## 2 Stress Mechanism of Major Crack Initiation and Development  
The occurrence of clay landslides are usually controlled by major cracks [28]. Cracks are the result of many factors, and tension is only one of the possible causes. At present, tensile failure is frequently the sole failure criterion employed by studies on soil cracking [44]. Soil belongs to a kind of friction material and mainly shows shear failure. Therefore, it is necessary to consider the tensile, compressive, and shear properties of soil during the initiation and development of clay cracks. When studying soil cracking from the perspective of stress, the first thing to do is to define the yield characteristics of the soils in the stress regions related to the cracking process. This study adopted the Mohr-Coulomb (M-C) criterion commonly used in stress perturbations caused by static loads. Critical state soil mechanics is often used to judge soil cracking [45]. When shear failure of the unsaturated soil reaches the limit state, the following applies [46]:  
\[\sigma_{1}-\sigma_{3}=\frac{6c^{\prime}\cdot\cos\varphi^{\prime}}{3-\sin \varphi^{\prime}}+\left[\left(\frac{\sigma_{1}+\sigma_{3}}{2}-u_{a}\right)+ \chi(u_{a}-u_{w})\right]\cdot\frac{6\sin\varphi^{\prime}}{3-\sin\varphi^{ \prime}} \tag{1}\]  
There is an extended M-C criterion for unsaturated soils when shear failure occurs:  
\[\sigma_{3}-u_{a}=\frac{1-\sin\varphi^{\prime}}{1+\sin\varphi^{\prime}}(\sigma _{1}-u_{a})-\frac{2\cos\varphi^{\prime}}{1+\sin\varphi^{\prime}}\big{[}c^{ \prime}+\chi(u_{a}-u_{w})\tan\varphi^{\prime}\big{]} \tag{2}\]  
Combining Equations (1) and (2) results in the following:  
\[\sigma_{3}-u_{a}=-\left[c^{\prime}/\tan\varphi^{\prime}+\chi(u_{a}-u_{w}) \right]=-t^{*} \tag{3}\]  
In Equations (1)-(3), \(\sigma_{1}\) and \(\sigma_{3}\) are the maximum and minimum principal stresses, respectively; \(c^{\prime}\) is the effective cohesion; \(\varphi^{\prime}\) is the effective internal friction angle; \(u_{a}\) and \(u_{w}\) are the pore air pressure and pore water pressure, respectively; and \(\chi\) is the parameter related to the matrix suction \((u_{a}-u_{w})\). \(t^{*}=c^{\prime}/\tan\varphi^{\prime}+\chi(u_{a}-u_{w})\) is the M-C criterion tensile strength of unsaturated soil. Under the condition of constant matrix suction, the normal stress \(\sigma\) in [47] can be replaced with the net normal stress \((\sigma-u_{a})\) to obtain Equation (4). This replacement is based on two considerations. First, [PERSON]'s analysis is based on normal stress; second, despite the fact that the failure surface shape of the M-C criterion for unsaturated soils is controlled by the dual stress variables \((u_{a}-u_{w})\) and \((\sigma-u_{a})\), \((u_{a}-u_{w})\) is a function of saturation and the permeability coefficient of the clay is very small. When the saturation of a certain place in the clay does not change significantly, that is, \((u_{a}-u_{w})\) remains basically unchanged, and the failure surface of the M-C criterion of unsaturated soil is projected on the shear stress and net normal stress plane, it still has the same form as the classic M-C criterion. \[\sigma_{3}-u_{a}=\frac{1-\sin{\varphi^{\prime}}_{m}}{\cos^{2}{\varphi^{\prime}}_ {m}}\big{[}(\sigma-u_{a})-c^{\prime}{}_{m}\cdot\cos{\varphi^{\prime}}_{m}\big{]} \tag{4}\]  
In Equation (4), \({\varphi^{\prime}}_{m}\) and \(c^{\prime}{}_{m}\) are the effective internal friction angle and effective cohesion after strength reduction, respectively, which are determined by \(\tan{\varphi^{\prime}}_{m}=\tan{\varphi^{\prime}}/f\) and \(c^{\prime}{}_{m}=c^{\prime}/f\), where \(f\) is the reduction factor. The net normal stress should not be less than the tensile strength:  
\[\sigma_{3}-u_{a}\geq-t \tag{5}\]  
The actual tensile strength \(t\) of the soil is not larger than \(t^{*}\), that is, \(0\leq t\leq t^{*}\), where \(t^{*}\) is determined by Equation (3).