# A Simple Method to Convert Cellular Polymers into Auxetic Metamaterials
## 3 Results and Discussion
### Auxetic Foams
#### 3.2.2 Poisson's Ratio
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Sample**} & \multirow{2}{*}{**Initial Density** (kg/m\({}^{3}\))} & \multirow{2}{*}{**Final Density** (kg/m\({}^{3}\))} & \multirow{2}{*}{**Compression** Ratio (D\({}_{\text{f}}\)/D\({}_{\text{o}}\))} & \multirow{2}{*}{**Minimum PR** (Tension)} & \multirow{2}{*}{**Minimum PR (Compression)**} \\ \cline{5-6}  & & & & & & \\ \hline \multirow{5}{*}{D16} & PE-D16-O & 15.5 & 15.5 & 1 & 0.38 (mean) & 0.10 (mean) \\  & PE-D16-P1.0 & 15.5 & 76.0 & 4.90 & \(-2.55\) & \(-0.61\) \\  & PE-D16-P1.3 & 15.5 & 109 & 7.03 & \(-2.83\) & \(-0.60\) \\  & PE-D16-P1.5 & 15.5 & 118 & 7.61 & \(-2.89\) & \(-0.65\) \\  & PE-D16-P1.8 & 15.5 & 120 & 7.74 & \(-0.89\) & \(-0.28\) \\ \hline \multirow{5}{*}{D21} & PE-D21-O & 20.9 & 20.9 & 1 & 0.34 (mean) & 0.08 (mean) \\  & PE-T105-D21-P1.5 & 20.9 & 88.0 & 4.21 & \(-1.52\) & \(-0.35\) \\  & PE-D21-P1.6 & 20.9 & 88.8 & 4.25 & \(-1.60\) & \(-0.32\) \\  & PE-D21-P1.9 & 20.9 & 98.4 & 4.71 & \(-2.55\) & \(-0.33\) \\  & PE-D21-P2.1 & 20.9 & 99.5 & 4.76 & \(-1.07\) & \(-0.31\) \\ \hline \multirow{5}{*}{D30} & PE-D30-O & 29.7 & 29.7 & 1 & 0.37 (mean) & 0.16 (mean) \\  & PE-D30-P1.5 & 29.7 & 116 & 3.91 & \(-1.60\) & \(-0.55\) \\ \cline{1-1}  & PE-D30-P1.8 & 29.7 & 120 & 4.04 & \(-0.97\) & \(-0.51\) \\ \cline{1-1}  & PE-D30-P2.0 & 29.7 & 126 & 4.24 & \(-1.52\) & \(-0.50\) \\ \cline{1-1}  & PE-T105-D30-P1.5 & 29.7 & 125.2 & 4.22 & \(-1.00\) & \(-0.49\) \\ \hline \multirow{5}{*}{D36} & D36-O & 35.6 & 35.6 & 1 & 0.40 (mean) & 0.13 (mean) \\ \cline{1-1}  & PE-D36-P2.1 & 35.6 & 108.7 & 3.05 & \(-0.49\) & \(-0.34\) \\ \cline{1-1}  & PE-D36-P1.8 & 35.6 & 89.0 & 2.50 & \(-0.72\) & \(-0.32\) \\ \cline{1-1}  & PE-D36-P1.5 & 35.6 & 81.5 & 2.29 & \(-1.50\) & \(-0.25\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance of auxetic LDPE foams. Figure 4: Poisson’s ratio as a function of tensile engineering strain for samples based on D16 (different vacuum conditions). For the other auxetic foams (D21, D30 and D36), the minimum NPR values obtained in tension were \(-2.55\), \(-1.60\) and \(-1.50\), respectively. The best compression ratios (D\({}_{\text{f}}\)/D\({}_{\text{o}}\)) were different for the D16, D21, D30 and D36 foams, these being 7.6, 4.7, 3.9 and 2.3, respectively. A higher mechanical pressure generated a higher foam density and a more re-entrant structure. However, too much compression resulted in cell collapse, completely destroying the re-entrant structure generated. This trend was also observed in our previous work [6]. The average compressive Poisson ratios were 0.10, 0.08, 0.16 and 0.13 for D16, D21, D30 and D36, respectively. This indicated that the Poisson ratio determined under compression was lower than that determined under tension (0.38, 0.34, 0.37 and 0.40, respectively). This phenomenon likely depended on the foam's cellular structure (anisotropy). The minimum compressive NPR values for the auxetic foams were \(-0.62\), \(-0.33\), \(-0.51\) and \(-0.34\) for D16, D21, D30 and D36, respectively. However, the minimum compressive NPR did not vary significantly with increasing the processing pressure. Very limited data is available in the literature on the compressive NPR of closed-cell foams. [PERSON] et al. reported very small minimum compressive Poisson's ratios of \(-0.026\), \(-0.090\) and \(-0.118\) based on 20, 30 and 45 expansion ratios, respectively, of LDPE closed-cell foams [15]. Similarly, [PERSON] et al. reported compressive NPR values of \(-0.1\) and \(-0.05\) for PU closed-cell foams at different compressive strain rates [19]. [PERSON] et al. obtained a better compressive NPR value (\(-0.34\)) for an auxetic foam based on open-cell PU foam [14]. [PERSON] et al. reported the best minimum NPR value (\(-0.85\)) for a compression of 3.33%, while the NPR value was substantially decreased (\(-0.12\)) for a 27 kg/m\({}^{3}\) PU open-cell foam with a compression ratio of 80% [31]. Thus, the minimum compressive NPR value of \(-0.65\) reported here was the lowest value for closed -cell PE foams.