# Effect of Compressive Strain Rate on Auxetic Foam
## 1 Introduction
Studies measuring NPR of foam during high strain rate compression have only tested open cell auxetic foams with relatively low magnitude NPRs (between \(0\) and \(-0.1\)) [36; 37; 52]. These studies did not control nor measure the applied strain rates, nor report that they were similar to those expected during sporting collisions and falls. Auxetic open cell foam is the most studied auxetic foam and it has been used to show the favourable impact response associated with NPR [4; 30; 53]. Auxetic open cell foam is typically fabricated by a thermomechanical process involving volumetric compression of conventional open cell foam in a mould, which buckles cell ribs to impose a re-entrant cell structure, then thermal fixing (heating then cooling) of the imposed structure [54]. Compression can be equal in all axes to give isotropic auxetic foam [54] or different in each of the principal directions to impart anisotropy [55; 56; 57]. Typically imposed uniform (triaxial) compression reduces the foam's volume by a factor of two to five [54; 56; 58], equating to \(20\) to \(40\%\) compressive strain in each direction. Open cell foams with compressive isotropic (same in all three axes) Poisson's ratios as low as \(-0.7\) have been reported [54; 59; 60]. Stretching foam in one direction during fabrication increases anisotropy [57] and can also increase the magnitude of Poisson's ratio [55]. Foams used in back protectors [9; 23], and closed cell foam in other PPE, like football and cricket pads [4; 6], are at least ten times stiffer than open cell foam [4; 56; 58]. Closed cell foams used in PPE typically have Young's moduli--the established measure of materials stiffness--close to or above \(1\) MPa [4; 6; 9; 23]. Closed cell auxetic foam can be fabricated using a bespoke pressure vessel [61] or by steam processing [62; 63]. Steam processing traps steam in closed cells, which condenses upon cooling to create a negative pressure differential, shrinking the foam without a mould while fixing the polymer to impart NPR [62; 63]. Auxetic foams fabricated by steam processing can have similar stiffness (~1 MPa) to conventional foams in PPE [62], so they should be more suitable for this application than open cell auxetic foam. Impact tests for certifying PPE, like back protectors (EN-1621-2 [24]), football shin guards (BS EN 13061:2009 [64]) and cricket thigh pads (BS 7928:2013 [65]), typically compress foam to densification (~80% of its thickness) in less than 5 ms [66; 66; 23; 5; 29], equating to a time-averaged engineering strain rate of ~200 s\({}^{-1}\). During such impact tests, a drop mass is decelerated by the energy absorbing material within the product, imparting variable strain rates up to and above 200 s\({}^{-1}\) that reduce to 0 s\({}^{-1}\) at maximum compression. Despite the wealth of research into auxetic open cell foam, most of the previous work measured [PERSON]'s ratio and Young's modulus quasi-statically (<1 s\({}^{-1}\), [34; 54; 60; 67; 68; 69; 70]), or at intermediate (<10 s\({}^{-1}\), [52]) strain rates. [PERSON]'s ratios of auxetic open cell foams, between 0 and \(-0.1\), have been measured using marker tracking during impact tests at higher, but variable, strain rates of ~100 s\({}^{-1}\)[36; 37]. Closed cell auxetic foam has only been characterised quasi-statically [61; 62; 63]. As such, it remains unclear how the desirable [PERSON]'s modulus of auxetic closed cell foam and the NPR--that could increase indentation resistance and decrease peak force under impact--could change during sporting collisions, falls, and other impacts. It is important to understand how [PERSON]'s modulus and [PERSON]'s ratio of auxetic foams may change at the high strain rates expected under impact, to inform the design, and help explain the findings, of high-speed indentation and impact studies. Here, we test auxetic open and closed cell foam at a range of compressive strain rates (0.05 to 200 s\({}^{-1}\)).