# Study on Mechanism of Static Blasting-Induced Hard Rock Fracture Expansion
## 2 Principles and Mechanisms of Hard Rock Fracturing Using Static Blasting Technology
### Mechanical Examination of Static Breakers' Suttering of Hard Rocks  
#### 2.1.1 Individual-Hole Cracking  
[PERSON] categorized the fracture damage process of single-pore rock under expansion pressure into three stages: the microfracture stage, the fracture development stage, and the breakage stage [36].  
1 Microfracture stage [36]  
Figure 2 illustrates the expansion stress analysis for each stage. In the microfracture stage, the expansion pressure of the crusher in the rock borehole gradually increases, resulting in radial compressive and tangential tensile stresses on the crushed rock. When the ultimate stress value is reached, crushing and cracking takes place. The damage zone, which is a region of microfractures caused by the initial tensile stress, exhibits a linear behavior at the beginning of this phase and becomes nonlinear towards the end.  
According to the thick-walled cylinder hypothesis, in the elastic phase and under the influence of internal pressure, the radial stress (\(\sigma_{r}\)) and tangential stress (\(\sigma_{\theta}\)) around the thick wall can be expressed as follows:  
\[\left\{\begin{array}{c}\sigma_{r}\\ \sigma_{\theta}\end{array}\right\}=P_{ex}\frac{R^{2}}{\left(R+d\right)^{2}-R ^{2}}\left[1\pm\frac{\left(R-d\right)^{2}}{r^{2}}\right] \tag{2}\]  
where \(P_{ex}\)--crusher expansion pressure, MPa; \(R\), \(d\)--respectively, the inner diameter and wall thickness of thick-walled cylinders, mm.  
As the crusher's expansion pressure increases, microfractures within the inelastic phase transform into the breakdown zone, that is, the elastic zone is \(c\leq r\leq b\). The damaged area \(R\leq r\leq c\). At this point, the equilibrium is found in Equation (3):  
\[\frac{d\sigma_{r}}{dr}+\frac{\sigma_{r}+\sigma_{\theta}}{r}=0 \tag{3}\]  
Figure 2: Analysis of single-hole rock fracturing’s mechanical action.  
Then, the rock damage condition is \(\frac{\sigma_{\theta}}{\sigma_{r}}-\frac{\sigma_{r}}{\sigma_{\theta}}=1\). Adding this to Equation (2) along with boundary condition \(\frac{\sigma_{r}}{r-R}=P_{ex}\) results in  
\[\sigma_{\theta}=\frac{\sigma_{t}}{\sigma_{c}}(\sigma_{t}+\sigma_{r}) \tag{4}\]  
\[k=1-\frac{\sigma_{t}}{\sigma_{c}}\]  
when \(r=b\), the stress, reaches the limit value, the inverse elastic zone (\(c\leq r\leq b\)), and at this moment, the stress distribution in Equation (5) is  
\[\left\{\begin{array}{l}\frac{\sigma_{r}}{\sigma_{c}}=\frac{(1-\frac{b^{2}}{ c^{2}})c^{2}}{2b^{2}}\\ \frac{\sigma_{b}}{\sigma_{t}}=\frac{(1+\frac{b^{2}}{c^{2}})c^{2}}{2b^{2}} \end{array}\right. \tag{5}\]  
2 Cleavage development stages [36]  
The stress intensity factor (\(K_{IC}\)) associated with the formation of fractures within the rock can be expressed as follows: Once the cracks propagate and reach the fractured area, multiple micro-fractures eventually merge and develop into macro-fractures. This phenomenon aligns with the principles of fracture mechanics.  
\[K_{IC}=FP\sqrt{\pi a} \tag{6}\]  
when \(\frac{a}{R}=1\sim 1.4\), \(F=0.1\sim 0.34\)  
Where \(a\)--fracture length, which is the measurement of the distance between the crack's tip and the hole's center; \(R\)--borehole radius.  
The condition of the crack extension at this moment is  
\[FP\sqrt{\pi a}\geq K_{IC} \tag{7}\]  
The necessary expansion pressure that corresponds to this is  
\[P_{ex}\geq\frac{K_{IC}}{FP\sqrt{\pi a}} \tag{8}\]  
3 Breakage stage  
When the fissure continues to grow until it reaches the free surface, the hard rock undergoes complete fracturing, resulting in the expansion cracking effect.