# Fractal Behavior of Size Distribution and Specific Surface Area of Blasting Fragments
## 3 Theory and Calculation Results
### Calculation of Fractal Dimensions of Blasting Fragmentation  
The size-cumulative mass fractal distribution curves for each experiment were obtained using Equation (10), as shown in Figure 7. The fractal dimensions of fragmentation and the statistical scale-free intervals are presented in Tables 4 and 5, respectively. As shown in Tables 4 and 5, \(R^{2}\) for the particle size-cumulative mass was greater than 0.88 for all the blasting experiments conducted at the four quarries. In general, a coefficient of determination (\(R^{2}\)) above 0.8 indicates a good fitting effect. In most cases, the coefficient of determination (\(R^{2}\)) of the particle size fractal curve was always greater than 0.839 [43]. This indicates that the blasting fragmentation distribution had distinct fractal characteristics. Comparative fitting was performed, and the blasting fragmentation distribution fractal curves of experiments #1-#7, #11-#13, and #17 could be linearly fitted with two line segments in the bilogarithmic plane, i.e., the blasting fragments obtained from experiments #1-#7, #11-#13, and #17 exhibited two scale-free intervals. Thus, the blasting fragments obtained from these experiments exhibited self-similarity within two scale-free intervals, indicating dual fractal characteristics. Figure 6: Power-law fractal distributions of blasting fragments obtained via the particle size–frequency method: (**a**) quarry \(\mathsf{Q}_{1}\); (**b**) quarry \(\mathsf{Q}_{2}\); (**c**) quarry \(\mathsf{Q}_{3}\); (**d**) quarry \(\mathsf{Q}_{4}\). \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Quarry**} & \multicolumn{3}{c}{**\(\boldsymbol{L}\)**} & \multicolumn{3}{c}{\(\boldsymbol{I}\)} & \multicolumn{3}{c}{\(\boldsymbol{S}\)} \\ \cline{2-11}  & \(\boldsymbol{D}\) & \(\boldsymbol{N_{0}}\) & \(\boldsymbol{R^{2}}\) & \(\boldsymbol{D}\) & \(\boldsymbol{N_{0}}\) & \(\boldsymbol{R^{2}}\) & \(\boldsymbol{D}\) & \(\boldsymbol{N_{0}}\) & \(\boldsymbol{R^{2}}\) \\ \hline \(\mathsf{Q}_{1}\) & 1.03 & 9176.3 & 0.99 & 1.00 & 4074.2 & 0.99 & 0.74 & 1340.5 & 0.98 \\ \(\mathsf{Q}_{2}\) & 1.12 & 6296.7 & 0.91 & 1.03 & 3252.4 & 0.91 & 0.83 & 1142.3 & 0.94 \\ \(\mathsf{Q}_{3}\) & 1.09 & 41,866 & 0.98 & 1.05 & 25,638 & 0.98 & 1.99 & 13,112 & 0.99 \\ \(\mathsf{Q}_{4}\) & 1.23 & 35,819 & 0.97 & 1.44 & 16,557 & 0.97 & 1.39 & 7470.3 & 0.97 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Fitting results for the power-law fractal distribution of blasting fragments based on the particle size–frequency method. On the basis of Figure 7 and Tables 4 and 5, the fractal dimensions of blasting fragmentation (\(D_{F}\)) for each experiment are presented in Figure 8. As indicated by Tables 4 and 5 and Figure 8, influenced by factors such as physical and mechanical properties of the rock mass, joint distribution, excavation method, and blasting parameters, the scale-free intervals and \(D_{F}\) values differed significantly among the experiments, revealing multiple fractal characteristics. As shown in Figure 8a, the \(D_{F}\) differences were substantial between the two scale-free intervals, with \(D_{F}\) values mostly exceeding 2.3 in the large scale-free interval and remaining below 2.3--mostly below 2.0--in the small scale-free interval. This variation pattern is consistent with the multifractal characteristics variation observed by [PERSON] et al. (2018) in the fault-gouge fragmentation process. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Quary Location** & **Lithology** & **Experiment Code** & **Scale-Free Interval (mm)** & **Slope** & \(\mathbf{R^{2}}\) & \(D_{F}\) \\ \hline \multirow{4}{*}{Q\({}_{1}\)} & & \#1 & 6–131 & 1.72 & 0.97 & 1.28 \\  & & \#1 & 131–1100 & 0.54 & 0.97 & 2.46 \\  & & & 6.8–123 & 1.60 & 0.99 & 1.40 \\  & & \#2 & 123–1400 & 0.56 & 0.94 & 2.44 \\  & & & 6.8–186 & 1.38 & 0.99 & 1.62 \\  & & & 186–1400 & 0.45 & 0.94 & 2.55 \\ \hline \multirow{2}{*}{Q\({}_{2}\)} & \multirow{2}{*}{Tuff} & \#4 & 3.2–70.1 & 0.71 & 0.94 & 2.19 \\  & & & 70.1–2030 & 0.71 & 0.91 & 2.30 \\ \hline \end{tabular}
\end{table}