# 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  
In a first step, the capability of the gradient-enhanced RDP model for predicting mode I failure is assessed. To this end, the experimental study of wedge splitting tests on Indiana limestone performed by [PERSON] and [PERSON] [40] is considered. The investigated specimen is illustrated in Figure 2.  
During the experimental test, the splitting force was applied by a vertically driven wedge, exerting a pressure against roller bearings that were mounted on both sides of the groove in the specimen. The vertical (machine) force and the crack mouth opening displacement (CMOD) were recorded, and the latter was controlled during the experimental test. The energy conjugate force to the CMOD, the splitting force \(F_{\mathrm{sp}}\), was calculated from the vertical force considering the geometry of the wedge and neglecting any frictional effects. In total, 5 tests were performed, but in [40] detailed load-displacement curves were presented only for Test 3. To investigate the degradation of the stiffness of the specimen during crack propagation, several loading/unloading cycles were performed. The experimental test is simulated by means of a two-dimensional finite element model assuming plane stress conditions. According to Figure 2, the specimen is supported in the vertical direction at the bottom center over a width of 10 mm and in the horizontal direction at midpoint. The splitting force \(F_{\mathrm{sp}}\), transmitted by the wedge, is approximated by the pressure \(p_{\mathrm{sp}}\) acting on the lateral groove faces. The simulation is performed in a displacement-driven manner by controlling the CMOD (i.e., the relative horizontal displacement between points (a) and (b) in Figure 2) during the loading and unloading cycles. The material parameters for representing the Indiana limestone were determined by a best fit with the recorded experimental results. In the present example of mode I failure, only few parameters have a significant influence on the results: \(E=22000\) MPa, \(\
u=0.15\), \(f_{\mathrm{cu}}=20\) MPa, \(f_{\mathrm{cy}}=2/3\,f_{\mathrm{cu}}=13.33\) MPa, \(m_{0}=6.5\), \(m_{\mathrm{g1}}=5\), \(A_{h}=5\times 10^{-3}\), \(C_{h}=20\), \(\varepsilon_{\mathrm{f}}=8\times 10^{-4}\), \(m=1.05\), and \(l=4\) mm. For the weighting parameter \(m\), any value larger than 1 is sufficient to ensure full  
Figure 2: Geometry and boundary conditions of the numerical model of the specimen for the wedge splitting test. regularization of the problem by avoiding spurious localization of plastic strain [33]. A discussion on the influence of the weighting parameter \(m\) may be found in [42] for an integral-type nonlocal model, where, however, for parameter \(m=2\), an overestimation of the energy dissipation close to a notch was demonstrated. This non-physical effect was also observed by the authors with increasing influence for larger values of \(m\). Thus, parameter \(m\) is chosen just slightly larger than 1, in accordance with proposed values from the literature [17; 43]. For the remaining model parameters \(e\), \(B_{h}\), \(D_{h}\), \(G_{h}\), \(A_{s}\), and \(B_{s}\), the default values summarized in Section 2 are employed. From Equation (7), the uniaxial tensile strength is calculated as \(f_{\mathrm{tu}}=3\,\mathrm{MPa}\). To investigate the influence of the finite element discretization on the predicted results, different meshes are employed: Three structured meshes with fully integrated 4-node quadrilateral elements and element sizes of \(5\,\mathrm{mm}\), \(1\,\mathrm{mm}\), and \(0.5\,\mathrm{mm}\) in the vicinity of the expected crack path, as well as one unstructured mesh with the same element type and an element size of \(1\,\mathrm{mm}\) along the crack path. Accordingly, the element size of \(5\,\mathrm{mm}\) is slightly larger compared to the assumed length scale \(l\) of \(4\,\mathrm{mm}\) for the coarse mesh and considerably smaller for the medium and fine mesh.