## 3 Results and Discussion
### Failure Mode and Stress Concentration Factor  
A comparison of the failure modes of the cracked specimens in the experiments and numerical simulations is shown in Figure 11. In general, the cracked specimens with different crack dip angles all exhibit specific characteristics of splitting failure, with a tendency for conjugate failure. When the crack dip angle is below \(45^{\circ}\), the crack direction tends to be more vertical, and the splitting features are more prominent. In the failure stage, the cracked specimens with crack dip angles of 15\({}^{\circ}\) and 30\({}^{\circ}\) show a larger FPZ (red cracks) in the model, indicating that when the crack dip angle is smaller, the damaged area of the rock is larger. At the same time, due to the damaged state of these areas, their subsequent load-bearing capacity is significantly reduced, resulting in many splitting cracks caused by tensile loads induced by the Poisson effect. As the crack dip angle increases, the area of FPZ falls, and the crack propagation direction is no longer concentrated in the vertical direction. The rock strength is not limited by tensile strength, and the shear strength can be further exerted, leading to an increase in the overall strength of the specimen. More branching cracks appear at this point, and the cracks exhibit conjugate patterns, consistent with the characteristics of shear failure. In studies focused on the macroscopic response of rock specimens, stress concentration factors provide insights into the localized stress amplification effects without delving into the detailed crack tip behavior. If the objective is to investigate crack initiation and propagation specifically, stress intensity factors may provide more detailed insights. The limitation of the stress concentration factors is that its calculation often relies on a linear elasticity assumption, which may not accurately represent the nonlinear behavior of rocks, especially in post-peak or post-failure conditions. To analyze the stress concentration in cracked specimens with different crack dip angles during loading, the stress concentration factors were calculated at 80% of the UCS. At this point, the loading-induced new microcracks have yet to develop extensively, allowing for an effective reflection of the stress concentration caused by pre-existing cracks. The maximum principal stress distribution at 80% of the UCS is shown in Figure 12. It should be noted that in Abaqus, the tensile stress direction is considered positive. Therefore, the maximum principal stress in uniaxial compression of the rock is the minimum principal stress in Abaqus. Additionally, in this model, the cohesive elements have a stress of 0 MPa when they fail, and a few cohesive elements fail at 80% of the UCS, resulting in a stress of 0 MPa. The calculation results of maximum principal stresses and stress concentration factors are shown in Table 1.  
The calculation results indicate that under uniaxial compression conditions, the stress concentration is related to the pre-existing crack, and the stress concentration factor decreases with the increase in the crack dip angles. When the crack dip angle is 15\({}^{\circ}\), the stress concentration factor is 5.91, dramatically decreasing to only 1.78 when the crack dip angle is 90\({}^{\circ}\). These calculation results also support the previous analysis of the overall strength and failure modes of the cracked specimens, namely, the smaller the crack dip angle, the higher the stress concentration near the pre-existing crack, making it more prone to plastic-yielding during loading. The opposite holds for the cracked specimens with larger crack dip angles.