## 3 Results
### Crack Stress Thresholds Using Existing Methods  
The crack stress thresholds of siltstone (Figure 6) and granite (Figure 7) were determined using various methods including the CVS, ASR, CCR, axial stiffness, volumetric stiffness, volumetric strain, AE, LSR, LSIR, and dissipated energy ratio methods. The crack stress thresholds for siltstone obtained from different methods widely fluctuated, with overlapping ranges (Figure 8a), while clear boundaries were observed for different crack  
Figure 5: Stress–strain curves of granite under triaxial step loading and unloading tests for different values of \(\sigma_{3}\): (**a**) 0 MPa, (**b**) 4 MPa, (**c**) 8 MPa, (**d**) 12 MPa, (**e**) 16 MPa, and (**f**) 20 MPa. stress thresholds in the case of granite (Figure 8b). At low confining pressures, \(\sigma_{\text{cc}}\) of the siltstone calculated by various methods are found to be similar. However, for confining pressure greater than 10 MPa, the \(\sigma_{\text{cc}}\) calculated using the ASR and CCR methods are lower than those obtained by employing other methods. The \(\sigma_{\text{ci}}\) obtained by the axial stiffness method is significantly higher than those obtained by other methods, gradually converging toward the average value as the confining pressure increases. The \(\sigma_{\text{cd}}\) of siltstone obtained using the volumetric strain method is lower than those obtained using other methods, particularly at confining pressures below 4 MPa, indicating that the volumetric strain method is unsuitable for siltstone under these conditions. Figure 8: Box diagrams of crack stress thresholds calculated by various methods for (**a**) siltstone and (**b**) granite. Figure 6: Crack stress thresholds for siltstone: (**a**) crack closure stress, (**b**) crack initiation stress, and (**c**) crack damage stress. Figure 7: Crack stress thresholds for granite: (**a**) crack closure stress, (**b**) crack initiation stress, and (**c**) crack damage stress. The stress threshold patterns of granite under different confining pressures are similar to those of siltstone. The \(\sigma_{\rm cc}\) calculated using the axial stiffness, CCR, and ASR methods, which are based on axial deformation, are lower than those obtained using CVS and volumetric stiffness methods. The \(\sigma_{\rm ci}\) of granite obtained by the axial stiffness method is also larger than those obtained using the other methods at low confining pressures. The LSR method yields the lowest \(\sigma_{\rm ci}\), whereas its modified LSIR method produces results closer to the average value. The varying results between siltstone and granite are closely related to the mesostructure of the rock types. Due to the high porosity of siltstone, the average spacing of rock particles gradually decreases with an increase in pressure, resulting in an evident plastic volumetric strain, an enhanced deformation modulus, and a significant nonlinear stress-strain curve [41]. The significant nonlinearity of the axial stress-strain curve of siltstone increases the \(\sigma_{\rm cc}\) calculated by the axial stiffness method. Conversely, it reduces the slope of the ASR and CCR reference lines, causing them to differ more from the slope of the linear section of the stress-strain curve and thereby decreasing the calculated results of the ASR and CCR [25]. Consequently, the \(\sigma_{\rm cc}\) obtained by the axial stiffness method for siltstone closely align with those obtained by the volumetric stiffness and crack volumetric strain methods. Similarly, the \(\sigma_{\rm cc}\) obtained by the axial stiffness method for granite closely match those obtained by the ASR and CCR methods. During the initial stage of axial loading, the stress-axial strain curves of siltstone and granite enter the linear phase earlier than the stress-lateral strain curves (Figures 4 and 5), indicating that cracks perpendicular to the axial direction are compacted earlier than cracks parallel to the axial direction [42].