# Review of Auxetic Materials for Sports Applications: Expanding Options in Comfort and Protection
## 2 Introduction to Sporting Protective Equipment  
Injuries in sport are common and place a significant burden upon participants and national economies [43; 44], estimated at $525 (\(\sim\)$2380) million per year in The Netherlands [45] for example. The main methods of intervention are elimination, modification (a reduction in severity or likelihood of injury) and reaction (i.e., medical) [44]. Preventative measures such as sports safety equipment, and/or rule changes, are more cost effective than reactive procedures [44] and successful products can increase a manufacturer's share of the sporting goods market (\(\sim\)$90/$66 billion in the USA in 2017 [46]). Sporting PE is intended to reduce risks and is typically either an addition to the playing field or environment (i.e., crash mat or barrier, known herein as PE) or PPE. Both equipment  
Figure 1: Lateral (vertical) deformation due to Poisson’s ratio during tensile axial (horizontal) loading for (**a**) a conventional material and (**b**) an auxetic material. Thick and thin arrows correspond to deformation due to loading and Poisson’s ratio respectively. types provide; (i) protection against impact, dissipating and absorbing energy while reducing peak forces/accelerations, pressures and in some cases such as helmets, impulses [43; 44; 47; 48; 49] and (ii) protection from penetration, abrasions and lacerations [18; 47; 48; 50]. Sporting PPE can also provide support to joints, muscles and the skeleton [18; 44; 47; 48; 51]. Sporting PPE (Figure 2a) often contains a shell--typically a stiff material or non-Newtonian fluid layer--to spread forces and reduce pressures [47; 48; 50; 52]. Impacts are typically attenuated by elastic [44; 52], viscoelastic and/or permanent deformation (i.e., crushable foam in cycling helmets) of a cushioning material with a lower compressive stiffness than the shell [47; 48; 50; 52]. Visco-elastic and permanent deformation reduce impulses, improving the level of protection [44]. Sporting PPE must sometimes cover large areas and should ideally be low in mass and bulk to reduce restriction of movement, fatigue and heat build-up. Foam is typically used for the cushioning layer in PPE, and with limited thickness to compress and decelerate impacting bodies material selection is crucial. In contrast, PE such as crash mats (where bulk, mass and ergonomics are less critical) are often large and thick, allowing more gradual deceleration over a greater distance through compression of relatively compliant foam [52]. For both sporting PE and PPE, deceleration (or force) ideally reaches a maximum yet safe value at low strains, before plateauing and deforming with no additional load to safely maximise energy absorption (the integral of decelerating force with respect to deformation) before foam densification (at high strains) causes high deceleration (or force) [53]. Crash mats are used in a variety of conditions; protecting skiers, snowboarders and mountain bikers (who can travel at high speeds) from hazards on a mountain (i.e., lift poles or trees), often in extreme and variable climates, to providing padding to gymnasts on flat surfaces at room temperature. Sport safety equipment is often certified according to tests specified in standards and regulations (i.e., [54; 55; 56; 57; 58]), which typically specify a maximum allowable peak force/acceleration under impact (i.e., Figure 2b). To perform well in these tests, a product should absorb or dissipate as much energy as possible to prevent force/acceleration from passing its allowable limit [59]. Criteria within standards (such as impact energies, velocities and masses) are not always well justified and tests typically feature fixed rigid anvils rather than biofelic (human like) surrogates, meaning they are not overly representative of the scenarios where the equipment may be required to perform [44]. In some cases there is no standard, so manufacturers certify products against a standard for another safety device, which can be misleading for consumers.