## 2 Methodology
### Constitutive Models Used
#### 2.1.2 Hardening Soil  
The Hardening Soil constitutive model represents stiff and soft soils' nonlinear and elastoplastic behavior under complex loading conditions, such as time-varying loads, non-uniform loads, and soil-structure interactions [10]. This model incorporated stress and strain hardening concepts, providing a more realistic description of soil behavior than simpler models like Mohr-Coulomb. The parameters governing the modeling are as follows:  
1. Stress-dependent stiffness, parameter m;
2. Plastic strain due to primary deviatoric loading, parameter \(E_{50}{}^{ref}\);
3. Plastic strain due to primary compression, parameter \(E_{ood}{}^{ref}\);
4. Elastic loading/unloading, parameter \(E_{ur}{}^{ref}\)  
These parameters can be measured from triaxial and oedometer tests. Figure 5a defines the modules Eurref and E50 ref for an effective principal stress of \(\sigma^{\prime}{}_{3}=-\)pref. The latter corresponds to the secant stiffness at 50% of the maximum deviatoric stress at a cell pressure equal to the reference stress pref. Figure 5b (adapted from [15]) defines E\({}_{\text{oedref}}\) as the tangent stiffness at a vertical stress of \(-\epsilon_{1}=\) pref. For the special case of soft soils, there are simplified equations between the modified compression and swelling indices and the moduli \(E_{ood}{}^{ref}\), \(E_{ur}{}^{ref}\), and \(E_{50}{}^{ref}\) as shown in Equations (9)-(11). \[E_{ood}{}^{ref}=\frac{p^{ref}}{\lambda^{*}} \tag{9}\]  
\[E_{ur}{}^{ref}=\frac{2p^{ref}}{k^{*}} \tag{10}\]  
\[E_{50}{}^{ref}=2E_{ood}{}^{ref} \tag{11}\]  
Figure 4: Relationship between volumetric strain and principal stress. Finally, the input parameters for the Hardening Soil model are as follows:  
* Secant stiffness (\(E_{\mathit{50}}{}^{\mathit{ref}}\));
* Tangent stiffness (\(E_{\mathit{oad}}{}^{\mathit{ref}}\));
* Loading/unloading stiffness (\(E_{\mathit{ur}}{}^{\mathit{ref}}\));
* Power for stress-level dependency of stiffness (m);
* Poisson's ratio for loading/unloading (v'ur);
* Reference stress for stiffnesses (\(p^{\mathit{ref}}\));
* Lateral earth pressure coefficient for normally consolidated soils (K0 nc);
* Failure ratio (Rf). On the other hand, Table 3 shows the parameters obtained that allow for a better fit to represent the experimental results for both saturated and unsaturated conditions. Calibration using a single set of parameters for both models ensures consistency. It simplifies the simulation process by avoiding discrepancies when using different parameters set under similar conditions. For the unsaturated condition, the soil-water characteristic curve (SWCC) plays a crucial role in determining the matric suction level based on the degree of saturation, facilitating a more accurate characterization of the material's hydromechanical behavior. Table 4 represents the matrix of the executed models, identifying all the curves extracted from Plaxis2D\({}^{\mathit{\textregistered}}\). The format used for identification includes the corresponding constitutive model (SS or HS), saturated or unsaturated condition, and the applied stress. For example, it is used as follows: SS/HS_Sat/Sr%_Applied Stress. Figure 5: (**a**) Definition of \(E_{\mathit{ur}}{}^{\mathit{ref}}\) and E50 ref for triaxial test results. (**b**) Definition of \(E_{\mathit{oad}}{}^{\mathit{ref}}\) for oedometer test results. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Hardening Soil** & **Soft Soil** \\ \hline \(E_{\mathit{50}}{}^{\mathit{ref}}\) (kN/m\({}^{2}\)) & 7500 & - \\ \(E_{\mathit{oad}}{}^{\mathit{ref}}\) (kN/m\({}^{2}\)) & 3950 & - \\ \(E_{\mathit{ur}}{}^{\mathit{ref}}\)(kN/m\({}^{2}\)) & 32,000 & - \\ \(p^{\mathit{ref}}\) (kN/m\({}^{2}\)) & 200 & - \\ M & 1 & - \\ Rf & 0.95 & - \\ v’ur & 0.17 & 0.17 \\ Ko\({}^{\mathrm{nc}}\) & 0.423 & 0.5658 \\ \(\lambda^{*}\) & - & \(3.86\times 10^{-2}\) \\ \(\kappa^{*}\) & - & \(4.348\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Input parameters for Soft Soil and Hardening Soil models.