## 4 Failure Pattern and Analysis of the Energy Consumption
### Fractal Dimension and Energy Dissipation Density of Fragments  
Water-saturated fissured sandstone specimens were damaged under different impact air pressures to generate fragments of different sizes, which were analyzed using sieve tests in conjunction with relevant fractal theory [34; 35; 36]. The fractal dimension analysis of the tested fragments was carried out, and the fractal dimension equation for the mass-equivalent size of the specimen fragments is shown in Equation (10).  
\[\left\{\begin{array}{l}b=\frac{\lg(M_{r}/M)}{\lg r}\\ D=3-b\end{array}\right. \tag{10}\]  
where \(b\) is the slope of the linear function in double \(\log\lg(M_{r}/M)-\lg r\) coordinates; \(M_{r}\) is the mass of the corresponding fragments under the characteristic size \(r\); \(M\) is the total mass of the specimen fragments; and \(D\) is the fractal dimension of the sandstone fragments.  
The double logarithmic curve for a typical sandstone is shown in Figure 14.  
Figure 14 shows that, when the strain rate is low, the cumulative mass percentage of sandstone fragments at each grain size is low, and with increasing strain rate, the mass percentage of sandstone fragments increases significantly, and the fragmentation gradient of sandstone fragments is more obvious. This implies that, with low strain rate, the sandstone is under low impact load and the energy absorption by the rock specimens is not sufficient to support the complete fracturing of the rock specimens, leading to no apparent crushing of the sandstone. However, when the strain rate increases, the energy absorbed by the rock sample increases, exceeding the bearing capacity of the sandstone and leading to an increase in the extent of rock fragmentation. Furthermore, the cumulative fragment mass and grain size data at each grain size of the sandstone were linearly fitted in double logarithmic coordinates to obtain the slopes of their linear fit functions to calculate the fractal dimensions of the sandstone fragments, as shown in Table 5. The variation in the fractal dimension of sandstone clasts under different strain rates is presented in Figure 15.  
Figure 15 illustrates the increase in the fractal dimension of sandstone fragments as the density of crushing energy dissipation increases. Combined with the crushing macroscopic morphology of typical sandstone under different strain rates, this indicates that a larger crushing energy dissipation density of the rock means that more energy is used for the rock crushing action; with deeper crushing of the rock, the number of fine-grained fragments will be larger, and the fractal dimension will be bigger as well. The fractal dimension of specimen fragments and the crushing energy consumption density satisfy the multiplicative power function relationship, and the fitting relationship is presented in Equation (11) with a positive correlation coefficient of 0.9981. Thus, the fractal dimension is suitable as a  
Figure 14: \(\lg(M_{r}/M)-\lg r\) curve of a typical specimen with different strain rates.  
quantitative index to characterize sandstone fragmentation, which can reflect the degree of sandstone fragmentation.  
\[D=2.6882E_{V}^{0.2745}(R^{2}=0.9344) \tag{11}\]  
where \(D\) is the fractal dimension of sandstone fragments and \(E_{V}\) is the energy dissipation density of the specimen, J-cm\({}^{-3}\).