## 2 Theoretical Analysis
### Case Study  
The selected mine had an average burial depth of 400 m, a mining height of 3 m, and a coal pillar width of 15 m. The cohesion between the layers was assumed to be 1.13 MPa, and the internal friction angle was 20.25\({}^{\circ}\) for the calculation. A formation density of 23.5 kN/m\({}^{3}\), a stress concentration factor of 2.7, and a support intensity of 0 were used in the analysis. Furthermore, the attenuation coefficient was set to 0.3. Finally, a blasthole measuring 12 m in length and 0.048 m in diameter was drilled, with its bottom located at \(x=3\) m and its orifice located at \(x=15\) m. The hoop stresses around the blasthole were obtained for lateral pressure coefficients of 1 and 0.5 as follows:  
\[\sigma_{\theta} = \left(15.98e^{0.3(3-x)}+9.4\right)\left(1+\frac{5.76\times 10^{-4}}{ \rho^{2}}\right)\quad x\geq 3,\lambda = 1 \tag{6}\] \[\sigma_{\theta} = \left(15.98e^{0.3(3-x)}+9.4\right)\left(0.75+\frac{4.32\times 10^{-4 }}{\rho^{2}}+\left(\frac{2.49\times 10^{-7}}{\rho^{4}}+0.25\right)cos2\theta \right)\quad x\geq 3,\lambda = 0.5 \tag{7}\]  
As seen in Figure 4, the hoop stress distribution around the blasthole was obtained based on Equations (6) and (7), where hoop compressive stress is positive. The distribution of hoop stress is depicted in Figure 4a at the blasthole wall for cross section I located at the orifice, where \(\sigma_{v}=10\) MPa; \(\lambda=1\), \(\sigma_{\theta}\) is compressive on the blasthole wall, exhibiting a circular distribution with a magnitude of \(2\sigma_{v}=20\) MPa; and \(\lambda=0.5\), \(\sigma_{\theta}\) is also compressive on the blasthole wall with a magnitude of \(2.5\sigma_{v}=25\) MPa in the horizontal direction (\(\theta=0^{\circ}\) and \(180^{\circ}\)) and \(0.5\sigma_{v}=5\) MPa in the vertical direction (\(\theta=90^{\circ}\) and \(270^{\circ}\)). Furthermore, as \(\theta\) changes from \(90^{\circ}\) to \(0^{\circ}\), the magnitude of \(\sigma_{\theta}\) on the blasthole wall gradually increases. Hoop compressive stress inhibits crack propagation [11]. Therefore, these results illustrate that the crack has a tendency to propagate in the maximum principal stress direction.  
Notably, since the magnitude of hoop stress on the blasthole wall is unrelated to the polar angle when the lateral pressure coefficient is 1, the hoop compressive stress distribution around the blasthole for cross section II is indicated in Figure 4b. The hoop compressive stress distribution on the surrounding area of blasthole for cross section II and III is presented in Figure 4c,d, respectively, for a lateral pressure coefficient of 0.5. Figure 4b,c shows that as \(\rho\) increases, the hoop compressive stress in the horizontal direction  
Figure 3: A force diagram in plan view of a blasthole with a radius of R under the influence of biaxial compressive stresses.  
declines sharply and then gradually approaches \(\sigma_{v}\) for the same \(x\), when \(\lambda=1\) or \(0.5\). As can be observed from Figure 4d, at a given \(x\), the hoop compressive stress that is aligned with the vertical axis initially increases slowly from \(0.5\,\sigma_{v}\) to \(0.7\,\sigma_{v}\), then decays back to \(0.5\,\sigma_{v}\) before finally remaining stable with the rising of \(\rho\), when \(\lambda=0.5\). In addition, for a given \(\rho\), the hoop compressive stress gradually decreases from the bottom of the blasthole to the orifice with the increase of \(x\). Therefore, as the NIS distributed along the blasthole axis decrease, the extent of rock damage gradually goes up in the axial direction owing to the hoop compressive stress.