# Experimental Study on the Dynamic Characteristics of Fractured Coal Under Cumulative Impact
## 3 Results and Analysis
### Fractal Characteristics of Fragmentation  
Macroscopic failure of fractured coal under dynamic loading arises from coalescence of microcracks, which nucleate from submicron-scale defects, exhibiting inherent self-similarity across multiple scales. The fractal theory provides a powerful framework for analyzing fragment size distribution, offering critical insights into failure mechanisms. The sieve analysis of post-impact fragments, as shown in Table 2, reveals systematic shifts in mass distribution with increasing impact cycles. Intact coal specimens without pre-existing cracks subjected to single impact loading exhibited residual coarse fragments with particle sizes exceeding 30 mm in post-SHPB test analysis. With increasing impact cycles, a progressive decrease in mass percentage of coarse fragments (>30 mm) was observed, accompanied by a corresponding increase in fine fragment proportion. The fractal dimension provides an intuitive and quantitative measure of the degree of fragmentation in fractured coal samples. A larger fractal dimension indicates a higher number of fragments, smaller fragment sizes and a greater degree of sample fragmentation. Fragment size distribution follows the fractal mass-size relationship, as in Formula (4) [46]:  
\[M(x)/M_{\rm T}=(x/x_{\rm m})^{3-D} \tag{4}\]  
Figure 5: Relationship diagram of wave velocity, disturbance impact energy and dynamic compressive strength. where \(x\) is the fragment size (mm); \(x_{\text{m}}\) is the maximum fragment size (mm); \(M(x)\) is the cumulative mass of fragments with size less than \(x\); \(M_{\text{T}}\) is the total fragment mass; \(D\) is the fractal dimension. Taking the logarithm of both sides of Equation (4) yields Formula (5):  
\[\lg(M(x)/M_{\text{T}})=(3-D)\lg(x/x_{\text{m}}) \tag{5}\]  
The relationship reveals that the slope \(k\) of the linear fit in the log-log coordinate system equals \((3-D)\), thus allowing the calculation of the fractal dimension \(D\) of fragmented coal, as in Formula (6). \[D=3-k \tag{6}\]  
As shown in Figure 6, the logarithmic curves of fragment mass distribution for fractured coal samples under cumulative impacts are presented. The SHPB test data points in Figure 6 demonstrate a linear relationship in log-log coordinates, with correlation coefficients exceeding 0.93, indicating strong statistical significance. This confirms that the fragmentation of fractured coal under cumulative impacts exhibits self-similarity across scales, satisfying fractal distribution characteristics. The slopes of fitting curves in Figure 6 are extracted to calculate fractal dimensions, with the relationship between fractal dimension and prefabricated fracture inclination plotted in Figure 7. Under identical impact cycles, intact specimens exhibit the lowest fractal dimensions between 1.6 and 2.1, while 30\({}^{\circ}\) inclined fractures show the highest values, ranging from 1.8 to 2.4. As the fracture angle increases, the fractal dimension of fractured coal samples decreases, demonstrating that the crack propagation path is more complex and the degree of fragmentation is more severe in coal samples with a fracture angle of 30\({}^{\circ}\). Furthermore, increasing fractal dimensions with cumulative impacts demonstrate progressive damage accumulation, rendering fractured coal more susceptible to comminution during SHPB failure. Combined with morphological analysis, the failure patterns under dynamic loading can be inferred from fragment geometries. Figure 8 displays typical fracture patterns of coal samples subjected to cumulative impacts. Post-SHPB specimens exhibit distinct fragmentation patterns: low-impact cycles (Figure 8a-d) predominantly yield large fragments (>10 mm), whereas high-impact cycles (Figure 8i-l) generate finer particles, demonstrating a transition from macro-fracture to pulverization with increasing impacts. Morphologically, low-impact fragments display elongated shapes, while high-impact debris exhibits polygonal geometries and powder-like particulates.