# Hydraulic Fracture Propagation in Fractured Rock Mass
## 3 Analysis the Failure Type of Crag Bridge
### Wing Crack-Shear Crack-Wing Crack Failure Mode  
This type of crag bridge connection mostly occurs when the prefabricated cracks have a large dip angle. According to the theory of maximum axial stress, the maximum crack initiation angle of type I stress intensity factor is about 70.5\({}^{\circ}\), and the crack initiation angle has nothing to do with the prefabricated crack angle. When the inclination angle of the prefabricated crack increases, the angle between the direction of wing crack initiation and the direction of maximum compressive stress increases, and the compressive stress has an obvious inhibitory effect on the propagation of wing crack, which makes it difficult to generate and develop the wing crack at the prefabricated crack tip of rock sample [23]. Therefore, in the loading process, the prefabricated crack tip produces wing cracks, moreover the wing cracks are difficult to expand in the direction of maximum compressive stress, meanwhile shear cracks occur between wing cracks, and the rock bridge is broken through eventually. In Figure 13, \(AB\) and \(EF\) are half of the upper and lower prefabricated cracks, respectively, and length is equal to \(a\), \(CB\) and \(DE\) are wing cracks, and \(CD\) is the crag bridge. \(\tau_{n}\) and \(\sigma_{n}\) are the normal stress and shear stress acting on the prefabricated crack. \(\tau_{CD}\) and \(\sigma_{CD}\) are the normal stress and shear stress acting on the bridge, and hydraulic pressure \(P\) acts on the two wings [36; 39; 40]. \[\sum F_{x}=0,\;\sum F_{y}=0 \tag{20}\] \[ \begin{cases}2r_{3}\sigma_{3}-2a(\tau_{n}\sin\psi+\sigma_{n}\cos \psi)-2r_{2}(\tau_{CD}\sin\theta+\sigma_{CD}\cos\theta)-2 PI=0\\ 2r_{1}\sigma_{1}+2a(\tau_{n}\cos\psi-\sigma_{n}\sin\psi)+2r_{2}(\tau_{CD}\cos \theta-\sigma_{CD}\sin\theta)=0\end{cases}  \tag{21}\]  
Figure 13: Characteristics of crag bridge shear failure (wing crack-shear crack-wing crack failure mode). (**a**) Stress analysis of crag bridge; (**b**) failure characteristics of crag bridge. In Equation (21):  
\[ \begin{cases}r_{1}=a\sin\psi+\frac{1}{2}\sqrt{D^{2}+h^{2}}\sin\theta\\ r_{2}=\frac{1}{2}\sqrt{D^{2}+h^{2}}-\frac{1}{\cos\theta}\\ r_{3}=a\cos\psi+\frac{1}{2}\sqrt{D^{2}+h^{2}}\cos\theta\\ \theta=\psi-\arctan\frac{D}{h}\end{cases}  \tag{22}\]  
Simultaneous Equations (20)-(22) can be obtained:  
\[\tau_{\text{CD}}=\frac{2A_{1}\tan\theta-2B_{1}}{(\sqrt{D^{2}+h^{2}}\cos\theta- 2l)\big{(}1+\tan^{2}\theta\big{)}} \tag{23}\]  
\[\sigma_{\text{CD}}=\frac{2A_{1}+2B_{1}\tan\theta}{(\sqrt{D^{2}+h^{2}}\cos \theta-2l)\big{(}1+\tan^{2}\theta\big{)}} \tag{24}\]  
\[ \begin{cases}A_{1}=r_{3}\sigma_{3}-a\big{(}\tau_{n}\sin\psi+\sigma_{n}\cos \psi\big{)}-Pl\\ B_{1}=r_{1}\sigma_{1}+a\big{(}\tau_{n}\cos\psi-\sigma_{n}\sin\psi\big{)}\end{cases}  \tag{25}\]  
With the extension of wing crack length, the shear strength between the crag bridge is weakened, which leads to the bridge being cut through. It is assumed that the crag bridge failure conforms to the Mohr-Coulomb strength criterion, then the failure conditions are as follows:  
\[\tau_{\text{CD}}-c-\sigma_{\text{CD}}\tan\varphi\geq 0 \tag{26}\]  
Simultaneous Equations (23)-(26) can be obtained:  
\[ \begin{array}{c}\Big{(}\frac{1}{2}\sqrt{D^{2}+h^{2}}\cos\theta-l\Big{)}c \tan^{2}\theta+(B_{1}\tan\varphi-A_{1})\tan\theta+A_{1}\tan\varphi+B_{1}\\ +\frac{1}{2}c\sqrt{D^{2}+h^{2}}\cos\theta-cl\leq 0\end{array}  \tag{27}\]  
If Equation (27) has roots that satisfy the root discriminant:  
\[\tan\theta\in\left[\frac{A_{1}-B_{1}\tan\varphi-\sqrt{\Delta}}{\sqrt{D^{2}+h^{ 2}}\cos\theta-2l)c},\frac{A_{1}-B_{1}\tan\varphi+\sqrt{\Delta}}{\sqrt{D^{2}+h ^{2}}\cos\theta-2l)c}\right] \tag{28}\]  
The crack length \(l\) increases with the increase of \(\theta\) angle.