# Evaluation of the Implicit Gradient-Enhanced Regularization of a Damage-Plasticity Rock Model
## 4 Numerical Study
### Modeling of Mode I Failure, Demonstrated by Analyzing Wedge Splitting Tests on Indiana Limestone
The purpose of the unstructured mesh is to investigate a potential bias of the crack pattern by following the grid lines of the finite element mesh, since mesh-biased crack paths have been observed for smeared crack models based on the crack band approach [44]. In Figure 3, the resulting load-displacement curves, i.e., splitting force \(F_{\mathrm{sp}}\) versus CMOD, are shown for the considered experimental test and the numerical simulations. It can be concluded that the qualitative shape of the experimentally obtained curve is approximated quite well in the numerical simulations. Regarding the influence of the finite element mesh on the computed load-displacement curves, a slight difference between the predicted response based on the structured coarse mesh and the one based on the structured medium mesh is visible. This is explained by the rough approximation of the gradient of the nonlocal field by the coarse mesh. In contrast, almost identical load-displacement curves are obtained for the structured medium and the structured fine mesh, confirming that the gradient of the nonlocal field can be resolved sufficiently by those meshes. The unstructured mesh results in a similar load-displacement curve, demonstrating that the gradient-enhanced approach is not biased by the orientation of the finite element mesh. The computed deformation of the specimen is depicted in Figure 4 for the three structured meshes. By increasing the splitting force during the numerical simulations, the tensile strength of the material is attained at first in the elements directly below the notch; consequently, damage is initiated. This leads to large, localized deformations in this region. Upon further loading, the increase of these deformations  
Figure 3: Splitting force vs. crack mouth opening displacement (CMOD) for the wedge splitting test: experimental results (data taken from [40]) and numerical results. reflects the opening of a crack, propagating towards the bottom of the specimen. This is represented by the gradient-enhanced damage-plasticity approach in a smeared manner. The width of the damage zone is related to the length scale parameter \(l\) (cf. (16)). Furthermore, it can be seen that an identical symmetrical response with respect to the vertical axis of symmetry is obtained for all three structured meshes. In Figure 5, the distribution of the damage variable \(\omega\) computed for the three structured meshes is plotted at a CMOD = 0.3 mm. It is visible that damage is also accumulated slightly above the notch tip. This is a consequence of the diffusive character of the gradient-enhanced formulation. Furthermore, the present gradient-enhanced RDP model with the constant length scale \(l\) predicts a rather broad zone of complete damage (red region in Figure 5). In fact, this behavior does not represent damage localizing into a discrete macro crack, and has been addressed in [45; 46]. However, comparison of the predicted damage distributions for the different meshes demonstrates mesh-insensitivity of the obtained failure patterns. Figures 3 and 6 show an identical load-displacement curve and identical deformation pattern and damage distribution computed by means of the unstructured mesh, which underlines the capability of the gradient approach to produce mesh-insensitive results. Modeling of Shear Failure, Demonstrated by Analyzing Triaxial Compression Tests on Innsbruck Quartz Phylltite  
In a second step, the performance of the gradient-enhanced RDP model for predicting shear failure under confined stress states attaining the compressive meridian of the yield surface is assessed. To this end, numerical simulations of triaxial compression tests on specimens of Innsbruck quartz phyllite performed by [PERSON] (1988) are conducted. For the experimental program, intact rock specimens with a diameter of 35 mm and a height of 70 mm were taken from drill cores sampled at the construction site of the Brenner Base Tunnel.