## 4 Numerical Simulation of Tunnel Excavation
### Influence of the Anisotropy Parameters  
For investigating the influence of the transversely isotropic behavior predicted by the TI-RDP model on the mechanical response due to tunnel excavation, a parameter study based on the variaton of the two anisotropy parameters \(\gamma\) and \(\delta\) (7) is performed. \(\gamma\) and \(\delta\) represent the ratio of the uniaxial compressive strength perpendicular and parallel to the foliation planes and the ratio of the shear strength in the planes of isotropy and  
Figure 8: Illustration of the 2D finite element model of the tunnel excavation: full model (**left**) and detailed view of the FE-mesh in the vicinity of the circular tunnel (**right**). in planes perpendicular to the latter, respectively. A horizontally foliated rock mass (\(\beta=0^{\circ}\)) is considered. As a benchmark for assessing the influence of the anisotropy parameters \(\gamma\) and \(\delta\) serves the simulation with the isotropic RDP model. For the latter, the parameters \(\gamma=1\), \(\delta=1\) and the isotropic elastic stiffness tensor in the stress-strain relation (1) with a [PERSON]'s modulus of \(E=E_{1}=E_{2}=\)45,000 MPa, a [PERSON]'s ratio of \(\
u=v_{12}=v_{23}=0.254\) and a shear modulus of \(G=G_{12}=E/\left(2\left(1+\
u\right)\right)\) are employed. Figure 9 shows the distribution of the damage variable \(\omega\) (left) and the displacement magnitude (right) in the final stage of the simulation before failure, i.e., for a release of the surface pressure of 97.5%. Due to the excavation, for a release of the surface pressure exceeding \(\sim\)90%, the rock mass is subjected to softening, which leads to localization of strains into multiple distinct shear bands, cf. Figure 9 left. In the simulation, strain localization is triggered by the weakened elements, mentioned in Section 4, breaking the otherwise expected axisymmetric behavior. A similar shape of the damage pattern was observed in experiments performed, for example, by [PERSON] et al. [54] or [PERSON] et al. [55]. In Figure 10, the distribution of the damage variable \(\omega\) (left) and the displacement magnitude (right) is shown for two different values of the anisotropy parameter \(\gamma=1.25\) (top) and \(\gamma=0.75\) (bottom) and a constant value of \(\delta=1\) in the final stage of the simulation before failure, at a release of the surface pressure \(p_{\rm i}\) of 96%. From Figure 10A,B follows that for \(\gamma=1.25\) the highly damaged zones are concentrated around the side walls of the tunnel, where the surface pressure \(p_{i}\) acts in the direction parallel to the foliation planes. This is attributed to the lower uniaxial compressive strength parallel to the foliation planes, \(f_{\rm cu}^{(2)}\), compared to the uniaxial compressive strength perpendicular to the foliation planes, \(f_{\rm cu}^{(1)}\). In Figure 10C,D the damage distribution and displacement pattern are shown for \(\gamma=0.75\), resulting in a lower value for \(f_{\rm cu}^{(1)}\) compared to \(f_{\rm cu}^{(2)}\). Compared to the side walls, this results in a concentration of the highly damaged zones and consequently larger displacements at the top and the bottom of the tunnel cross section. Here the surface pressure at the excavation boundary acts in the direction perpendicular to the foliation planes. Figure 9: Distribution of the damage variable \(\omega\) (**left**) and the displacement magnitude (**right**) (displacement scale factor 3) in the final stage of the tunnel simulation using the **isotropic** RDP model. Figure 11 shows the distribution of the damage variable \(\omega\) (left) and the displacement magnitude (right) obtained for different values of \(\delta=1.25\) (top) and \(0.75\) (bottom) and a constant value of \(\gamma=1.0\) in the final stage of the simulation before failure at a release of the surface pressure \(p_{\rm i}\) of \(97.5\%\).