# Theoretical Study of Large Uncoupling Coefficient Loading for Surface Blasting
## 3 Critical Uncoupling Coefficient of Large Diameter Drilled Holes
### Calculation of Large Uncoupling Coefficient
#### 4.3.3 Hole Spacing to Be Less Than the Length of the Burst Fracture  
1. Blasting hole spacing: In accordance with the theory of the joined action of stress waves and explosive gas, blast hole spacing can be determined as \[2r_{b}p_{b}=(a-2r_{K})S_{t}\] (36) The required blast hole spacing is \[a=2r_{K}+d_{b}P_{b}/S_{t}\] (37) \[a=2\left(\frac{bP_{4}^{\
u}}{S_{t}}\right)^{\frac{1}{a}}r_{b}+d_{b}P_{b3}/S_{t}\] (38) (2) Inter-hole burst crack propagation: During light surface blasting in the rock, due to the presence of adjacent blast holes, the stress state near the blast hole wall is changed, resulting in the concentration of tensile stress in the direction of the blast hole center line. The cracks are generated and developed  
Figure 4: Tensile stress concentration coefficient curve ([PERSON] et al. [35]). by the impact of gas expansion on the blast hole wall. When the strength factor in the crack tip region is less than the fracture toughness value of the rock, the crack extension will stop. Assuming that the depth of gas wedging into the rock is equal to the crack extension length and considering gas leakage and energy consumption, the pressure change on the blast hole wall will be  
\[\sigma_{r(\theta)}=\mp p_{b}\left(\frac{r_{b}}{l+r_{b}}\right)^{\alpha} \tag{39}\]  
where \(l\) is crack extension length (m); \(\alpha\) is stress decay index. The fracture toughness of a rock can be defined as its ability to stop the destabilizing expansion of a crack. It is independent of the shape and size of the fracture and external forces. It is only related to the rock composition, which is a characteristic of the rock itself. Two cases can be considered as follows: for \(K_{1}>K_{IC}\), the crack is destabilizing, expanding, and brittle; for \(K_{1}<K_{IC}\), the stress intensity factor of the fracture cut-off is ([PERSON] et al. [36])  
\[K_{1}=\sigma_{r(\theta)}\sqrt{\pi(l+r)} \tag{40}\]  
According to the [PERSON] strength theory, the fracture toughness of rock under plane strain conditions is  
\[K_{IC}=\sqrt{\frac{2 ET}{1-\
u^{2}}} \tag{41}\]  
where \(E\) is the modulus of elasticity (kg/cm\({}^{2}\)); \(T\) is unit surface energy; \(\
u\) is Poisson's ratio. According to [PERSON]'s strength theory, fracture toughness \(K_{IC}\) under plane strain conditions is  
\[\sigma_{t}=6.68K_{IC} \tag{42}\]  
Under quasi-static gas action, the tension on the fracture surface can be considered to be equal to the gas pressure in the blast hole. Because of the guiding effect between adjacent blasting holes, the phenomenon of tensile stress concentration is generated in the radial direction of the blasting hole. By making full use of energy, it can be concluded from Equations (16) and (17) that  
\[K_{1}\geq K_{IC} \tag{43}\]  
That is  
\[K_{\theta}\sigma_{r(\theta)}\sqrt{\pi(l+r)}\geq K_{IC} \tag{44}\]  
Substituting Equation (15) into (17) gives  
\[a=2r_{K}+d_{b}P_{b}/S_{t}\]0  
Substitution of Equation (5) into Equation (19) gives  
\[a=2r_{K}+d_{b}P_{b}/S_{t}\]1  
For adjacent blast holes, the value of the blast hole spacing is equal to the blast crack in the rock extension crack length. For a single shell hole, the crack length generated by the blast load is equal to half of the blast hole spacing:  
\[a=2r_{K}+d_{b}P_{b}/S_{t}\]2  
\[a=2r_{K}+d_{b}P_{b}/S_{t}\]3  
Therefore, all three requirements must be met when determining the axial uncoupling factor of the air cushion blast charge. The radial uncoupling factor of the charge structure is taken as the maximum value between Equations (32) and (33). The values of axial uncoupling factor and blast hole spacing are determined using Equations (34), (35) and (48).