## 3 Results and Discussion  
In a typical SWCC (Figure 4a) [20], the AEV can also be expressed as \(\psi_{b}\) (bubbling pressure); it is a turning point on the soil-water characteristic curve. In the boundary-effect phase, almost all the pores in the soil are filled with water. After the suction reaches \(\psi_{b}\), the water content decreases drastically with the increase in suction. In a traditional way [1], this point is often obtained by extending the portion with a constant slope to intersect the suction axis at saturation. At this stage, the slope of the curve in the semi-logarithmic coordinate system is approximately constant, and this phase is referred to as the transition section. As the suction increases, the connectivity of pore water continues to decrease as the suction value increases, reaching the residual moisture content \(\theta_{r}\). It can be considered as the moisture content at the time when the liquid phase begins to become discontinuous. When the soil is dehumidified to this moisture content, it will become more difficult to discharge pore water by the increase in the suction force.  
Through the geometry of a SWCC in semi-logarithmic coordinates, the characteristic parameters of a SWCC can be calculated as follows (Equations (3)-(7) are quoted from reference [20]).  
Fit the SWCC with the Fredlund model [10]:  
\[\theta=\frac{\theta_{s}}{\left\{\ln\left[e+\left(\frac{\pi}{a}\right)^{n}\right] \right\}^{n}} \tag{2}\]  
where \(\theta_{s}\) is the saturated-water-content coefficient, which represents the starting position of the Fredlund fitting curve, and the \(m\), \(n\), and \(a\) are the soil parameters affecting the soil-water characteristic curve. Referring to Figure 4, in the semi-logarithmic coordinate system, the slope at the inflection point and the constant slope portion of the curve is defined as \(s_{1}\) (defined as positive). Since \(\psi=\text{a}\) at the inflection point,  
\[\begin{array}{l}s_{1}=\left|\frac{d\theta}{d\log\left(\theta\right)}\right|_ {\psi-a}\right|\\ =\left|\frac{\left(-m\right)\times\theta_{s}}{\left\{\ln\left[e+\left(a/a \right)\right]^{n}\right\}^{n+1}}\times\frac{n}{e+\left(a/a\right)^{n}}\times \left(\frac{a}{a}\right)^{n-1}\times\frac{a}{a}\times\ln 10\right|\\ =\frac{m\times\theta_{s}}{1.31^{m+1}}\times\frac{n}{37.2}\times\ln 10\end{array} \tag{3}\]  
AEV obtained by geometric relational operation,  
\[\psi_{b}=a\times 0.1\frac{3.72+1.31^{n+1}(1-e^{-\frac{m}{38N}})}{m\times n\times \ln 10} \tag{4}\]  
Figure 4: (**a**) Typical SWCC in the semi-logarithmic coordinate diagram [20]. (**b**) SWCCs of low-liquid-limit silty clays at three densities without freeze-thaw cycles.  
Similarly, when you take a residual segment point (\(\psi^{\prime}\), \(\theta^{\prime}\)), the slope of the residual segment can be represented by \(s_{2}\).  
\[s_{2}=\frac{m\times\theta_{s}}{\left\{\ln[e+(\psi^{\prime}/a)]^{n}\right\}^{m+ 1}}\times\frac{n}{e+(\psi^{\prime}/a)^{n}}\times\left(\frac{\psi^{\prime}}{a} \right)^{n-1}\times\frac{\psi^{\prime}}{a}\times\ln 10 \tag{5}\]  
According to the geometric relation, the residual water content and residual suction can be obtained by combining the above equations as follows:  
\[\psi_{r}=10^{\frac{\theta_{s}-\theta^{\prime}+s_{1}\times\log(a)-s_{2}\times \log(\psi^{\prime})}{s_{1}-s_{2}}} \tag{6}\]  
\[\theta_{r}=\theta_{i}-s_{1}\times(\log\psi_{r}-\log a) \tag{7}\]  
The results consistent with the traditional construction can be obtained. The characteristic parameters calculated by the number of different F-T cycles under three dry densities are summarized in Table 2.