## 4 Discussion
### Influence of Debris Fractal Characteristics on Energy Evolution and Bursting Liability  
The degree of debris after fragmentation can directly reflect the failure characteristics, stress state, and energy change law of coal [42; 43]. The expression of the fractal dimension is as follows [44; 45]:  
\[D=3-\alpha \tag{13}\]  
\[\alpha=\frac{\lg(M(L)/M)}{\lg L} \tag{14}\]  
Figure 7: Crack propagation forms in coal with different JIAs. In the figure, \(T\) represents tension crack, \(S\) represents shear crack, and \(F\) represents tension–shear mixed failure. where \(D\) is the fractal dimension of the debris, \(\alpha\) is the slope of \(\lg(M(L)/M)-\lg L\) in the double logarithmic coordinate, \(L\) is the characteristic size of the equivalent particle size in the statistical interval, and \(M\) is the total mass of the debris. The debris collected and screened after coal failure with different JIAs was sorted into nine sizes. Taking a 60\({}^{\circ}\) JIA coal sample as an example, the side length of each grid was 10 mm; the screening results are shown in Figure 8. The sieved debris was measured and weighed to calculate the fractal dimension using Formulas (13) and (14), resulting in a fractal dimension fitting curve of coal failure debris with different JIAs (Figure 9). The correlation coefficient \(R^{2}\) of coal with different JIAs was greater than 0.88, indicating a high correlation between \(\lg(M(L)/M)\) and \(\lg L\), demonstrating that the size distribution of debris after coal failure exhibited obvious fractal characteristics. Figure 10 shows that the fractal dimension of coal with different JIAs after failure was between 1.7 and 2.4. When the JIA was 15\({}^{\circ}\), the minimum average fractal dimension was 1.79, and when the JIA was 90\({}^{\circ}\), the maximum average fractal dimension was 2.29. Studies have shown that the fractal dimension of intact rock ranges from 1.7 and 2.0 [46], while the fractal dimensions of coal containing pores fall between 2.2 and 2.4 [47]. Therefore, it can be observed that the existence of joints leads to the expansion of the fractal dimension  
Figure 8: Distribution of clastic particles in coal destruction. Figure 9: \(\lg(M(L)/M)\)—\(\lg L\) fitting curve of coal with different JIAs. interval of coal failure, that is, the fracture degree of coal with different JIAs varies greatly, and the distribution after fracture is not uniform. The fractal dimension is closely related to the JIA. The process of coal failure effects many fractures, resulting in more thorough fragmentation. The proportion of small particle debris increases, resulting in an increase in fractal dimension. The larger the JIA, the more elastic strain energy is accumulated before failure; as shown in Figure 4a, as the dip angle of the joint increased from \(0^{\circ}\) to \(90^{\circ}\), the average elastic energy increased from \(22.03\%\) to \(89.65\%\). At the moment of failure, coal can release energy, which is transformed into surface energy \(U_{a}\), kinetic energy \(U_{k}\), thermal energy \(U_{n}\), and radiation energy \(U_{s}\). This is illustrated in Equation (15):  
\[U_{d}=U_{a}+U_{k}+U_{n}+U_{s} \tag{15}\]  
where \(U_{a}\) is surface energy, \(U_{k}\) is kinetic energy, \(U_{n}\) is thermal energy, and \(U_{s}\) is radiation energy. Thermal energy \(U_{n}\) and radiation energy \(U_{s}\) are relatively small, and the kinetic energy \(U_{k}\) released under dynamic loading is generally less than \(7\%\) of the total energy [48]. Under the condition of static uniaxial compression, the released kinetic energy \(U_{k}\) is smaller, so the dissipated energy is approximately equal to the surface energy \(U_{a}\) of the new fracture surface [49]:  
\[U_{d}\approx U_{a}=A_{c}S_{a} \tag{16}\]  
where \(A_{c}\) is the area of new crack surface caused by coal failure and \(S_{a}\) is the energy required per unit crack area.