## 2 Materials and Methods
### FE Simulation of the Interaction between Soil and the Potato Digging Shovel
#### 2.3.2 Setup of the FE Simulations  
The 3D model of potato digging shovels and the soil box are created and then imported into ABAQUS software (Dassault Systemes Simulia Corp., Providence, RI, USA) to set up their FE models. For the convenience of modeling, the potato digging shovel and the connecting frame are immerged into a single part. The potato digging shovel is set as a rigid body with a reference point (RP) located at its tip. Shovels are meshed with C3D4 elements (4-node linear tetrahedron element) and the element size is 10 mm, while C3D8 elements (8-node linear brick element) are employed for soil boxes. The soil box in the FE model for the interaction between soil and a single shovel is a cuboid with a dimension of 1500 mm \(\times\) 1000 mm \(\times\) 600 mm (length \(\times\) width \(\times\) height), as shown in Figure 3a,b. The dimension of the shank' cross section is shown in Figure 3c. To balance the accuracy and calculation time in FE simulations, a finer mesh with an element size of 10 mm is used for the domain in the middle of the top layer, while the element size for the remainder in the soil box is 50 mm, as shown in Figure 3a,b. The bottom left-side wall and right-side wall in the soil box are totally fixed, while other surfaces are not constrained. Meanwhile, the potato digging shovels are controlled to move only along the X direction with different speeds. In order to reach the needed speed gradually to avoid the divergence in calculations, the speed is multiplied by an amplitude function. The \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Density of soil, \(\rho\) (kg/m\({}^{3}\)) & 1520 \\ Young’s modulus of soil, \(E\) (MPa) & 10.87 \\ Fracture strain of soil, \(\varepsilon_{s}\) & 0.15 \\ Displacement at failure, \(\varepsilon_{1}\) (mm) & 9 \\ Poisson’s ratio of soil, \(\mu\) & 0.36 \\ Internal friction angle of the extended Drucker–Prager & 39.99 \\ constitutive model, \(\beta\) (\({}^{\circ}\)) & 35 \\ Dilation angle, \(\psi\) (\({}^{\circ}\)) & 0.78 \\ Flow stress ratio, \(k\) & 0.78 \\ Density of the shovel, \(\rho_{1}\) (kg/m\({}^{3}\)) & 8000 \\ [PERSON]’s modulus of the shovel, \(E_{1}\) (MPa) & \(2.1\times 10^{5}\) \\ Poisson’s ratio of the shovel, \(\mu_{1}\) & 0.28 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters used in the FE simulations. contact between shovel's rigid surfaces and soil nodes is defined as a surface-to-surface contact using a penalty function method, and the rigid surfaces are selected as master surfaces and the deformable soil nodes in the finer mesh domain as slave surfaces. The normal behavior between contacted surfaces is defined as hard contact, while the tangential behavior is defined as the penalty friction formulation with a friction coefficient of 0.42. Figure 4 shows the FE models for the interactions between soil and grouped shovels, which are employed to predict the field working performance of shovels. According to the requirements of ridge planting for potatoes in Gansu province, a ridge (red line in Figure 4a,b) is added on the soil surface (green line in Figure 4a,b). The top width, bottom width and height of the ridge are 650 mm, 900 mm and 150 mm, respectively. Thus, the soil box in the FE models for the interactions between the soil and grouped shovels is a cuboid (length \(\times\) width \(\times\) height = 1000 mm \(\times\) 1300 mm \(\times\) 450 mm) with a ridge on its top, as shown in Figure 4a,b. Similarly, the whole soil box is partitioned into two layers by a partition plane (black line in Figure 4a,b), and a finer mesh with an element size of 10 mm is used for the top layer, while the element size for the bottom layer is 50 mm. Moreover, the interaction parameters used in the FE models for the interactions between soil and grouped shovels are kept identical with those of the FE models for the interactions between soil and a single shovel. The bottom of the soil box is totally fixed, while other surfaces are not constrained.