## 4 Model Comparison and Validation
### Comparison Results of the Proposed Models  
The ring shear test data were fitted by the shear constitutive model with different damage statistical distribution functions. Series A of Figure 9 present the experiment and model result of slip zone soil from the Huangtupo landslide, which represents weak softening shear curves (Type A curves). In contrast, series B of Figure 10 presents the experiment and model result of slip zone soil from the Shizibao landslide, which represents the strong softening shear curves (Type B curves). The values of 1-8, as shown in Figures 9 and 10, respectively, denote the results fitted by the constitutive models based on eight kinds of damage distribution, and the parameters of the fitted curves for various models are also presented in Figures 9 and 10. However, the slip zone soil of the Huangtupo landslide presents strain hardening characteristic under 400 kPa normal stress; therefore, the shear curve of the Huangtupo landslide under 400 kPa normal stress is not fitted, and only the shear deformation curves of 100 kPa, 200 kPa, 300 kPa normal stress were fitted. Test parameters \(K_{\text{s}}\), \(u_{\text{y}}\), and \(\tau_{\text{r}}\) can be obtained from the ring shear tests, and the specific values are shown in Table 1.  
To measure the applicability of various constitutive models based on different damage distributions for different types of curves, the correlation coefficient \(R^{2}\) was invited to evaluate the model quality and can be formulated as follows:  
\[R^{2}=1-\frac{\sum\limits_{i=1}^{n}(\tau_{i}-\tau_{mi})^{2}}{\sum\limits_{i=1} ^{n}(\tau_{i}-\overline{\tau})^{2}} \tag{23}\]  
where \(\tau_{i}\) is the measured data of shear test, \(\tau_{\text{mi}}\) is the fitting value of the model, \(\overline{\tau}\) is the average value of the measured data, and \(n\) is the number of data obtained from the test. Figure 11 shows the correlation coefficient of each constitutive model, the average value of the correlation coefficient under different normal stresses, and the standard deviation. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Slip Zone Soil**} & **Normal Stress Level** & **Shear Stiffness \(K_{\text{s}}\)** & **Shear Displacement at** & **Residual Strength \(\tau_{\text{r}}\)** \\  & **(kPa)** & **(kPa/mm)** & **Yield Point \(u_{\text{y}}\) (mm)** & **(kPa)** \\ \hline \multirow{4}{*}{A: Huangtupo} & 100 & 165.390 & 0.294 & 45.853 \\  & 200 & 154.600 & 0.303 & 67.689 \\ \cline{1-1} \cline{2-5} \cline{4-5} \  
Figure 9: Fitting results of eight constitutive models on weaking softening type of shear curves. Series A presents the model curves and experimental data of the slip zone soil from Huangtop landslide representing weak softening type. (**A-1**) Weibull-constitutive model; (**A-2**) logistic-constitutive model; (**A-3**) normal-constitutive model; (**A-4**) lognormal-constitutive model; (**A-5**) exponential-constitutive model; (**A-6**) Gamma-constitutive model; (**A-7**) hyperbolic-constitutive model; (**A-8**) Gumbel-constitutive model. Figure 10: Fitting results of eight constitutive models on strong softening type of shear curves. Series B presents the model curves and experimental data of the slip zone soil from Shizibao landslide representing the strong softening type. (**B-1**) Weibull-constitutive model; (**B-2**) logistic-constitutive model; (**B-3**) normal-constitutive model; (**B-4**) lognormal-constitutive model; (**B-5**) exponential-constitutive model; (**B-6**) Gamma-constitutive model; (**B-7**) hyperbolic-constitutive model; (**B-8**) Gumbel-constitutive model. Combined with the fitting results and correlation analysis of each model, the test results were analyzed as follows:  
1. In the shear constitutive model based on Weibull distribution (Simplified as Weibull-constitutive model, and other models below were similar), the peak displacement of the model curve under high normal stress (400 kPa, 500 kPa) in type B deviated to the left compared with the test data (Figure 10(B-1)).