## 3 Results
### Failure Mode  
Table 2 displays the splitting failure modes observed in circular specimens with pre-existing cracks at different inclination angles. Under the splitting test conditions, the crack propagation and failure modes of specimens with pre-existing cracks are changed to a varying degree due to the change in the crack inclination angle. As shown in Table 2, when the inclination angle \(\alpha\) of the prefabricated crack is \(0^{\circ}\), the loading direction is perpendicular to the crack surface, and the tensile failure of the sandstone specimen with through-going fissures is parallel to the loading direction. Furthermore, the tension cracks that occur during the failure pass through the center of the prefabricated crack, i.e., the center of the disk, resulting in a higher measured tensile strength. Secondary cracks extend from the tip of the prefabricated crack toward the edge of the disk, exhibiting symmetry about the height of the disk, indicating a higher degree of homogeneity within the sandstone. When the inclination angle of the prefabricated crack is between 30 and \(60^{\circ}\), at the macroscopic level, the failure mode of the sandstone specimens exhibits highly similar fracturing patterns. The main pattern is characterized by the formation of two through-going cracks with approximately the same orientation (primary) and secondary cracks. The cracks extend from the tip of the prefabricated crack toward the loading end and gradually become parallel to the loading direction. When the inclination angle of the prefabricated crack is \(90^{\circ}\), the loading direction is parallel to the crack surface, and the failure mode of the sandstone specimen is similar to that of an intact specimen. The through-going crack is parallel to the loading direction, and secondary cracks are also generated during the extension of the through-going crack in the upper part of the specimen. The upper part of the specimen is severely fragmented, and the vertical crack on the left side of the through-going crack may result from local tensile stress during the extension of the through-going crack at the center of the disk. Based on the particle flow theory method, the PFC simulation was used to investigate the sample fracture process under different dip angles of joints. The simulation results exhibited good agreement with the experimental failure morphology, particularly in the through-crack region, further validating the experimental findings. The cracks that appear during the rock sample's failure mainly include three forms: tensile, shear, and combined cracks. Based on the morphology and formation mechanism of the specimen's failure cracks, the failure modes can be classified into the following categories: tensile failure along the center of the disk (Type I), tensile failure along the tip of the prefabricated crack (Type II), and tensile-shear combined failure along the tip of the prefabricated crack (Type III), as shown in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Specimen Number** & **Failure Mode** & **Numerical Simulation Results** & **Failure Pattern** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Failure mode and classification of failure patterns for specimens with pre-existing cracks. As shown in Table 2, due to the stress concentration caused by the prefabricated crack, the failure mode of the specimen is mainly in the form of a combination of cracks. Specifically, at a crack inclination angle of 0\({}^{\circ}\), the failure mode predominantly comprises Type I and Type III, due to the presence of the horizontally oriented prefabricated crack, the specimen experiences a tensile-shear combined failure along the crack tip. In the range of crack inclination angles between 30 and 60\({}^{\circ}\), a consistent Type III combined failure mode is observed, indicating intensified internal rock damage that leads to frictional sliding along particle boundaries during the fracturing process.