## 4 Failure Pattern and Analysis of the Energy Consumption
### Analysis of the Energy Dissipation  
The incident, reflected, transmitted, and absorbed energies of the rock specimen during the impact test using the SHPB test device can be calculated using Equation (5) [33]:  
\[\left\{ \begin{array}{l}E_{I}(t)=E_{0}C_{0}A_{0}\int_{0}^{t}\varepsilon_{I}^{2}(t )dt\\ E_{R}(t)=E_{0}C_{0}A_{0}\int_{0}^{t}\varepsilon_{R}^{2}(t)dt\\ E_{T}(t)=E_{0}C_{0}A_{0}\int_{0}^{t}\varepsilon_{T}^{2}(t)dt\\ E_{D}(t)=E_{I}(t)-E_{R}(t)-E_{T}(t)\end{array} \right. \tag{5}\]  
where \(E_{I}(t),E_{R}(t),E_{T}(t),E_{D}(t)\) are the incident, reflected, transmitted, and absorbed energies, respectively; \(\varepsilon_{I},\varepsilon_{R},\varepsilon_{T}\) are the incident, reflected, and transmitted strains, respectively; \(E_{0}\) is the elastic modulus of the compression bar; \(C_{0}\) is the longitudinal wave velocity of the compression bar; and \(A_{0}\) is the cross-sectional area of the compression bar. The energy values of the water-saturated fissured sandstone specimens during the test at different impact air pressures are shown in Table 3. It can be seen that the incident, reflected, transmitted, and absorbed energies of the specimen change as the impact air pressure and strain rate increase. Figure 11: Failure patterns of specimen at different strain rates. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Impact Air Pressure /MPa** & **Specimen** & **Incident** & **Reflected** & **Transmitted** & **Absorbed** & **Effective Volume** \\
**Dynamic** & **Number** & **Energy** & **Energy** & **Energy** & **Energy** & **of Specim** \\
**Dynamic** & **E\({}_{I}\)/J** & **E\({}_{R}\)/J** & **E\({}_{T}\)/J** & **E\({}_{D}\)/J** & **\(V_{S}\)/cm\({}^{3}\)** \\ \hline
0.300 & PJ42-01 & 76.74 & 42.77 & 9.28 & 24.68 & 96.23 \\  & PJ42-02 & 71.99 & 34.98 & 9.66 & 27.36 & 96.18 \\
0.325 & PJ42-04 & 90.95 & 47.56 & 10.02 & 32.37 & 96.40 \\  & PJ42-05 & 87.74 & 42.74 & 10.47 & 34.54 & 96.38 \\
0.350 & PJ42-07 & 100.75 & 47.85 & 11.61 & 41.29 & 96.27 \\  & PJ42-08 & 106.34 & 55.71 & 10.85 & 39.78 & 96.22 \\  & PJ42-10 & 120.05 & 65.97 & 11.74 & 42.34 & 96.34 \\
0.375 & PJ42-11 & 118.11 & 60.71 & 11.11 & 46.29 & 96.19 \\  & PJ42-13 & 133.97 & 76.33 & 12.74 & 44.91 & 96.33 \\
0.400 & PJ42-14 & 134.43 & 68.33 & 12.29 & 53.80 & 96.31 \\  & PJ42-16 & 139.34 & 76.93 & 12.65 & 49.76 & 96.21 \\  & PJ42-17 & 139.91 & 69.40 & 14.30 & 56.21 & 96.30 \\ \hline \end{tabular}
\end{table} 
Table 3: Energy values of water-saturated fissured sandstone specimens. Figure 10: Crack propagation diagram. To minimize the influence of specimen scale in energy dissipation, the crushing energy dissipation density is considered to reflect the amount of energy absorbed by the specimen, that is, the amount of energy dissipated per unit volume of rock absorbed due to crushing, as shown in Equation (6). \[E_{V}=E_{D}/V_{S} \tag{6}\]  
where \(E_{V}\) is the energy dissipation density of the specimen, J-cm\({}^{-3}\) and \(E_{D}\) is the energy dissipation of the specimen, J. \(V_{S}\) is the effective volume of the specimen, cm\({}^{3}\). The relationship between the crushing energy dissipation density and the strain rate for the water-saturated fissured sandstone specimens is shown in Figure 12.  
From Figure 12, it is clear that the crushing energy dissipation density of the water-saturated fissured sandstone specimens basically tends to increase quadratically with increasing strain rate. As the air pressure is raised, the incident energy increases and the sandstone specimen gradually transitions from the larger block state to the broken and crushed state. In this process, more energy needs to be absorbed for the germination, expansion and penetration of cracks inside the specimen, so the crushing energy dissipation density shows a gradual increase as the strain rate rises. Equation (7) displays the correlation between the crushing energy dissipation density of the specimen and the strain rate with a positive correlation coefficient of 0.9942.