## 3 Triaxial Compression Calibration Test and Verification
### Edinburgh Elasto-Plastic Adhesion Model (EEPA)  
The interaction among a granular system like wet sand is highly complex, influenced by factors such as surface topography, chemical properties, and the characteristics of the interstitial medium [39]. The E EPA model is an extension of the linear viscoelastic model [40]. This model can effectively simulate the nonlinear force-displacement behavior during compression and incorporates both viscous damping and rolling resistance mechanisms, similar to a linear rolling resistance model. In this study, E EPA (Figure 5) is employed to describe the elastoplastic wet sand, and PartsNet calibrates the key parameters of E EPA. EEPA is influenced by seven parameters: the initial loading stiffness \(k_{1}\), unloading and reloading stiffness \(k_{2}\), contact plasticity ratio \(\lambda_{p}\), constant pull-off force \(f_{0}\), adhesion energy parameter \(\Delta\gamma\), stiffness exponent \(n\), and adhesiveness exponent \(\chi\). The stiffness exponent \begin{table}
\begin{tabular}{c c} \hline
**Hyperparameters** & **Triaxial Test** \\ \hline Batch Size & 40 \\ Hidden Size & 72 \\ Number of Heads & 4 \\ Number of layers & 1 \\ Sparsity & 0.49 \\ Initial Learning Rate & 0.00056 \\ Optimizer & Adam \\ Loss Function & MES loss \\ Q-Quantile Value & 0.75 \\ Training Epochs & 600 \\ \(N\) (Sequence Length) & 99 \\ GDR-Net Latent\_dim & 3 \\ GDR-Net Epochs & 1000 \\ \hline \end{tabular}
\end{table} 
Table 1: List of PartsNet hyperparameters. Figure 5: E EPA normal loading–overlapping displacement diagram. n takes two values, 1 and 1.5, with \(n=1\) indicating whether the model is equivalent to a linear elastic model. In order to define the nonlinear behavior during loading and unloading, this study adopts a value of 1.5 for \(n\). The overall normal force \(F_{N}\) is composed of two parts, the hysteresis spring force \(f_{\text{{hys}}}\) and the normal damping force \(f_{n}^{d}\):  
\[F_{N}=\left(f_{\text{{hys}}}+f_{n}^{d}\right)\mathbf{u} \tag{16}\]  
Here, \(\mathbf{u}\) is the unit normal vector pointing from the contact point towards the center of the particle. The mathematical expression for the normal hysteresis spring force \(f_{\text{{hys}}}\) can be represented as follows:  
\[f_{\text{{hys}}}=\left\{ \begin{array}{cl}f_{0}+k_{1}\delta^{n}&if\,\left( \delta^{n}-\delta_{p}^{n}\right)\geq k_{1}\delta^{n}\\ f_{0}+k_{2}\left(\delta^{n}-\delta_{p}^{n}\right)&if\,k_{1}\delta^{n}>k_{2} \left(\delta^{n}-\delta_{p}^{n}\right)>-k_{adh}\delta^{n}\\ f_{0}-k_{adh}\delta^{n}&f-k_{adh}\delta^{n}>k_{2}\left(\delta^{n}-\delta_{p}^{n }\right)\end{array} \right. \tag{17}\]  
The normal damping force \(f_{n}^{d}\) can be expressed as follows:  
\[f_{n}^{d}=-2\sqrt{\frac{5}{6}}\beta_{\text{{NL}}}\sqrt{K_{n}m^{*}}v_{n}^{ \rightarrow} \tag{18}\]  
Here, \(\delta^{n}\) represents the normal overlap, \(\delta_{p}^{n}\) represents the normal plastic overlap, \(k_{adh}\) represents the adhesive stiffness, and \(v_{n}^{\rightarrow}\) represents the relative normal velocity generated during particle contact. \(k_{1}\) is defined by the Hertzian contact theory, and its magnitude depends on the equivalent radius of particles, \(R^{*}\), and the equivalent Young's modulus, \(E^{*}\), in the particle system. \[k_{1}=\frac{4}{3}\sqrt{R^{*}}E^{*} \tag{19}\]  
\(\lambda_{p}\) is defined based on the ratio of two-stage stiffness, and a value of 0 indicates the elastic contact of particles. \[\lambda_{p}=1-\frac{k_{1}}{k_{2}} \tag{20}\]  
\(K_{n}\) represents the Hertz normal stiffness, which depends on the equivalent radius, [PERSON]'s modulus, and the normal overlap quantity \(\delta_{n}\). \[K_{n}=2E^{*}\sqrt{R^{*}\delta_{n}} \tag{21}\]  
\(\beta_{NL}\) represents the normal damping coefficient of the nonlinear model, and its calculation formula is provided by [PERSON] [31].