# Hydraulic Fracture Propagation in Fractured Rock Mass
## 3 Analysis the Failure Type of Crag Bridge
### Wing Crack-Shear Crack Failure Mode  
During the loading process, the prefabricated crack generates a wing crack, which expands stably in the direction of maximum compressive stress. When the wing crack expands to a certain extent, it is connected with another shear crack generated by the prefabricated crack, and finally the crag bridge is connected to form a macroscopic fracture surface. \[\sum F_{x}=0,\;\sum F_{y}=0 \tag{36}\]  
\[ \begin{cases}r_{6}\sigma_{3}-2a(\tau_{n}\sin\psi+\sigma_{n}\cos\psi)-r_{5}( \tau_{CD}\sin\theta+\sigma_{CD}\cos\theta)-Pl=0\\ r_{4}\sigma_{1}+2a(\tau_{n}\cos\psi-\sigma_{n}\sin\psi)+r_{5}(\tau_{CD}\cos \theta-\sigma_{CD}\sin\theta)=0\end{cases}  \tag{37}\]  
In Equation (37):  
\[ \begin{cases}r_{4}=2a\sin\psi+\sqrt{D^{2}+h^{2}}\sin\theta\\ r_{5}=\sqrt{D^{2}+h^{2}}-\frac{l}{\cos\theta}\\ r_{6}=2a\cos\psi+\sqrt{D^{2}+h^{2}}\cos\theta\end{cases}  \tag{38}\]  
Simultaneous Equations (36)-(38) can be obtained:  
\[\tau_{CD}=\frac{2A_{2}\tan\theta-2B_{2}}{(\sqrt{D^{2}+h^{2}}\cos\theta-l) \big{(}1+\tan^{2}\theta\big{)}} \tag{39}\]  
\[\sigma_{CD}=\frac{2A_{2}+2B_{2}\tan\theta}{(\sqrt{D^{2}+h^{2}}\cos\theta-l) \big{(}1+\tan^{2}\theta\big{)}} \tag{40}\]  
\[ \begin{cases}A_{2}=\frac{1}{2}r_{3}\sigma_{3}-a(\tau_{n}\sin\psi+\sigma_{n} \cos\psi)-\frac{1}{2}Pl\\ B_{2}=\frac{1}{2}r_{1}\sigma_{1}+a(\tau_{n}\cos\psi-\sigma_{n}\sin\psi)\end{cases}  \tag{41}\]  
With the extension of wing crack length, the shear strength between crag bridge is weakened, which leads to the bridge being cut through. It is assumed that crag bridge failure conforms to the Mohr-Coulomb strength criterion, then the failure conditions are as follows:  
\[\tau_{CD}-c-\sigma_{CD}\tan\varphi\geq 0 \tag{42}\]  
Simultaneous Equations (39)-(42) can be obtained:  
\[ \begin{cases}\frac{1}{2}\sqrt{D^{2}+h^{2}}\cos\theta-\frac{1}{2}l \bigg{)}c\tan^{2}\theta+(B_{2}\tan\varphi-A_{2})\tan\theta+A_{2}\tan\varphi+ B_{2}\\ +\frac{1}{2}c\sqrt{D^{2}+h^{2}}\cos\theta-\frac{1}{2}cl\leq 0\end{cases}  \tag{43}\]  
If Equation (43) has roots that satisfy the root discriminant:  
\[\tan\theta\in\bigg{[}\frac{A_{2}-B_{2}\tan\varphi-\sqrt{\Delta}}{\sqrt{D^{2}+h ^{2}}\cos\theta-l)c},\frac{A_{2}-B_{2}\tan\varphi+\sqrt{\Delta}}{\sqrt{D^{2}+h ^{2}}\cos\theta-l)c}\bigg{]} \tag{44}\]The crack length \(l\) increases with the increase of \(\theta\) angle. When the crag bridge inclination angle reaches \(\theta_{c}\) and the crack growth length reaches critical value \(l_{c}\), then:  
\[\theta_{c}=\arctan\frac{A_{2}-B_{2}\tan\varphi+\sqrt{\Delta}}{\Big{(}\sqrt{D^{2} +h^{2}}\cos\theta-l_{3c}\Big{)}c} \tag{45}\]  
It can be seen from the geometric relationship of the element body in Figure 14 that:  
\[\tan\theta_{c}=\frac{\sqrt{D^{2}+h^{2}}\sin\theta}{\sqrt{D^{2}+h^{2}}\cos \theta-l_{3c}} \tag{46}\]  
Simultaneous Equations (45) and (46) can be obtained:  
\[l_{3c}=\frac{E_{2}-\sqrt{E_{2}^{2}-(c+P\tan\theta)F_{2}}}{(c+P\tan\varphi)} \tag{47}\]  
In Equation (47):  
\[ \begin{array}{c}E_{2}=\frac{1}{2}r_{6}\sigma_{3}\tan\varphi-a(\tau_{n}\sin \varphi+\sigma_{n}\cos\varphi)\tan\varphi+B_{2}+\\ \frac{1}{2}\sqrt{D^{2}+h^{2}}[(2c-P\tan\varphi)\cos\theta+P\sin\theta]\end{array}  \tag{48}\]  
\[ \begin{array}{c}F_{2}=\big{(}D^{2}+h^{2}\big{)}c+2\sqrt{D^{2}+h^{2}}\Big{[} \big{(}\frac{1}{2}r_{3}\sigma_{3}-a\tau_{n}\sin\psi-a\sigma_{n}\cos\varphi \big{)}\\ (\cos\theta\tan\varphi-\sin\theta)+B_{2}(\sin\theta\tan\varphi+\cos\theta)] \end{array}  \tag{49}\]  
It is assumed that crack \(l_{5}(t)\) is a function related to time \(t\). When \(l_{5}(t)<l_{3c}\), the crag bridge is not connected at time \(t\). When \(l_{5}(t)=l_{3c}\), the stress intensity factor \(K_{I}=K_{Ic}\), the crag bridge is broken through. Figure 14: Characteristics of crag bridge shear failure (wing crack-shear crack failure mode). (**a**) Stress analysis of crag bridge; (**b**) failure characteristics of crag bridge. In simultaneous Equations (11), (12), and (47), the stress intensity factor of the crack tip is obtained when the wing crack reaches the critical length.