# Effect of Compressive Strain Rate on Auxetic Foam
## 4 Discussion  
Negative Poisson's ratio of auxetic foam was observed with full-field strain measurement at higher applied strain rates of up to 200 s\({}^{-1}\) than in previous work [58, 68, 69, 70]. NPR was present at the highest strain rates, in agreement with previous tests at imposed rates of 10 s\({}^{-1}\)[52] and impacts imparting variable strain rates of \(\sim\)100 s\({}^{-1}\)[36], but approximately halved at the highest strain rate (Figure 5). Established benefits to quasi-static indentation resistance from NPR [32, 41, 42, 43, 44, 45] may, then, apply during impacts of auxetic foams, but should be confirmed in further work. Measured strain rates (80 s\({}^{-1}\)) were lower than the maximum applied value of 200 s\({}^{-1}\), due to higher compression close to the plates (Figure 4c,d) than in the central region where [PERSON]'s ratio was measured (Figure 4a,b). Obtained NPRs here were of higher magnitude (v \(-\)0.1 to \(-\)2 Figure 5) than reported in previous studies measuring NPR quasi-statically and under impact (v \(>\)\(-\)0.1 [36, 52]), where strain rates were neither constant nor measured. The anisotropic open cell foam exhibited high magnitude NPR below \(-\)1 (Figure 5a). Previous work on high aspect ratio (i.e., highly anisotropic) cell structures in foams also showed high anisotropy in mechanical properties; which typically have high Young's moduli and high magnitudes of Poisson's ratios parallel to the elongated cell axis [55, 57, 62]. Figure 3b shows the elongated cells along the \(y\)-axis (the stretched direction during conversion). [PERSON] and [PERSON]'s flexure model [47], considering flexing of the slightly re-entrant long cell ribs (Figure 3b), would give higher strains along the perpendicular (lateral) x- or z-axes during compression or tension in the long cell \(y\)-axis, and hence high magnitude NPR. Steam processed auxetic closed cell foam had quasi-static Poisson's ratio of \(-\)0.12 (Figure 5b) up to 2% compression. The quasi-static Young's modulus (\(\sim\)1 MPa, Figure 6d) of this auxetic closed cell foam was at the lower end of the reported range for foam in snow-sports back protectors (1 to 10 MPa [9, 23]), football shin guards [6, 66] and cricket thigh pads [5], to name a few. Young's modulus of the auxetic closed cell foam approximately doubled while NPR halved between minimum (0.005 s\({}^{-1}\)) and maximum strain rates (200 s\({}^{-1}\), Figure 6d). This doubling of [PERSON]'s modulus as strain rate increased by 40,000 times is lower than the increase by a factor of five with increased strain rate of only 50 times reported for dilatant foam in soft-shell back protectors [9]. The linear relationship between Young's modulus and strain rate (Supplementary Figure S2) suggests the closed cell foams tested here were viscoelastic, rather than dilatant [20]. Recent work with the parent foam used here has produced higher magnitude anisotropic NPRs, approaching \(-\)1, in larger sheets of auxetic closed cell foam [62]. Further work could test whether such high magnitude NPR auxetic closed cell foam in isolation, or in combination with dilatant foam, could pass certification tests and ultimately enhance protection during sporting collisions and falls. The sample sizes of 13 to 20 mm for closed cell and 20 to 25 mm for open cell foam, due to the various conversion methods, caused differences in applied strains (45 to 80%, Figure 4e,f) and strain rates (120 to 250 s\({}^{-1}\), Supplementary Figure S1) during compression testing. Sample sizes were otherwise accounted for in stress and strain calculations. The relative increase in strain rate (lowest to highest) was caused by differences in applied compression rate and so was the same between samples, despite the different sizes. We have demonstrated that NPR was maintained at the high strain rates expected under impact using 2D DIC, but NPR approximately halved in magnitude at the highest compression rates (Figures 4 and 5). Further work could use 3D DIC to confirm the Poisson's ratios measured here.