# 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
A series of triaxial compression tests with different levels of confining pressures \(p^{(0)}=0\) MPa (uniaxial compression), 12.5 MPa, 25.0 MPa, and 37.5 MPa was conducted. The experiments were performed in a sequential manner: Firstly, the confining pressure was applied, resulting in an initial hydrostatic stress state in the specimen, and subsequently, an axial pressure was applied displacement-driven up to failure. The mechanical response in the post-peak regime was also recorded. Finite element simulations of triaxial compression tests are often performed in an approximate manner. Commonly, such tests are simplified as single element tests in which the non-homogeneous deformation of the specimen observed in the experiments cannot be captured, or by two-dimensional models in which three-dimensional effects are neglected, e.g., [PERSON] (1988); [PERSON] (1988); [PERSON] (1988); [PERSON] (1988). By contrast, in the present study, a three-dimensional finite element model is employed in order to capture the three-dimensional deformations in a realistic manner. The numerical model with the prescribed boundary conditions and loads is illustrated in Figure 7. Since the failure mode is expected to be symmetric with respect to a vertical plane through the center axis of the specimen, symmetry is exploited by considering only one half of the specimen. By analogy to the experimental tests, the numerical simulations are performed in two sequential steps: firstly, the specimen is supported vertically at the bottom face and the confining pressure is applied; secondly, the axial loading is applied by imposing a uniform vertical displacement at the top of the specimen. The specimens are discretized with 20-node hexahedral elements employing reduced numerical integration. For both the displacement field and the nonlocal field, quadratic shape functions are used. To analyze potential mesh-sensitivity of the gradient-enhanced RDP model, three different structured finite element meshes are examined. A coarse mesh with 828 elements (an element size of 4 mm), a medium mesh with 5950 elements (an element size of 2 mm), and a fine mesh with 13,160 elements (element size of 1.5 mm) are employed. To trigger localized failure, at the center of the specimen, a small zone of slightly weakened elements is introduced, as indicated in Figure 7. In the numerical simulations, snapback behavior may occur, i.e., a simultaneous decrease of the load and the displacement after attaining the peak load. At this stage, displacement-controlled experiments become unstable. To overcome potential snapback behavior in the numerical simulations, the indirect  
Figure 6: Deformed specimen with a displacement scale factor of 100 (**left**) and distribution of the damage variable \(\omega\) (**right**) for the unstructured mesh (9504 elements) at a CMOD of 0.3 mm. displacement control technique [50; 51] is employed. For the present example, this technique can be applied by enforcing a monotonic decrease of the vertical distance between two nodes, with each node located at one boundary of the expected shear band. The material parameters required for the elastic-plastic part of the RDP model were identified from single element simulations, as discussed in [52]. The additional parameters for the softening regime \(A_{s}\), \(\epsilon_{\mathrm{f}}\), \(I\), and \(m\) are calibrated from the present numerical simulations for a best fit with the experimental results for the confining pressure of \(p^{(0)}=37.5\,\mathrm{MPa}\). The employed parameters for Innsbruck quartz phyllite are \(E=56670\) MPa, \(\
u=0.2\), \(f_{\mathrm{cu}}=42\,\mathrm{MPa}\), \(f_{\mathrm{cy}}=29.5\,\mathrm{MPa}\), \(m_{0}=12.0\), \(m_{\mathrm{g1}}=9.9\), \(A_{\mathrm{h}}=0.0045\), \(C_{\mathrm{h}}=8.8\), \(A_{\mathrm{s}}=4\), \(\epsilon_{\mathrm{f}}=4\times 10^{-4}\), \(m=1.05\), \(l=2\,\mathrm{mm}\). For the remaining model parameters \(e\), \(B_{\mathrm{h}}\), \(D_{\mathrm{h}}\), \(G_{\mathrm{f}}\), and \(B_{\mathrm{s}}\), the default values summarized in Section 2 are employed.