## 4 Numerical Simulation Results and Analysis
### Failure Mode Analysis  
The initiation and propagation of microcracks inside the rock are essential factors leading to the final failure. Table 6 gives the initiation locations of microcracks in the hole-containing sandstone under different fissure morphologies. In contrast, C and H in the table indicate two forms of initiation, fissure tip initiation, and hole wall initiation. It is shown in the table that the change in the crack inclination makes the microcrack initiation location evolve from the crack tips to the hole wall. This is because the stress concentration location in the specimen evolves gradually from the fissure tip to the hole wall when the fissures are gradually parallel to the principal axial stress, which causes the microcrack initiation location to change. The final failure modes of the hole-containing sandstone at different fissure dip angles are given in Figure 11, from which it can be seen that the failure modes of the specimens are mainly divided into four types as the fissure dip angle increases with the fissure length of 8 mm:  
1. At \(\alpha\) = 0\({}^{\circ}\), 15\({}^{\circ}\), the prominent cracks all crack along the direction perpendicular to the symmetrical fissures and then respectively expand obliquely up (down) to the specimen boundary, showing mixed shear-tensile failure. In addition, secondary cracks sprouted on the upper and lower sides of the fissure tips and hole wall, and the secondary cracks all expanded roughly in the direction of the principal axial stress, which aggravated the tensile failure of the specimen. 2. At \(\alpha\) = 30\({}^{\circ}\), 45\({}^{\circ}\), the proper main crack obliquely extends upward (downward) to the specimen boundary, forming a triangular fragment. Moreover, the secondary cracks  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameter** & \(\alpha\) = 0\({}^{\circ}\) & \(\alpha\) = 15\({}^{\circ}\) & \(\alpha\) = 30\({}^{\circ}\) & \(\alpha\) = 45\({}^{\circ}\) & \(\alpha\) = 60\({}^{\circ}\) & \(\alpha\) = 75\({}^{\circ}\) & \(\alpha\) = 90\({}^{\circ}\) \\ \hline
2a = 4 mm & C & C & C & C & C & C & H \\
2 aall crack from the fissure tip along the direction perpendicular to the fissure, and then the extension direction turns toward the axial main stress direction. Compared with \(\alpha=0^{\circ}\), \(15^{\circ}\), the specimen's tensile failure degree is weakened, and the degree of shear failure is enhanced. 3. At \(\alpha=60^{\circ}\), the main crack is co-linear with the symmetric fissures, and the specimen eventually produces shear failure along the diagonal. 4. At \(\alpha=75^{\circ}\) and \(90^{\circ}\), the failure modes of the specimens are similar to those of the single-hole fissure-free specimen, with the prominent cracks starting from the right (left) side of the hole wall and then expanding obliquely upward (downward), leading to shear failure of the specimens, however, due to the compressive stress, some of the tensile cracks penetrate the fissure tip to the hole wall, which is also reflected in the \(\alpha\) = \(45^{\circ}\) specimen, and the tensile failure of the specimens is slightly enhanced. Figure 12 illustrates the relationship between fissure length and failure modes of specimens with a \(45^{\circ}\) fissure dip angle. The plot shows specimens experience shear failure along the diagonal when the fissure length is small (2a = 4 mm). At 2a = 8 mm and 12 mm, the primary cracks form a triangular fragmentation with the specimen boundary. Secondary cracks perpendicular to the symmetric fracture are generated, and tensile cracks penetrate the rock bridges between the fracture tips and the hole. Tensile failure is more significant in the 2a = 12 mm specimen, intensifying the fragmentation compared to the 2a = 8 mm specimen. When 2a = 16 mm, the number of cracks substantially decreases, and the main crack extends obliquely upward (or downward) from the fissure tip to the boundary of the specimen.