## 2 Numerical Methods
### Erosion Scheme
Figure 4: Simulation results for PDC cutting at different cutting depths: (**a**) tensile and compressive damage fields, and the cutting force for \(d=0.3\,\mathrm{mm}\) (red dots mark the peaks included in calculating the average cutting force); (**b**) tensile and compressive damage fields, and the cutting force for \(d=0.9\,\mathrm{mm}\); (**c**) tensile and compressive damage fields, and the cutting force for \(d=1.5\,\mathrm{mm}\). Figure 3: Finite element mesh for cutting simulations (523,928 linear tetrahedrons). Figure 6 shows the predicted cutting force vs. cutting length response, compared to the experimental one. The predicted and the experimental cutting force vs. length curves, have similar qualitative features, but the oscillations are somewhat larger in the predicted curve. Moreover, the average of the cutting force is 565 N for the predicted and 761 N for the experimental. ### Effect of the Stiffness Recovery Parameter \(w_{\mathrm{c}}\)

The effect of the \(w_{\mathrm{c}}\) parameter, which controls the stiffness recovery effect on the tensile damage variable (Equation (13)), was tested. The model was calibrated by adjusting this parameter so that the experimental cutting force at the cutting depth of 0.9 mm was matched (the predictions are compared to experiments in Section 4). The simulation results setting \(w_{\mathrm{c}}=1\) (full recovery) are shown in Figure 7. With full recovery, the tensile damage variable has no effect in compression, and hence the cutting force is substantially larger than in the case of \(w_{\mathrm{c}}=0.15\). However, the volume of removed (ghost) elements (Figure 7b) is only 2.4% larger with full recovery. Therefore, the parameter \(w_{\mathrm{c}}\) seems to have a minor effect on the failure mode. Notwithstanding, the number of elements with low values of tensile damage is much larger here than it is in Figure 4b. Figure 5: Simulation results for PDC cutting at different cutting depths: removed elements and shape of cut grooves at (**a**) \(d=0.3\) mm, (**b**) \(d=0.9\) mm, and (**c**) \(d=1.5\) mm. Figure 6: Simulation results for PDC cutting (\(d=0.9\) mm, \(\beta=20^{\circ}\)): (**a**) predicted cutting force vs. cutting length; (**b**) experimental cutting force vs. cutting length (graph reproduced by estimation from the original graph in [4]). ### Effect of Rake Angle

The effect of rake angle, \(\beta\), (see Figure 2) was tested next. Other parameters were kept unaltered. Figure 8 shows the results in the case where \(d=0.9\) and \(\beta=10^{\circ}\). The damage patterns in Figure 8 do not exhibit major differences from the case with \(\beta=20^{\circ}\), and the volume of removed elements differs by only 2.3%. However, the cutting forces in Figure 8c show some deviations. ### Effect of Confinement

The final simulations concern deep drilling conditions, where high pressures, due to the overburden rock and/or the drilling mud pillar, as well as horizontal tectonic stresses, prevail. Two cases are tested. In the first (Conf1), the overburden pressure (\(p_{v}\) in Figure 2) is 50 MPa and the tectonic, horizontal pressure is 25 MPa; while in the second case (Conf2) \(p_{v}=20\) MPa and \(p_{h}=50\) MPa. Referring to Figure 3, the overburden pressure is applied to the \(y\)-direction and the horizontal pressure in the \(x\)- and \(z\)-directions. According to the simulations results, in Figure 9, the removed volumes are 106.9 mm\({}^{2}\) and 94.7 mm\({}^{2}\) in the confined cases Conf1 and Conf2, respectively, while 91.9 mm\({}^{3}\) of rock

Figure 8: Simulation results for PDC cutting with \(\beta=10^{\circ}\) (\(d=0.9\) mm, \(w_{\rm c}=0.15\)): (**a**) tensile and compressive damage fields; (**b**) removed elements; (**c**) cutting force vs. cutting length. Figure 7: Simulation results for PDC cutting with \(w_{\rm c}=1\) (\(d=0.9\) mm, \(\beta=20^{\circ}\)): (**a**) tensile and compressive damage fields; (**b**) removed elements; (**c**) cutting force vs. cutting length. removal is predicted in the unconfined case.