Materials Science and Engineering A 423 (2006) 214–218

Negative Poisson’s ratios in cellular foam materials

Joseph N. Grima a,∗, Ruben Gatt a, Naveen Ravirala b,Andrew Alderson b, K.E. Evans c

a Department of Chemistry, University of Malta, Msida MSD 06, Maltab Centre for Materials Research and Innovation, University of Bolton, Bolton BL3 5AB, UK

c Department of Engineering, University of Exeter, Exeter EX4 4QF, UK

Received 21 July 2005; accepted 29 August 2005

Abstract

Materials with negative Poisson’s ratios (auxetic) get fatter when stretched and thinner when compressed. This paper discusses a new explanationfor achieving auxetic behaviour in foam cellular materials, namely a ‘rotation of rigid units’ mechanism. Such auxetic cellular materials can beproduced from conventional open-cell cellular materials if the ribs of cell are slightly thicker in the proximity of the joints when compared to thecentre of the ribs with the consequence that if the conventional cellular material is volumetrically compressed (and then ‘frozen’ in the compressedcmio©

K

1

tn1aaawmtssVrbtt

0d

onformation), the cellular structure will deform in such a way which conserves the geometry at the joints (i.e. behave like ‘rigid units’) whilst theajor deformations will occur along the length of the more flexible ribs which form ‘kinks’ at their centres as a result of the extensive buckling. It

s proposed that uniaxial tensile loading of such cellular systems will result auxetic behaviour due to re-unfolding of these ‘kinks’ and re-rotationf the ‘rigid joints’.

2006 Published by Elsevier B.V.

eywords: Auxetic; Foams; Negative Poisson’s ratios; Mechanical properties

. Introduction

Materials with negative Poisson’s ratios (auxetic) [1] exhibithe unusual property of becoming fatter when stretched and thin-er when compressed. This unusual property was first reported in944 when iron pyrites single crystals were described as havingnegative Poisson’s ratio, a phenomenon, which was regarded

s an anomaly and attributed to twinning defects [2]. Since then,nd most particularly in the last two decades, auxetic behaviouras predicted, discovered, or deliberately introduced in variousaterials ranging from molecular level systems (e.g. nanos-

ructured and liquid crystalline polymers [1,3–7], metals [8],ilicates [9,10] and zeolites [11]) to microstructured materialsuch as foams [12–15] and microstructured polymers [16–18].arious auxetic structures have also been proposed including

e-entrant [19–21] and chiral honeycombs [22–24] and modelsased on rigid ‘free’ molecules [25–27]. In all of these systems,he negative Poisson’s ratios are a consequence of the way thathe geometry of the structure (the nano/microstructure in the

∗ Corresponding author. Tel.: +356 23402274; fax: +356 21330400.

case of materials) changes when uniaxial mechanical loads areapplied.

Auxetic materials are known to exhibit various enhancedphysical characteristics over their conventional counterpartsranging from increased indentation resistance [28,29] toimproved acoustic damping properties [30,31]. These enhancedcharacteristics make auxetic materials perform better in manypractical applications.

A class of auxetics, which has attracted considerable atten-tion in recent years is that of auxetic foams (see Fig. 1). Thesefoams were first manufactured by Lakes [12] and can be pro-duced from commercially available conventional foams througha process involving volumetric compression, heating beyond thepolymer’s softening temperature and then cooling whilst remain-ing under compression. For example, Smith et al. [15] reportthat they convert samples of commercially available reticulated30 ppi polyurethane foams (‘Filtren’ by Recticel Ltd.) whichoriginally exhibited Poisson’s ratio of ca. +0.85 for loading inthe ‘rise direction’ to an auxetic form which exhibits Poisson’sratios of ca. −0.60 trough a process of compression in volume byca. 30%, heating at 200 ◦C and then cooling in the compressed

E-mail address: joseph.grima@um.edu.mt (J.N. Grima). shape.

921-5093/$ – see front matter © 2006 Published by Elsevier B.V.oi:10.1016/j.msea.2005.08.229

J.N. Grima et al. / Materials Science and Engineering A 423 (2006) 214–218 215

Fig. 1. SEM images of (a) conventional polyurethane foams, and (b) auxetic polyurethane foam.

2. Existing models for explaining the occurrence ofauxetic behaviour in foams

Various two-dimensional models, which represent a cross-section of conventional and auxetic foams have been proposedaimed at explaining the experimentally measured values of thePoisson’s ratios in terms of the microstructure of the foams.For example it has been proposed that conventional foams canbe modelled using hexagonal [20,21] and diamond-shaped [15]honeycombs (see Figs. 2a and 3a) whilst auxetic foams can bemodelled through modified versions of these honeycombs (seeFigs. 2b and 3b). In the case of hexagonal honeycomb model,the modification needed to convert the conventional honeycombto an auxetic form is to transform the ‘Y’ shaped joints to arrowshaped joints as illustrated in Fig. 2. This process of inversion ofjoints results in the formation of a re-entrant honeycomb, whichcan exhibit auxetic behaviour if it deforms through changes inthe angles between elements of the honeycomb (see Fig. 2b)or flexure of their ribs. In the case of the diamond-shaped [15]honeycombs (see Fig. 3a), the transformation process requires

Ff

a procedure involving the selective removal of ribs in a regularmanner (see Fig. 3b) so as to obtain a ‘chiral model’, whichexhibits negative Poission’s ratios [15,24].

Reports have also been made of three-dimensional modelswhere the foam cells are represented by a dodecahedron [13,32]or tetrakaidecahedron [33–36]. In these models, it was assumedthat the conversion from a conventional to an auxetic foam wascaused by changes in the shape of the joints, a process whichresults in the formation of re-entrant cells (i.e. a transformationwhich is similar to what is illustrated in Fig. 2 for the two-dimensional hexagonal honeycombs).

Although these models can independently or concurrentlyexplain and reproduce the experimentally measured values of thePoisson’s ratios [15] and probably play some role in modellingthe auxetic behaviour in these foams, one may argue that there isnot enough experimental evidence to justify the assumption thateither of these models represents the main structural modifica-tions which result in the observed auxetic effect. For example,whilst there is experimental evidence that there are ‘broken ribs’

Ft

ig. 2. (a) The conventional hexagonal honeycomb model for conventionaloams, and (b) the re-entrant hexagonal honeycomb model for auxetic foams.

ig. 3. (a) The conventional diamond-shaped honeycomb model for conven-ional foams and (b) the ‘missing ribs’ model for auxetic foams.

216 J.N. Grima et al. / Materials Science and Engineering A 423 (2006) 214–218

on the surface of the auxetic foams (see Fig. 1b), it is still unclearwhether ‘broken ribs’ are also present in the bulk of the foamsor whether they are only present at the surface. Furthermore,one could argue that the removal of ribs in a regular fashion isunlikely to occur in the existing foam conversion process. In thecase of the re-entrant models, there is no clear experimental evi-dence that a majority of the ‘Y-shaped’ joints in the conventionalfoam are converted to an ‘arrow shaped’ joints during the com-pression/heat treatment process. In fact, one may argue that thistransformation is unlikely to occur since one usually observesthat the ribs of open cell foams are slightly thicker in the prox-imity of the joints than at the centre of the ribs, and hence itis unlikely that the majority of the deformations occurs at thejoints (a requirement for a successful conversion of a Y-shapedjoint to an arrow shaped joint).

3. The new model: rotation of rigid units

We propose an alternative explanation for the presence of thenegative Poisson’s ratios in auxetic foams. This new model isbased on the hypothesis that it is more likely that changes in themicrostructure during the compression/heat treatment processwill conserve the ‘geometry of the joints’ and the major defor-mations will occur along the length of the ribs, which buckle (thefoam is being subjected to ca. 30% compressive strain along eachaxis). The model also assumes that the ‘additional thickness’ inttbtmpttrmdut

aaftnaeo‘pur

‘eil

Fig. 4. The newly proposed model: (a) the conventional hexagonal honeycombmodel for conventional foams, (b) the ‘rotation of rigid units (joints)’ model forauxetic foams, and (c) an idealised representation of the ‘rotation of rigid joints’model where the joints are represented by perfectly rigid equilateral triangles.

nor regular in shape. This is consistent with the fact that thePoisson’s ratios in the heat-treated auxetic polyurethane foamproduced by Smith et al. was experimentally measured to be ca.−0.60 [15].

4. Evidence for the new ‘rotation of rigid units’ model

Evidence that this new model is physically realistic can beobtained by examining SEM images of the auxetic foams, suchas the one in Fig. 1b. From these images one may clearly observethat the ribs appear to have extensively buckled with the resultthat ‘kinks’ have formed along the length of the ribs whilstthe joints appear to have retained their shapes but re-orientedthemselves to form a more compact foam. These structural mod-ifications results in a foam with a microstructure, which has theessential features to be able to exhibit auxetic behaviour throughthe ‘rotation of joints’ mechanism proposed here. In this respect,it is important to note that in these images there is no evidencethat the majority of joints have been inverted as one would haveexpected from the ‘re-entrant models’ (see Fig. 2), and althoughthere are some ‘cuts’ in the ribs, these ‘cuts’ appear to be mainlypresent at the surface of the foam and not in the inside.

Having established that the microstructure of the auxeticfoams can be described in terms of ‘rigid joints at an anglewith each other’, we have attempted to simulate through finite-ehmoi

he proximity of the joints will make it possible for the jointso behave as ‘rigid joints’ with the result that the foam maye described in terms of ‘rigid units’ (the joints) which duringhe foam manufacture process rotate relative to each other. The

icrostructure of foam then ‘freezes’ in this much more com-act form which contains the necessary structural features forhe ‘rotating rigid units’ mechanisms to operate and generatehe experimentally observed auxetic behaviour (i.e. connectedigid units which can rotate relative to each other to generate aore open structure [37–39]). The extent of the auxeticity will

epend, amongst other things, on rigidity of the joints (the ‘rigidnits’) when compared to the flexibility of the units connectinghem (the ribs).

An illustration, which exemplifies how this mechanism oper-tes were given in Fig. 4, which shows how a conventional ide-lised two-dimensional hexagonal model for the conventionaloam (Fig. 4a) can be converted through the compression/heatreatment process into an auxetic form (Fig. 4b) without theeed of inversion of the ‘Y-joints’ or rib breakage. Here it isssumed that during the heating/compression process, there isxtensive buckling in along the length of ribs, which form ‘kinks’r ‘folds’ at their centre whilst the ‘Y-shaped joints’ behave likerigid triangles’, which rotate relative to each other. This processroduces a more compact microstructure (Fig. 4b) which whenniaxially loaded will exhibit auxetic behaviour as a result of re-otation of the ‘triangular joints’ and unfolding of the ‘kinks’.

In an idealised scenario where the joints are assumed to beequilateral triangles’ and ‘perfectly rigid’, the system wouldxhibit in-plane Poisson’s ratios of −1 (Fig. 4c) [39]. However,n real systems, the Poisson’s ratios would be expected to beess negative as in reality the joints will not be perfectly rigid,

lements modelling (FEM) the effect of uniaxial loading on aoneycomb system such as the one illustrated in Fig. 4b whichay be considered as an idealised 2D model of a cross-section

f the auxetic foams. In these simulations, the system illustratedn Fig. 5a was constructed in ANSYS (release 5.7.1) using plane

J.N. Grima et al. / Materials Science and Engineering A 423 (2006) 214–218 217

Fig. 5. (a) The idealised 2D system as constructed in ANSYS showing thevarious constrains applied; (b) A plot showing how |Y|, the size of the systemin the Y-direction changes when |X|, the size of the system in the X-directionchanges. Note that |Y| = |EF| whilst |X| = average of |AB| and |BC|. (c) The shapeof the system at different strains in the x-direction as simulated by ANSYS.

elements (Plane 82) and was solved linearly for deformations Ux

which correspond to 5%, 10%, . . ., 30% changes in the distancesAB and CD (i.e. strains of 5%, 10%, . . ., 30% in the X-direction).From these simulations it could be confirmed that the systembecame larger in the Y-direction (i.e. the distance EF increased)as a result of an applied strain the X-direction (i.e. as the dis-tances AB and CD increased), see Fig. 5b, hence suggesting thatthis system exhibits negative Poisson’s ratio. Furthermore, thesesimulations confirm the hypothesis being proposed here since avisual analysis of the deformed structures also confirms that the‘joints’ have rotated with respect to each other hence confirm-ing that ‘rotation of the joints’ is taking place. However, thesesimulations also show that the ‘joints’ are not perfectly rigid andchange shape as the system is strained in the X-direction. Thisresults in a lowering of the auxetic effect which means that thePoisson’s ratio will be less negative than the theoretical valueof νxy = −1 as predicted from for the ‘idealised rotating trian-gles’ model [39]. As stated above, these effects are likely to beobserved in the real foams where Poisson’s ratios of −0.6 wereexperimentally measured.

5. Discussion and conclusion

Through this work, we have shown that the experimentallymeasured negative Poisson’s ratios in auxetic open-cell foamsmanufactured through a process involving compression of con-veTnotf

(i.e. ‘joints’ which have been rotated with respect to each otherthough the formation of ‘kinks’ along the lengths of the ribs).Furthermore, FE modelling of an idealised 2D representationof a cross-section of the foams suggests that these systems canindeed exhibit negative Poisson’s ratios through a mechanismof ‘rotation of the rigid joints’.

It is worthwhile to mention that in reality, the ‘rotation ofrigid units’ mechanisms operating in real foams are likely to bemuch more complex. For example, in this model we have onlyconsidered ‘two-dimensional’ joints, which rotate in the planeof the system. In the real foams, the ‘rigid units’ will be three-dimensional and one would expect that the rotations will occurin different planes. Furthermore, as stated above, one would alsoexpect that the ‘rigid units’ will not be perfectly rigid and thiswill influence the extent of the auxetic effect.

We also envisage that in reality this new mechanism willwork in parallel to other mechanisms such as the ones presentedin Figs. 2 and 3. However, in view of the evidence presentedhere, and in view of the fact that such ‘rotation of rigid units’ areresponsible for auxetic behaviour in various other molecular-level systems (e.g. zeolites [11,38–40] and silicates [40,41]),we are confident that this mechanism is one of predominantmechanisms that is responsible for the observed auxetic effectin foams.

References

[

[

[[

[

[[

[[

[[

[[[

entional open-cell foams at an elevated temperature may bexplained through a model based on rotation of rigid units.his explanation is distinct from previous models as it doesot require any structural modifications at the joints of the cellr rib-breakage. This new explanation is supported by the facthat SEM images of the auxetic foams contain the geometriceatures that are required fro this is new mechanism to operate

[1] K.E. Evans, M.A. Nkansah, I.J. Hutchinson, S.C. Rogers, Nature 353(1991) 124.

[2] A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity, 4thed., Dover, New York, 1944.

[3] R.H. Baughman, D.S. Galvao, Nature 365 (1993) 735–737.[4] J.N. Grima, K.E. Evans, Chem. Commun. 16 (2000) 1531–1532.[5] J.N. Grima, J.J. Williams, K.E. Evans, Chem. Commun. (2005)

4065–4067.[6] C.B. He, P.W. Liu, A.C. Griffin, Macromolecules 31 (1998) 3145–3147.[7] C.B. He, P.W. Liu, P.J. McMullan, A.C. Griffin, Phys. Stat. Sol. (b) 242

(2005) 576–584.[8] R.H. Baughman, J.M. Shacklette, A.A. Zakhidov, S. Stafstrom, Nature

392 (1998) 362–365.[9] N.R. Keskar, J.R. Chelikowsky, Nature 358 (1992) 222–224.10] A. Yeganeh-Haeri, D.J. Weidner, J.B. Parise, Science 257 (1992)

650–652.11] J.N. Grima, R. Jackson, A. Alderson, K.E. Evans, Adv. Mater. 12 (2000)

1912–1918.12] R. Lakes, Science 235 (1987) 1038–1040.13] K.E. Evans, M.A. Nkansah, I.J. Hutchinson, Acta Metall. Mater. 42

(1994) 1289–1294.14] N. Gaspar, C.W. Smith, E.A. Miller, G.T. Seidler, K.E. Evans, Phys.

Stat. Sol. (b) 242 (2005) 550–560.15] C.W. Smith, J.N. Grima, K.E. Evans, Acta Mater. 48 (2000) 4349–4356.16] B.D. Caddock, K.E. Evans, J. Phys. D: Appl. Phys. 22 (1989)

1877–1882.17] K.L. Alderson, K.E. Evans, J. Mater. Sci. 28 (1993) 4092–4098.18] A.P. Pickles, K.L. Alderson, K.E. Evans, Polym. Eng. Sci. 36 (1996)

636–642.19] F.K. Abd el-Sayed, R. Jones, I.W. Burgens, Composites 10 (1979) 209.20] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, Perg-

amon, Oxford, 1988.21] I.G. Masters, K.E. Evans, Compos. Struct. 35 (1996) 403–422.22] D. Prall, R.S. Lakes, Int. J. Mech. Sci. 39 (1997) 305–314.23] A. Spadoni, M. Ruzzene, F. Scarpa, Phys. Stat. Sol. (b) 242 (2005)

695–709.

218 J.N. Grima et al. / Materials Science and Engineering A 423 (2006) 214–218

[24] N. Gaspar, X.J. Ren, C.W. Smith, J.N. Grima, K.E. Evans, Acta Mater.53 (2005) 2439–2445.

[25] K.W. Wojciechowski, Mol. Phys. 61 (1987) 1247–1258.[26] K.W. Wojciechowski, A.C. Branka, Phys. Rev. A 40 (1989) 7222–7225.[27] K.W. Wojciechowski, J. Phys. A-Mat. Gen. 36 (2003) 11765–11778.[28] A. Alderson, Chem. Ind. 10 (1999) 384–391.[29] R.S. Lakes, K. Elms, J. Compos. Mater. 27 (1993) 1193–1202.[30] F. Scarpa, F.C. Smith, J. Intel. Mater. Syst. Str. 15 (2004) 973–979.[31] F. Scarpa, W.A. Bullough, P. Lumley, P. I. Mech. Eng. C-J. Mech. Eng.

Sci. 218 (2004) 241–244.[32] F.R. Attenborough, Ph.D. Thesis, University of Liverpool, UK, 1997.[33] J.B. Choi, R.S. Lakes, J. Compos. Mater. 29 (1995) 113–128.

[34] J.B. Choi, R.S. Lakes, Int. J. Mech. Sci. 37 (1995) 51–59.[35] C.P. Chen, R.S. Lakes, Cell. Polym. 14 (1995) 186–202.[36] H. Zhu, J.F. Knott, N.J. Mills, J. Mech. Phys. Sol. 45 (1997) 319–343.[37] J.N. Grima, A. Alderson, K.E. Evans, J. Phys. Soc. Jpn. 74 (2005)

1341–1342.[38] J.N. Grima, K.E. Evans, J. Mater. Sci. Lett. 19 (2000) 1563–1565.[39] J.N. Grima, K.E. Evans, J. Mater. Sci., in press.[40] J.N. Grima, A. Alderson, K.E. Evans, Phys. Stat. Sol. (b) 242 (2005)

561–575.[41] J.N. Grima, R. Gatt, A. Alderson, K.E. Evans, Paper Presented at the 9th

International Conference on Advanced Materials, July 2005, Singapore,2005, p. E-P035.