# Review of Auxetic Materials for Sports Applications: Expanding Options in Comfort and Protection
## 4 Expected Characteristics and Supporting Evidence  
Beneficial characteristics of auxetic materials include increased shear modulus, indentation resistance [10; 30], dynamic [274; 275] and static compressive energy absorption [8; 37] and decreased bulk modulus [10; 30]. Increased indentation resistance [30] and compressive energy absorption [2; 8; 33; 37; 204] have been shown experimentally in comparisons of auxetic and conventional foams. Lateral expansion due to axial tensile loading makes auxetic structures/materials ideal candidates for straps in apparel, increasing area to spread increasing loads and prevent 'digging in' [276]. Many of the characteristics which provide unique enhancements for auxetic materials come from Elasticity theory [10; 277]. For isotropic materials experiencing elastic deformation, [PERSON]'s modulus (\(E\)) and shear modulus (\(G\)) are related (Figure 5a, Equation (1)), as are [PERSON]'s modulus and bulk modulus (\(K\), Figure 5b, Equation (2)) [278]:  
\[G=\frac{E}{2(1+\upsilon)} \tag{1}\]  
\[K=\frac{E}{3(1-2\upsilon)} \tag{2}\]  
Elasticity theory states that Poisson's ratio must be between \(-1\) and 0.5 for 3D isotropic materials [118; 279; 280], and between \(-1\) and +1 for 2D isotropic materials [281]. From Equations (1) and (2), as Poisson's ratio tends towards \(-1\) both shear (Equation (1)) and bulk (Equation (2)) modulus are driven towards extremely high or low values (respectively) in a 3D isotropic material. Figure 5: Schematic showing (**a**) shear deformation and (**b**) bulk/volumetric deformation. Black arrows show shear loading, red arrows show loading due to hydrostatic pressure, dashed lines show original (**a**) shape and (**b**) volume. Indentation resistance is a measure of the load required to indent a material (Figure 6a,b). From elasticity theory, Hertzian indentation resistance (\(H\), Equation (3)) for an isotropic material compressed with a uniform indenter depends on Poisson's ratio, Young's modulus and a constant (\(x\)) related to the shape of the indenter [31]. \[H\propto\left(\frac{E}{(1-v^{2})}\right)^{x} \tag{3}\]  
A material with large indentation resistance, but low Young's modulus, could be used in a protective pad that will deform by a similar amount for high or low area (i.e., flat surface vs. studded) impacts. Such a pad would be more versatile, reacting to different surfaces to optimise its resistance to deformation [282]. Assuming a constant Young's modulus, as Poisson's ratio tends towards \(-1\), a material should exhibit infinitely higher resistance to shear deformation (Figure 5a, Equation (1)) and high indentation resistance (Figure 6, Equation (3)), while becoming increasingly easier to deform volumetrically (Figure 5b, Equation (2)). Infinite shear modulus and maximum indentation resistance and volumetric compressibility can only be achieved elastically and isotropically with auxetic materials [10], and increased indentation resistance has been shown experimentally [30; 283; 31]. Equation (3) for Hertzian indentation resistance comes from elastic properties and assumes; (i) the surfaces are continuous and have non-conforming profiles; (ii) the area of contact (Figure 6) is much smaller than the characteristic dimensions of the contacting bodies; (iii) the strains are small and purely elastic and (iv) the surfaces are frictionless at the contact interface. These four assumptions for Hertzian indentation are not always held, although Equation (3) is often referred to and discussed in the context of non-Hertzian indentations [284; 285]. Sporting PPE typically has a low thickness, so does not often meet assumption (ii). A finite thickness model has been developed for soft and thin cushion materials where Hertzian theory is expected to become invalid, and auxetic cushions were found to reduce the contact pressure on the buttocks (indenter) [286].