# Evaluation of the Implicit Gradient-Enhanced Regularization of a Damage-Plasticity Rock Model
## 4 Numerical Study
### Numerical Simulation of Localizing Deformations in Deep Tunnel Excavation  
Finally, the performance of the gradient-enhanced RDP model for predicting the formation of multiple shear bands during the excavation of a deep tunnel is assessed. This benchmark is derived from a stretch of the Brenner Base Tunnel constructed by the drill, blast, and secure procedure, which has already been the subject of investigations in previous publications [52; 54]. An analogy to the present problem can be found in the context of petrol engineering, where the formation of shear bands has been observed and reported for borehole breakout [47; 55; 56]. Since in this contribution the major focus is on the assessment of the gradient-enhanced rock model, the tunnel excavation is approximated by means of a simplified two-dimensional model. Supporting measures like a shotcrete shell or rock anchors are neglected. Potential time-dependent effects of the mechanical behavior of rock mass due to the excavation procedure are not considered. For the excavation of the tunnel profile by means of drill and blast, either a full-face or a sequential excavation procedure can be employed. The chosen excavation sequence may have a considerable impact on the stability of the tunnel. In this numerical model, the worst case scenario is considered by assuming full-face excavation of the circular tunnel profile without any supporting measures, which results in the maximum loading of the rock mass in the vicinity of the tunnel. The IBVP of tunnel excavation with its geometry, initial conditions, and boundary conditions is illustrated in Figure 14. Within the discretized domain of rock mass, a hydrostatic geostatic stress state characterized by a pressure of \(p_{\mathrm{i}}^{(0)}=25.7\,\mathrm{MPa}\) is assumed, corresponding to the overburden at the tunnel axis of \(950\,\mathrm{m}\). In the numerical simulations, initial equilibrium is established by applying the geostatic stress state together with the internal pressure \(p_{\mathrm{i}}^{(0)}\) acting on the excavation boundary. Figure 12: Computed load–displacement curves for triaxial extension tests for different levels of confining pressure \(p^{(0)}\). Figure 13: Distribution of the vertical displacement \(u_{z}\) (deformation scale factor 10) at an applied top displacement of \(0.1\,\mathrm{mm}\) in the triaxial extension test with \(p^{(0)}=37.5\,\mathrm{MPa}\) for the three finite element meshes: coarse (**left**), medium (**center**), and fine (**right**). The excavation procedure is simulated by gradually decreasing the internal pressure \(p_{\rm i}^{(0)}\) to zero. Since the analyzed tunnel section is located in Innsbruck quartz phyllite rock mass, most of the material parameters of Section 4.2 are adopted. The material behavior of rock mass in contrast to intact rock is considered by empirical down-scaling factors based on the geological strength index \(GSI\) and the disturbance factor \(D\) proposed by [PERSON] [27] accounting for the influence of distributed discontinuities. They are taken from the geological survey, reported in [52]. The additional material parameters for Innsbruck quartz phyllite rock mass are \(GSI=40\), \(D=0\), \(A_{\rm s}=15\), \(\varepsilon_{\rm f}=7\times 10^{-4}\), and \(l=50\,\)mm. To trigger the formation of shear bands in spite of the axisymmetric problem, non-uniformly distributed rock mass properties are employed by introducing zones in which the strength of the rock mass is slightly weakened (indicated in Figure 14). It was verified that these weakened zones do not affect the predicted mechanical response before the onset of strain softening. For investigating the influence of the finite element mesh on the predicted results, three structured meshes with fully integrated and reduced integrated 8-node quadrilateral elements with element sizes of 300 mm (8078 elements), 140 mm (21,414 elements), and 70 mm (55828 elements) in the close vicinity of the tunnel are employed. Both the displacement field and the nonlocal field are approximated by quadratic shape functions.