## 4 Developed Constitutive Model
### Constitutive Parameters
The constitutive relationships from Stage I to Stage II can be described using Equation (8). \[\left\{ \begin{array}{l}0=\frac{\mu-1}{E}\mathrm{d}(\sigma_{h}-\sigma_{h0})+ \frac{\mu}{E}\mathrm{d}(\sigma_{v}-u_{a})\\ \mathrm{d}(\varepsilon_{v}-\varepsilon_{mv})=-\frac{\mathrm{d}(\varepsilon_{v }-u_{a})}{E}+\frac{2\mu}{E}\mathrm{d}(\sigma_{h}-\sigma_{h0})\end{array} \right., \tag{8}\]  
\[E=\frac{\left(1-\mu-2\mu^{2}\right)\mathrm{d}(\sigma_{v}-u_{a})}{(1-\mu) \mathrm{d}(\varepsilon_{mv}-\varepsilon_{v})}, \tag{9}\]  
where (\(\varepsilon_{v}-\varepsilon_{mv}\)) represents the reduction in vertical strain resulting from net normal stress. Equation (9) is derived from Equation (8) to estimate the elastic modulus (\(E\)) with respect to the net normal stress. Once the net normal stress is loaded onto the specimen, it remains constant. Thus, the increment in net normal stress is zero, namely \(\mathrm{d}(\sigma_{v}-u_{a})=0\). The first term on the right-hand side of Equation (7) equals zero, rendering this equation identical to Equation (4). This indicates that the vertical swelling strain of the unsaturated expansive soil is due to a moisture increment consisting of two parts. In addition to the vertical strain resulting from the change in matric suction itself, the lateral swelling strain would also occur if it were not laterally confined. The lateral confinement results in the representation of this part in the vertical direction. However, as mentioned in Section 2.2, the maximum radial strain of the oedometer ring is 0.032%. Therefore, the latter part could be extremely small compared to the previous part and can be ignored. Hence, the vertical swelling strain in OVS (\(\varepsilon_{v,OVS}\)) and the vertical swelling strain in OHS (\(\varepsilon_{v,OHS}\)) can be described as shown in Equation (10):  
\[\left\{ \begin{array}{l}\mathrm{d}\varepsilon_{v,OVS}=\frac{\mathrm{d}s}{H_ {b}}\\ \mathrm{d}\varepsilon_{v,OHS}=\frac{\mathrm{d}s}{H_{b}}\end{array} \right., \tag{10}\]  
\[H_{p}=\frac{\mathrm{d}s}{\mathrm{d}\varepsilon_{v,OVS}}, \tag{11}\]  
\[H_{h}=\frac{\mathrm{d}s}{\mathrm{d}\varepsilon_{v,OHS}}, \tag{12}\]  
Equations (11) and (12) are derived from Equation (10) to estimate the vertical and horizontal elastic moduli (\(H_{v}\) and \(H_{h}\)) with respect to matric suction, respectively. For the determined unsaturated expansive soil, the change in vertical swelling strain is caused by the coupling effect of the net normal stress and matric suction. These three elastic parameters are all influenced by the changes in net normal stress and matric suction, demonstrating the nonlinear elastic behavior during wetting. However, Equation (9) would be invalid if the soil is inundated without vertical net normal stress, whereas the elastic modulus should be a property of the material itself. Although it may change under the influence of external factors (stress state and boundary conditions, etc. ), it should always have an initial value. Equation (9) would need to be further modified based on the test results to estimate the E without the net normal stress. Furthermore, Poisson's ratio is also essential for the constitutive model. Unsaturated expansive soil undergoes significant volume changes after saturation, leading to substantial changes in dry density. Soils with smaller dry densities are more likely to deform, and those with larger Poisson ratios are more susceptible to this deformation. However, recent standards do not provide methods for measuring the Poisson's ratio of unsaturated soils,nor do they specify the size of specimens and the testing conditions of stress and strain that are required for determining Poisson's ratio. In engineering practice, Poisson's ratio is often assumed to be a constant value. The suggested range of values for Poisson's ratio ranges from 0.2 (dry sand) to 0.5 (saturated clay tested under undrained conditions) [10; 33].