## 2 Shear Constitutive Models of Slip Zone Soil  
The strain softening phenomenon often occurs in the process of large shear deformation of landslides; that is, the shear stress decreases with displacement when the shear stress exceeds the peak. According to a large number of ring shear test results, the whole process of shear deformation and failure of slip zone soil can be divided into five stages (Figure 2): the pore compaction stage (Stage 1 in Figure 2); the elastic deformation stage (Stage 2 in Figure 2); the plastic hardening stage (Stage 3 in Figure 2); the strain softening stage (Stage 4 in Figure 2); and the residual strength stage (Stage 5 in Figure 2). The shear characteristics of the above stages should be taken into consideration in the establishment of the shear constitutive model of slip zone soil. During the shearing process of the slip zone, the slip zone soil is progressively damaged. In this process, the shear stress of soil is jointly undertaken by the damaged part and the intact part. According to the strain equivalent hypothesis [21], the strength relationship of the microelement of the slip zone soil based on the statistical damage can be obtained:  
\[\tau=\tau^{\prime}(1-D)+\tau^{\prime\prime}D \tag{1}\]  
where \(\tau^{\prime\prime}\) is the shear stress of the damaged part, \(\tau^{\prime}\) is the shear stress of the intact part and \(\tau\) is the total shear stress of the soil; \(D\) is the damage variable of soil, which changes with the process of shear failure and deformation. Figure 2: Shear stress–shear displacement curve and damage evolution curve of the slip zone soil in the whole shear process. The damage variable \(D\) can define the damage degree of the soil. The damage variable of 0 indicates that the soil is not damaged, and the damage variable of 1 indicates that the soil is completely damaged. There are many methods to define the damage variable \(D\). In the shear process of slip zone soil, the damage variable \(D\) can be defined as the ratio of the damaged part area and the total area of the shear body (Figure 3):  
\[D=\frac{N_{\text{D}}}{N} \tag{2}\]  
where \(N_{\text{D}}\) is the area of damage microelement and \(N\) is the total area of microelement soil. When the soil is not damaged, the damage element \(N_{\text{D}}\) is 0, and the damage variable \(D\) is 0. When the soil is completely destroyed, the intact elements are totally transformed into the damaged elements, and the damage variable \(D\) is 1, as shown in Figure 2.  
The soil microelements can be divided into two parts: intact element and damaged element, as shown in Figure 3. According to the mechanical response model of soil microelement (Figure 4), the soil element is an elastic component with shear stiffness \(K_{\text{s}}\) in the stage without damage, which maintains elastic deformation and follows [PERSON]'s law, i.e., \(\tau^{\prime}=K_{\text{s}}u\). When the displacement of the element exceeds the critical displacement \(u_{\text{y}}\) in the shear process, a part of the intact element is transformed into the damaged element, which obeys a certain random distribution in the soil. Because the strength of the soil decreases to the residual strength when it is completely damaged, the shear stress of the damaged element is equal to the residual strength, i.e., \(\tau^{\prime\prime}=\tau_{\text{r}}\). Therefore, the strength of the soil is provided by both the intact parts and damaged parts and can be expressed as:  
\[\tau=K_{\text{s}}u(1-D)+\tau_{\text{r}}D \tag{3}\]  
where \(K_{\text{s}}\) is the slope of the linear elastic stage of the shear curve; \(\tau_{\text{r}}\) is the residual strength; \(K_{\text{s}}\) and \(\tau_{\text{r}}\) can be obtained from shear test data; \(u\) is the shear displacement. Figure 3: Damage model of soil microelement in shear process of slip zone soil. According to the statistical damage, the microelement strength level of the material obeys certain random distribution characteristics (Figure 3).