# Review of Auxetic Materials for Sports Applications: Expanding Options in Comfort and Protection
## 4 Expected Characteristics and Supporting Evidence
Studies report auxetic foams with lower initial Young's modulus than their conventional counterparts [189; 274; 295]. Reduced Young's modulus has been attributed to the presence of buckled ribs in the auxetic foam being easier to deform than the straighter ribs of the conventional foam [295]. Elasticity theory (Equation (1), Figure 5a) also supports a reduction in Young's modulus as Poisson's ratio decreases if shear modulus remains constant. Auxetic foams typically have a higher density than their open cell parent foam, so the reduction in Young's modulus is contrary to the usual expectation of an increase in Young's modulus with increased density [118]. Note, though, that [PERSON] and [PERSON] refer to a density increase caused by thicker ribs, whereas in the auxetic foam fabrications density increases due to changes in rib orientation. [PERSON] and [PERSON]'s cellular solid theory actually indicates that [PERSON]'s modulus can either increase or decrease when moving from a hexagonal to a re-entrant cell geometry, characteristic of auxetic foams [118]. The increased Young's modulus is allowed by elasticity theory--materials with the same bulk modulus (Equation (2), Figure 5b) will have increased Young's modulus as Poisson's ratio decreases. Increases [2; 190; 204] as well as the aforementioned decreases in [PERSON]'s modulus [9; 274; 295] have been reported in auxetic foam fabrications. The re-entrant structures in auxetic foams typically give an initially quasi-linear compressive stress-strain curve, with hardening as pores close at higher compression levels (>-50%) [9; 183; 275; 282; 285]. Conventional open cell foams exhibit a low-stiffness plateau region due to buckling of cell ribs between \(\sim\)5% and 80% strain (Figure 7a) [118]. Both of these cases have been explained numerically and validated experimentally [23; 118]. It should be noted that the relatively linear and plateauingstress-strain relationships only apply to specific forms of cellular materials that adapt a re-entrant structure, including PU foams [9; 23] and 2D honeycombs [96]. Exceptions have been recently presented; re-entrant auxetic PU foams with a plateau region [23; 195]. For a linear elastic isotropic material subject to a uniaxial stress (\(\sigma\)), the total strain energy density (\(U_{l}\)) is the sum of the volumetric strain energy density (\(U_{v}\)) and the distortional strain energy density (\(U_{d}\)), which are related to the Young's modulus and Poisson's ratio (i.e., [278]):  
\[U_{v}=\ \frac{(1-2v)\sigma^{2}}{6E} \tag{5}\]  
\[U_{d}=\ \frac{(1+v)\sigma^{2}}{3E} \tag{6}\]  
In Figure 7b, an applied compressive uniaxial stress and Young's modulus are arbitrarily equated to 1 kPa for simplicity. Plotting these values in Equations (5) and (6) between elastic limits of \(-1\) and \(0.5\) for Poisson's ratio (Figure 7b) shows an increase in volumetric strain energy density and a reduction in distortional strain energy density as Poisson's ratio tends towards \(-1\). Auxetics material's tendency to volumetric (rather than distortional deformation) could effectively increase indentation resistance (Figure 6c, Equations (3) and (4)). As Poisson's ratio decreases, so does the stress concentration at a crack's tip, preventing crack propagation and increasing toughness [296; 297]. [PERSON] and maximum shear stress theory both define failure when distortional strain energy exceeds a maximum value. The reduction in distortional strain energy (to zero, Figure 7b) as Poisson's ratio reduces to \(-1\)[279] is, then, expected to lead to an increase in toughness. A natural example of where this may be exploited may be found in the nacre layer of certain seashells. Nacre has a reported tensile NPR of the order of \(\sim\)\(-0.1\)[93] and \(\sim\)\(-0.4\)[298] which is thought to increase volumetric strain energy density by \(\sim\)eleven times while more than halving distortional strain energy density, allowing the system to absorb more energy before failure [298].