# 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 experimental results for confining pressures of \(p^{(0)}=0\,\mathrm{MPa}\), \(12.5\,\mathrm{MPa}\), and \(25\,\mathrm{MPa}\), which have not been used for calibration, serve for validation of the numerical model. Figure 8 shows the load-displacement curves obtained from the experiments and the numerical simulations for the different confining pressures. Note that the non-zero axial force at the beginning is the consequence of the initial hydrostatic stress state due to the applied confining pressure. Expectedly, for \(p^{(0)}=37.5\,\mathrm{MPa}\) and for the uniaxial compression test (\(p^{(0)}=0\,\mathrm{MPa}\)), the experimentally obtained peak load is represented very well since the test results were used for calibration. For \(p^{(0)}=12.5\,\mathrm{MPa}\) and \(25\,\mathrm{MPa}\), the peak loads are predicted satisfactorily, slightly underestimating the experimental results. The qualitative shape of the softening branch is also represented quite well. For the uniaxial compression test, no meaningful experimental results after the peak stress were recorded during the experiments. This unstable behavior is also manifested in the numerical simulations, for which strong snapback behavior is observed. The numerical results computed on the basis of the different finite element meshes reveal the capability of the gradient-enhanced RDP model to regularize the underlying IBVP: Once the mesh is sufficiently fine, mesh-insensitive load-displacement curves are obtained. Figure 9a shows the deformed specimen with the confining pressure \(p^{(0)}=37.5\,\mathrm{MPa}\) in the final stage of the triaxial compression tests, computed by means of the three different meshes. While the displacements in the lower and the upper part of the specimen are almost uniform, the displacements localize into a single inclined shear band in the center part of the specimen. In the experiments, localization into a shear band was found to be the dominant failure mode, as shown in Figure 9b for a specimen tested with \(p^{(0)}=37.5\,\mathrm{MPa}\). The formation of the shear band is also reflected by the distribution of the damage variable \(\omega\) shown in Figure 10. Comparing the predicted damage  
Figure 7: Geometry and boundary conditions of the specimen for the triaxial compression test. distributions for the medium and the fine mesh confirms this distribution as insensitive with respect to the discretization. The influence of the level of confining pressure on the inclination of the shearing zone is demonstrated in Figure 11 based on the predicted distribution of the vertical displacement for \(p^{(0)}=0\,\mathrm{MPa},12.5\,\mathrm{MPa},\,25\,\mathrm{MPa},\,\mathrm{and}\, 37.5\,\mathrm{MPa}\), respectively. For the uniaxial compression test (\(p^{(0)}=0\,\mathrm{MPa}\)), the zone of localized displacements is strongly inclined. With increasing confining pressure, the inclination angle of the shear band is decreasing, which is best visible by comparing  
Figure 8: Load–displacement curves (axial force versus axial displacement) for triaxial compression tests: experimental and numerical results for different levels of confining pressures \(p^{(0)}\). Figure 10: Distribution of the damage variable \(\omega\) in the final stage of the triaxial compression test with \(p^{(0)}=37.5\,\mathrm{MPa}\) for the three finite element meshes: coarse (**left**), medium (**center**), and fine (**right**). Figure 2:0 (**a**) Distribution of the vertical displacement \(u_{z}\) (scale factor 5) in the final stage of the triaxial compression test with \(p^{(0)}=37.5\,\mathrm{MPa}\) for the three finite element meshes: coarse (**left**), medium (**center**), and fine (**right**). (**b**) Corresponding deformed rock specimen after a triaxial compression test with \(p^{(0)}=37.5\,\mathrm{MPa}\), reproduced with permission from [PERSON] taken from the report [41]. the results for \(p^{(0)}=0\,\)MPa and \(p^{(0)}=12.5\,\)MPa. Upon further increase of the confining pressure, the inclination of the shear band becomes slightly smaller.