# Calculation of Blasting Damage Zone Radius of Different Charge Structures in Burnt Rock
## 5 Radius Calculation of Burnt-Rock-Blasting Damage Area
### Calculation of Fracture Zone Radius of Burnt Rock
#### 5.2.1 Coupled Charge  
(1) The range of the fracture zone under the action of stress wave From Formulas (20)-(31):  
\[(1-D)\sigma_{td}=\frac{1}{\sqrt{2}}\cdot\sigma_{R}\cdot B\cdot\left(\frac{R_{2(1)} ^{1}}{R_{1}^{1}}\right)^{-(\frac{\theta_{2}}{1-D})} \tag{32}\]  
The initial radius \(R_{2(1)}^{1}\) of the crack zone caused by the stress wave with the coupled charge structure is  
\[R_{2(1)}^{1}=R_{1}^{1}\cdot\left[\frac{B\sigma_{R}}{\sqrt{2}\sigma_{td}}\right] ^{\frac{1-D}{42}}=R_{1}^{1}\cdot\left(\frac{\sigma_{cd}}{\sigma_{td}}\right)^{ \frac{1-D}{2-R_{c}}} \tag{33}\]  
(2) The range of fracture zone under the action of explosive gas  
With the coupled charge structure, the detonation gas penetrates into the cracks generated by the stress wave. At this time, the pressure \(P_{d}^{1}\) on the borehole wall is  
\[P_{d}^{1}=\frac{1}{2}P \tag{34}\]  
Because the rock mechanics parameters of the fractured rock mass are quite different from those of the homogeneous rock, the damage effect of the shock wave again leads to serious changes in the rock mechanics parameters at this time. Therefore, according to the theory of damage mechanics, the damage factor of the fracture zone at this time also needs to consider the damage caused by the shock wave to the rock, so the damage factor at this time should be  
\[D^{\prime}=D+D_{0} \tag{35}\]  
In this formula, \(D^{\prime}\)--the sum of the damage, \(D\)--initial damage and \(D_{0}\)--the damage of the stress wave done to the rock formation.  
The damage \(D_{0}\) of stress wave done to the rock can be obtained according to the following formulas:  
\[D_{0}=1-\frac{1}{1+Ak\theta^{m}a^{3}} \tag{36}\]  
\[A=\frac{16\big{(}1-\mu_{c}^{2}\big{)}}{9(1-2\mu_{c})} \tag{37}\]  
\[a=\frac{1}{2}\Bigg{(}\frac{\sqrt{20}K_{IC}}{\rho\mathcal{C}_{p}\hat{\theta}_{ \max}}\Bigg{)}^{\frac{2}{3}} \tag{38}\]  
In these formulas, \(k\) and \(m\)--Weibull coefficients of rock; \(\theta\)--volume tensile strain; \(a\)--average radius of microcracks, m; \(\mu_{c}\)--dynamic [PERSON]'s ratio of fractured rock; \(K_{IC}\)--fracture toughness; and \(\hat{\theta}_{\max}\)--maximum volume strain rate.  
The radial stress and tangential stress under the action of the quasi-static stress field are  
\[\sigma_{rd}=\frac{\sigma_{r}}{1-D^{\prime}}=\frac{P_{d}^{1}}{1-D^{\prime}} \Bigg{(}\frac{R_{2(1)}^{1}}{r}\Bigg{)}^{2} \tag{39}\]  
\[\sigma_{bd}=\frac{\sigma_{\theta}}{1-D^{\prime}}=-\frac{P_{d}^{1}}{1-D^{\prime }}\Bigg{(}\frac{R_{2(1)}^{1}}{r}\Bigg{)}^{2} \tag{40}\]  
When \(\sigma_{bd}>S_{te}\), the crack propagation is strengthened, which can be expressed by the following formula:  
\[\frac{\sigma_{dd}}{1-D^{\prime}}\geq S_{te} \tag{41}\]To sum up, combining (34)-(41) gives  
\[S_{te}(1-D^{\prime})=\frac{P_{d}^{1}}{1-D^{\prime}}\cdot\left(\frac{R_{2(1)}^{1}}{ R_{2(2)}^{1}}\right)^{2} \tag{42}\]  
Therefore, the radius \(R_{2(2)}^{1}\) of the fracture zone of the burnt rock with the coupling charge structure is  
\[R_{2(2)}^{1}=\frac{1}{(1-D^{\prime})}\cdot\sqrt{\frac{P_{d}^{1}}{S_{te}}}\cdot R _{2(1)}^{1} \tag{43}\]