## 4 Numerical Simulation
### Energy Characteristics  
To investigate the energy characteristics of the coal specimens, the strain energy and kinetic energy were recorded during the impact tests. In the DEM numerical models, the strain energy of the specimen refers to the sum of the strain energy stored in all contacts in the specimen and the kinetic energy refers to the sum of the kinetic energy of all particles in the specimen. In this study, energy density is defined as the energy in the specimen per unit volume. Figure 5 presents the curves of strain energy density vs. \(k_{\mathrm{s}}\) for coal specimens at three typical impact speeds. Figure 6 further gives the maximum strain energy per unit volume in coal specimens subjected to different pre-stress levels and impact speeds. It can be seen that the impact speed has a remarkable influence on the strain energy release characteristics. When the impact speed is only 4 m/s, the strain energy release is relatively slow. At the end of the impact test (impact time = 600 us), all specimens have residual strain energy inside them. For specimens under the impact speed of 8 m/s, the strain release rate is much higher than that under the impact speed of 4 m/s. Furthermore, at the end of the impact test, the residual strain energy is also much lower. For specimens at high pre-stress levels (\(k_{\mathrm{s}}\geq 0.7\)), the residual strain energy is close to zero, which means that the broken specimens in such cases lose almost their whole energy storage capacity. When the impact speed reaches 14 m/s, the strain energy drops sharply after the peak point in all cases, indicating that specimens are suffering violent energy release, which may cause serious failure and fragment ejection. Moreover, there is no residual strain energy of all specimens under such a high impact speed; namely, these specimens are all completely broken.  
Generally, the strain energy release characteristics directly associate with the impact speed; a higher impact speed contributes to more strain energy stored in the specimen and a faster energy release rate during the instability stage of the specimen. Compared with the impact speed, pre-stress has less influence on the strain energy. For the specimen under a low impact speed of 4 m/s, the strain energy release rate rises gradually with the increase in the pre-stress coefficient (Figure 5a). However, when the impact speed increases to 8 m/s or 14 m/s, the influence of the pre-stress coefficient on strain energy release is not clear anymore. For specimens at a lower impact speed, the relation between pre-stress  
Figure 5: The curves of strain energy per unit volume vs. \(k_{\mathrm{s}}\) for coal specimens subjected to different pre-stress levels and impact speeds.  
and the strain energy per unit volume is approximately linear, but this relation gradually becomes unclear with the increase in the impact speed.  
When a coal burst happens, it is always characterized by violent fragment ejection with high kinetic energy. Therefore, it is of great importance to determine the kinetic energy response characteristics of the coupled static and dynamic loads. Figure 7 presents the maximum kinetic energy per unit volume of coal specimens subjected to different pre-stress levels and impact speeds during impact tests. It can be seen from Figure 7a that with the increase in the impact speed, the maximum kinetic energy per unit volume first increases slowly and then begins to increase rapidly when the impact speed exceeds about 6 m/s. As shown in Figure 7b, however, for a given impact speed, the value of the maximum kinetic energy per unit volume fluctuates without a clear law when the pre-stress coefficient increases from 0.0 to 0.8.