## 2 Numerical Scheme
### Archard's Wear Law  
Archard's wear law (Archard [7; 8]) was used to evaluate the amount of wear caused by the collision between the structure and the soil particles. This equation defines complex  
Figure 2: Configurations of particle contact forces: (**a**) HLS normal force model; (**b**) LSCL tangential force model.  
and nonlinear wear phenomena through the relationship between the normal force, sliding distance, and material properties.  
\[\mathrm{V}=\mathrm{k}\frac{F_{\mathrm{N}}S}{H} \tag{9}\]  
where V is the total volume lost due to wear, \(F_{n}\) is the normal force, \(S\) is the sliding distance, \(H\) is the material hardness, and k is the wear coefficient.  
Equation (9) is a numerical model to describe sliding wear. It may be reasonable to express the load caused by the behavior of sliding as a tangential force. In addition, many studies ([PERSON] et al. [24]; [PERSON] et al. [25]; [PERSON] et al. [26]) have been conducted to define appropriate k values according to the shape of the load or the surface state of the material. [PERSON] et al. [27] state that this phenomenological law relates the volume loss of a material to the work performed by the frictional force on the surface of the material. In Equation (9), \(F_{n}S_{T}\) is proportional to the work done by the friction forces. In this study, [PERSON]'s wear law is defined as the relationship between the shear work and volume lost. Equation (10) explains the amount of wear suggested by [PERSON]'s wear law.  
\[\mathrm{V}=\mathrm{k}\frac{F_{\mathrm{T}}S}{H} \tag{10}\]  
where \(F_{T}\) is the tangential force, \(F_{T}\) and \(S\) are values that can be calculated through DEM analysis, and \(H\) and k are material property values. In Equation (10), the product of \(F_{T}\) and \(S\) is defined as the shear work as in Equation (11).  
\[\Delta W_{shear}=F_{T_{\mathrm{T}}}S \tag{11}\]  
where \(\Delta W_{shear}\) is the shear work generated in one time step. If the volume of material lost by the calculated \(\Delta W_{shear}\) is defined as the variable C, the unit is defined as \(\mathrm{m}^{3}/\mathrm{J}\). The variable C is defined as the ratio of \(H\) and \(k\) in Equation (12).  
\[\mathrm{C}=\frac{k}{H} \tag{12}\]  
The amount of wear due to friction can be rearranged as in Equation (13).  
\[\Delta\mathrm{V}=\mathrm{C}\Delta W_{shear} \tag{13}\]  
In cases in which the amount of wear is not large, the deformation caused by material loss does not significantly affect the flow of particles. However, when the material loss increases due to wear, the flow of particles changes nonlinearly. Therefore, when material loss occurs due to wear, the process of changing the shape of the surface must be reflected. \(\Delta\mathrm{V}\) calculated through Equation (13) was applied to the structure model every 0.005 s to reflect shape change due to material loss. Since the area of the grid composed on the surface of the structure model is determined, the calculation of the lost volume was considered as a method of moving the grid in the direction of material loss. The wear shape evaluation method is shown in Figure 3.  
Figure 3: Mechanism of shape change by wear: (**a**) at time \(t\); (**b**) at time \(t+\Delta t\).