## 2 Methodology  
The methodology was structured into several stages, each designed to enable a comparative analysis between the NS and HSS constitutive models against experimental data from undrained triaxial tests, as shown in Figure 1.  
The first stage involved a theoretical analysis of both constitutive models. The NS model is widely used to simulate the behavior of liquefiable sands under quasi-static loading conditions, owing to its ability to capture stress-state-dependent behavior and the evolution of the state parameter. Conversely, the HSS model is an extension of the Hardening Soil model that incorporates small-strain stiffness, allowing for improved representation of the initial response of granular materials. A sensitivity analysis of the model parameters was conducted using the SoilTest module in PLAXIS 2D. This analysis identified the most influential parameters affecting the model response, which was critical for calibration. Prioritizing the accurate estimation of these key parameters enhances the reliability of the model predictions. On the other hand, a Python script was developed to automate the curve-fitting process using an optimization approach based on the RMSE. This tool was essential for achieving better alignment between the stress-strain curves generated by SoilTest and the experimental data derived from undrained triaxial tests. The relevance of this approach has been highlighted in previous studies, such as [PERSON] (2007) [20], which emphasized the importance of validating constitutive models to ensure their applicability in real geotechnical contexts. Also, using the calibrated parameters, an axisymmetric model was constructed in PLAXIS 2D to replicate the triaxial test. This step enabled the validation of the behavior observed in the SoilTest module, confirming that the numerical model accurately reproduces the dilative behavior observed in the experimental curves. An automated parametric calibration procedure was implemented through a Python script connected to the SoilTest module of PLAXIS 2D. The constitutive model parameters were iteratively adjusted using an algorithm that overwrites the input values and runs consecutive simulations, evaluating the degree of fit against the experimental data. Validation of each parameter set was performed by comparing the numerical response with the curves obtained from CIU triaxial tests, specifically the deviatoric stress versus axial strain relationship and the excess pore pressure versus axial strain evolution. A valid calibration was considered one that exhibited physical consistency and an acceptable qualitative and quantitative correspondence with the behavior observed in the laboratory tests. Finally, the performance of each model was assessed to determine which one more faithfully represented the actual behavior of the material. This evaluation was based on quantitative error metrics and qualitative observation of key features such as dilatancy and peak strength in the stress-strain response. The comparison provided insights into the predictive capabilities and limitations of each model under undrained loading conditions. The samples used in the CIU triaxial tests were obtained from fresh copper tailings sourced from a sulfide concentrator plant located in northern Chile. The specimens were prepared using the moist damping method to achieve a density corresponding to 81% of the Standard Proctor maximum dry density. Additionally, according to the geotechnical  
Figure 1: Geotechnical Characterization of the Material Used. design firm responsible for the project, the water content of the samples was approximately 16% [21]. The main physical properties of the material are presented below in Table 1. The soil under study corresponds to a low-plasticity silt. Figure 2 shows the particle size distribution curve of the material, as reported by [PERSON] et al. [22].