## 3 Verification with Numerical Solutions  
### Numerical Modeling  
[PERSON] and [PERSON] [25] verified the analytical solution of soil thrust through numerical analysis and model experiments on a flat ground surface in their study; following this, this study performed numerical analyses to validate the proposed theoretical solution for a track system on a clay slope. The interaction between the track and the ground was simulated using ABAQUS Explicit. Table 1 lists the material properties of both the ground and the track system considered in the numerical analyses. The track system was modeled as a linear elastic material employing the material properties of steel. In the numerical analysis, this study described the mechanical responses of the saturated clay using a linear  
Figure 4: Soil thrust \(F_{x}\) versus the undrained shear strength \(c_{u}\) of clay with fixed dimensions of a track system (height \(H\), length \(L\), and width \(D\)), as referred to in [PERSON] et al. [22]; ground slope \(\beta\) and weight \(W_{g}\) transmitted from the upper vehicle. elastic-perfect plastic constitutive model coupled with the Tresca yield criterion (Mohr-Coulomb yield criterion with zero friction angle). The elastic behavior was modeled as a linear elastic model with [PERSON]'s modulus \(E\) = 20,000 kPa and [PERSON]'s ratio \(\
u\) = 0.49 (to describe the undrained responses), while its plastic behavior was described using the Tresca model with its yield strength corresponding to the undrained shear strength of clay assuming the perfect plastic condition. To examine the influence of the undrained shear strength \(c_{u}\) of clay, this study conducted the numerical analyses by varying its values to 5, 7.5, 10, 15, 20, and 25 kPa. Figure 5 depicts the geometric modeling of a track system and subgrade clay ground in the numerical analyses of this study. To improve the efficiency of the numerical analysis, a half section of the entire domain considering x-symmetricity was modeled, as shown in Figure 5. In Figure 5, the ground slope angle \(\beta\) was set at \(10^{\circ}\); regarding the track system, its height, length, and width were set to 0.035, 0.124, and 0.062 m, respectively. To speed up the numerical analyses, a mass scaling factor of 400 was applied to all simulations. The outer boundaries of the ground (except the symmetric boundary) were modeled using infinite elements to minimize the disturbance caused by reflection. The numerical analysis consists of two stages: loading; and pushing. In the loading stage, the surface traction (\(\sigma_{c}\) = 11.2 kPa = \(W_{g}/(LD)\)) was uniformly applied as traction on the upper surface of the track. The perpendicular and tangential components of traction with respect to the surface were \(\sigma_{c}\) sin\(\beta\) and \(\sigma_{c}\) cos\(\beta\), respectively. The interface between the track and clay was modeled as the frictionless contact model. In the pushing stage, the front panel of the track was pushed at 1 mm/s velocity parallel to the ground surface (that had an angle \(\beta\) with the horizontal surface); this stage was maintained for 7.5 s, and the final displacement of the track was 7.5 mm. To verify the convergence of the numerical simulations, this study carried out multiple simulations under consistent conditions while adjusting the mesh density. Figure 6 illustrates three different mesh configurations considered in this study. The section around the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Clay Ground** & **Track System** \\ \hline Density \(\rho\) (t/m\({}^{3}\)) & 2.0 & 7.8 \\ [PERSON]’s modulus \(E\) (kPa) & 20,000 & \(210\times 10^{6}\) \\ [PERSON]’s ratio \(\
u\) & 0.49 & 0.3 \\ Plasticity model & Tresca & - \\ Cohesion \(c_{u}\) (kPa) & 5, 7.5, 10, 15, 20, 25 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters in the numerical analyses in this study. Figure 5: Geometrical modeling of a track system over clay slope for the numerical simulations in this study.