## 3 Mechanical Experiment
### Analysis of the Experimental Results
#### 3.2.2 Different LRs  
There are three SR loading conditions based on the LR, namely, static (0.1-0.5 mm\(\cdot\)min\({}^{-1}\)), quasi-dynamic (1-5 mm\(\cdot\)min\({}^{-1}\)), and dynamic (10-50 mm\(\cdot\)min\({}^{-1}\)). The fitted Formula (4) and \(R^{2}\) values for the UCS and axial peak strain under these three states are also obtained, respectively. From Figure 7, it can be seen that the stress-strain curves of the specimens in the three SR loading states have different trends. However, during the original fracture compaction stage, the strain of the specimen was higher under static loading conditions, especially at 0.1 mm\(\cdot\)min\({}^{-1}\), indicating that the loading rate is low enough to allow a more fully integrated expansion of the internal cracks.  
\[\left\{\begin{array}{l}U_{1}=60.75x^{2}-33.325x+8.105\\ U_{2}=0.24583x^{2}-1.4975x+7.41167\\ U_{3}=0.00254x^{2}-0.11625x+6.13833\\ A_{1}=0.39616x+0.98769\\ A_{2}=-0.18231x+2.01615\\ A_{3}=0.000325x^{2}-0.02075x+1.495\end{array}\right. \tag{4}\]  
where \(U_{1}\) and \(A_{1}\) are the UCS value and peak axial strain when the SR in the static phase (0.1-0.5 mm\(\cdot\)min\({}^{-1}\)), respectively; \(U_{2}\) and \(A_{2}\) are the UCS value and peak axial strain when the SR in the quasi-dynamic phase (1-5 mm\(\cdot\)min\({}^{-1}\)), respectively; \(U_{3}\) and \(A_{3}\) are the UCS value and peak axial strain when the SR in the dynamic phase (10-50 mm\(\cdot\)min\({}^{-1}\)), respectively; \(x\) is the LR.  
From Figure 8a, it can be seen that the UCS of sandstone-like specimens with the same HDR does not increase or decrease monotonically under different LRs. The fitting  
Figure 7: Stress-strain curves of the specimens under different LRs: (**a**) static (0.1–0.5 mm\(\cdot\)min\({}^{-1}\)); (**b**) quasi-dynamic (1–5 mm\(\cdot\)min\({}^{-1}\)); (**c**) dynamic (10–50 mm\(\cdot\)min\({}^{-1}\)).  
was carried out according to each of the three loading states, which fitted well with a \(R^{2}\) of 0.9999, 0.9999 and 0.9999, respectively. The trend for each SR phase is approximately the same: first decrease and then increase. During the static phase, the lowest value of UCS, 3.47 MPa, is obtained at 0.2 mm\(\cdot\)min\({}^{-1}\). In the quasi-dynamic stage, a similar pattern of variation is observed, with the lowest value of UCS occurring at 2 mm\(\cdot\)min\({}^{-1}\) with 5.4 MPa. However, the UCS of the specimens in the quasi-dynamic phase are generally greater than those in the static phase. In the dynamic loading phase, the UCS values of the specimens with LRs between 10 and 20 mm\(\cdot\)min\({}^{-1}\) are smaller, and the lowest value appear at 20 mm\(\cdot\)min\({}^{-1}\). However, at a LR of 50 mm\(\cdot\)min\({}^{-1}\), the value of UCS is 6.68 MPa, which is the maximum across all phases. Additionally, all error bars are small, indicating a low discretization.  
Axial Peak Strain \(\varepsilon\) he trend in \(\varepsilon\) varies at different phases of SRs (Figure 8b). The fitting parameter \(R^{2}\) are 0.99964, 0.98909 and 0.99998, respectively. During the static loading phase, as the LR increases, the \(\varepsilon\) rises slowly with a slope rate of 0.39616. There is a sharply rise trend during the transition from the static phase to the quasi-dynamic loading phase. The \(\varepsilon\) tends to decrease rapidly during the quasi-dynamic loading process. However, the values are greater than those observed during the static loading phase. In the dynamic loading phase, the curve of \(\varepsilon\) first slowly decrease and then quickly increase as the LR increases. These changing characteristics are not consistent with UCS, indicating that different LRs have some effect on the elastic modulus of the specimens. The elastic modulus reflects a rock's ability to resist deformation and also represents the strength of the bonds between microscopic particles within the rock. This shows that different LR environments have a pronounced effect on the internal structure of the same specimen as well as on the bonding ability between microscopic particles.