## 3 Test Results and Discussion
### Energy Characteristics  
The dynamic energy release leads to some form of rock failure [50]. The deformation of rock elements under external loads can be regarded as a closed system, and as a result, the total input energy density \(U\) produced by the external load can be converted into two kinds of energy [51,52]. One is the released elastic energy \(U_{\rm e}\), directly related to the elastic modulus and Poisson's ratio of the rock. The release of elastic energy can cause structural failure of the specimen at the end of the loading. The other is the dissipated energy \(U_{\rm d}\), manifested by the micro damage and irreversible deformation of the specimen during the loading process and accompanied by the strength deterioration [35].  
Figure 8 illustrates a typical stress-strain curve and the relationship between the released elastic energy density and the dissipated energy density, which are shown in the regions of vertical lines (red) and oblique lines (yellow), respectively. Based on the two assumptions [51,52], the total energy density input from the external work can be written as:  
\[U=U_{\rm e}+U_{\rm d}=\int_{0}^{\varepsilon_{0}}\sigma d\epsilon \tag{11}\]  
where \(\sigma\) represents the stress and \(\epsilon\) indicates the strain of the rock specimen during the loading process; \(\varepsilon_{0}\) denotes the strain value corresponding to the peak stress point.  
The released elastic energy density (\(U_{\rm e}\)) can be obtained as:  
\[U_{\rm e}={\sigma_{0}}^{2}/2E \tag{12}\]  
where \(\sigma_{0}\) represents the SUCS and \(E\) represents the elastic modulus of the specimen.  
Consequently, the dissipated energy density (\(U_{\rm d}\)) of the specimen can be obtained as:  
\[U_{\rm d}=U-U_{\rm e} \tag{13}\]  
Figure 7: The (**a**) strength and the (**b**) elastic modulus influenced by the dynamic loading rate.  
Figure 9 reveals the effect of the static pre-stress on the released elastic energy density and dissipated energy density. When the dynamic loading rate is constant, the released elastic energy density decreases linearly with increasing static pre-stress, similar to the SUCS trend. There is no clear trend between the dissipated energy density and the static pre-stress. In general, the dissipated density increases as the static pre-stress increases. Combining Figure 9a,b, it can be seen that when the static pre-stress is low (40% of the UCS), the dynamic loading rate plays a major role in the energy partitioning, with high released elastic strain energy and low dissipated energy.  
Figure 10 shows the effect of dynamic loading rate on the released elastic energy density and dissipated energy density. When the static pre-stress is 40% of the UCS, the released elastic energy increases logarithmically. When the static pre-stress is greater than or equal to 50% of the UCS, the logarithmic increasing function for the released elastic energy no longer exists, but a general increasing trend still holds. For the dissipated energy density, there is no direct correlation between dissipated energy density and dynamic strain rates.  
Figure 8: The relationship between the released elastic energy density and the dissipated energy density.  
Figure 9: Effect of static pre-stress on (**a**) the released elastic energy density and (**b**) the dissipated energy density.  
Table 3 list details of the energy characteristics of specimens under superimposed static and dynamic loads. The total input energy density \(U\) varies from 218 to 330 kJ/m\({}^{3}\) under different combinations of static and dynamic loads, with an average value of 264.8 kJ/m\({}^{3}\).