 0.8 \\ \hline \end{tabular}
\end{table} 
Table 1: Relevant material parameters used in the finite element analysis. Figure 6: Grid independence study. the rock, reflecting the cutting force of the cutting tooth under actual working conditions. CFN2 values are extracted at various time points (i) after contact between the tooth and the rock to determine the average cutting force. The average cutting force is then calculated using the following equation:  
\[F_{h}=\overline{C_{FN2}}=\frac{1}{n}\sum_{i=1}^{n}C_{FN2i} \tag{18}\]  
\[\sigma_{X}=\sqrt{\frac{\sum\limits_{i=1}^{n}\left(C_{FN2i}-\overline{F_{h}} \right)^{2}}{n}} \tag{19}\]  
Figure 8 below shows the simulated cutting forces followed by the change in back-rake angle. When the rear back-rake angle ranges from \(5^{\circ}\) to \(15^{\circ}\), the average cutting force increases as the angle risk increases. Within this range, the standard deviation of the cutting force initially increases and then decreases, with an overall minimal difference. This behavior is attributed to the fact that as the back-rake angle increases, the contact area between the cutting teeth and the rock also increases. Consequently, the rock becomes more tightly compressed, increasing friction between the cutting teeth and the rock. With the same cutting speed and depth of cut, more excellent cutting and axial forces are required to break the rock, leading to a natural increase in the cutting force. Within the 5-\(15^{\circ}\) range, the back-rake angle causes more significant fluctuations in the cutting forces when the cutting teeth of the drill break the rock. Although the rock becomes easier to break, the cutting teeth are subjected to more severe impact loads. This scenario is not conducive to drilling and coring stability. Therefore, a back-rake angle within this range is not recommended, despite the lower average cutting force and higher efficiency it may offer under the same rock-breaking conditions. Figure 7: Relationship between cutting force and time; the cutting back-rake angle was set as \(15^{\circ}\).