## 5 Determination of the Elastic Parameters  
### Elastic Modulus with Respect to Net Normal Stress  
The elastic modulus with respect to the net normal stress (\(E\)) characterizes the compressibility of the soil skeleton. It is required as a parameter within the proposed constitutive model relating stress-strain responses under changing net normal stresses or suction conditions.  
As mentioned above, both the denominator and the numerator on the right-hand side of Equation (9) would become zero when no net normal stress is loaded, rendering it unable to estimate the elastic modulus \(E\) in this state. The elimination method can be carried out if there is a certain relationship between the denominator and the numerator. Figure 7 illustrates the variation curves of the denominator (\(\varepsilon_{v}-\varepsilon_{mv}\)) with respect to the numerator (\(\sigma_{v}-u_{a}\)).  
Figure 7 indicates that under the condition of constant matric suction, the increase in (\(\varepsilon_{mv}-\varepsilon_{v}\)) gradually increased with the increase in (\(\sigma_{v}-u_{a}\)), and the increment gradually decreased, tending to stabilize. The stress-strain relationship can be fitted in the form of a hyperbolic function, as shown in Equation (13). Table 3 shows that parameters \(a\) and \(b\) exhibited a strong correlation \(>\)0.95, validating the approach.  
\[\varepsilon_{mvi}-\varepsilon_{vi}=\frac{a(\sigma_{vi}-u_{a})}{1+b(\sigma_{vi} -u_{a})}, \tag{13}\]  
where \(a\) and \(b\) are the fitting parameters.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Matrix Suction (kPa)** & \(a\) & \(b\) & \(R^{2}\) \\ \hline
200 & 0.0008 & 0.0524 & 0.9914 \\
100 & 0.0014 & 0.0532 & 0.9908 \\
50 & 0.0020 & 0.0545 & 0.9501 \\
20 & 0.0039 & 0.0779 & 0.9829 \\
10 & 0.0052 & 0.0882 & 0.9792 \\
5 & 0.0067 & 0.0981 & 0.9814 \\
0 & 0.0088 & 0.1070 & 0.9906 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Fitting results of Equation (13).  
Substituting Equation (13) into Equation (9):  
\[E=\frac{\big{(}1-\mu-2\mu^{2}\big{)}}{(1-\mu)a/\big{[}1+b(\sigma_{yi}-u_{a})\big{]}}, \tag{14}\]  
Comparing Equation (14) with Equation (9), Equation (14) is still valid in terms of estimating the elastic modulus \(E\) even in the absence of a vertical net normal stress. A series of elastic moduli E under different net normal stresses and matric suctions was estimated using Equation (14).  
Figure 8 shows the variations in the calculated \(E\) values with changing net normal stress and suction states. Several trends were observed. \(E\), with respect to net normal stress, decreased as matric suction decreased, showing a rapid reduction when the matric suction exceeded 20 kPa under a net normal stress of 50-100 kPa, while it gradually slowed down as the soil became increasingly saturated. With the decrease in net normal stress, the rapid reduction in \(E\) disappeared. Furthermore, under low net normal stress, matric suction had less impact on \(E\) than net normal stress. With an increase in net normal stress, the elastic modulus \(E\) under the same matric suction also increased, indicating that the influence of matric suction on \(E\) became much stronger. At the same matric suction level, a higher net normal stress led to a greater \(E\). These results also highlight the fact that unsaturated expansive soil under low matric suction is more prone to deformation under external loads, and higher external loads enhance its resistance to deformation.