# Effect of Compressive Strain Rate on Auxetic Foam
## 1 Introduction  
Sports participants often wear personal protective equipment (PPE) to reduce injury risk during collisions and falls [1]. Personal protective equipment, including back and limb protectors, traditionally use foam for cushioning and a stiff shell to spread impacts by concentrated loads [2, 3, 4, 5, 6]. These stiff shells can hinder movement and make the protective equipment bulky and uncomfortable [7, 8, 9, 10]. Indeed, some sports participants choose not to wear PPE if it is uncomfortable [7, 8, 11, 12, 13, 14, 15, 16]. Contemporary protective equipment can use dilatant foams [17, 18, 19], which are like viscoelastic foams in that they have an effective viscosity, except this viscosity increases with shear strain rate [20, 21, 22]. Dilatant foams can be flexible during normal use but then stiff during impacts, so uncomfortable stiff shells [7, 8, 9, 10] are not needed to pass certification tests [9, 10, 23]. Soft-shell dilatant foam back protectors intended for sports use have been certified to EN-1621-2 [24] for motorbiking, as there is no application specific standard [2, 3, 9, 10, 23, 25]. Other types of sporting PPE are also designed and certified under test conditions that may not represent infield collisions and falls [26, 27, 28, 29]. The stiffness of dilatant foams can vary exponentially when impacted and compressed atthe various temperatures and speeds possible during sporting collisions and falls [9; 23]. Applying such rate dependent materials to PPE means products can change in performance between certification tests and actual use, with implications for injury risk, including users overestimating how well they are protected [3; 26; 29]. Developing PPE that is less likely to see such changes in performance between certification tests and infield collisions and falls could, therefore, reduce the occurrence of sporting injuries. Auxetic materials have a negative Poisson's ratio (NPR), a characteristic that could improve PPE [4; 30; 31]. Poisson's ratio is the negative of the ratio of lateral to axial strain, so auxetic materials contract laterally during axial compression and expand in tension. NPR can give benefits such as increased vibration damping [32; 33] and reduced peak force under impact [34; 35; 36; 37; 38], as detailed in review papers [4; 30; 31]. The established benefits of NPR that are most relevant to soft-shell PPE are (i) improved equipment comfort [39; 40] and (ii) increased indentation resistance [32; 41; 42; 43; 44; 45]. According to Equation (1) from Hookean indentation theory, maximum indentation resistance (H, [31; 42; 44]) is achieved for a given Young's modulus (E) as the magnitude of Poisson's ratio (v) approaches one [4]:  
\[H\;\propto\;\left(\frac{E}{1-\
u^{2}}\right)^{x} \tag{1}\]  
whereby '\(x\)' relates to indenter shape. Equation (1) has been tested for auxetic and conventional materials, including foams, using hemispheres (\(x=2/3\)) and the flat end of a cylinder (\(x=1\)), with NPR shown to increase indentation resistance [32; 41; 42; 43; 44; 45]. For isotropic materials, [PERSON]'s ratio is between \(-1\) and \(0.5\)[46], so indentation resistance should be highest for materials with NPR between \(-0.5\) and \(-1\). Replacing foam in PPE with auxetic foam with a higher magnitude of [PERSON]'s ratio (i.e., ideally between \(-0.5\) and \(-1\)), which is typically between \(0.1\) and \(0.4\) for conventional foam [47; 48; 49], could increase indentation resistance. As such, auxetic foam could offer high indentation resistance to protect against penetrating loads, and low Young's modulus flexibility during normal use or impacts with distributed loads, reducing the need for stiff shells. Combining auxetic and dilatant foams [50; 51], or using auxetic foam in isolation, could improve the comfort of soft-shell PPE and reduce changes in performance between certification tests and actual use. The increased indentation resistance of auxetic foam could only improve soft-shell PPE if NPR is present when the material is compressed rapidly during impacts.