Auxetic materials and structures in the automotive industry: Applications and insights

Abstract

Auxetic materials have gained prominence due to their ability to absorb and dissipate energy, making them valuable for enhancing safety and performance in automotive components and crash protection systems. This article aims to provide an overview of the current and potential applications of auxetic materials and structures in the automotive industry. The exploration began with an elucidation of concepts and the characteristic cellular structure of auxetic materials. It then delved into the common forms of auxetic materials, encompassing auxetic foams, auxetic composites, and auxetic textiles. Subsequently, specific applications of auxetic structural materials in the automotive realm were enumerated, such as non-pneumatic tyres, crash boxes, front bumper systems, jounce bumpers, and vehicle doors. Finally, a detailed discussion addresses the challenges and future potential of auxetic textiles in the automotive industry, with a focus on innovations like auxetic seat belt webbings, airbag fabrics, and car seat covers. The findings aim to inspire further research and development in this field.

Graphical Abstract

Keywords

Automotive industry auxetic materials and structures auxetic textiles negative Poisson’s ratio

Introduction

The Poisson’s ratio ( $v$ ) is a parameter that defines the ratio between the transverse strain ( $ε_{t}$ ) and the longitudinal strain ( $ε_{l}$ ) of the material in the loading direction.¹ Most normal materials exhibit a positive Poisson’s ratio. However, materials with negative Poisson’s ratio (NPR) also exist. They are also known as ‘auxetic materials’. Under axial loading, they become thin under compression and fat under tension, instead of becoming fat under compression and thin under tension as normal materials (Figure 1). Origin-wise, auxetic materials fall into two main categories: natural auxetic materials and artificial auxetic materials. Natural auxetic materials comprise mineral crystals like arsenic single crystal, pyrite, and natural phosphorus, as well as biomaterials such as human cancellous bone, cork, cat skin, and cow nipple. Artificial auxetic materials are very diverse and can be divided into two main groups: artificial synthetic polymers and artificial structure design. In which the design of artificial structures, including auxetic foams, auxetic composites, and auxetic textiles, constitutes a significant proportion. These materials are formed from cellular structures such as re-entrant honeycomb structures, chiral structures, rotating structures, and so on.^2–11

Figure 1.

The behaviours under direct load: (a) normal material; (b) auxetic material.²⁴

The exploration of cellular auxetic structures began with Gibson and colleagues in 1982, who studied a silicon rubber re-entrant honeycomb structure.^12,13 Since then, the concave re-entrant honeycomb structure, serving as a representative model, has been extensively researched through a combination of numerical simulations, theoretical analyses, and experiments to understand its deformation behaviours and properties. The first is research on Young’s modulus (E) and Poisson’s ratio.¹⁴ Poisson’s ratio is calculated according to equation (1). Young’s modulus can be calculated based on the stress–strain curve (Figure 2)^15,16 from the stresses and strains at two points on the line OA, such as M and N (equation (2)). These are two important mechanical parameters of the elastic materials because from them, it is possible to calculate shear modulus G (equation (3)) and bulk modulus K (equation (4)). Subsequent research includes dynamic mechanical behaviours,¹⁷ impact stress,¹⁸ deformation modes under impact, and the dynamics of explosive loads.^19–22 These studies are also based on the stress–strain curve to determine the TDE, TPM, ADF, and PDF indexes.²³ The TDE refers to the total deformation energy generated in plastic region (equation (5)), where F(ε) represents the direct force and ε denotes the sample deformation. The TPM is determined by dividing the TDE by the mass of the sample (m) under test (equation (6)). The ADF is determined as the average direct force over the deformation length (equation (7)). And, the PDF, also known as the peak direct force, represents the highest recorded value of the direct force throughout the test. These results deepen our understanding of the impact resistance and efficient energy absorption capabilities of auxetic materials and structures, highlighting their immense potential across various applications such as aerospace, military, and notably, automotive. According to statistics from the WHO in 2023, around 1.19 million deaths occur annually as a result of road accidents, which is alarming. Therefore, the use of auxetic materials and structures in automotive components and crash protection systems is essential to enhance occupant safety.

v = - \frac{ε_{t}}{ε_{l}}

(1)

E = \frac{σ_{N} - σ_{M}}{ε_{N} - ε_{M}}

(2)

G = \frac{E}{2 (v + 1)}

(3)

K = \frac{E}{3 (1 - 2 v)}

(4)

T D E = \int_{0}^{ε} F_{(ε)} d_{ε}

(5)

T P M = \frac{D T E}{m}

(6)

A D F = \frac{1}{ε} \int_{0}^{ε} F_{(ε)} d_{ε}

(7)

Figure 2.

Stress–strain curve of the elastic materials.

This article offers a comprehensive overview of auxetic materials and structures, emphasizing their applications within the automotive industry. The discussion is structured into fine key sections: An introduction to the mechanical designs of typical auxetic cellular structures, including re-entrant honeycomb structures, chiral structures, and rotating structures; an exploration of common forms of auxetic materials and structures, such as auxetic foams, auxetic composites, and auxetic textiles; a presentation of typical automotive industrial applications of auxetic materials and structures, encompassing components like non-pneumatic tyres, crash boxes, front bumper systems, jounce bumpers, and vehicle doors. These components are manufactured in foams and composites form and play critical roles in energy absorption and enhancing occupant safety during various types of collisions; finally, identification of technical challenges and potential applications of auxetic textile structure in the automotive industry, with special attention given to areas like auxetic seatbelt webbings, auxetic car seat covers, and auxetic airbag fabrics production, deemed crucial focal points for future research endeavours.

Cellular structures

Cellular structures such as re-entrant honeycomb, chiral, and rotating models serve as the fundamental building blocks for the production of common forms of auxetic materials and structures. The following section explores the most prevalent types of cellular auxetic materials and structures in detail.

Re-entrant honeycomb structure

The introduction of the 2D re-entrant honeycomb structure (Figure 3) by Gibson et al.¹³ in 1982 marked a significant breakthrough in cellular auxetic design. Subsequently, Masters et al. published a theoretical model to predict elastic constants by considering honeycomb cell deformation through stretching, flexure, and hinging.^25,26 When a tensile load is applied along the axial direction, the diagonal ribs rotate vertically, causing the structure to expand laterally and achieve the NPR effect. In 2007, the theory proposed by Gibson was again proved by Bezazi et al., who employed both analytical methods and finite element based-methods to determine the in-plane Poisson’s ratio and Young’s modulus of a novel centre-symmetric honeycomb structure subjected to single-axis loading.²⁷ Later, in 2017, Hu et al. derived equations to compute the Poisson’s ratio and Young’s modulus for a 2D NPR honeycomb structure, presented as equations (8) and (9).²⁸ Additionally, Larsen et al.²⁹ explored compliant micro-mechanisms and structures with NPR using numerical topology optimization. This approach enabled engineers to specify elastic properties and achieve optimal structures, as evidenced by an optimized 2D re-entrant triangular model of Saxena et al.³⁰

Figure 3.

Re-entrant honeycomb structures: (a) The unit cell²⁵; (b) in resting state; and (c) in expanding state.²⁶

The transition from 2D to 3D re-entrant honeycomb structures was investigated. Schwerdtfeger et al.³¹ crafted a 3D structure featuring a hexagonal super-lattice pattern, demonstrating NPR in various directions. Evans et al.³² reported the insights into the auxetic behaviour of three-dimensional open-celled foams by the finite element model. Later, Yang et al.³³ combined finite element and experimental analysis demonstrated that the modulus, Poisson’s ratios and yield strength of the 3D re-entrant cellular structure control via geometrical designs. Hengsbach et al.³⁴ also introduced a new method to produce the 3D re-entrant structure (Figure 4), which is direct laser writing. Wang et al.³⁵ used the finite element models of this structure to develop the cylindrical NPR structure, with the purpose of application in damping devices. Li et al.³⁶ introduced a versatile 3D augmented re-entrant cellular structure, allowing for a wide adjustment of the Poisson’s ratio from negative to positive. Harkati et al.³⁷ presented a multi-entrant NPR honeycomb structure, conducting a parametric investigation into the structure’s behaviour with variable stiffness and Poisson’s ratio effects. In addition, the enhancement of 3D auxetic lattice structure has been studied by integrating narrow struts into a 3D re-entrant lattice structure (Figure 5). The outcomes of these researches present promising possibilities for applications requiring auxetic materials with both negative Poisson’s ratio and high stiffness.^38–41

v_{21} = \frac{\sin θ (\frac{h}{l} + \sin θ)}{\cos^{2} θ}

(8)

E_{1} = \frac{E_{s} {(\frac{t}{l})}^{3} (\frac{h}{l} + \sin θ)}{\cos^{3} θ}

(9)

Figure 4.

The 3D re-entrant model: (a) general structure; (b) zoomed-in image.³⁴

Figure 5.

The 3D unit cell structure: (a) re-entrant; (b) enhanced.³⁸

Chiral structure

The chiral NPR structure comprises a core node and the spokes. When apply force on the structure, the winding and unwinding of the spokes around the core node, each cell structure undergoes a torsional effect, leading to either contraction or extension of the entire structure, thus producing the auxetic effect.⁴² The inaugural chiral structure, as illustrated in Figure 6, was introduced by Prall and Lakes.⁴³ Their study showed that the in-plane Poisson’s ratio (v) was −1, and the in-plane Young’s modulus (E) is calculated according to equation (10), with t as thickness and L as the length of the ligament, while r is the radius of the ring, and Es is the intrinsic Young’s modulus of the cell walls. This seminal research served as a catalyst for subsequent explorations in this domain. Alderson and collaborators⁴⁴ delved into the study of various chiral structures, including three-ligament chiral structures, three-ligament anti-chiral structures, four-ligament anti-chiral structures, and six-ligament anti-chiral structures (Figure 7). They established the connection among elastic modulus, Poisson’s ratio, and the geometric parameters of these structures based on their elastic properties. In 2008, Grima et al.⁴⁵ conducted a new chiral auxetic structures, termed meta-tetrachiral, constructed using chiral building blocks and illustrated in Figure 8. In another study about anti-tetrachiral system, Gatt et al.⁴⁶ demonstrated that the rigidity of a structure can be modified without affecting the Poisson’s ratio, with these changes being influenced by the geometry and material properties of the structure. Subsequently, Gatt et al.⁴⁷ explored the connection mode between core node and spokes in anti-tetrachiral structures using finite element methods. Additionally, Dilek and Buket recently conducted a study on curved composite sandwich materials with cores designed in hexachiral, tetrachiral, and anti-tetrachiral configurations. The results revealed that the impact energy absorption capabilities of the hexachiral structure are superior.⁴⁸

Figure 6.

Hexachiral structure with the node radius r, the cell wall thickness t, and the length straight ligaments L.⁴³

Figure 7.

Chiral structure: (a) tri-chiral; (b) anti-tri-chiral; (c) tetra-chiral; and (d) anti-tetra-chiral.⁴⁴

Figure 8.

Meta-tetrachiral systems with: (a) six spokes in node; (b) four spokes in node; (c) three spokes in node⁴⁵.

In addition to 2D chiral models, 3D models have also been studied. For instance, Ha et al.⁴⁹ proposed three-dimensional chiral lattices with an NPR (Figure 9). These chiral lattices, featuring numerous cubical nodules, underwent finite element analysis, revealing a Poisson’s ratio ranging from negative to zero, dependent on specific geometry. In a recent development, Huang et al.⁵⁰ presented a three-dimensional anti-tetrachiral model that exhibiting NPR behaviour. The research indicated a strong correlation between the Young’s modulus, Poisson’s ratio, and the structure’s geometry.

E = E_{s} {(\frac{L}{r})}^{2} {(\frac{t}{L})}^{3}

(10)

Figure 9.

The chiral lattice configuration.⁴⁹

Rotating structure

Another typical cellular structure of auxetic materials and structures is the rotating structure. Auxetic rotating structures are constructed using models that incorporate both stiff and partially stiff (rigid and semi-rigid) design, where these units are connected at specific vertices. Upon the application of a tensile load, torque is produced in the stiff units, causing them to rotate and lateral movement in the partially stiff hinges, resulting in the extension of the structure and obtaining the NPR effect.^26,51 Grima et al.⁵² introduced a configuration of rigid squares (Figure 10). The Poisson’s ratio (v) and Young’s modulus (E) of the model can be expressed using equations (11) and (12), respectively. Subsequently, the author and collaborators have also proposed rotating auxetic models of triangles (Figure 11), rectangles (Figure 12), rhombi, and parallelograms.^53–55 Extending their exploration, Grima et al.⁵⁶ introduced a semi-rigid squares model. The result showed in significant alterations to the mechanical properties of the model.

v_{12} = v_{21} = - 1

(11)

E_{1} = E_{2} = K \frac{8}{l^{2}} . \frac{1}{(1 - \sin θ)}

(12)

Figure 10.

Rotating square model⁵².

Figure 11.

Rotating triangle model model.⁵³

Figure 12.

The rotating rectangles model.⁵⁴

Research conducted by Alderson, Rafsanjani, Dudek, and Kim, has also contributed to the growth of rotating models. Alderson et al.,⁵⁷ explored NPR performance in the α-cristobalite model (Figure 13). They considered rotation, maintenance of shape with size change, and concurrent rotation and dilation, all contributing to auxetic responses. Drawing inspiration from historical geometric patterns, Rafsanjani et al⁵⁸ introduced a new planar mechanical metamaterial (Figure 14) that can adjust its expandability, stiffness, and bistability by manipulating the geometry of the fundamental cut profiles. Using a dynamics methodology, Dudek KK et al.⁵⁹ investigated the deformation behaviour of NPR hierarchical rotating square models. Research demonstrates that by manipulating the resistance to rotational movement of hinges within the system, the mechanical properties, deformation behaviour, and configuration of the structure for a specified applied force can be controlled (Figure 15). This allows for increased tunability without the need to alter the geometry of the system. Kim J et al.⁶⁰ proposed 3D models created with regular polygonal prisms include triangular, square and hexagonal (Figure 16). The study formulated Poisson’s ratio equations for each model and demonstrated, through analysis and numerical methods, that the models display NPR characteristics under specific geometrical specifications.

Figure 13.

The α-cristobalite rotating tetrahedral model.⁵⁷

Figure 14.

Ancient geometric motifs and bistable auxetic mechanical metamaterials.⁵⁸

Figure 15.

The two-level hierarchical auxetic model.⁵⁹

Figure 16.

3D rotating structure.⁶⁰

Common forms

Auxetic materials are produced in three main forms: auxetic foam, auxetic composite, and auxetic textiles. Auxetic foam is produced using a specialized cellular network structure, while auxetic composites combine various cellular auxetic materials. Both forms are used in the automotive industry for applications in anti-collision and sound insulation. Auxetic textiles, which include fibres, yarns, woven, knitted, and nonwoven fabrics, are crucial for creating highly flexible and adaptable auxetic materials. These auxetic textiles hold potential applications in the automotive industry.

Auxetic foams

In 1987 Lakes⁷ successfully fabricated auxetic foam. The study produced a polyurethane foam with an NPR of approximately −0.7, through a thermo-mechanical methodology. Lakes’ success paved the way for the birth and rapid development of many new design methods. We can mention the foam materials theory of Gibson and Ashby⁶¹ Foam materials theory helps predict and understand the basic mechanical properties of foam materials based on their underlying structure, thereby optimising the design and use of foam materials in a variety of applications.

The research of Caddock and Evans^81,82 presented the morphology and mechanical behaviour of a microporous material made from extended poly-tetrafluoroethylene foam that, when specially treated, has auxiliary properties, with Poisson’s ratio that can reach −12. At the same time, the study has proposed a theoretical model for this structure founded on a linked network of anisotropic particles. This is an excellent explanation for the connection between the difference of Poisson’s ratio with tensile strain, and changes in material morphology. Subsequent typical studies have also successfully converted polyvinyl chloride foam,⁸³ silicone rubber foam,⁸⁴ closed-cell polyurethane foam,⁸⁵ and copper foam^86,87 into auxetic foam. Besides, scientists have further improved the preparation process as well as providing the planar and volumetric configurations of auxetic foams of many new auxiliary materials (Table 1), improving the performance of auxetic polymer and foam materials.^{45,62,63,70,88–91} This is an important premise for upcoming studies in developing new auxetic structures.

Table 1.

The planar and volumetric configurations of auxetic foams.^62,63

No.	Category model	Cite
1	Re-entrant hexagonal honeycomb	^7,64–68
2	Missing rib	⁶⁹
3	Hexagonal based missing rib	^70,71
4	Rigid triangle	^53,72–74
5	Triangles-honeycomb lattice	⁷⁵
6	Random cellular solid	⁷⁶
7	Idealized re-entrant	⁷⁷
8	Kelvin minimum area tetrakaidecahedron	^32,78,79
9	Dodecahedron	⁸⁰

Auxetic composites

Similar to composite materials, auxetic composites are crafted to blend the benefits of auxetic materials with other substances, aiming to enhance mechanical properties, bolster impact resistance, trim weight, and widen application versatility. In 1984, Herakovich⁹² conducted pioneering research on composite laminates. The study suggested that Poisson’s ratios could be positive or negative based on fibre arrangement. Further studies focus on research to investigate and enhance the properties of auxetic composites, such as investigate the viscoelastic properties,⁹³ examine the elastic properties,⁹⁴ eliminate delamination and brittleness,⁹⁵ improve the Young’s modulus,⁹⁶ change in the conductivity,⁹⁷ restore fractured and cracks,⁹⁸ and enhance energy absorption, tensile strength, and compression properties.⁹⁹

In addition, research on applying new methods to fabricate auxetic composite materials has also been conducted. Tatlier et al.¹⁰⁰ introduced a modelling approach to explore the NPR properties of compressed fused fibrous networks, highlighting the critical influence of compression and anisotropy on auxetic behaviour in such materials. Valentini et al.¹⁰¹ utilized carbon dioxide gas produced during yeast fermentation and the gelation process of the liquid rubber matrix to create durable auxetic rubbers. This composite material can alter its Poisson’s ratio from negative to positive under tensile stress. Boldrin et al.,¹⁷ employing comprehensive techniques such as full-scale Finite Elements, Component Mode Synthesis sub-structuring, 3D printing, and scanning laser vibrometry, investigated the dynamic behaviour of auxetic gradient composite hexagonal honeycombs. Cristiano Veloso et al. developed a MATLAB algorithm tailored for designing auxetic laminates in both in-plane and through-the-thickness configurations. This algorithm is based on the micro and macro-mechanic relationships derived from Maxwell’s method for lamina and Kirchhoff’s plate theory. The results were compared with benchmark cases from the literature, demonstrated high accuracy and consistency.¹⁰²

Besides, new auxetic composite materials are also created. Zorzetto et al.¹⁰³ created a novel composite material by integrating two fundamental cellular structures with distinct Poisson’s ratios. The results revealed that even a small fraction of re-entrant components could significantly enhance stiffness at a consistent overall relative density. Bubert et al.¹⁰⁴ developed a passive one-dimensional morphing aircraft skin, incorporating an elastomer fibre-composite surface layer advocated by a flexible honeycomb structure, both demonstrating near-zero in-plane Poisson’s ratios. In the study by Chen et al.,¹⁰⁵ a flexible skin made of fibre-reinforced composite with in-plane auxetic behaviour was fabricated and examined. Freshly, Hu et al.⁹⁸ produced innovative hydrogel-elastomer NPR composites, observing that hydrogel integration could provisionally restore fractured ligaments and cracks in the composite. Silva et al.¹⁰⁶ ingeniously employed recycled rubber from discarded tyres to craft re-entrant auxetic structures and performed initial examinations. This innovative approach enhances the toughness and mechanical strength of polymeric composites when incorporating waste tyres, aligning with the ongoing trend of sustainable development.

Auxetic textiles

Auxetic fibres

Auxetic fibres are a type of polymeric material known for their remarkable mechanical properties, which surpass those of other polymers in both strength and flexibility.^28,107 These fibres are particularly well-suited for use in fibre-reinforced composites, making them an excellent material choice for various applications.^108–114 In 2002, Alderson and his colleague developed polypropylene (PP) fibres that showcased auxetic behaviour by utilizing a modified melt spinning technique. The NPR obtained in their experiment was −0.6. ± 0.05.¹¹⁵ To achieve large-scale production and ensure consistent quality, optimizing production parameters is crucial. Therefore, Alderson et al. have been producing auxetic PP fibres using large-scale extruders under different conditions, including extrusion temperature, screw speed, feed rate, and cooling conditions. The aim is to study the effects of these processing parameters on the fibres.¹¹⁶ Then, Alderson’s research team¹¹⁷ successfully achieved the first industrial-scale production of auxetic PP fibres using an extruder. The study demonstrated that the produced fibres exhibited excellent reproducibility and auxetic behaviour across an expanded strain range of up to 5%, exceeding the limits formerly documented for lab-scale extrusions performed at reduced extruder thermal conditions. The highest observed negative Poisson’s ratio was −0.86. Moreover, the optimized manufacturing technique for polypropylene fibres can be adapted to other polymer-based fibres, such as polyester and polyamide, with the potential to exhibit NPR behaviour.^118,119

Auxetic yarns

Auxetic yarns can be created using a special yarn structure with non-auxetic fibres. The first structure of an auxetic multifilament construction was proposed by Hook and Evans,¹²⁰ which was a helical auxetic yarn (HAY) structure. The construction consists of a thin, stiff yarn that is helically wrapped around a thicker, soft-core yarn. When the HAY is elongated along its length, the stiff wrapping yarn unbends, which causes the soft-core yarn to coil helically encircling it,^121–123 as illustrated in Figure 17. Then, Miller et al.¹¹¹ developed a new structure called Double Helix Auxetic Yarns (DHY), which resembles the structure of HAY. This yarn is created by combining two types of yarn: a stiffer and thinner wrap yarn alongside a thicker, straightforward elastomeric core yarn. When the DHYs are stretched, the core-yarn’s elastic property causes it to expand laterally. Once the stretching force is removed, the DHY returns to its original helix shape, as depicted in Figure 18. Miller and co-worker¹²⁴ also created the DHY with the same structure by employing carbon yarn as the outer layer and a stretched nylon yarn as the central strand. In another study conducted by Ullah et al.,¹²⁵ multifilament yarns made of nylon and polypropylene were used as the core while wrapping yarns made of Dyneema and Kevlar were adopted. Numerous studies on HAY have shown that a higher core-to-wrap diameter ratio, a greater modulus ratio between the elements, and a smaller initial wrapping angle of the wrap yarn lead to a greater maximum NPR.^{122,125–130} Additionally, Bhattacharya et al.¹²⁶ discovered that the modulus ratio of the elements must be sufficiently high to produce an NPR behaviour, yet sufficiently low to prevent the occurrence of the indentation issue. Their findings indicate that indentation reduces the maximum negative Poisson’s ratio of HAYs. To address this issue, they recommended using a multifilament wrap element rather than a monofilament wrap element because of their differing deformation mechanisms. These parameters can be modified based on the intended application to optimize the NPR performance of HAY.

Figure 17.

HAY model: (a) under no strain and (b) under peak strain.⁶²

Figure 18.

DHY model: (a) Under no strain and (b) under peak strain.¹¹¹

A significant challenge for HAY and DHY lies in ensuring a uniform wrap angle throughout both the manufacturing process and their usage. This problem arises because of the tendency of the wrap ply to easily slip over the core ply, as shown in Figure 19, which stems from the management of tension utilized on both plies.^124,131,132 To address these shortcomings, Lim has developed a new type of yarn, known as semi-auxetic yarn (SAY), which is a modified version of the auxetic helical yarn, as outlined in Figure 20. This was achieved by stitching a thin cord in a triangular pattern through an elastic fat cord. The results of the experiment showed that the Poisson’s ratio of the yarn was affected by two factors: the first half angle of the thin cord and the Poisson’s ratio of the thick cord.¹³⁰ At the same time, Zhang et al¹³³ developed a three-part NPR yarn. This yarn comprised a rigid wrap yarn as the first part, helically coiled around an elastomeric core yarn as the second part, and enclosed by a sheath as the third part, as presented in Figure 21. The thickness of the coating on the third component has an impact on the auxetic behaviour of this yarn type. This could be utilized as a fresh configuration parameter to adjust both the Poisson’s ratio and modulus of this innovative composite reinforcement for various applications. However, the structure of the yarn may be limited by the silicone sheath. Moreover, the auxetic behaviour reduces as the sheath coating thickness increases. An alternative form of NPR yarn is the helical auxetic plied yarn (APY), introduced by Ge et al.¹³⁴ The APY was created using a matching, uniform number of soft and stiff yarns, arranged in a specific configuration. A common example of this is a four-ply yarn structure, comprising two soft yarns and two stiff yarns with varying diameters and modulus, designed to achieve NPR behaviour, as indicated in Figure 22. In 2022, Razbin et al.¹³⁵ presented an innovative design for producing auxetic yarn, referred to as the “double-core helical auxetic yarn.” This design is an adaptation of HAY and SAY structures, utilizing braiding technology. It comprises two identical soft yarns as core parts and a stiff yarn as the wrapping part, as observed in Figure 23.

Figure 19.

Poor conformance between the wrap and the core using conventional spinning technology.⁶²

Figure 20.

Three-component auxetic structure.¹³⁰

Figure 21.

Schematic illustrations of the side and top view of a semi-auxetic yarn: (a) no elongation, (b) with slight elongation, (c) with moderate elongation, and (d) with extensive elongation.¹³⁰

Figure 22.

4-Ply NPR yarn.¹³⁴

Figure 23.

Double-core helical NPR yarn.¹³⁵

Auxetic fabrics

Auxetic woven fabrics (AWFs)

Auxetic fabrics can be created using specially designed auxetic yarns, which are woven in a conventional pattern. These fabrics have unique mechanical properties, which make them suitable for filtration or energy-absorbing applications.⁶² In a study conducted by Sum Ng. and Hu in 2018, auxetic plied yarns were utilized to create auxetic woven fabrics. The research involved the incorporation of four-ply auxetic yarns into various woven fabrics with distinct design parameters in investigating their NPR behaviour, as depicted in Figure 24.¹³⁶ In 2020, Gao et al.¹³⁷ conducted a study to investigate the prospective applications of auxetic textiles. They produced HAY and AWFs to investigate the impact of yarn and fabric parameters on the auxetic properties of the engineered fabrics. The HAYs were made by wrapping a conventional polyamide multifilament ply over a highly elastic polyurethane multifilament ply, with different parameters used to optimize the auxeticity and structural stability. The HAY was then selected as the warp and weft yarns for weaving the plain-woven fabric. AWFs structure and Surface of AWFs after the impact test are shown in Figure 25.

Figure 24.

Images of woven NPR fabric created from plied auxetic yarns.¹³⁶

Figure 25.

AWFs configuration and it´s surface after impact test.¹³⁷

Later, in 2022, Gao and Chen¹³⁸ developed woven fabrics using HAY and conducted a study on the key parameters affecting Poisson’s ratio while stretched. For the auxetic yarn, a highly elastic polyurethane multifilament was used as the core, wrapped in conventional nylon 6,6. Different wrapping angles of HAY were chosen for the weft, while nylon 6,6 was used as the binder for the HAY in the warp during the weaving of a plain-woven fabric. The findings of this study indicated that increased float lengths in weave structures, along with smaller wrapping angles and reduced diameters of the auxetic yarn, and decreased tensile stiffness of the warp yarn resulted in improved NPR behaviour. This research provides valuable insights for future studies aimed at optimizing the factors for fabricating NPR woven fabrics.

Auxetic woven fabrics can also be created by blending traditional yarns with auxetic geometry. Researchers have developed bi-stretch auxetic woven fabric using foldable and re-entrant hexagonal geometry. Until now, the manufactured AWFs using these geometries have found applications in clothing where improved shape fit and comfort are desired.⁶² Specifically, Cao et al. (2019) introduced a novel category of bi-stretch woven fabrics with NPR properties, utilizing traditional elastic and non-elastic yarns and standard looms. Bi-stretch auxetic woven fabrics were initially devised using a foldable geometry, as presented in Figure 26.¹³⁹ Subsequently, Kamrul et al.¹⁴⁰ also introduced an AWFs derived from a double-directional parallel in-phase zig-zag foldable geometry, as shown in Figure 27. Additionally, Zulifqar and Hu^141,142 suggested a different geometric model for auxetic woven fabrics derived from re-entrant hexagonal geometry (AWF-REH) and formed a relationship linking Poisson’s ratio and the tensile deformation exerted on the fabrics. This structure was initially created using re-entrant hexagonal geometry, blending loose and tight weaves, as seen in Figure 28. Following this, the AWF-REH was produced using both non-elastic and elastic yarns on a rapier weaving machine, depicted in Figure 29.

Figure 26.

Bi-stretch NPR woven with combination of plain weave and: (a) Satin 4/1; (b) Twill 3/1; and (c) Twill 2/2.¹³⁹

Figure 27.

The unit cell of Bi-stretch NPR woven with foldable geometry: (a) the diagram, and (b) the the outlines.¹⁴⁰

Figure 28.

Assembly of re-entrant hexagonal geometry: (a) weave in part A; (b) weave in part B; (c) weave in part C; (d) distribution of loose and tight weaves.¹⁴¹

Figure 29.

A bi-stretch AWFs designed with REH geometry: (a) the fabric surface, (b) the hexagonal unit cell, and (c) the basic unit of the fabric.¹⁴²

Auxetic knitted fabrics (AKFs)

The folded knitted NPR fabrics are designed on the basis of an origami structure, achieved through the combination of the face and back loops within the fabric’s structural configuration, as illustrated in Figure 30. Even though the unique knitting pattern starts off as flat due to weft-knitting technology, the fabric is inclined to curl and adopt a three-dimensional shape after the knitting process. This occurs because of the structural imbalance between the face and reverse loops. When the fabric is stretched in a single axis, the folded structure unfolds and may extend in the perpendicular direction, creating an in-plane NPR behaviour, as demonstrated in Figure 31. Additionally, researchers have created various alternative forms of weft-knitted NPR fabrics, such as those with the rhombus-shaped grid re-entrant,¹⁴³ re-entrant hexagon,¹⁴⁴ and rotating and double-headed arrow structures.¹⁴⁵

Figure 30.

Origami geometrical structure.¹⁴⁶

Figure 31.

Auxetic weft-knitted fabric in foldable structure with different phases: (a) no tension, (b) under tension, (c) completely extended.¹⁴⁶

Along with weft-knitted, warp-knitted designs may likewise be employed to create auxetic structures. Wang et al. (2014, 2017) developed 3D warp-knitted fabrics using spacer structures, as illustrated in Figure 32. These garments demonstrated NPR behaviour across all planes and the thickness dimension. However, the highest NPR effect was observed when the fabric was stretched in the weft orientation, as demonstrated in Figure 33. Furthermore, these fabrics exhibited exceptional moldability and compressive properties, rendering them particularly well-suited for use in protective clothing for humans.^147,148

Figure 32.

Auxetic warp-knitted spacer fabric: (a) the muti-layer fabric, (b) structure of face layers (b), (c) structure of unit cell.¹⁴⁷

Figure 33.

Comparison between conventional and auxetic spacer fabrics.¹⁴⁷

Auxetic textile composites

Auxetic textile preforms, which include auxetic fibres, yarns, and fabrics, may serve to create NPR textile composites with a range of different properties. To enhance the auxetic effect and mechanical properties in both AWFs and AKFs for use in industrial sectors, researchers have proposed using advanced-performance yarns, optimizing the production process of AWFs and AKFs, and utilizing these AWFs and AKFs for composite fabrication. In this section, we will also introduce auxetic textile composites composed of AWFs and AKFs. Miller et al. (2009) were the first to create a composite laminate with an NPR using auxetic woven fabrics constructed from double helical yarns and a silicone rubber gel matrix. In their study, the wrapped yarn was Ultrahigh Molecular Weight Polyethylene, while the core yarn was made of polyurethane. The designs were constructed using a plain weave, with the DHY used for the weft and meta-aramid yarn used for the warp.¹¹¹ Subsequently, in 2012, Miller et al. developed NPR composite laminates by saturating AWFs made from DHY with polyester resin. To create the DHY, they used a core made of monofilament nylon yarn wrapped in a low tow count carbon yarn.¹²⁴ Liu (2019) used Miura-origami patterns featuring a unique arrangement of the face loops and back loops in a zig-zag form to fabricate auxetic knitted fabrics for impact-protective applications.¹⁴⁹ Three types of yarns, specifically aramid, polyester multifilament, and low-melting-point polyester multifilament yarns, were employed. Steffens et al. (2020) examined and analysed the effect behaviour based on the energy approach of weft-knitted configurations, including a jersey and an NPR composite using advanced-performance yarns such as para-aramid and hybrid designs. They effectively transferred the NPR from the reinforcing NPR textile to the composite, demonstrating enhanced total energy absorption under various impact loads compared to jersey-reinforced composites. Furthermore, incorporating high-stiffness filaments in the manufacture of the NPR weft-knitted garments significantly improved the NPR in the composite, outperforming hybrid materials.¹⁵⁰

Applications of auxetic materials and structures in the automotive industry

This section presents researched applications of auxetic materials in the automotive industry, namely non-pneumatic tyres, crash boxes, font bumper systems, jounce bumpers, and vehicle doors. The auxetic structure is integrated into the non-pneumatic tyre structure to provide greater flexibility and adaptability on different road surfaces. This not only enhances driving safety but also improves the driving experience by reducing vibrations and noise from the road. In the realm of crash safety, auxetic materials find their place in the construction of crash boxes and front bumper systems. These materials absorb impact energy more efficiently than conventional materials, mitigating collision damage and safeguarding vehicle occupants. Moreover, auxetic materials are utilized in jounce bumpers, contributing to smoother rides by effectively dampening shocks and vibrations, thus enhancing overall vehicle comfort and handling. Additionally, vehicle doors featuring auxetic materials benefit from increased resistance to impact and deformation, bolstering structural integrity and ensuring passenger safety in the event of a side collision.

Non-pneumatic tyre

Dunlop introduced the first commercial pneumatic bicycle tyre in 1888. While pneumatic tyres are lightweight, energy-efficient, and have low stiffness and contact pressure, they face issues like susceptibility to flats, maintenance needs, and complex manufacturing. As a result, engineers have developed non-pneumatic tyres (NPTs), which replace air with elastomers or polygon-like spokes that endure cyclic tension-compression loading.¹⁵¹ These spokes require high stiffness and elasticity, which conventional materials struggle to provide. However, materials displaying an NPR offers a promising solution to overcoming these challenges.

The first to mention is the study of Ju et al.¹⁵² The study employed finite element methods to design honeycomb spokes for NPTs as a replacement for air in traditional tyres. They tested six honeycomb configurations, including regular and auxetic types (Figure 34), finding that the proportion of the unit cell’s length to its height (l/X₁) significantly affected NPTs flexibility (Figure 35). Regarding the cell angle, it can be divided into two cases. In the case where honeycomb geometries have the identical unit cell wall thickness (t), the NPR spokes exhibited a force-deformation behaviour similar to conventional counterparts. An enhancement in the cell angle θ extent induced in-plane adaptability, leading to a reduced reaction force. Building on this foundation, Shankar P et al.¹⁵³ optimized the meso-structure of NPTs to meet specific design requirements. They developed a new S-type structure (Figure 36) and applied optimization algorithms such as particle swarm, genetic, and FAST–SIMPLEX. The results show that the optimised S-type satisfies the criteria of 10% shear flexure at a 10 MPa equivalent shear modulus with a wall thickness of 0.411 mm and a peak value of 0.608. Besides, the study also showed that FAST–SIMPLEX algorithm effectively identified the parametric scenarios that optimise the shear flexibility for a specified equivalent shear modulus across meso-structures.

Figure 34.

The regular (Types A, B, C) and auxetic (Types D, E, F) honeycomb spokes.¹⁵²

Figure 35.

The hexagonal honeycombs unit cell.¹⁵²

Figure 36.

Four meso-structure configurations.¹⁵³

Advancing this research, Yoo S. et al.¹⁵⁴ used a heat-mechanics model to assess heat increase in an auxetic with re-entrant hexagonal spokes (Figure 37). By combining deformation-related hysteresis with thermal transfer modelling, they successfully developed a numerical tool capable of evaluating both rolling energy dissipation and the resulting internal thermal production in a non-pneumatic tyre. Finally, Wu TY et al.¹⁵⁵ introduced a NPTs design incorporating a gradient anti-tetra-chiral structure, replacing traditional pneumatic components. Through theoretical and simulation analyses, they confirmed that this structure has a high load-carrying capacity and can support various engineering applications (Figure 38). Together, these studies progressively enhanced the structural, optimization, and thermal analysis aspects of NPTs, establishing a solid foundation for the practical design and application of non-pneumatic tyres.

Figure 37.

The NPTs with a re-entrant hexagonal spoke.¹⁵⁴

Figure 38.

The gradient anti-tetra-chiral structure model.¹⁵⁵

Crash box

The crash box is a key component in vehicle crashworthiness, significantly impacting energy absorption. While traditional optimization methods for crash box design face limitations due to restricted design space, recent studies have introduced innovative designs. These new designs leverage the unique deformation and energy absorption characteristics of materials with a negative Poisson’s ratio, improving the energy absorption performance of conventional crash boxes. In the research conducted by Zhou et al.,^156,157 a ground breaking NPR crash box was designed using a multi-objective genetic algorithm. This design integrates an NPR structure-filled core with a traditional crash box (Figure 39). A parameterized model was created for efficient optimization, validated against a traditional finite element model. Using the non-dominated sorting genetic algorithm-II, the team optimized the NPR cell structure, significantly improving crash box performance. The study demonstrates that combining parameterized modelling with multi-objective algorithms effectively enhances crashworthiness.

Figure 39.

FEM model: (a) the traditional crash box; (b) 3D NPR cell structure; and (c) the NPR crash box.¹⁵⁷

Other studies have also applied auxetic structures in crash box designs to improve crashworthiness. Wang et al.¹⁵⁸ developed a crash box with a 3D NPR core based on a hexagonal cellular structure. Simulations showed that this design enabled smooth, controllable deformation for effective energy absorption, optimized through a tailored algorithm. Similarly, Fengxiang Xu et al.¹⁵⁹ created an NPR honeycomb crash box inspired by horn structures, known for impact resistance. Their finite element analysis indicated that this biomimetic design, when compared to traditional shapes, significantly improved crashworthiness. Further advancing the field, Ziyu S. et al.¹⁶⁰ introduced a 3D double-arrow structure based on the hexagonal crystal system (Figure 40). Using experimental, theoretical, and numerical methods, they examined deformation, fracture, rebound, and energy absorption. The findings revealed that the hexagonal dual-arrow configuration featuring NPR showed increased stiffness and plateau strength, allowing for superior capacity to absorb energy during axial compressive loading, despite reduced deformation and plateau enhancement area. Moreover, comparison with conventional crash boxes via trolley collision experiments demonstrated the auxetic crash box’s exceptional ability to dissipate energy and notable enhancement in crashworthiness.

Figure 40.

The auxetic crash box model in the frontal bumper.¹⁶⁰

Font bumper system

Statistically, front collisions are the most common in automobile accidents.¹⁶¹ In these situations, the bumper system, positioned at the vehicle’s forefront, serves as the initial point of impact with an object to protect both the vehicle and pedestrians. Therefore, the bumper system needs to exhibit sufficient stiffness and strength to ensure it absorbs kinetic energy from the collision. However, meeting all these requirements simultaneously proves challenging for traditional bumper systems. Recognizing this limitation, Wang CY et al. studied an innovative bumper system comprising a auxetic bar and an auxetic absorber (Figure 41). The auxetic bar aims to enhance the vehicle’s frontal impact resistance, incorporating numerous inner hexagonal cellular structures arranged periodically. Simultaneously, the auxetic absorber, positioned between the bumper bar and the outer panel, is designed to safeguard pedestrians’ lower legs in collision accidents. Simulation results indicate that the overall efficiency of the auxetic bumper system, optimised through electronic search algorithms, has been notably enhanced, showcasing excellent performance in both pedestrian safety and vehicle impact resistance.^156,162,163

Figure 41.

The modelling of auxetic absorber: (a) unit cell, (b) blank, (c) thin shells; and The modelling of auxetic bar: (d) unit cell, (e) blank, (f) thin shells; (g) auxetic bumper; and (h) auxetic front bumper system model.¹⁶³

Jounce bumper

The jounce bumper (Figure 42(a)), initially designed to prevent full compression of suspension components during sudden impacts from heavy loads, potholes, curbs, or road obstacles, has become a key component of modern vehicle suspension systems. Typically made from hyper-elastic materials like rubber or polyurethane, these bumpers are essential for absorbing impact energy and improving noise, vibration, and harshness performance. They can also function as auxiliary compression springs when approaching the maximum load.^164,165 Achieving an optimal load-displacement curve is critical for effective suspension design. However, traditional jounce bumpers often fail to meet these requirements, prompting the exploration of materials with an NPR as a potential solution.

Figure 42.

(a) Traditional jounce bumper,¹⁶⁶ (b) NPR jounce bumper, and (c) Two-dimensional double-arrow auxetic structure¹⁶⁷.

Wang et al. published a series of studies on this problem. First, they used both numerical and experimental methods to fabricate NPR jounce bumpers with double-arrow structures (Figure 42(b) and (c))^166,167. Then, Wang et al. investigated how design features and material properties affect the load-deformation relationship of NPR jounce bumpers. They concluded that factors such as the number of cells and layers notably influence the displacement at which hardening occurs, with all factors impacting the structural rigidity across various deformation phases.¹⁶⁸ In another study, they incorporated NPR jounce bumpers into digital models of double wishbone and multi-link suspension systems for single-wheel motion tests. The findings showed that NPR jounce bumpers produced a more favourable wheel force versus jounce height profile without the need to adjust free travel, optimizing suspension space.¹⁶⁹ Additionally, comfort tests under bump road conditions were conducted using an optimization approach combining a hybrid Genetic algorithm and Sequential Quadratic Programming. Virtual ride simulation results show reduced maximum vertical acceleration, improving the vehicle’s overall ride comfort.¹⁷⁰

Vehicle door

The standard car door, made of a window, outer and inner panel, frequently faces issues of unstable deformation and insufficient energy dissipation during the explosion impact event. This can lead to excessive penetration of the inner panel into the cabin, resulting in severe passenger injuries. Traditional solutions, such as structural optimization or adding energy-absorbing materials like aluminium foam or honeycomb, have limitations due to design space and material constraints. To address this challenge, Wang et al.¹⁷¹ introduced the concept of an NPR explosion-proof door. This design incorporates an NPR material with a graded composition the inner core section among the outer and inner panels, with varying thicknesses in different gradient layers to optimise energy absorption and impact resistance (Figure 43). Mathematical models were developed to optimise objectives such as inner panel penetration, energy dissipation of the NPR core structure, and the mass of the door. Then, utilising an adaptive hybrid multi-objective particle swarm optimization algorithm, a comprehensive optimization process was carried out for the new explosion-resistant door. Simulation results demonstrated significant improvements over traditional doors, with reduced inner panel intrusion by 9.5%, increased total energy absorption by 3.5%, decreased inner panel acceleration by 34.1%, and enhanced energy absorption of the NPR inner core structure by 9.1%. Additionally, a slight reduction in the door’s mass was achieved post-optimization. These results provide important guidance for the development and optimization of explosion-proof doors.

Figure 43.

Modelling of (a) auxetic internal structure; (b) auxetic explosion- resistant door; and (c) explosion simulation using FEM.¹⁷¹

Challenges and potential applications of auxetic materials and structures in the automotive industry

The challenges

Through the aforementioned research, it is clear that auxetic materials and structures possess significant for automotive applications. However, there are some challenges that arise for the application of auxetic materials in automobiles as show below.

For existing research

Current studies on auxetic materials remain fragmented and lack cohesive integration. Research efforts on Non-Pneumatic Tyres, crash boxes, and bumper systems are still in their infancy when it comes to industrial implementation. Most other studies are confined to theoretical exploration, simulations, or conceptual frameworks. This limitation stems from the complexity of auxetic material production technologies, which, during the initial stages of industrialization, require substantial financial and human resources.

For potential applications

Auxetic textiles hold significant potential for applications like seatbelt webbing, airbag fabrics, and car seat covers. However, they have not been fully explored in these contexts yet. This is largely because traditional textile materials have been used to manufacture these components and have generally performed adequately, leading to a reluctance to adopt new materials without clear advantages. As a result, the challenge lies not only in developing manufacturing technologies for auxetic textile products but also in convincing investors, who may lack sufficient motivation to adopt these innovations. This underscores the need for robust scientific research to demonstrate the feasibility and cost-effectiveness of such investments compared to the potential benefits.

The potential applications

Despite these challenges, they also offer opportunities to guide future research directions. At the same time, with advancements in production technologies, optimizing costs for the industrial-scale manufacturing of auxetic materials is entirely feasible. Some development directions and potential applications of auxetic materials and structures in the automotive industry can be highlighted as follows.

(1) Prioritizing research on technology and production processes to bridge the gap between theoretical understanding and practical implementation, such as casting, welding and mechanical assembly processes.

(2) Conducting further theoretical and experimental investigations into various types of negative Poisson’s ratio structures to elucidate their energy absorption and deformation mechanisms, going beyond the current focus on the honeycomb re-entrant structure.

(3) Textile materials, including knitted, woven, and nonwoven fabrics, play a significant role in the automotive industry by being applied to various surfaces of the car interior. Textile materials find extensive use in automobiles in the following areas: upholstery (front seat, rear seat, X-seat), carpets, pre-assembled interior components including door kick panels, boot linings roof linings and insulation, tyres, safety service (seatbelt and airbag), filter materials, and acoustic absorption in the engine area.^172,173 Many applications require distinct physical and mechanical characteristics, and exploiting the benefits of auxetic textiles in various applications will create new opportunities for the vehicle industry.

Auxetic seat belt webbings

The first thing to mention is the seatbelt webbings, an important part of seat belt systems, often known as safety belts, have become a standard feature in all types of cars and other vehicles. The seatbelt webbings have crucial features such as good retraction behaviour, high load-bearing capacity, abrasion resistance, temperature resistance, the ability to be removed and reinstalled, UV resistance, low weight, flexibility, and extensibility. Especially, webbing em|ployed in seat belts for vehicles must possess strong absorption or attenuation of the impact force imparted to the person during an accident.¹⁷⁴ Therefore, the seat belt systems can keep occupants stable in their seats when the vehicle stops suddenly or experiences an impact, preventing sudden backward movements and reducing the risk of injury. According to statistics from the National Highway Traffic Safety Administration, in 2017 seat belts saved approximately 14,955 lives and could have saved an additional 2,549 lives if individuals had been wearing seat belts at the time of the accident.

However, despite their benefits, safety belt webbings have been known to cause injuries to occupants. In accidents, the seatbelts are tightened, leading to a decrease in the width of the webbing, leading to a decrease in the load-bearing area, and a sharp increase in the force exerted on the body of the sitter and driver, causing mechanical injuries such as chest fractures, shoulder fractures, and internal organ damage. This weakness of current webbings can be overcome by replacing traditional materials with materials with negative Poisson’s coefficient, a material with energy absorption properties and it will expand in the original square direction with the direction of force applied to it. That counterintuitive deformation increases the contact area between the seat belt and the occupants’ body, helping to disperse force and reduce injury. As in the study by Chang Q. and colleagues¹⁷⁵ proposed a design of three-point seat belt webbings with auxetic characteristics (Figure 44). The study uses a finite element analysis model with a seatbelt webbing model and a simple structure of a human chest to simulate the occupant protection performance of the auxetic seatbelt and compare it with the performance of a traditional seat belt webbings. The above simulation results show that auxetic webbing properties are better able to protect vehicle occupants in a collision than traditional webbings. Under the condition of equal resistance, the auxetic seat belt becomes wider under tension, the contact area with the chest-shaped impact object increases, and the contact stress decreases, thus minimising compression deformation.

Figure 44.

3D auxetic seatbelt model: (a) in a rest state; (b) in expanded state.¹⁷⁶

While research on NPR seatbelts is currently limited to simulation, it is clear that the seatbelt material used does not correctly reflect real-world uses, which commonly use polyester and nylon fibre. As a result, NPR seatbelt designs may struggle to maintain durability, longevity, and seamless interaction with other vehicle components. Therefore, it is essential to consider using auxetic textile structures in the design of NPR seatbelt webbings.

Auxetic airbag fabrics

Next, airbags offer additional protection and are one of the most important automotive safety features. They are primarily intended to deploy in crashes to protect the driver and passengers and limit injuries. However, injuries can still occur when a person is exposed to an airbag at a high speed and with a high impact force while the bag is completely inflated. To avoid catastrophic injuries, it is necessary to improve the performance of airbag cushions. As a result, the textile material used in the airbag is critical to achieve this. To function properly, airbag materials must have high mechanical qualities and meet specific quality standards. These parameters include having a small fabric thickness, low specific fabric weight, high tensile strength, high tear strength, high elongation, minimal air permeability, high heat capacity, good resistance to ageing, good folding behaviour, enhanced energy absorption, superior coating adhesion, reliability in hot and cold environments, compact packaging, minimized skin abrasion, high temperature heat stability, clear of knots, splices, spots, and broken ends.^177–179 If the auxetic structure can be converted into the textile material used to manufacture the airbag, it will greatly reduce possible injuries thanks to the special features of auxetic textiles. Thus, airbags are another prospective application area for auxetic textiles. Exploring the possibilities of auxetic textiles in airbags could lead to substantial advances in car safety.

Auxetic car seat covers

Lastly, passenger safety is not the only important factor when it comes to car travel. Comfort is also crucial, and textile fabrics contribute greatly to achieving this. Car seats that are designed to be ergonomic and keep the body in a proper seating posture are essential, but it is also important to ensure that passengers do not experience bodily fatigue caused by discomfort from prolonged sitting.^180,181 The comfort of interaction between the body and seat is largely dependent on the cushioned area, which is usually made up of fabric, and leather. Car seat cover materials must be light and UV resistant, high tear strength, mould resistance, low water porosity for easy cleaning, amazing resilience, and crease resistance to meet the product’s usage requirements.^182–185 Furthermore, in addition to the seat cushion, the seat cover fabric must have anti-indentation ability to ensure that passengers do not become tired after sitting in a car seat for an extended period and that the seat surface is not distorted, extending the seat’s life. This problem can be rectified by incorporating auxetic structures into seat cushions and upholstery fabrics. This is a promising use for auxetic textiles, particularly for seat cover spacer fabric.

Conclusions

This article seeks to offer a structured overview of the exciting world of auxetic materials and structures in the automotive industry. Outstanding results are listed as follows:

(i) First, the article has outlined the relationship between mechanical parameters and the properties of elastic materials, namely, E, v, G, and K. In particular, the study has highlighted indices such as the TDE index, TPM index, ADF index, and PDF index for assessing the energy absorption capacity and impact resistance of auxetic structures in car accidents.

(ii) Second, the study thoroughly presented the structural parameters and deformation mechanisms of typical auxetic cellular structures, including re-entrant, chiral, and rotating structures. This establishes a foundation for the mechanical designs of practical applications and serves as the basis for optimising auxetic structure algorithms.

(iii) Third, the common forms of auxetic materials and structures, including auxetic foams, auxetic composites, and auxetic textiles, are extensively discussed regarding their development history and contemporary advancements. Each form possesses distinct advantages in terms of hardness, elasticity, flexibility, and weight. Hence, comprehending these common forms is pivotal as it provides a foundation for their suitable application.

(iv) Fourth, the practical applications of auxetic materials and structures in the automotive industry were explored. By illustrating specific examples, the article delves into decoding how auxetic materials and structures are integrated into key structural parts of cars, such as non-pneumatic tyres, crash boxes, font bumper systems, jounce bumpers, and vehicle doors.

(v) Finally, the article points out the potential prospects for further applications of auxetic textiles in the automotive industry, including seat belt webbings, airbag fabrics, and car seat covers.

Footnotes

Author contributions

D. Truong: Writing – original draft, Validation, Methodology, Investigation, Conceptualization. H. Nguyễn: Writing – original draft, Validation, Methodology, Investigation, Data curation. R. Fangueiro: Supervision, Writing – review & editing. F. Ferreira: Supervision, Methodology, Writing – review & editing, Formal analysis. Q. Nguyễn: Supervision, Conceptualization, Methodology, Writing – review & editing, Formal analysis.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work has been supported by the European Union under the Next Generation EU, through a grant of the Portuguese Republic’s Recovery and Resilience Plan (PRR) Partnership Agreement, within the scope of the project GreenAuto: Green Innovation for the Automotive Industry – “Agenda Mobilizadora GreenAuto,” aiming the mobilization of the production technologies industry towards of the reindustrialization of the manufacturing industrial fabric (Project ref. nr. 02 – C05-i01/2022; Total project investment: 118.461.005,00 Euros; Total Grant: 2.269.549,89 Euros).

ORCID iDs

Dat Van Truong

Hoa Nguyễn

Quyền Nguyễn

Data Availability Statement

The data can be provided upon request.*

References

Hemp

. Foundations of solid mechanics. J R Aeronaut Soc. 1966; 70: 453–454. DOI: 10.1017/S0001924000058632.

Bhullar

. Three decades of auxetic polymers: a review. E-polymers 2015; 15: 205–215. DOI: 10.1515/epoly-2014-0193.

Christensen

Kadic

Wegener

, et al. Vibrant times for mechanical metamaterials. Mrs Communications 2015; 5: 453–462. DOI: 10.1557/mrc.2015.51.

Frolich

LaBarbera

Stevens

. Poisson's ratio of a crossed fibre sheath: the skin of aquatic salamanders. J Zool 1994; 232: 231–252. DOI: 10.1111/j.1469-7998.1994.tb01571.x.

Greaves

Greer

Lakes

, et al. Poisson's ratio and modern materials. Nat Mater 2011; 10: 823–837. DOI: 10.1038/nmat3134.

Kolpakov

. Determination of the average characteristics of elastic frameworks. J Appl Math Mech 1985; 49: 739–745. DOI: 10.1016/0021-8928(85)90011-5.

Lakes

. Foam structures with a negative Poisson's ratio. Science 1987; 235: 1038–1040. DOI: 10.1126/science.235.4792.1038.

Lakes

. Advances in negative Poisson's ratio materials. Hoboken, NJ: Wiley Online Library, 1993.

Milton

. Composite materials with Poisson's ratios close to—1. J Mech Phys Solid 1992; 40: 1105–1137. DOI: 10.1016/0022-5096(92)90063-8.

10.

Yeganeh-Haeri

Weidner

Parise

. Elasticity of agr-cristobalite: a silicon dioxide with a negative Poisson's ratio. Science 1992; 257: 650–652. DOI: 10.1126/science.257.5070.650.

11.

Veronda

Westmann

. Mechanical characterization of skin—finite deformations. J Biomech 1970; 3: 111–124. DOI: 10.1016/0021-9290(70)90055-2.

12.

Gibson

Ashby

. The mechanics of three-dimensional cellular materials. Proceedings of the royal society of London A Mathematical and physical sciences 1982; 382: 43–59. DOI: 10.1098/rspa.1982.0088.

13.

Gibson

Ashby

Schajer

, et al. The mechanics of two-dimensional cellular materials. Proceedings of the Royal Society of London A Mathematical and Physical Sciences 1982; 382: 25–42. DOI: 10.1098/rspa.1982.0087.

14.

Mukhopadhyay

Adhikari

. Effective in-plane elastic properties of auxetic honeycombs with spatial irregularity. Mech Mater 2016; 95: 204–222. DOI: 10.1016/j.mechmat.2016.01.009.

15.

Zhang

Ruan

, et al. Tensile behavior of an auxetic structure: analytical modeling and finite element analysis. Int J Mech Sci 2018; 136: 143–154. DOI: 10.1016/j.ijmecsci.2017.12.029.

16.

Zhang

Wang

, et al. Large deformation of an auxetic structure in tension: experiments and finite element analysis. Compos Struct 2018; 184: 92–101. DOI: 10.1016/j.compstruct.2017.09.076.

17.

Boldrin

Hummel

Scarpa

, et al. Dynamic behaviour of auxetic gradient composite hexagonal honeycombs. Compos Struct 2016; 149: 114–124. DOI: 10.1016/j.compstruct.2016.03.044.

18.

Hou

Deng

Zhang

. Dynamic crushing strength analysis of auxetic honeycombs. Acta Mech Solida Sin 2016; 29: 490–501. DOI: 10.1016/S0894-9166(16)30267-1.

19.

Dong

Zhao

, et al. Experimental and numerical studies on the compressive mechanical properties of the metallic auxetic reentrant honeycomb. Mater Des 2019; 182: 108036. DOI: 10.1016/j.matdes.2019.108036.

20.

Chen

Xiao

, et al. The Dynamic response of shallow sandwich arch with auxetic metallic honeycomb core under localized impulsive loading. Int J Impact Eng 2020; 137: 103442. DOI: 10.1016/j.ijimpeng.2019.103442.

21.

Xiao

Chen

, et al. The structure response of sandwich beams with metallic auxetic honeycomb cores under localized impulsive loading-experiments and finite element analysis. Mater Des 2019; 176: 107840. DOI: 10.1016/j.matdes.2019.107840.

22.

Xiao

Kang

, et al. Insight into the negative Poisson's ratio effect of metallic auxetic reentrant honeycomb under dynamic compression. Materials Science and Engineering: A 2019; 763: 138151. DOI: 10.1016/j.msea.2019.138151.

23.

Jackowski

Posuniak

Zielonka

, et al. Experimental testing of energy-absorbing structures used to enhance the crashworthiness of the vehicles. Energies 2023; 16: 2183. DOI: 10.3390/en16052183.

24.

Novak

Vesenjak

Ren

. Auxetic cellular materials-a review. Strojniški vestnik-Journal of Mechanical Engineering 2016; 62: 485–493. DOI: 10.5545/sv-jme.2016.3656.

25.

Masters

Evans

. Models for the elastic deformation of honeycombs. Compos Struct 1996; 35: 403–422. DOI: 10.1016/S0263-8223(96)00054-2.

26.

Liu

. A review on auxetic structures and polymeric materials. Sci Res Essays 2010; 5: 1052–1063.

27.

Bezazi

Scarpa

Remillat

. A novel centresymmetric honeycomb composite structure. Compos Struct 2005; 71: 356–364. DOI: 10.1016/j.compstruct.2005.09.035.

28.

Zhang

Liu

. Auxetic textiles. 1 ed. Amsterdam, The Netherlands: Elsevier Ltd: Woodhead Publishing, 2019.

29.

Larsen

Signund

Bouwsta

. Design and fabrication of compliant micromechanisms and structures with negative Poisson's ratio. J Microelectromech Syst 1997; 6: 99–106. DOI: 10.1109/84.585787.

30.

Saxena

Das

Calius

. Three decades of auxetics research− materials with negative Poisson's ratio: a review. Adv Eng Mater 2016; 18: 1847–1870. DOI: 10.1002/adem.201600053.

31.

Schwerdtfeger

Heinl

Singer

, et al. Auxetic cellular structures through selective electron‐beam melting. Phys Status Solidi 2010; 247: 269–272. DOI: 10.1002/pssb.200945513.

32.

Evans

Nkansah

Hutchinson

. Auxetic foams: modelling negative Poisson's ratios. Acta Metall Mater 1994; 42: 1289–1294. DOI: 10.1016/0956-7151(94)90145-7.

33.

Yang

Harrysson

West

, et al. Mechanical properties of 3D re-entrant honeycomb auxetic structures realized via additive manufacturing. Int J Solid Struct 2015; 69-70: 475–490. DOI: 10.1016/j.ijsolstr.2015.05.005.

34.

Hengsbach

Lantada

. Direct laser writing of auxetic structures: present capabilities and challenges. Smart Mater Struct 2014; 23: 085033. DOI: 10.1088/0964-1726/23/8/085033.

35.

Wang

, et al. Parametric analysis of a cylindrical negative Poisson’s ratio structure. Smart Mater Struct 2016; 25: 035038. DOI: 10.1088/0964-1726/25/3/035038.

36.

Yang

, et al. Directions dependence of the elastic properties of a 3D augmented re-entrant cellular structure. Mater Des 2017; 134: 151–162. DOI: 10.1016/j.matdes.2017.08.024.

37.

Harkati

Daoudi

Bezazi

, et al. In-plane elasticity of a multi re-entrant auxetic honeycomb. Compos Struct 2017; 180: 130–139. DOI: 10.1016/j.compstruct.2017.08.014.

38.

Xue

Gao

Zhou

, et al. An enhanced three-dimensional auxetic lattice structure with improved property. Materials 2020; 13: 1008. DOI: 10.3390/ma13041008.

39.

Lim

T-C

. Metamaterials with Poisson’s ratio sign toggling by means of microstructural duality. SN Appl Sci 2019; 1: 176. DOI: 10.1007/s42452-019-0185-1.

40.

Huang

Zhang

Scarpa

, et al. Shape memory polymer-based hybrid honeycomb structures with zero Poisson’s ratio and variable stiffness. Compos Struct 2017; 179: 437–443. DOI: 10.1016/j.compstruct.2017.07.091.

41.

Chen

Xie

, et al. Re-entrant auxetic lattices with enhanced stiffness: a numerical study. Int J Mech Sci 2020; 178: 105619. DOI: 10.1016/j.ijmecsci.2020.105619.

42.

Evans

Alderson

. Auxetic materials: functional materials and structures from lateral thinking. Adv Mater 2000; 12: 617–628.

43.

Prall

Lakes

. Properties of a chiral honeycomb with a Poisson's ratio of—1. Int J Mech Sci 1997; 39: 305–314. DOI: 10.1016/S0020-7403(96)00025-2.

44.

Alderson

Attard

, et al. Elastic constants of 3-4-and 6-connected chiral and anti-chiral honeycombs subject to uniaxial in-plane loading. Compos Sci Technol 2010; 70: 1042–1048. DOI: 10.1016/j.compscitech.2009.07.009.

45.

Grima

Gatt

Farrugia

. On the properties of auxetic meta‐tetrachiral structures. Physica Status Solidi (B) 2008; 245: 511–520. DOI: 10.1002/pssb.200777704.

46.

Gatt

Attard

Farrugia

, et al. A realistic generic model for anti‐tetrachiral systems. Physica Status Solidi (B) 2013; 250: 2012–2019. DOI: 10.1002/pssb.201384246.

47.

Gatt

Brincat

Azzopardi

, et al. On the effect of the mode of connection between the node and the ligaments in anti‐tetrachiral systems. Adv Eng Mater 2015; 17: 189–198. DOI: 10.1002/adem.201400120.

48.

Atilla Yolcu

Okutan Baba

. Experimental investigation on impact behavior of curved sandwich composites with chiral auxetic core. Compos Struct 2024; 329: 117749. DOI: 10.1016/j.compstruct.2023.117749.

49.

Plesha

Lakes

. Chiral three-dimensional lattices with tunable Poisson’s ratio. Smart Mater Struct 2016; 25: 054005. DOI: 10.1088/0964-1726/25/5/054005.

50.

Huang

Wong

Chou

. Design and properties of 3D‐printed chiral auxetic metamaterials by reconfigurable connections. Phys Status Solidi B 2016; 253: 1557–1564. DOI: 10.1002/pssb.201600027.

51.

Cho

Seo

Kim

. Mechanics of auxetic materials. Handbook of mechanics of materials 2019: 733–757.

52.

Grima

Evans

. Auxetic behavior from rotating squares. J Mater Sci Lett 2000; 19: 1563–1565.

53.

Grima

Evans

. Auxetic behavior from rotating triangles. J Mater Sci 2006; 41: 3193–3196. DOI: 10.1007/s10853-006-6339-8.

54.

Grima

Alderson

Evans

. Negative Poisson's ratios from rotating rectangles. CMST 2004; 10: 137–145. DOI: 10.12921/cmst.2004.10.02.137-145.

55.

Grima

Farrugia

Gatt

, et al. On the auxetic properties of rotating rhombi and parallelograms: a preliminary investigation. Physica Status Solidi (B) 2008; 245: 521–529. DOI: 10.1002/pssb.200777705.

56.

Grima

Zammit

Gatt

, et al. Auxetic behaviour from rotating semi‐rigid units. Physica Status Solidi (B) 2007; 244: 866–882. DOI: 10.1002/pssb.200572706.

57.

Evans

Alderson

. Rotation and dilation deformation mechanisms for auxetic behaviour in the α-cristobalite tetrahedral framework structure. Phys Chem Miner 2001; 28: 711–718. DOI: 10.1007/s002690100209.

58.

Rafsanjani

Pasini

. Bistable auxetic mechanical metamaterials inspired by ancient geometric motifs. Extreme Mechanics Letters 2016; 9: 291–296. DOI: 10.1016/j.eml.2016.09.001.

59.

Dudek

Gatt

Mizzi

, et al. On the dynamics and control of mechanical properties of hierarchical rotating rigid unit auxetics. Sci Rep 2017; 7: 46529. DOI: 10.1038/srep46529.

60.

Kim

Shin

Yoo

D-S

, et al. Regularly configured structures with polygonal prisms for three-dimensional auxetic behaviour. Proc Math Phys Eng Sci 2017; 473: 20160926. DOI: 10.1098/rspa.2016.0926.

61.

Lee

. Cellular solids, structure and properties. Mater Sci Technol 2000; 16: 233.

62.

Nguyễn

Fangueiro

Ferreira

, et al. Auxetic materials and structures for potential defense applications: an overview and recent developments. Textil Res J 2023; 93: 5268–5306. DOI: 10.1177/00405175231193433.

63.

Jiang

Ren

Wang

, et al. Manufacturing, characteristics and applications of auxetic foams: a state-of-the-art review. Compos B Eng 2022; 235: 109733. DOI: 10.1016/j.compositesb.2022.109733.

64.

Duncan

Allen

Foster

, et al. Fabrication, characterisation and modelling of uniform and gradient auxetic foam sheets. Acta Mater 2017; 126: 426–437. DOI: 10.1016/j.actamat.2017.01.004.

65.

Scarpa

Yates

Ciffo

, et al. Dynamic crushing of auxetic open-cell polyurethane foam. Proc IME C J Mech Eng Sci 2002; 216: 1153–1156. DOI: 10.1243/095440602321029382.

66.

Alderson

Rasburn

Evans

, et al. Auxetic polymeric filters display enhanced de-fouling and pressure compensation properties. Membr Technol 2001; 2001: 6–8. DOI: 10.1016/S0958-2118(01)80299-8.

67.

Choi

Lakes

. Nonlinear analysis of the Poisson's ratio of negative Poisson's ratio foams. J Compos Mater 1995; 29: 113–128. DOI: 10.1177/002199839502900106.

68.

Chan

Evans

. The mechanical properties of conventional and auxetic foams. Part II: shear. J Cell Plast 1999; 35: 166–183. DOI: 10.1177/0021955X9903500205.

69.

Lim

Alderson

. Experimental studies on the impact properties of auxetic materials. Phys Status Solidi B 2014; 251: 307–313. DOI: 10.1002/pssb.201384249.

70.

Smith

Grima

Evans

. A novel mechanism for generating auxetic behaviour in reticulated foams: missing rib foam model. Acta Mater 2000; 48: 4349–4356. DOI: 10.1016/S1359-6454(00)00269-X.

71.

Gaspar

Ren

Smith

, et al. Novel honeycombs with auxetic behaviour. Acta Mater 2005; 53: 2439–2445. DOI: 10.1016/j.actamat.2005.02.006.

72.

Grima

Gatt

Ravirala

, et al. Negative Poisson's ratios in cellular foam materials. Materials Science and Engineering: A 2006; 423: 214–218. DOI: 10.1016/j.msea.2005.08.229.

73.

Grima

Chetcuti

Manicaro

, et al. On the auxetic properties of generic rotating rigid triangles. Proc R Soc A 2012; 468: 810–830. DOI: 10.1098/rspa.2011.0273.

74.

Chetcuti

Ellul

Manicaro

, et al. Modeling auxetic foams through semi‐rigid rotating triangles. Phys Status Solidi B 2014; 251: 297–306. DOI: 10.1002/pssb.201384252.

75.

Pozniak

Smardzewski

Wojciechowski

. Computer simulations of auxetic foams in two dimensions. Smart Mater Struct 2013; 22: 084009. DOI: 10.1088/0964-1726/22/8/084009.

76.

Dong

Yin

, et al. Negative Poisson’s ratio in 2D Voronoi cellular solids by biaxial compression: a numerical study. J Mater Sci 2016; 51: 7029–7037. DOI: 10.1007/s10853-016-9992-6.

77.

Doyoyo

Wanhu

. Plastic failure analysis of an auxetic foam or inverted strut lattice under longitudinal and shear loads. J Mech Phys Solid 2006; 54: 1479–1492. DOI: 10.1016/j.jmps.2005.12.007.

78.

Choi

Lakes

. Analysis of elastic modulus of conventional foams and of re-entrant foam materials with a negative Poisson's ratio. Int J Mech Sci 1995; 37: 51–59. DOI: 10.1016/0020-7403(94)00047-N.

79.

Zhu

Knott

Mills

. Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells. J Mech Phys Solid 1997; 45: 319–343. DOI: 10.1016/S0022-5096(96)00090-7.

80.

Chan

Evans

. Microscopic examination of the microstructure and deformation of conventional and auxetic foams. J Mater Sci 1997; 32: 5725–5736. DOI: 10.1023/A:1018665617008.

81.

Caddock

Evans

. Microporous materials with negative Poisson's ratios. I. Microstructure and mechanical properties. J Phys D Appl Phys 1989; 22: 1877–1882. DOI: 10.1088/0022-3727/22/12/012.

82.

Evans

Caddock

. Microporous materials with negative Poisson's ratios. II. Mechanisms and interpretation. J Phys D Appl Phys 1989; 22: 1883–1887. DOI: 10.1088/0022-3727/22/12/013.

83.

Fan

Qiu

, et al. Novel method for preparing auxetic foam from closed-cell polymer foam based on the steam penetration and condensation process. ACS Appl Mater Interfaces 2018; 10: 22669–22677. DOI: 10.1021/acsami.8b02332.

84.

Friis

Lakes

Park

. Negative Poisson's ratio polymeric and metallic foams. J Mater Sci 1988; 23: 4406–4414. DOI: 10.1007/BF00551939.

85.

Chan

Evans

. Fabrication methods for auxetic foams. J Mater Sci 1997; 32: 5945–5953. DOI: 10.1023/A:1018606926094.

86.

Chen

Lakes

. Holographic study of conventional and negative Poisson's ratio metallic foams: elasticity, yield and micro-deformation. J Mater Sci 1991; 26: 5397–5402. DOI: 10.1007/BF00553637.

87.

Dong

Lakes

. The properties of copper foams with negative Poisson's ratio via resonant ultrasound spectroscopy. Physica Status Solidi (B) 2013; 250: 1983–1987. DOI: 10.1002/pssb.201384229.

88.

Bezazi

Scarpa

. Mechanical behaviour of conventional and negative Poisson’s ratio thermoplastic polyurethane foams under compressive cyclic loading. Int J Fatig 2007; 29: 922–930. DOI: 10.1016/j.ijfatigue.2006.07.015.

89.

Grima

Attard

Gatt

, et al. A novel process for the manufacture of auxetic foams and for their re‐conversion to conventional form. Adv Eng Mater 2009; 11: 533–535. DOI: 10.1002/adem.200800388.

90.

Grima J

Alderson

Evans

. An alternative explanation for the negative Poisson's ratios in auxetic foams. J Phys Soc Jpn 2005; 74: 1341–1342. DOI: 10.1143/JPSJ.74.1341.

91.

Rodney

Gadot

Martinez

, et al. Reversible dilatancy in entangled single-wire materials. Nat Mater 2016; 15: 72–77. DOI: 10.1038/nmat4429.

92.

Herakovich

. Composite laminates with negative through-the-thickness Poisson's ratios. J Compos Mater 1984; 18: 447–455. DOI: 10.1177/002199838401800504.

93.

Chen

Lakes

. Viscoelastic behaviour of composite materials with conventional-or negative-Poisson's-ratio foam as one phase. J Mater Sci 1993; 28: 4288–4298. DOI: 10.1007/BF01154934.

94.

Nkansah

Evans

Hutchinson

. Modelling the effects of negative Poisson's ratios in continuous-fibre composites. J Mater Sci 1993; 28: 2687–2692. DOI: 10.1007/BF00356204.

95.

Zahra

Dhanasekar

. Characterisation of cementitious polymer mortar–Auxetic foam composites. Constr Build Mater 2017; 147: 143–159. DOI: 10.1016/j.conbuildmat.2017.04.151.

96.

Wei

Edwards

. Auxeticity windows for composites. Phys Stat Mech Appl 1998; 258: 5–10. DOI: 10.1016/S0378-4371(98)00220-9.

97.

Polpaya

Rao

Varughese

. Strain induced insulator-to-conductor transition in conducting polymer composites from the auxetic behaviour of hierarchical microstructures. Phys Chem Chem Phys 2017; 19: 19377–19385. DOI: 10.1039/C7CP02742G.

98.

Zhou

Liu

, et al. Pattern switching in soft cellular structures and hydrogel-elastomer composite materials under compression. Polymers 2017; 9: 229. DOI: 10.3390/polym9060229.

99.

Arshad

Ali

. 3D woven auxetic composites using onloom shaped reinforcement. Polym Bull 2024; 81: 13137–13154. DOI: 10.1007/s00289-024-05332-8.

100.

Tatlier

Berhan

. Modelling the negative Poisson's ratio of compressed fused fibre networks. Physica Status Solidi (B) 2009; 246: 2018–2024. DOI: 10.1002/pssb.200982032.

101.

Valentini

Bittolo Bon

Pugno

. Graphene and carbon nanotube auxetic rubber bionic composites with negative variation of the electrical resistance and comparison with their nonbionic counterparts. Adv Funct Mater 2017; 27: 1606526. DOI: 10.1002/adfm.201606526.

102.

Veloso

Mota

Cunha

, et al. Analytical design of in-plane and through-the-thickness auxetic composite laminates. Composites Part C: Open Access 2024; 15: 100500. DOI: 10.1016/j.jcomc.2024.100500.

103.

Zorzetto

Ruffoni

. Re-entrant inclusions in cellular solids: from defects to reinforcements. Compos Struct 2017; 176: 195–204. DOI: 10.1016/j.compstruct.2017.05.039.

104.

Bubert

Woods

Lee

, et al. Design and fabrication of a passive 1D morphing aircraft skin. J Intell Mater Syst Struct 2010; 21: 1699–1717. DOI: 10.1177/1045389X10378777.

105.

Chen

Scarpa

Farrow

, et al. Composite flexible skin with large negative Poisson’s ratio range: numerical and experimental analysis. Smart Mater Struct 2013; 22: 045005. DOI: 10.1088/0964-1726/22/4/045005.

106.

Silva

Panzera

Brandao

, et al. Preliminary investigations on auxetic structures based on recycled rubber. Physica Status Solidi (B) 2012; 249: 1353–1358. DOI: 10.1002/pssb.201084225.

107.

Bera

. 17 - auxetic textiles. In: Maity

Singha

Pandit

(eds). Functional and technical textiles. Cambridge: Woodhead Publishing, 2023, pp. 543–614.

108.

Asad

Dhanasekar

Zahra

, et al. Impact mitigation of masonry walls with carbon fibre and Auxetic fibre composite renders–A numerical study. Structures. Amsterdam, The Netherlands: Elsevier, 2020, pp. 2733–2751.

109.

Evans

Donoghue

Alderson

. The design, matching and manufacture of auxetic carbon fibre laminates. J Compos Mater 2004; 38: 95–106. DOI: 10.1177/0021998304038645.

110.

Jayanty

Crowe

Berhan

. Auxetic fibre networks and their composites. Physica Status Solidi 2011; 248: 73–81. DOI: 10.1002/pssb.201083973.

111.

Miller

Hook

Smith

, et al. The manufacture and characterisation of a novel, low modulus, negative Poisson’s ratio composite. Compos Sci Technol 2009; 69: 651–655. DOI: 10.1016/j.compscitech.2008.12.016.

112.

Neelakantan

Bosbach

Woodhouse

, et al. Characterization and deformation response of orthotropic fibre networks with auxetic out-of-plane behaviour. Acta Mater 2014; 66: 326–339. DOI: 10.1016/j.actamat.2013.11.020.

113.

Simkins

Alderson

Davies

, et al. Single fibre pullout tests on auxetic polymeric fibres. J Mater Sci 2005; 40: 4355–4364. DOI: 10.1007/s10853-005-2829-3.

114.

Yang

Chalivendra

Kim

. Fracture and impact characterization of novel auxetic Kevlar®/Epoxy laminated composites. Compos Struct 2017; 168: 120–129. DOI: 10.1016/j.compstruct.2017.02.034.

115.

Alderson

Smart

, et al. Auxetic polypropylene fibres: Part 1 - manufacture and characterisation. Plast Rubber Compos 2002; 31: 344–349. DOI: 10.1179/146580102225006495.

116.

Alderson

Davies

, et al. The effect of processing parameters on the mechanical properties of auxetic polymeric fibers. J Mater Sci 2007; 42: 7991–8000. DOI: 10.1007/s10853-006-1325-8.

117.

Alderson

Nazaré

Alderson

. Large‐scale extrusion of auxetic polypropylene fibre. Phys Status Solidi 2016; 253: 1279–1287. DOI: 10.1002/pssb.201600079.

118.

Ravirala

Alderson

Davies

, et al. Negative Poisson’s ratio polyester fibers. Textil Res J 2006; 76: 540–546. DOI: 10.1177/0040517506065255.

119.

Ravirala

Alderson

, et al. Expanding the range of auxetic polymeric products using a novel melt‐spinning route. Phys Status Solidi 2005; 242: 653–664. DOI: 10.1002/pssb.200460384.

120.

Sloan

Wright

Evans

. The helical auxetic yarn–a novel structure for composites and textiles; geometry, manufacture and mechanical properties. Mechanics of Materials. 2011; 43: 476–486. DOI: 10.1016/j.mechmat.2011.05.00.

121.

Darja

Tatjana

Alenka

P-Č

. Auxetic textiles. Acta Chim Slov 2013; 60: 715–723.

122.

Sloan

Wright

Evans

. The helical auxetic yarn – a novel structure for composites and textiles; geometry, manufacture and mechanical properties. Mech Mater 2011; 43: 476–486. DOI: 10.1016/j.mechmat.2011.05.003.

123.

Hook

Uses of auxetic fibers. United States Patent. USA 2011. https://patents.google.com/patent/US8002879B2/en.

124.

Miller

Ren

Smith

, et al. A negative Poisson’s ratio carbon fibre composite using a negative Poisson’s ratio yarn reinforcement. Compos Sci Technol 2012; 72: 761–766. DOI: 10.1016/j.compscitech.2012.01.025.

125.

Ullah

Ahmad

Nawab

. Development of helical auxetic yarn with negative Poisson’s ratio by combinations of different materials and wrapping angle. J Ind Textil 2020; 51: 2181S–2196S. DOI: 10.1177/1528083720941116.

126.

Bhattacharya

Zhang

Ghita

, et al. The variation in Poisson’s ratio caused by interactions between core and wrap in helical composite auxetic yarns. Compos Sci Technol 2014; 102: 87–93. DOI: 10.1016/j.compscitech.2014.07.023.

127.

McAfee

Faisal

. Parametric sensitivity analysis to maximise auxetic effect of polymeric fibre based helical yarn. Compos Struct 2017; 162: 1–12. DOI: 10.1016/j.compstruct.2016.11.077.

128.

Zhang

Ghita

Lin

, et al. Varying the performance of helical auxetic yarns by altering component properties and geometry. Compos Struct 2016; 140: 369–377. DOI: 10.1016/j.compstruct.2015.12.032.

129.

Zhou

. Effects of structural parameters and performance on Poisson's ratio of auxetic yarn. In: Advanced materials research. Baech, Switzerland: Trans Tech Publ, 2013, pp. 252–258.

130.

Liu

Pan

Zheng

, et al. Study on the structure formation and heat treatment of helical auxetic complex yarn. Textil Res J 2019; 89: 1003–1012. DOI: 10.1177/0040517518760754.

131.

Wright

Burns

James

, et al. On the design and characterisation of low-stiffness auxetic yarns and fabrics. Textil Res J 2012; 82: 645–654. DOI: 10.1177/0040517512436824.

132.

Gao

Chen

Studd

. Experimental and numerical study of helical auxetic yarns. Textil Res J 2021; 91: 1290–1301. DOI: 10.1177/0040517520977194.

133.

Zhang

Ghita

Evans

. The fabrication and mechanical properties of a novel 3-component auxetic structure for composites. Compos Sci Technol 2015; 117: 257–267. DOI: 10.1016/j.compscitech.2015.06.012.

134.

Liu

. A novel plied yarn structure with negative Poisson’s ratio. J Text Inst 2016; 107: 578–588. DOI: 10.1080/00405000.2015.1049069.

135.

Razbin

Jamshidi Avanaki

Asghariyan Jeddi

, et al. Double-core helical auxetic yarn: a Novel structure, geometrical modeling and experimental verification. J Text Inst 2022; 113: 1256–1269. DOI: 10.1080/00405000.2021.1923928.

136.

. Woven fabrics made of auxetic plied yarns. Polymers 2018; 10: 226. DOI: 10.3390/polym10020226.

137.

Gao

Liu

, et al. Manufacture and evaluation of auxetic yarns and woven fabrics. Phys Status Solidi 2020; 257: 1900112. DOI: 10.1002/pssb.201900112.

138.

Gao

Chen

. A study of woven fabrics made of helical auxetic yarns. Appl Compos Mater 2022; 29: 109–119. DOI: 10.1007/s10443-021-09952-5.

139.

Cao

Zulifqar

Hua

, et al. Bi-stretch auxetic woven fabrics based on foldable geometry. Textil Res J 2019; 89: 2694–2712. DOI: 10.1177/0040517518798646.

140.

Kamrul

Dong

Zulifqar

, et al. Deformation behavior of auxetic woven fabric made of foldable geometry in different tensile directions. Textil Res J 2020; 91: 87–99. DOI: 10.1177/0040517520936289.

141.

Zulifqar

. Development of Bi-stretch auxetic woven fabrics based on Re-entrant hexagonal geometry. Phys Status Solidi 2018; 256. DOI: 10.1002/pssb.201800172.

142.

Zulifqar

. Geometrical analysis of bi-stretch auxetic woven fabric based on re-entrant hexagonal geometry. Textil Res J 2019; 89: 4476–4490. DOI: 10.1177/0040517519836936.

143.

Sun

Wei

, et al. Stab-resistance of auxetic weft-knitted fabric with Kevlar fibers at quasi-static loading. J Ind Textil 2021; 50: 1384–1396. DOI: 10.1177/1528083719865044.

144.

Magalhães

Rana

Fangueiro

, et al. Development and characterization of re-entrant auxetic fibrous structures for application in ballistic composites. International Journal of Materials and Metallurgical Engineering 2018; 12: 446–451. DOI: 10.5281/zenodo.1474449.

145.

Sun

Raji

, et al. Design and fabrication of novel auxetic weft-knitted fabrics with Kevlar yarns. J Text Inst 2018; 110: 1257–1262. DOI: 10.1080/00405000.2018.1557359.

146.

Liu

Lam

JKC

, et al. Negative Poisson's ratio weft-knitted fabrics. Textil Res J 2009; 80: 856–863. DOI: 10.1177/0040517509349788.

147.

Wang

. 3D auxetic warp‐knitted spacer fabrics. Phys Status Solidi B 2014; 251: 281–288. DOI: 10.1002/pssb.201384239.

148.

Wang

. Tensile and forming properties of auxetic warp-knitted spacer fabrics. Textil Res J 2017; 87: 1925–1937. DOI: 10.1177/0040517516660889.

149.

Liu

. Auxetic self-folding knitted fabrics for protective application. Proceedings of the 19th world textile conference-Autex 2019. Crossroads, Ghent, Belgium: Ghent University, 2019, p. 4.

150.

Steffens

Oliveira

Fangueiro

. Energy absorption from composite reinforced with high performance auxetic textile structure. J Compos Mater 2021; 55: 1003–1013. DOI: 10.1177/0021998320964552.

151.

Rhyne

Cron

. Development of a non-pneumatic wheel. Tire Sci Technol 2006; 34: 150–169. DOI: 10.2346/1.2345642.

152.

Kim

D-M

Kim

. Flexible cellular solid spokes of a non-pneumatic tire. Compos Struct 2012; 94: 2285–2295. DOI: 10.1016/j.compstruct.2011.12.022.

153.

Shankar

Fazelpour

Summers

. Comparative study of optimization techniques in sizing mesostructures for use in nonpneumatic tires. J Comput Inf Sci Eng 2015; 15: 041009. DOI: 10.1115/1.4031828.

154.

Yoo

Uddin

Heo

, et al. Thermoviscoelastic modeling of a nonpneumatic tire with a lattice spoke. Proc Inst Mech Eng - Part D J Automob Eng 2017; 231: 241–252. DOI: 10.1177/0954407016656287.

155.

Zhu

, et al. Research on non-pneumatic tire with gradient anti-tetrachiral structures. Mech Adv Mater Struct 2021; 28: 2351–2359. DOI: 10.1080/15376494.2020.1734888.

156.

Zhou

Z-D

, et al. Design optimization of a NPR structure based on HAM optimization method. Struct Multidiscip Optim 2016; 53: 635–643. DOI: 10.1007/s00158-015-1341-x.

157.

Zhou

Z-D

, et al. Design optimization of a novel NPR crash box based on multi-objective genetic algorithm. Struct Multidiscip Optim 2016; 54: 673–684. DOI: 10.1007/s00158-016-1452-z.

158.

Wang

Zou

Zhao

. Multi-objective optimization of a novel crash box with a three-dimensional negative Poisson’s ratio inner core. Proc Inst Mech Eng - Part D J Automob Eng 2019; 233: 263–275. DOI: 10.1177/0954407017741780.

159.

Hua

, et al. Crashworthiness design of crash box filled with negative Poisson's ratio based on horn structure. Mech Adv Mater Struct 2022; 29: 6403–6420. DOI: 10.1080/15376494.2021.1978594.

160.

Song

Liang

Ding

, et al. Structure design and mechanical properties of a novel anti-collision system with negative Poisson's ratio core. Int J Mech Sci 2023; 239: 107864. DOI: 10.1016/j.ijmecsci.2022.107864.

161.

Moxey

Johnson

McCarthy

, et al. Advanced protection for vulnerable road users: a case study. Proc Inst Mech Eng - Part D J Automob Eng 2006; 220: 723–734. DOI: 10.1243/09544070D16004.

162.

Hou

Yang

Zhang

, et al. Design of energy-dissipating structure with functionally graded auxetic cellular material. Int J Crashworthiness 2018; 23: 366–376. DOI: 10.1080/13588265.2017.1328764.

163.

Wang

Zhao

, et al. Structure design and multi-objective optimization of a novel NPR bumper system. Compos B Eng 2018; 153: 78–96. DOI: 10.1016/j.compositesb.2018.07.024.

164.

Zhao

Wang

Zhao

, et al. Research on the multi-disciplinary design method for an integrated automotive steering and suspension system. Proc Inst Mech Eng - Part D J Automob Eng 2015; 229: 1249–1262. DOI: 10.1177/0954407014559565.

165.

Zhao

Wang

, et al. Integrated optimisation of active steering and semi-active suspension based on an improved Memetic algorithm. Int J Veh Des 2015; 67: 388–405. DOI: 10.1504/IJVD.2015.070411.

166.

Wang

, et al. A negative Poisson's ratio suspension jounce bumper. Mater Des 2016; 103: 90–99. DOI: 10.1016/j.matdes.2016.04.041.

167.

Wang

, et al. Finite element analysis of a jounce bumper with negative Poisson’s ratio structure. Proc IME C J Mech Eng Sci 2017; 231: 4374–4387. DOI: 10.1177/0954406216665415.

168.

Wang

Zhao

Zhou

, et al. Parametric design strategy of a novel cylindrical negative Poisson’s ratio jounce bumper for ideal uniaxial compression load-displacement curve. Sci China Technol Sci 2018; 61: 1611–1620. DOI: 10.1007/s11431-017-9194-2.

169.

Wang

Zhao

Zhou

, et al. Suspension mechanical performance and vehicle ride comfort applying a novel jounce bumper based on negative Poisson's ratio structure. Adv Eng Software 2018; 122: 1–12. DOI: 10.1016/j.advengsoft.2018.04.001.

170.

Wang

Zhao

Zhou

, et al. Optimization of an auxetic jounce bumper based on Gaussian process metamodel and series hybrid GA-SQP algorithm. Struct Multidiscip Optim 2018; 57: 2515–2525. DOI: 10.1007/s00158-017-1869-z.

171.

Wang

Zou

Zhao

, et al. Multi-objective explosion-proof performance optimization of a novel vehicle door with negative Poisson’s ratio structure. Struct Multidiscip Optim 2018; 58: 1805–1822. DOI: 10.1007/s00158-018-2026-z.

172.

Stegmaier

Mavely

Schweins

, et al. 3 - woven and knitted fabrics used in automotive interiors. In: Shishoo

(ed). Textile advances in the automotive industry. Cambridge: Woodhead Publishing, 2008, pp. 43–62.

173.

Mukhopadhyay

Partridge

. Automotive textiles. Textil Prog 1999; 29: 1–125. DOI: 10.1080/00405169908688876.

174.

Behera

. Automotive textiles and composites. High performance technical textiles. Hoboken, NJ: John Wiley & Sons, 2019, pp. 353–371.

175.

Chang

Wenzi

Shu

. FE simulation of occupant collision protection performance of negative Poisson's ratio seat belt webbing (负泊松比安全带织带乘员碰撞保护性能的 FE 仿真). Journal of Automobile Safety and Energy Conservation 2013; 4: 215. DOI: 10.3969/j.issn.1674-8484.2013.03.003.

176.

Wenzi

Study of the occupant crash protection performance of the negative Poisson's ratio seat belt webbing (车用负泊松比安全带织带乘员保护效能研究). [Master Thesis]. Dalian: Dalian University of Technology, 2013.

177.

Partridge

Mukhopadhyay

. Mathematical predictions of gas permeability for automotive air-bags. J Appl Polym Sci 2000; 77: 2104–2112. DOI: 10.1002/1097-4628(20000906)77:10<2104::AID-APP2>3.0.CO;2-G.

178.

Shaikh

Chaudhari

Rasania

. Air bag: a safety restraint system of an automobile. Int J Eng Res Afr 2013; 3: 615–621.

179.

Mankodi

Kodinaria

. Airjet textured yarn fabrics for airbag: an innovative approach. J Textil Sci Eng 2020; 10: 1–5. DOI: 10.37421/jtese.2020.10.423.

180.

Schwarz

Kovačević

Kos

. Physical–mechanical properties of automotive textile materials. J Ind Textil 2014; 45: 323–337. DOI: 10.1177/1528083714532113.

181.

Horrocks

Anand

. Handbook of technical textiles. Amsterdam, Netherlands: Elsevier, 2000.

182.

Ceken

Pamuk

. Properties of automotive seat fabrics. Tech Textilien 2006; 49: 29–31.

183.

Bilisik

Demiryurek

Turhan

. Mechanical characterization of flocked fabric for automobile seat cover. Fibers Polym 2011; 12: 111–120. DOI: 10.1007/s12221-011-0111-7.

184.

Alam

SMM

Kibria

Islam

. Investigation of the basic properties of car seat fabrics applied in automotive textiles. J Textile Eng Fashion Technol 2021; 7: 92–96. DOI: 10.15406/jteft.2021.07.00273.

185.

Jerkovic

Pallares

Capdevila

. Study of the abrasion resistance in the upholstery of automobile seats. Autex Res J 2010; 10: 14–20. DOI: 10.1515/aut-2010-100103.