## 3 Results  
The results obtained after running the models described in Section 2 will be hereby presented. Please note that compressive stresses are negative. In particular, the interest is in the possible spalling phenomenon occurring during the tunnel excavation. Figure 8 shows the initial principal maximum stress configuration around the excavation at the instant in which the core is removed at time step 100,000 (the initial time step with the tunnel completely excavated, thus reproducing the face at the instant in which the Young's modulus of the core is set to zero), while Figures 9-11 exhibit the evolution of the crack propagation due to the stress-induced damage, starting from the same time step to the last computed time step 600,000.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Stress Ratio \(\mathbf{k_{0}}\) = \(\sigma_{\mathbf{h}}\)/\(\sigma_{\mathbf{v}}\)** & \(\sigma_{1}\) = \(\sigma_{\mathbf{h}}\) = \(\sigma_{\mathbf{xx}}\) (MPa) & \(\sigma_{3}\) = \(\sigma_{\mathbf{v}}\) = \(\sigma_{\mathbf{yy}}\) (MPa) \\ \hline A & 1.0 & 66.4 & 66.4 \\ B & 1.5 & 99.6 & 66.4 \\ C & 2.0 & 132.8 & 66.4 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Initial in situ stress conditions adopted for the different cases considered.  
Figure 8 underlines the presence of high in situ stress, as the cracking, except for the hydrostatic condition, is triggered immediately after the excavation. Regarding Figures 9-11, at the last time step 600,000, spalling occurring around the tunnel is quite extended already with the lowest in situ stress (k\({}_{0}\) = 1). This evidence points out the necessity of considering a support to be installed to prevent the fall of blocks from the contour of the cavity. Considering the conditions with k\({}_{0}\) = 1.5 and k\({}_{0}\) = 2, the extension of the spalling phenomenon increases, as expected, in the crown and the invert of the tunnel, as in the remaining parts the tangential stress \(\sigma_{\theta}\) is not increasing. The results also reveal the need to consider enough time steps to capture crack propagation and show the full depth of the unstable blocks.  
Figure 8: Principal maximum stress around the excavation at the instant of the core removal.  
Figure 9: Simulated evolution of displacement contour and fracture pattern for k\({}_{0}\) = 1.0 (video provided in the Supplementary Materials).  
Regarding the simulation including the simplified lining layer, the results are presented in Figure 12. In this case, there is evidence of the contribution of the reinforcement around the opening, which ideally minimises the deformations occurring in the contours. On the other hand, the same picture highlights the price of such a contribution in terms of stress applied: the maximum compressive stress induced in the elastic lining ranges from around 190 MPa for the hydrostatic in situ stress case to 340 MPa with k\({}_{0}\) = 2.0. These values represent a theoretical threshold that cannot be reached with the usual technologies (a typical stiff concrete support can hold some tens of MPa in compression). Thus, in this case, the maximum compressive stress induced in the elastic lining ranges is \(\approx\) 1.5.  
Figure 11: Simulated evolution of displacement contour and fracture pattern for k\({}_{0}\) = 2.0 (video provided in the Supplementary Materials).  
Figure 10: Simulated evolution of displacement contour and fracture pattern for k\({}_{0}\) = 1.5 (video provided in the Supplementary Materials).  
it appears to be more realistic to apply a different kind of support (for example, bolting and shotcreting), which can allow enough relaxation to the rock mass to reduce the stress applied to the support itself.