## 4 Numerical Simulation Results and Analysis
### Displacement Field Analysis  
To study the effect of symmetric fissure morphology on the relative displacement of particles around a hole, specimens with 2a = 8 mm were selected to analyze the displacement field of particles near the hole with different fissure dip angles. As illustrated in Figure 9, the small arrow's color indicates the particle displacement's magnitude, while the large red arrow denotes its direction. Based on the size and direction of the particle displacement, it can be identified that the specimens primarily exhibit tensile and shear-tensile cracks. Figure 6: Stress-strain curve of symmetrical fissure-hole sandstone. **(a)** The stress-strain curve of 2a = 8 mm specimen; **(b)** the effect of fissure dip angle on stress drop amplitude. Figure 7: Effect of symmetrical fissure dip on strength parameters of hole-containing sandstone. **(a)** Peak strength; **(b)** elastic modulus. Tensile cracks can be classified into two categories based on their causes. The first category is tensile cracks attributed to the displacement difference in the direction of particle movement, represented by Tension(I) in the diagram. These cracks are typically more minor in size and occur mainly around the hole of the specimen with small fissure dip angles (\(\alpha=0\)-45\({}^{\circ}\)). The second category is tensile cracks caused by the opposite direction of particle transverse displacement, represented by Tension(II) in the diagram. These cracks primarily originate from the fissure tips, are usually larger, and can cause partial tensile failure of the specimen. Additionally, they are responsible for the hole and fissure tip penetration in specimens with fissure dip angles of 45\({}^{\circ}\), 75\({}^{\circ}\), and 90\({}^{\circ}\). Shear-tensile cracks, indicated by Shear-Tension in the diagram, primarily result from particles displaced in opposite directions, with the angle between the particle displacement directions being obtuse. This often leads to particle slip and tearing, although overall shear-slip is still predominant. Moreover, the prominent cracks that eventually cause damage are all macroscopic shear-tensile cracks in the specimens. The dip angle of fissures directly influences the deformation and failure of a hole. Particle displacement around the hole changes continuously with an increase in \(\alpha\), resulting in three primary forms of deformation and failure. At \(\alpha=0^{\circ}\) and 15\({}^{\circ}\), compressive stress causes particles around the hole to move toward its interior, leading to shrinkage. At \(\alpha\) values between 30\({}^{\circ}\) and 60\({}^{\circ}\), shear failure is enhanced, and particle displacement on both sides of the hole is parallel and reversed, resulting in shear deformation. Finally, when \(\alpha\) is at 75\({}^{\circ}\) and 90\({}^{\circ}\), stress concentration on the hole wall causes particles to flake off towards the inside under compressive stress, leading to failure similar to that of a single hole non-fissure  
Figure 8: Effect of symmetrical fissure length on strength parameters of hole-containing sandstone. (a) Peak strength; (b) elastic modulus. Figure 9: Displacement vector diagram of specimens with different fracture dip angles (2a = 8 mm). specimen. Therefore, it can be concluded that the effect of symmetric fissure dip angle on particle displacement around the hole decreases as \(\alpha\) increases. To further investigate the effect of fissure length on the internal displacement field of the hole-containing sandstone, the displacement fields near the hole of specimens with different fissure lengths at \(\alpha\) = 45\({}^{\circ}\) were selected for analysis, as shown in Figure 10. When 2a = 4 mm and 16 mm, particles around the hole and fissures are displaced in opposite directions on the fissure direction, the hole and fissures are distorted by shear slip, and shear-tension cracks sprout from the fissure tip. When 2a = 12 mm, the counterclockwise twisting of the particles near the hole wall produced tensile cracks through the hole and the fissure tips.