## 2 Materials and Methods
### Description of the Case Study  
The case study considered in this paper is the Turin-Lyon tunnel, which will be part of a new railway line to connect the two cities. Such a line is an essential component of the Mediterranean corridor that will connect Algeciras, in Spain, with Budapest, in Hungary, and which, in turn, is part of the TEN-T project, the European Union's high-speed rail network. The project will have to be suitable for transalpine freight traffic, as it is intended in particular to limit road traffic in these sensitive areas from an environmental point of view. The connection between Lyon and Turin will extend for 270 km and will also have great relevance for the transport of passengers since it will connect the Italian and French high-speed networks, therefore reducing travel times between the two frontier regions, namely Piedmont and the Savoy. The main infrastructure of the Turin-Lyon connection is the Mont Cenis Base Tunnel (Figure 5), which will allow crossing the Alps at the plain level by reducing time and cost for transporting goods and people. It is a twin-tube, single-track tunnel 57.5 km long (12.5 km in Italian territory and 45 km in French territory) with a slope of less than 0.12% and an overburden of more than 2000 m. The tunnel's diameter is 10.4 m. The tunnel connects the international stations of Saint-Jean-de-Maurienne and Susa, points of connection to their respective national lines in France and Italy, and it is equipped with four access points, railway service areas, and ventilation shafts. The Base Tunnel route passes through an extremely complex geological context in terms of lithological variety and geomechanical-hydrogeological aspects [33]. As seen in Figure 6, the formations crossed from west to east will start with the loose deposits of the Saint-Julien alluvial fan, followed by the flysch lithotypes of the Zone ultradauphinoise (External Alps). Once the Pennine Front is crossed, the Zone subrianconnraise, the Zonehouillere, the Zone brianconnraise (Vanoise), the Zone piemontraise (Schistes lustres), the Nappe des gypes, the Zone brianconnraise (Massif d'Ambin), and the Zone piemontraise (calscchists) will be intersected. These formations are increasingly deformed from west to east and are separated from each other by tectonised horizons [33]. These very complex geological conditions were recognised with more than 40 deep boreholes from the surface, about 500 boreholes while advancing, and 17 km of access a  
Figure 5: Summary view of the Mont Cenis Base Tunnel (modified from [33]). In particular, the tunnel will reach its maximum depth (and temperatures) below the Ambin Massif. Here, the prevailing lithology features hard rock such as micaschists and gneiss, and water is present in small quantities [34; 35]. Ref. [35] reports that in geomechanical terms, the rock mass varies between fair and good quality. Because of the high in situ stresses in this zone and the typical lithology described above, problems induced by brittle failure are to be expected. For this reason, a 2D cross-section of the tunnel at a depth of 2500 m in the Ambin Massif was selected for the numerical analyses described in this paper. As described in [34], the stiffness and strength properties of the Ambin Massif microsachists and gneiss (Table 1) can be derived from the characterisation of the F2 tunnel excavated in the same massif along the Val Clarea for the Pont Ventoux-Susa Hydropower Project according to [36], as well as the in situ state of stress known at lower depths thanks to in situ testing. The Ambin Massif is known to be characterised by a stress state in the horizontal plane greater than the vertical stress. Y  
Given such properties, numerical analyses are performed and described in the following in order to simulate deep, circular tunnel excavation in the Ambin Massif zone at a depth of 2500 m and to investigate the effectiveness of the FDEM in reproducing spalling behaviour.