# Theoretical Study of Large Uncoupling Coefficient Loading for Surface Blasting
## 3 Critical Uncoupling Coefficient of Large Diameter Drilled Holes
### Calculation of Large Uncoupling Coefficient  
3.1 The Initial Peak Radial Stress in the Surrounding Rock of the Blast Hole Lower Than the Compressive Strength of the Rock  
(1) The axial uncoupling factor based on the impact pressure generated by a shock wave:  
For the radial and axial uncoupled charge, there are no evident cracks on blast hole walls. To ensure that no compressive damage occurs in the surrounding rock of the blast hole, especially in the loaded section, it is necessary that the peak initial radial stress acting on the rock is lower than its compressive strength. Since the values of the uncoupling coefficient, \(K_{1}\), and \(K_{d}\) are greater than 1, \(P_{4}^{\prime\prime}<P_{4}^{\prime}\). Therefore, Equation (26) for large uncoupling factor is modified to satisfy a smaller value as  
\[\bigg{[}\bigg{(}\frac{2n\rho_{a}}{k+1}\bigg{)}\frac{(3k-1)^{2}\big{[}2Q/\big{(} k^{2}-1\big{)}\big{]}}{K_{b}S_{c}}\bigg{]}^{\frac{3}{4}}\leq K_{d} \tag{32}\]  
(2) The axial uncoupling factor based on impact pressure generated by gas expansion:  
Similarly, for the gas expansion pressure on the blast hole wall, since the uncoupling factors, \(K_{1}\) and \(K_{d}\), are greater than 1, \(P_{b3}<P_{b2}\). This implied that the charging section can be assumed to be a radially uncoupled continuous charge. Bringing Equation (30) into the transformation solution yields  
\[\bigg{(}\frac{P_{K}}{S_{c}}\bigg{)}^{\frac{3}{6}}\bigg{(}\frac{P_{C}}{P_{K}} \bigg{)}^{\frac{1}{6}}\leq K_{d} \tag{33}\]  
3.2 The Peak Initial Tangential Stress in the Surrounding Rock of the Blast Hole Higher Than the Tensile Strength of the Rock  
(1) The axial uncoupling factor based on the impact pressure generated by a shock wave:  
To ensure formation of radial cracks in the surrounding rock of the blast hole, the peak initial tangential tensile stress acting on the rock should be higher than the tensile strength of the rock. Due to the guiding effect between adjacent blast holes, the concentration of tensile stress will occur in the direction of the blast hole centerline. This effect would be the strongest at the blast hole wall, thus increasing the initial tangential tensile stress value in the surrounding rock of the blast hole. Based on \(K_{\theta}\sigma_{\theta}>S_{t}\), since the values of the uncoupling coefficients (shown in Figure 4), \(K_{1}\) and \(K_{d}\), are greater than 1, \(P_{4}\arcsec<P_{4}\arcmin\). Therefore, Equation (27) is brought into the transformation to find  
\[{K_{d}}^{2}K_{1}\leq\left[\left(\frac{K_{b}b2n\rho_{a}}{k+1}\right)\frac{(3k-1) ^{2}\big{[}2Q/\big{(}k^{2}-1\big{)}\big{]}}{K_{b}S_{t}}\right]^{\frac{3}{2}} \tag{34}\]  
2. The axial uncoupling factor based on the impact pressure generated by gas expansion: Similarly, for the gas expansion pressure acting on the blast hole wall, since the values of the uncoupling coefficients, \(K_{1}\) and \(K_{d}\), are greater than 1, \(P_{b2}<P_{b1}\). This indicates that the charge section can be assumed as a radial uncoupled continuous charge. The quasi-static pressure in the axial air-spaced section is less than that in the charge section; therefore, \(P_{b1}>P_{b2}>S_{t}\). As long as the quasi-static pressure in the axial air spacing section is greater than the tensile strength of the rock, it can meet the conditions. Thus, the uncoupling coefficient can be calculated as  
\[K_{d}^{2}K_{1}\leq\left(\frac{bK_{b}P_{k}}{S_{t}}\right)^{\frac{3}{2}}\left( \frac{p_{c}}{p_{k}}\right)^{\frac{1}{3}} \tag{35}\]