## 3 Results and Discussion
### Case Studies
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & \multicolumn{8}{c}{\(\eta\)**1**} \\ \hline  & \multicolumn{8}{c}{**Bottom Surface**} & \multicolumn{8}{c}{**Top Surface**} \\ \hline \(\Theta\) [\({}^{\circ}\)] & **1** & **1.25** & **1.5** & **1.75** & **2** & **1** & **1.25** & **1.5** & **1.75** & **2** \\ \hline \(-75\) & 0.0336 & 0.0298 & 0.0268 & 0.0243 & 0.0223 & 0.0359 & 0.0318 & 0.0286 & 0.0260 & 0.0238 \\ \(-60\) & 0.1473 & 0.1299 & 0.1161 & 0.1050 & 0.0959 & 0.1572 & 0.1387 & 0.1240 & 0.1122 & 0.1024 \\ \(-45\) & 0.3943 & 0.3439 & 0.3049 & 0.2739 & 0.2486 & 0.4210 & 0.3672 & 0.3256 & 0.2925 & 0.2655 \\ \(-30\) & 0.9516 & 0.8157 & 0.7137 & 0.6344 & 0.5710 & 1.0161 & 0.8710 & 0.7621 & 0.6774 & 0.6097 \\ \(-15\) & 2.7222 & 2.2711 & 1.9483 & 1.7059 & 1.5171 & 2.9067 & 2.4251 & 2.0804 & 1.8215 & 1.6199 \\ \(0\) & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 & 0.0000 \\ \(15\) & \(-4.6234\) & \(-3.4572\) & \(-2.7609\) & \(-2.2980\) & \(-1.9681\) & \(-4.9367\) & \(-3.6916\) & \(-2.9480\) & \(-2.4538\) & \(-2.1015\) \\ \(30\) & \(-2.8549\) & \(-1.9032\) & \(-1.4274\) & \(-1.1419\) & \(-0.9516\) & \(-3.0484\) & \(-2.0322\) & \(-1.5242\) & \(-1.2193\) & \(-1.0161\) \\ \(45\) & \(-2.2979\) & \(-1.2397\) & \(-0.8488\) & \(-0.6454\) & \(-0.5206\) & \(-2.4536\) & \(-1.3237\) & \(-0.9064\) & \(-0.6891\) & \(-0.5558\) \\ \(60\) & \(-2.0510\) & \(-0.7156\) & \(-0.4334\) & \(-0.3109\) & \(-0.2423\) & \(-2.1900\) & \(-0.7641\) & \(-0.4628\) & \(-0.3319\) & \(-0.2587\) \\ \(75\) & \(-1.9374\) & \(-0.2324\) & \(-0.1236\) & \(-0.0842\) & \(-0.0638\) & \(-2.0 Figure 9 presents the evolution of the D\({}_{22}\) coefficient for the sets of angles and \(\eta 1\) values considered, being similar for both of the functionally graded auxetic materials, although the higher values assumed occur for the Al-WC material as those that occurred for D\({}_{11}\). However, in the present case, these higher values appear for the angle \(0^{\circ}\). The increase in \(\eta 1\) produces a significant decreasing effect in D\({}_{22}\). To illustrate more briefly the effect of these parameters on the transverse shear stiffness coefficients, Figure 10 depicts the Al-WC auxetic material and the trends observed in the same discrete domains of \(\theta\) and \(\eta 1\). When \(\eta 1\) increases, both transverse shear stiffnesses decrease. For A\({}_{55}\), the values assumed for the different \(\eta 1\) are very close when the angle is between \(-75^{\circ}\) and \(45^{\circ}\). However, for higher angle values, the A\({}_{55}\) decrease is very visible when this ratio increases. Concerning the other stiffness coefficient, A\({}_{44}\), the effect of \(\eta 1\) demonstrates a visible effect in a greater range of angles. Figure 8: Bending stiffness D\({}_{12}\) [N.m] (a) Al-WC, (b) Al-SiC. Figure 7: Bending stiffness D\({}_{11}\) [N.m] (a) Al-WC, (b) Al-SiC. Figure 10: Transverse shear stiffness of Al-WC. (**a**) A\({}_{55}\), (**b**) A\({}_{44}\). Figure 9: Bending stiffness D22 [N.m] (**a**) Al-WC, (**b**) Al-SiC. 2.2 Influence of Re-Entrant Hexagonal Honeycomb Geometrical Aspect Ratios on the Shear Correction Factor  
With this case study, it is intended to investigate the influence of the ratio's \(\eta 1\) and \(\eta 3\), as well as the angle \(\theta\), on the shear correction factors, as determined for each FGM auxetic structure considered in the present work. To illustrate these influences, one has selected an Al-WC FGM beam with an aspect ratio of a/h = 20. Figures 11 and 12 illustrate the shear correction factors (equal) obtained considering the sets of angles \(\{-75^{\circ}\), \(-60^{\circ}\), \(-45^{\circ}\), \(-30^{\circ}\), \(-15^{\circ}\), \(15^{\circ}\), \(30^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), \(75^{\circ}\)] and for \(\eta 1\) and \(\eta 3\): \(\{1,1.25,1.5,1.75,2\}\) and \(\{0.01,0.015,0.02,0.025,0.03,0.035,0.04,0.045,0.05\}\), respectively. The illustration of the shear correction factor as a function of these parameters is split into two figures to enable an easier visualization.