# Improved Block Element Method for Simulating Rock Failure
## 5 Numerical Simulation and Discussion  
### Comparison Between Simulation and Experimental Results of Rock Failure  
The failure patterns of rocks under compressive loading conditions exhibit significant diversity, which contrasts sharply with the singularity of tensile failure. Based on extensive experimental observations [41; 42; 43; 44], rock failure under compression is primarily controlled by mesoscale tensile-shear mixed failure, compressive-shear mixed failure, and other complex forms. In this section, numerical simulations are conducted with reference to uniaxial compression experiments performed using the MTS 815 servo-controlled rock mechanics testing system by the Institute of Crustal Dynamics, China Earthquake Administration [39]. As shown in Figure 10, the specimen is an Alxa granite sample with dimensions of 50 mm \(\times\) 100 mm. The bottom of the model is fixed, while displacement-controlled loading is applied at the top. For model discretization, a Voronoi mesh composed of 800 block elements and 2279 layer elements is used as the computational grid. The material parameters used in the simulation are listed in Table 1. The elastic modulus and Poisson's ratio are directly derived from the experimental data [39]. The shear strength is assumed to be 6.9 times the tensile strength, and the shear fracture energy is assumed to be 10 times the normal fracture energy. The macroscopic axial stress-strain curve obtained using the improved block element method proposed in this study is shown in Figure 11, in which the numerical simulation results are represented by the black solid line, and the experimental data from Reference [39] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**E/GPa** & \(\mu\) & \(f_{t}\)**/MPa** & \(f_{g}\)**/MPa** & \(G_{m}\)**/(N\(\cdot\)m\({}^{-1}\))** & \(G_{s}\)**/(N\(\cdot\)m\({}^{-1}\))** \\ \hline
53.7 & 0.236 & 15 & 104 & 80 & 800 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters used in numerical simulation. Figure 10: Schematic diagram of rock sample loading scheme. Figure 9: Double-logarithmic relationship between element area and perimeter with different iteration numbers under 800 block elements. are shown as colored dashed lines. The results demonstrate that the simulated stress-strain curve agrees well with the experimental data, particularly during the softening stage, for which the model captures both the steep stress drop and the distinct plateau segment. The residual stress corresponding to the plateau is notably low. Specifically, the simulated compressive strength is 196.1 MPa, which is in good agreement with the experimentally measured range of 174.8-194.2 MPa with a mean value of 183.5 MPa. These results validate the effectiveness and accuracy of the proposed method in capturing the quasi-brittle compressive failure behavior of rock materials. To further investigate the failure mechanism of the rock specimen under compression, three typical points (A, B, and C) are selected along the stress-strain curve shown in Figure 11. The corresponding fracture and failure patterns at these points are illustrated in Figure 12. As shown, the failure process of the rock specimen under uniaxial compression proceeds as follows:  
1. In the initial stage of failure, due to material heterogeneity and the influence of the randomized mesh, microcracks begin to appear at random locations within the structure;
2. As loading progresses, the microcracks propagate and begin to coalesce, gradually forming macroracks that span across the specimen;
3. Eventually, the macrocrack fully develops, and the specimen is crushed. As expected, the cracks are not perfectly parallel to the loading direction. Instead, they form at an inclined angle, resulting in the overall splitting-type failure mode of the specimen. Figure 11: Comparison between experimental and simulation results of stress–strain curves.