## 4 Developed Constitutive Model
### Constitutive Parameters  
In comparison with the original model, this modified constitutive model includes three elastic moduli, \(E\), \(H_{\text{\tiny{E}}}\), and \(H_{\text{\tiny{H}}}\), all of which can be determined through modified unsaturated consolidation tests. In the modified unsaturated consolidation test, the specimen is subjected to constant net normal stress and gradually reduced matric suction, while always being subjected to lateral confinement, as shown in Figure 6.  
Assuming that the direction of the swelling is positive, Stage I demonstrates the stress state changes that occur in the soil elements of the specimen swelled under zero net normal stress from its initial matric suction to a specific matric suction level with lateral confinement. The lateral confining pressure without certain net normal stress is generated in the horizontal direction, and the maximum vertical swelling strain occurs in the vertical direction. The constitutive relationship at Stage I can be derived from Equation (1) and described by Equation (2):  
\[\left\{ \begin{array}{l}0=\frac{\mu-1}{E}\text{d}(\sigma_{h0}-u_{a})+\frac{ \text{d}s}{H_{\text{\tiny{H}}}}\\ \text{d}\varepsilon_{mv}=\frac{2\mu}{E}\text{d}(\sigma_{h0}-u_{a})+\frac{ \text{d}s}{H_{\text{\tiny{E}}}}\end{array} \right., \tag{2}\]  
\[\text{d}(\sigma_{h0}-u_{a})=\frac{E}{(1-\mu)H_{\text{\tiny{H}}}}\text{d}s, \tag{3}\]  
\[\text{d}\varepsilon_{mv}=\frac{2\mu}{1-\mu}\cdot\frac{\text{d}s}{H_{\text{ \tiny{H}}}}+\frac{\text{d}s}{H_{\text{\tiny{E}}}}, \tag{4}\]  
where \(\sigma_{h0}\) is the lateral confining pressure without certain net normal stress, and \(\varepsilon_{mv}\) is the maximum vertical swelling strain. Equations (3) and (4) are derived by solving the previously derived equations to estimate the lateral confining pressure without certain net normal stress and the maximum vertical swelling strain at Stage I. The lateral confining pressure actually corresponds to the lateral swelling pressure associated with free swelling without certain net normal stress. In Stage II, the stress state changes occur in the soil elements of the specimen swelled under certain net normal stress from its initial matric suction to a specific matric suction level with lateral confinement. The lateral confining pressure under certain net normal stress is generated in the horizontal direction, and the vertical swelling strain under certain net normal stress occurs in the vertical direction. The constitutive relationship at Stage II can also be derived from Equation (1) and described by Equation (5):  
\[\left\{ \begin{array}{l}0=\frac{\mu-1}{E}\text{d}(\sigma_{h}-u_{a})+\frac{ \mu}{E}\text{d}(\sigma_{v}-u_{a})+\frac{\text{d}s}{H_{\text{\tiny{E}}}}\\ \text{d}\varepsilon_{v}=-\frac{\text{d}(\sigma_{v}-u_{a})}{E}+\frac{2\mu}{E} \text{d}(\sigma_{h}-u_{a})+\frac{\text{d}s}{H_{\text{\tiny{E}}}}\end{array} \right., \tag{5}\]  
\[\text{d}(\sigma_{h}-u_{a})=\frac{\mu}{1-\mu}\text{d}(\sigma_{v}-u_{a})+\frac{ E}{(1-\mu)H_{\text{\tiny{H}}}}\text{d}s, \tag{6}\]  
\[\text{d}\varepsilon_{v}=-\frac{\left(1-\mu-2\mu^{2}\right)\text{d}(\sigma_{v} -u_{a})}{(1-\mu)E}+\frac{2\mu}{1-\mu}\cdot\frac{\text{d}s}{H_{\text{\tiny{H}}} }+\frac{\text{d}s}{H_{\text{\tiny{E}}}}, \tag{7}\]  
Figure 6: Stress state change of soil elements. Comparing Equation (3) with Equation (6) and Equation (4) with Equation (7), it can be observed that these two pairs of equations have similar forms, with the difference being the presence of vertical net normal stress. In terms of elasticity, it can be understood that a vertical net normal stress (\(\sigma_{v}-u_{a}\)) is further loaded on the top of the specimen at Stage I. Under its influence, the vertical swelling strain decreases from the maximum value to the vertical swelling strain under surcharge, and the horizontal confining pressure generated in Stage I would further experience an increment and increase to (\(\sigma_{h}-u_{a}\)). During this period, the increment of matric suction is zero, namely \(\mathrm{d}s=0\).