## 2 Principle and Intensity of Effective Stress of Unsaturated Soil  
### Principle of Effective Stress in Unsaturated Soil  
In its natural state, unsaturated soil is under the effect of moistening and stress fields. Typical unsaturated soil consists of three phase media: soil particles and water and air in the pores. The phase media share the load and interact with each other. The principle of effective stress in unsaturated soil is shown in Figure 1. On any section \(A\)-\(A\) of the soil with cross-sectional area \(A\), a total load \(\sigma\) is applied, and the water content is increased by \(\Delta A_{\text{sw}}\). Assuming the total area of soil particles is \(A_{\text{s}}\), \(A_{\text{s}w0}\) and \(A_{\text{s}w1}\) represent the cross-sectional area of soil particles in pore water before and after the change in the soil water content, respectively, and \(A_{\text{s}a0}\) and \(A_{\text{s}a1}\) denote the cross-sectional area of soil particles in the pore air before and after the change of soil water content, respectively. In addition, \(A_{\text{s}}=A_{\text{sw}0}+A_{\text{sa}0}=A_{\text{sw}1}+A_{\text{sa}1}\). Assuming the total pore area is \(A_{\text{v}}\), \(A_{\text{w}0}\) and \(A_{\text{w}1}\) denote the cross-sectional area of pore water before and after the change in the soil water content, respectively, and \(A_{a0}\) and \(A_{a1}\) denote the cross-sectional area of pore air before and after the change of soil water content, respectively. In addition, \(A_{\text{v}}=A_{\text{w}0}+A_{\text{a}0}=A_{\text{w}1}+A_{\text{a}1}\). Then, \(A=A_{\text{s}}+A_{\text{v}}\). The soil skeleton stresses before and after the change of soil water content are \(\sigma_{\text{s}v0}\) and \(\sigma_{\text{s}v1}\), the pore water pressure is \(u_{\text{w}0}\) and \(u_{\text{w}1}\), and the pore air pressure is \(u_{\text{a}0}\) and \(u_{\text{a}1}\). The relationship between the soil particle cross-sectional area before and after the change in water content is:  
\[\left\{ \begin{array}{l}A_{\text{sw}1}\ =\ A_{\text{sw}0}+\Delta A_{\text{sw}}\\ A_{\text{sa}1}\ =\ A_{\text{sa}0}-\Delta A_{\text{sw}}\end{array} \right. \tag{1}\]  
Figure 1: Principle of effective stress in unsaturated soil. The pore water cross-sectional area and pore air cross-sectional area before and after the change in water content have the following relationships:  
\[\left\{ \begin{array}{l}A_{\text{w1}}\ =\ A_{\text{v}}-A_{\text{a0}}\Big{(}1- \frac{\Delta A_{\text{sw}}}{A_{\text{sa0}}}\Big{)}\\ A_{\text{a1}}\ =\ A_{\text{a0}}\Big{(}1-\frac{\Delta A_{\text{sw}}}{A_{\text{sa0}}} \Big{)}\end{array} \right. \tag{2}\]  
Assuming that the pore water and pore air are connected, and that the three-phase media are homogeneous in the soil, then:  
\[\left\{ \begin{array}{l}\frac{A_{\text{sw0}}}{A_{\text{a}}}\ =\ \frac{A_{\text{sw}}}{A_{ \text{v}}}\ =\ S_{\text{r0}}\\ \frac{A_{\text{sw1}}}{A_{\text{s}}}\ =\ \frac{A_{\text{sw1}}}{A_{\text{v}}}\ =\ S_{\text{r1}}\end{array} \right. \tag{3}\]  
where \(S_{\text{r0}}\) and \(S_{\text{r1}}\) denote the saturation of the soil before and after the change in the water content, respectively, and \(S_{\text{r1}}=S_{\text{r0}}+\Delta S_{r}\). According to Figure 1 and the above analysis, the equilibrium equation after the change of water content in the \(A\)-\(A\) section is:  
\[\sigma A\ =\ \sigma^{\prime}{}_{\text{sv}}A_{\text{s1}}+u_{\text{w1}}(A_{ \text{sw1}}+A_{\text{w1}})+u_{\text{a1}}(A_{\text{sa1}}+A_{\text{a1}}) \tag{4}\]  
It can be deduced from Equations (1)-(4) that:  
\[\sigma\ =\ \sigma^{\prime}{}_{\text{sv}}\frac{A_{\text{s1}}}{A}+u_{\text{w1}}S _{\text{r1}}+u_{\text{a1}}(1-S_{\text{r1}}) \tag{5}\]  
where \(n\) denotes the porosity of the soil. \(\Delta S_{r}\) denotes the incremental saturation. Assuming that \(\sigma^{\prime}\ =\ \sigma^{\prime}{}_{\text{sv}}\frac{A_{\text{s1}}}{A}\ =\ \sum p^{ \prime}{}_{\text{sv}}/A\), where \(\sigma^{\prime}\) denotes the effective stress, it can be brought into Equation (5) to give the following after sorting and transformation.