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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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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 Poissons 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 Poissons 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 Poissons 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 Poissons
ratio. After four years, materials with negative Poissons ratio were coined as auxetics or auxetic
materials by Evans et al. [22] for the sake of simplifying the long description of negative Poissons
ratio.
Typical mechanical metamaterials are materials with negative indexes, e.g., negative Poissons 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 rner 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
3
1cos )sin( blh
kE
3
)(l
t
bEk s
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.
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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 Poissons 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
Poissons 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
 
 
sin1 18
2
21
l
KEE h
13
000
011
011
1
000
0
0
2221
1211
E
SS
SS
S
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; (ef) 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.
18
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
20
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 Poissons 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
22
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 Poissons
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 Poissons 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 Poissons 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 Poissons 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 Poissons 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 Poissons 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, Poissons ratio, and microstructure and
estimated neural differentiation of pluripotent stem cells, the authors concluded that the microstructure
and Poissons ratio of auxetic scaffolds may enhance neural differentiation. By utilizing the dynamic
optical projection stereolithography (DOPsL) system, Warner et al. [163] fabricated non-positive
Poissons 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 Poissons 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 Poissons 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 dualmaterial 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 Poissons 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 Poissons 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
Poissons 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 Poissons
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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