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2011of Achievements in Materialsand Manufacturing Engineeringof Achievements in Materialsand Manufacturing Engineering
Workings of auxetic nano-materialsY.T. Yao a, M. Uzun a,b,*, I. Patel a,c
a Institute for Material Research and Innovation, The University of Bolton, BL3 5AB, Bolton, UK b Department of Textile Educations, Marmara University, 34722, Goztepe, Istanbul, Turkec British University of Egypt, Cairo, Suez Desert Road, El Sherouk City, 11837, Egypt* Corresponding author: E-mail address: [email protected]
Received 19.10.2011; published in revised form 01.12.2011
Materials
AbstrAct
Purpose: The human mind is consistently interested in new materials having unique properties. Recently, a relatively new field is being investigated which exhibits a negative Poisson’s ratio (NPR), and consequently are termed auxetic materials. Design/methodology/approach: One of the main reason for interest in auxetic materials is due to the possibility of enhanced mechanical properties such as shear modulus, plane strain fracture toughness and indentation resistance compared to non auxetic material. Findings: Auxetic materials were described concerning their classification, characteristic, properties and potential applications.Research limitations/implications: The paper is an overview the modelling structure and deformation mechanisms of auxetic nano-materials.Originality/value: The paper shows the possibilities of auxetic materials application resulting from their mechanical properties.Keywords: Auxetic; Nano-materials; Negative Poisson’a ratio
Reference to this paper should be given in the following way: Y.T. Yao, M. Uzun, I. Patel, Workings of auxetic nano-materials, Journal of Achievements in Materials and Manufacturing Engineering 49/2 (2011) 585-593.
1. Introduction
Auxetic materials are typically divided into two classes in terms of their scale, which are the common auxetic materials (macro-scale) and nano-scale auxetic materials. This paper, mainly concentrates on study of materials having auxetic behaviour at the molecular level and investigating their deformation mechanisms.
When a material is stretched in one direction, it tends to contract (or, rarely, expand) in the other two directions. Conversely, when a sample of material is compressed in one direction, it tends to expand (or rarely, contract) in the other two
directions. A measure of this dimensional change can be defined by Poisson’s ratio ( ),
y
xyx directionloadingtheinstrain
directionlateraltheinstrain
(1) where x and y are the strains in the x and y directions, respectively.
In fact, for most materials this value is positive (Poisson’s ratios ranging between 0 and 0.5) and reflects a need to conserve
Journal of Achievements in Materials and Manufacturing Engineering
Y.T. Yao, M. Uzun, I. Patel
Volume 49 Issue 2 December 2011
volume. Negative Poisson’s ratio (NPR) material expands (or contracts) laterally when stretched (or compressed), in contrast to ordinary materials. These new types of materials were named “auxetic” by Evans [1]. “Auxetic” comes from the Greek word auxetos, meaning “that which may be increased”. Consider, as an example, auxetic ultra high molecular weight polyethylene (UHMWPE) proposed and fabricated by Alderson & Evans, 1992 to give a general idea as to what is the difference between auxetic and conventional material and to visualize the deformation mechanism (See Fig. 1) [2].
Fig. 1. Schematic diagram of structural changes observed in microporous UHMWPE undergoing tensile loading in the longitudinal direction: Non-auxetic (A) and auxetic (B) deformation due to fibril hinging in a nodule-fibril microstructure [2]
Fig. 1(a) on the top shows schematically the geometry and deformation of a common material undergoing lateral contraction for loading with a tensile longitudinal stress. Fig. 1(b) on the bottom shows auxetic behaviour in which the undeformed material (left) responds to the tensile longitudinal stress with lateral expansion (right).
2. Auxetic materials properties
Recently, the investigation of auxetic materials has attracted significant attention. One of the reasons for interest in auxetic materials comes from the fact that negative Poisson’s ratio can lead to enhancement in other mechanical properties, including: Shear modulus [3,4], Indentation resistance [5,6], Dynamic properties [7,8], Fracture toughness [9].
Shear modulus
Auxetic materials have higher resistance to shear strain, which can be qualitatively explained by the relationships between shear (or rigidity) modulus G, Young’s modulus E, bulk modulus K (the inverse of the compressibility) and Poisson’s ratio . For isotropic materials the relationships between these constants are:
)3
23(21
GKGK
(2)
)1(2EG
(3)
)21(3EK
(4)
GKKGE
39
(5)
The Poisson's ratio of a stable isotropic material cannot be less than -1.0 or greater than 0.5 due to the requirement that the Young’s modulus, shear modulus and bulk modulus have positive values [10]. However, the Poisson’s ratio can exceed that limits in certain direction for anisotropic materials [11].
For isotropic materials, when Young’s modulus (E) remains unchanged, the values of the shear modulus (G) and the bulk modulus (K) can be altered through the changes in Poisson’s ratio ( ) (Eqs. 3 and 4). E is at least twice the value of G for positive Poisson’s ratio materials. Conversely, E reduces to below 2G when is negative. For example, E=G when =-0.5 and E 0 when -1. In other words, when decreasing to -1, the shear modulus tends to very high values and the Young’s modulus decreases to low values, which makes a solid difficult to shear but easy to deform. K also decreases as -1, which means the material becomes hard to shear but highly compressible. When approaches +0.5, the shear modulus (G) is greatly exceeded by the bulk modulus (K), which makes the material incompressible but easy to shear. Indentation response
For isotropic materials, the indentation resistance (or hardness) (H) is inversely proportional to (1- 2) for a given pressure, defined as:
)1( 2
EH (6)
where the value of relates to the theoretical analysis used. 1stands for uniform pressure distribution [12] and 3/2
is for Hertzian indentation [13].
Eq. 6 shows the indentation resistance tends to infinity with increase in the magnitude of Poisson’s ratio ( ) for a given value of Young’s modulus (E). As already noted, the range of Poisson’s ratios for isotropic materials is -1< <0.5, and so auxetic isotropic material show enhancements in indentation resistance when -1< < -0.5. As approaches -1, the indentation resistance towards infinity. Enhanced indentation resistance of auxetic materials has been demonstrated through investigation of synthetic auxetic materials (i.e. polymeric and metallic foams [5,6]; and microporous polymers [14]). A schematic of indentation response is illustrated in Fig. 2 for both non-auxetic and auxetic materials subjected to compressive impact loading [15].
>0
<0
>0
<0
>0
<0
A
B
Oin thperpedefinthe omatethe iminden
Fig. (b) mlatera
3. app 3.1
Wstrucsynthlevelclasswith mateor Yo
Asynthmicroultra polyp[21,2naturand z Appl
AdemoThercan b
Fthrou
On the non-auxethe direction of imendicular to and ne the lateral ‘floother hand, the rial also contractmpact, leading tntation resistance
2. An object material in the al ‘flow’ of mate
Auxetiplications
Classificati
Within the last thtures have bee
hesised, rangingl. Fig. 3 shows tses of materials
auxetic properterials are known oung’s modulus Auxetic materiahesized, such as oporous polytet
high moleculpropylene (PP), 22], several typrally occurring mzeolites [27].
lications
As stated in thonstrated enhaefore, a range obe envisaged. For example, auugh thermomech
tic material, the mpact and the maway from the d
ow’ of material)vertical impact ts laterally-mateto increased dene [15].
hitting a non-vertical directio
erials in each cas
ic mate
ions and typ
hirty years, a vaen discovered, from the molecthat there are ex(polymers, comties, and that naover several ord[16].
als have been polyurethane an
tra-fluoroethylenlar weight polhighly anisotrop
pes of rocks wimolecular auxeti
he above sectiancements in of potential app
uxetic foam withanical processin
force compressematerial spreadsdirection of the i). For the auxeti
compression merial ‘flows’ into sification, theref
-auxetic (a) anon. The arrowsse
erials p
pes
ariety of auxetic designed, man
cular level up toxamples of each
mposites, metals atural and man-ders of magnitud
discovered, fand polyethylene ne (PTFE) [19]lyethylene (UHpic composites [ith microcracksics, e.g. –cristo
ion, auxetic mother physica
plications of aux
th a re-entrant mng was establish
es the material s in directions mpact (arrows ic material, on
means that the the vicinity of
fore, enhanced
nd an auxetic s indicate the
potential
materials and nufactured or o macroscopic h of the major and ceramics) -made auxetic de of stiffness,
abricated and foams [17,18],
] microporous HMWPE) and
20], laminates [23,24], and obalite [25,26],
materials have al properties. xetic materials
microstructure hed to provide
dispersion ohave potent
Fig. 3. Strumaterials fr
Auxetic
fasteners anof the auxethe auxeticfastener is more tightlyprocessing cushions asfoams. Wanbeneficial iand in redu[32]. AuxetUS6412593features in air leaks) an
volume. Negative Poisson’s ratio (NPR) material expands (or contracts) laterally when stretched (or compressed), in contrast to ordinary materials. These new types of materials were named “auxetic” by Evans [1]. “Auxetic” comes from the Greek word auxetos, meaning “that which may be increased”. Consider, as an example, auxetic ultra high molecular weight polyethylene (UHMWPE) proposed and fabricated by Alderson & Evans, 1992 to give a general idea as to what is the difference between auxetic and conventional material and to visualize the deformation mechanism (See Fig. 1) [2].
Fig. 1. Schematic diagram of structural changes observed in microporous UHMWPE undergoing tensile loading in the longitudinal direction: Non-auxetic (A) and auxetic (B) deformation due to fibril hinging in a nodule-fibril microstructure [2]
Fig. 1(a) on the top shows schematically the geometry and deformation of a common material undergoing lateral contraction for loading with a tensile longitudinal stress. Fig. 1(b) on the bottom shows auxetic behaviour in which the undeformed material (left) responds to the tensile longitudinal stress with lateral expansion (right).
2. Auxetic materials properties
Recently, the investigation of auxetic materials has attracted significant attention. One of the reasons for interest in auxetic materials comes from the fact that negative Poisson’s ratio can lead to enhancement in other mechanical properties, including: Shear modulus [3,4], Indentation resistance [5,6], Dynamic properties [7,8], Fracture toughness [9].
Shear modulus
Auxetic materials have higher resistance to shear strain, which can be qualitatively explained by the relationships between shear (or rigidity) modulus G, Young’s modulus E, bulk modulus K (the inverse of the compressibility) and Poisson’s ratio . For isotropic materials the relationships between these constants are:
)3
23(21
GKGK
(2)
)1(2EG
(3)
)21(3EK
(4)
GKKGE
39
(5)
The Poisson's ratio of a stable isotropic material cannot be less than -1.0 or greater than 0.5 due to the requirement that the Young’s modulus, shear modulus and bulk modulus have positive values [10]. However, the Poisson’s ratio can exceed that limits in certain direction for anisotropic materials [11].
For isotropic materials, when Young’s modulus (E) remains unchanged, the values of the shear modulus (G) and the bulk modulus (K) can be altered through the changes in Poisson’s ratio ( ) (Eqs. 3 and 4). E is at least twice the value of G for positive Poisson’s ratio materials. Conversely, E reduces to below 2G when is negative. For example, E=G when =-0.5 and E 0 when -1. In other words, when decreasing to -1, the shear modulus tends to very high values and the Young’s modulus decreases to low values, which makes a solid difficult to shear but easy to deform. K also decreases as -1, which means the material becomes hard to shear but highly compressible. When approaches +0.5, the shear modulus (G) is greatly exceeded by the bulk modulus (K), which makes the material incompressible but easy to shear. Indentation response
For isotropic materials, the indentation resistance (or hardness) (H) is inversely proportional to (1- 2) for a given pressure, defined as:
)1( 2
EH (6)
where the value of relates to the theoretical analysis used. 1stands for uniform pressure distribution [12] and 3/2
is for Hertzian indentation [13].
Eq. 6 shows the indentation resistance tends to infinity with increase in the magnitude of Poisson’s ratio ( ) for a given value of Young’s modulus (E). As already noted, the range of Poisson’s ratios for isotropic materials is -1< <0.5, and so auxetic isotropic material show enhancements in indentation resistance when -1< < -0.5. As approaches -1, the indentation resistance towards infinity. Enhanced indentation resistance of auxetic materials has been demonstrated through investigation of synthetic auxetic materials (i.e. polymeric and metallic foams [5,6]; and microporous polymers [14]). A schematic of indentation response is illustrated in Fig. 2 for both non-auxetic and auxetic materials subjected to compressive impact loading [15].
>0
<0
>0
<0
>0
<0
A
B
Oin thperpedefinthe omatethe iminden
Fig. (b) mlatera
3. app 3.1
Wstrucsynthlevelclasswith mateor Yo
Asynthmicroultra polyp[21,2naturand z Appl
AdemoThercan b
Fthrou
On the non-auxethe direction of imendicular to and ne the lateral ‘floother hand, the rial also contractmpact, leading tntation resistance
2. An object material in the al ‘flow’ of mate
Auxetiplications
Classificati
Within the last thtures have bee
hesised, rangingl. Fig. 3 shows tses of materials
auxetic properterials are known oung’s modulus Auxetic materiahesized, such as oporous polytet
high moleculpropylene (PP), 22], several typrally occurring mzeolites [27].
lications
As stated in thonstrated enhaefore, a range obe envisaged. For example, auugh thermomech
tic material, the mpact and the maway from the d
ow’ of material)vertical impact ts laterally-mateto increased dene [15].
hitting a non-vertical directio
erials in each cas
ic mate
ions and typ
hirty years, a vaen discovered, from the molecthat there are ex(polymers, comties, and that naover several ord[16].
als have been polyurethane an
tra-fluoroethylenlar weight polhighly anisotrop
pes of rocks wimolecular auxeti
he above sectiancements in of potential app
uxetic foam withanical processin
force compressematerial spreadsdirection of the i). For the auxeti
compression merial ‘flows’ into sification, theref
-auxetic (a) anon. The arrowsse
erials p
pes
ariety of auxetic designed, man
cular level up toxamples of each
mposites, metals atural and man-ders of magnitud
discovered, fand polyethylene ne (PTFE) [19]lyethylene (UHpic composites [ith microcracksics, e.g. –cristo
ion, auxetic mother physica
plications of aux
th a re-entrant mng was establish
es the material s in directions mpact (arrows ic material, on
means that the the vicinity of
fore, enhanced
nd an auxetic s indicate the
potential
materials and nufactured or o macroscopic h of the major and ceramics) -made auxetic de of stiffness,
abricated and foams [17,18],
] microporous HMWPE) and
20], laminates [23,24], and obalite [25,26],
materials have al properties. xetic materials
microstructure hed to provide
dispersion ohave potent
Fig. 3. Strumaterials fr
Auxetic
fasteners anof the auxethe auxeticfastener is more tightlyprocessing cushions asfoams. Wanbeneficial iand in redu[32]. AuxetUS6412593features in air leaks) an
For fibre reinfore mechanism. Aailure resistance rted the successfuce the first auxulated that comced fibre pull-ouugh maintainingson’s ratios of th38,39]. Subsequ performed uspropylene fibre ancrease) and eonstrate for the aThere are manyposite materials,bination with othquipment, e.g. inf vest, shin paulated for aux
material, a bandaet.
Limitations
As mentioned ae range of applicue behaviour. Hve their behavioure less stiff thantic materials hative Poisson’s rages” to flex, or “y auxetic materications. A further reasohanisms responshe developmentblished benefits posites containin
Auxetic ma
Auxetic materialcategories: naturdly discussed in
ecular level aux
A negative Poisspyrites [41] as alar geometric arre then negative le crystal phaseested to have a carbon nitride wBaughman & Gson’s ratio is ve
orced compositeAuxetic materials
properties of coful development xetic polymeric
mposites with aut, potentially hg the interface he matrix and fi
uently, comparatsing positive anand enhancemenenergy adsorptiauxetic fibre systy potential appli, such as vehicleher materials fon a crash helmetad, knee pad oxetic fibres inage or wound p
s
above, auxetic cations and are aHowever, most ur from structuren the solids fromave intrinsic poratio (i.e. auxetic “nodules” to sprerials have limit
on for developiible for auxetic
t of auxetic nanofor auxetic mac
ng auxetic nano-
aterials
ls at the molecural and syntheticthe following.
xetics I: Natural
son’s ratio was fa result of crystalrangement and aPoisson’s ratios
es. For examplnegative Poisso
would be harder tGalvao 1993 havery common in
es, fibre pull-ous are also predictmposites. Aldersof a melt-extrus(polypropylene)
auxetic fibres helping to resist
by careful maibre expansion dtive single-fibrend negative Pont in maximum lion (factor of tems [40]. ications for auxe body or car bu
or personal protet, projectile-resior glove. Othernclude fishnet, ressure pad, and
materials potenattracting intere
manmade auxe at micro or mam which they arrosity in order tmaterials need
ead out and so oted potential in
ng an understabehaviour at theo-fibres to realiro-fibres in high
-fibre.
ular level can bec nanoauxetics.
l nanoauxetics
first reported in ls having a tighta twinned crystals have been reple, carbon nitron’s ratios [42]. than diamond. ve also found thcubic crystals w
ut is a major ted to enhance son et al. 2002 sion process to ) fibre. It was could display crack growth, tching of the during loading e pull-out test oisson’s ratio load (factor of
f 3 increase)
xetic fibres in umper; and in ective clothing stant or bullet r applications
replacement d as a seal or
ntially have a st due to their etic materials
acro levels and re made. Most to achieve the space to allow
on). Therefore, n load-bearing
anding of the e nano-scale is se the already h performance
e divided into These will be
single crystal tly constrained l arrangement.
ported in other ide has been It is believed
hat a negative when they are
stretched almechanismstretched alcauses atomto move clatom 1 and along the [However, t
axis cases ato the ribsdirection. Hthe structurleading to a
For tmonocrystain its basabehaviour ialso pointeauxetic beh
Fig. 4. The ratios. (001[43]. [110] principal ax
The maYeganeh-Hsingle cryst
–cristobalframework
The sinanisotropic it has beenwould alsobeen observframework and Cheliko
Auxeticconcurrent axes in the edges) of framework been also d
long the [110] am is illustrated in
long the [110] dms 2 and 4 to seoser together al
d 3). This then co110] and [001] the moving toge
atoms 5 and 6 toconnecting the
Hence, there is anre, accompanyina negative value transversely ialline structure),al plane [44]. is predicted by ed out that 25haviour based on
nanostructure o1) refers to the
and [1 1 0] are xis
ain breakthroughHaeri et al 1992
tal Poisson’s ratite. Fig. 5 shoof corner-sheari
ngle-crystal –cand by employ
n calculated that be auxetic. Neved in -quartz, of SiO4 tetrahe
owsky 1992 [47c behaviour in
dilation and cx-y plane, passthe SiO4 mo
structure. Baseddeveloped for –
and [1 1 0] direFigure 4. Cubic
direction (axis beparate along [1long the [001] dorresponds to exaxes and so Poi
ether of atoms 1
o move apart alse atoms aligninn expansion of t
ng the expansionfor Poisson’s raisotropic zinc the auxetic behAn orthorhombRovati 2003. M% of monoclin
n theoretical anal
of cubic metals wdirection along directions at a
h in inorganic a2, measured abntios (up to -0.5 ows that –crising SiO4 tetrahedristobalite elastiing suitable avean isotropic po
egative Poisson’which also possdral (See Fig. 6)]. –cristobalite is tooperative rotating through the
olecular tetrahed on this ideal, acristobalite and
ction. The deforc elemental metabetween atom 2 10], and atoms direction (axis b
xtension and conisson’s ratio is p and 3 along th
ong the [110] ang towards the
the [1 1 0] dimenn in the [110] diatio [43]. c (with hexhaviour has beenbic alloy with
Meanwhile, Rovanic crystals dislysis [45,46].
with negative Poa cubic princip
90° angle from
auxetics occurrenormally high nfor some directi
stobalite consistdron. ic constants are
eraging techniquolycrystalline ags ratios also ha
sesses a corner-s) as reported by
thought to be dution (about tetrmidpoints of op
edral making an analytical moextended to –q
rmation als were and 4), 1 and 3 between ntraction positive. he [001]
ma et al. 200ulations. Experimequently been hez-Valle et al.
Apart from thoe are also organihave been very example, crystala et al. in 2006 e for loading aloacterized by x-rarted that cellulosa uniaxial tensiction) [49].
ecular level aux
The re-entrant hbeen identifiedrming through ytically by Abde
f 2x2 -cristobal
of 2x2 -quartz
her important clseveral idealizedossess negative
00 using forcemental confirmat
reported for Z2005 [48].
ose naturally ocic materials withrecently foundline cellulose I as having negatong the z-axis (ay diffraction. Nse II has auxeticile load in the
s, predicted by ar modelling behaviour has (natrolite) by
nic materials, ystal structure
etic properties. en reported by atios in the x-z
direction), as oworkers 2004 e (1 1 0) plane ellulose chain
etics
structure that on’s ratio by een modelled [50].
Fig. 7(ahoneycombunder stretcand open uratio.
Recent synthesis ofof organic a
Generalapproacheddownscalin1991 succepropose andsingle crysstructure hasuch a strucas -0.94) balso proposthe mechflexyne/reflentrant hexconcurrent molecular nn acetylene a)
b)
Fig. 7. (a) honeycomby-direction.reflexyne [1
While
provide a would lead a result of these particprocessed [in one plancrystallites
a) shows the debs. For a re-entch in the y direcup in the x dire
attention has ff auxetic structuauxetics have belly, the design
d by first designng these to the messfully employd demonstrate t
stalline networksaving auxetic prcture was showny force-field typ
sed a simple 2 dhanical properlexyne (n, m is n
xagons) networkstretching, hing
network. The velinks, respective
Two-dimensionb structure, wh. (b) Auxetic m1]
the re-entrant hpolymer with t
d to a high meltintheir highly cro
cular materials a[55]. Additionalne, and a bulk m
may be not auxe
y
x
y
x
y
x
CCC
C
C
C
C
C
C
C CC
formation mechtrant hexagonal ction the cells election, leading t
focused on the ures on a molecueen proposed. n of molecula
ning auxetic macmolecular or nanyed such a dowhrough modellins based on the roperties (See Fn to be auxetic (pe calculations dimensional analrties of thenumber of triple s. Deformation ging and flexingrtical and diagonely.
nal deformation hich are subjectmolecular honey
honeycomb struthe desired auxng temperature ss-linked structure not likely to bly, the auxetic b
material consistinetic.
C CC
C CC
C
C
C
C
hanism of the regeometry (Fig
ongate along theto a negative Po
theoretical desiular level, and a
ar auxetics is cro structures, anno level [51,52]wnscaling technng polyphenylac
re-entrant honeig. 7(b)). For ex(Poisson’s ratios[53]. Evans et alytical model to ir so-called bond) (conventiwas assumed tog of the ‘arms’nal arms contain
mechanisms in ted to loading
ycomb network:
ucture can in pxetic properties, and poor tractabure [54]. Conseqbe easily synthebehaviour only
ng of randomly o
C
C
C
C
C
C
C
C
-entrant g. 7(a)), e y-axis oisson’s
ign and number
being nd then . Evans
nique to cetylene eycomb xample, s as low al. have predict (n,m)-
For fibre reinfore mechanism. Aailure resistance rted the successfuce the first auxulated that comced fibre pull-ouugh maintainingson’s ratios of th38,39]. Subsequ performed uspropylene fibre ancrease) and eonstrate for the aThere are manyposite materials,bination with othquipment, e.g. inf vest, shin paulated for aux
material, a bandaet.
Limitations
As mentioned ae range of applicue behaviour. Hve their behavioure less stiff thantic materials hative Poisson’s rages” to flex, or “y auxetic materications. A further reasohanisms responshe developmentblished benefits posites containin
Auxetic ma
Auxetic materialcategories: naturdly discussed in
ecular level aux
A negative Poisspyrites [41] as alar geometric arre then negative le crystal phaseested to have a carbon nitride wBaughman & Gson’s ratio is ve
orced compositeAuxetic materials
properties of coful development xetic polymeric
mposites with aut, potentially hg the interface he matrix and fi
uently, comparatsing positive anand enhancemenenergy adsorptiauxetic fibre systy potential appli, such as vehicleher materials fon a crash helmetad, knee pad oxetic fibres inage or wound p
s
above, auxetic cations and are aHowever, most ur from structuren the solids fromave intrinsic poratio (i.e. auxetic “nodules” to sprerials have limit
on for developiible for auxetic
t of auxetic nanofor auxetic mac
ng auxetic nano-
aterials
ls at the molecural and syntheticthe following.
xetics I: Natural
son’s ratio was fa result of crystalrangement and aPoisson’s ratios
es. For examplnegative Poisso
would be harder tGalvao 1993 havery common in
es, fibre pull-ous are also predictmposites. Aldersof a melt-extrus(polypropylene)
auxetic fibres helping to resist
by careful maibre expansion dtive single-fibrend negative Pont in maximum lion (factor of tems [40]. ications for auxe body or car bu
or personal protet, projectile-resior glove. Othernclude fishnet, ressure pad, and
materials potenattracting intere
manmade auxe at micro or mam which they arrosity in order tmaterials need
ead out and so oted potential in
ng an understabehaviour at theo-fibres to realiro-fibres in high
-fibre.
ular level can bec nanoauxetics.
l nanoauxetics
first reported in ls having a tighta twinned crystals have been reple, carbon nitron’s ratios [42]. than diamond. ve also found thcubic crystals w
ut is a major ted to enhance son et al. 2002 sion process to ) fibre. It was could display crack growth, tching of the during loading e pull-out test oisson’s ratio load (factor of
f 3 increase)
xetic fibres in umper; and in ective clothing stant or bullet r applications
replacement d as a seal or
ntially have a st due to their etic materials
acro levels and re made. Most to achieve the space to allow
on). Therefore, n load-bearing
anding of the e nano-scale is se the already h performance
e divided into These will be
single crystal tly constrained l arrangement.
ported in other ide has been It is believed
hat a negative when they are
stretched almechanismstretched alcauses atomto move clatom 1 and along the [However, t
axis cases ato the ribsdirection. Hthe structurleading to a
For tmonocrystain its basabehaviour ialso pointeauxetic beh
Fig. 4. The ratios. (001[43]. [110] principal ax
The maYeganeh-Hsingle cryst
–cristobalframework
The sinanisotropic it has beenwould alsobeen observframework and Cheliko
Auxeticconcurrent axes in the edges) of framework been also d
long the [110] am is illustrated in
long the [110] dms 2 and 4 to seoser together al
d 3). This then co110] and [001] the moving toge
atoms 5 and 6 toconnecting the
Hence, there is anre, accompanyina negative value transversely ialline structure),al plane [44]. is predicted by ed out that 25haviour based on
nanostructure o1) refers to the
and [1 1 0] are xis
ain breakthroughHaeri et al 1992
tal Poisson’s ratite. Fig. 5 shoof corner-sheari
ngle-crystal –cand by employ
n calculated that be auxetic. Neved in -quartz, of SiO4 tetrahe
owsky 1992 [47c behaviour in
dilation and cx-y plane, passthe SiO4 mo
structure. Baseddeveloped for –
and [1 1 0] direFigure 4. Cubic
direction (axis beparate along [1long the [001] dorresponds to exaxes and so Poi
ether of atoms 1
o move apart alse atoms aligninn expansion of t
ng the expansionfor Poisson’s raisotropic zinc the auxetic behAn orthorhombRovati 2003. M% of monoclin
n theoretical anal
of cubic metals wdirection along directions at a
h in inorganic a2, measured abntios (up to -0.5 ows that –crising SiO4 tetrahedristobalite elastiing suitable avean isotropic po
egative Poisson’which also possdral (See Fig. 6)]. –cristobalite is tooperative rotating through the
olecular tetrahed on this ideal, acristobalite and
ction. The deforc elemental metabetween atom 2 10], and atoms direction (axis b
xtension and conisson’s ratio is p and 3 along th
ong the [110] ang towards the
the [1 1 0] dimenn in the [110] diatio [43]. c (with hexhaviour has beenbic alloy with
Meanwhile, Rovanic crystals dislysis [45,46].
with negative Poa cubic princip
90° angle from
auxetics occurrenormally high nfor some directi
stobalite consistdron. ic constants are
eraging techniquolycrystalline ags ratios also ha
sesses a corner-s) as reported by
thought to be dution (about tetrmidpoints of op
edral making an analytical moextended to –q
rmation als were and 4), 1 and 3 between ntraction positive. he [001]
ma et al. 200ulations. Experimequently been hez-Valle et al.
Apart from thoe are also organihave been very example, crystala et al. in 2006 e for loading aloacterized by x-rarted that cellulosa uniaxial tensiction) [49].
ecular level aux
The re-entrant hbeen identifiedrming through ytically by Abde
f 2x2 -cristobal
of 2x2 -quartz
her important clseveral idealizedossess negative
00 using forcemental confirmat
reported for Z2005 [48].
ose naturally ocic materials withrecently foundline cellulose I as having negatong the z-axis (ay diffraction. Nse II has auxeticile load in the
s, predicted by ar modelling behaviour has (natrolite) by
nic materials, ystal structure
etic properties. en reported by atios in the x-z
direction), as oworkers 2004 e (1 1 0) plane ellulose chain
etics
structure that on’s ratio by een modelled [50].
Fig. 7(ahoneycombunder stretcand open uratio.
Recent synthesis ofof organic a
Generalapproacheddownscalin1991 succepropose andsingle crysstructure hasuch a strucas -0.94) balso proposthe mechflexyne/reflentrant hexconcurrent molecular nn acetylene a)
b)
Fig. 7. (a) honeycomby-direction.reflexyne [1
While
provide a would lead a result of these particprocessed [in one plancrystallites
a) shows the debs. For a re-entch in the y direcup in the x dire
attention has ff auxetic structuauxetics have belly, the design
d by first designng these to the messfully employd demonstrate t
stalline networksaving auxetic prcture was showny force-field typ
sed a simple 2 dhanical properlexyne (n, m is n
xagons) networkstretching, hing
network. The velinks, respective
Two-dimensionb structure, wh. (b) Auxetic m1]
the re-entrant hpolymer with t
d to a high meltintheir highly cro
cular materials a[55]. Additionalne, and a bulk m
may be not auxe
y
x
y
x
y
x
CCC
C
C
C
C
C
C
C CC
formation mechtrant hexagonal ction the cells election, leading t
focused on the ures on a molecueen proposed. n of molecula
ning auxetic macmolecular or nanyed such a dowhrough modellins based on the roperties (See Fn to be auxetic (pe calculations dimensional analrties of thenumber of triple s. Deformation ging and flexingrtical and diagonely.
nal deformation hich are subjectmolecular honey
honeycomb struthe desired auxng temperature ss-linked structure not likely to bly, the auxetic b
material consistinetic.
C CC
C CC
C
C
C
C
hanism of the regeometry (Fig
ongate along theto a negative Po
theoretical desiular level, and a
ar auxetics is cro structures, anno level [51,52]wnscaling technng polyphenylac
re-entrant honeig. 7(b)). For ex(Poisson’s ratios[53]. Evans et alytical model to ir so-called bond) (conventiwas assumed tog of the ‘arms’nal arms contain
mechanisms in ted to loading
ycomb network:
ucture can in pxetic properties, and poor tractabure [54]. Conseqbe easily synthebehaviour only
ng of randomly o
C
C
C
C
C
C
C
C
-entrant g. 7(a)), e y-axis oisson’s
ign and number
being nd then . Evans
nique to cetylene eycomb xample, s as low al. have predict (n,m)-
oisson’s ratios as between the dn mechasnism wf auxetic zeoliteauxetic respon
re’ molecular unet al, 2005 hapolymers based
hese were proture shown in haviour. As wiular network s
d. one of the mo
syntheticaly accequid crystallineed by Griffin e
f polytriangles-2olytriangles-7-yn
ios have been ons and maximum
zy=-0.96), and exing triangles. Inare due to a variifferent layers. Twas also used tes [57,58]. Othense through rotanits [48]. ave also proposd on calix[4] areoduced by dowFig. 10, whichith the polypheshown in Fig.
ost simple and essible moleculae polymer coet al 1995; Gri
-yne; (b) the mine at z=0.0 GPa
obtained in 3 dirm auxeticity is fxplained by coopn the plane x-y aiation in the amThe ‘rotating trito interpret the r zeolites are pr
ation of corner-
ed a range of ene molecular bwnscaling the h is known to nylacetylene ne
10(b) is not
promising appar-level auxetic poncept, proposeiffin & Liu 19
inimum a [57]
rections found in perative and x-z,
mount of iangles’ auxetic
redicted -sharing
auxetic building
folded display
etworks, easily
roaches polymer ed and 998 and
showbasedchain‘rigidSomemoletensithe latensichain
Tprincalignforcemate a)
b)
Fig. theormanumole
wn in Fig. 11. Td on site-connecn to achieve auxd’ rod molecule e rigid rods are
ecular, while othle stress, full exaterally attachedle axis to a posns further apart (The main advaciple, as it is liqn the chains, leade transfer [59].erials have as yet
10. (a) The auxretical moleculaufactured in steecular ‘double ca
The liquid crystctivity driven roxetic behaviourinterconnected b attached at the
her rigid rods havxtension of the pd rods from a poition normal to (See Fig. 11). antage of this pquid crystalline,ding to more cov Unfortunately,t failed to measu
xetic macrostrucar system is baeel and marketealix[4]’ network
talline polymersod reorientation . The main chaby ‘flexible’ spa
eir ends to the fve lateral attachmpolymer main chosition roughly it and push the
polymer netwo, it should be mvalent than non-, mechanical te
ure a negative Po
cture (30×30 cmased. The macred as an ‘egg ranalogues [59]
s are designed in their main
ain consists of acer molecule. flexible spacer ment. Under a hain will force parallel to the neighbouring
rk is that in much easier to covalent bond ests on these oisson’s ratio.
m) on which a rostsructure is rack’; (b) the
Fig. 11. (Asynthesisedbased terphpossible spsiloxane (Xproposed malign with sa concomita
Referen [1] K.E.
Molec[2] K.L. A
polyet33/20
[3] J.B. CcellulaMater
[4] J.B. CcellulaMater
[5] R.S. LnegatiMater
[6] N. Chand au231-2
[7] C.P. CmaterPoissoScienc
[8] C.P. Cdynamratio Techn
[9] J.B. Cfoam and a73-83
[10] H. GeJourna1-13.
A
B1
B2
) An example od by Griffin anhenyl transversepacers: methylenX=Si(CH3)2O). mechanism resusurrounding liquant lateral expan
nces
Evans, M.A. Ncular network deAlderson, K.E. Ethylene having (1993) 4435-44
oisson’s ratios as between the dn mechasnism wf auxetic zeoliteauxetic respon
re’ molecular unet al, 2005 hapolymers based
hese were proture shown in haviour. As wiular network s
d. one of the mo
syntheticaly accequid crystallineed by Griffin e
f polytriangles-2olytriangles-7-yn
ios have been ons and maximum
zy=-0.96), and exing triangles. Inare due to a variifferent layers. Twas also used tes [57,58]. Othense through rotanits [48]. ave also proposd on calix[4] areoduced by dowFig. 10, whichith the polypheshown in Fig.
ost simple and essible moleculae polymer coet al 1995; Gri
-yne; (b) the mine at z=0.0 GPa
obtained in 3 dirm auxeticity is fxplained by coopn the plane x-y aiation in the amThe ‘rotating trito interpret the r zeolites are pr
ation of corner-
ed a range of ene molecular bwnscaling the h is known to nylacetylene ne
10(b) is not
promising appar-level auxetic poncept, proposeiffin & Liu 19
inimum a [57]
rections found in perative and x-z,
mount of iangles’ auxetic
redicted -sharing
auxetic building
folded display
etworks, easily
roaches polymer ed and 998 and
showbasedchain‘rigidSomemoletensithe latensichain
Tprincalignforcemate a)
b)
Fig. theormanumole
wn in Fig. 11. Td on site-connecn to achieve auxd’ rod molecule e rigid rods are
ecular, while othle stress, full exaterally attachedle axis to a posns further apart (The main advaciple, as it is liqn the chains, leade transfer [59].erials have as yet
10. (a) The auxretical moleculaufactured in steecular ‘double ca
The liquid crystctivity driven roxetic behaviourinterconnected b attached at the
her rigid rods havxtension of the pd rods from a poition normal to (See Fig. 11). antage of this pquid crystalline,ding to more cov Unfortunately,t failed to measu
xetic macrostrucar system is baeel and marketealix[4]’ network
talline polymersod reorientation . The main chaby ‘flexible’ spa
eir ends to the fve lateral attachmpolymer main chosition roughly it and push the
polymer netwo, it should be mvalent than non-, mechanical te
ure a negative Po
cture (30×30 cmased. The macred as an ‘egg ranalogues [59]
s are designed in their main
ain consists of acer molecule. flexible spacer ment. Under a hain will force parallel to the neighbouring
rk is that in much easier to covalent bond ests on these oisson’s ratio.
m) on which a rostsructure is rack’; (b) the
Fig. 11. (Asynthesisedbased terphpossible spsiloxane (Xproposed malign with sa concomita
Referen [1] K.E.
Molec[2] K.L. A
polyet33/20
[3] J.B. CcellulaMater
[4] J.B. CcellulaMater
[5] R.S. LnegatiMater
[6] N. Chand au231-2
[7] C.P. CmaterPoissoScienc
[8] C.P. Cdynamratio Techn
[9] J.B. Cfoam and a73-83
[10] H. GeJourna1-13.
A
B1
B2
) An example od by Griffin anhenyl transversepacers: methylenX=Si(CH3)2O). mechanism resusurrounding liquant lateral expan
nces
Evans, M.A. Ncular network deAlderson, K.E. Ethylene having (1993) 4435-44
Journal of Achievements in Materials and Manufacturing Engineering
Y.T. Yao, M. Uzun, I. Patel
Volume 49 Issue 2 December 2011
[11] R.S. Lakes, Design considerations for negative Poisson's ratio Materials Journal of Mechanical Design 115 (1993) 696-700.
[12] H. Hertz, Über die Berührung fester elastischer Körper, Journal of Maths (Crelle J) 92 (1881) 156.
[13] S.P. Timoshenko, Theory of elasticity, McGraw-Hill, New York, 1970.
[14] K.L. Alderson, A.F. Fitzgerald, K.E. Evans, The strain dependent indentation resilience of auxetic microporous polyethylene, Journal of Materials Science 35 (2000) 4039-4047.
[15] A. Alderson, A triumph of lateral thought, Chemistry and Industry 10 (1999) 384-391.
[16] K.E. Evans, A. Alderson, Auxetic materials: Functional materials and structures from lateral thinking, Advanced Materials 12/9 (2000) 617-628.
[17] R.S. Lakes, Foam structures with a negative Poisson’s ratio, Science 235/4792 (1987) 1038-1040.
[18] B. Brandel, R.S. Lakes, Negative Poisson’s ratio polyethylene foams, Journal of Materials Science 36 (2001) 5885-5893.
[19] B.D. Caddock, K.E. Evans, Microporous materials with negative Poisson's ratios. II, Mechanisms and interpretation, Journal of Physics D: Applied Physics 22 (1989) 1883-1887.
[20] A.N. Sherbourne, M.D. Pandey, Proceedings of the “Engineering Mechanics Specialty Conference” ASCE’1991, Columbus, 1991, 841.
[21] R.G. Zhang, H.L. Yeh, H.Y. Yeh, A preliminary study of negative Poisson's ratio of laminated fiber reinforced composites, Journal of Reinforced Plastics and Composites 17/18 (1998) 1651-1664.
[22] R.G. Zhang, H.L. Yeh, H.Y. Yeh, A discussion of negative Poisson's ratio design for composites, Journal of Reinforced Plastics and Composites 18/17 (1999) 15461556.
[23] F. Homand-Etienne, R. Houpert, Thermally induced microcracking in granites: characterization and analysis, International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 26/2 (1989) 125-134.
[24] A. Nur, G. Simmons, The effect of saturation on velocity in low-porosity rocks, Earth and Planetary Science Letters 7/2 (1969) 183-193.
[25] A. Yeganeh-Haeri, D.J. Weidner, J.B. Parise, Elasticity of -cristobalite: A silicon dioxide with a negative Poisson's ratio, Science 2575070 (1992) 650-652.
[26] A. Alderson, K.E. Evans, Rotation and dilation deformation mechanisms for auxetic behaviour in the -cristobalite tetrahedral framework structure, Physics and Chemistry of Minerals 28/10 (2001) 711-718.
[27] J.N. Grima, R. Jackson, A. Alderson, K.E. Evans, Do zeolites have negative Poisson's ratios, Advanced Materials 12/24 (2000) 1912-1918.
[28] P.J. Stott, R. Mitchell, K.L. Alderson, A. Alderson, A growing industry, Materials World 8 (2000) 12-14.
[29] A. Alderson, J.A. Rasburn, K.E. Evans, An auxetic filter, A tuneable filter displaying enhanced sizes electivity or defueling properties, Industrial and Engineering Chemistry Research 39 (2000) 654.
[30] J.B. Choi, R.S. Lakes, Design of a fastener based on negative Poisson's ratio foam, Cellular Polymers 10 (1991) 205-212.
[31] M.A. Loureiro, R.S. Lakes, Scale-up of transformation of negative Poisson's ratio foam: Slabs, Cellular Polymers 16 (1997) 349-363.
[32] Y.C. Wang, R. Lakes, Analytical parametric analysis of the contact problem of human buttocks and negative Poisson's ratio foam cushions, International Journal of Solids and Structures 39 (2002) 4825-4838.
[33] M. Burke, A stretch of the imagination, New Scientist 154 (1997) 36-39.
[35] X. Huang, S. Blackburn, Developing a new processing route to manufacture honeycomb ceramics with negative Poisson’s ratio, Key Engineering Materials 206-213 (2001) 201-204.
[36] M. Avellanads, P.J. Swart, Calculating the performance of 1-3 piezo composites for hydrophone applications: An effective medium approach, Journal of the Acoustical Society of America 103/3 (1998)1449-1467.
[37] W.A. Smith, US Patent No. 5334903, 1994. [38] K.L Alderson, A. Alderson, G. Smart, V.R. Simkins,
P.J. Davies, Auxetic polypropylene fibres, Part 1, Manufacture and characterisation, Plastics, Rubber and Composites 31/8 (2002) 344-349.
[39] K.E. Evans, Tailoring the negative Poisson’s ratio, Chemistry and Industry 20 (1990) 654-657.
[40] V.R. Simkins, N. Ravirala, P.J. Davies, A. Alderson, K.L. Alderson, An experimental study of thermal post-production processing of auxetic polypropylene fibres, Physica Status Solidi 245 (2008) 598-605.
[41] A.E.H. Love, A Treatise on the mathematical theory of elasticity, Dover, New York, 1944.
[42] Y. Guo, I.W. Goddard, Is carbon nitride harder than diamond? No, but its grith increases when stretched (Negative Poisson’s ratio), Chemical Physics Letters 237 (1995) 72-76.
[43] R.H. Baughman, D.S. Galvao, Crystalline network with unusual predicted mechanical and thermal properties, Nature 365 (1993) 735-737.
[44] V.A. Lubarda, M.A. Meyers, On the negative Poisson ratio in monocrystalline zinc, Scripta Materialia 40/8 (1999) 975-977.
[45] M. Rovati, On the negative Poisson's ratio of an orthorhombic alloy, Scripta Materialia 48 (2003) 235-240.
[46] M. Rovati, Directions of auxeticity for monoclinic crystals, Scripta Materialia 51 (2004) 1087-1091.
[47] N.R. Keskar, J.R. Chelikowsky, Negative Poisson ratios in crystalline SiO2 from first-principles calculations, Nature 358 (1992) 222-224.
[48] C. Sanchez-Valle, S.V. Sinogeikin, Z.A. Lethbridge, R.I. Walton, C.W. Smith, K.E. Evans, J.D. Bass, Ab initio structural, elastic, and vibrational properties of carbon nanotubes Physical Review B 59 (1999) 12678-12688.
[49] K. Nakamura, M. Wada,S. Kuga, T. Okano, Poisson's ratio of cellulose I and cellulose II, Journal of Polymer Science Part B: Polymer Physics 42/7 (2004) 1206-1211.
[50] F.K. Abdel-Sayed, R. Jones, I.W. Burgens, Theoretical approach to the deformation of honeycomb based composite-materials, Composites 10/4 (1979) 209-214.
[51] K.E. Evans, A. Alderson, F.R. Attenborough, Auxetic two-dimensional polymer networks: An example of tailoring
geometry for specific mechanical properties, Journal of the Chemical Society, Faraday Transactions 91 (1995) 2671-2680.
[52] F.R. Attenborough, Ph.D. Thesis, University of Liverpool, UK, 1997.
[53] M.A. Nkansah, K.E. Evans, I.J. Hutchinson, Modelling the mechanical properties of an auxetic molecular network, Modelling and Simulation in Materials Science and Engineering 2 (1994) 337-352.
[54] A.C. Griffin, S. Kumar, P.J. McMullan, NTC Project, M04-GT21, http://www.ptfe.gatech.edu/faculty/mcmullan/paper.php.
[55] P. Aldred, S.C. Moratti, Dynamic simulations of potentially auxetic liquid-crystalline polymers incorporating swivelling mesogens, Molecular Simulation 31/13 (2005) 883-887.
[56] G.Y. Wei, Design of auxetic polymer self-assemblies Physica Status Solidi 242/3 (2005) 742-748.
[57] J.N. Grima, K.E. Evans, Auxetic behaviour from rotating squares, Journal Of Materials Science Letters 19 (2000) 1563-1565.
[58] U.H.F. Bunz, Y. Rubin, Y. Tobe, Polyethynylated cyclic pi-systems: scaffoldings for novel two and three-dimensional carbon networks, Chemical Society Reviews 28/2 (1999) 107-119.
[59] J.N. Grima, R. Gatt, A. Alderson, K.E. Evans, On the auxetic properties of ‘rotating rectangles’ with different connectivity, Journal of the Physical Society of Japan 7/104 (2005) 2866-2867.
[11] R.S. Lakes, Design considerations for negative Poisson's ratio Materials Journal of Mechanical Design 115 (1993) 696-700.
[12] H. Hertz, Über die Berührung fester elastischer Körper, Journal of Maths (Crelle J) 92 (1881) 156.
[13] S.P. Timoshenko, Theory of elasticity, McGraw-Hill, New York, 1970.
[14] K.L. Alderson, A.F. Fitzgerald, K.E. Evans, The strain dependent indentation resilience of auxetic microporous polyethylene, Journal of Materials Science 35 (2000) 4039-4047.
[15] A. Alderson, A triumph of lateral thought, Chemistry and Industry 10 (1999) 384-391.
[16] K.E. Evans, A. Alderson, Auxetic materials: Functional materials and structures from lateral thinking, Advanced Materials 12/9 (2000) 617-628.
[17] R.S. Lakes, Foam structures with a negative Poisson’s ratio, Science 235/4792 (1987) 1038-1040.
[18] B. Brandel, R.S. Lakes, Negative Poisson’s ratio polyethylene foams, Journal of Materials Science 36 (2001) 5885-5893.
[19] B.D. Caddock, K.E. Evans, Microporous materials with negative Poisson's ratios. II, Mechanisms and interpretation, Journal of Physics D: Applied Physics 22 (1989) 1883-1887.
[20] A.N. Sherbourne, M.D. Pandey, Proceedings of the “Engineering Mechanics Specialty Conference” ASCE’1991, Columbus, 1991, 841.
[21] R.G. Zhang, H.L. Yeh, H.Y. Yeh, A preliminary study of negative Poisson's ratio of laminated fiber reinforced composites, Journal of Reinforced Plastics and Composites 17/18 (1998) 1651-1664.
[22] R.G. Zhang, H.L. Yeh, H.Y. Yeh, A discussion of negative Poisson's ratio design for composites, Journal of Reinforced Plastics and Composites 18/17 (1999) 15461556.
[23] F. Homand-Etienne, R. Houpert, Thermally induced microcracking in granites: characterization and analysis, International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 26/2 (1989) 125-134.
[24] A. Nur, G. Simmons, The effect of saturation on velocity in low-porosity rocks, Earth and Planetary Science Letters 7/2 (1969) 183-193.
[25] A. Yeganeh-Haeri, D.J. Weidner, J.B. Parise, Elasticity of -cristobalite: A silicon dioxide with a negative Poisson's ratio, Science 2575070 (1992) 650-652.
[26] A. Alderson, K.E. Evans, Rotation and dilation deformation mechanisms for auxetic behaviour in the -cristobalite tetrahedral framework structure, Physics and Chemistry of Minerals 28/10 (2001) 711-718.
[27] J.N. Grima, R. Jackson, A. Alderson, K.E. Evans, Do zeolites have negative Poisson's ratios, Advanced Materials 12/24 (2000) 1912-1918.
[28] P.J. Stott, R. Mitchell, K.L. Alderson, A. Alderson, A growing industry, Materials World 8 (2000) 12-14.
[29] A. Alderson, J.A. Rasburn, K.E. Evans, An auxetic filter, A tuneable filter displaying enhanced sizes electivity or defueling properties, Industrial and Engineering Chemistry Research 39 (2000) 654.
[30] J.B. Choi, R.S. Lakes, Design of a fastener based on negative Poisson's ratio foam, Cellular Polymers 10 (1991) 205-212.
[31] M.A. Loureiro, R.S. Lakes, Scale-up of transformation of negative Poisson's ratio foam: Slabs, Cellular Polymers 16 (1997) 349-363.
[32] Y.C. Wang, R. Lakes, Analytical parametric analysis of the contact problem of human buttocks and negative Poisson's ratio foam cushions, International Journal of Solids and Structures 39 (2002) 4825-4838.
[33] M. Burke, A stretch of the imagination, New Scientist 154 (1997) 36-39.
[35] X. Huang, S. Blackburn, Developing a new processing route to manufacture honeycomb ceramics with negative Poisson’s ratio, Key Engineering Materials 206-213 (2001) 201-204.
[36] M. Avellanads, P.J. Swart, Calculating the performance of 1-3 piezo composites for hydrophone applications: An effective medium approach, Journal of the Acoustical Society of America 103/3 (1998)1449-1467.
[37] W.A. Smith, US Patent No. 5334903, 1994. [38] K.L Alderson, A. Alderson, G. Smart, V.R. Simkins,
P.J. Davies, Auxetic polypropylene fibres, Part 1, Manufacture and characterisation, Plastics, Rubber and Composites 31/8 (2002) 344-349.
[39] K.E. Evans, Tailoring the negative Poisson’s ratio, Chemistry and Industry 20 (1990) 654-657.
[40] V.R. Simkins, N. Ravirala, P.J. Davies, A. Alderson, K.L. Alderson, An experimental study of thermal post-production processing of auxetic polypropylene fibres, Physica Status Solidi 245 (2008) 598-605.
[41] A.E.H. Love, A Treatise on the mathematical theory of elasticity, Dover, New York, 1944.
[42] Y. Guo, I.W. Goddard, Is carbon nitride harder than diamond? No, but its grith increases when stretched (Negative Poisson’s ratio), Chemical Physics Letters 237 (1995) 72-76.
[43] R.H. Baughman, D.S. Galvao, Crystalline network with unusual predicted mechanical and thermal properties, Nature 365 (1993) 735-737.
[44] V.A. Lubarda, M.A. Meyers, On the negative Poisson ratio in monocrystalline zinc, Scripta Materialia 40/8 (1999) 975-977.
[45] M. Rovati, On the negative Poisson's ratio of an orthorhombic alloy, Scripta Materialia 48 (2003) 235-240.
[46] M. Rovati, Directions of auxeticity for monoclinic crystals, Scripta Materialia 51 (2004) 1087-1091.
[47] N.R. Keskar, J.R. Chelikowsky, Negative Poisson ratios in crystalline SiO2 from first-principles calculations, Nature 358 (1992) 222-224.
[48] C. Sanchez-Valle, S.V. Sinogeikin, Z.A. Lethbridge, R.I. Walton, C.W. Smith, K.E. Evans, J.D. Bass, Ab initio structural, elastic, and vibrational properties of carbon nanotubes Physical Review B 59 (1999) 12678-12688.
[49] K. Nakamura, M. Wada,S. Kuga, T. Okano, Poisson's ratio of cellulose I and cellulose II, Journal of Polymer Science Part B: Polymer Physics 42/7 (2004) 1206-1211.
[50] F.K. Abdel-Sayed, R. Jones, I.W. Burgens, Theoretical approach to the deformation of honeycomb based composite-materials, Composites 10/4 (1979) 209-214.
[51] K.E. Evans, A. Alderson, F.R. Attenborough, Auxetic two-dimensional polymer networks: An example of tailoring
geometry for specific mechanical properties, Journal of the Chemical Society, Faraday Transactions 91 (1995) 2671-2680.
[52] F.R. Attenborough, Ph.D. Thesis, University of Liverpool, UK, 1997.
[53] M.A. Nkansah, K.E. Evans, I.J. Hutchinson, Modelling the mechanical properties of an auxetic molecular network, Modelling and Simulation in Materials Science and Engineering 2 (1994) 337-352.
[54] A.C. Griffin, S. Kumar, P.J. McMullan, NTC Project, M04-GT21, http://www.ptfe.gatech.edu/faculty/mcmullan/paper.php.
[55] P. Aldred, S.C. Moratti, Dynamic simulations of potentially auxetic liquid-crystalline polymers incorporating swivelling mesogens, Molecular Simulation 31/13 (2005) 883-887.
[56] G.Y. Wei, Design of auxetic polymer self-assemblies Physica Status Solidi 242/3 (2005) 742-748.
[57] J.N. Grima, K.E. Evans, Auxetic behaviour from rotating squares, Journal Of Materials Science Letters 19 (2000) 1563-1565.
[58] U.H.F. Bunz, Y. Rubin, Y. Tobe, Polyethynylated cyclic pi-systems: scaffoldings for novel two and three-dimensional carbon networks, Chemical Society Reviews 28/2 (1999) 107-119.
[59] J.N. Grima, R. Gatt, A. Alderson, K.E. Evans, On the auxetic properties of ‘rotating rectangles’ with different connectivity, Journal of the Physical Society of Japan 7/104 (2005) 2866-2867.