# 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
Upon decreasing the internal pressure, initially the rock mass behavior remains in the linear elastic regime, followed by the formation of plastic zones emerging from the excavation boundary. Once the strength of the rock mass is attained, strain softening is initiated. From the onset of strain softening, the strains are localizing into narrow zones of the rock mass, and, eventually, large displacements accumulate. Finally, at a certain release level of the internal pressure, equilibrium is lost. Figure 15 shows the deformed rock mass at the level of 12% of the internal pressure for the three meshes with fully integrated elements. For the medium and the fine mesh, the localization of displacements into narrow zones has already reached a very progressed stage close to failure, and the formation of shear bands is clearly visible. The non-axisymmetric displacement field indicates the transition from an initially quasi-continuous rock mass to quasi-discontinuous rock mass. The latter is characterized by blocks of rock mass moving towards the tunnel center, so potential failure of the tunnel is imminent. Concerning the influence of the finite element discretization, for the medium and the fine mesh, an almost identical displacement field is obtained. Slight differences between those two solutions can be observed only for the upper right quadrant. Figure 14: 2D initial boundary value problem of deep tunnel excavation: full model (**left**) and the detail center view (**right**). Figure 16 shows the load-displacement curves, i.e., the normalized internal pressure versus the mean displacement magnitude along the excavation boundary computed for each mesh. The load-displacement curves predicted by the three meshes are very close to each other, and, in particular, the results for the medium and the fine mesh are almost identical. The results for the coarse mesh are slightly different due to the already discussed required mesh size for a sufficient resolution of the gradient of the nonlocal field. Figure 17 depicts the damaged zones in the rock mass at the level of 12% of the initial internal pressure for the three meshes employing full numerical integration. The highly damaged zones correspond to large shear deformations, which were shown previously in Figure 15. A similar shape of failure zones was obtained by [PERSON] et al. [57] in laboratory tests on a bore hole in weak sandstone. Hence, the potential of the gradient-enhanced approach to predict the onset of failure of a structure was demonstrated in spite of the rather complex failure mode. Figure 16: Load–displacement curves: normalized internal pressure versus the mean displacement magnitude at the tunnel surface for the three finite element meshes with full and reduced numerical integration: total view (**left**), detailed view (**right**). Figure 15: Distribution of the magnitude of the displacement vector in the vicinity of the tunnel surface at the level of 12 % of the initial internal pressure with a displacement scale factor of 10 for the three finite element meshes with full numerical integration: coarse (**left**), medium (**center**), and fine (**right**).