## 2 Stress Mechanism of Major Crack Initiation and Development
Substituting Equation (5) into Equation (4) produces the following:  
\[\sigma-u_{a}\geq-\bigg{[}\cos{\varphi^{\prime}}_{m}\cdot\bigg{(}t\cdot\frac{ \cos{\varphi^{\prime}}_{m}}{1-\sin{\varphi^{\prime}}_{m}}-c^{\prime}{}_{m} \bigg{)}\bigg{]}=-T\]  
where \(T=T(c^{\prime},\varphi^{\prime},t,f)=\cos{\varphi^{\prime}}_{m}\cdot[t\cdot \cos{\varphi^{\prime}}_{m}/(1-\sin{\varphi^{\prime}}_{m})-c^{\prime}{}_{m}]\). The condition for the intact soil to start to appear with a tensile crack is as follows [48]:  
\[\sigma-u_{a}\rightarrow-T \tag{6}\]  
We assume that the compressive stress is positive according to the soil mechanics sign convention. From the viewpoint of continuum mechanics, \(\sigma-u_{a}=0\) or \(\Delta(\sigma-u_{a})=0\) exists in the continuous changes in stresses from \(\sigma-u_{a}>0\) to \(\sigma-u_{a}\rightarrow-T<0\). The shear strain field of intensely fissured clays is very similar to that of unfissured clays [49, 50]. Shortly before the peak deviator stress, the shear band is formed and cracks begin to occur; thereafter, the shear band develops further and the cracks gradually extend and connect; long after the peak stress, shear failure occurs. When shear deformation develops to a certain extent, a shear band characterized by shear element will appear [51]. Taking into account the softening of geomaterials, the shear stress-strain relationship of the M-C criterion can be simplified to a linear form [52]. When the shear element is in a plastic state, the following applies [53]:  
\[\frac{d\tau}{d\varepsilon_{s}}=G_{s}\bigg{(}\frac{\sigma-u_{a}}{p_{a}}\bigg{)} ^{n}\left\{ \begin{array}{ll}>0&(\tau_{r}<\tau<\tau_{f}^{-})\\ <0&(\tau_{r}<\tau<\tau_{f}^{+})\end{array} \right. \tag{7}\]  
where \(G_{s}\) is a small amount that tapers to zero; \(p_{a}\) is the standard atmospheric pressure; \(n\) is the compression index, which is a positive value; \(\varepsilon_{s}\) is the shear strain; \(\tau_{r}\) is the residual shear strength; \(\tau_{f}^{-}\) and \(\tau_{f}^{+}\) are the pre- and post-peak shear stress, respectively. Equation (7) modified the formula in [53], replacing \(\sigma\) with \(\sigma-u_{a}\) and dividing the peak shear stress \(\tau_{f}\) into two parts (\(\tau_{f}^{-}\) and \(\tau_{f}^{+}\)) to determine the value of \(d\tau/d\varepsilon_{s}\). The reason for dividing \(\tau_{f}\) into two parts is because the simplified linear form of the shear stress-strain differential relationship is not continuous at \(\tau_{f}\), so the value at \(d\tau/d\varepsilon_{s}\) does not exist. The replacement of stress is based on similar (but not the same) nonlinear shear stress-strain relationship with saturated soil and the extended M-C criterion projected on the plane of shear stress and net normal stress. Although the shear stress-strain curve is nonlinear, the constitutive equation of soil must satisfy the continuous and smooth conditions [35]. Thus, the curve is approximately smooth at \(\tau_{f}\) and \(d\tau/d\varepsilon_{s}=0\), thereby \(\Delta\tau=0\) or \(\Delta(\sigma-u_{a})=0\) exists when it is extremely close to \(\tau_{f}\) (or shear cracks generate). Cracks in soil occur successively rather than simultaneously [14, 54]. When cracks in soil develop with depth, tensile cracks will eventually stop developing due to increase in effective stress. At a certain depth inside the soil, the tip of the tensile crack will change from a tensile state to a compressive state. If the shear crack occurs at this time, \(\Delta(\sigma-u_{a})=0\) will still exist. Considering the stress conditions that exist immediately after the crack is formed [55], the stress distribution on the walls of a crack still has the abovementioned situation. In summary, the principal stress-based \(\Delta(\sigma-u_{a})=0\) can be set as the stress mechanism of the initiation and development of major cracks. This also means that there will be a sudden change in stress near the crack.