## 2 Materials and Methods
### Construction of the Numerical Model  
The construction of the numerical model has to consider that fracture trajectories are restricted along the edges of finite elements and that larger elements will require more significant energy to fracture. For this reason, the mesh setup, beyond affecting the elastic response like any other Finite Element (FE) code, also influences strength response and fracture pattern and has to be designed appropriately with a sufficiently small size, especially in areas where high stress/strain gradients and fracturing are expected, and the adoption of unstructured grids with random triangulation. Given its simplicity, the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Properties** & **Value** & **Properties** & **Value** \\ \hline E (GPa) & 35 & s\({}_{\text{P}}\) (-) & 0.0558 \\ v (-) & 0.25 & s\({}_{\text{r}}\) (-) & 0.0054 \\ GSI (-) & 90 & c\({}_{\text{P}}\) (MPa) & 26.5 \\ \(\sigma_{\text{ti}}\) (MPa) & 10 & \(\Phi_{\text{P}}\) (-) & 15.8 \\ \(\sigma_{\text{ci}}\) (MPa) & 144 & c\({}_{\text{r}}\) (MPa) & 0.6 \\ \(\text{mp}\) (-) & 0.39 & \(\Phi_{\text{r}}\) (\({}^{\circ}\)) & 55.0 \\ \(\text{m}_{\text{r}}\) (-) & 8 & \(\gamma\) (kg/m\({}^{3}\)) & 2500 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ambin Massif rock mass stiffness and strength parameters ([PERSON]’s modulus E, Poisson’s ratio \(\
u\), Geological Strength Index GSI, intact rock tensile strength \(\sigma_{\text{ti}}\), intact rock unconfined compressive strength \(\sigma_{\text{ci}}\), Hoek and Brown parameters for peak and residual condition \(\text{mp}\), \(\text{m}_{\text{r}}\), \(\text{sp}\), \(\text{sp}\), \(\text{sp}\), \(\text{sp}\), \(\text{sp}\), peak and residual cohesion \(\text{c}_{\text{p}}\), \(\text{c}_{\text{r}}\), peak and residual friction angle \(\Phi_{\text{P}}\), \(\Phi_{\text{r}}\), density \(\gamma\)) [34; 36]. Figure 6: Sketch of the longitudinal geological cross-section along the Mont Cenis Base Tunnel (modified from [[https://chantier.smp4.eu/travaux/](https://chantier.smp4.eu/travaux/)]([https://chantier.smp4.eu/travaux/](https://chantier.smp4.eu/travaux/)), accessed on 3 January 2024). model geometry and mesh were built directly into the Irazu User Interface, as shown in Figure 7. External boundaries are placed far enough from the area of interest (10 times the diameter) not to constrain the simulated displacement field. As depicted in Figure 7, the model considers a 10.4 m diameter circular tunnel at the centre of a 200 m \(\times\) 200 m square domain; the model mesh comprises 60,678 elements and 30,420 nodes. The average size of the elements in the immediate surroundings of the excavation is set at 0.1 m, while outside, it is 5 m. Based on an automatic mesh quality check, the minimum angle, the edge length, and the aspect ratio fall into acceptable ranges. Since the model is two-dimensional, a plain strain condition is assumed. About boundary conditions, rollers with zero velocity in the vertical direction are assigned to the nodes belonging to the horizontal external boundaries; rollers with zero velocity in the horizontal direction are assigned to the nodes belonging to the vertical external boundaries; pins are applied to the four external vertices to prescribe zero velocity in all directions. The domain is homogeneous. Elastic properties are assigned to the constant strain triangular finite elements pertaining to both the rock mass and the tunnel core to be excavated (see Table 1, considering a unit viscous damping). Instead, strength properties and penalty values are assigned to the cohesive crack elements placed between all finite elements throughout the model. The values pertaining to these properties are also derived from [34] and are summarised in Table 2.  
Based on the above-mentioned observation of an anisotropic in situ state of stress, the domain is subjected to a uniform pre-excavation stress field with a horizontal stress higher