## 3 Results
### ANOVA Investigation of Parameter Effects of E, \(\sigma\) and \(\
u\). To understand the influence of unit cell parameters on E, \(\sigma\) and \(\
u\), the results were analysed using analysis of variance (ANOVA), using Minitab 19 (Minitab LLC, Chicago, USA). Due to the half factorial DOE model, the analysis was restricted to study only the main effects of the unit cell parameters without interaction. Level of significance (\(\alpha\)) of 0.05 was chosen for this study. The strength of the main effects (F-factors) and significance level (P-Values) are given in Table 4 (E\({}_{\rm x}\) and E\({}_{\rm y}\)), Table 5 (\(\sigma_{\rm x}\) and \(\sigma_{\rm y}\)) and Table 6 (\(\
u_{\rm xy}\) and \(\
u_{\rm yx}\)). The magnitude of the effect of each parameter is given in Table 7.  
Figure 18: Poisson’s ratio for compression of auxetic structures along x and y axes. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Region**} & **Poisson’s Ratio (\(\
u\))** & **Poisson’s Ratio (\(\
u\))** \\  & **Compression in x Direction** & **Compression in y Direction** \\ \hline A & \(-0.6\pm 0.5\) & \(0.0\pm 0.1\) \\ \hline B & \(-0.7\pm 0.6\) & \(-0.1\pm 0.1\) \\ \hline C & \(-1.2\pm 0.8\) & \(-0.2\pm 0.1\) \\ \hline D & \(-1.3\pm 0.7\) & \(-0.2\pm 0.1\) \\ \hline Average & \(-0.9\pm 0.7\) & \(-0.1\pm 0.1\) \\ \hline SD & 0.34 & 0.09 \\ \hline Error & 18\% & 43\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: [PERSON]’s ratio values for different unit cell locations in sample 6. There is only one significant parameter controlling E, that is, \"strut thickness (t)\" for E\({}_{x}\) and \"re-entrant strut length (L)\" for E\({}_{y}\). From Table 7, t is seen to have a positive correlation with E\({}_{x}\), increasing stiffness by an average of 0.536 GPa when increasing nominal strut thickness from 0.8 to 1.0 mm. Re-entrant strut length (L), on the other hand, has a significant negative effect on E\({}_{y}\), reducing stiffness by 2.274 GPa when moving from an L of 2.5 to 3.5 mm. For \(\sigma\), there are two significant controlling parameters, t and L, with t being slightly more dominant for \(\sigma_{x}\) and L having the larger effect for \(\sigma_{y}\). For both \(\sigma_{x}\) and \(\sigma_{y}\), increasing L from 2.5 mm to 3.5 mm has a negative correlation with \(\sigma\), reducing it by 22.27 MPa (\(\sigma_{x}\)) and 28.71 MPa (\(\sigma_{y}\)). Conversely, t has a positive correlation with both \(\sigma_{x}\) and \(\sigma_{y}\), increasing them by 28.32 and 24.15 MPa, respectively. For Poisson's ratio (\(\
u\)), there is one factor only that is dominant, although for \(\
u_{xy}\), it is L, and for \(\
u_{yx}\), it is t. L has a large negative correlation with \(\
u_{xy}\), reducing it by 1.583 when moving from 2.5 to 3.5 mm in length. t has a small positive correlation \begin{table}
\begin{tabular}{c c c c c} \hline  & \multicolumn{2}{c}{**Poisson’s Ratio (\(\
u_{xy}\))**} & \multicolumn{2}{c}{**Poisson’s Ratio (\(\
u_{yx}\))**} \\  & \multicolumn{2}{c}{**x Direction**} & \multicolumn{2}{c}{**y Direction**} \\ \hline Parameter & F-Value & \(p\)-Value & F-Value & \(p\)-Value \\ \hline H & 2.15 & 0.239 & 0.02 & 0.895 \\ \hline L & 32.39 & 0.011 & 0.02 & 0.895 \\ \hline \(\theta\) & 0.05 & 0.837 & 4.59 & 0.122 \\ \hline t & 1.07 & 0.377 & 12.76 & 0.038 \\ \hline \end{tabular} Grey box indicates statistical significance. R\({}^{2}\) = 92.24\(\%\) (\(\
u_{xy}\)), 85.29\(\%\) (\(\
u_{yx}\))  
\end{table} 
Table 6: ANOVA output for \(\
u_{xy}\) and \(\
u_{yx}\).