## 3 Experimental Results
### Propagation Velocity Analysis of the Deformation Waves at the Elastic Stage  
Deformation waves are generated during the loading process of the red sandstone samples. Some researchers have studied the propagation of deformation waves and estimated their propagation velocity range. However, the relationships between the propagation velocities of the deformation waves and the loading rates have not been fully clarified. In our study, such relationships were determined based on the experimental data. Deformation waves have one unique feature: they propagate along the sample's axis. The loading diagram is shown in Figure 9. Since the samples were loaded at a steady loading rate, an approximate dependence \(\varepsilon\sim t\) was assumed. The deformation waves' strain peaks (i.e., peaks and troughs of the deformation waves) were plotted on the \(X\)-axis. The slope of the straight line representing the relationship of \(X\) vs. \(t\) was calculated. On this basis, the motion velocities of the points with the maximum and minimum values of strain, that is, the propagation velocities of the deformation waves were estimated. Taking the case with a loading rate of 0.5 mm/min as an example. After the microscopic characterization of the specimen, the deformation diagram of the strain component \(\varepsilon_{xx}\) on the centerline at 8 time moments was selected in the elastic stage (Figure 10). According to the treatment method of local strain maximization, the points of local strain maximum are chosen for comparative analysis. The results are shown in Figure 11.  
Figure 9: The loading scheme of rock samples. Figure 11: Variation trend of the maximum value of localized strain along the central line. Figure 10: Deformation waves along the central line of the samples from 640 s to 920 s.  
After data processing, the deformation wave's propagation velocities were estimated as 5.4 \(\times\) 10\({}^{-5}\), 6.0 \(\times\) 10\({}^{-5}\), and 6.3 \(\times\) 10\({}^{-5}\) m/s in the three groups of experiments under the loading rate of 0.5 mm/min. After averaging, the strain peak's propagation velocity (of deformation waves) was 5.9 \(\times\) 10\({}^{-5}\) m/s in the samples. It exceeded the axial boundary advance propagation velocity of 0.8 \(\times\) 10\({}^{-6}\) m/s (equivalent to the loading rate of 0.5 mm/min) by one order of magnitude. Therefore, the propagation velocities of the deformation waves were significantly higher than those of the matter particles. The propagation velocity of the slow deformation waves was calculated under the loading rate of 0.5 mm/min. Under this loading rate, stable regions of localized deformation were formed due to the slow deformation waves. These regions did not move over time. It can be observed from the dashed box in Figure 10 that at 60-100 mm on the sample surface, the peaks and troughs' positions were relatively fixed, and the peak and the basic shape of the wave only change slightly. However, the waveform and the peaks' positions changed significantly for the regions formed at an earlier stage, indicating the normal propagation and fluctuations. This situation persisted until the last diagram. The above region was stable and no longer experienced dramatic plastic deformation over time. The propagation of the slow deformation wave occurs at the initial stage of loading, that is, the elastic stage and the compaction stage. At the initial stage, the damage of the sample develops uniformly, and the slow deformation wave propagation is produced. As the loading process goes on, the plastic localization occurs gradually, and the propagation of the slow deformation wave no longer appears. At this stage, the plastic deformation is concentrated to the adjacent weak points, thus showing the plastic localization, and then these weak points are gradually connected, resulting in the fracture of a plane.