## 3 Experimental Results and Analysis
### Strain Rate Effects on Characteristic Stresses  
The compressive deformation and failure of rock materials intrinsically represent a continuum of progressive evolution, characterized by the nucleation, propagation, and coalescence of internal microcracks [6]. Classical theoretical models [31] delineate the stress-strain response into five characteristic phases, as illustrated in Figure 5. The initial compaction stage manifests a concave-upward curve morphology where the tangent modulus progressively increases with axial loading, indicative of the gradual closure of pre-existing fractures. This transitions into a linear elastic deformation phase marked by a near-ideal proportional relationship between stress and strain, the slope of which quantifies the rock's elastic modulus [32]. Upon reaching \(\sigma_{\rm ci}\), the system enters a stable crack propagation regime characterized by randomly distributed nascent microcracks lacking dominant growth trajectories. Exceeding \(\sigma_{\rm cd}\) triggers unstable crack growth, exhibiting preferential orientation alignment and concomitant volumetric dilation. Macroscopic fracture plane formation occurs subsequent to attaining \(\sigma_{\rm p}\) of the stress-strain curve. These critical thresholds--\(\sigma_{\rm ci}\) governing crack nucleation, \(\sigma_{\rm cd}\) signifying damage acceleration, and \(\sigma_{\rm p}\) demarcating ultimate compressive strength--collectively define brittle behavioral attributes. Additionally, the characteristic stress ratios serve as pivotal indices for evaluating rock brittleness and failure mode transitions. Figure 4: Stress–strain curves of granite specimens in triaxial compression under different strain rates ((a) \(10^{-6}\) s\({}^{-1}\); (b) \(10^{-5}\) s\({}^{-1}\); (c) \(10^{-4}\) s\({}^{-1}\); (d) \(10^{-3}\) s\({}^{-1}\); (e) \(10^{-2}\) s\({}^{-1}\)). The \(\sigma_{\rm cd}\) corresponds to the peak stress on the axial stress-volumetric strain curve, where the volumetric strain is determined by Equation (1):  
\[\varepsilon_{\rm v}=\varepsilon_{1}+2\varepsilon_{3} \tag{1}\]  
where \(\varepsilon_{\rm v}\) denotes volumetric strain, \(\varepsilon_{1}\) represents axial strain, and \(\varepsilon_{2}\) is lateral strain. There are many methods to ensure the value of \(\sigma_{\rm ci}\), including the crack volumetric strain method [32], AE method [16], and lateral strain response method (LSR method) [33]. In this paper, the LSR method was applied to determine \(\sigma_{\rm ci}\), and the detailed procedures were as follows:  
1. The peak point of the axial stress-volumetric strain curve is the volumetric dilatancy point (point A in Figure 6), and the corresponding axial stress value is \(\sigma_{\rm cd}\);
2. Assuming the point of \(\sigma_{\rm cd}\) in the curve of the axial stress--ateral strain and connecting it to the curve's origin as the reference line, as shown in Figure 6;
3. Drawing the curve based on the difference value between the curve of axial stress-lateral strain and the reference line, and the peak point of this curve corresponding to the axial stress is \(\sigma_{\rm ci}\), as shown in Figure 6.  
The characteristic stresses calculated using the aforementioned methodology are systematically presented in Table 2. Figure 7 illustrates the relationships between the logarithm of strain rate and three characteristic stresses. The results demonstrate positive correlations between these characteristic stresses and the strain rate across the investigated strain rate range of \(10^{-6}\) to \(10^{-2}\) s\({}^{-1}\) under various confining pressures. For instance, at a confining pressure of 20 MPa, increasing the strain rate from \(10^{-6}\) to \(10^{-2}\) s\({}^{-1}\) elevates \(\sigma_{\rm ci}\) from 161.63 MPa to 196.51 MPa, \(\sigma_{\rm cd}\) from 220.12 MPa to 282.44 MPa, and \(\sigma_{\rm p}\) from 289.39 MPa to 354.03 MPa, corresponding to relative increases of 21.58%, 28.31%, and 22.34%, respectively.