## 4 Uniaxial Loading of Critically Damaged Coal Samples
### Fractal Characteristics  
The fragments of critically damaged coal samples were collected under uniaxial compression at different confining pressure unloading spans, and the damage effects and damage patterns of critically damaged coal samples at different confining pressures were analyzed using the mass fractal principle. To visually analyze the degree of fragmentation of the coal samples, standard sieves with apertures of 40, 20, 12.5, 8, 6, 4, 2, 1, and 0.5 mm were selected to sort the fragmented coal samples one by one according to their fragmentation characteristics, as shown in Figures 10 and 11. lg (MLeq/M) and LG (Leq) fitting curves and calculation of the fractal dimension were employed to further analyze the fragmentation trend of the coal samples. Figure 9: Cumulative number of impacts and the first occurrence time of \(b\)-value in different groups. To calculate the fractal dimension \(D\) of the coal sample using the fragment mass-equivalent edge length relationship, quantitatively characterize the distribution of clastic rock as:  
\[D=3-\alpha \tag{5}\]  
\[\alpha=\frac{\log_{10}(M_{Leg}/M)}{\log_{10}Leq} \tag{6}\]  
where \(D\) is the fractal dimension of the fragment; \(M_{Leg}/M\) is the cumulative proportion of fragments with an equivalent edge length less than Leo: \(M_{Leg}\) is the fragment mass at an equivalent edge length of Leo: \(M\) is the fragment mass within the calculated particle size. According to Equations (5) and (6), the fractal dimension D was calculated for each group of experiments as shown in Figure 5; \(\log_{10}(M_{Lag}/M)\) and \(\log_{10}Leq\) fitting curves were plotted by using Origin software ([[https://www.originlab.com/](https://www.originlab.com/)]([https://www.originlab.com/](https://www.originlab.com/))), and the fitting effects all reached more than 85%, which indicated that there was a strong correlation between the two. Based on \(D\) in Table 3 and Figure 12, it can be seen that there are large differences  
Figure 11: Uniaxial damage debris. Figure 10: Screen. in the fractal characteristics of coal samples under different groups. The larger the fractal dimension, the more complex the fragmentation and the more fragments produced, and the more regular the distribution of fragments of different sizes. Firstly, the figures in the table visually show that the overall trend in each particle size range from the control group to the 5 MPa-60% group to the 10 MPa-60% group and finally to the 15 MPa-60% group exhibits a decreasing trend and a corresponding decrease in fragmentation. For example, in the particle size range of 0-0.5 mm, 1.03%, 0.58%, 0.32%, and 0.30% of the total weight were accounted for in that order. Secondly, in terms of fractal dimension, the values also decreased from 2.08, 1.97, 1.85 to 1.77, one by one, from undamaged coal samples to peritectic pressure of 5 MPa to 10 MPa and finally to 15 MPa. After a single uniaxial compression, the distribution of fragments was mainly concentrated above 8 mm, and the mass of fragments above 8 mm accounted for 85.28%, 85.93%, 87.09%, and 91.25% of all fragments in the four groups, respectively, with 78.25%, 71.43%, 81.02%, and 86.06% of fragments above 12.5 mm in size. The analysis of the peak intensity and acoustic emission signals above shows that as the unloading span of the peritectic compression decreases, the degree of damage to the critically damaged coal sample increases in turn, with the primary small cracks within the coal sample gradually decreasing and gradually being replaced by cracks after the