# Experimental and Numerical Investigations on the Slate Shearing Mechanical Behavior
## 3 Test Results  
### Shear Strength Parameters  
The shear stress-displacement curve of slate under different \(\beta\) and \(\sigma_{\mathrm{n}}\) is shown in Figure 4. The overall variation in shear stress-shear displacement curves for all samples is similar before reaching \(\tau_{\mathrm{p}}\). Initially, the curve exhibits a slight slope during loading, with the shear displacement rapidly increasing and the shear stress increasing slowly. This is attributed to a specific gap between the shear box and the sample; the initial shear stress has decreased these gaps. After reaching a certain shear displacement value, the slope of the curve increases significantly, and shear displacement rises at a slower. Notably, when \(\beta=30^{\circ}\), higher \(\sigma_{\mathrm{n}}\) values correspond to a longer linear deformation stage [32]. After reaching the \(\tau_{\mathrm{p}}\), the slope of the curve shifts from positive to negative, and the stress decline phenomenon appears multiple times. As shear stress decreases to a specific level, the slope suddenly becomes smooth. With increasing shear displacement, the shear stress gradually enters the residual strength phase. Figure 3: (**a**) Ture triaxial loading system and shear box and (**b**) schematic of principal of the shear box [31]. In general, when \(\beta\) = 45\({}^{\circ}\), \(\tau_{\mathrm{p}}\) is the largest, and it is the smallest at 0\({}^{\circ}\). The \(\tau_{\mathrm{p}}\) of different \(\beta\) values correspond to distinct tangential displacements. Specifically, for \(\beta\) values of 0\({}^{\circ}\) and 90\({}^{\circ}\), the tangential displacement at peak load is relatively minor, ranging from 0.7 mm to 1.0 mm and 0.8 mm to 1.8 mm, respectively. For \(\beta\) values of 45\({}^{\circ}\) and 60\({}^{\circ}\), the spans of tangential displacement (1.3-2.9 mm and 1.2-2.8 mm, respectively) are similar. Brittle shear failure occurs only at 5 MPa for \(\beta\) = 0\({}^{\circ}\) and 90\({}^{\circ}\). The brittle shear failure is more pronounced for \(\beta\) = 45\({}^{\circ}\) and \(\sigma_{\mathrm{n}}\) < 20 MPa. At \(\beta\) = 60\({}^{\circ}\), there are more shear-slip phenomena, particularly evident when \(\sigma_{\mathrm{n}}\) = 15 MPa and the displacement corresponding to \(\tau_{\mathrm{p}}\) is the smallest at \(\sigma_{\mathrm{nmax}}\). With the increase in \(\sigma_{\mathrm{n}}\), samples with different \(\beta\) values exhibit a shear-slip phenomenon. All samples exhibit residual stress. Table 1 presents the mean and standard deviation (SD) ratio of residual stress to peak stress ratio (\(\tau_{\mathrm{r}}/\tau_{\mathrm{p}}\)). The variation in shear strength concerning \(\beta\) and \(\sigma_{\mathrm{n}}\) is shown in Figure 5, while the correlation between the ratio (\(\tau_{\mathrm{r}}/\tau_{\mathrm{p}}\)) and \(\beta\) is depicted in Figure 6. As \(\beta\) increases, the ratio initially decreases and then increases. The highest corresponding value (0.7) is observed when \(\beta\) is 90\({}^{\circ}\), while the lowest (0.5) occurs at \(\beta\) = 45\({}^{\circ}\). The stress drop associated with a vertical bedding plane is the most significant, indicating brittle failure, followed by horizontal bedding, with smaller stress drops corresponding to oblique bedding (0.5-0.6). Based on the Mohr-Coulomb shear strength criterion [33], the shear strength parameters of slate with different \(\beta\) values were calculated using Equation (1) and given in Table 1. The variations in cohesion (\(c\)) and internal friction angle (\(\varphi\)) are illustrated in Figure 7.  
\[\tau\ =\ \sigma_{\mathrm{n}}tan\varphi+c \tag{1}\]  
where \(c\) and \(\varphi\) represent the cohesion and internal friction angle, respectively, while \(\tau\) and \(\sigma_{\mathrm{n}}\) denote the shear and normal stresses. As \(\beta\) increases, cohesion exhibits a linear increase, whereas the internal friction angle undergoes an opposite change.