## 5 Numerical FE Modelling of Auxetic Systems' Mechanical Behavior  
Using the experimental testing results presented in Section 4 as a reference, finite element models were defined as a means of replicating the auxetic response of the fabricated samples. For this purpose, the 2D auxetic system with PLA and the 3D auxetic system with PA12 were modelled by means of implicit FE analysis, using the commercially available software ABAQUS/Standard of Dassault Systemes [47]. Details of the model geometry are shown in Figure 9, which consists of the auxetic sample, bottom plate, and top plates. The auxetic unit cell has identical geometrical characteristics as that of 3D printed models. For both cases (2D and 3D auxetic systems), the samples were meshed with the element \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Auxetic Sample** & \begin{tabular}{c} **Stiffness** \\ **(N/mm)** \\ \end{tabular} & **Max. Force (N)** & \begin{tabular}{c} **Aver. Poisson’s** \\ **Ratio (-)** \\ \end{tabular} & \
\begin{tabular}{c} **Young’s Modulus** \\ **(N/mm2)** \\ \end{tabular} &
\begin{tabular}{c} **Tensile Strength** \\ **(N/mm2)** \\ \end{tabular} \\ \hline \multicolumn{5}{c}{2D Auxetic Samples made by FDM} \\ \hline PLA & 28.5 & 480 & \(-\)0.43 & 2636 & 46.6 \\ PET & 16.3 & 320 & \(-\)0.47 & 1472 & 31.9 \\ TPU & 0.5 & 65 & \(-\)0.59 & 9.4 & 29.0 \\ \hline \multicolumn{5}{c}{3D Auxetic Sample made by SLS} \\ \hline PA12 & 1.6 & 34 & \(-\)0.22 & 1700 & 48.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The results of the average stiffness and average maximum force for the auxetic samples compared to the bulk mechanical properties of filament and powder materials. Figure 8: Poisson’s ratio variation as a function of strains captured by image analysis of each sample at different displacements. C3D8R (an 8-node linear brick, reduced integration), and the two plates were simulated using discrete rigid surfaces with a reference point at their center. A mesh sensitivity analysis was performed to ensure that the simulations' results were insensitive to the mesh size (convergence study). The auxetic sample was modeled as an elastic-perfectly plastic material (von Mises) by defining its elastic modulus \(E\), Poisson's ratio \(\
u\), and yield point Y values, based on the properties of PLA and PA12 taken from the literature for FDM and SLS processing, respectively [48; 49]. General contact conditions were defined between the two plates and the sample, ensuring an accurate calculation of contact stresses at each node. The contact between them introduces moving boundary conditions, which are often discontinuous, and solving the contact requires iterations for updating the model stiffness at every load increment. The contact formulation includes the use of a constrained enforcement method for the pair surfaces of the master (plates)-slave (auxetic sample) and accounts for finite strain, rotations, and sliding. Furthermore, the reference point of the top plate was subjected to a displacement load along the z-direction, while all degrees of freedom of the reference point of the bottom plate were fixed. Regarding all of the simulated studies, it is assumed that the loading rate is slow enough such that static friction can securely model the interface response. The force-displacement curve was calculated by defining the vertical punch compression motion as a function of time-step and the reaction forces at the bottom plate. The FE analysis was conducted for a compressive deformation for which no contact or self-contact occurred between or within unit cells of the auxetic systems. The rendered simulation results for the two auxetic systems are presented in Figures 10 and 11. The calculated deformed shape for the 2D auxetic system with PLA (left) as well as for the 3D auxetic system with PA12 (right) demonstrated alignment with the experimental results for progressive compression displacements with a maximum deviation less than 5%.