## 3 Materials and Methods
### Numerical Analysis  
To conduct numerical analysis, two 3D CAD models for the MFAU cell and MFA composite bridge bearing using a sandwich core (\(5\times 5\times 3\) unit cells) were generated by Fusion 360 software, as shown in Figure 4. Then, a 3D finite element method-based approach using Siemens NX 12.0 Nastran (Siemens, Munich, Germany) software was followed in order to simulate the analytical buckling analysis on the two column components (vertical and diagonal) of the MFAU cell, as shown in Figure 5c.  
Two 3D column finite element models with the same boundary condition used in the analytical analysis (free-fixed) were produced, with 1925 and 5872 tetra10 elements for the vertical and diagonal columns, respectively. These two solid element models had a very refined mesh (0.2 mm element size). The compressive load of 1 N was applied to each component in the vertical direction, as shown in Figure 5a,b. When the simulation of the eigenvalue problem was carried out, the obtained buckling load factor, \(\delta\), was used for calculating critical loads for the two components. To obtain the critical loads, the initial compressive loads were multiplied by the buckling load factor, \(\delta\). Furthermore, the numerical investigation included the overall compressive behaviour of the MFAU cell under the highest local critical buckling load obtained. In terms of bridge bearing applications, a full three-dimensional model consisted of 13,319 and 2,301,690 tetra10 solid elements for the MFAU cell and MFA composite bridge bearing model, respectively. Their entire three-dimensional tetrahedral method is presented in Figure 5c,d. Both solid elements models had the same element size, 1 mm. The highest local critical buckling load face was imposed on the top surface of the MFAU cell model; for the other model, the enforced displacement constraint of 4 mm was applied to the top surface of a thin metallic facet connected to the top surface of the auxetic core by using a glue surface-to-surface contact. The base of each model was fixed.  
Validation of the simulations was carried out by comparing the numerical critical buckling loads of the two column components of the MFAU cell with the analytical ones under the free-fixed condition, as shown in Figure 6. Under compression, the diagonal column showed an analytical critical buckling load of 9.68 N and a numerical one of 9.74 N (relative error 0.62 per cent), while in the case of the vertical column it presented an analytical critical buckling load of 108.28 N and a numerical one of 109.55 N (relative error 0.70 per cent). In fact, the relative error of the vertical column would be higher if the actual critical buckling load were calculated from the Johnson formula instead of the Euler formula, due to the slenderness ratio of the TPU material being less than the critical one.  
Figure 4: A three-dimensional CAD model for the meta-functional auxetic unit cell model (**a**) and composite bridge bearing model (**b**), respectively.  
Figure 5: A 3D solid element model of the vertical column (**a**), the diagonal column (**b**), the MFAU unit cell (**c**), and the MFA composite bridge bearing (**d**).