# Evaluation of the Implicit Gradient-Enhanced Regularization of a Damage-Plasticity Rock Model
## 5 Conclusions  
The present contribution addressed the prediction of complex failure modes in numerical simulations by means of a new implicit gradient-enhanced damage-plasticity model for intact rock and rock mass. It was derived from the damage-plasticity rock model presented by [PERSON] et al. [24; 25] and extended by adopting the implicit gradient-enhancement proposed in [17]. For the assessment of the model, a comprehensive numerical study was presented. In particular, numerical simulations of wedge splitting tests for evaluating the representation of mode I failure, simulations of triaxial compression and extension tests for evaluating the representation of shear failure under confined stress states, and finally simulations of deep tunneling for examining the prediction of failure mechanisms of a complex structure were performed. From the obtained results of the numerical study, the following conclusions can be drawn:  
* In numerical simulations of wedge splitting tests, the formation of a crack propagating from the notch is modeled by the implicit gradient-enhancement in a smeared manner over a width related to the assumed length scale parameter. The capability of the gradient-enhanced damage-plasticity rock model of representing the experimentally observed material behavior was realistically demonstrated. * Finite element analyses with both structured and unstructured meshes confirmed the regularizing effect of the implicit gradient-enhancement in mode I failure and thus revealed mesh-insensitive results. In particular, it was demonstrated that the crack direction is not biased by the orientation of the finite element mesh. * Regarding the simulations of triaxial compression tests, a proper representation of shear failure under confined stress states, attaining the compressive meridian of the yield surface, was shown. After attaining peak strength upon initiation of damage, localization into a distinct shear band was observed. Furthermore, reasonable agreement with experimental results, and in particular the experimentally observed increasingly ductile material behavior with increasing confining pressure was obtained. * The simulations of triaxial extension tests demonstrated that failure under confined stress states attaining the tensile meridian of the yield surface is represented reasonably well, i.e., the localization of damage into a distinct shear band is predicted. In contrast to the triaxial compression tests, a more brittle structural response was observed. * For both the triaxial compression and extension tests, a mesh study confirmed mesh-insensitive results for shear failure under confined stress states. * In the numerical simulations of deep tunnel excavation, the rock mass was subjected to softening material behavior due to the excavation procedure, which leads to localization of strains into multiple distinct shear bands. In spite of the complex structural failure mechanism involving the  
Figure 17: Distribution of the damage variable \(\omega\) in the rock mass in the vicinity of the tunnel surface at the level of 12 % of the internal pressure for the three finite element meshes employing full numerical integration: coarse (**left**), medium (**center**), and fine (**right**). formation of multiple shear bands, mesh insensitive load-displacement curves were obtained. By analyzing the convergence of the damage patterns upon mesh refinement, only slight differences were recognized. * For the adopted formulation of the implicit gradient-enhanced rock model, the length scale \(l\) is assumed as a constant parameter. Thus, a rather broad zone of completely damaged material is predicted by the model. Possible remedies were proposed in the literature, for instance, based on variable length scale parameters [58; 59] or the so-called concept of decreasing interactions [45], which is motivated by a physically based micromechanical homogenization theory [60], cf. [46] for a discussion of these approaches.