# Fractal Behavior of Size Distribution and Specific Surface Area of Blasting Fragments
## 4 SSA and Fractal Dimensions  
Studies indicated that the fracture surfaces of rock particles also exhibit fractal structures [28]. For rock particles, the surface area \(S_{F}\) is related to the particle size \(d\) and the surface fractal dimension \(D_{\text{s}}\) as follows:  
\[S_{F}\propto d^{D_{\text{s}}}. \tag{12}\]  
By applying the Griffith energy balance criterion, [PERSON] [45] analyzed the variations in the potential energy release rate and fracture-specific energy during the rock-particle fragmentation process, proposing a relationship between the SSA of rock particles and their surface fractal dimension:  
\[\text{SSA}\propto d^{D_{\text{s}}-3}. \tag{13}\]  
Figure 11: Average blasting fragmentation probability in each blasting experiment. Figure 10: High-speed photography images of the crack distribution of the bench surface during the blasting fragmentation of granite in quarry Q\({}_{3}\): **(a)** experiment #5; **(b)** experiment #11. Drawing on the research of [PERSON] [45], [PERSON] et al. [46] used the Walker-Lewis energy equation to derive the relationship between the surface fractal dimension \(D_{\rm s}\) and the fractal dimension of the particle size distribution \(D\):  
\[D_{\rm s}=\frac{D+3}{2}. \tag{14}\]  
Substituting Equation (14) into Equation (13) yields  
\[\rm{SSA}\propto d^{\frac{D-3}{2}}. \tag{15}\]  
[PERSON] et al. [47] presented the relationship between the SSA of the fragmented material and the fragmentation fractal dimension \(D_{F}\):  
\[\rm{SSA}\propto d^{D_{F}-3}. \tag{16}\]  
Using a cubic model, [PERSON] [28] established the relationship between the SSA of rock particles and the particle size:  
\[\rm{SSA}\propto d^{D-3}. \tag{17}\]  
The exponents in Equations (13) and (15)-(17) can be regarded as the SSA exponents of rock particles. Detailed measurements and values of \(D_{\rm s}\) and SSA for the blasting fragments randomly collected from the four quarries are presented in references [3; 40]. Using the mean particle size distribution fractal dimension \(D\) and fragmentation fractal dimension \(D_{F}\) reported in Sections 3.1 and 3.2, the SSA exponents of blasting fragments randomly collected from the four quarries were computed using Equations (13) and (15)-(17). The results are shown in Figure 12.  
As shown in Figure 12, the SSA exponents calculated using the surface fractal dimension \(D_{\rm s}\), fragmentation fractal dimension \(D_{F}\), and particle size distribution fractal dimension \(D\) differed from the measured values. The errors are presented in Figure 13.  
As shown in Figure 13, the SSA exponents calculated using Equation (16) and the fragmentation fractal dimension \(D_{F}\) exhibited the smallest differences from the measured values. Additionally, the findings presented in Section 3.3 indicate that the fragmentation fractal dimension is closely related to the probability of blasting fragmentation, which is associated with the newly generated surface area during the fragmentation process. Hence, the SSA, fragmentation energy, and energy utilization efficiency calculated according to the fragmentation fractal dimension align more closely with practical observations. \[\text{SSA}\propto d^{D_{\text{s}}-3}. \tag{13}\]0 Comparison of SSA exponents. According to the above analysis and the SSA presented in references [3,40], a predictive formula for calculating the SSA of blasting fragments using the fragmentation fractal dimension \(D_{F}\) and median particle size \(I\) is as follows:  
\[\text{SSA}\propto d^{D_{\text{s}}-3}. \tag{13}\]1  
where \(\lambda_{1}\) and \(\lambda_{2}\) are constants related to the lithology, rock fragmentation process, etc.