## 2 Materials and Methods
### Construction of the Numerical Model
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value or Type** \\ \hline Friction coefficient, \(f_{t}\) & 0.7 \\ Type of fracture model & Isotropic \\ Cohesion, \(c\) & 23 MPa \\ Tensile strength, \(f_{t}\) & 10 MPa \\ Mode I fracture energy, \(G\!f_{l}\) & 32.5 N/m \\ Mode II fracture energy, \(G\!f_{l}\) & 650 N/m \\ Fracture penalty, \(p_{t}\) & 17.5 GPa \\ Normal penalty, \(p_{n}\) & 35 GPa\(\cdot\)m \\ Tangential penalty, \(p_{t}\) & 8.5 GPa/m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Strength properties and penalty values assigned to the cohesive crack elements in the numerical model. Figure 7: Geometry of the FDEM model. than the vertical one. The in situ state of stress is based on the model condition described in [34], by considering the vertical component \(\sigma_{\rm v}\) equal to 66.4 MPa. The calculation then examines three cases with k\({}_{0}\) equal to 1, 1.5, and 2. The different initial in situ stress conditions applied are shown in Table 3, with reference to the three independent values of [PERSON]'s stress tensor in a 2D \(xy\) system of reference \(\sigma_{\rm x}\), \(\sigma_{\rm y}\), and \(\sigma_{\rm xy}\) (the latter is null). Since the tunnel is deep, gravitational gradients are neglected. The calculation is carried out, enabling the absorbing boundary condition to speed up the convergence of the in situ stage. Moreover, the gravity option is not enabled, so the displacements shown refer only to the effect due to the spalling. The time step size selected for the equation of motion integration is 1 \(\times\) 10\({}^{-7}\) s, and the total number of time steps is 600,000. The output frequency of results is set at 20,000 time steps (30 outputs). First, an FE analysis is run where crack elements are not generated, exactly as in a FE model. After time-stepping until elastic equilibrium, the initial state of stress is applied. The nodal coordinates calculated by the FEM solver (deformed mesh) are input for the following FDEM simulation run. The number of steps selected for the FE simulation was successfully verified by observing nodal velocities, i.e., to check whether equilibrium had been reached. Tunnel excavation is simulated by using a preliminary core replacement procedure. Therefore, the initial Young's modulus of the tunnel core material was progressively softened by two orders of magnitude over 75,000 time steps, from time step 25,000 to time step 100,000, in a linear way. This was carried out to mimic the gradual 3D excavation process and the progressive advancement of the tunnel face. Indeed, the sudden removal of the core material could result in dynamic artefacts and unrealistic fractures. Then, the inner core elements are removed by setting them to \"excavated\", i.e., the corresponding triangular FEs are instantaneously removed from the simulation loop, and the state of the surrounding crack elements is set to \"broken\". The models describing the three cases A, B, and C are simulated, considering no lining installed and examining the possibility of installing a simplified lining around the opening. This stage consists of inserting a 400-millimetre-thick layer of concrete in contact with the contour of the tunnel. The layer is made of an elastic material with Young's modulus equal to 30 GPa, which is installed just one time step after the complete excavation, deriving from the above-mentioned core replacement procedure. This methodology represents a simplification, as the concrete is installed immediately after the simulated core excavation; therefore, the deformations are minimal, whereas the stresses on the lining are very high. In this case, the results are helpful in simulating and identifying crack propagation in the presence of a hypothetical reinforcement intervention.