# Review of Auxetic Materials for Sports Applications: Expanding Options in Comfort and Protection
## 4 Expected Characteristics and Supporting Evidence
In another approach, Equation (3) has been adapted for thin sheets of rubber [287], to include a correctional multiplier based on a ratio of contact area between the sheet and indenter (\(A_{t}\), Figure 6) and the sheet's thickness (\(a_{t}\)) (Equation (4)). As thickness decreases towards zero, contact area/thickness increases and the correction tends towards unity. The force required to indent rubber to a specific depth increases as thickness decreases, but it is unclear if the same trend applies to NPR materials. The final assumption of zero friction has been shown through FEM and continuum mechanics to be invalid in simulations of infield situations [288; 289; 290; 285]. In these simulations, friction was found to enhance NPR's contribution to indentation resistance. \[Correction=1-exp^{-\frac{A_{t}}{a_{t}}} \tag{4}\]  
Figure 6: Simplified 2D indentation showing (**a**) Axial deformation; (**b**) Lateral deformation described in Equation (3); (**c**) Effect of large shear modulus (Equation (1)) on indentation area (- - -) and lateral mass flow, both un-accounted for in Equation (3). The red line shows contact area (Equation (4)). Equations (3) and (4) are for instantaneous, linear values. They do not, therefore, account for the different amounts of densification and possible hardening caused by lateral deformation due to Poisson's ratio (Figure 6c). Auxetic foam (with a relatively high shear modulus, Equation (1)) should resist shear deformation more than its conventional counterpart. Auxetic foam's upper surface would, therefore, compress as a larger, flatter area (represented by the dashed line, Figure 6c) as shown in cylindrical indentations [30] and using FEM [291]. FEM and analytical models also report a reduction in contact area for simulations with NPR materials [292], suggesting that deformation occurred over a larger radius, rather than the foam wrapping around the indenter. The opposite effect (auxetic foam wrapping around the indenter) has been predicted in FEM simulations of low strain indentation of 2D linear-elastic isotropic blocks [183] and also observed during impacts with a hemispherical dropper onto samples covered with 1 to 2 mm thick polypropylene sheets [3]. In the FEM study, a lower Young's modulus as well as NPR was employed for the auxetic foam, providing a shear modulus almost a factor of 2 lower than the conventional foam (Equation (1)), which possibly explains the discrepancy. In the experimental study, the added complexity when considering multi-material systems (e.g., featuring a stiff shell and compliant foam), hemispherical or studded indenters (rather than cylindrical) and high strain rates caused by impact could explain the observed differences. The relationship between the shell and foam's elastic moduli (as discussed in relation to coatings [293; 294]), the foam's Poisson's ratio [292; 294] and synclastic/anti-synclastic (domed or saddled) bending in the upper surface may affect indentation resistance. Hertzian indentation requires corrections for high strain indentations, impacts of non-linear materials or multi-material systems typical in sporting PPE [47; 48] and, therefore, testing of auxetic foams for sports applications [285; 288]. As noted above, friction (assumption iv) is expected to amplify increases in indentation resistance caused by NPR [285; 288], and a correction factor has been applied [288]. Experimental indentations of auxetic materials, particularly foams, are not common [7; 30], although some limitations of Hertzian indentation theory have been noticed and discussed [30]. Further significant modelling and experimental research examining NPR's effect on indentation responses of conforming, non-linear, anisotropic materials subject to a range of indenter sizes and shapes is therefore required. One of the difficulties in testing NPR's effect on expected benefits (i.e., impact force attenuation or indentation resistance) when using foam is changes to Young's modulus and stress-strain relationships following fabrication [9; 295].