# Hydraulic Fracture Propagation in Fractured Rock Mass
## 3 Analysis the Failure Type of Crag Bridge
### Shear Crack-Shear Crack Failure Mode  
This cra bridge failure mode occurs in the case of coplanar prefabricated cracks. With the increase of load, shear cracks appear in the inner ends of the two precast cracks. When the cracks grow to a certain length, the rock bridge breaks through, forming a macroscopic shear fracture surface (see Figure 15).  
\[\sum F_{x}=0,\;\sum F_{y}=0 \tag{52}\]  
\[\begin{cases}2r_{9}\sigma_{3}-2a(\tau_{n}\sin\psi+\sigma_{n}\cos\psi)-2r_{8}( \tau_{CD}\sin\theta+\sigma_{CD}\cos\theta)=0\\ 2r_{7}\sigma_{1}+2a(\tau_{n}\cos\psi-\sigma_{n}\sin\psi)+2r_{8}(\tau_{CD}\cos \theta-\sigma_{CD}\sin\theta)=0\end{cases} \tag{53}\]  
In Equation (53):  
\[\begin{cases}r_{7}=a\sin\psi+\frac{1}{2}\sqrt{D^{2}+h^{2}}\sin\theta\\ r_{2}=\frac{1}{2}\sqrt{D^{2}+h^{2}}\\ r_{3}=a\cos\psi+\frac{1}{2}\sqrt{D^{2}+h^{2}}\cos\theta\\ \theta=\psi\end{cases} \tag{54}\]  
Simultaneous Equations (52)-(54) can be obtained:  
\[\tau_{CD}=\frac{2A_{3}\tan\theta-2B_{3}}{\sqrt{D^{2}+h^{2}}\cos\theta\left(1+ \tan^{2}\theta\right)} \tag{55}\]  
Figure 15: Characteristics of cra bridge shear failure (shear crack-shear crack failure mode). (**a**) Stress analysis of crag bridge; (**b**) failure characteristics of crag bridge.  
\[\sigma_{\rm CD}=\frac{2A_{3}+2B_{3}\tan\theta}{\sqrt{D^{2}+h^{2}}\cos\theta\big{(}1+ \tan^{2}\theta\big{)}} \tag{56}\]  
\[\left\{\begin{aligned} A_{3}&=r_{9}\sigma_{3}-a(\tau_{n }\sin\psi+\sigma_{n}\cos\psi)\\ B_{3}&=r_{7}\sigma_{1}+a(\tau_{n}\cos\psi-\sigma_{n} \sin\psi)\end{aligned}\right. \tag{57}\]  
With the extension of wing crack length, the shear strength between crag bridge is weakened, which leads to the crag 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:  
\[\tau_{CD}-c-\sigma_{CD}\tan\varphi\geq 0 \tag{58}\]  
Simultaneous Equations (55)-(58) can be obtained:  
\[\begin{aligned} \Big{(}\frac{1}{2}\sqrt{D^{2}+h^{2}}\cos \theta\Big{)}& c\tan^{2}\theta+(B_{3}\tan\varphi-A_{3})\tan\theta +A_{3}\tan\varphi+B_{3}\\ &+\frac{1}{2}\text{c}\sqrt{D^{2}+h^{2}}\cos\theta\leq 0\end{aligned} \tag{59}\]  
If Equation (59) has roots that satisfy the root discriminant:  
\[\tan\theta\in\left[\frac{A_{3}-B_{3}\tan\varphi-\sqrt{\Delta}}{\sqrt{D^{2}+h^{ 2}}\cos\theta c},\frac{A_{3}-B_{3}\tan\varphi+\sqrt{\Delta}}{\sqrt{D^{2}+h^{2 }}\cos\theta c}\right] \tag{60}\]  
The maximum crack angle of shear crack-shear crack failure mode is as follows:  
\[\theta_{c}=\arctan\frac{A_{3}-B_{3}\tan\varphi+\sqrt{\Delta}}{\sqrt{D^{2}+h^{ 2}}\cos\theta c} \tag{61}\]  
According to [31], when the prefabricated fracture dip angle is equal to the rock bridge dip angle, and with the increase of prefabricated fracture angle, this failure mode occurs with more difficulty. Therefore, it can be seen that shear crack-shear crack failure mode occurs less easily when rock crag inclination is \(\psi\) than \(\theta_{c}\).  
It can be seen that for type II cracks (shear cracks), the crack expansion in shear. From Equations (55) and (56), it is known that \(K_{II}\) is independent of the osmotic pressure. Therefore, if the crack occurs as a type II shear crack, it is not related to the osmotic pressure. Hydraulic pressure only has a splitting effect on the wing crack. Compared with the four failure modes, the wing crack-shear crack-wing crack mode has a greater impact on the rock failure and the rock crag is broken through in less time.