# Study on Strength of Slip Soil-Tuff Contact Surfaces in Tuff Landslide Based on Model Test
## 3 Analysis of Shear Test Results
### Shear Strength of Slip Soil-Rigid Contact Surfaces Considering Roughness  
The test data of the peak shear strength of the slip soil-concrete contact surface, the contact surface roughness, and the vertical stress were fitted in a spatial coordinate system. These points form a surface, which is the spatial surface where the peak strength of the contact surface varies with roughness and vertical stress, as shown in Figure 10. It can be seen that the peak shear strength of the slip soil-concrete contact surface increased with the increase in vertical stress and contact surface roughness. The peak shear strength reached the maximum state when the roughness and vertical stress reached the maximum value. In addition, in the direction of change in the vertical stress, the magnitude of change in the peak shear strength was more significant than that along the contact surface roughness.  
Figure 9: Failure pattern diagrams for contact surfaces with different roughness levels. (**A**) For R > R0. (**B**) For R = R0. (**C**) For R < R0.  
This suggests that vertical stresses have a more significant effect on the peak shear strength of the contact surface.  
To further explore the extent to which vertical stress and contact surface roughness affect the peak strength of the contact surface, a mathematical analysis of the surface enveloped by the spatial points of peak strength was fitted.  
According to previous studies [23; 24; 25; 26], the failure of the contact surface of the soil with a rigid object obeys the Moore-Cullen law. The steps involved in the method are as follows:  
\[\tau=\sigma\tan\phi+c \tag{3}\]  
In the formula, \(\tau\) is the contact surface shear strength, \(\sigma\) is the contact surface vertical stress, \(\varphi\) is the contact surface internal friction angle, and c is the contact surface cohesion. Considering the number of grooves on the surface of the test piece to take the limit, the contact surface roughness reaches the maximum roughness state: a calculated \(R_{max}=45.216\) mm. The definition of the roughness influence coefficient for the contact surface roughness and the maximum roughness state of the contact surface of the ratio is as follows:  
\[\lambda=\frac{R}{R_{\text{max}}} \tag{4}\]  
According to (4), the corresponding roughness influence coefficients are 0, 0.125, 0.25, 0.375, and 0.5 at contact surface roughness levels of 0 mm, 5.652 mm, 11.304 mm, 16.956 mm, and 22.608 mm, respectively. The experimental results of [PERSON] et al. [26] show that the cohesion of a clay-hydraulic soil contact surface can be approximately taken as the clay's cohesion. Therefore, in this study, it is assumed that the contact surface cohesion is not affected by the contact surface roughness and is only controlled by the nature of the soil. The contact surface roughness has a significant effect on the contact surface internal friction angle, i.e.,:  
\[\tau=\sigma F(\lambda)+c \tag{5}\]  
\(F(\lambda)\) is the tangent of the roughness influence coefficient as a function of the contact surface roughness. Based on the test results, nonlinear curve fitting was performed, and the fitting function results yielded the following equations:  
\[\tau=\sigma e^{(0.55\lambda-0.95)}+82.5 \tag{6}\]  
Figure 10: Spatial envelope of shear strength, roughness, and vertical stress on the contact surface.  
\[F(\lambda)=e^{(0.55\lambda-0.95)} \tag{7}\]  
Thus, the formula for calculating the shear strength between the slip soil and the rigid interface considering roughness was finally obtained, as shown in Equation (8).  
\[\tau=\sigma e^{(0.55\frac{R}{R_{\text{max}}}-0.95)}+82.5 \tag{8}\]  
The data obtained from the test were substituted for correlation coefficient analysis. The correlation coefficient \(R^{2}\) of the fitted Equation (8) was 0.982, which indicates a better fitting result; the fitted surface is shown in Figure 11.