## 2 Numerical Methods
### Rock Failure Model  
The rock is described as an isotropic homogeneous material up to the elastic limit, after which the material softens, and its strength and stiffness degrade. Temperature effects are, as usual in numerical studies on PDC cutting, neglected. However, it should be noted that considerable heat generation may occur due to friction, especially in rock cutting with conical picks [8]. In cutting, on the one hand, the rock material undergoes large deformations. On the other hand, rock, as a brittle material, cannot withstand large deformations, at least in normal conditions. Therefore, it seems to be enough if the constitutive model can describe finite rigid body rotations. For this reason, the St. Venant-Kirchhoff material model is applied to write the elasticity part of the rock model as:  
\[\sigma =\mathbf{C}_{\mathrm{e}}:\big{(}\epsilon_{\mathrm{GL}}-\epsilon_{ \mathrm{vp}}\big{)} \tag{1}\] \[\epsilon_{\mathrm{GL}} =\frac{1}{2}(\mathbf{C}-1),\,\mathbf{C}=\mathbf{F}^{\mathrm{T}} \mathbf{F} \tag{2}\]  
where \(\sigma\) is the stress tensor, \(\mathbf{C}_{\mathrm{e}}\) is the elasticity tensor, \(\epsilon_{\mathrm{GL}}\) is the Green-Lagrange strain tensor, \(\epsilon_{\mathrm{vp}}\) is the viscoplastic strain tensor, and \(\mathbf{C}\) is the right Cauchy-Green deformation tensor, defined by the deformation gradient \(\mathbf{F}\), and \(1\) is the second order identity tensor. The colon (:) in (1) denotes the double contraction operator for tensors, i.e., \(A:A=A_{ij}A_{ij}\) where repeated indices imply summation. The stress in (1) should be the 2 nd Pidal-Kirchhoff stress, \(\mathbf{S}\), but under the present small deformation framework, all stress measures reduce to the same numerical values. The softening process, in both compression and tension, is governed by a phenomenological damage-viscoplasticity model. For present purposes, the viscoplasticity part is formulated with the linear Drucker-Prager surface, cut by the rounded Rankine criterion, to match the tensile strength. Mathematically, these surfaces read:  
\[f_{\mathrm{DP}}\big{(}\sigma,\dot{\kappa}_{\mathrm{DP}}\big{)}=\sqrt{I_{2}}+ \alpha_{\mathrm{DP}}(\varphi)I_{1}-k_{\mathrm{DP}}(\varphi)c\big{(}\dot{ \kappa}_{\mathrm{DP}}\big{)} \tag{3}\]  
\[g_{\mathrm{DP}}(\sigma)=\sqrt{I_{2}}+\beta_{\mathrm{DP}}(\psi)I_{1} \tag{4}\]  
\[f_{\mathrm{MR}}\big{(}\sigma,\dot{\kappa}_{\mathrm{MR}}\big{)}=\sqrt{\sum_{i= 1}^{3}\langle\sigma_{i}\rangle^{2}}-\sigma_{\mathrm{t}}\big{(}\dot{\kappa}_{ \mathrm{MR}}\big{)} \tag{5}\]  
\[c\big{(}\dot{\kappa}_{\mathrm{DP}}\big{)}=c_{0}+s_{\mathrm{DP}}\dot{\kappa}_{ \mathrm{DP}},\,\sigma_{\mathrm{t}}\big{(}\dot{\kappa}_{\mathrm{MR}}\big{)}= \sigma_{\mathrm{t}0}+s_{\mathrm{MR}}\dot{\kappa}_{\mathrm{MR}} \tag{6}\]  
where the symbol meanings are as follows: \(I_{1}\) and \(J_{2}\) are the first and the second invariants of the stress tensor \(\sigma\), respectively; \(c_{0}\) and \(\sigma_{\mathrm{t}0}\) are the intact cohesion and tensile strength; \(\sigma_{i}\) is the \(i\)th principal stress, from which only the positive part is considered through McAuley brackets in the MR criterion (5); \(\dot{\kappa}_{\mathrm{DP}},\,\dot{\kappa}_{\mathrm{MR}}\) are the rates of the internal variables in compression and tension; \(s_{\mathrm{DP}},s_{\mathrm{MR}}\) are the constant viscosity moduli; \(\alpha_{\mathrm{DP}}=2\sin\phi/(3-\sin\phi)\), \(k_{\mathrm{DP}}=6\cos\phi/(3-\sin\phi)\) are DP parameters defined with the internal friction angle \(\phi\), so as to match the uniaxial compressive strength. Equation (4) is the viscoplastic potential, where \(\beta_{\mathrm{DP}}\) is a parameter, controlling the dilation, similar to \(\alpha_{\mathrm{DP}}\) but defined with the dilation angle \(\psi\). Thermodynamic consistency is guaranteed for \(\psi\leq\phi\)[28], while associated viscoplasticity is recovered when \(\psi=\phi\). Equation (6) accounts for the strain rate dependency of the rock. The damage part of the model has separate scalar damage variables for tensile and compressive stress regimes.