## 3 Results and Discussion
### Case Studies  
The case studies presented in this sub-section intend to characterize the influence of each geometric and material parameter of the re-entrant hexagonal auxetic structure on the shear correction factor and the maximum static deflection of simply supported beams. As mentioned in the introductory section, the beams are only constituted by the honeycomb auxetic structure, with no outer layers typical of a sandwich configuration. This is because it is only the auxetic structure that one aims to study without the typical reinforcement effect of the sandwich configuration's outer layers. The beam length is the same for all the case studies, and its cross-section is square. In all of the case studies, the beam boundary conditions and the applied transverse loading are the same and correspond to Figure 4.  
The materials used and the relevant properties for this work are presented in Table 3.  
For simplification purposes, the materials will be mentioned in the text by the designation presented in the second column of Table 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & & & \(E_{c}/E_{m}\) & & & \\ \hline
**N** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline
0 & 0.8333 & 0.8333 & 0.8333 & 0.8333 & 0.8333 & 0.8333 & 0.8333 \\
0.2 & 0.8333 & 0.8137 & 0.7921 & 0.7699 & 0.7478 & 0.7262 \\
0.4 & 0.8333 & 0.7973 & 0.7589 & 0.7214 & 0.686 & 0.6532 \\
0.6 & 0.8333 & 0.7843 & 0.7338 & 0.6865 & 0.6437 & 0.6051 \\
1 & 0.8333 & 0.7654 & 0.6993 & 0.6409 & 0.5902 & 0.5464 \\
2 & 0.8333 & 0.7395 & 0.6552 & 0.5855 & 0.5281 & 0.4805 \\
4 & 0.8333 & 0.7184 & 0.6215 & 0.5451 & 0.4844 & 0.4355 \\
6 & 0.8333 & 0.7094 & 0.6075 & 0.5286 & 0.4670 & 0.4178 \\ \hline \end{tabular}
\end{table} 
Table 2: Shear correction factors for sandwich structure as a function of the power-law exponent and the relationship between the ceramic and metallic materials’ Young’s modulus. 2.1 Characterization of [PERSON]'s Ratios and Stiffness Coefficients of Re-Entrant Hexagonal Honeycomb Auxetic FGM Beams  
With this first case, it is shown how the present auxetic material's Poisson ratios and bending stiffness coefficients vary considering as design parameters, the angle \(\theta\), and the unit cell ratio between the length of the horizontal and the inclined segments \(\left(\eta 1=\frac{d}{7}\right)\). To illustrate these profiles, it was assumed that, in this case, the honeycomb auxetic beam has a unit length (\(a=1\) m) and an aspect ratio \(a/\hbar=10\). The relation between the unit cell thickness and the inclined segment is \(\eta 3=0.01\). The functionally graded materials selected are the Al-WC and Al-SiC. The evolution profiles of the Poisson's ratios for these auxetic FGM can be observed in Figure 5 for the \(v_{12}\) at the layer's lower and upper surface. As one observes from Figure 5a), where the constituent materials have close values of Poisson's ratios, the evolution of the Poisson's ratio \(v_{12}\) for the range of angles analyzed is very similar, whether one is considering the values assumed at the bottom (inf) or top (sup) surfaces of the beam. However, for the other FGM auxetic beam, where the constituent materials have very different values of Poisson's ratios, the evolution profile of \(v_{12}\) differs depending on the thickness coordinate considered. Overall, it is possible to observe that higher absolute values of \(v_{12}\) are achieved when the angle \(\theta\) assumes the values of \(-75^{\circ}\) and \(75^{\circ}\), which correspond to more compact configurations of the re-entrant cells, as these angles characterize more flattened cells. Regarding the effect of the cell aspect ratio \(\eta 1\), it shows a direct influence on \(v_{12}\), i.e., when the relation between the cell horizontal segment and the inclined one increases, this Poisson's ratio also increases.