## 3 Analysis of Rock Damage Characteristics
### Mesoscopic Analysis of Rock Damage Characteristics
When \(f(\alpha)\)'s slope is minimal on the right end, the corresponding horizontal coordinate is defined as the minimum fluctuation singularity index \(\alpha_{\min}\). The width of the multifractal spectrum \(\Delta\alpha\), defined as the difference between \(\alpha_{\min}\) and \(\alpha_{\max}\), represents the curve's opening size and quantitatively characterizes the heterogeneity of the probability measure of the fractal structure. The larger the value of \(\Delta\alpha\), the more pronounced the multifractal characteristics of the image, and the greater the heterogeneity of the probability measure. Figure 10 shows the multifractal spectra of CT scan images for the different slice positions of the sandstone after blasting under 0 MPa in situ stress. Slice images were taken at \(h\) = 10 mm, \(h\) = 20 mm, \(h\) = 30 mm, \(\ldots\), \(h\) = 100 mm. The difference between the x-coordinates of \(\alpha\) at the two ends represents \(\Delta\alpha\), which can be used to characterize the damage of the sandstone specimens. The larger the value of \(\Delta\alpha\), the greater the relative damage. Using the aforementioned method, \(\Delta\alpha\) values at different positions of sandstone after blasting under various in situ stress conditions were calculated, resulting in the curves of \(\Delta\alpha\) variation with slice positions under different in situ stress conditions, as shown in the figure below. Figure 9: Multifractal spectrum curve. Table 1 lists the \(\Delta\alpha\) values at different slice positions under various in situ stress conditions. From Table 1 and Figure 11, it can be seen that the slice position significantly affects the blasting damage. Regardless of the in situ stress, \(\Delta\alpha\) reaches its maximum value at \(h=100\) mm (located in the stemming segment), indicating the highest damage, and its minimum value at \(h=10\) mm (located at the bottom segment), indicating the lowest damage. Under zero in situ stress, \(\Delta\alpha\) decreases from 0.3956 at \(h=100\) mm (the stemming segment) to 0.3736 at \(h=60\) mm (the charge segment), and finally to 0.3516 at \(h=10\) mm (the bottom segment). Under 0.5 MPa in situ stress, \(\Delta\alpha\) decreases from 0.3410 at \(h=100\) mm to 0.3010 at \(h=60\) mm, and finally to 0.2415 at \(h=10\) mm. Similarly, under 1 MPa in situ stress, \(\Delta\alpha\) decreases from 0.3067 to 0.2865, and finally to 0.2322. Under 2 MPa in situ stress, \(\Delta\alpha\) decreases from 0.2967 to 0.2764, and finally to 0.2322. Under 3 MPa in situ stress, \(\Delta\alpha\) decreases from 0.2806 to 0.2622, and finally to 0.2235. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Slice Position/\(h\)**} & \multicolumn{3}{c}{**Bottom Segment**} & \multicolumn{3}{c}{**Charge Segment**} & \multicolumn{3}{c}{**Stemming Segment**} \\  & **10 mm** & **20 mm** & **30 mm** & **40 mm** & **50 mm** & **60 mm** & **70 mm** & **80 mm** & **90 mm** & **100 mm** \\ \hline
0 MPa & 0.3516 & 0.3546 & 0.3572 & 0.3588 & 0.3695 & 0.3736 & 0.3786 & 0.3881 & 0.3895 & 0.3956 \\
0.5 MPa & 0.2415 & 0.2434 & 0.2477 & 0.2582 & 0.2817 & 0.3010 & 0.3237 & 0.3360 & 0.3390 & 0.3410 \\
1 MPa & 0.2322 & 0.2337 & 0.2381 & 0.2505 & 0.2721 & 0.2865 & 0.2913 & 0.2980 & 0.3015 & 0.3067 \\
2 MPa & 0.2322 & 0.2379 & 0.2409 & 0.2504 & 0.2673 & 0.2764 & 0.2784 & 0.2837 & 0.2886 & 0.2967 \\
3 MPa & 0.2235 & 0.2323 & 0.2349 & 0.2398 & 0.2489 & 0.2622 & 0.2639 & 0.2710 & 0.2753 & 0.2806 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The \(\Delta\alpha\) values at different slice positions of sandstone specimens under various in situ stress conditions. Figure 11: Curves of variation with slice positions under different in csitu stress conditions. Figure 10: Multifractal spectra at different slice positions of the sandstone specimens under 0 MPa in situ stress. The curves of \(\Delta\alpha\) at different slice positions under varying in situ stress conditions are shown below. From Table 1 and Figure 12, it is evident that the in situ stress significantly affects the blasting damage.