# Calculation of Blasting Damage Zone Radius of Different Charge Structures in Burnt Rock
## 4 Attenuation Law of Stress Wave in Burnt Rock  
The strong shock wave generated in the rock after the explosion of the explosive propagates over a distance at a supersonic speed, where the propagation distance is generally 3-7 R. With the propagation of the shock wave, the stress amplitude and wave velocity decrease gradually, and gradually decay into a compressive stress wave. The propagation distance of the compressive stress wave is generally 120-150 R. When the strength of the compressive stress wave attenuates to the point where it can no longer damage the rock, it will be transformed into seismic waves. The formation and attenuation processes are shown in Figure 5 below. The attenuation curve of the stress wave is shown in Figure 6 below. The shock wave in the rock will rapidly attenuate into a stress wave. The attenuation law of radial stress caused by any point in the rock mass within the range of the shock wave can be expressed by the following formula:  
\[\sigma_{\rm r}=\mathrm{Pf}^{-\alpha} \tag{18}\]  
\[\alpha=2\pm\frac{\mu_{d}}{1-\mu_{d}} \tag{19}\]  
In these formulas, \(\overline{\mathrm{r}}\)--the ratio of the distance from the gun hole and the radius of the gun hole; \(\sigma_{\rm r}\)--radial stress, MPa; \(\alpha\)--stress wave attenuation coefficient; \(\pm\)--shock wave area/stress wave zone; and \(\mu_{d}\)--[PERSON]'s ratio, \(\mu_{d}=0.8\mu\). By introducing the damage variable \(D\), the attenuation law of radial stress and tangential stress caused by any point in the burnt rock can be obtained:  
\[\sigma_{\rm re}=\mathrm{Pf}^{-\frac{\alpha}{1-D}} \tag{20}\]  
Figure 5: Evolution process of explosion stress wave. Figure 6: Theoretical simulated stress wave attenuation curve. \[\sigma_{\theta e}=-b_{e}\sigma_{re} \tag{21}\] \[\sigma_{ze}=\mu_{c}(\sigma_{re}+\sigma_{\theta e})=\mu_{c}(1-b_{e}) \sigma_{re}\] (22) \[\alpha= \begin{cases}\alpha_{1}:\ 2+b_{e},\ Shock\ wave\ zone\\ \alpha_{2}:\ 2-b_{e},\ Stress\ wave\ zone\end{cases} \] (23) \[b_{e}=\frac{\mu_{c}}{1-\mu_{c}}=\frac{\mu_{d}\Big{(}1-\frac{16}{9}C_{d} \Big{)}}{1-\mu_{d}\Big{(}1-\frac{16}{9}C_{d}\Big{)}} \tag{24}\]  
In these formulas, \(\sigma_{re}/\sigma_{\theta e}\)--radial stress and tangential stress in the burnt rock, MPa, and \(b_{e}\)--lateral stress coefficient. Under the action of the explosion load, any point is in a three-dimensional stress state, and its strength is  
\[\sigma_{i}=\frac{1}{\sqrt{2}}\Big{[}(\sigma_{r}-\sigma_{\theta})^{2}+(\sigma_ {\theta}-\sigma_{z})^{2}+(\sigma_{z}-\sigma_{r})^{2}\Big{]}^{\frac{1}{2}} \tag{25}\]  
Arranged:  
\[\sigma_{i}=\frac{1}{\sqrt{2}}\sigma_{re}\sqrt{(1+b_{e})^{2}-2\mu_{c}(1-b_{e}) ^{2}(1-\mu_{c})+(1+b_{e}^{2})} \tag{26}\]  
Since \(\sqrt{(1+b_{e})^{2}-2\mu_{c}(1-b_{e})^{2}(1-\mu_{c})+(1+b_{e}^{2})}\) is a constant in the formula and in order to facilitate the derivation of the subsequent formula, this is defined as a constant \(B\). It can be seen from the analysis that the crushing area in the blasting failure area is compressed and the fracture area is stretched. Therefore, according to the [PERSON] criterion, when the stress \(\sigma_{i}\) at any point in the failure area satisfies the following conditions, the rock is destroyed. \[\frac{\sigma_{i}}{1-D}\geq\sigma_{0} \tag{27}\]  
\[\sigma_{0}= \begin{cases}\sigma_{cd}(Crushing\ Circle)\\ \sigma_{td}(Fissured\ Circle)\end{cases}  \tag{28}\]  
In these formulas, \(\sigma_{0}\)--rock failure strength, MPa; \(\sigma_{cd}\)--dynamic compressive strength of rock, \(\sigma_{cd}=S_{c}\sqrt[3]{\epsilon}\), MPa; \(\sigma_{td}\)--dynamic tensile strength, \(\sigma_{td}=\sigma_{t}\), MPa; and \(S_{c}\)--static compressive strength, MPa. The difference between the dynamic compressive and tensile strengths of the rock and the static value is not fixed. It is affected by many factors, including the type, composition, structure and test conditions of the rock. The following is a general description of the difference between the dynamic and static compressive and tensile strengths of rock:  
1.