## 4 Discussion  
### Local Stress Characteristics under Crack Initiation Stress  
In the previous section, the crack initiation and deformation field of the sample model with different flaw inclination angles were analyzed. The results show that the crack initiation process of samples with different flaw inclination angles has different characteristics. Besides, the crack initiation process of fractured rock or rock-like materials is obviously related to the change of stress field around the flaw [39; 51]. Figure 10 shows the parallel bond contact force distribution around the flaw just before reaching the crack initiation stress. In this Figure, the contact force is represented by line segments; green represents tensile force, and red represents compressive force. The width and direction of the line represent the magnitude and direction of the force respectively. The parallel bond contact force of the intact sample before the crack initiation stress is uniformly distributed in the sample, such as Figure 10a. For the samples with flaws, the distribution of parallel bond force is not uniform, and the distribution characteristics have a great relationship with the flaw inclination angle. When the flaw inclination angle \(\beta=15^{\circ}\) in the sample model, the tensile stress is concentrated in the middle of the upper and lower surfaces of the flaw, and the compressive stress is concentrated at the tip of the flaw, as shown in Figure 10b. With the increase of flaw inclination angle, the area of tensile stress concentration on the flaw surface decreases and gradually transfers to the tip of the flaw, while the area of compressive stress concentration at the tip of the flaw gradually extends to the middle of the flaw, as shown in Figure 10c-e. When the flaw inclination angle \(\beta=75^{\circ}\), there is almost no tensile stress concentration on the flaw's surface, and all of them are compressive stress concentration areas, such as in Figure 10f. It shows that the concentration of tensile stress around the flaw decreases with the increase of flaw inclination angle. Due to the stress-shielding effect, there is a low-level parallel bond contact force area between the two flaws, and with the increase of flaw inclination angle, this area gradually decreases until it almost disappears at \(\beta=75^{\circ}\). Figure 9: The number of cracks in sample models with different flaw inclination angles under peak stress. Figure 10 reveals the stress distribution around the flaw intuitively, but it still stays at the qualitative level. To realize the quantitative description, the specific stress value should be calculated. In PFC\({}^{\text{2D}}\), the local stress distribution around the flaw can be obtained by setting a series of measuring circles to monitor the average stress in the area to be measured. Because of the discreteness of the PFC\({}^{\text{2D}}\) model, the diameter of the measuring circle is too large or too small, which will cause the error of stress measurement. According to the research results of [PERSON] et al. (2018) and [PERSON] et al. (2018), the measured stress value is more accurate when each measurement circle area contains about 15 particles. Because the cracks and loading conditions are symmetrical in the experiment, the upper cracks were mainly studied here. In this model, a total of 12 measuring circles with a radius of 1.27 mm was set, which were distributed around the flaw. Their positions and numbers are shown in Figure 11.  
Figure 11: Layout of measuring circles. Figure 10: Parallel bond force distribution of different models just before crack initiation stress. (**a**) Intact model; (**b**) \(\beta=15^{\circ}\); (**c**) \(\beta=30^{\circ}\); (**d**) \(\beta=45^{\circ}\); (**e**) \(\beta=60^{\circ}\); (**f**) \(\beta=75^{\circ}\). In PFC\({}^{\text{2D}}\), the average stress monitored by measuring circles can only be expressed in the form of components.