# Theoretical Study of Large Uncoupling Coefficient Loading for Surface Blasting
## 3 Critical Uncoupling Coefficient of Large Diameter Drilled Holes
### Radial Axial Uncoupling Equivalence Assumption  
Because the diameter of the drill rig is not consistent with that of the explosive charge, smooth blasting has an axial uncoupled charge and radial uncoupled charge. In order to study the impact pressure acting on the blast hole wall under the combined action of the two and ensure the generation of a smooth wall, the axial radial uncoupled charge is taken as an equivalent to the radial uncoupled continuous charge. The equivalence process can be divided into two types: one is based on the shape of the hole being unchanged, and the other is based the shape of the explosive being unchanged, while the volume of the blast hole is the same. The above equivalence satisfies the following assumptions:  
(1) The charge quality of blasting hole is certain, and the equivalent process does not consider the critical diameter value of explosive detonation. (2) The explosive density during the charging process is uniform and constant. When the radial and axial uncoupled charge is equivalent to the radial uncoupled continuous charge, the equivalent charge diameter may be less than the critical diameter of the explosive detonation. In reality, the explosive diameter is greater than the critical diameter, and assumption (1) can be met. As only the charge structure of the blasting hole is changed, and the size and composition of the explosives used in each hole remain unchanged, assumption (2) basically holds. Considering that the shape of the blast hole remains the same and the explosives are equivalent, the cross-sectional view of an equivalent blast hole for the axial uncoupled charge is shown in Figure 1.  
The radially and axially uncoupled charge is equivalent to a radial uncoupled continuous charge, wherein the explosive mass is equal, and the explosive density is constant. This leads to  
\[\rho_{0}\frac{\pi}{4}{d_{c}}^{2}l_{c}=\rho_{0}\frac{\pi}{4}{d_{x}}^{2}(l_{c}+l _{a}) \tag{15}\]  
Figure 1: Cross-sectional view of equivalent uncoupled charge in the blast hole. The equivalent diameter for a radially uncoupled continuous charge is obtained from Equation (15) as  
\[d_{cx}=d_{c}\sqrt{\frac{l_{c}}{l_{c}+l_{a}}} \tag{16}\]  
Substituting Equation (16) in Equation (3), the impact pressure \(P_{3}\) acting on the blast hole wall for the case of an axially and radially uncoupled charge is obtained as follows:  
\[P_{3}=\frac{\rho_{0}{D_{1}}^{2}}{8}(\frac{d_{c}}{d_{b}})^{6}(\frac{l_{c}}{l_{c }+l_{a}})^{3}n \tag{17}\]  
The radial uncoupling coefficient \(K_{d}\) (\(K_{d}=d_{b}/d_{c}\) and axial uncoupling coefficient \(K_{1}\) (\(K_{1}=(l_{a}+l_{c})/l_{c}\)) can be obtained by substituting into Equation (17):  
\[P_{3}=\frac{\rho_{0}{D_{1}}^{2}}{8}K_{d}{}^{-6}{K_{1}}^{-3}n \tag{18}\]  
Considering that the shape of the explosive remains the same and the volume of the blast hole is equivalent, the cross-sectional view of an equivalent blast hole for the radially and axially uncoupled charge is shown in Figure 2.  
For a fixed blast hole, the radial axial uncoupled charge is equivalent to the radial uncoupled continuous charge, as shown in Figure 2. The equivalent process of blast hole volume remains unchanged. This leads to  
\[\frac{\pi}{4}{d_{b}}^{2}(l_{c}+l_{a})=\frac{\pi}{4}{d_{x}}^{2}l_{c} \tag{19}\]  
The diameter of an equivalent blast hole for a radially uncoupled continuous charge is obtained from Equation (4) as  
\[d_{bx}=d_{b}\sqrt{\frac{l_{c}+l_{a}}{l_{c}}} \tag{20}\]  
\[\rho_{0}\frac{\pi}{4}{d_{c}}^{2}l_{c}=\rho_{0}\frac{\pi}{4}{d_{x}}^{2}(l_{c}+l _{a}) \tag{15}\]0 Cross-sectional view of an equivalent blast hole for a radially and axially uncoupled charge. Substituting Equation (20) in Equation (3), the impact pressure acting on the wall of the axially and radially uncoupled charge is obtained from Equation (18).