# Fractal Behavior of Size Distribution and Specific Surface Area of Blasting Fragments
## 3 Theory and Calculation Results
### Calculation of Fractal Dimensions of Blasting Fragmentation

Table 4: Fractal dimensions of blasting fragments with dual fractal characteristics. Figure 7: Relationship between the particle size and cumulative mass of blasting fragments in different quarries: (**a**) quarry Q\({}_{1}\) (experiments #1–#3); (**b**) quarry Q\({}_{3}\) (experiments #5–#7 and #11–#13); (**c**) quarry Q\({}_{3}\) (experiments #8–#10 and #14–#16); (**d**) quarries Q\({}_{2}\) (experiment #4) and Q\({}_{4}\) (experiment #17). The lithology of both quarries Q\({}_{1}\) and Q\({}_{4}\) is limestone. However, quarry Q\({}_{4}\) had a lower rock compressive strength, and experiment #17 employed a coupled explosive charging structure using mixed-porous granular ammonium nitrate. Under the action of blast loads, quarry Q\({}_{4}\) consumed more explosive energy near the boreholes, resulting in more pronounced fragmentation, which caused the small scale-free interval in experiment #17 to be narrower than those in experiments #1-#3. However, within the small scale-free interval, the \(D_{F}\) value in experiment #17 exceeded those in experiments #1-#3, indicating a higher level of fragmentation. Moreover, as shown in Figure 8a, despite significant differences in the large scale-free intervals between the two quarries, the variation in \(D_{F}\) remained minor. The lithology of quarry Q\({}_{3}\) is granite. When the bench height was 12 m (experiments #5-#7 and #11-#13), the rock fragmentation exhibited multifractal characteristics. However, \begin{table}
\begin{tabular}{c c c c c c} \hline
**Quarry Location** & **Lithology** & **Experiment Code** & **Scale-Free Interval (mm)** & **Slope** & \(\mathbf{R^{2}}\) & \(\mathbf{D_{F}}\) \\ \hline \multirow{9}{*}{Q\({}_{3}\)} & & \#5 & 3.2–215 & 1.31 & 0.98 & 1.69 \\  & & \#5 & 215–2400 & 0.37 & 0.94 & 2.63 \\  & & \#6 & 3.3–389 & 0.80 & 0.98 & 2.20 \\  & & \#6 & 389–2480 & 0.24 & 0.92 & 2.76 \\  & & \#7 & 3.2–310 & 1.12 & 0.98 & 1.88 \\  & & \#7 & 310–1500 & 0.40 & 0.97 & 2.60 \\  & Granite & & 3.2–68 & 2.49 & 0.99 & 0.51 \\  & & \#11 & 68–1600 & 0.53 & 0.98 & 2.47 \\  & & \#12 & 3.2–68 & 1.12 & 0.97 & 1.88 \\  & & \#12 & 68–1600 & 0.63 & 0.96 & 2.37 \\  & & \multicolumn{1}{c}{} & 3.2–100 & 2.25 & 0.98 & 0.75 \\  & & \#13 & 100–1840 & 0.47 & 0.98 & 2.53 \\ \hline \end{tabular}
\end{table} 
Table 4: _Cont._  
Figure 8: Fractal dimensions of blasting fragmentation in each experiment. (**a**) fractal dimensions of blasting experiments with dual fractal characteristics; (**b**) fractal dimensions of blasting experiments with single fractal characteristics. when the bench height was 8 m (experiments #8-#10 and #14-#16), multifractal characteristics were not observed during the fragmentation process. When the bench height was 12 m, experiments #5-#7 and #11 employed blasting techniques involving shock-reflection devices and water-decked devices, respectively. These two excavation methods increased the transmission and reflection of stress waves within the rock, enhancing the fragmentation in the small scale-free interval and resulting in higher fragmentation levels and larger \(D_{\text{F}}\) values within the small scale-free interval than within the large interval. In addition, Figure 8b shows that when the bench height was 8 m, the blasting with shock-reflection devices (experiments #8-#10) and the conventional bench blasting technique (experiments #14-#16) yielded similar \(D_{\text{F}}\) values. However, the granite in this area had significant weathering; thus, further research into the effects of excavation methods on rock fragmentation characteristics is needed.