## 4 Results
### Numerical Results  
The results of the finite element model calculations in this paper are based on linear Euler buckling theory. Obviously, the numerical results of the critical buckling load analysis for the diagonal column are in good agreement with the analytical solutions, as shown in Table 2, while in the case of the vertical column there is a large difference in the critical buckling load analysis between the numerical and analytical results. This is because the critical buckling load obtained from the FEM tends to overestimate Euler's critical load when the critical slenderness ratio of a material is higher than that of a column. To address this issue, the nonlinear Euler buckling analysis should be conducted and compared to the critical buckling load obtained from the analytical calculation using the Johnson equation, which is more accurate for the calculation in this case. It is important to mention that in this paper we focus on the linear buckling analysis in order to identify the local failure modes of a meta-functional auxetic unit cell for bridge bearing using a linear isotropic material, which at low strain is TPU. Furthermore, Figures 7 and 8 demonstrate the numerical buckling behaviour (including displacement and stress distribution) of the vertical and diagonal columns with a free-fixed boundary condition under compression, respectively. Therefore, it can be concluded that the local buckling behaviours of the MFAU cell using a TPU material with rubber-like properties for bridge bearing applications are dominated by the diagonal column at the critical buckling load of 9.74 N, and that local buckling failure occurs in both columns before yielding. Furthermore, the results of the overall behaviours of the MFAU cell and the MFA composite bridge bearing both exhibit auxetic behaviour (contraction) under compression and have a local buckling failure located on the vertical and diagonal columns, as shown in Figures 9 and 10. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Vertical Column** & **Diagonal Column** \\ \hline Buckling load factor, \(\delta\) & 109.55 & 9.74 \\ \(F_{applied}\) (N) & 1 & 1 \\ \(F_{cri}\) (N) & 109.55 & 9.74 \\ Relative error (\%), compared to analytical results & overestimation & 0.62 \\ Type of failure & Buckling & Buckling \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Numerical buckling results for both column components. Figure 7: Displacement (**a**) and stress distribution (**b**) of the vertical column with free–fixed boundary condition in the buckling analysis. For the bridge bearing applications, we found that the previous MFAU cell model using a TPU material was not suitable for the applications due to its low compressive stiffness. Therefore, in this paper, the selection of an optimised material for the MFA composite bridge bearing model was considered, especially for Young's modulus, in order to act as a typical bridge bearing. The variation of the mechanical properties used in this model is given in Table 3. Surprisingly, the auxetic sandwich core used in the MFA composite bridge bearing model (given in Figure 11) was still stable in the vertical direction under compression, while the core showed auxetic behaviour. In addition, Figure 12 demonstrates that the appearance of individual area corresponds to the mechanisms of deformation and the interactions among these novel unit cells at different strain regions (0, 20, and 40 per cent of strain, respectively), which has been widely investigated in many studies. Figure 8: Displacement (**a**) and stress distribution (**b**) of the diagonal column with free–fixed boundary condition in the buckling analysis. Figure 10: Overall deformation (**a**) and von Mises stress distribution (**b**) of the meta-functional auxetic composite bridge bearing model under compression at vertical displacement of 4 mm. Figure 9: Overall deformation (**a**) and stress distribution (**b**) of the meta-functional auxetic unit cell under compression at the highest critical buckling load.