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Article published in

Smart Materials and Structures, Vol. 27, 2018, 023001

https://doi.org/10.1088/1361-665X/aaa61c

Auxetic metamaterials and structures: A review

Xin Ren 1, Raj Das 2, Phuong Tran 3, Tuan Duc Ngo 3 and Yi Min Xie 2,4,*

1 College of Civil Engineering, Nanjing Tech University, Nanjing, Jiangsu Province 211816, P.R. China

2 Centre for Innovative Structures and Materials, School of Engineering, RMIT University, Melbourne 3001, Australia

3 Department of Infrastructure Engineering, University of Melbourne, Melbourne 3010, Australia

4 XIE Archi-Structure Design (Shanghai) Co., Ltd., Shanghai 200092, China

* Corresponding author. Email: mike.xie@rmit.edu.au

Abstract

Materials and structures with negative Poisson’s ratio exhibit a counter-intuitive behaviour. Under

uniaxial compression (tension), these materials and structures contract (expand) transversely. The

materials and structures that possess this feature are also termed as ‘auxetics’. Many desirable properties

resulting from this uncommon behaviour are reported. These superior properties offer auxetics broad

potential applications in the fields of smart filters, sensors, medical devices and protective equipment.

However, there are still challenging problems which impede a wider application of auxetic materials.

This review paper mainly focuses on the relationships among structures, materials, properties and

applications of auxetic metamaterials and structures. The previous works of auxetics are extensively

reviewed, including different auxetic cellular models, naturally observed auxetic behaviour, different

desirable properties of auxetics, and potential applications. In particular, metallic auxetic materials and

a methodology for generating 3D metallic auxetic materials are reviewed in details. Although most of

the literature mentions that auxetic materials possess superior properties, very few types of auxetic

materials have been fabricated and implemented for practical applications. Here, the challenges and

future work on the topic of auxetics are also presented to inspire prospective research work. This review

article covers the most recent progress of auxetic metamaterials and auxetic structures. More

importantly, several drawbacks of auxetics are also presented to caution researchers in the future study.

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Keywords: Auxetic, negative Poisson’s ratio, metamaterial, structure

1. Introduction

The concept of metamaterials (meta means “beyond” in Greek) was originally defined as novel artificial

materials with unusual electromagnetic properties that are not found in naturally occurring materials

[1]. These superior properties created an avenue for the research field of transformation optics [2],

which have many applications ranging from cloaking [3] to subdiffraction imaging [4] and super lens

[5]. Recently, the concept of metamaterials has been extended to a class of materials whose effective

properties are generated not only from the bulk behaviour of the materials which produce it, but also

from their internal structuring [6]. Metamaterials possess superior and unusual properties in the aspects

of static modulus, density [7], energy absorption [8-10], acoustic and phononic performance [11-13],

heat transport performance [14, 15], smart materials and negative Poisson’s ratio [16, 17].

Poisson’s ratio, denoted ν and named after Siméon Denis Poisson (1787-1840), as a measure of the

Poisson’s effect, is employed to characterize a material, which is the property of materials to expand

(contract) in directions perpendicular to the direction of compression (tension). Poisson [18] defined

the ratio ν between transverse strain () and longitudinal strain in the elastic loading directions as

. For isotropic materials, ν can also be presented using bulk modulus B and the shear

modulus G, which relate to the change in size and shape respectively [19]:

. This formula defines numerical limits of Poisson’s ratio for isotropic bulk materials as,

. The corresponding numerical range is illustrated in Figure 1 where ν

is plotted as a function of for many materials. Starting with compact, nearly incompressible

materials, such as liquids and rubbers, where stress primarily results in shape change and ν is close to

0.5 [20]. For most well-known bulk materials, the Poisson’s ratio is in the range of 0 - 0.5, e.g., metals

and polymers, . Glasses and minerals are more compressible, and for these For

gases and cork, . Re-entrant polymer foams and some metallic crystals can exhibit .

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Figure 1. Numerical range of Poisson’s ratio ν, from –1 to 0.5, plotted as a function of the ratio of the bulk and shear

moduli B/G for a wide range of isotropic classes of materials. [20]

Dating back to 1987, Lakes [21] reported a re-entrant foam structure which exhibits negative Poisson’s

ratio. After four years, materials with negative Poisson’s ratio were coined as “auxetics” or “auxetic

materials” by Evans et al. [22] for the sake of simplifying the long description of “negative Poisson’s

ratio”.

Typical mechanical metamaterials are materials with negative indexes, e.g., negative Poisson’s ratio,

negative compressibility, or negative normal stress [7]. As a most studied branch of mechanical

metamaterials, auxetic materials exhibit counter-intuitive deformation behaviour during deformation.

To be more specific, under uniaxial compression (tension), conventional materials expand (contract) in

the directions orthogonal to the applied load. In contrast, auxetic materials contract (expand) in the

transverse direction, as shown in Figure 2. Numerous desirable properties resulting from this unusual

behaviour have been attracting an increasing number of researchers to the field of auxetics materials

and structures, which could be easily seen from the number of publications (Scopus search engine) as

shown in Figure 3. From only 1 publication in 1991 to around 165 publications in 2016, the number has

increased to 165 times in only 25 years which clearly demonstrates that the topic of auxetics has become

of significant interest.

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Figure 2. Behaviours in tensile and compressive load: (a) non-auxetic material; (b) auxetic material. [23]

Figure 3. Number of publications on the topic of auxetics per year since 1991. (The data is obtained from Scopus search

engine; The bracked numbers indicates the number of review papers published in that year)

Accompanied by uncommon deformation pattern under compression and tension, auxetic materials and

structures are endowed with many desirable material properties, such as superior shear resistance [24],

indentation resistance [25, 26], fracture resistance [27], synclastic behaviour [21], variable permeability

[28] and better energy absorption performance [8, 9, 29-34].

These aforementioned advantages of auxetic metamaterials make them potential candidates for

applications that include but not limited to prostheses [35], auxetic textiles [36-41], smart sensors [42-

44], indentation and fatigue resistance [45-48], smart filters [28, 49], magnetic auxetic system [50, 51],

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molecular sieves [52], seat cushions [53], superior vibration dampers [54] and acoustic isolators [55].

All the above-mentioned properties draw many researchers to engage in the studies of auxetic materials

and structures, however, it is still a challenging work to embed these superior properties into real

applications which could benefit the majority of people.

This article aims to present the most comprehensive review of auxetics in terms of the relationship

among materials, structures, properties and applications. Different cellular models of auxetic materials

and structures are firstly reviewed in section 2. Then, the auxetic materials and structures which exist

in nature are briefly discussed in section 3. As a primary objective of this review, a detailed introduction

of metallic auxetic materials and structures is presented in section 4. Afterwards, multi-material auxetic

materials and structures are stated in section 5. In section 6, the properties of auxetic materials and

structures are extensively illustrated. After that, the potential applications of auxetic materials and

structures are thoroughly discussed in section 7. A comprehensive summary of advantages and

disadvantages of previous studies is given as the last section of this review.

2. Cellular auxetic materials and structures

Compared with solid materials, cellular materials have numerous superior mechanical and thermal

properties, such as low density, high energy absorption, high acoustic isolation and damping, filters etc.

Apart from the above-mentioned advantages, cellular auxetic materials and structures also possess

another special property when compared with most conventional materials and structures, they could

exhibit counter-intuitive behaviour, i.e., shrinking (expanding) under perpendicular loading direction

of compression (tension).

The Milton-Ashaby map of the auxetic materials (bulk modulus (B), shear modulus (G) and mass

density (ρ)) could be utilized to demonstrate the relationship between ordinary solids and the auxetic

materials, as shown in Figure 4. The black ellipses represent the property space of ordinary solids and

red space shows the property domain of auxetic cellular materials [56]. The properties of low-density

materials mainly depend on their cellular configuration and the properties of the base material. An ultra-

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lightweight metallic lattice with a density of 10 mg cm-3 was fabricated and reported by Schaedler et al.

in 2011 [57]. In the work of Körner and Liebold-Ribeiro [58], a systematic approach to identifying

auxetic cellular materials was proposed based on eigenmode analysis.

In order to illustrate properties of cellular auxetic materials and structures with different cellular

architecture, in the following sections, based on the difference of geometrical configuration of cellular

auxetic materials and structures, six kinds of models are comprehensively reviewed.

Figure 4. Milton–Ashby map of auxetic materials. B represents bulk modulus, G represents shear modulus, and ρ

represents mass density. [56]

2.1 Re-entrant models

Gibson et al. [59] firstly proposed the traditional cellular structure in the form of re-entrant honeycombs

in 1982. The typical honeycomb with 2D re-entrant hexagons is shown in Figure 5. Ideally, the

externally vertical diagonal ribs are moved outwards when the re-entrant honeycomb is stretched along

the horizontal direction. However, the flexure of the diagonal ribs also occurs and cannot be avoided

for most of the honeycombs with re-entrant cellular configuration. Auxetic behaviour could also be

attributed to the flexure of the ribs for the re-entrant hexagonal honeycomb system [30]. Masters et al.

[60] developed a theoretical model for 2D re-entrant structures which could predict the elastic constants

of honeycombs based on the deformation of the honeycomb cells by flexure, stretching and hinging.

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Figure 5. Illustration of auxetic behaviour on re-entrant structures. [61]

Gibson et al. [62] provided a traditional two-dimensional model to illustrate the behaviour of

conventional and auxetic honeycombs and foams. The Poisson’s ratio and Young’s modulus along the

loading direction are presented as below:

2

12 cos )sin(sin

lh

(1)

3

1cos )sin( blh

kE

(2)

3

)(l

t

bEk s

(3)

where h, l, b, θ are as defined in Figure 6,

s

E

is the intrinsic Young’s modulus of the material forming

the cell walls.

Figure 6. Hexagonal unit cell of Masters and Evans. [60]

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Larsen et al. [63] conducted the pioneering work of designing compliant micromechanisms and

structures with negative Poisson’s ratio using a numerical topology optimization method. The method

enabled the user to specify the elastic properties of materials of compliant mechanisms and obtain

optimal structures. Besides, engineers can conveniently interpret the resulting topologies. One typical

optimized 2D re-entrant triangular model is shown in Figure 7, and the auxetic performance of the model

can be tuned by the length of the ribs and angle between the ribs.

Figure 7. The 2D re-entrant triangular model. [64]

Unlike the 2D re-entrant triangular model as shown in Figure 7, the re-entrant honeycomb structures

shown in Figure 6 could be easily patterned into 3D structures with sufficient unit cell connections and

auxetic behaviours in multiple principal directions. By extending the concept of 2D re-entrant auxetic

structure, Schwerdtfeger et al. [65] designed a 3D structure with a hexagonal super-lattice pattern which

exhibits negative Poisson’s ratio in multiple directions. The first orthotropic 3D re-entrant honeycomb

auxetic structure as shown in Figure 8b was reported by Evans et al. [66] which could examine the

behaviour of open-celled foams. Yang et al. [67] conducted a further analytical investigation, and the

3D re-entrant structure was presented by the unit cell as shown in Figure 8a. It was found that the

mechanical properties of the re-entrant honeycomb auxetic structure could be controlled by the

characteristic strut ratio and re-entrant angle.

9

Figure 8. The 3D re-entrant models: (a) unit cell; (b) 3D lattice. [67]

A variant of 3D re-entrant cellular structure, shown in Figure 9, was reported by Hengsbach et al. [68].

A promising approach to the development of auxetic metamaterials and devices using direct laser

printing was presented in this study. In a recent study of Fu et al. [69], by embedding a rhombic

configuration into a normal re-entrant hexagonal honeycomb (NRHH), a novel auxetic honeycomb was

generated and reported to have higher in-plane stiffness and critical buckling strength than NRHH.

Figure 9. A 3D re-entrant variant: (a) overall configuration; (b) magnified image with more details. [68]

Wang et al. [70] developed a cylindrical auxetic structure using a 3D re-entrant triangle as shown in

Figure 10. The effective Young’s modulus

*E

and the effective Poisson’s ratio

*

along the vertical

direction were calculated as below:

3222

3

sin)(sin2

*KKK HKE

E

y

(4)

10

sin2sin3sin

sincoscossin

*223

22

L

K

L

L

K

L

L

L

H

L

H

L

H

H

KL

H

H

y

x

(5)

Some variables are illustrated in Figure 10a. In addition, K and β are length ratios, where

and

; α is the cell wall thickness to length ratio, and

. In these two formulas,

most of variables have been illustrated in Figure 10a. Besides, K and β are length ratios, where

LMK

and

LN

; α is the wall thickness to length ratio and

MTLT ML

.

Star-shaped models could be regarded as variants of re-entrant models and a typical work regarding the

cellular materials and structures with star-shape units was made by Grima et al. [71]. In this study, a

technique based on force-field based methods (the EMUDA technique) was employed to explore the

mechanical performance of the star-shape systems where the stars have rational symmetry of order 3, 4

or 6 as shown in Figure 11. This work is of importance mainly based on the following two points: Firstly,

this study provided the convincing evidence that star-shape systems have a potential for auxetic

behaviour and the magnitudes of the Poisson’s ratio could be tuned by the stiffness of the hinges and

the rod elements of the structure; Secondly, this work also demonstrated that the behaviour of periodic

structures under applied loads could be easily investigated using the EMUDA technique, and in

particular to distinguish auxetic and non-auxetic systems. Recently, Wang et al. [72] reported that re-

entrant cell shapes could not guarantee auxetic behaviour, and the auxetic angles should be larger than

20° in order to give rise to the auxetic behaviour of the models proposed in their work.

Li et al. [73] proposed a novel 3D augmented re-entrant cellular structure. The Poisson’s ratio of the

proposed model could be adjusted in a wide range from negative to positive. Grima et al. [74] proposed

hexagonal honeycombs with zero Poisson’s ratios and enhanced stiffness. Sun et al. [75] introduced a

concept of a novel active honeycomb configuration based on inflatable tubes and an auxetic

centresymmetric cellular topology for morphing wingtips. Harkati et al. [76] proposed a multi-entrant

auxetic honeycomb and the behaviour of the proposed auxetic structure with variable stiffness and

Poisson’s ratio effects were parametrically investigated. Recently, based on energy method, Wang et

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al. [77] proposed an analytical model of a 3D re-entrant auxetic cellular structure and found that when

the struts were slender enough, the bending of the struts was the decisive factor on the deformation of

the structure. In a recent work of Huang et al. [78], a novel auxetic honeycomb design composed of re-

entrant configurations was proposed. Peel et al. [79] discussed wing skin, actuator and actuator

attachment development, as well as possible auxetic skin behaviour. Lira and Scarpa [80] investigated

gradient auxetics and found that the gradient configuration of the cellular structure provided additional

complexity and the possibility of tailoring design properties. Later on, Lira et al. [81] investigated

gradient cellular auxetics as potential cores for aeroengine fan blades. Hou et al. [82] investigated the

bending and failure of sandwich structures with auxetic gradient cellular cores and found that the aspect

ratio and the extent of gradient have a significant influence on the flexural properties of the structures.

Figure 10. 3D re-entrant triangular: (a) the mechanical analytic model of a 2D re-entrant cell; (b) a layer of cylindrical

auxetic structure, (c) cylindrical axuetic structure. [70]

Figure 11. Various star-shape systems with different rotational symmetry of order: (a) auxetic honeycomb; (b) Star-3

system; (c) Star-4 system; (d) Star-6 system. [71]

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2.2 Rotating polygonal models

Rotating polygonal model is another typical category of auxetics. Grima [83-87] carried out extensive

research on the rotating polygonal models. Alderson et al. [88] and Rafsanjani et al. [89] also

contributed to the development of rotating polygonal models.

Grima et al. [83] proposed a novel mechanism to achieve a negative Poisson’s ratio based on an

arrangement with rigid squares connected together at their vertices by hinges, and the unit cell of this

mechanism is shown in Figure 12. This geometry could also be regarded as a projection of a specific

plane of a three-dimensional structure or a two-dimensional arrangement of squares.

Figure 12. Unit cell of rotating square model. [83]

When two assumptions are made, i.e., the squares are assumed to be non-deformable along loading

directions; the rotating square model is unable to shear. Hence, the Poisson’s functions, Young’s

modulus and compliance matrix of the model could be presented using three formulas respectively as

below:

1

2112

(6)

sin1 18

2

21

l

KEE h

(7)

13

000

011

011

1

000

0

0

2221

1211

E

SS

SS

S

(8)

where

h

K

is the stiffness constant of the hinges, l is the length of the square and the θ is the angle

between the squares and S is the compliance matrix as shown in Figure 13.

Figure 13. Rotating triangle model. [84]

Apart from the rotating square model, Grima [84] also conducted a theoretical analysis, in which he

concluded that the ‘rotating triangles’ mechanism can be a very effective way of introducing negative

Poisson’s ratios in real materials. The formulas of Poisson’s ratio, Young’s modulus and the compliance

matrix for the rotating triangle model shown in Figure 12 are presented as below:

1

1

1212

(9)

3

cos1

34

2

21 l

KEE h

(10)

14

000

011

011

1

000

0

0

2221

1211

E

SS

SS

S

(11)

where

h

K

is the stiffness constant of the hinges, l is the length of sides of the triangle.

Figure 14. The system composed of hinged 'rotating rectangles' of size (a × b) with a rectangular unit cell of dimensions

(X1 × X2). [85]

Considering the squares are special rectangles, Grima et al. [85] proposed a more general auxetic model

of rotating rectangles as shown in Figure 14. Hence, the formulas of Poisson’s functions, Young’s

modulus and compliance matrix of the rotating square model shown in Formulas (6)-(8) should be

modified using three formulas respectively as below:

2

sin

2

cos

2

cos

2

sin

2222

2222

1

1221

ba

ba

(12)

15

2

2

2

1

2

sin

2

cos

2

sin

2

cos

2

cos

2

sin

8

2

cos

2

sin

2

cos

2

sin

2

sin

2

cos

8

baba

ba

KE

baba

ba

KE

h

h

(13)

000

0

1

0

1

1

000

0

0

21

12

2

21

1

2221

1211

EE

EE

E

SS

SS

S

(14)

where

h

K

is the stiffness constant of the hinges, a and b are the two sides of the rectangles.

Another subsequent work in terms of rotating polymeric models was reported by Grima et al. [86]. In

this study, the rotating systems were constructed from either connected rhombi or connected

parallelograms. Various rotating variants were generated and investigated as shown in Figure 15, the

Poisson’s ratio of these systems can be positive or negative, is anisotropic and depend on the

configuration of the parallelograms (rhombi) and the degree of openness of the system.

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Figure 15. The rotating rhombi and rotating parallelograms systems: (a) rotating rhombi of Type α; (b–c) rotating

parallelograms of Type I α and Type II α respectively; (d) rotating rhombi of Type β; (e–f) rotating parallelograms of Type

I β and Type II β respect. [86]

Alderson et al. [88] explored the rotation and dilation deformation mechanisms for auxetic behaviour

in the α-cristobalite tetrahedral framework structure as shown in Figure 16. Three types of deformation

mechanisms are assumed and analysed in this study. For the first one, the tetrahedral model is assumed

to be rigid and free to rotate, and the auxetic response is caused by rotation (RTM). For the second one,

the tetrahedral model is assumed to maintain shape and orientation but free to change size, and the

auxetic response is caused by tetrahedral dilating (DTM). Tetrahedral rotation and dilation are assumed

to act concurrently (CTM), the auxetic response is caused by both of RTM and DTM to act in a

concurrent manner.

Figure 16. The rotating tetrahedral model. [88]

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A new model system of ‘semi-rigid’ squares was also proposed by Grima et al. [87] to further extend

the previous work of rigid rotating models. In this work, a simple modification of the idealised ‘rotating

rigid squares’ model was made to allow the squares to deform by giving an additional degree of freedom.

This small modification leads to significant changes in the mechanical properties of the proposed model,

which makes the model more suitable for presenting the Poisson’s ratio for many real materials, such

as zeolite and SiO2.

Inspired by ancient geometric motifs, Rafsanjani et al. [89] proposed bistable auxetic mechanical

metamaterials which exhibit auxeticity and structural bistability simultaneously. One typical model of

their work is shown in Figure 17.

Figure 17. A typical model: (a) ancient geometric motif; (b) a bistable auxetic mechanical metamaterial before deformation;

(c) a bistable auxetic mechanical metamaterial after deformation. [89]

2.3 Chiral models

Chiral models are another kind of widely investigated cellular auxetic materials, and the word ‘chiral’

originally means a molecule that is non-superimposable on its mirror image. However, this term is often

used to present a physical property of spinning. The basic chiral units are formed by connecting straight

ligaments (ribs) to the central nodes as shown in Figure 18. Lakes [90] firstly reported that a chiral

hexagonal microstructure can exhibit auxetic behaviour. Prall et al. [91] conducted a theoretical and

experimental investigation on a two-dimensional chiral honeycomb, and the result indicated that the in-

plane Poisson’s ratio was -1.

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Figure 18. Chiral structure with a highlighted unit cell. [91]

Grima et al. [92] analysed a novel class of structures (named as ‘meta-chiral’) which belongs to the

class of auxetics constructed using chiral building blocks. The meta-chiral is also regarded as an

intermediate structure between the ‘chiral’ and ‘anti-chiral’. Some examples of meta-chiral systems are

shown in Figure 19. It should be noted that for all the systems, the ligaments are always attached

tangentially to the nodes in a way that they protrude out from the circles in the same direction to form

the ‘chiral’ sub-units but the ligaments are not attached to the rods in a rotationally symmetric manner

where the order is equal to the number of rods.

Figure 19. Meta-chiral systems with different number of ribs attached to each node: (a) six ribs; (b) four ribs; (c) three ribs.

[92]

Another important work regarding the auxetic chiral models was conducted by Gatt et al. [93], where

the on-axis mechanical properties of the general forms of the flexing anti-tetrachiral system were

investigated through both analytical and finite element methods. The results indicated that the geometry

and material properties of the constituent materials have a significant impact on the mechanical

properties of the flexing anti-tetrachiral system. To be more specific, the Poisson’s ratio of the general

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flexing anti-tetrachiral depends on the ratio of the ligament lengths and the thickness. Also, Gatt et al.

[93] concluded that the rigidity of the anti-tetrachiral system can be adjusted without affecting the

Poisson’s ratio in a form of changing the relative stiffness of the ligaments.

After that, Gatt et al. [94] explored the mode of connection between the nodes and ligaments in the anti-

tetrachiral structure using finite element method. The amount of gluing material used to attach the

ligaments to the node has little effect on the Poisson’s ratio but has a huge influence on the stiffness of

the structure. Besides, the stiffness of the glue is proved to have a significant effect on the mode of

deformation of the chiral system.

Rossiter et al. [95] presented a novel shape memory auxetic deployable chiral structure which could

deform without external actuation mechanism. The contraction of chiral structures is shown in Figure

20. When the structure was heated, the shape memory alloy beams transitioned from the programmed

curled shape (as shown in Figure 20b) into straight beams (as shown in Figure 20a), resulting in a large

overall expansion. Figure 20c and Figure 20d present the maximum expansion and fully compressed

state when the thickness of the connecting beam is regarded as negligible. The linear expansion ratio

here is

rd 2

or

rrl 2)4( 2/122

, where l is the length of the connecting beam.

Figure 20. Contraction of chiral structures: (a) expanded triangular element; (b) compressed triangular element; (c) a single

structural element at maximum extension; (d) at maximum contraction. [95]

Mizzi et al. [96] carried out a pioneering study on the influence of translational disorder on the

mechanical properties of hexachiral honeycomb systems through a finite element approach. The type

of disorder was found to have minimal effect on the Poisson’s ratios of these systems when the ligament

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length to thickness ratio is large enough and the overall length to width ratio of the disordered system

is same as that of its ordered counterpart.

Ha et al. [97] proposed a chiral three-dimensional lattices with tuneable Poisson’s ratio as shown in

Figure 21. These chiral lattices were developed with a lot of cubical nodules and finite element analysis

was employed. The chiral 3D lattices exhibit stretch-twist coupling that increases with relative

slenderness of ribs. The Poisson’s ratio of the chiral 3D lattices could be negative to zero and this value

depends on the specific geometry.

Figure 21. Unit cell of chiral lattice structure (The aspect ratio is defined to be L/a, where L/a > 1). [97]

Recently, Huang et al. [98] proposed a design of 3D chiral metamaterials which exhibit auxetic

behaviour. The compressive test was conducted with good agreements with finite element analysis both

in deformation patterns and Poisson’s ratio. Jiang et al. [99] defined and characterized a parameter for

auxetic chiral structures of internal rotation efficiency, and they found that Poisson’s ratio of their

proposed auxetic chiral structures was linearly related to this parameter. Recently, based on 2D cross

chiral structures and evolvement of star structures, Lu et al. [100] proposed a novel 3D cross chiral

structure with the uniform auxetic behaviour in all three principal directions, as shown in Figure 22.

21

Figure 22. The analytical cell of 3D cross structure with the uniform auxetic behaviour. [100]

Chiral models have been widely investigated in the past few years, particularly for the application of

morphing structures. Airoldi et al. [101, 102] produced chiral honeycombs made of thin composite

laminates and manufactured the morphing ribs for a variable camber wing-box. In the work of Bornengo

et al. [103], a concept of hexagonal chiral honeycomb was proposed as a truss-like internal structure

for adaptive wing box configurations. Budarapu et al. [104] proposed a framework to design an aircraft

wing structure and analyse a morphing airfoil with chiral structure. Gong et al. [105] proposed a novel

zero Poisson’s ratio honeycomb structure that could achieve deformations along two orthogonal

directions without increasing the effective stiffness in the morphing direction.

2.4 Crumpled sheets models

Crumpled sheets model could be regarded as a variant of planar sheet model which has drawn a

considerable attention recently. Alderson et al. [106] proposed a novel procedure for manufacturing

thin auxetic flat sheet and curved foams by uniaxial compression. Auxetic behaviour was found as a

consequence of a crumpled through the thickness microstructure using the detailed optical microscopy

and Poisson’s ratio measurements.

Zhang et al. [107] fabricated and studied an auxetic structure made of tubes and corrugated using a

conventional method. The structural parameters and the deformation of the auxetic structure are shown

in Figure 23. The equations for calculating Poisson’s ratio ν of the structure can be obtained as shown

in formula (15) based on the following assumptions: the auxetic structure is a perfectly periodic

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structure; the shape of each corrugated sheet is formed by connecting straight ligaments with circular

arcs; the straight ligaments of corrugated sheets are always kept straight under loading conditions; the

effect of elastic deformation of the structure is not considered; the thickness of corrugated sheets h is

ignorable; tubes are firmly connected with corrugated sheets so that no slippage takes place between

pipes and corrugated sheets. In this study, the authors concluded that the auxetic behaviour of the

proposed structure depends on its geometrical parameters. The auxetic behaviour could be increased

when the crimped effect of the corrugated sheets is decreased, and the stability of the structure at the

initial deformation is also reduced.

Figure 23. Auxetic structure with corrugated sheets. [107]

0000

0000 sin1coscossinsin sin1cossin1cossin

bab bb

a

y

x

(15)

2

22

011

arcsin abaab

(16)

a and b are non-dimensional parameters defined as

(17)

23

By mixing single-walled and multi-walled nanotubes, Hall et al. [108] found that the in-plane Poisson’s

ratio of carbon nanotube sheets (buckypaper) can be tuned from positive to negative with the

consequence of a substantial increase in the density-normalized sheet toughness, strength and modulus.

Scarpa et al. [109] conducted a study on the effective elastic mechanical properties of single layer

graphene sheets using analytical and numerical methods. They concluded that the shear loading seems

to imply an equivalent auxetic behaviour for the bonds, with a significant negative Poisson’s ratio value

when the bond material is regarded as an equivalent isotropic material. Grima et al. [110] conducted

extensive molecular dynamics simulations which demonstrated that the conformation of graphene can

be modified through the introduction of defects so as to exhibit a negative Poisson’s ratio. Typical

images of a crumpled sheet of paper and a graphene sheet, shown in Figure 24, have illustrated the

auxetic effect of a crumpled sheet conformation. A similar work was conducted by Tan et al. [111],

where a non-porous smooth curve sheet was reported to exhibit auxetic behaviour and a low-stress

concentration factor.

Figure 24. Typical images of a crumpled sheet conformation: (a) a crumpled sheet of paper at different levels of applied

strain; (b) a graphene sheet with 3.0% defects at different levels of applied strain. [110]

The Japanese art of kirigami has inspired many researchers to engage in the studies of auxetic materials

and structures. Eidini [112] created a kind of one-degree of freedom cellular mechanical metamaterials.

The unit cell of the patterns was made of two zigzag strips surrounding a hole with a parallelogram

shape. The authors concluded that the dislocating zigzag strips of the Miura-ori along the joining ridges

24

preserved and tune the properties of the Miura-ori. Liu et al. [113] carried out extensive finite element

analysis and experimental studies on the Miura-ori patterned sheets, particularly its deformation under

three types of tests: out-of-plane compression, three-point-bending and in-plane compression. Although

it is widely known that auxetic responses in paper structures are related to the cellulose fibre network

structure in the sheet, how the materials and processing variables affect auxetic behaviour is still needed

to explore. Verma et al. [114] proposed a mathematical model aiming to explain auxetic behaviour in

an idealised arrangement of fibres in the paper. Javid et al. [115] proposed a novel class of non-porous

auxetic materials with periodically arranged dimples. It was found that the auxetic behaviour resulted

from a novel mechanism whereby the out-of-plane deformation of the spherical dimples was exploited.

A typically numerical result of the proposed non-porous auxetic materials under uniaxial tension is

shown in Figure 25. Utilising Kirigami techniques, Hou et al. [116] manufactured and investigated

graded conventional/auxetic honeycomb structures and found that graded core configuration composed

of dual conventional-negative Poisson’s ratio cellular structure showed interesting capabilities

regarding flatwise compression and edgewise loading. Based on Kirigami design principles, Neville et

al. [117] investigated the Poisson’s ratios of a family of cellular metamaterials and the authors claimed

that the proposed mechanical metamaterials had potential for shape change applications, i.e., morphing

structures. Using Kirigami techniques from polyetheretherketone films, Chen et al. [118] investigated

curved SILICOMB cellular structures with zero Poisson’s ratio for large deformation and morphing.

25

Figure 25. A typically numerical result of tensile deformation: (a) before deformation; (b) after deformation. [115]

Bouaziz et al. [119] fabricated crumpled aluminium thin foils as shown in Figure 26, and carried out a

compression test on it. Comparing with the compression result of other cellular materials, the crumpled

materials exhibit a hybrid mechanical behaviour, between foams and entangled fibrous materials. These

materials exhibit a remarkable plasticity and a low hysteresis similar to conventional foams but have no

plateau beyond the yield stress.

Figure 26. Micrographs of a typical crumpled aluminium foil: (a) 3D image; (b) 2D segmented image. [119]

Duncan et al. [120] conducted an extensive study on the fabrication, characterisation and modelling of

uniform and gradient auxetic foam sheets. Several sheets were fabricated with uniform triaxial

compression, with and without through-thickness pins, and also with different compression regimes in

opposing quadrants. The quadrant of graded foam exhibited positive and negative Poisson’s ratios in

tension and compression, respectively, accompanied by high and low in-plane tangent modulus.

2.5 Perforated sheets models

Perforated sheets models, as a novel kind of cellular auxetic materials and structures, have attracted

considerable attentions in the last several years. Grima et al. [121] firstly reported that conventional

materials containing diamond or star-shaped perforations can exhibit auxetic behaviour in both tension

and compression, and the mechanism could also be regarded as an extension based on rotating rigid

units. Some typical examples of perforated sheets are shown in Figure 27. The Poisson’s ratio function

can be represented as below:

26

2

cos

2

sin

2

sin

2

cos

2222

2222

1

ba

ba

yxxy

(18)

where a and b are the sides of the rectangles, and θ represents the angle between the rectangles.

Figure 27. Examples of perforated sheets which may have negative Poisson’s ratio: (a) the angle between the direction of

diamond-shaped inclusions and the corresponding sides of the square is 0°; (b) the angle between the direction of

diamond-shaped inclusions and the sides of the square is 45°; (c) with two different sizes diamond-shaped inclusions; (d)

with a star-shaped inclusion. [121]

Through the finite element and analytic methods, Grima et al. [122] also reported the process of

fabricating star or triangular shaped perforated sheets to exhibit Poisson’s ratio. Mizzi et al. [123]

proposed a novel approach to design auxetic metamaterials using the patterned slit perforations. The

maximum negative value of Poisson’s ratio of the proposed auxetic metamaterial can reach -13 as

shown in Figure 28. Slann et al. [124] proposed a cellular structure configuration with a rectangular

perforation topology exhibiting auxetic in-plane behaviour and found that auxetic perforations could be

used to achieve structures with compliant solids for multiphysics and morphing characteristics.

27

Figure 28. A typical auxetic perforated sheet: (a) Diagram of a typical auxetic perforated sheet at different degrees of

strain in the y-direction; (b) Poisson’s ratio against engineering strain of the same sheet. [123]

Another significant study was contributed by Grima et al. [125], where they proposed a novel class of

perforated systems containing quasi-random cuts demonstrating the auxetic behaviour. Through finite

element simulations and experiments, as shown in Figure 29, the authors concluded that despite the

disorder and randomness in the orientation, the proposed perforated system still maintains auxetic

properties. More importantly, this study indicated that a high degree of symmetry is not necessary for

the system to exhibit auxetic behaviour, which tremendously released the design freedom for generating

auxetic materials. Recently, Carta et al. [126] proposed a porous material with isotropic negative

Poisson’s ratio, which was validated using experimental tests and numerical simulations. The planar

auxetic and isotropic behaviour of the considered porous medium was observed on three specimens,

characterized by a 45° rotation of the pores disposition relative to each other and loaded in the same

direction. The parametric study indicated that the Poisson’s ratio is strongly influenced by the relative

orientation of the pores. Recently, Tang et al. [127] proposed a design of cut unit geometry in

hierarchical kirigami-based auxetic metamaterials which could demonstrate a high stretchability and

compressibility, as shown in Figure 30.

Figure 29. Images from finite element analysis and experiment: (a) numerical model with alternative perpendicular

perforated slits; (b) numerical model with maximal rotating angle of perforated slits equal ±30°; (c) numerical model with

28

maximal rotating angle of perforated slits equal ±5°; and (d) experiment with maximal rotating angle of perforated slits

equal ±30°. [125]

Figure 30. Demonstration of high stretchability and compressibility: (a) 2d auxetic materials with the unit cell of circle as

void; (b) 2d auxetic materials with the unit cell of modified re-entrant structure. [127]

2.6 Other models

Apart from the five types of cellular auxetic models which have been reviewed in the above, there are

several auxetic models geometrically difficult to classify appropriately. Alderson et al. [128] proposed

a nodule-fibril model to explain the auxetic microporous polymers employing concurrent fibril hinging

and stretching deformation mechanisms. Smith et al. [129] proposed a novel mechanism of missing rib

model, as shown in Figure 31a. Through comparing the conventional auxetic model and experimental

data, the authors found that the missing rib model is superior in predicting the Poisson’s function and

slightly better at predicting the stress-strain behaviour of the experimental data. Grima et al. [130]

proposed a simple analytical model based on a simplified and idealised system having the ‘egg rack’

geometry, as shown in Figure 31b, to explain the behaviour of this system when in-plane force is loaded.

29

Figure 31. Auxetic models: (a) The missing rib model [129]; (b) Egg rack model [130]; (c) Generalized tethered nodule

model [131]; (d) Hexatruss model [132]; (e) ball link model [133]; (f) entangled wire model [134].

Gaspar et al. [131] proposed a new three-dimensional tethered-nodule model which could provide a

better understanding towards the underlying principles of many three-dimensional auxetic models, as

shown in Figure 31c. The auxetic behaviour of the model is mainly caused by the bending of beams in

re-entrant angles which imposes a limitation on number of planes in which auxetic behaviour can be

formed. Dirrenberger et al. [132] proposed a hexatruss model, as shown in Figure 31d, has good results

regarding indentation strength, particularly in the scenario when volume fraction or density is the key

parameter for designing a structure.

Pasternak et al. [133] proposed an auxetic ball link model, as shown in Figure 31e which has a

theoretical negative Poisson’s ratio of -1, however, the experimental tests indicated that the Poisson’s

ratio value of the ball link model was slightly different from the theoretical value. Rodney et al. [134]

reported that a simple three-dimensional architected material made of a self-entangled single long coiled

wire exhibited a considerable and reversible dilatancy in both tension and compression, as shown in

Figure 31f. Recently, 3D fibre networks in the form of needle-punched nonwoven materials were

reported to have out-of-plane auxetic behaviour in large deformations by Rawal et al [135].

30

In the work of Babaee et al. [136], a 3D soft metamaterial with bucklicrystal structure was reported to

exhibit auxetic behaviour in a large compression strain up to 0.3, as shown in Figure 32. Employing

selective laser sintering technique, Yuan et al. [137] fabricated a similar 3D soft auxetic lattice structure

using porous thermoplastic polyurethane (TPU).

Figure 32. Collection of bucklicrystals with different crystal systems (i.e., body-centred cubic (bcc), face-centered cubic (fcc)

and simple cubic (sc)): (a) building blocks with 6, 12 and 24 holes, which are shown in red, green and blue, respectively;

(b) representative volume elements (RVE) for the bucklicrystal before deformation; and (c) buckled configurations for the

RVEs under uniaxial compression. [136]

3. Natural auxetic materials and structures

3.1 Molecular auxetics

Many scientists expressed skeptical attitude towards the concept of auxetic behaviour at the beginning,

although the existence of auxetic materials was proposed more than 160 years ago through classical

elasticity theory [138]. Natural materials were found to exhibit auxetic properties later, e.g., iron pyrite

31

monocrystal, some cubic elemental metals, cancellous bone and cat skin. The case of iron pyrite

monocrystal was reported using experiments on the twisting and bending of mineral rods [139] in 1882.

Until 1944, Love [140] proved the existence of iron pyrite monocrystal and an estimated Poisson’s ratio

of -1/7 was given. Although the negative Poisson’s ratio is theoretically possible, this property was

generally believed to be rare in crystalline solids [141]. In contrast to this belief, Baughman et al. [142]

concluded that negative Poisson’s ratio is a common feature of many cubic metals, and 69% of the

cubic elemental metals have a negative Poisson’s ratio when stretched along the [110] direction, as

depicted in Figure 33a. In the work of Wang et al. [143], a 2D material of δ-Phosphorene was reported

to have a highly negative Poisson’s ratio as shown in Figure 33b, and this 2D material was regarded to

have huge potential for manufacturing low dimensional electromechanical devices. Through

investigating the elastic behaviour of α-cristobalite and other forms of silica with first-principles and

classical interatomic potentials, Keskar et al. [141] concluded that α-quartz, the most common form of

crystalline silica, also exhibits a negative Poisson’s ratio under large uniaxial tension.

Figure 33. Microscopic structures: (a) the structural origin of a negative Poisson’s ratio and a giant positive Poisson’s ratio

for the case of a rigid-sphere body-centred cubic solid [142]; and (b) the top and side views of the relaxed δ-Phosphorene

(the directions of two basic vectors a and b of the unit cell are indicated) [143].

By utilising the force-field-based molecular simulations, Grima et al. [52] reported that some idealized

zeolitic cage structures possess negative Poisson’s ratios. The combination of framework geometry and

simple deformation mechanisms can explain the auxetic behaviour for most idealized molecular

structures. By “off-axis analysis” of experimental and simulated elastic constants for natrolite, Williams

32

et al. [144] reported that the Poisson’s ratio

xy

and

yx

could be negative if the zeolite material were

subjected to the stress at 45° along either the x or y crystallographic axes. The first direct experimental

evidence for on-axis auxetic behaviour in a synthetic zeolite structure was reported by Sanchez-Valle

et al. [145] through measuring the single-crystal elastic properties of MFI-silicalite using Brillouin

scattering. Kimizuka et al. [146] carried out a molecular-dynamics study for investigating the

mechanisms for the negative Poisson’s ratios over the α-β transition of cristobalite (SiO2). It was found

that the mechanisms differ between the α and β phases. In the cubic β phase, among the adiabatic elastic

constants (Cij) of SiO2, C44 has a value close to C11 rather than C12 which is in contrast to the Cauchy

relation. Recently, through first-principle calculations, Kou et al. [147] reported that under a tensile

strain in the armchair direction, a synthesized atomically thin boron sheet (i.e, borophene) exhibits an

unexpected negative Poisson’s ratio results from its special triangle hinge structure and the related hinge

dihedral angle variation. In a recent work of Yu et al. [148], a novel class of auxetic single-layer 2D

materials was reported to exhibit an intrinsic in-plane negative Poisson’s ratio. Unlike previously

reported auxetic materials, its auxetic behaviour resulted from electronic effects. Qin et al. [149]

reported that a semi-fluorinated graphene could exhibit auxetic behaviour. More importantly, unlike

conventional rigid mechanical models of auxetics, this new auxetic mechanism resulted from chemical

functionalization with fluorine atom which was believed to be a promising method to manufacture novel

auxetic nanomaterials. Through molecular dynamics simulations, Qin et al. [150] found that a rippled

graphene could demonstrate an auxetic behaviour. Employing first-principles density functional theory

calculation, Mortazavi et al. [151] found that Mo2C sheets also exhibit an auxetic behaviour. Gao et al.

[152] reported that interface structures with fully mechanically controllable thermal perforce could be

achieved through tailoring auxetic and contractile graphene, which is believed to be a significant work

to design and manufacture 2D functional materials with superior mechanical and thermal performance.

Employing molecular dynamics simulations, Deng et al. [153] found that graphene could be converted

from normal to auxetic at certain strains. Using first-principles density functional theory calculations,

Mortazavi et al. [151] found that Mo2C sheets could exhibit negative Poisson’s ratio which enabled the

Mo2C sheets could be useful for the application of nanodevices. Recently, using first-principles

33

calculations, Zhou et al. [154] reported that 2D rectangular materials of titanium mononitride (TiN)

could exhibit auxetic behaviour. Using a structure-matching procedure, Dagdelen et al. [155] reported

that negative or near-zero Poisson’s ratio is a common feature of materials with cristobalite-like

structures. Han et al. [14] investigated the thermal transport of auxetic carbon crystals and found that

thermal transport of carbon-based materials could be significantly enhanced by simple stretching. The

unusual feature of auxetic carbon crystals made them promising candidates for the application of

nanoelectronic devices.

3.2 Auxetic biomaterials

Auxetic biomaterials are biomaterials which could demonstrate negative Poisson’s ratio under certain

loading conditions. Although it is very difficult to obtain the accurate elastic properties of naturally

occurring auxetic biomaterials, some classical examples have still been reported, e.g., Williams et al.

[156] observed that cancellous bone from the proximal tibial epiphysis exhibits negative Poisson’s ratio.

Veronda et al. [157] examined the mechanical characterization of cat skin and found that the cat skin is

auxetic under a finite deformation. Lees et al. [158] conducted experiments on cow teat skin in uniaxial

and biaxial strain and found that the cow teat skin can present negative Poisson’s ratio at low strains.

All the above mentioned auxetic effects are believed to result from the fibrillar structures at

microstructure level.

Apart from the auxetic behaviour found in the macroscopic biomaterials mentioned above, auxetic

behaviour was also found at the microscopic scale of cells. Baughman [159] reported that the

membranes found in the cytoskeleton of red blood cells demonstrate negative Poisson’s ratio. Wang

[160] reported that the nuclei of embryonic stem cells (ESCs) extracted from mouse were found to be

auxetic during the transitioning towards differentiation, as shown in Figure 34. However, the

mechanism which drove this auxetic phenotype was not presented. Pagliara et al. [161] reported a

similar finding that the nuclei of ESCs are auxetic. They also concluded that the auxetic phenotype of

transition ESC nuclei is driven at least partly by global chromatin decondensation. Auxeticity could be

34

a significant element in mechanotransduction through the regulation of molecular transformation in the

differentiating nucleus by external forces. In a recent work of Yan et al. [162], through fabricating

auxetic polyurethane scaffolds with various elastic modulus, Poisson’s ratio, and microstructure and

estimated neural differentiation of pluripotent stem cells, the authors concluded that the microstructure

and Poisson’s ratio of auxetic scaffolds may enhance neural differentiation. By utilizing the dynamic

optical projection stereolithography (DOPsL) system, Warner et al. [163] fabricated non-positive

Poisson’s ratio (NPPR) scaffold which could be used in stretch-mediated cell differentiation

applications. The auxetic properties of the fabricated tissue-scale scaffolds could be adjusted through

fabrication parameters. Through theoretical prediction, Yamamoto and Schiessel [164] reported that

chromatin gels could exhibit negative Poisson’s ratio due to cooperative nucleosome assembly and

disassembly dynamics.

Figure 34. Nucleus of an embryonic stem cell in the transition state (T-ESC) expands when stretched. [160]

4. Metallic auxetic materials and structures

Materials and structures that could demonstrate auxetic behaviour, and whose base materials are metal,

are defined as metallic auxetic materials and structures in this section. Although microscopically, 69%

of the cubic elemental metals were reported to have a negative Poisson’s ratio when stretched along the

[110] direction [142], very few metallic auxetic materials and structures have been reported in the

macroscopic scale, especially for metallic auxetic materials and structures which could demonstrate

negative Poisson’s ratio behaviour in a large strain range.

35

The base materials of the majority of the existing literature related to auxetic material and structures are

polymeric materials, which tremendously restricts the application for auxetics where enhanced

mechanical properties can be provided by metallic materials, such as strength and stiffness. The

pioneering work on metallic auxetics was a non-periodic copper foam with a negative Poisson’s ratio

reported by Friis et al. [165] in 1988. Li et al. [166] extended this work using resonant ultrasound

spectroscopy to explore the properties of copper foams with negative Poisson’s ratio, as shown in Figure

35. The minimal Poisson’s ratio of around -0.7 was observed for the tested sample with a permanent

compression strain in this study.

Figure 35. Microstructures (optical, reflected light) of copper foam with different volumetric compression ratio: (a) 1; (b)

4.34; (c) 4.94. [166]

The auxetic behaviour of crystalline solids was believed to be rare, however, Baughman et al. [142]

stated when stretched along the [110] direction, the 69% of the cubic elemental metals exhibit a negative

Poisson’s ratio. Zhang et al. [167] explored the auxetic properties of iron-gallium and iron-aluminium

alloys using both theoretical and experimental approaches. A good agreement between the experiment

and theory indicated the validity and effectiveness of using the density functional calculations for

determining auxetic properties. The negative Poisson’s ratios resulted from elastic anisotropy. In

addition, Fe75Ge25 was predicted to have a significant negative Poisson’s ratio value as low as -0.9

through the proposed theoretical approach. Schwerdtfeger et al. [168] conducted a thorough

investigation of the mechanical properties of a non-stochastic cellular auxetic structure. The base

material of Ti-6Al-4V was employed to build experimental samples using the selective electron beam

melting method. It was found that the Poisson’s ratio of the structure strongly depends on the relative

36

density of the structure. Using selective electron beam melting, Mitschke et al. [169] fabricated a linear-

elastic cellular structure from Ti-6Al-4V alloy whose linear-elastic properties were measured by tensile

tests and a negative Poisson’s ratio of -0.75 was observed. Zhang et al. [107] proposed an auxetic

structure made of tubes and corrugated sheets, and a real structure was fabricated using aluminium

material. The compression test exhibited that the proposed structure has an auxetic effect and a wide

range of application due to its easy fabrication. Dirrenberger et al. [170] conducted full-field simulations

of an auxetic microstructure and concluded that the auxetic effect persists and becomes even stronger

with plastic yielding. Besides, it was also found that the effect of plasticity on auxeticity reduces with

the expansion of the plastic zone. Recently, Taylor et al. [171] demonstrated that low porosity metallic

periodic structures exhibited a negative Poisson’s ratio. The experiments were performed on aluminium

cellular plates, manufactured using the CNC machine as shown in Figure 36.

Figure 36. Samples comprised of different cavities in the undeformed configuration: (a) with circular holes; (b) with

elliptical holes. [171]

Auxetic effects of metamaterials and structures were previously regarded to be mainly dependent on

the geometrical configuration rather than chemical composition or base material properties. In contrast,

the previous work of Ren et al. [16, 172] offered solid evidence that the base material has significant

effect towards the auxetic behaviour of 3D auxetic metamaterials and 3D tubular structures. It was

found that when the base material of elastomer was replaced with the metallic base material, the auxetic

effect of the buckling-induced 3D auxetic metamaterial and auxetic tubular structure would disappear.

This finding has been validated both numerically and experimentally for the 3D auxetic metamaterial

and numerically for the 3D auxetic tubular structure.

In order to regain the auxetic effect for the buckling-induced 3D auxetic metamaterial and auxetic

tubular structure when the base material of elastomer is replaced by metallic material, a novel

37

methodology for generating 3D metallic auxetic metamaterials and auxetic tubular structures have been

developed, and its procedures can be summarized as four steps, as shown in Figure 37. These steps are:

designing buckling-induced 3D auxetic metamaterial or auxetic tubular structure; conducting buckling

analysis of the original finite element model on 3D auxetic metamaterial or auxetic tubular structure

with linear elastic base material; identifying the desirable buckling mode for the 3D auxetic

metamaterial or tubular structure; and altering the geometry of the representative volume element (RVE)

using the desirable buckling mode and repeating the altered RVE in three axial directions to form a 3D

metallic auxetic metamaterial, or altering the initial tubular structure using the desirable buckling mode

to form an auxetic tubular structure.

The proposed 3D metallic auxetic metamaterials and tubular structures possess superior properties

compared to conventional auxetics. One of the most significant advantages of the 3D auxetic

metamaterial and 3D auxetic tubular structure is that the mechanical properties of these structures could

be easily tuned by one single parameter of pattern scale factor (PSF). In addition, these models could

exhibit auxetic behaviour both in compression and tension. When a certain value of PSF (~60% for the

proposed models [16, 172]) is imported to the initial geometry, the proposed models could even

demonstrate an approximately identical auxetic performance in compression and tension which is

crucial for some applications such as sensors and smart filters. Another advantage of the proposed 3D

metallic auxetic metamaterial and 3D auxetic tubular structure is that the base material is metal which

makes these 3D metallic auxetics much stronger and stiffer with respect to mechanical performance.

Hence, the proposed 3D metallic auxetics have potentials to be used as protective devices to absorb

impact energy and reduce impact loading. In addition, the proposed 3D metallic auxetics could maintain

auxetic effect in a large effective strain range which means these auxetics could demonstrate a large

auxetic deformation that is beneficial for biomedical applications such as oesophageal stent and blood

vessel. Lastly, good symmetric geometry is an apparent merit of the proposed auxetics which is vital

when a symmetrical and precise auxetic effect is required, such as 3D sensors.

38

Figure 37. Methodology of generating 3D metallic auxetic metamaterials and 3D metallic tubular structures: (a) 3D metallic

auxetic metamaterial; [16] (b) 3D metallic auxetic tubular structure. [172]

5. Multi-material auxetics and auxetic composites

Multi-material auxetics are auxetics which composed of more than one single base material, and

auxetic composites are composed of auxetic materials and non-auxetic materials. Most of the existing

literature in terms of auxetics are focused on one single base material. However, it is promising to

investigate multi-material auxetics and auxetic composites which enable us to combine the desirable

auxeticity with preferred properties which are not possessed by one single base material. Some of the

works on multi-material auxetics and auxetic composites are discussed in this section.

5.1 Multi-material auxetics

39

Hiller et al. [173] conducted a pioneering work to answer how the auxetic effect would be if multiple

materials are used to generate auxetic materials and structures. In this work, using digital material and

inclusion of hierarchical voxel substructures, a novel auxetic composite with dense combinations of

common materials was obtained, as shown in Figure 38a, Theoretically, a negative Poisson’s ratio can

be obtained by combining any stiff and flexible materials. A digital material made of 6 × 8 × 2 voxel

base unit consisting of 68% aluminium, 48% acrylic and 8% voids was observed to produce a minimal

Poisson’s ratio of -0.63 as shown in Figure 38b. Using the struts of three different types of crystal

structures (FCC, BCC and simple cubic configurations) with different elastic moduli, Hughes et al.

[174] generated a periodic truss structure with tuneable auxetic properties. They also concluded that it

is possible to design an auxetic truss structure with a specific Poisson’s ratio, shear modulus and tensile

modulus. Employing the computer-aided design and dual-material 3D printing techniques, Wang et al.

[175] designed and fabricated some dual–material auxetic metamaterials which are remarkably different

compared to the of traditional single-material based auxetic metamaterials. In this work, the effects of

two novel design parameters introduced by the dual-material nature, e.g., the material selections and

fraction of the stiff region were explored both computationally and experimentally. With the

introduction of another material, researchers could adjust the mechanical properties of bulk

metamaterials without changing the overall geometry of an auxetic unit cell. In the work of Vogiatzis

et al. [176], topology optimization method was employed to generate auxetics with multi-material, and

both numerical simulations and physical experiments of their study proved that the achieved design

demonstrated a desirable auxetic behaviour.

40

Figure 38. Images of a digital material with negative Poisson’s ratio: (a) a digital auxetic material; (b) the resulting Poisson’s

ratio as a function of strain for material with random and auxetic voxel structure. [173]

5.2 Composite auxetics

The main purpose of research on composite auxetics is to combine the advantages of composites and

auxetics to expand the potential applications. The existing researches on composite auxetics are based

on the four aspects [64]: 1) generating negative Poisson’s ratio through sequential piling of angle ply-

reinforced laminates; 2) producing auxetic composites by introducing auxetic inclusions or using the

auxetic matrix; 3) exploring and evaluating the properties of auxetic composites; and 4) manufacturing

of auxetic composites.

As a pioneer, Herakovich [177] reported an early work on composite laminates with negative through-

the-thickness Poisson’s ratios in 1984. In this study, he concluded that the laminate dilatation can be

positive or negative, mainly depending on fibre orientation. In 1989, Miki et al. [178] investigated the

unique behaviour of the Poisson’s ratio of laminated fibrous composites and observed a minimal value

of Poisson’s ratio of -0.369 exists on unbalanced bi-directional laminates. Milton [179] reported that a

kind of two-dimensional, two-phase, composite materials with hexagonal symmetry exhibits a

Poisson’s ratio close to -1. He also concluded that by layering the component materials together in

different directions on widely separated length scales, elastically isotropic two and three-dimensional

composites with a Poisson’s ratio close to -1 could be easily generated. Chen et al. [180] conducted an

experimental investigation of viscoelastic properties of composites made of traditional and re-entrant

auxetic copper foam as a matrix. In the work of Zorzetto et al. [181], a new architectured composite

41

material was generated by combining two basic cellular structures with contrasting Poisson’s ratio and

they found that a small fraction of re-entrant inclusions (around 12%) was enough to generate a

significant augmentation in stiffness (300%) at a constant overall relative density. By employing finite

element methods, Nkansah et al. [182] examined the elastic properties of continuous-fibre reinforced

composites, and they found that an auxetic material can be used as the matrix in a continuous-fibre

composite to increase the value of the transverse composite modulus without decreasing the

longitudinal modulus. By conducting a series of experiments and comparing the laminate theory, Clarke

et al. [183] concluded that laminate theory could make accurate predictions of the composite properties

and validate the assumption of homogeneous strain. Using homogenization theory, Gibiansky et al.

[184] proposed optimal piezocomposites for hydrophone applications, and they found that the optimal

matrix is highly anisotropic and is characterized by negative Poisson’s ratios in certain directions.

Theocaris et al. [185] investigated the variation of Poisson’s ratio in fibre composites using

homogenization method. In this study, they concluded that the shape and the ratio of shear-to-bending

of the beams have a significant influence on the value of Poisson’s ratio, and this conclusion is still

valid for continua with voids, composites with irregular inclusions. Zahra et al. [186] reported that

delamination and brittleness suffered by the cementitious composites could be eliminated through

auxetic material embedment. Wei et al. [187] proposed that the relatively low Young’s modulus of

existing auxetics could be improved by embedding an elastic material with sufficiently high modulus

with auxetics. This novel idea was theoretically validated, and they found that such composite materials

exhibit auxetic behaviour when the inclusion volume fraction of embedded elastic material is below a

certain value. Zhang et al. [188] reported that the ply orientations for achieving maximum transverse

strain in a composite laminate are close to [70°/20°]s. Using a specially designed software, Evans et al.

[189] reported that researchers could match the mechanical properties of laminates with predicted

negative Poisson’s ratio to those with similar mechanical properties but positive Poisson’s ratio. The

fabricated samples and experiments demonstrated a good agreement with theoretical predictions.

Alderson et al. [45] presented two different methods to fabricate auxetic composite. The first one was

to use off-the-shelf prepregs and, by variations of the stacking sequence employed to design an auxetic

composite with a through-the-thickness or in-plane negative Poisson’s ratio. The second one was to use

42

auxetic constituents as part of the composite, and they concluded that the fibre pullout was resisted due

to the auxetic deformation of the fibres. Tatlier et al. [190] proposed a modelling method to investigate

the auxetic behaviour of compressed fused fibrous networks and they found that compression and

anisotropy are the critical parameters that result in auxetic behaviour for these materials. Sigmund et al.

[191] designed 1-3 piezo-composites with optimal performance for hydrophone applications by

employing a topology optimization method. Subramani et al. [192] explored the development of auxetic

structures from composite materials, and the mechanical properties of these auxetic structures were also

characterized. Based on the experimental and analytical result, they concluded that auxetic behaviour

and tensile characteristics of the proposed structures significantly depend on the initial geometric

configuration. A Poisson’s ratio range of -0.30 to -5.20 was observed for the proposed auxetic structures.

Zhang et al. [193] explored the Poisson’s ratio behaviour of a further development of the helical auxetic

yarn. The proposed three-component auxetic yarn was based on a stiff wrap fibre helically wound

around an elastomeric core fibre coated by a sheath, as shown in Figure 39. They concluded that the

coating thickness can be used as a new design parameter to tune both the Poisson’s ratio and modulus

of this novel composite reinforcement.

Figure 39. Three-component auxetic yarn. [193]

Recently, Jiang et al. [9] conducted a study on low-velocity impact response of multilayer orthogonal

structural composite with auxetic effect, and they concluded that the auxetic composite had better

energy absorption performance in medium strain range. Another interesting work was conducted by

Valentini et al. [194], where a biogenic successful method was reported to transform conventional

silicone rubber composites to auxetic robust rubbers. Cicala et al. [195] reported a truss-core structure

made of hemp/epoxy biocomposite based on a topology with auxetic characteristics and found that

hexachiral biocomposite truss core exhibited specific shear modulus and higher strength. Also, the use

43

of biocomposites as cores for cellular structures has potential to generate novel truss-core configurations

with superior sustainability, lightweight and stiffness characteristics for many structural applications.

In a recent work of Ghaznavi et al. [196], through a finite-element-based global-local layerwise theory

and algorithm, the authors found that auxeticity of the core could significantly stiffen the core and plate

of composites and reduce the lateral deflections of the plate. Michelis and Spitas [195] fabricated high-

strength auxetic triangular cores by utilising directionally reinforced integrated single-yarn. In the work

of Chen and Feng [197], the nonlinear dynamic behaviour of a thin laminated plate embedded with

auxetic layers was investigated. It was found that the natural frequency of the thin laminated plate

increased with the absolute value of Poisson’s ratio. Bubert et al. [198] designed and fabricated a

passive one dimensional morphing aircraft skin. The combined system included an elastomer-fiber-

composite surface layer that is supported by a flexible honeycomb structure, each of which demonstrate

a near-zero in-plane Poisson’s ratio. Through experimental investigation, Polpaya et al. [199] reported

that auxetic behaviour may contribute to changes in conductivity of polymer composites. In the work

of Chen et al. [200], a class of fiber-reinforced composite flexible skin with in-plane negative Poisson’s

ratio behaviour was manufactured and investigated. Recently, Hu et al. [201] fabricated novel hydrogel-

elastomer auxetic composite materials and they found that gel inclusion can temporarily recover

fractured ligaments and cracks on the matrix material. As an innovative and sustainable solution for

engineering applications, Silva et al. [202] utilised recycled rubber to fabricate re-entrant auxetic

structures and conducted preliminary investigations. Filho et al. [203] carried out failure analysis and

Taguchi design of auxetic recycled rubber structures. Utilising numerical and experimental methods,

Boldrin et al. [204] investigated dynamic behaviour of auxetic gradient composite hexagonal

honeycombs.

6. Properties of auxetic materials and structures

Auxetic materials and structures possess counter-intuitive deformation behaviour which makes these

auxetics endowed with many superiors comparing with conventional materials. The primary properties

of auxetic materials and structures are presented in the following sections.

44

6.1 Shear resistance

Under shear forces, auxetic materials are known to be more resistant than regular materials. According

to the classical theory of elasticity for three-dimensional isotropic solids, the elastic behaviour of a body

can be presented by two of the four constants [205]: the Young’s modulus (E), the shear modulus (G),

the bulk modulus (K) and the Poisson’s ratio (ν) [206]. In three-dimensional cases, the relationship

between these constants can be presented by two equations as below:

12 213K

G

(19)

)1(2

E

G

(20)

It can be clearly seen that the value of the shear modulus increases when the Poisson’s ratio decreases,

resulting in a consequent enhancement for shear resistance. The range of elastic modulus

corresponding to instability and stability under different conditions is shown in the map in Figure 40.

This map indicates that the Poisson’s ratio of the isotropic solid has to be in the range of -1 to 0.5.

When the value of Poisson’s ratio approaches -1, the shear modulus would be infinity.

45

Figure 40. Map of elastic material properties corresponding to different values of bulk modulus K and shear modulus G.

[207]

6.2 Indentation resistance

Under an indentor local compression, the conventional material would spread in the direction

perpendicular to the applied load [48] as shown in Figure 41a. In contrast, an indentation would occur

if the same compression is applied on an isotropic auxetic material, and the material flows into the

immediate region of an impact as shown in Figure 41b. Using holographic interferometry, Lakes et al.

[208] reported that the indentation resistance of foams, both of conventional structure and of the re-

entrant structure resulting in auxetic behaviour. According to the classical theory of elasticity, the

indentation resistance is closely related to the material hardness (H), which could be correlated to the

Poisson’s ratio by the following equation:

2

1E

H

(21)

where E is Young’s modulus, ν is the Poisson’s ratio of the base materials and γ is assumed to be 1 or

2/3 in the scenario of uniform pressure distribution or hertzian indentation, respectively. As can be

clearly seen from equation (21), when the value of ν approaches -1, the indentation resistance tends to

46

infinity [209]. When the value of ν reaches the maximal limit for 3D isotropic solids of 0.5, the

indentation resistance would be much lower. However, because the maximal value of ν is 1 for the 2D

isotropic system [210, 211], the materials with such positive Poisson’s ratio would possess an infinite

hardness as well. Argatov et al. [25] conducted a theoretical work which was regarded as the first step

towards an indentation and impact analysis of real auxetic materials. In the work of Coenen et al. [26],

enhancements in indentation resistance were seen for the auxetic laminates with smaller, more localized

damage areas for the two larger diameter indentors where delamination was concluded as the main

failure mechanism. Dirrenberger et al. [132] conducted a series of numerical simulations on cylindrical

and spherical elastic indentation tests to investigate effective elastic properties of auxetic

microstructures. They concluded that auxetics can be superior to honeycomb cells in terms of

indentation strength under certain conditions.

Figure 41. Indentation behaviours: (a) conventional material; (b) auxetic material. [212]

6.3 Fracture resistance

Materials which exhibit negative Poisson’s ratio are reported to have a better fracture resistance than

conventional materials [21, 213]. These auxetic materials were also reported that have low crack

propagation [24]. Through an experimental investigation, Donoghue et al. [214] concluded that more

energy was required to propagate a crack in the auxetic laminate. Maiti et al. [215] observed crack

growth as shown in Figure 42.

47

Figure 42. Schematic of crack extension manners in a cellular solid: (a) through the bending failure mode of the non-

vertical cell elements; (b) through the tensile fracture of the vertical cell elements. [215]

Liu [216] presented a detailed work on discussing the fracture mechanics side of auxetic materials. The

non-singular stress field, at the distance r for a crack of 2a with crack tip radius rtip and stress intensity

factor KI, is given [27, 215-217]:

r

r

r

K

r

Ktip

ll 2

22

(22)

Then, the force acting on the cell rib is:

drr

r

r

r

K

r

K

Ftip

t

r

rtip

II

tip

tip

)

2

(

)

2

(2

22

(23)

Furthermore, with the thickness of the rib being t, and the first order of the Taylor expansion,

equation (23) can be simplified to:

l

tl

KF I

*

38.2

(24)

48

where

*

I

K

is the stress intensity of the conventional foam, and l is the rib length. The stress results

from the bending moment is given by:

3

12.2 t

Fl

F

(25)

Substituting equation (24) and the stress becomes:

2

*1

05.5

t

l

KF I

(26)

The crack propagation will occur when

f

, where

f

is the fracture strength of the cell rib. The

critical stress intensity factor or the fracture toughness can therefore be calculated as:

2

*20.0

l

t

lK fI

(27)

Because

n

slt )/(/

*

the stress intensity factor of conventional foams is proportional to the

normalized density:

s

f

Il

K

*

19.0

*

(28)

For the fracture toughness of re-entrant structure, the similar equations (27) and (28) become:

s

f

IC l

K

*

2cos12

sin1

10.0

*

(29)

49

where

*

IC

K

is fracture toughness of re-entrant foams and

is the rib angle as shown in Figure 42.

Choi et al. [27] conducted an experimental investigation. According to their observation, the following

equation could be generated:

2cos1 2

sin1

53.0

*

IC

r

IC

K

K

(30)

In the work of Bhullar et al. [218], they concluded that when comparing with non-auxetic materials,

auxetic materials have almost twice crack resistance to fracture. Similar to this conclusion, in a recent

work of Yang et al. [219], they stated that auxetic composites showed approximately two times fracture

toughness than conventional composites.

6.4 Synclastic behaviour

When subjecting an out-of-plane bending moment, conventional materials exhibit a saddle shape and

auxetic materials demonstrate a dome-shape, as shown in Figure 43. The dome-shape deformation

pattern shown in Figure 43b can be also called synclasticity. This property is reported to be very useful

[220] based on the fact that it provides a way to fabricate a dome-shaped structure with using damaging

techniques nor additional machining [221].This uncommon property is believed to have wide potential

to be used in the medical areas.

50

Figure 43. Deformation patterns for non-auxetic and auxetic materials under out-of-plane bending: (a) saddle shape

(non-auxetic); (b) dome shape (auxetic). [222]

6.5 Variable permeability

Because most of the well-known auxetic materials possess porous microstructures and the sizes of the

pores of auxetic materials could vary during the compressive and tensile deformation, utilizing this

behaviour, auxetic materials are believed to have a significant potential for the application of filters.

This behaviour can be illustrated in the Figure 44. In 2001, Alderson et al. [223] reported a pioneering

work which illustrates that how auxetic materials offer improved filter performance from the macro-

scale to the nano-scale due to their unique pore-opening properties and characteristics. As an extension

of their previous work, Alderson et al. [224] conducted glass bead transmission tests on auxetic

polyurethane foams. They confirmed the benefits in mass transport applications because of auxeticity

persists in 3D macroscale filters, and in 3D sieves at any scale which exhibit a significant tortuosity in

the pore structure.

Figure 44. Smart filters to demonstrate the variable permeability. [225]

6.6 Energy absorption

In terms of the performance of energy absorption, auxetic materials are reported to be superior to

conventional non-auxetic materials. Chen et al. [226] conducted an investigation on the in-plane elastic

buckling of hierarchical honeycomb materials. The study on the stress/strain law and deformation

energy indicated that specific energy absorption would be enhanced when the hierarchical level n is

51

increased. Yang et al. [227] carried out an extensive numerical simulations on ballistic resistance of

sandwich panels with aluminium foam and auxetic honeycomb cores. They found that the auxetic-cored

sandwich panel is far superior to the aluminium foam-cored panel in ballistic resistance due to the

material concentration at the impacted area resulting from the auxetic behaviour. Mohsenizadeh et al.

[10] conducted a comprehensive study both in simulations and experiments to investigate the

mechanical properties of the auxetic foam-filled tube under quasi-static axial loading. They found that,

in terms of all studied crashworthiness indicators, the auxetic foam-filled square tube is superior to

empty and conventional foam-filled square tubes. Imbalzano et al. [8] reported that, under blast, auxetic

composite panels could absorb double the amount of impulsive energy via plastic deformation, and

reduce up to 70% of the back facet’s maximum velocity compared with monolithic ones. In another

work, Imbalzano et al. [34] also compared the blast-resistance performance of a re-entrant auxetic

composite panel with an equivalent honeycomb composite panel. It was found that a considerable

improvement in the shockwave mitigation and impulsive load absorption by using the auxetic panel.

Recently, Qi et al. [228] reported that auxetic panels were superior to conventional honeycomb panels

of the same dimension, areal density and material. Employing numerical simulations, Jin et al. [229]

reported that the auxetic re-entrant cell honey-comb sandwich structures have a better performance than

the hexagonal cell honeycomb sandwich structures. Scarpa et al. [230] investigated the acoustic

properties of iron particle seeded auxetic polyurethane foam and they found that the auxetic

polyurethane foam possesses intrinsic higher acoustic absorption properties comparing with

conventional open-cell foams. In the work of Ruzzene et al. [231], the attenuation capabilities of the

auxetic lattice and their design flexibility were demonstrated. Through numerical and experimental

investigation on a planar auxetic metamaterial (PAM), He and Huang [11] found that the proposed

structure composed of the PAMs demonstrated great tenability and significant advantages over the

regular materials for controlling sound wave propagation and filtering sound waves in a certain

frequency ranges.

7. Applications of auxetic materials and structures

52

Because of the counter-intuitive behaviour which auxetic materials and structures exhibit during

deformation, many desirable properties are offered to these smart materials which make them have a

huge potential in many applications. Prawoto [217] summarized the applications of auxetic materials

from various work [30, 142, 165, 212, 232, 233], as shown in Table 1.

Table 1: Summary of the applications of the auxetic materials [217]

Field

Application and rationale

Aerospace

Vanes for engine, thermal protection, aircraft nose-cones, wing panel, vibration absorber

Automotive

Bumper, cushion, thermal protection, sounds and vibration absorber parts, fastener

Biomedical

Bandage, wound pressure pad, dental floss, artificial blood vessel, artificial skin

drug-release unit, ligament anchors. Surgical implants

Composite

Fibre reinforcement (because it reduce the cracking between fibre and matrix)

Military (Defence)

Helmet, bullet proof vest, knee pad, glove, protective gear (better impact property)

Sensors / actuators

Hydrophone, piezoelectric devices, various sensors

Textile (Industry)

Fibres, functional fabric, colour-change straps or fabrics, threads

7.1 Medical application

In the aspect of medical application, auxetics are reported to be of importance to act as foldable devices

such as angioplasty stents [234-236], annuloplasty rings [237] and oesophageal stents [238, 239].

Oesophageal cancer is regarded as one of the ninth most vital cancer in the world, the patients who have

this cancer may have tumours in their esophaguses and block the food go through. The auxetic stent

could be employed to expand their esophagus and release the pain of the patients resulting in a longer

lifespan, as shown in Figure 45. In the work of Ali et al. [238], an auxetic structure film was designed

and manufactured and this film was configured as an auxetic stent for the palliative treatment of

oesophageal cancer, and for the prevention of dysphagia. Later on, Ali et al. [239] discussed the

manufacture of a small diameter auxetic oesophageal stent and stent-graft. The tensile test of the auxetic

polyurethane film exhibits a Poisson’s ratio of -0.87 to -0.963 at different uniaxial tensile load values.

It was found that the diameter of auxetic oesophageal stent expanded from 0.5 to 5.73 mm and the

length of the stent extended from 0.15 to 1.83 mm at a certain pressure from the balloon catheter. Based

on the rotating rigid units, a new class of hierarchical auxetics was reported by Gatt et al. [240]. The

53

proposed stent with two levels hierarchical rotating square geometry, as shown in Figure 46, was

believed can reduce inflammation occurring through reducing the actual surface area of the solid portion.

Also, this hierarchical system was also reported to be very suitable for making skin grafts due to the

relieved pressure on the swelling area. In a recent work of Bhullar et al. [241], auxetic patterned thin

nanofiber membrane samples demonstrated almost ten times increase in their elongation capacity

compared with control sample of non-auxetic nanofiber membranes. Therefore, the auxetic patterned

thin nanofiber membrane was believed to have various biomedical applications, e.g., tissue engineering.

Figure 45. An auxetic tube (stent) can be employed to hold open a narrowed portion of the esophagus [242].

Figure 46. Hierarchical stent. [240]

54

7.2 Protective devices

Auxetic materials also possess a huge potential to be used for sports protective devices, e.g., pads,

gloves, helmets and mats [64]. Using auxetic materials in impact protector devices could offer better

conformability for support, and improved energy absorption for lighter and thinner components. Wang

et al. [53] reported that auxetic cushions could reduce the pressure which could bring more comfort.

The work of Michalska et al. [243] further proofed that auxetic materials could reduce contact stress

concentrations by conducting numerical simulations on a seat with an auxetic polyamide spring skeleton.

In the work of Sanami et al. [244], through numerical simulations, a new type of auxetic honeycomb

was reported to have potential in helmet applications, along with indentation test of auxetic and non-

auxetic foams for evaluating the applications of protective pads and running shoes. Also, auxetics made

by perforated plates and auxetic foams are potential candidates for protective devices.

7.3 Smart sensor and filter

Another promising application of auxetic materials is manufacturing piezoceramic sensors, and it is

reported that auxetic materials may improve the performance of piezoelectric actuators by more than

an order of magnitude [42]. The piezoelectric composites including piezoelectric ceramic rods within a

passive polymer matrix are functioning by converting a mechanical stress into an electrical signal and

vice versa, as shown in Figure 47. An auxetic smart material with magneto-elastic properties, iron-

gallium, known as Galfenol, has been investigated by Raghunath et al. [245, 246], Schurter et al. [247]

and Zhang et al. [167].

Shape memory, as a desirable property of a material, can remember its original shape despite plastic

deformation [248]. Bianchi et al. [248] found that the shape memory effect had a significant impact in

the mechanical behaviour of the auxetic foams. Hassan et al.[249] proposed a new functional structure

combining the chiral honeycomb topology and shape memory alloys (SMA) as a novel concept of smart

cellular solid. Inspired by an auxetic structure, Jacobs et al. [250] designed, manufactured and tested a

55

deployable SMA antenna. Rossiter et al. [95] proposed and investigated some new shape memory

auxetic deployable structures that needed no external actuation mechanisms. Hassan et al. [251]

investigated in-plane tensile behaviour of SMA honeycombs with positive and negative Poisson’s ratio.

Figure 47. Auxetic piezoelectric sensor. [212]

The desirable property of various permeability which possessed by auxetic material, can be used to

fabricate smart filters. In the work of Alderson et al. [28], simple experiments were performed which

demonstrate that auxetic materials have superior properties than conventional materials in filter

defouling and controlled pore size applications. Passage pressure could be controlled using smart filters

[223].

7.4 Auxetic nails

Back to the year of 1991, Choi et al. [252] reported that an auxetic fastener could be easier to insert and

harder to pull out. Based on a similar concept that auxetic nails become thinner when knocked in and

become fatter when pulled out, as shown in Figure 48, Grima et al. [253] mentioned that auxetic nails

could be a potential application for auxetic materials. However, no work regarding auxetic nails has

ever been conducted. Until recently, the first auxetic nails were designed, fabricated and experimentally

investigated by Ren et al. [254], as shown in Figure 49. It was found that auxetic nails do not always

exhibit superior push-in and pull-out performance to conventional nails. Designing and fabricating

metallic auxetic nails with better push-in and pull-out performance is still a challenging work.

56

Figure 48. Illustration of auxeticity for auxetic nails: (a) during push-in; (b) during pull-out. (The nails in grey and red

colour represent the configurations of the nails before and after deformation, respectively) [254]

Figure 49. 3D printed twelve different types of nails in four nail groups using brass and stainless steel materials: (a) auxetic

nails (ANs); (b) nails with circular holes (CNs); (c) solid nails (SNs). (The nails with gold colour and silver colour are printed

using brass and stainless steel, respectively) (scale bar: 10mm) [254]

7.5 Textile

Auxetic textile materials are increasingly popular because they could provide comfort, higher energy

absorption, high volume change, wear resistance and drapeability [255]. The auxetic textile materials

are mainly generated through two methods. The first one is to use auxetic based fibres to knit and weave

textiles directly [256]. The second method to produce auxetic textiles is to use conventional fibres to

weave or knit in a way which could make the textile production to be auxetic [255]. Recently, a novel

57

kind of 3D auxetic fabric was reported to have auxetic behaviour in all the fabric plane directions. In

the commercial market, auxetic textile has occurred, e.g., GoreTex and polytetrafluorethylene [209],

and some sports shoes recently released by Under Armour company and Adidas company are shown in

Figure 50.

Figure 50. Sport shoes with: (a) auxetic skin; auxetic sole.

8. Concluding remarks

8.1 Conclusions

This paper aims to provide a comprehensive review on auxetic materials and structures, including

various types of cellular auxetics, natural and artificial auxetics, metallic auxetics, multi-material and

composite auxetics. Superior and unusual properties of auxetics are presented and some existing or

potential applications are summarized in details. It should be noted that although we have made best

effort to cover most literature in the field of auxetic metamaterials and structures, due to the limited

space, not all the works regarding auxetics are included in this review.

Remarkable progress has been made in the past three decades in the field of auxetics, including

theoretical analysis, finite element simulations and experiments. However, many important and

interesting problems still require further investigations. First of all, constrained by traditional

manufacturing techniques, most of the previous studies of auxetics are based on simple 2D models.

58

Besides, although some 3D auxetic materials have been reported, most base materials of these 3D

auxetics are rubber-like materials which could only sustain a very limited loading force and impact.

Apart from that, most of the existing 3D auxetic materials only exhibit negative Poisson’s ratio in a

small effective strain range which greatly limits applications of these novel materials. More importantly,

the geometries of the majority of the existing 3D auxetics are predesigned which often creates difficulty

to tune their mechanical properties. Lastly, all reported auxetic tubular structures may not demonstrate

auxetic behaviour under compression which also constrains a wider application for auxetic tubular

structures. In order to fill the research gap of auxetics, a novel and recent methodology for generating

3D auxetic metallic materials is particularly reviewed, and successful cases are presented with

numerical simulations and experiments.

8.2 Challenges of auxetic materials and structures

Firstly, the cost of manufacturing auxetic materials is still too high. For most of the 3D auxetic materials

with reliable auxetic behaviour, the experimental specimens are fabricated using 3D printing technique.

Although 3D printing technique is convenient in fabrication and enables engineers to concentrate on

the design itself without considering too much of the manufacturing procedure, it is still a crucial

problem to realize mass production with a lower cost which significantly constrains the wide application

for auxetics.

More importantly, all of the reviewed auxetic materials and auxetic structures have a substantial

porosity in their geometrical configuration which inevitably reduces their mechanical capability when

sustaining a load or impact. That is to say, obtaining some desirable properties of auxetic materials,

such as enhanced shear resistance, improved indentation resistance and superior energy absorption, is

actually at the cost of sacrificing the mechanical performance in the beginning when compared with

solid materials. Therefore, in the most scenarios, the obtained auxetic behaviour could not compensate

for the loss of the mechanical performance resulting from the porous microstructure of auxetic materials

which significantly decreases the advantages of the auxetic behaviour, especially for the application of

59

energy absorption or protective devices. However, the shortage of the porous microstructure of auxetic

materials is rarely discussed.

Although the properties of auxetics are very desirable, it is not very practical to bring them into the

practical stage, especially when considering the cost performance. Also, these desirable properties of

auxetic materials are actually not indispensable. This may explain that although a certain amount of

auxetic materials have been designed and fabricated, very few of them have been used in the practical

stage.

8.3 Future work

For the last three decades, since Lakes reported the first re-entrant foam structures exhibit negative

Poisson’s ratio in 1987 which is regarded as a breakthrough in the fields of auxetics, a significant

progress has been made. Although many potential applications of auxetics have been proposed and

some preliminary application has been reported, most of the reported auxetics still maintain in their

infancy, and very few auxetics have been reported to the stage of practical application. Therefore, it is

worthy for more researchers to make efforts towards exploring the applications of auxetics and enable

normal people could witness and experience the advantages of auxetics rather than just maintaining at

the stage of laboratory study.

Recently, hierarchical materials and structures which demonstrate auxetic behaviour have attracted

considerable attention. Sun et al. [257] reported that hierarchical tubes exhibit auxetic behaviour under

longitudinal axial tension through theoretical method. Based on rotating units mechanism, Gatt et al.

[240] reported a hierarchical 2D system with auxetic behaviour by employing the numerical method.

Tang et al. [258] proposed a design of hierarchically cut hinges. The auxetic behaviour of this design

was validated both numerically and experimentally. A hierarchical configuration comprising of

rectangular perforation that exhibits auxetic behaviour was reported by Billon et al. [259]. However, all

of their works are mainly limited to the 2D cases, and it would be an interesting and vital work to design

a 3D hierarchical system which could demonstrate auxetic behaviour under deformation.

60

In addition, most of the researchers only focus on auxetic behaviour either in compression or tension

and most of the reported auxetic materials and structures could not demonstrate auxetic behaviour both

in compression and tension, let alone an auxetic material and structure which could exhibit an identical

auxetic behaviour both in compression and tension. In our previous work [172, 260], using our proposed

methodology of generating 3D auxetic metallic materials and structures [16, 172], when PSF value of

60% was chosen, the 3D auxetic metamaterial and 3D auxetic tubular structure could demonstrate a

nearly identical auxetic behaviour both in compression and tension, as shown in Figure 51. When this

auxetic material is compressed and stretched at the same vertical strain, the deformed horizontal strain

are the same. Therefore, we believe that this methodology could inspire more researchers to design and

investigate more auxetic materials and structures which have identical auxetic behaviour both in

compression and tension which could be very useful for the applications of biomedical devices and 3D

smart sensors.

Figure 51. Auxetic material with identical auxetic behaviour both in compression and tension.

Another interesting and promising work is to combine the auxetic property with other negative indexes

such as negative compressibility (NC) or negative thermal expansion (NTE) to generate some novel

and functionally advanced materials which satisfy the requirements for multifunctional and

multipurpose devices. A pioneering work was conducted by Ai et al. [261], and they reported that a

metallic metamaterial with biomaterial star-shaped re-entrant planar lattice structure demonstrates an

61

auxetic NTE behaviour through numerical simulations. Recently, through numerical simulation and

analytic method, Ng et al. [15] carried out a pioneering work to combine NTE with negative Poisson’s

ratio (NPR) on a dual-material re-entrant cellular metamaterial. As mentioned in the work of Huang et

al. [7], further investigations of the inter-relations among NPR, NC, and NTE are necessary, and the

relationships of these negative indexes are presented in Figure 52. In addition, in a recent work of

Hewage et al. [262], a mechanical metamaterial exhibiting negative stiffness and NPR was presented.

Figure 52. Relationship among NPR, NTE, and NC, and materials with bi-/trifold negative indexes (indicated by the

shadowed regions). [7]

In summary, most of the existing designs of auxetic materials are mainly based on the experience of

engineers. Although we proposed a novel methodology of generating 3D metallic auxetic materials and

structures, and the mechanical performance of the designed auxetics could be easily tuned by only one

single parameter of PSF, there are still some limitations for this methodology. The largest constraint is

that this methodology has to start with a geometrical configuration with the buckling-induced

mechanism. Therefore, the design freedom is tremendously constrained. However, using topology

optimization to design auxetic materials with preferred performance is a promising direction in the field

of auxetics [263]. Several successful optimized auxetic materials have achieved in recent years [264-

267], but all of these designs still remain in 2D. More 3D auxetic materials with preferable and superior

performance could be designed by using advanced topology optimization algorithms, such as

62

evolutionary structural optimization (ESO) [268, 269] and bi-directional evolutionary structural

optimization (BESO) [270, 271].

Acknowledgements

This work was supported by the Australian Research Council (DP160101400, LP150100906), the China

Scholarship Council (201306370057).

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