# A Macro-Micro Damage Model for Rock under Compression Loading
## 2 Wing Crack Model Analysis
### Comparative Analysis of Models  
In order to verify the effectiveness of the modified stress intensify factor \(K_{\text{I}}\)wing proposed in this paper, the calculation results of each model (Equations (1), (4), (5) and (15)) are compared and analyzed. Since Equation (5) is only applicable to the situation that the wing crack have a fixed orientation parallel to maximum principal stress (i.e., \(\theta=\pi/2-\beta\)), the other three equations are compared first. Figure 4 shows the relationship between the dimensionless stress intensity factor at the crack tip and the orientation \(\theta\) of straight wing crack. The crack friction coefficient \(\mu=0.3\) and \(K_{0}=\sigma_{1}\sqrt{\pi a}\) in this study. (1) Main crack inclination angle \(\beta\): \(0.2\pi\) and \(0.4\pi\);  
Figure 4: Stress intensity factors computed by different models; (**a**) \(\lambda\)=0, \(1/a\)=0.0001; (**b**) \(\lambda\)=0.2, \(1/a\)=0.0001; (**c**) \(\lambda\)=0, \(1/a\)=1; (**d**) \(\lambda\)=0.2, \(1/a\)=1. 2. Stress ratio \(\lambda=\sigma_{3}/\sigma_{1}\): 0 and 0.2 (uniaxial compression and biaxial compression);
3. The equivalent crack propagation length \(l/a\): 0.0001 and 1 (rather short wing crack and long wing crack). The following conclusions can be drawn from Figure 4: 1. When \(l/a=0.0001\), that is, the wing crack length is rather short. The calculation results of the S model and current model are basically consistent. Of course, deviation of the results given by the HN model is also small. However, it can be seen from Figure 4a,b that when \(\theta\) is equal to about 70.5\({}^{\circ}\), the stress intensity factors predicted by the S model and the current model in this study all reach the maximum value except for the HN model. This is consistent with the theoretical cracking angle \(\theta\)c (i.e., \(\theta\)c = 70.5\({}^{\circ}\)) of wing crack [3], while the predicted cracking angle of HN model is 90\({}^{\circ}\). 2. When \(l/a=1\), that is, the wing crack length is long. It can be seen from Figure 4c,d that the calculation results of the HN model and current model are basically consistent. The results given by S model are generally too large, indicating that S model is not suitable for the case of long wing crack. With the increase of confining pressure and the main crack inclination angle \(\beta\), the deviation of the result given by the S model becomes more obvious. However, the large deviation mainly occurs when the stress intensity factor is negative. The negative stress intensity factor indicates that the wing crack has been closed and would not propagate. When the external load continues to increase, the stress intensity factor at the wing crack tip increases. When the stress intensity factor exceeds the fracture toughness, the wing crack will continue to grow and quickly bend in the direction which is consistent with the maximum principal stress. It is assumed that the winged cracks are aligned in the direction of the principal compressive stress (i.e., \(\theta=\pi/2-\beta\)) in order to be convenient for computation and implementation in practice. The above assumptions have been accepted by many scholars to study the effect of microcrack growth on progressive damage and failure of rock, and can achieve good results from the perspective of engineering application [14; 15; 21]. Based on this, the relationship between the dimensionless stress intensity factor at the crack tip and the equivalent crack propagation length is showed in the Figure 5. As can be observed in Figure 5, all models predict that with the increase of the equivalent crack propagation length, the dimensionless stress intensity factor becomes smaller. When there is no confining pressure, the calculation results of HN model and the current model are almost consistent. However, when the equivalent crack propagation length is short (\(l/a<1\)), the results given by LK model are obviously larger.