## 2 Numerical Methods
### Erosion Scheme
The similarity of the cut grooves and the volume in the unconfined and these confined cases, means that no significant cracking events outside the cutter face area occurred during the simulations (confinement usually suppresses spall cracks). Moreover, in Conf2, the cutting force peaks are generally higher than in Conf1. Therefore, it seems that Conf2 is more detrimental. In confined conditions, a much higher cutting force is required to remove the same volume of material. ## 4 Discussion

The results are discussed and compared to the experiments. First, the results are quantified in Table 2, with some calculated drilling technical indexes. More specifically, the work done by the cutter, \(w_{\text{bit}}\), is the area under the cutting force vs. cutting length curve. In addition, the MSE is the mechanical specific energy, i.e., \(E_{\text{MS}}=w_{\text{bit}}/V_{\text{crater}}\)[11]. The cutting force, \(F_{\text{c}}\), is defined, not as the average of the force vs. length curve, but as the average of the peaks (as in [4]), marked by red dots in Figure 4a. It should be noted that the peaks included to calculate the cutting forces start at the cutting length of 2 mm, while for \(E_{\text{se}}\), the whole response is used, i.e., starting from zero cutting length. The model predicts the cutting force at higher cutting depths with an acceptable accuracy. The best prediction is naturally obtained at the cutting depth of 0.9 mm, because the model was calibrated to that case (by adjusting \(w_{\text{c}}\)). At the smallest cutting depth, the error, 48%, is unacceptably large, but at 1.5 mm, the error is acceptable at both tested rake angles. No clear data in [4] was provided on cutting forces under confined conditions. Anyway, confinement results in a substantial increase, \(\sim\)50% here, in the cutting force. Moreover, according to the present predictions, the case with higher horizontal than vertical pressure is more detrimental, i.e., requires a higher cutting force to remove the same volume of rock. As for the other rock cutting parameters, the work done on the rock by the cutter naturally increases as the cutting depth increases (higher force moves the same cutting length). Interestingly, the MSE does not deviate much at the three tested cutting depths. The reason for this is that, in the present simulations, the volume of removed rock increases in proportion to the work done into the rock, so that the ratio \(w_{\text{bit}}/V_{\text{crater}}\) is almost the same in each case. Confinement increases the MSE, which means that drilling under confinement is inefficient. \begin{table}
\begin{tabular}{c|c c c c c c} \hline \(d\) [mm] & \(F_{\text{c}}\)/Exper [N] & \(F_{\text{c}}\) [N] & \(Error\) [\%] & \(w_{\text{bit}}\) [J] & \(E_{\text{ms}}\) [MPa] & Info \\ \hline
0.3 & 225 & 333 & 48 & 2.5 & 77.1 & \(\beta=20^{\circ}\) \\
0.9 & 1065 & 1083 & 1.7 & 7.1 & 77.4 & \(\beta=20^{\circ}\) \\
1.5 & 2231 & 1922 & 14 & 13.3 & 79.4 & \(\beta=20^{\circ}\) \\
0.9 & 1025 & 922 & 13.4 & 5.6 & 59.2 & \(\beta=10^{\circ}\) \\
0.9 & - & 1539 & - & 10.4 & 97.1 & Conf1 \\
0.9 & - & 1701 & - & 9.5 & 100.0 & Conf2 \\ \hline \end{tabular}
\end{table} 
Table 2: Simulation results compared to experiments in [4] (gray cells). Figure 9: Simulation results for PDC cutting (\(d=0.9\) mm, \(\beta=20^{\circ}\), \(w_{\text{c}}=0.15\)): (**a**) removed elements in Conf1; (**b**) removed elements in Conf2; (**c**) cutting force vs. cutting length curves. The weaknesses of the present approach include the inability to properly describe cracking, and, especially, fragmentation. The standard continuum model, based on damage and viscoplasticity theories, describes a crack in the smeared (integral) sense, smoothing the crack tip effects. Moreover, the PDC cutter itself is idealized to a single node with rigid virtual geometry. Therefore, the only information the present approach yields for the cutter is the cutting force. However, cutter geometry, cutting speed, rake angle, and cutting depth can be modelled.