## 4 Numerical Simulation of Tunnel Excavation
### Influence of the Inclination of the foliation planes  
The influence of the inclination of the foliation planes on the mechanical response of the 2D tunneling simulation is investigated by considering four different values of the inclination angles of the foliation planes \(\beta=0^{\circ},30^{\circ},60^{\circ}\) and \(90^{\circ}\). In Figure 12, the distribution of the damage variable \(\omega\) is shown for the four inclination angles \(\beta\) in the final stage of the simulation before failure at a release of the surface pressure \(p_{\rm i}\) of \(97.5\%\). It can be seen that the failure pattern, manifested by the distribution of the highly damaged zones, strongly depends on the inclination angle of the layered rock mass. Considering the mechanical properties of _Innsbruck quartz-phyllite_, cf. Table 1  
Figure 11: Distribution of the damage variable \(\omega\) (**left–A,C**) and the displacement magnitude (**right–B,D**) (scale factor = 3) in the final stage of the tunnel simulation using the **transversely isotropic** RDP model with values of \(\delta=1.25\) (**top–A,B**) and \(0.75\) (**bottom–C,D**) and a constant value \(\gamma=1\). (\(\gamma=1.529\), \(\delta=1.201\), \(E_{2}/E_{1}=0.87\)), it can sustain higher loads for loading perpendicular to the foliation planes compared to loading parallel to the foliation planes. Thus, the highly damaged zones are localized where the predominant loading direction, i.e., the direction of the acting surface pressure, is aligned with the foliation planes (cf. Figure 12A-D). Note that for the present axisymmetric tunnel geometry and the hydrostatic initial geostatic stress, the damage patterns for the different inclinations of the foliation planes are not only obtained by a rotation around the tunnel center but are also slightly affected by the randomly distributed weakened zones. In Figure 13, the release of the normalized surface pressure \(p_{\rm i}/p_{\rm i}^{(0)}\) is depicted in terms of the displacement magnitude for three characteristic points, i.e., top (M1), left (M2) and right side wall (M3) of the tunnel (cf. Figure 12A) and the four inclination angles \(\beta=0^{\circ},30^{\circ}\), \(60^{\circ}\) and \(90^{\circ}\). In addition, the corresponding results from the simulation using the isotropic RDP model are shown. Up to a stress release of 90%, i.e., \(p_{\rm i}/p_{\rm i}^{(0)}=10\%\), the influence of the inclination of the foliation planes on the displacement magnitude is negligible for the three considered points M1 to M3. However, from the onset of softening behavior, i.e., for a stress release  
Figure 12: Distribution of the damage variable \(\omega\) in the final stage of the simulation, determined by the **transversely isotropic** RDP model with the calibrated material parameters of Section 3 for four different inclination angles \(\beta\) of the foliation planes, i.e., \(\beta=0^{\circ}\) (**A**), \(\beta=30^{\circ}\) (**B**), \(\beta=60^{\circ}\) (**C**) and \(\beta=90^{\circ}\) (**D**). exceeding 90%, the displacement magnitude varies significantly since the strains localize into narrow zones of the rock mass, leading to the formation of shear bands in the vicinity of the tunnel. Since no securing measures are considered in this study, equilibrium is lost beyond the stress release of approximately 97.5%. By employing the **isotropic** RDP model (dashed line), the displacement magnitude in the final stage of the simulation is in the range of 117 to 120 mm at the top and the side  
Figure 13: Normalized surface pressure \(p_{\mathrm{i}}/p_{\mathrm{i}}^{(0)}\) as a function of the displacement magnitude \(u_{\mathrm{mag}}\) for three characteristic points, i.e., Top: M1 (**top**), Left: M2 (**middle**) and Right: M3 (**bottom**) and the four inclination angles of the foliation planes \(\beta=0^{\circ}\) (orange), \(30^{\circ}\) (red), \(60^{\circ}\) (grey) and \(90^{\circ}\) (blue) compared to the isotropic (dashed) simulation. walls of the tunnel cross section (cf. Figure 13 (M1) to (M3)).