# Double-Borehole Superimposed Effect of a New Non-Explosive Directional Rock-Breaking Method
## 2 Methodology
### The Theoretical Model of Double-Borehole Directional Rock-Breaking by IESF
#### 2.2.2 Crack Propagation Criterion of IESF  
The stress intensity factor at the fracture tip is the superposition of stress intensity factors induced by in situ stress and gas pressure. Therefore, the mechanical model for crack propagation in rock fragmentation by IESF can be decomposed into a fracture tip extension model under in situ stress and a crack propagation model with gas pressure effects in the absence of in situ stress.  
The relationship between the in situ stress (\(\sigma_{1}\) and \(\sigma_{3}\)) and \(\sigma_{x}\), \(\sigma_{y}\), \(\tau_{xy}\) on the fracture surface can be expressed as  
\[\left\{\begin{array}{l}\sigma_{x}=\frac{\sigma_{1}+\sigma_{3}}{2}+\frac{ \sigma_{1}-\sigma_{3}}{2}\cos 2\theta\\ \sigma_{y}=\frac{\sigma_{1}+\sigma_{3}}{2}-\frac{\sigma_{1}-\sigma_{3}}{2} \cos 2\theta\\ \tau=\frac{\sigma_{1}-\sigma_{3}}{2}\sin 2\theta\end{array}\right. \tag{7}\]  
where \(\theta\) is the angle between the fracture surface and the x-direction. For the points on the fracture surface, since the fracture length is much larger than the fracture width, the value of \(\theta\) is extremely small and can be approximated as \(\theta\to 0\), then there is \(\sigma_{y}=\sigma_{3}\). For points on the crack surface, \(\sigma_{y}\) is in the same direction as the gas pressure, so the total stress on points on the crack surface is  
\[\sigma_{t}=p(x,t)+\sigma_{3} \tag{8}\]  
where \(l\) is the fracture length. \(F_{1}\) is related to \(\sigma_{1}/\sigma_{3}\) and \(l/r_{d}\).  
The stress intensity at the fracture tip induced by gas pressure can be expressed as  
\[K_{I}^{\sigma}=F_{\sigma}\int_{r}^{r+l}{[p(x,t)+\sigma_{3}]dx\sqrt{\pi(r+l)}} \tag{9}\]  
where \(F_{\sigma}\) is the shape correction factor, which varies with fracture size and shape, geometric parameters of the fracture structure, and boundary conditions.  
Substituting Equation (6) into Equation (9) can be obtained by  
\[K_{I}^{\sigma}=F_{\sigma}\int_{r}^{r+l}{\bigg{[}\frac{Tp_{0}kmt}{T_{0}[\pi r^ {2}L+l(t)dL-nb]}-\frac{an^{2}}{V^{2}}-\sigma_{3}\bigg{]}[1-\frac{x-r}{l(t)}]dx \sqrt{\pi(r+l)}} \tag{10}\]  
The dynamic fracture toughness of a rock during crack propagation can be expressed as [39; 40]  
\[K_{lod}=\frac{(c_{R}-v_{a})}{(c_{R}-0.75v_{a})}K_{lc} \tag{11}\]  
where \(c_{R}\) is the Rayleigh wave velocity of the rock material, \(v_{a}\) is the crack extension velocity, and \(K_{lc}\) is the static fracture toughness of the rock. When the fracture is unaffected by adjacent ones, the crack propagation condition requires that the stress intensity factor at the fracture tip is not less than the dynamic fracture toughness of the rock,  
\[K_{l}^{\sigma}\geq K_{lcd} \tag{12}\]  
It can be also obtained as  
\[F_{\sigma}\int_{r}^{r+l}\bigg{[}\frac{Tp_{0}kmt}{T_{0}[\pi r^{2}L+l(t)dL-nb]}- \frac{an^{2}}{V^{2}}-\sigma_{3}\bigg{]}[1-\frac{x-r}{l(t)}]dx\sqrt{\pi(r+l)} \geq\frac{(c_{R}-v_{a})}{(c_{R}-0.75v_{a})}K_{lc} \tag{13}\]  
When the fracture expands to the point affected by the extension of neighboring ones, the strength factor at the fracture tip needs to be added to \(K_{l}^{\sigma\prime}\), so  
\[K_{ld}=K_{l}^{\sigma}+K_{l}^{\sigma\prime} \tag{14}\]  
Theoretically, the stress intensity factor at fracture tips is identical to adjacent ones, and its effect on extension \(K_{l}^{\sigma\prime}\) is \(K_{l}^{\sigma}\) for n times, so  
\[K_{l}^{\sigma\prime}=nK_{l}^{\sigma} \tag{15}\]  
where \(n\leq 1\).  
Then, the criterion for the occurrence of fracture tip extension is  
\[K_{ld}=K_{l}^{\sigma}+K_{l}^{\sigma\prime}\geq K_{lcd} \tag{16}\]  
It also be also obtained as  
\[K_{ld}=(1+n)K_{l}^{\sigma}\geq K_{lcd} \tag{17}\]  
Therefore, the criterion is finally obtained by  
\[(1+n)F_{\sigma}\int_{r}^{r+l}\bigg{[}\frac{Tp_{0}kmt}{T_{0}[\pi r^{2}L+l(t)dL- nb]}-\frac{an^{2}}{V^{2}}-\sigma_{3}\bigg{]}[1-\frac{x-r}{l(t)}]dx\sqrt{\pi(r+l)} \geq\frac{(c_{R}-v_{a})}{(c_{R}-0.75v_{a})}K_{lc} \tag{18}\]