## 2 Model Development and Numerical Computation
### Simulation Process for a Conventional Triaxial Drained Shear Test  
Based on the aforementioned calculation principles and modeling approaches, a discrete element method (DEM) triaxial test numerical simulation was conducted under flexible membrane boundary conditions. The triaxial test is divided into three principal stages: (1) conventional consolidation to ensure that these S-RMs achieve the desired consolidated state; (2) application of flexible boundary conditions, replacing the traditional rigid boundary; and (3) shear loading, during which shear stresses are incrementally applied for testing. * Conventional Consolidation: Employing the inherent servo control mechanism in conventional triaxial testing, the entire specimen undergoes initial servo-controlled consolidation to attain a fully consolidated state consistent with the required conditions. * Flexible Consolidation: Prior to loading, a multi-particle wall servo was employed using a model with flexible membrane boundary conditions. Initially, the rigid wall was replaced by a particle membrane boundary. Alterations in boundary conditions may introduce potential disturbance to the specimens. To mitigate this, constant peripheral pressure was incrementally applied through iterative fine-tuning of the servo mechanism, thereby facilitating the specimen's gradual transition to its initial consolidation state. For detailed information on the establishment of particle membrane boundaries, refer to [PERSON] et al. [60]. * Loading Stage: The particles in contact with the upper and lower loading plates at the flexible boundary interface are arranged in an adhesive contact configuration. To ensure these particles achieve the same velocity as the loading plates, synchronous velocity control is essential. To maintain the entire model in a quasi-static state [61], the inertial number \(\mathbf{I}\) must be less than \(10^{-3}\)[62]. The formula for calculating the inertial number \(\mathbf{I}\) is presented below: \[\mathbf{I}=\hat{\epsilon}\overline{d}\sqrt{\overline{\rho_{g}}/p}\] (1) where \(\overline{\rho_{g}}\) and \(\overline{d}\) denote the bulk density of the specimen and the average diameter, respectively; \(\hat{\epsilon}\) is the strain rate; and \(p\) is the mean effective stress. Based on the foregoing, all simulations in this study were conducted at a strain rate of 0.04 m/s. This loading speed was sufficiently slow in the vertical direction of the specimen to ensure that these models remained in a quasi-static condition. During the shearing \begin{table}
\begin{tabular}{c c c c c} \hline
**Reference** & \(\mathbf{k_{n}}\) **(N/m)** & \(\mathbf{k_{t}}\) & **Friction** & **Damping Factor** \\ \hline [PERSON] et al. [42] & \(7.5\times 10^{7}\)–\(4.6\times 10^{8}\) & \(k_{t}=(4/3)\,k_{n}\) & 0.5 & 0.7 \\ [PERSON] [34] & \(5\times 10^{7}\)–\(1.08\times 10^{8}\) & \(k_{t}=k_{n}\) & 0.7 & 0.8 \\ [PERSON] et al. [37] & \(1.0\times 10^{8}\) & \(k_{t}=(4/3)\,k_{n}\) & 0.5 & 0.7 \\ [PERSON] [38] & 29 GP & - & 0.01, 0.1, 0.25 & - \\ [PERSON] et al. [39] & 28 GP & \(k_{t}=0.25\,k_{n}\) & 0.3 & 0.7 \\ [PERSON] et al. [40] & \(3\times 10^{8}\) & \(k_{t}=0.8\,k_{n}\) & 0.7 & - \\ [PERSON] et al. [41] & \(2.5\times 10^{8}\) & \(k_{t}=k_{n}\) & 1.0 & - \\ \hline \end{tabular}
\end{table} 
Table 2: The values of contact stiffness in the literature. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Particle** & **Density** & **Contact Stiffness (N/m)** & \multicolumn{2}{c}{**Bond Strength (N)**} & \multirow{2}{*}{**Friction Coefficient**} \\  & **(kg/m\({}^{3}\))** & **Normal** & **Tangent** & **Normal** & **Tangent** \\ \hline Gravel clump & 2650 & \(1.0\times 10^{8}\) & \(1.0\times 10^{8}\) & - & - & 0.80 \\ Sand ball & 1900 & \(2.0\times 10^{6}\) & \(1.0\times 10^{6}\) & \(1.0\times 10^{2}\) & \(1.0\times 10^{2}\) & 0.40 \\ Gravel–sand & - & \(3.9\times 10^{6}\) & \(2.0\times 10^{6}\) & - & - & 0.40 \\ Membrane particle & 1000 & \(1.0\times 10^{5}\) & \(1.0\times 10^{5}\) & \(1.0\times 10^{300}\) & \(1.0\times 10^{300}\) & 0.0 \\ \hline \end{tabular}
\end{table}