## 2 Numerical Methods
### Erosion Scheme
Moreover, the viscosity moduli given are chosen, on the one hand, to be small enough so as to not cause any significant strain rate effects, and, on the other hand, to be large enough that they have a stabilizing effect on the stress return mapping at the material point level. Furthermore, the parameters \(w_{\mathrm{t}}\) and \(w_{\mathrm{c}}\), controlling the degree of stiffness recovery upon stress reversals, are chosen so as to match the cutting experiments. Finally, the cutting speed is a compromise between computational labour of the explicit time integration and the requirement of keeping it as close as possible to what was used in the experiments, 5 mm/s. The stable time step for the present simulations is 3.15 \(\times\) 10\({}^{-8}\) s, so that simulating a 10 mm stroke of cutting at 5 mm/s would require more than 600 million time steps. As performing the analyses of a single time step takes \(\sim\)0.4 s of CPU time, using such a cutting speed is not feasible. This is a common problem in modelling cutting problems with explicit codes, as exemplified by [PERSON] et al. [20], who used a cutting speed of 4 m/s in their finite element simulations. The finite element mesh, made of 523,928 linear tetrahedrons, is shown in Figure 3. The mesh has a 10 mm long initial groove, roughly representing a cut groove. The purpose is to avoid excessive edge effects and, thus, to reach stabilized conditions earlier in the cutting process. ### Validation Simulations

The validation simulations concern comparing the cutting force predicted with the present method, against the experiments from [4], at cutting depths of 0.3, 0.9, and 1.5 mm. Moreover, the rake angle is 20\({}^{\circ}\), while the radius of the cutter is 9.5 mm. Figure 4 shows the results after a 12 mm stroke. In each case, there are heavily damaged, both in tensile and compressive (shear) mode, regions, representing the grooves of removed rock. Most of the elements in these regions are eroded (ghost) elements, having interaction neither with the cutter nor with the intact rock. The related cutting force graphs exhibit huge initial peaks, reaching 3.6 kN even at the smallest cutting depth, due to the smooth initial contact surface causing massive fracture events at its edge. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \(E\) [GPa] & \(\
u\) & \(\sigma_{\mathrm{t0}}\) [MPa] & \(c_{0}\) [MPa] & \(\varphi\) [\({}^{\circ}\)] & \(\psi\) [\({}^{\circ}\)] & \(G_{\mathrm{Ic}}\) [N/m] & \(G_{\mathrm{Ic}}\) [N/m] & \(\rho\) [kg/m\({}^{3}\)] \\ \hline
35 & 0.28 & 11 & 30 & 50 & 5 & 40 & 1000 & 2600 \\ \hline \(A_{\mathrm{t}}\) & \(A_{\mathrm{c}}\) & \(s_{\mathrm{RM}}\) [MPa\(\cdot\)s] & \(s_{\mathrm{DP}}\) [MPa\(\cdot\)s] & \(w_{\mathrm{t}}\) & \(w_{\mathrm{c}}\) & \(\omega_{\mathrm{mesh}}\) & \(v_{0}\) [m/s] & \(\mu\) \\ \hline
0.9998 & 0.9998 & 0.01 & 0.01 & 0 & 0.15 & 0.99 & 1 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties (from [4]) and model parameters. After the initial stage, oscillations in the cutting force are much milder, representing more local fracture events. The actual removed or ghost elements are plotted in Figure 4. The volume of removed rock material, represented as the volume of the elements where both damage variables exceed the threshold 0.99, increases linearly as a function of the cutting depth, as can be easily calculated from the values given in Figure 5. Unfortunately, the authors in [4] do not provide data to enable comparison to their experiments. However, the circular cross-sectional shape of the removed element volumes in Figure 5, and the bottom shape of the cut grooves in the rock, agree with the experimental grooves cut into granite with traditional circular PDC cutters in [11].