## 3 Methods: Finite Element Modelling
### Type Model
#### 3.1.2 2D and 3D Tyre Modelling  
This study focused on the contact interaction between the rubber tread material and the different grain size aggregates when the tyre is running in the longitudinal direction; ignoring the transverse tyre tread pattern had little effect on the analysis results. Meanwhile, to save computational time, the tyre modelling ignored the transverse tyre tread pattern and retained only the two longitudinal grooves. As the work of tyre modelling is two-dimensional (2D), the 3D tyre finite element model was obtained by rotating the 2D tyre  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multirow{2}{*}{
\begin{tabular}{c} **Neo-Hookean Model** \\ **C\({}_{\textbf{10}}\) (MPa)** \\ \end{tabular} } & \multicolumn{2}{c}{**\(\begin{array}{c}\textbf{Do}_{\textbf{1}}\) (MPa\({}^{\textbf{-1}}\))** \\ \end{tabular}} & \multirow{2}{*}{**Density (g/cm\({}^{\textbf{3}}\))**} \\ \hline Tread rubber & 0.835 & 0.024 & 1.12 \\ Crown rubber & 0.869 & 0.04 & 1.12 \\ carcass rubber & 1.0 & 0.02 & 1.15 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Rubber material parameters for the 175/80 R14 radial tyre.  
Figure 4: Material composition of the tyre.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \multirow{2}{*}{\begin{tabular}{c} **Cross-Sectional** \\ **Area (mm\({}^{\textbf{2}}\))** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Spacing** \\ **(mm)** \\ \end{tabular} } & \multirow{2}{*}{**Angle (\({}^{\circ}\))**} & \multirow{2}{*}{\
\begin{tabular}{c} **Young’s Modulus** \\ **(10\({}^{\textbf{5}}\) MPa)** \\ \end{tabular} } & \multirow{2}{*}{**Poisson’s Ratio**} & \multirow{2}{*}{
\begin{tabular}{c} **Density** \\ **(g/cm\({}^{\textbf{3}}\))** \\ \end{tabular} } \\ \hline Belt 1 & 0.212 & 1.16 & 70 & 1.722 & 0.3 & 5.9 \\ Belt 2 & 0.212 & 1.16 & 110 & 1.722 & 0.3 & 5.9 \\ carcass cord & 0.421 & 1.0 & 0 & 9.87 & 0.3 & 1.5 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Cord skeleton material parameters for the 175/80 R14 radial tyre.  
model. To reduce the modelling workload, only half of the tyre profile was selected for modelling. The specific modelling process employed was as follows:  
First, we drew a 2D profile of the tyre based on its size and material composition using AutoCAD and exported the *.iges file format to the HyperMesh 2021 pre-processing software for meshing. Considering that mesh quality has a major influence on the accuracy and convergence of the model results, the rubber material was divided into quadrilateral CGAX4 cells with triangular CGAX3 cells used only at the sharp corners of the material boundaries, and the cord skeleton material was represented by the SFMGAX1 cell type. In ABAQUS 2021, the material mechanical parameters were subsequently set based on the intrinsic tyre model, and a uniform pressure was applied to the inside of the tyre to simulate its inflation pressure. The resulting 2D tyre finite element model is shown in Figure 5.  
Second, the 2D finite element model of the tyre was rotated 360\({}^{\circ}\) around its rotation axis in ABAQUS 2021 to obtain half of the 3D finite element model, as shown in Figure 6. As the use of a dense tyre mesh can increase model calculation overhead but the use of a sparse tyre mesh can affect the accuracy of the results, mesh encryption was performed only in the contact interval between the tyre and pavement, and the mesh sizes of the tyre and pavement were matched to the greatest extent possible.  
Figure 5: Two-dimensional finite element tyre model.  
Figure 6: Half of the 3D finite element tyre model.  
Third, half of the 3D finite element model was mirrored to obtain the completed 3D finite element model of the tyre, as shown in Figure 7. An analytical rigid body was used at the inner edge of the tyre bead to form the rim.