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Seeing auxetic materials from the mechanics point of view: A structural review

on the negative Poissons ratio

Yunan Prawoto

Faculty of Mechanical Engineering, Universiti Teknologi Malaysia, 81310 UTM, Skudai, Johor, Malaysia

a r t i c l e i n f o

Article history:

Received 28 December 2011

Received in revised form 7 February 2012

Accepted 7 February 2012

Keywords:

Poissons ratio

Auxetic materials

Periodic microstructure

Disordered microstructure

Homogenization

a b s t r a c t

This paper summarizes research work related to materials with zero, or negative Poissons ratio, materi-

als which are also referred to as auxetic materials. This review puts an emphasis on computations and

aspects of their mechanics. It also considers diverse examples: from large structural, to biomedical appli-

cations. It is concluded that auxetic materials are technologically and theoretically important. While the

development of the research has been dominated by periodic/ordered microstructures, the author pre-

dicts that future research will be in the direction of disordered microstructures utilizing the homogeni-

zation method.

2012 Elsevier B.V. All rights reserved.

1. Introduction

Although normal materials contract when they are stretched,

auxeticmaterials areopposite: theyexpandlaterally whenstretched

longitudinally. In a book published in 1944, Love described a mate-

rial with negative Poissons ratio [1]. According to the knowledge of

the author, that is the first engineering mechanics fact finding

recorded, although materials with either negative or zero Poissons

ratio may have been known to exist more than a 100 years ago.

The next documented evidence of an auxetic material was found

38 years after that, in 1982 by Gibson[2,3]. He realized the auxetic

effect in the form of the two-dimensional silicone rubber or alumi-

num honeycombs deforming by flexure of the ribs.

The intentional development of the concept was first pub-lished inSciencemagazine in 1987 by Lakes 5 years after Gibsons

finding[4,5]. In his publication, Lakes did not use the term aux-

etic to refer to these materials. This terminology came 4 years

later, in 1991. The word is derived from the word atvgsijo1(read: auxetikos), which means that which tends to increase

and which has its roots in the word atvg1i1 (read: auxesis),which is the noun form of increase. This terminology was

coined by Evans et al., when they first fabricated the microphor-

ous polyethylene with negative Poissons ratio [68]. The concept

development and subsequently the fabrication of this prototype

provided the momentum for the modern day auxetic material,

which is the object of this review.

In his publication in 1999, Alderson claimed to have introduced

the novel elastic property with a negative Poissons ratio that char-

acterizes an auxetic material. Such a material becomes thicker

widthwise when stretched lengthwise, and thinner when com-

pressed. This is apparently contradictory to the response of many

common materials, which become thinner when stretched.

In 2004, Yang et al.[9]focused their review on molecular auxet-

ics, payingattentionmore to moleculardesignswitha view towards

nanotechnology. As a complement to their reviews and also some

other available reviews, e.g. [10], which reviews the subject with

the intention of the incorporation into the undergraduate curricu-

lum, the present paper covers more a general understanding from

the mechanics point of view and especially incorporating develop-

ments since their publications [9,10]. Alderson also revieweddevelopments in the modeling, design, manufacturing, testing,

and potential applications of auxetic cellular solids, polymers, com-

posites, and sensor/actuator devices in aerospace engineering[11].

Although not directly discussing auxetic materials, the most re-

cent review on this subject was published inNature Materials, enti-

tled Poissons ratio and modern materials. It was published at a

time when the present manuscript was being submitted for publi-

cation [12]. It mainly discusses the Poissons ratio. The auxetic

material is only part of the example of the larger topics of Poissons

ratio. That review accentuates the importance of auxetic materials

in modern engineering applications.

Reviews on this subject are available from long ago, e.g.

[12,13,911]. However, none of them focuses on its mechanics.

In contrast with the available reviews, the present review focuses

on the mechanics of auxetic materials.

0927-0256/$ - see front matter 2012 Elsevier B.V. All rights reserved.

doi:10.1016/j.commatsci.2012.02.012

Tel.: +60 167 279048; fax: +60 755 66159.E-mail address:[email protected]

Computational Materials Science 58 (2012) 140153

Contents lists available at SciVerse ScienceDirect

Computational Materials Science

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c o m m a t s c i


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The aim of this short review is to give some bases for the under-

standing of the mechanics of auxetic materials, and report on the

research works over the past two decades, especially their

mechanical and computational aspects. It hopes to stimulate more

research works on this subject matter.

2. Poissons ratio in the mechanics of materials

Poissons Ratio is so named because it was invented by Simeon

Dennis Poisson (21 June 178725 April 1840)[14], a mathematical

genius who was born in Pithiviers, France. It is defined as the ratio

of transverse contraction strain to longitudinal extension strain

with respect to the direction of stretching force applied. Tensile

deformation is considered positive and compressive deformation

is considered negative. The definition of Poissons ratio contains a

minus sign, so that normal materials have a positive ratio. It is usu-

ally represented by the lower case Greek nu, m .Almost all common engineering materials have positive Pois-

sons ratio, having a figure of close to 0.3 for most materials and

slightly less than 0.5 for rubbery materials [12,15]. It is logical tothink that Poissons ratio is typically positive. This agrees with

the microstructural point of view, whereby the reason for the usual

positive Poissons ratio is that inter-atomic bonds realign with

deformation. In order to do that, the system tends to maintain its

density, resulting in contraction in the lateral direction. The corre-

lation between the Poissons ratio and the atomic packing density

is known: for materials that are atomically dense, such as gold,

typically their Poissons ratio ? isotropic upper limit, which is

0.5, while crystalline metals that are atomically less densely

packed, such as steels, have their Poissons ratio 0.3. Naturally,

to compare common crystal structures, face centered cubic, hexag-

onal closed packed, body centered cubic, and cubic diamond, their

Poissons ratios would be in sequence:mfcc,hcpP mbccP mcd[16]. The

same reason and mechanism make the typical value of Poissonsratio for ceramics, glass, and semi-conductors become 0.250.42

[1720].

Most engineering materials have a higher shear modulus G

than their bulk modulus K. By changing the microstructure of a

material in such a way that the Poissons ratio m is lower, the val-ues of K and G can be altered if E is kept constant. Naturally,

decreasing the value ofm to zero or below would result in a highshear modulus G relative to the bulk modulus K that can be

obtained.

From the continuum mechanics point of view, most materials

resist a change in volume as determined by the bulk modulus K

more than they resist a change in shape, as determined by the

shear modulus G . Meanwhile, the relation between E, G, and Kis

as follows [21,19]:

G E21 m 1

and

K E31 2m 2

Combining Eqs.(1) and (2), the following is obtained:

1 m1 2m

3K2G

3

A graphical depiction of this relationship is shown in Fig. 1. For

conventional structural engineering materials, the values ofKare

typically larger than the values ofG, which leads to

1 m1 2m

P3

2 4

This restricts a conventional structural material to have its Pois-sons ratio to bemP 1/8. For a Poissons ratio to bem 6 0, the valueof the bulk modulus must be much less than the shear modulus,

K G. Meanwhile, Eq.(3)can also be expressed as:

2G1 m K1 2m 5or

m 3K 2G2G 6K 6

Eq.(6)is graphically depicted inFig. 2. The figure also shows the

contour line ofm= 0. While classical mechanics treats Poissons ra-tio as a static component, a dynamic approach is also available. An-

other point of view for expressing Poissons ratio is by the speeds of

sound[19]:

md1

2

VtVl

2 1

VtVl

2 1

7

where Vtis the transversal speed or shear wave velocity, Vl is the

longitudinal speed traveling in the particular material, and md isthe dynamic Poissons ratio. As an example, a liquid with a transver-

sal speed ofVt= 0 would have a Poissons ratio of 0.5, according to

Eq. (7). Another restriction using the concept of Rayleigh wave

length is also available. The ratio of the Rayleigh wave velocity to

Fig. 1. Graphical description of the relation between the Poissons ratio and the

value of 1m12m

h ifor conventional structural materials.

Fig. 2. Graphical description of the relation between the Poissons ratio and the

value ofKand G.

Y. Prawoto / Computational Materials Science 58 (2012) 140153 141


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the shear wave velocity, a VRVt

, depends only on the value of Pois-

sons ratio of the medium, and it can be found as the admissible, real

and positive root of the following equation [22]:

a6 8a4 8 2 md1

md a

2 8

1

md

0 8

Again, this concept prevents Poissons ratio from being negative.

In general, Poissons ratio does not have an effect on the distri-

bution of stresses in plane elasticity problems that do not involve

body forces. For anisotropic materials, multiple Poissons ratios

need to be expressed. Furthermore, for orthotropic elastic materi-

als, six Poissons ratios are defined, see Fig. 3 [2325]. Although

elasticity theory does not impose limits on Poissons ratios for such

materials, there is a specific inequality derived from energy consid-

erations. For Poissons ratios defined in other directions, some unu-

sual values may be obtained. In fact, Ting and Chen have

theoretically shown that Poissons ratio for anisotropic materials

can have either a positive or a negative value as long as the strain

energy density is positive [26,27].

While skeptics may say that thermodynamics restricts the com-pressibility of an elastic material to be positive for stability and as a

result Poissons ratio can never be negative, the fact is that many

researchers have proven experimentally and computationally that

it is possible to have a material with negative or zero Poissons ra-

tio, e.g. [6,7,26,27,8,12,2831]. Despite its controversial values and

concepts of Poissons ratio, one thing is sure: Poissons ratio is a

very interesting property for researchers and engineers alike who

work with applied mechanics. Furthermore, Poissons ratio has also

inspired material scientists and engineers to create new materials,

including auxetic ones.

3. Natural and man-made auxetic materials

3.1. Are there any natural auxetic materials?

This has been the common question over the past decades start-

ing in 1944, when Love reported a natural auxetic material, which

was quite controversial[1]. Although he suggested that this special

paradoxical value might be due to twinning of particular crystals,

some researchers say there are, while others say there are not.

The first group includes researchers who believe that auxetic mate-

rials are: cancellous bones, living cat skin, cow teat skin, some nat-

ural minerals such as a-Cristobalite (SiO2), pyrolytic graphites,

single crystals such as pyrite (FeS2), and some types of zeolites

such as siliceous zeolite MFI-Silicalites [32,33,28,3439]. However,

the majority of the experts in classical mechanics still belong to the

latter group because of the reason discussed previously, e.g., see

Eqs.(7) and (8).

While the existence of natural auxetic materials remains con-troversial from the perspective of traditional mechanics, man-

made auxetic materials have been produced for some time. Cellu-

lar solids such as polymer or metallic foams with inverted or re-en-

trant cell structures, and anisotropic fibrous composites, are

among the examples of man-made auxetic materials. They are

microstructurally engineered [30,8,12,4,13,10,6,7], and are also

proven to be highly resistant to shear deformations but easy to de-

form volumetrically, i.e., the shear modulus, G, is much greater

than the bulk modulus,K, see Eqs.(1) and (2).Fig. 4shows a basic

depiction of examples of the auxetic materials extracted from sev-

eral publications[4045].

4. Classification of auxetic materials based on mechanics and

microstructural morphology

4.1. Mechanism and structure

The typical mechanism of man-made auxetic materials is

shown in Fig. 5. When a load is applied to the structure in one

direction (e.g., vertically), the structure expands in the perpendic-

ular direction. Therefore, the structure gets fatter, resulting in a

negative Poissons ratio.

Various auxetic materials have been discovered and fabricated

over the past decades, ranging from the macroscopic to micro-

scopic and to the molecular levels. The following classification is

mainly a classification via mechanical considerations. The author

classifies also for computational purposes; almost all of them are

based on this simple mechanism that is treated as a unit cell lead-ing to a global stiffening effect. Readers interested in a classifica-

tion based on the materials are advised to refer elsewhere

[46,10,47].

4.2. Re-entrant structure

The adjective re-entranthere means (of an angle in a polygon)

greater than 180o (negative angle) and thus pointing inwards, sim-

ilar to that used in mathematics [48]. Gibson, Ashby, Evans, and

Fig. 3. Definition of the elastic moduli and Poissons ratio in isotropic and orthotropic materials [23].

142 Y. Prawoto / Computational Materials Science 58 (2012) 140153


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Alderson are among the pioneers working on this geometry[49,2].

Fig. 6shows the basic principle of the structure. Theoretically, the

alignment of the diagonal ribs along the horizontal direction, when

stretched longitudinally, causes them to move apart along the ver-

tical direction, resulting in the expansion of the lateral movement.

Referring toFig. 6, Poissons ratio and Youngs modulus in the load-

ing direction are given by

m12sin h h

l sin h

cos2 h 9

and

E1 jhl sin h

b cos3 h 10

whereh,l,b, and h are as defined inFig. 6,j =Esb(t/l)3 andEis the

intrinsic Youngs modulus.

While the early development of this type was dominated by

analytical approaches, several researchers also added some com-

putational approaches to their analytical ones, especially in more

recent publications. Lira et al. are among those that incorporated

their analytical technique with finite element analysis[50]. They

evaluated the transverse shear properties of a centersymmetrichoneycomb structure using analytical and finite element models.

The cellular structure features a unit cell geometry that allows

in-plane auxetic deformations, and multiple topologies to design

the honeycomb for multifunctional applications. The out-of-plane

properties are calculated using a theoretical approach based on

Voigt and Reuss theories. Their honeycomb topology provides five

sets of geometric parameters, enabling the material designer to

engineer optimum and multifunctional cellular cores.Fig. 7shows

the unit cell of the re-entrant structure evaluated. Their analytical

formulas can be used to perform a parametric analysis for the de-

sign of the cores in classical and multifunctional sandwich con-

structions [51,50]:

m12sin h

asin h

2

csin

ucos hcos h 2c cos u 11

By setting the anglehto be a re-entrant (negative) angle, Bezazi

and Lira carried out their analytical approaches, which were sup-

ported by computation.

4.3. Polymeric structure

The characteristics of this type of structure can be interpreted

by a simple 2D model, as shown inFig. 8 [6,7,52]. They consist of

interconnected networks of nodules and fibrils. If a tensile load is

applied, the fibrils cause lateral nodule translation, leading to a

negative Poissons ratio. A series of auxetic polymeric materials

have been produced in the form of foams, fibers, and composites.

Many molecular-level auxetic polymers have also been synthe-sized.Fig. 9shows one of the examples of polyurethane foam, both

regular and auxetic[43]. Referring toFig. 9, Grima explains that the

auxetic foam consists of chains of rigid rod molecules connected by

Fig. 4. Examples of several auxetic structures extracted from several publications

[4045].

Fig. 5. The basic mechanism of man-made auxetic materials.

Fig. 6. The basic hexagonal unit cell used to describe some re-entrant geometry

[2,116].

Fig. 7. The basic unit cell for re-entrant geometry analyzed by Lira et al. [50].



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flexible spacer groups along the chain lengths. The flexible spacer

groups are attached to the ends of some of the rigid rods connected

terminally or laterally. In the relaxed state, all the rigid rods are

oriented along the chain directions. Auxetic behavior occurs when

there is a rotation of the laterally attached rods upon stretching of

the foam. This agrees with the basic concept highlighted by He

et al.[53]. Interestingly, this type of foam can be produced easilyin any laboratory from commercially available conventional foams

through a process involving volumetric compression, heating it be-

yond its softening temperature, and then cooling it while under

compression[43], also see the Appendix for the technique formu-

lated by Lakes back in 1984[4]. Analytical and computational ap-

proaches to this type of structure are difficult to find if available at

all.

4.4. Chiral structure

The adjectivechiralhere originally meant a molecule that is not

superimposable on its mirror image, or a particle surrounded by

unique groups attached to it [48]. The original etymology is jiq

(kheir) meaning hand. However, researchers in this area usethe terminology to mean a physical property of spinning. Based

on the authors observations, this type of structure is the most

actively researched, especially from the mechanical and computa-

tional point of views. According to the theoretical and

experimental investigations performed by Prall and Lakes [54],

the Poissons ratio of a chiral structure for in-plane deformations

can be tailored to be around 1, seeFig. 4bottom left. Their ana-

lytical basis is summarized inFig. 10 [54]. With simple mechanics,

they analyzed the deformation,e, to be:

e r sin /e1 r/ cos he2

r/

sin h

12

in which, for small deflection, r sin/ r/ and h = 30o. By using the

elementary beam theory, if the thickness is t, the deflection eventu-

ally becomes:

/ TL6EsI

13

or

T Est3d

2L / 14

whereI 112

t3d. The strains, therefore, become:

e1 /rR

e2 /rR

15

Poissons ratios can then be calculated as:

m12 e2e1m21 e1e2

16

Another pioneer in this structure is Wojciechowski, who pub-

lished his first research on this subject back in 1987 [55]. He and

his group have been very actively researching this material even

until today[5667]. They have been followed by other researchers

such as Ishibashi and Iwata[68], Bornengo et al.[69], Spadoni et al.[70], Vasiliev[71], Grima[72], and Milton[73].

The chiral based structure has developed into various shapes

from around 2005. Grima et al. published their research on auxetic

Fig. 8. The basic mechanism of polymer auxetic materials [6,7,52].

Fig. 9. Example of polyurethane foams (left: conventional and right: auxetic) [43].

Fig. 10. Basic model of the unit cell for chiral structure analysis used by Prall and

Lakes[54].



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behaviour from rotating rigid units [41,43], Attard and Grima

published from a rotating rhombi [74], and rotating tetrahedra

[75]. They made an analysis of the role of the tetrahedra found in

the frameworks of the predicted auxetic zeolites natrolite (NAT),thomsonite (THO), and edingtonite (EDI) for generating auxeticity.

This related the theoretical human-made chiral with the natural

zeolites. Fig. 11 shows their major findings, where the three-

dimensional tetrahedra in the zeolite framework are perfectly rigid

and simply rotate relative to each other. Then their 2D projected

behavior in the (001) plane becomes equivalent to the idealized

two-dimensional rotating squares model with a Poissons ratio of

1. Their works are significant because they were the first to be

able to provide a link between the modeling approach, which con-

siders only the 2D projected framework of NAT in the auxetic

(001) plane, with a more realistic approach involving the tetrahe-

dra, thus showing that modeling approaches are complementary

and not conflicting with each other.

Although more research works on this chiral structure are avail-able, the above publications are representative of those from the

mechanical point of view.

4.5. Star-shape structure

Theocaris was among the first to propose the star-shaped

microstructure. He used the numerical homogenization approach

in his research[76]. Other researchers working on this structures

are Grima, Gatt, Ellul, and Chetcuti [77,78]. Fig. 12 shows the

mechanism of the auxetic characteristics of this type of structure

[77]. Referring to the figure, the Poissons ratio is 1, while their

analytical Youngs modulus is:

E 4 ffiffiffi3

p Kh

a2 1 cos p3 h 17

wherea is the length of the sides, h is the hinging angle, and Kh is

the force constant due to hinging. Both the Poissons ratio m and

the Youngs modulusEare independent of the direction of the load-

ing. Furthermore, they developed star-shaped structures from gen-

eral isosceles triangles, see Fig. 13. For their systems, Poissons

ratios for loading in the Ox1 or Ox2 are:

m21 m1

12 b cos a

h2 acos h2tan 2a h

2

bsin a h

2

a sin h2

18

This equation shows that one can adjust the Poissons ratio by

adjusting the values of botha and h. Eq.(18)can be easily derivedfrom the trigonometric relation of:

a cos1 b2a

19

X1 2b sin a h2

2a sin h

2 20

and

X2 2a cos a h2 21

and

mij dejdei

22

wherea, X1, X2are as shown in Fig. 12.

4.6. Other (ordered and disordered) structures

The ones belonging to this group are: square, triangular and

rectangular or their combination. Grima, Manicaro, and Attard

are among the ones working on the ordered structures involving

different-sized squares and rectangles [44]. The same group also

works on rectangles with different connectivities [44]. Recently,

there has also been work on disordered structures. Blumenfeldand Edwards are among the very few researchers working on dis-

ordered structures [79]. In the authors opinion, the disordered

structure approach will be the main topic of interest for research

in the near future, e.g.[80].

Of more than a hundred papers read by the author, most of the

modeling of auxetic materials have been mainly based on ordered

structures, despite the existence of auxetic behavior in disordered

structures. Therefore, in this subsection, more emphasis is put on

disordered structures: the former structures (periodic/ordered tri-

angle, rectangle, square and their combination) are here omitted.

The work of Blumenfeld is based on three auxetons shown in

Fig. 14. Their global auxetic behaviour is the result of local folding

and unfolding of auxetons when stressed.Fig. 15depicts the math-

ematical model of the disordered combination of the auxetons[79]. In doing so, they use the fabric tensor Qij, which plays an

important role in modeling auxetic strains, and also vectors for

each auxetons edge, rcg, and Rcg, a vector that extends from the

centroid of each auxeton. They further show that the tensor

appearing in the isostaticity stress equations are the symmetrical

part ofCcg, summed over the cells:

Qg 12

1 X

Ccg CcgT

23

where Ccgij r

cgi Rcg

j and is the p

2rotation matrix in the plane (the

Levi- Civita matrix) andCT is the transpose ofC. Eventually, the total

strain can be written as:

ecgij

Egijklrgkl

Qgijklhkl

rg

24

where the first term Eg

ijklrg

klis the expansion strain and the second

term Qgijklhklrg is the strain due to rotation [79]. Subsequently,

Poissons ratio can be calculated conveniently.

Fig. 11. (a) The structures of NAT, EDI and THO in the (00 1) plane (shown here are

the conformation with the rigid tetrahedral) and the off-axis plots for (b) Poissons

ratios, (c) Youngs moduli and (d) shear moduli in the (001) plane for NAT

frameworks of different rigidity[75].



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5. The use of auxetic materials

Taking advantage of the characteristics of auxetic materials,

suitable applications of the auxetic materials are enormous. Stud-

ies and experiments have demonstrated that auxetic materials can

improve mechanical properties, including shear resistance, hard-ness value (indentation resistance), fracture toughness, fatigue

crack propagation, etc., as compared to the conventional materials

from which they are made. In general, applications of auxetic

materials are good mainly in the field, where either one of the fol-

lowings are needed:

Poissons ratio being negative or zero.

Large shear resistance.

Hardness improvement.

Lower fatigue crack propagation.

Large toughness and modulus resilience.

Vibration absorption.

Table 1summarizes the applications summarized from variousresearch works [81,11,82,83,13,84,29]. In the textile industry,

Alderson proposed and tested the use of an auxetic fabric formed

by auxetic filaments or yarns to deliver active agents that could

lead to intelligent textiles having anti-inflammatory and anti-odor

effects [81]. Avellanada has used it for piezoelectric sensors [82].

Friis et al. see its potential in surgical implants [83]. It also has

more structural potential, such as in vanes for aircraft gas turbine

engines[29,84], or other military devices from helmets to sonar

receivers [85,86]. He reported in his patent that auxetic polymer

is preferred to a non-auxetic for piezoelectric composites

consisting of piezoelectric ceramic rods, due to the auxetic matrixs

converting the compressive planar stress into a compressive

longitudinal stress and therefore strengthens the incident vertical

compressive stress. Skeptics may say that there are not many other

applications related to mechanics, but most applications are less

used for stress carrying materials so far. However, in the authors

opinion its potential is quite promising.

5.1. Limitations

Generally, auxetic materials need substantial porosity. There-

fore, this type of material is less stiff than the solids from which

they are made. Eventually, this causes limitations on the structural

applications of the materials with negative Poissons ratio [87].

Consequently, for applications that require substantial load-bear-

ing, they are not the best choice.

6. Analytical and computational aspects

6.1. Constants influencing computational approach

6.1.1. Elastic moduli

The consequences of the Poissons ratios being negative include

significant changes in Youngs modulusEand the Shear modulusG.

Fig. 12. Example of star-shaped auxetic structure. Stretching of these systems result in an increase in the angle h between the triangles which results in a more open

structure. Note that if these systems are viewed from the perspective of the empty spaces (unshaded) between the triangles (shaded), then these systems can be described

either in terms of their star shaped perforations, or triangular shaped perforations in the special case when the angles between the triangles is 60 (the middle structure)[77].

Fig. 13. A more general connected triangles system constructed out of isosceles

triangles rather than equilateral triangles[77].

Fig. 14. auxetons made of three-contact building blocks used by Blumenfeld. Each

auxeton can expand and rotate when forces are applied to its ends[79].

Fig. 15. A section of a disordered auxetic structure, made of joining auxetons at

their contacts. The contacts are joined by straight lines (blue dashed) into a triangle

[79].



9/15


Referring to Eq.(1), one can immediately obtain the higher resis-

tance to shear strain, G, caused by twisting or tearing forces

[88,21,19]. Practically, the material would become highly com-

pressible but difficult to shear; low E, highG.

Choi was among the first to explicitly calculate the elastic mod-

ulus of an auxetic material. He used the model of the re-entrant

foam as shown inFig. 16 [89]. Starting from the simple principle

of Egli [90], he used the relationship between the density ratio

and Youngs modulus ratio, E

Es/ q

qs

n, where the subcript (s) is

the solid, and the superscript () means the cellular. With some

mathematical manipulations, he was able to use it via the volume

change ratio.

Meanwhile, the Castigliano principle was also used:

dp @U@P

Xn

i

Zz

MiEI

@Mi@P

dz 25

whereUis the strain energy, Mi is the bending moment exerted at

each beam, E is Youngs modulus, Iis the area moment of inertia,

and n is the number of beams in the structure. Based on the freebody diagram shown inFig. 16, the bending moment of each com-

ponent was then calculated. Using Eq.(25), Youngs modulus of the

conventional foam becomes:

EcEs

0:88 qcqs

226

For the re-entrant foam in the same figure, the value becomes

[89]:

ErEc

0:05 HuJu

1

1 sin p2u qcqs

227

whereH(u), J(u) are geometrical functions of the particular re-en-trant form.

6.1.2. Material hardness

The traditional hardness test is based on the resistance to

indentation. The following equation describes their relation:

H E1 m2 c

28

where cis a constant to which the load is applied. For uniform pres-sure,c is 1. When a non-auxetic material is subjected to hardnesstesting, the force compresses the material, and the material com-

pensates by spreading in the directions perpendicular to and away

from the direction of the impact. However, when the hardness

indentor is applied to an auxetic material, the auxetic material will

contract laterally. The material flows into (compresses towards) the

vicinity of the impact, as shown inFigure 17. This creates an area ofdenser material, which is more resistant to indentation. Therefore,

the hardness of an auxetic material is higher. Fig. 17 shows the

graphical depiction of this mechanism [10,47]. Experimental

investigation showed that re-entrant foams had higher yield

strengthrY and less stiffness Ethan conventional foams with thesame original relative density. It has also been further proven that

re-entrant foams density increases under indentation due to the in-

crease in shear stiffness[91].

6.1.3. Fracture mechanics characteristics

Fig. 18shows the crack growth observation done by Maiti [92].

Liu discussed in detail the fracture mechanics side of auxetic mate-

rials by citing Maitis works [52]. The non-singular stress field at

the distance r for a middle crack of 2a with crack tip radius rtipand stress intensity factorKIis [52,93,94]:

r KIffiffiffiffiffiffiffiffiffi2pr

p KIffiffiffiffiffiffiffiffiffi2pr

p rtip2r

29

Table 1

Summary of the applications of the auxetic materials (in alphabetical order) [81,11,82,83,13,84,29].

Field (Existing and potential) Application and the rationale

Aerospace Vanes for gas turbine engine, thermal protection, aircraft nose-cones, wing panel, sounds and vibration absorber, rivet

Automotive Bumper, cushion, thermal protection, sounds and vibration absorber parts that need shear resistant, fastener

Biomedical Bandage, wound pressure pad, dental floss, artificial blood vessel (the wall thickness increases when a pulse of blood flows throughit), artificial skin, drug release unit, ligament anchors. Surgical implants (similar to that of bone characteristics)

Composite Fiber reinforcement (because it reduce the cracking between fiber and matrix)

Military

(defence)

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

Sensors/

actuators

Hydrophone, piezoelectric devices, various sensors (the low bulk modulus makes them more sensitive to hydrostatic pressure)

Textile

Industry

Fibers, functional fabric, color-change straps or fabrics, threads

Fig. 16. The basic free body diagram (FBD) assumption for regular tetrakaideca-

hedron (left) and re-entrant unit cell (right) proposed by Choi [89].



10/15


Subsequently, the force acting on the cell rib is:

FZ rtip

2t

rtip2

KIffiffiffiffiffiffiffiffiffi2pr

p KIffiffiffiffiffiffiffiffiffi2pr

p rtip2r

rtip dr 30

Furthermore, with the thickness of the rib being t, and the first

order of the Taylor expansion, Eq.(30)is simplified to:

F 2:38 KIffiffi

lpffiffiffiffip

p tl

31

KI is the stress intensity of the conventional foams and l is the rib

length. The stress due to the bending moment is given by:

F 2:12 Flt3

: 32

Substituting Eq.(31)the stress becomes:

F 5:05 KI1ffiffiffiffip

p lt

233

The crack propagation takes place whenrP rf, whererfis thefracture strength of the cell rib. The critical stress intensity factor

or the fracture toughness can therefore be calculated as:

KI 0:20rf

ffiffiffiffiffipl

p tl

234

Using the relation ofq

qs/ t

l

n, the stress intensity factor is pro-

portional to the normalized density:

KIrf

ffiffiffiffiffipl

p 0:19 q

qs

35

For the re-entrant structure, a similar value becomes:

KIc

rf ffiffiffiffiffipl

p 0:10ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 sin p2u q

1 cos 2uq

qs

36

whereKIcis the fracture toughness of re-entrant foams and uis therib angle (seeFig. 18).

Experimentally, Choi showed that for his re-entrant foam, the

following was observed:

KrIcK

Ic

0:53ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 sin p2u q

1 cos 2u 37

whereKrIc is fracture toughness of the re-entrant foam.

6.2. Numerical analysis with hierarchical structure

There is a moderately high number of papers discussing compu-

tational approaches to the study of auxetic materials, e.g. [50,95

97]. Here, the author limits the review to multi-level ones, because

it is believed that this type of approach will become the major

trend in the near future.

6.2.1. Unit cell

Almost all computational approaches reviewed use the unit cell

concept, directly or indirectly, for calculation. Many of them are in

the form of testing the analytical approaches they developed. Take

Scarpas work for example[97].Fig. 19shows the unit cell Scarpa

used. In his research, he combined analytical, numerical, and

experimental analysis on the compressive strength of hexagonalchiral honeycombs due to elastic buckling of the unit cells under

flatwise compressive loading. His analytical elastic collapse for

the hexagonal chiral cell was:

bEc

p31 m2c

4b2

8p 3ffiffiffiffiffiffiffiffi

3a2p Kbb0:8 Kaa 38

where Kb is 7.2 and Ka is 0.08. The equation was then compared

with the computational results as shown in Fig. 20. The result of

the unit cell can then be used as data to perform a larger scale com-

putation, provided they have the correct boundary condition. This is

the basic idea of homogenization.

6.2.2. HomogenizationSeveral introductory variations of the homogenization theory

are available, but they are beyond the scope of this review. How-

ever, for the sake of the readers convenience, part of the authors

previously published introduction is simplified and used [98], see

the Appendix. Theocaris, Lee, and Choi were among the first to

use homogenization theory in auxetic materials research. Lee and

Choi applied homogenization using commercial code in their re-

search using regular and re-entrant structures, while Theocaris

used it for star-shaped structures [99,76]. Similar to Eqs. (A-3)

and (A-11), Choi formulated his equation into:

lime!0

ZX

Ux;ydX ZX

1

jYjZ

Y

Xx;ydY

dX 39

They concluded that the microscale integration could be re-

placed by an average value integration for a general Yperiodic

functionU. Recalling the weak form of linear elasticity problem fi-

nite element:ZX

Eijkl@uk@xi

@vi@xj

dX ZX

bivi dXZC

tivi dC 40

They then obtained the homogenized elasticity tensor EH (see

also Appendix for the basic Homogenization method for auxetic

materials):

Eijkl 1

jY

j ZYEijkl Eijkl

@vklp@yq

!dY 41

wherevklp is the microscale parameter. Conveniently, the effectiveelastic modulus, Ee, and Poissons ratio, me, can be obtained fromthe plane stress assumption as follows:

Fig. 17. The basic mechanism deformation behaviors near the hardness indentor

tips[10,47].

Fig. 18. Crack propagation observed by Maiti. (a) Is through the bending failure on

the non-vertical cell elements and (b) is through the tensile fracture of the vertical

cell elements[92].



11/15


EH1111

EH1122

0

EH2211

EH2222

0

0 0 EH1212

264

375 E

1 m2e

1 me 0

me 1 0

0 0 1me

2

264

375

Based on their studies on using a homogenization technique,

they concluded that matrix material properties do not significantly

affect Poissons ratio of the regular and re-entrant honeycomb

structure. Youngs modulus of the regular honeycomb structure in-

creases with volume fraction. The regular honeycomb structure

has a decreasing Poissons ratio, with an increasing volume frac-

tion. The re-entrant structure has itsm value dependent on the in-verted angle of the cell edge. Youngs modulus of the re-entrant

honeycomb structure decreases with an increase in the inverted

angle.

Another work worth presenting here is that of Dirrenberger

[95]. He analyzed three auxetic periodic lattices. The elastic moduli

were computed and its anisotropy was investigated by using the

finite element method combined with the numerical homogeniza-

tion technique. Similar to the work of Choi, Dirrenberger used

homogenization in the form of macroscopic stress and strain

tensorsR and Edefined by the spatial averages:

R 1jVjZ

V

rdV 42

and

E 1jVjZ

V

edV 43

Periodic boundary condition over the unit-cell leads displace-

ment field u such as:

u Ex v8x 2 V 44with va periodic fluctuation. It takes the same value at two homol-

ogous points on opposite faces of V, whereas the traction vector

t=r ntakes opposite values,n being the normal vector. By apply-ing either macroscopic strain or stress, one can compute the effec-

tive elastic moduli fourth-rank tensor Cand compliance tensor Sof

materials:

R C :E E S: R 45Dirrrenberger evaluated the elastic moduli of three periodic

auxetic lattices: hexachiral, rotachiral, and tetra-antichiral. The

hexachiral was found to possess high in-plane elastic moduli and

a Poissons ratio close to 1. With its circular (or elliptic) liga-

ments, the auxetic rotachiral lattice provides a parameter for tun-

ing the microstructure for specific absorption properties. This

lattice can exhibit a highly negative Poissons ratio when loaded

out-of-plane. The orthotropy of the tetra-antichiral lattice was

found to have a higher stiffness E in the principal directions of

the cell. For this microstructure, the auxetic effects in the plane

are restricted to short angle intervals around the principal

directions.Pasternak and Dyskin also used the homogenization concept in

their research[100]. They showed that a multiscale distribution of

spherical inclusions with a Poissons ratio having a different sign

from that of the elastic isotropic matrix can increase (up to two

orders of magnitude) the effective Youngs modulus considerably,

even when the Youngs moduli of the matrix and the inclusions

are the same.

The author believes that the homogenization concept is very

useful when pursuing a computational approach in this area,

regardless of the fact that many researchers did not use it

intentionally.

7. Experimental approach

The main experimental approaches in auxetic material research

remain the challenge of prototype making and its mechanical char-

acterization. The first experimental approach used was that of

Fig. 19. Unit cell used by Scarpa. left: based for analytical approach, right: brick element to model the computation [97].

Fig. 20. Comparison between FE and analytical non-dimensional collapse stress

used by Scarpa[97].



12/15


Lakes[4]. For the readers convenience, his technique is appended

to this article as the Appendix.

In 1991, Alderson and Evans also published their method of fab-

rication of microporous polyethylene having a negative Poissons

ratio [101]. Their method involved a thermo-forming processing

route consisting of three separate stages: compaction, sintering,and extrusion. It can produce an expanded UHMWPE (ultra high

molecular weight polyethylene) microstructure that possesses

negative Poissons ratios. They proved that their material produced

was homogeneous and continuous, and most importantly, has

Poissons ratio values varying from 0 to 1.24, depending on the

applied strain, in the radial direction, and approximately 0 in the

axial direction.Fig. 21shows their representative works.

Ten years after Lakes first published his fabrication method,

Chan and Evans published another method for auxetic foam

[102]. Bianchi et al. produced auxetic open cell foams for curved

and arbitrary shapes[103]. These researchers based their fabrica-

tion on polyethylene/polymeric foam. Their typical results are

anisotropic structures. Other researchers in this category are

Donoghue, Kettle, Neale, Pickles, Bezazi, Scarpa, Larson, Remillat,Sigmund, Bouwstra, Simkins, Ravirala, Davies, etc. [104

106,51,107]. In terms of the manufacturing process, besides the

heat treatment, thermo-forming, and compaction methods, there

are also rapid prototyping techniques such as micro machining

and fusion mold deposition (FDM) and HP-PA powder stereo-

lythography.

Other groups of materials fabricated are in textile, e.g.[107,108]

and carbon fibers, e.g.[109]. Details of the manufacturing methods

are beyond the scope of this review.

Experimental methods that are related to mechanics involve

characterization and functional experiments. Starting with experi-

ments that are as simple as what Gaspar et al. often used, simple

experimental tools to support their mathematical modeling

[110112], to Clarke, who uses mechanical/conventional charac-

terization [113], to Lethbridge et al., who use direct, static mea-

surement of the single-crystal Youngs moduli of the zeolite

natrolite, and to using mechanical to Brillouin scattering and ultra-

sound[114].

Tee, Spadoni, Scarpa, and Ruzzene researched wave propagation

in auxetic tetrachiral [115]. They performed numerical and

experimental investigations on flexural wave propagation. A wave

approach is applied to the representative unit cell of the honey-

comb to calculate its dispersion characteristics and phase constant

surfaces varying the geometric parameters of the unit cell. The

modal density of thetetrachiral lattice and of a sandwich panel hav-

ing the tetrachiral as core is extracted from the integration of the

phase constant surfaces, and compared with the experimental ones

obtained from measurements using scanning laser vibrometers.

Other experimental works are available. However, the ones re-

lated to our scope have been sufficiently reviewed here.

8. Final remarks

Auxetic materials have a lot of potential applications from bio-

medical to automotive and defense industries. Also, these materi-

als could potentially be used for completely new structures with

special functions. However, more research work needs to be done

for further understanding of these materials and their applications

to real components. From the mechanical and computational point

of view, it is predicted that research on this subject will be domi-

nated by the unit cell concept, hierarchical, homogenization of

both periodic/ordered and disordered cells. Until recently, almost

all relevant papers have been based on ordered structures, which

is convenient for the purpose of analysis. However, in the near fu-

ture, the work presumably will be dominated by study of disor-

dered structures since that analysis is closer to real materialstructures. The author believes that such materials, with little

existing understanding of deformations in the presence of disorder,

will require a newer theory and computational approach that

could go beyond the currently available homogenization concept.

Acknowledgments

Funding by Ministry of Higher Education (MOHE) through FRGS

program titled: Fundamental study of auxetic materials: Analyti-

cal, Computational, and Experimental Approaches is gratefully

acknowledged.

Appendix A

A.1. Technique for producing auxetic polymeric foams

Note: The following procedure is adapted from Lakes published

paper[4].

The mould: Aluminum square tube, 100 square, for a mould. If

too large a mould is used, heat transfer will be poor, and only

the outer portion of the foam will be transformed.

Preheat furnace to about 160170C.

Either measure or mark foam for later determination of strains.

Mark foam in all 3 orthogonal directions, i.e., two adjacent

corners and down one side.

(optional: this is not necessary if sufficient care is taken in

removing wrinkles) Lubricate sides of square aluminum tubewith vegetable oil. Spray cooking oil (PAM) can also be used,

but does not seem to work better. DO NOT use a petroleum dis-

tillate base lubricant; it will smell terrible when heated.

Stuff the foam in the tube. It works well to start the foam

slightly by hand and then work it up gently with a tongue

depressor to remove wrinkles.

Pull the foam a little on both ends to get rid of creases created

by stuffing the material. This procedure will result in a pre

stretched sample in the tube. The actual original length of the

sample must be used when determining the amount of pre

compression to apply.

Place the compression device and end plates on the stuffed tube.

If the desired specimen length is less than the square tube size,

select the correct length of cut tubing [pipe] within the mould

to compress the foam longitudinally by the same amount as

transversely. Alternatively, cut the foam proportionally longer

than the square tube length and do not use pipe.

Fig. 21. Micrograph of the auxetic ultra high molecular weight polyethylene

(UHMWPE) developed by Alderson and Evans [101].



13/15


Push the pipe down on the loose end plate such that the foam is

compressed evenly at the end. Try not to push too fast; this may

contribute to the uneven distribution of compression along the

length of the specimen.

Gently tighten down the side screws to hold the cut pipe in

place. Place assembly in centre of furnace or oven. A kitchen oven is

sufficient.

Leave the foam in the oven for a predetermined amount of time.

The gray polyester foams transform better at a slightly lower

temperature for a longer amount of time, about 20 min maxi-

mum. The white/cream colored polyether foam seems to be

more sensitive with respect to melting together; 1718 min. is

appropriate.

Remove and cool the specimen completely. Taking the speci-

men out of the mould before complete cooling may result in

premature release of the pre compression. It may be helpful

to release foam ribs which have stuck together: stretch the

specimen gently in each of three directions. Congratulations!

You have made negative Poissons ratio foam (also called anti-rubber, dilational material, or auxetic material).

Measure the amount of permanent compression retained by the

specimen by either measuring the new distance between the

marks or by measuring the size of the transformed sample.

Other kinds of moulds are possible and have been used success-

fully by others.

A.2. Homogenization theory applied for auxetic computation

This subsection is extracted from authors previously pub-

lished works and some other publications, e.g. [98,99]. In this

theory, the local constitutive structure is thought to have a locally

specified periodic unit cell as shown in Fig. A-1. Hence, every

physical variable field on this material support can be expressed

by the combination of the averaged part with the local distur-

bance as depicted in part (b) of the figure. Assuming that the

periodicity, , should be sufficiently small, every physical field

u(x,y) in total can be represented by the following asymptotic

expansion in :

ux;y u0x u1x;y 122u2x;y . . . . . . for y

x= A-1or,

ux;y u0x u1x;y A-2where {u0, u1} are functions for the averaged part and disturbance in

the coordinates {x,y}, respectively. Owing the periodicity of func-

tionu in they-coordinates, the following two equations are power

tool for formulation:

@

@xiux;y @u

@xi 1

@u

@yiA-3

and

limx!0

ZV

ux;ydVZ

V

1

jYjZ

Y

ux;ydY

dV A-4

where Ydenotes for a volume of unit cell. These two equations or

similar forms of those two were mainly used for computational ap-

proach in auxetic materials too. To use the principle in FEA to com-

pute the elastic modulus, recall the weak form of linear elasticity

problem finite element:ZX

Eijkl@uk@xl

@vi@xj

dX ZX

bivi dXZC

tivi dC A-5

and, using the homogenization principle, we use x and y in micro-

scale coordinate (seeFig. A-1),

ux;y u0x u1x;y 122u2x;y . . . . . . for y

x=

A-6

and

vx;y v0x v1x;y 122v2x;y . . . . . . for y

x= A-7or

ux;y u0x u1x;y A-8and

vx;y v0x v1x;y A-9Meanwhile, the gradient ofu and vare:

5 ux;y 5xu0

x 5xu1

x;y 5yu1

x;y5 vx;y 5xv0x 5xv1x;y 5yv1x;yA-10

Combine Eq.(A-5)with the above equations gives:ZX

Eijkl@u0k@xl

@u1k

@yl

@v0i@xj

@v1i

@yj

!dX

ZX

Eijkl@u1k@xi

@v0i@xj

@v1i

@yj

! @u

0k

@xl @u

1k

@yl

@v1i@xj

" #dX

2ZC

Eijkl@u1k@xl

@v1i@xj

dX

ZX

bi v0

i v1i dX

ZCti v

0

i v1i dC A-11

when ? 0, it becomes:ZX

Eijkl@u0k@xl

@u1

k

@yl

@v0i@xj

@v1i

@yj

!dX

ZX

biv0

i dXZC

tiv0

i dC A-12

can be separated to:

limx!0

ZX

Eijkl@u0k@xl

@u1k

@yl

@v0i@xj

dX

limx!0

ZX

biv0

i dXZC

tiv0

i dC

A-13

and

limx!0

ZX

Eijkl@u0k@xi

@u1

k

@yi

@v1i@yj

dX 0 A-14

Using our Eq.(A-11), we can rewrite into

limx!0

ZX

Ux;y dX ZX

1

jYjZ

Y

Ux;ydY

dX A-15

Eqs.(A-13) and (A-14)eventually become:ZX

1

jYjZ

Y

Eijkl@u0k@xl

@u1k

@yl

@v0i@xj

dY dX

ZX

biv0

i dXZC

tiv0

i dC A-16

andZX

1

jYjZ

Y

Eijkl@u0k@xi

@u1k

@yi

@v1i@yj

dY dX 0 A-17



14/15


Introducing a separation of variables to satisfy this based on lin-

ear elasticity

u1i x;y vpqi y@u0p@xq

x A-18

where vklp is the microscale parameter. This can be obtained by com-bining Eqs.(A-18) and (A-17):

ZX

1

jYjZ

Y

Eijkl Eijpq@vpqi@yj

!@v1i@yj

dY@u0k@xl

0 A-19

Therefore, our weak form becomes:ZX

EHijkl@u0k@xl

@v0i@xj

dX

ZX

biv0

i dXZC

tiv0

i dC A-20

where the homogenized tensor of elasticity is:

EHijkl 1

jYjZ

Y

Eijkl Eijkl@vklp@yq

!dY A-21

Conveniently, the effective elastic modulus, Ee, and Poissons

ratio, me, can be obtained from the plane stress assumption asfollows:

EH1111

EH1122

0

EH2211

EH2222

0

0 0 EH1212

264

375

E

1 m2

e

1 me 0

me 1 0

0 0 1me2

264

375

Therefore, the homogenization concept is highly useable in

auxetic material computational approach.

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seeing auxetic materials from the mechanics point of view: a structural review on the negative...

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