Mechanical metamaterials for sports helmets: structural mechanics, design optimisation, and performance

The idealised goal of impact protective equipment is to absorb induced impact energies without exceeding measures associated with injury risk. For a consistent impact scenario, like a certification test, the selection process for an energy-absorbing material is established. The challenge with helmets is the diverse variety of impact scenarios. There are various helmet designs available for different sports [35, 36, 217], with two main categories. The first category is single-impact helmets, which crush under impact and are designed to protect the head against one severe (high energy) impact. These include motorsports, cycling, and alpine sports helmets. After such an impact, these helmets should be replaced as they are damaged, and offer limited protection [35, 217]. The other category is multi-impact helmets, which are designed to maintain their impact performance over a (typically) long service life, e.g. several years. These are used for American Football, ice hockey, and lacrosse [35, 217], to name a few.

Helmets typically have at least three layers. A stiff (polymer or composite) outer shell (figure 1(A)) prevents penetration [35, 36, 84, 217, 255, 256], absorbs the initial shock [84, 257], and helps to hold the helmet together during or after an impact [84]. A compressible foam or lattice liner absorbs energy through deformation (figure 1, [217, 256]). Most single-impact helmets, particularly those for cycling, use crushable, expanded polystyrene (EPS) foam [36, 258]. Vinyl nitrile (VN) (figure 1(A)) and expanded polypropylene (EPP) (figure 1(B)), are often used in multi-impact helmets [35, 151, 259–261]. Many helmets also include a comfort liner, often a compliant foam [262], as shown in figure 1(A). New materials and components are also being added to helmets, generally intended to exceed minimum requirements in certification tests (e.g. [30, 31, 33, 263–269]).

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Inspecting example compressive stress (σ) vs. strain () relationships (figure 2), the area under the curve is the energy absorbed per unit volume (W [270]). The compressive stress vs. strain relationships of cellular materials, such as foams, can often be divided into three sections: (i) linear elasticity, up to ∼5% strain; (ii) plateau, elastic or plastic buckling of the cell walls; and (iii) densification, where cell walls self-contact and the constituent material is compressed [270].

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Energy absorption efficiency (W/σ) is highest during the plateau region [271], which can be tailored by modifying the constituent material or foam relative density [270]. An ideal foam for a given impact (e.g. curve 0.03 in figure 2) absorbs the induced energy during the plateau region, without densifying [35, 270]. Energy absorption before densification increases with liner thickness. However, overly large helmets are uncomfortable [35, 270] and can increase rotational accelerations by increasing torque applied to the head [35]. As such, helmet liner thickness is generally limited. So, combining layers of foam of varying relative density, and hence stiffness, may broaden the range of manageable impacts, but will give lower maximal efficiency [36, 272].

Various approaches have been taken to make helmets more effective over a wider range of impacts. Shear-thickening fluids (STFs) and polymers (STPs) are non-Newtonian (figure 3). The viscosity of these materials increases with shear strain rate [273–276]. STFs include suspensions [273] and gels [277], while STPs (which are more commonly used in consumer products) are viscoelastic solids [275, 276, 278]. STPs adapt to impact severity [279, 280]; they can be flexible and elastic during normal use and minor impacts, or stiffen and increase damping during severe impacts [273, 281]. The viscosity change is reversible, providing an alternative to crushable foam over multiple impacts [282, 283], with slow recovery potentially reducing rebound, impact duration, and various measures of injury risk [125, 133–136]. So, foamed STPs are used in PPE [273, 275, 277, 279], including helmets liners (figure 1(C)) [34]. STPs can also be formed into a structure and used within helmets to reduce rotational kinematics [18, 25].

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A low-friction layer, placed between the helmet's liner and shell (figure 2(B) [21, 284]), or between layers of foam [265], allows relative rotation between components. This relative rotation has been shown to reduce the rotational kinematics of headforms during oblique impact tests [32, 230, 245, 284–288]. A well-known example of this technology is the multi-directional impact protection system (MIPS) [264]. The inclusion of anisotropic helmet liners has also been shown to reduce rotational acceleration during certain oblique impacts [19–21, 289]. Such liners may have fibrous columns [21], or elongated cells [20], making them stiff through thickness but transversely compliant, lowering shear stiffness [290, 291]. Salomon's EPS 4D helmet liner uses a similar principal, whereby columns of EPS foam appear to be designed to shear during oblique impacts [268]. These examples are relatively standard, using conventional materials and manufacturing methods.

Patents have been filed featuring concepts related to the application of mechanical metamaterials in helmets. These include helmets, or helmet liners, based on structured polymers such as lattices (e.g. [292–297]), sprung/suspended inserts (e.g. [298]), modular/custom fit structured components (e.g. [299, 300]), foamed/structured shear thickening materials (e.g. [301, 302]), bulk shear thickening materials (e.g. [303]), and fluidic properties (e.g. [304, 305]). Many of these innovations feature in commercially available helmets (e.g. [30, 31, 263, 269]).

Poisson's ratio is the negative product of the ratio of lateral to axial strain. Auxetic materials undergo transverse expansion when stretched axially, and contract transversely in compression [11, 285, 306]. So, they form a dome shape under bending, and are used in a helmet liner for this reason (i.e. flat sheets can fit into domed helmets [32, 263]).

Poisson's ratio is one of the basic elastic constants, and (with Young's modulus/stiffness) affects shear modulus, bulk modulus, and indentation resistance [368]. As detailed elsewhere [368], Poisson's ratio increases resistance to penetration by concentrated loads [369–373], and shear modulus [374–378]. The increased tendency of materials with a low or negative Poisson's ratio to deform volumetrically, rather than in shear (figure 4), may also increase strength. With lower shear strain, the likelihood of failure close to a crack tip or an ellipsoid reduces, according to Von-Mises, Tresca, and crack propagation theories [379, 380]. Without the presence of stress concentrations caused by a crack or ellipsoid, Von-Mises and Tresca criterion are unaffected [368]. So, auxetic helmet sections may fail less readily, reducing waste and severe head injury risk.

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The re-entrant-like cellular structure of auxetic foams is imparted by compressing conventional foam to buckle cell ribs [306, 307, 381]. So, while there is some uncertainty over whether these foams meet the requirement for the precisely defined topology of some metamaterial definitions [7–9], they are still a related medium. Readers interested in auxetic foam manufacturing are referred to Jiang et al's review [307]. Auxetic foams have been shown to increase vibration damping [382, 383], and to exhibit peak impact forces up to ten times lower than their conventional counterparts [291, 384–388]. It should be noted that peak force during the typically stiff anvil and impactor impacts is not a scalar measure. Indeed, peak force increases exponentially as the foam densifies (see figure 2) and 'bottoms out' under impact. Further, auxetic foams made from expanded foam, as typically used in helmet liners, have not been reported.

Auxetics may provide benefits in helmet liners: (i) The ability to adapt to the shape of impacting bodies—remaining soft when impacting a relatively flat surface, but effectively stiffening under concentrated loads. (ii) High vibration damping, redistributing vibrations transversely. (iii) A tendency to bulk over shear deformation, reducing the likelihood of failure. Conversely, the early densification strain caused by the tendency to bulk deformation may shorten the stress vs. strain plateau in cellular solids—causing densification at lower strain ([291], figure 2). With the ability to include a stiff shell, the benefit of the high indentation resistance possible for auxetics is unclear and has not been empirically demonstrated in helmets. Flexible shell helmets (e.g. [227, 228]) may, however, benefit from the increased indentation resistance of auxetic materials. Further, the increase in shear modulus with negative Poisson's ratio goes against the broad strategy of reducing liner shear stiffness to reduce rotational acceleration [25, 32, 263, 264, 284, 286–288]. So, the application of auxetic materials in helmet liners requires careful design based on justifiable benefits, such as increasing indentation resistance to facilitate lower stiffness liners.

An unstudied, potentially useful topic is auxetic helmet shells. Fibre-polymer composites can be auxetic, with the negative Poisson's ratio achieved by fibre alignment [389, 390]. Due to the use of conventional fibres and pre-preg, auxetic fibre-polymer composites can be made with standard composite manufacturing methods [391]. These auxetic composites have been shown to resist back face damage under impacts [392]. Such increased resistance to back face damage could increase the lifespan of multi-impact helmets featuring composite shells, particularly those with flat sections that are more susceptible to back face damage.

Advances in additive manufacturing, and moulding methods [25], have allowed mass production of lattice and honeycomb mechanical metamaterials (e.g. [31, 33, 263, 393, 394]). These methods allow precise manufacturing of complex geometries [16, 395–397], expanding the range of available properties to meet complex requirements, such as those seen in helmets. 2D extruded cellular structures, such as honeycombs (figure 5(A)), repeat periodically in two directions [398]. Honeycomb and tubular structures are studied frequently as energy-absorbing elements in sports equipment and helmets [23, 399–403], such as in Koroyd's helmet liner [31]. 3D periodic cellular structures, such as lattices, consist of unit cells repeating in three directions, increasing degrees of freedom during design, but also increasing manufacturing complexity and hence costs [395, 397, 398, 404–409].

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Exemplary unit cell designs include hexagonal/re-entrant (figure 5(A)) [410, 411]), square/cubic (figure 5(B)) [412]), or chiral/antichiral (figure 5(C)) [413–417]. Unit cell design degrees of freedom include; varying rib orientation (figures 5(A) and (B)), length, or thickness—slender ribs are often less stiff, varying rib form (figure 5(B))—Eigenmode example); varying the number of ribs (figures 5(A) and (B)), or adding and combining shapes and features (figure 5(C)). Unit cell patterning also affects topology, and so metamaterial properties; unit cells can be mirrored (figures 5(A-i) and (C)), linearly repeated (figures 5(A), (B) and (C-iii)), or rotated (figure 5 (C-v)).

When creating honeycombs, some variation can be applied in the extruded direction, as is the case of Miura-ori-inspired structures (figure 5 (B-ii)) [418, 419]. Gradient metamaterials can also be developed by spatially varying unit cell parameters and properties [402, 403, 420, 421]. These variations can be continuous or discrete (i.e. gradual or abrupt change) [422–424]. 3D periodic cellular structures are being used in helmets (e.g. ice hockey [33] and American football [393]). Such liners, or inserts, can also feature some through thickness variation, and can be made from STPs [30].

Concerning some common topologies, hexagonal honeycombs are relatively stiff, with low density, for compression parallel to their extruded dimension [402, 425–428]. During impacts in the extruded direction, cell walls crumple and buckle [31, 426], with densification occurring at ∼75% compression [402]. When impacted or compressed perpendicular to their extruded dimensions, honeycomb stiffness is lower [28, 402, 427, 429, 430]. Re-entrant hexagonal honeycombs can be more compliant than equivalent density regular hexagonal ones (at low strains), due to the extra junction for rib hinging to occur around [423, 429, 431, 432]. Buckling may not occur with these re-entrant unit cells, causing an almost linear compressive stress vs. strain response, i.e. a less pronounced plateau and densification region [369, 423, 429]. Such re-entrant unit cells may not be optimal within a target impact severity (e.g. as in figure 2), but are less prone to stark peak force increases during severe impacts [29, 291, 385]. It is possible to design tall, slender stiff re-entrant cells that undergo buckling [271, 422].

Structures made of solid rotating shapes are often stiff in compression, as the internal shapes undergo self-contact/densification at low strains, making them less suited for sporting impacts [433–436]. These rotating shapes have been made as lattices with hollow cells [437], or cut from foams, to tune out of plane properties while relying on foam characteristics through thickness [438, 439]. High energy absorption, low initial crushing peak forces, large densification strain, and low strain rate sensitivity can be achieved with folded (kirigami or origami) structures [332, 333, 335, 440, 441]. By including such folds through the thickness of these structures, it is possible to tune buckling regions, and the length of the compressive stress plateau [17] (figure 2). Kirigami structures made from paper have been used in a recyclable cycling helmet [442].

For application in helmets, periodic structures must be patterned to fill an often complex/nearly spherical space. Such patterning, using conventional computer-aided design software, can be time-consuming and inefficient. Algorithmic-based design software, such as those marketed by nTopology [443], Hyperganic [444], or Rhino3d [445] can make the process of generating such geometries more efficient. Noting that impact vectors are usually non-centric, further challenges arise. Where there are enough unit cells, the response at various angles can be calculated based on orthotropic, strain-dependent properties, using standard elasticity tensor transformation [423, 446, 447]. So, response to off-axis impacts can be designed by tuning the out-of-plane properties. Where there are too few unit cells to homogenise the material properties, as is often the case for periodic structures, the off-axis response must be obtained by higher order material approximations [448], microstructurally faithful simulations [25], or experimentally [25, 32].

Advances in fabrication methods have allowed realisation of a wide range of unusual and potentially beneficial properties. Mechanical metamaterials with force–torque coupling (known herein as twist) have seen increasing interest, since their rational design was shown in 2017 [363]. Like (negative) Poisson's ratio, twist translates axial deformation to transverse deformation—increasing resistance to indentation [360]. So, twisting mechanical metamaterials may resist penetration by concentrated loads, without shortening the stress plateau under compression (as seen for auxetics—see section 5.1).

The development of these metamaterials could also facilitate more efficient analysis and design of lattices. The twisting response is not included in classical (Cauchy) continuum mechanics, but it is in micro-morphic continuum mechanics (where a uniform load causes internal strain gradients [448]). Eringen presented some special cases in micro-morphic theories [448]. These include micropolar—where the gradients occur by rotation of rigid unit cells (sand provides an intuitive example), and micro-stretch—where the gradients occur by unit cell volume change, without shape change (picture the bronchi). Each of these allows some simplification (over the micro-morphic continuum)—reducing the amount of information required to approximate the response of a metamaterial—but may also cause some loss in precision.

Clearly, in the case of foams and lattices, which have relatively large internal features (when compared to the scale of external loads), micro-morphic continuum theories often apply [360, 449, 450]. Where classical continuum theories cannot be used, typical approaches to designing mechanical metamaterials for impact protection are experimental, or by using microstructurally faithful numerical models [17, 26, 28, 29, 415, 416, 419, 451], which are less efficient. So, developing and applying these micro-morphic continuum theories, including their viscoelastic and visco-plastic formulations, could facilitate more efficient mechanical metamaterial analysis, design, and application in single or multi-impact helmets [448].

Snap-through elements cause negative stiffness behaviour, corresponding to a drop in force as applied deformation increases [349–355]. Negative stiffness can be achieved by the buckling of an end constrained/preconditioned beam (figure 6). The beam snaps from one state of equilibrium to the next following the application of a perpendicular load (often via a connecting rib) [349, 350, 355]. The negative stiffness region is present for a segment of the force vs. compression relationship, corresponding to when the beam snaps through (figure 6(E)). Increasing the diagonal angle of the symmetric beam (ϕ, figure 6(B)) increases the onset, and length, of the region [353]. Making the beam less slender increases the critical buckling load and hence the magnitude of both positive and negative stiffness [353]. Negative stiffness has been shown to improve protection during impacts [452], balancing the positive stiffness of neighbouring unit cells to flatten and elongate the stress plateau (figure 2). Designing and manufacturing relatively unstable negative stiffness inclusions within helmets could bring added complexity, and further work applying these concepts to helmets is needed.

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Precise control over topology allows the design of a desired response to various loading conditions. An efficient approach to topology optimisation is to optimise a unit cell, and homogenise (expand to an effective bulk material) using a set of boundary conditions (i.e. periodic symmetry [451, 453–458]). Readers interested in homogenisation theory could refer to refs [459–461]. Such an approach is not widely applied during sporting impacts, causing high strains and strain rates, meaning degrees of freedom surrounding unit cell boundaries are influential and challenging to predict. For example, end loading for buckling beams, such as cell ribs, can vary with neighbouring cell deformation, meaning a multi-cell optimisation is needed [462–464]. Instead, whole metamaterial samples are often optimised or iteratively improved [17, 22, 26, 32]. Developing and applying micro-morphic theories [448] to lattices under impact may facilitate the prediction of rib constraints, and efficient (unit-cell) topology optimisation [465]. Open data approaches (such as the meta-genome [466]) could help develop these new homogenisation methods.

While shear thickening polymers can adapt their stiffness to various collision types, more degrees of design freedom, such as deformation mode switching, are possible. Deformation mode switching can be achieved by including an adaptive material, such as a viscoelastic one, that activates a topological instability [14]. For example, a beam's buckling mode (e.g., direction) can be designed to switch at a given strain rate. When two laterally connected beams of different stiffness (i.e. bi-beams) are axially compressed (figure 7(A)), the stiffer one drives the buckling direction while the other provides support, causing buckling towards the stiffer side. So, in figure 7(A), the deformation direction will switch at the strain rate when the viscoelastic beam becomes stiffer than the hyperelastic one. Bi-beams positioned like those in figure 7(A) will buckle away from each other at low compression rates, and towards each other, causing stiffening via self-contact, at high compression rates. Negative effective viscoelasticity can also be achieved with such a system of bi-beams [365], if the order in figure 7(A) is reversed, so bi-beams effectively soften by buckling away from each other at high compression rates. Obtaining negative effective viscoelasticity with highly viscoelastic materials demonstrates the level of control available via topology that could be useful when designing helmets. The response of these bi-beams, while previously shown to be retained for off-axis deformation angles of ∼10° [365], is unknown during oblique impacts. Flexing of the beams may provide a desirable low shear modulus, as in similar tests of long-cell anisotropic foam liners [20, 289], and should be studied further.

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More stable adaptive metamaterial systems can also be designed. The dominant deformation mode in hexagonal honeycombs and lattices is cell rib flexure [429, 447] (figure 7(B-i)). Such flexure reduces the distance to the neighbouring junction, respectively reducing and increasing the magnitude of positive and negative compressive Poisson's ratios. Placing viscoelastic material in the cell ribs could switch the dominant mode, increasing the magnitude of positive Poisson's ratio, and hardness [368], during more severe impacts (figure 7(B-ii)). Conversely, placing viscoelastic material in the junction of auxetic, re-entrant honeycombs or lattices could have a similar effect, by amplifying the dominance of rib flexure to draw neighbouring junctions inward. These concepts have been demonstrated using dual materials of different stiffness, but not viscoelastic ones, and present notable options for future research, particularly related to applications in helmets [467, 468]. Interestingly, such switching mechanisms and changes to Poisson's ratio can be achieved with one material, based on local changes in strain rate and stiffness; shifting the point of deformation [469–471].

The use of embedded electronics as active adaptive mechanisms is emerging [14, 472, 473]. These include piezoelectric inclusions, which stiffen when an electrical field is applied [472], or embedded electromagnets [352]. Such systems allow a controlled response, with the potential for embedded electronics to sense environmental changes, like impact severity or strain rate, fall initiation, temperature or relative humidity, and micro-controllers to define a programmed response or adaption [14, 472, 473]. Challenges to application are associated with the manufacture of sufficiently small and robust components [14, 352].