## 2 Unsaturated Subgrade Filler Triaxial Testing System
### Analysis of Experimental Results
#### 3.4.4 Analysis of Constitutive Behavior  
The Barcelona basic model (BBM) is one of the earliest and most widely used constitutive models for unsaturated soils. It accurately describes the mechanical behavior of soils at different moisture contents and the impact of moisture changes on the stress-strain relationship. In recent years, many scholars have calibrated the parameters of this model using triaxial test data from samples with varying moisture contents [52; 53; 54; 55]. The BBM model parameters were calibrated using the third (of three) sampling set of triaxial test data (the specific coefficients are presented in Table 5), and then we directly compared the experimental values to the theoretical model predictions in the three-dimensional space of mean net stress, suction, and specific volume, as illustrated in Figure 15a. The yield curve of the sample was plotted, as shown in Figure 15b. Figure 15a shows high consistency between the experimental and predicted values. The trend of specific volume variation remains consistent across different mean net stress levels, with a small gap between the experimental and predicted values. For instance, at a mean net stress of 200 kPa, the experimental specific volume is 1.63, while the predicted value is 1.62, resulting in a small difference of only 0.61%, indicating that the model can accurately predict the specific volume. However, in some regions, such as at a stress of 400 kPa, the difference between the predicted and experimental values is about 12.57%, while at lower stress levels, such as 100 kPa, the difference is 11.83%. This trend indicates that as stress increases, the prediction error of the model also increases, particularly in regions of higher suction. This suggests that the BBM model demonstrates good predictive accuracy in the medium-low stress range, but its predictive ability may be limited under high stress or high suction conditions. In regions of higher suction, the variation in specific volume is more pronounced, especially in the low stress range, where the experimental values show more significant changes. In contrast, the predicted values of the theoretical model show smoother changes in this region, suggesting that the model may not fully capture the subtle variations of the sample under low stress and high suction conditions. In conclusion, the theoretical model proposed in this study effectively simulates the mechanical behavior of unsaturated soils, particularly in the medium-low stress range. However, as stress and suction increase, the predictive accuracy of the model decreases. Future research should consider incorporating additional soil properties or more complex nonlinear behaviors to enhance predictive capabilities under high-stress conditions. Figure 14: Mohr–Coulomb circles and failure envelope at the shear failure point of the specimen. Figure 15: Comparison of constitutive models for specimens with different moisture contents under isotropic loading conditions: (**a**) differences between experimental data and theoretical predictions, (**b**) yield curves. \begin{table}
\begin{tabular}{c c} \hline
**Coefficient** & **Optimal Fit** \\ \hline \(p^{c}\) (MPa) & 0.121 \\ \(\kappa\) & 0.0077 \\ \(\beta\) (MPa\({}^{-1}\)) & 7.999 \\ \(r\) & 0.377 \\ \(Q(0)\) & 0.667 \\ \(\lambda(0)\) & 0.054 \\ \(\kappa_{s}\) & 0.0116 \\ \hline \end{tabular} Note: \(\kappa\) represents the gradient of the unloading-reloading curve corresponding to the average net stress; \(\kappa_{s}\) indicates the gradient of the unloading-reloading curve related to soil suction; \(Q(0)\) refers to the specific volume when \(p\) equals \(p^{c}\) and \(s\) is zero; \(\lambda(0)\) also denotes the gradient of the virgin compression curve related to the average net stress when \(s\) equals zero; \(\kappa\) is a parameter that regulates the steepness of the virgin compression curve; \(\beta\) is a parameter that influences the slope of the virgin compression curve when \(s\) is not equal to zero; and \(p^{c}\) is the reference stress.  
\end{table}