A review of the manufacture, mechanical properties and potential applications of auxetic foams

Phys. Status Solidi B, 1–20 (2013) / DOI 10.1002/pssb.201248550 p s sb

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A review of the manufacture,mechanical properties and potentialapplications of auxetic foams

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Richard Critchley, Ilaria Corni*, Julian A. Wharton**, Frank C. Walsh, Robert J. K. Wood,and Keith R. Stokes

National Centre of Advanced Tribology at Southampton (nCATS), University of Southampton, University Road, Southampton,

SO17 1BJ, UK

Received 14 November 2012, revised 11 March 2013, accepted 14 March 2013

Published online 24 April 2013

Keywords auxetic, impact, Poisson’s ratio, reticulated foams, smart materials

* Corresponding author: e-mail [email protected], Phone: þ44(0)2380592890, Fax: þ44(0)2380593016** e-mail [email protected], Phone: þ44(0)2380592890, Fax: þ44(0)2380593016

Auxetics are a modern class of material fabricated by altering

the material microstructure. Unlike conventional materials,

auxetics exhibit a negative Poisson’s ratio when subjected to

tensile loading. These materials have gained popularity within

the research community because of their enhanced properties,

such as density, stiffness, fracture toughness and dampening.

This paper provides a critical oversight of the auxetic field with

particular emphasis to the auxetic foams, due to their low price,

easy availability and desirable mechanical properties. Key

areas discussed include the fabrication method, the effects

played by different parameters (temperature, heating time, cell

shape and size and volumetric compression ratio), micro-

structural models, mechanical properties and potential appli-

cations.

� 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Poisson’s ratio (n) is defined as theratio between the longitudinal expansion and the lateralcontraction of a material during loading [1]. Conventionalmaterials present a positive Poisson’s ratio and their cross-section becomes larger in compression and smaller in tension(Fig. 1a) [2, 3]. Thermodynamic considerations of strainenergy in the theory of elasticity demonstrate that thePoisson’s ratio for a homogeneous solid isotropic materialcould be between�1 and 0.5, thus allowing the existence ofmaterials with a negative Poisson’s ratio [1, 4–6]. This classof materials is identified with the term ‘auxetics’, that derivesfrom the Greek word ‘auxetikos’ which means ‘that tends toincrease’ and Poisson’s ratios are as low as �0.7 forpolymers and �0.8 for metals [7, 8]. Auxetic materials arecharacterised by a counterintuitive behaviour, which isevident by applying a tensile load in one direction theyexpand in all directions (Fig. 1b) [9, 10], or more simply theybecome fatter, laterally, when stretched lengthwise.

Auxetic materials constitute a new class of materialsthat can be found in nature, i.e. cubic elemental metals,a-cristobalite (high temperature polymorphic mineral),and biological tissues, i.e. cat skin and cow teat skin

[11, 12]. These materials demonstrate unique and enhancedmechanical properties and for this reason numerousresearches have been carried out to understand themechanisms that render a material auxetic and to reproducethese mechanisms and properties in man-made materials.To date a wide range of auxetic materials, such as polymers,metals, ceramics, composites, laminates and fibres havebeen manufactured (see Fig. 2a) using a particular fabrica-tion process that results in a change of the material structure[11–14]. A timeline for the discovery and the areas in whichman-made and natural auxetic materials are available (withtheir length-scale) are reported in Fig. 2a and b, respectively.The increased interest in the research and applications ofauxetic materials is demonstrated by the increased number ofpatent filed (Fig. 3a) and research paper published (Fig. 3b)since the late 1980s. Auxetic materials can be currentlyfound in commercially available products such as polytetra-fluorethylene (PTFE) and GoreTex [15]; since their initialproposal by Love, numerous organisations includingToyota, Yamaha, Mitsubishi, AlliedSignal Inc., BNFL andthe US Office of Naval Research have filed numerous patents[15, 16].


2 R. Critchley et al.: Manufacture, mechanical properties and applications of auxetic foamsp

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Figure 1 Differences between (a) a positive and (b) a negative Poisson’s ratio material (adapted from Ref. [11]).

Several reviews on auxetic materials have been pub-lished dedicating particular attention to known auxeticmaterials, different types of auxetic microstructures and theirapplications [11, 13, 17–21]. The aim of this paper is toprovide critical overview of auxetic foams, their processingmethods, deformation mechanisms and applications. Auxe-tic foams have been chosen from all the various auxeticmaterials since the auxetic effect and the mechanismsinvolved have been widely studied using open cellpolyurethane foam; their low price, easy availability andfabrication method further increase their attractiveness asmaterials. The mechanical properties and the possibleapplications of auxetic foams have also been reviewed,while new areas of study that have yet to be explored areproposed.

2 Methods to convert conventional foams intoauxetic The manufacturing of auxetic foams was firstreported in 1987 by Lakes [3] (although the concept ofauxetic foams were proposed in 1985 by Kolpakov) [23],who suggested two different manufacturing procedures: onefor polymeric and one for metallic foams.

2.1 Methods for the conversion of polymericfoams For polymeric materials, Lakes started with aconventional open-cell foam and made it auxetic by makingthe ribs of each cell protrude inward producing a re-entrantstructure [3]. This change is achieved in a three-step processin which the open cell foam is tri-axially compressed, heatedto a temperature slightly higher than the softening tempera-ture of the material and cooled at room temperature to storepotential energy in the compressed ribs [3, 13, 24]. Thecompression stage of the process changes the shape ofthe cellular ribs and the heating and cooling stages softenand fix the ribs in a new position.

The main problems that have been identified asdependent on the auxetic conversion process are: long-terminstability with the samples reverting back to their originalshape and structure, severe surface creasing and inability toproduce large samples. For example Bianchi et al. [5] noticedthat following the auxetic conversion, within hours, thesamples would naturally attempt to return to their originaldimensions. This behaviour was also reported by Scarpaet al. [10], who noticed that the final volumetric compression


ratio (the change in volume from the sample originaldimensions to the new dimensions obtained after theconversion expressed as a ratio) could differ up to 52%from the originally applied value while the sample densitywas susceptible to a time-dependant change of up to 30%after a week. This behaviour is most likely the result of creep,which occurs as a consequence of the long-term exposure tostresses and not sufficient heating times employed duringthe fabrication process. The surface creasing and wrinklesobserved in the more deformable regions of the foamspecimen are due to the volumetric compression ratioapplied during fabrication [24, 25]. In order to resolve thisproblem, two solutions have been suggested: lubrication ofthe mould (the lubricant should not be an oil-derivative or adistilled oil due to their instability at high temperatures andthe production of unpleasant smells) [6, 24, 26] and the use ofwires or tweezers inside the moulds to pull the foam insteadof pushing it [10]. A further solution would be to redesignthe moulds so that the tri-axial compression is applied at auniform rate and no creasing areas form. Another solutionhas been reported by Chan and Evans in 1997 [24], theyattempted to resolve the formation of wrinkles on the samplesurface by applying the volumetric compression in severalstages in order to produce a more homogeneous auxeticmaterial.

Since 1987, this production process has been applied bynumerous investigations and a number of modifications havebeen reported [2–5, 10, 24–32]. While each has been alteredeither to produce differences within the polymeric materialsor to optimise the process the overall principle has remainedthe same, i.e. volumetric compression followed by heatingand cooling [4, 24, 31]. These modifications are describedbelow.

2.2 Multi-phase auxetic fabrication Bianchi et al.[5] further modified the fabrication process by incorporatinga re-conversion back to conventional foam via a shapememory polymer (SMP) process followed by a secondauxetic conversion. Auxetic materials can be considered asan SMP even if the fabrication process is usually intended asa permanent structural change. Indeed it has been observedthat the application of an external trigger, such as exposureto a solvent or heat, reverts the auxetic foam back to itsconventional state and dimensions. The re-conversion

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Figure 2 (a) Timeline for the discovery of natural and man-made auxetic materials, (b) areas where auxetic materials have been discoveredand their size (adapted from Ref. [11]).

www.pss-b.com � 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


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Figure 3 (a) Number of patents filed (information taken from Ref.[22]) and (b) number of auxetic papers published since the discoveryof the auxetic materials where bracketed numbers indicate thenumber of review papers published that year.

process is undertaken by heating the unconstrained auxeticsamples to a temperature near to the auxetic conversiontemperature (200 8C) and allowing them to relax. Theauthors observed that at around 90 8C the samples began toexpand towards their original dimensions, requiring onlya couple of seconds to regain their original size (at atemperature of 135 8C the auxetic samples completelyrecovered their original dimensions) [5, 28]. The samplethat returned to the conventional state had lost all the auxeticproperties and behaviour and presented a positive Poisson’sratio, these were regained again when the sample was re-converted to auxetic [5].

A similar study was also reported by Grima et al. [33]that re-converted auxetic samples into conventional samplesby placing them in acetone followed by air-drying (duringacetone exposure the foam re-expanded in all directions).The re-converted samples presented a positive Poisson’sratio and a conventional honeycomb structure.


2.2.1 Solvent-based auxetic fabrication Morerecently Grima et al. [33] presented a novel chemo-mechanical process for the fabrication of auxetic foams;conventional polyurethane samples were mechanicallycompressed, wrapped in filter paper, placed in acetone for1 h and then air dried. After being removed from the moulds,the samples fabricated with this methodology retained thecompressed shape and exhibited auxetic behaviour. Whencompared to thermo-mechanical fabricated auxetic samples,it was found that both sample types exhibited negativePoisson’s ratios of approximately�0.3 and presented highlyconvoluted auxetic microstructure.

2.2.2 Vac-bag auxetic fabrication Bianchi et al.[34] recently presented a novel manufacturing methodologyusing a vac-bag system. Unlike classical manufacturingmethods that are limited to rectangular or cylindricalgeometries and to a volume of only a couple of cubiccentimetres, the vac-bag fabrication is capable of producingsamples with complex shapes, such as arcs, and largevolumes (30 cm� 16 cm� 3 cm).

The vac-bag method requires a sheet of conventionalfoam to be placed upon a semi-circular mould and layeredover with a non-porous fluorinated ethylene propylenerelease film followed by a medium weigh polyester non-woven breather blanket which covers both the mould and thefoam sample. After sealing the bag, a vacuum pump isemployed to reduce the internal pressure down to 0.7 barcausing the foam to be drawn into the mould and to gain themould curvature. The mould and foam are then placed into afurnace at 200 8C and heated for 30 min. After been removedfrom the mould, the foam displayed auxeticity and thenatural curvature of the mould. SEM analysis furtherconfirmed the auxeticity of the foam although each side ofthe sample presented different configurations. Throughtensile and cyclic loading testing, it was found that themaximum negative Poisson’s ratio achieved by this foamwas�1.26 at 5% strain and the maximum energy dissipationwas 1.42 mJ cm�3. The Poisson’s ratio of around �1indicates that the sample produced was anisotropic.

2.2.3 Dual density auxetic fabrication Bianchiet al. [35] have also recently presented a more classicalexample of auxetic fabrication resulting in large changes ofthe density of the auxetic foams. Manufacturing saw twobatches of foam samples of varying dimensions placed into ametallic tube mould and compressed. The mould was thenplaced into an oven and heated at 200 8C for 15, 45 and60 min to temperatures of 135, 150 and 170 8C, respectively.Samples heated for 15 min were cooled at room temperature(20 8C) for 5 min, whilst the 45 and 60 min samples wereinstantly removed from the moulds, stretched then allowed tocool at room temperature.

The removal of the foam from the mould instantly afterthe heating step creates a differentiated microstructurebetween the external and internal part of the specimen: astiff outer layer and a less dense core. This behaviour is

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suggested to be directly caused by the non-homogenoustemperature distribution in the foam body, where onlythe higher temperatures penetrate the external regions of thesample, and not the interior. Densities were also found tobe affected based upon the time a sample spent waiting tobe removed, with those waiting longer exhibiting higherdensities. This feature is likely to be the result of the samplesbeing tri-axially constrained whilst cooling, thus stoppingthe ribs from protruding towards their original structure.Samples fabricated through this novel route were also foundto only exhibit a negative Poisson’s ratio under compressiondue to the non-homogenous microstructure caused inmanufacture. The greatest negative Poisson’s ratio achievedin this instance was �0.34.

2.3 Methods for conversion of metallicfoams For metallic materials, Lakes stated that reticulatedmetal foam specimens were transformable into auxetics bycompressing the material successively in each of the threeperpendicular directions at room temperature (18–23 8C).The foams subjected to this procedure exhibited a re-entrantcell structure [3]. This methodology has also been applied byother authors, with little to no known changes beingintroduced [2, 36]. However, it has been suggested by Friiset al. [2] that, in principle, the thermal transformationtechnique used for thermoplastic foams could be applied alsoto metallic foams, although this may be difficult due to thehigher and sharper melting points of metals.

3 Parameters affecting manufacture of auxeticfoams From the literature [4, 5, 28, 29, 32], it is clear thatthere are many variables influencing the manufacture ofauxetic foams, such as the composition of the material, itsrelative density and cell size, the processing temperature andheating-time, the humidity and the volumetric compressionratio applied [29]. There is still much debate regarding whichare the key factors influencing the process. For exampleWang et al. [32] stated that the main physical parametersinfluencing the auxetic transformation process are thevolumetric compression ratio, the processing temperatureand the heating time. However, other authors [4, 5, 28]consider only the volumetric compression ratio as the mainparameter responsible for a successful conversion, with theother parameters acting as secondary influence. Althoughan exact relationship between the effects that each of theparameters play in the transformation is still unknown, theinfluences that the different process parameters have onthe final results are reviewed below. There have been alsoattempts to identify the relationships between the manufac-turing parameters and the final material properties (density,Poisson’s ratio, stiffness and energy dissipation) [4].

3.1 Influence of temperature and heatingtimes Lakes reported that for a polyester open cell foamthe ideal temperature to be applied in the conversion processwas between 163 and 171 8C, just above the softeningtemperature of the material [3]. This statement was then

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followed by numerous studies attempting to determine themethodology and the justification behind this choice oftemperature. In 1997, Chan and Evans employed athermocouple placed in the centre of a foam specimen todetermine the softening temperature of the material when thecell ribs collapsed [24]. This experiment suggested that thetemperature employed for the auxetic conversion shouldbe between 5 and 20 8C lower than the softening temperatureof the material. Conversely Evans and Alderson [2, 11],supported by Friis et al. [2] employed a heating temperatureabove the softening point and Wang et al. [32] suggested thatthe conversion temperature should be equal or greater thanthe softening temperature of the material. From the resultsreported, it is reasonable to assume that Chan and Evans’suggestion of keeping the conversion temperature below thematerials softening temperature (e.g. 180 8C) was incorrectand instead the temperature employed should in fact be equalor above the softening temperature of the conventionalpolymeric foam, as this allows for the softening of thepolymeric ribs to occur [24].

Whilst attempting to determine the reasons behind thetemperature values published by Lakes [3], Chan and Evansreported several interesting observations [24]. Theyobserved that if the heating time was too long the foamwould either decompose or melt with the cell ribs stickingtogether to form a dense block of material, and if the heatingtime was too short the foam could not be ‘set’ into its new re-entrant structure and would soon begin to expand back to itsoriginal size [24]. Choi and Lakes and Bianchi et al. [4, 29]further confirmed this behaviour with the rate at which thenegative Poisson’s ratio was lost fluctuating from a few daysto a few months. Furthermore, Chan and Evans noticed thatthe heating time needed to turn a conventional porousmaterial into an auxetic one depended on the exact type offoam and hence initial base material and porosity [24]. Wanget al. confirmed these observations and further suggested thatthe cell size was also an important factor influencing theauxetic fabrication. It was found that polyurethane foamshaving a smaller cell size required higher heating tempera-tures and heating times compared to foams with larger poresize. It was speculated that this behaviour could be due toeither the surface tension effects within smaller cells or to apossible difference in the chemistry of the material [32].

Other notable effects of the temperature on the auxeticconversion process have been reported when studying theearly stages of heating and cooling. During the early stages ofheating at temperatures of 100–135 8C, Bianchi et al. [28]noticed using thermogravimetric analysis that conventionalfoams experience an approximate 2% loss in weight. It issuggested that the loss in weight could potentially explain thedifference in mechanical properties between conventionaland auxetic materials during the multi-phase conversion(phase 1: conventional, phase 2: first auxetic, phase 3:returned conventional and phase 4: second auxetic). Duringcooling, numerous studies have shown that the specimensshould be cooled at room temperature, as set out by Lakes [3,24, 29]. In 2008, Bianchi et al. [4] introduced an alternative



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Figure 4 Diagramshowingthedifferent time–temperatureprofilesfor auxetic fabrication reported in the literature.

cooling method in which samples were cooled in water for5 min. Specimens cooled in water exhibited unusual densityvalues compared to samples that were cooled in air. Thenumerical differences are unknown as Bianchi et al. [4] didnot reported the actual numerical values. It was further notedthat by employing water cooling a more uniform temperature

Table 1 Examples of foams employed by various research groups.

authors year material

Chan and Evans 1997 closed-cell polyester urethane foamChan and Evans 1997 reticulated polyester urethane foamChan and Evans 1997 open-cell polyether urethane foamChan and Evans 1997 open-cell polyether urethane foamChan and Evans 1997 open-cell polyether urethane foamBianchi et al. 2008 conventional grey open-cell polyurethaneScarpa et al. 2002 open-cell polyurethane foamWang et al. 2001 open-cell polyurethane foamWang et al. 2001 open-cell polyurethane foamWang et al. 2001 open-cell polyurethane foamScarpa et al. 2005 conventional grey open-cells polyurethanBianchi et al. 2010 grey-coloured open-cell polyurethane-basBianchi et al. 2010 conventional light-blue-coloured open-ceBianchi et al. 2010 grey-coloured open-cell polyurethane-basBianchi et al. 2010 conventional light-blue-coloured open-ceScarpa et al. 2004 open cell polyurethane grey foam


distribution was obtained. It was speculated that uniformtemperature distribution and alternative density values occuras the water jet induces a significant thermal shock to thesample.

Many authors have studied the effects of temperature onthe fabrication of auxetic materials but there are stilldiscrepancies associated with determining an optimumtemperature and heating time to be employed. Thesediscrepancies are likely the result of the variance in materialsand equipment that each individual researcher has employedfor the experiments. Numerous temperature–time combi-nations have been reported in the literature and aresummarised in Fig. 4, where the heating time varied from 6to 60 min [24, 35] but the majority of the time–temperaturecombinations were less than 20 min. The temperatureapplied varied between 130 and 220 8C [4, 24, 28, 32], thiswide range of temperatures can be explained considering thefindings of Wang et al. [32] that foams with smaller cell sizeneed higher heating temperatures and shorter processingtimes compared to foams with larger pore size. The wideranges in which both the heating time and the temperaturehave been varied seem extreme and therefore more studiesare needed to understand the effect that these two variableshave on the auxetic conversion process.

3.2 Influence of cell shape Since the auxeticmaterials were first produced [3] the most commonlyreported auxetic foams produced were thermoplastic (poly-ester urethane, polyether urethane), thermosetting (siliconerubber) and metallic (copper) [11]. However, the foam mostemployed has been the open-cell polyurethane foam due toits availability and ease of application [9] as shown inTable 1. Even though polyurethane has been widely studiedonly a few studies have investigated the effect that pore sizeon the auxetic conversion process [2–5, 8, 18–22, 24–27].

pore size(pores perlinear inch)

density(kg m�3)

reference

60 39.9 [24]60 33.7 [24]10 24.1 [24]30 24.5 [24]60 21.7 [24]30–35 27.2 [4]not given 32 [26]20 30 [32]65 30 [32]100 33 [32]

e 30–35 27 [10]ed foam 30–35 27.2 [5]ll polyurethane-based foams 52–57 27 [5]ed foams 30–35 27.2 [28]ll polyurethane-based foams 52–57 27 [28]

30–35 32 [27]

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One of the more comprehensive studies on the effect thatcell shape and size have on the properties of the foam wasreported by Chan and Evans [37]. They specifically studiedthe microstructure and the deformation mechanisms ofconventional and auxetic open-cell foams and observed thatthe cell size was not the dominating factor in determining themechanical properties, while the cell shape and geometrywere found to be greater contributor, as also reported in Ref.[38]. The cell geometry is influenced by the manufactureprocess of the conventional foam that is formed by injectinggas in the semi-liquid material, the gas dissipates throughoutthe material then the liquid is cooled to the solid stateretaining the air pockets formed by the gas. The air pocketsare somewhat elongated and therefore the foam could beanisotropic [37, 39]. By assuming that the mechanicalproperties of the foams are similar to those of the solidmaterial employed for their production, the main variableaffecting the foam mechanical properties is its cellulargeometry [37]. It should be noted that although the foams canbe treated similarly to their solid counterparts, they areconsiderably less stiff [1, 7, 40].

Chan and Evans studied and modelled the cellgeometries and volumes in two and three dimensions forconventional (see Fig. 5) and auxetic foams (see Fig. 6),where nearly 100 specimens were measured to achieve asufficient statistical analysis of the data set [37]. Theyobserved that once the conventional foam has beenconverted to an auxetic structure the cell geometry no longerresembles that of an ellipsoid with the smaller side within thedirection of rise (Fig. 5), but becomes more reminiscent of

Figure 5 Schematic representation of (a) a three-dimensional (3-D)geometry of the same pore type (adapted from Ref. [37]).

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bow-tie re-entrant microstructure (Fig. 6). Both models forthe auxetic and conventional pores are described in moredetail below.

3.3 Conventional model For conventional foamsreported in Fig. 5, the cell length can be calculated using thefollowing equations [37]:

geom

Maximum: AD ¼ hþ 4L sinu; (1)

Minimum: BC ¼ hþ 2L sinu; (2)

Average: y ¼ hþ 3L sinu; (3)

and the cell width at any point can be taken as:

Normal: EF ¼ 2L cosu; (4)

Average: x ¼ 2L cosu: (5)

3.4 Auxetic model For auxetic foams the cell lengthcan be given by the following equations (all the parametersare specifically related to Fig. 6):

Maximum: GE ¼ h; (6)

Minimum: BC ¼ h� 2L sinu; (7)

Average: H ¼ h� L sinu; (8)

the auxetic cell width can be obtained by:

Normal: EF ¼ 2L cosu; (9)

etry for a conventional foam and (b) a two-dimensional (2-D)



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Figure 6 Diagrammatic representation of (a) 3-D geometry of an auxetic pore and (b) 2-D geometry of the same auxetic pore (adapted fromRef. [37]).

and the auxetic mid-plane waist height is calculated using:

� 20

Maximum: JK ¼ h� 2L sinu; (10)

Minimum: AD ¼ h� 4L sinu; (11)

Average: W ¼ h� 3L sinu: (12)

Figure 7 Comparison of the 100 PPI samples manufactured indifferent conditions for the screening process to test the effects ofpore size, adapted from Ref. [32].

It is important to underline that although these modelsallow estimating the cell geometry, real samples containnumerous cellular and structural anomalies. A foam speci-men contains a variety of cell shapes with four to six sidesand a wide range of cell sizes, some of which are so small thatdeformation is not possible and therefore act as junctionpoints within the foam [37]. They also observed that thefoams tested were slightly elongated due to the foammanufacturing process. While each of these anomaliesdiscredits somewhat the proposed mathematical model,it should be underlined that the model should not bedisregarded, since it can still provide insightful informationfor the cell geometry.

One area that Chan and Evans failed to address in thisstudy was to understand how the cellular size affected theoverall negative Poisson’s ratio of the foam [37]. Wang et al.[32] found that for a number of polymeric foam specimens,the lower the number of pores per inch (PPI), i.e. the largerthe cell size, then the greater the negative Poisson’s ratio.

3.5 Effect of the cell size The material porosity andtherefore the cell-size is a further factor affecting the auxeticconversion process. Wang et al. [32] tested three poly-urethane foams with porosity of 20, 65 and 100 PPI andin order to explore the effects that the pore size has on thePoisson’s ratio, Scott reticulated 100 PPI white foam was

13 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

subjected to a screening process where samples weremanufactured at temperatures between 120 and 280 8C andprocessing time between 5 and 21 min at a volumetriccompression ratio of approximately 2.5, yielding a total of48 samples, the data is shown in Fig. 7.

It was found that for the 100 PPI foam, the optimalprocessing parameters were a heating time between 8 and12 min, a temperature between 210 and 230 8C. Whereas thebest conditions for the 20 and 60 PPI foam specimens were aheating times of 17 and 13 min and a temperature of 170 and190 8C, respectively. These results show that smaller cellsize foams prefer higher temperatures and shorter heatingtime.

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Figure 8 Model describing the rib behaviour under tensile loadingadapted from Lakes et al. [43].

3.6 Effect of applied volumetric compressionratio The volumetric compression ratio (Rc) is defined asthe ratio between the original volume (Vo) and the finalvolume (Vf) after the auxetic conversion of the specimen[24]. The volumetric compression ratio plays a critical role inthe auxetic conversion process and it has been observed thatfailure to apply a value within the optimum range can resultin an unsuccessful conversion process. It has been reportedin the literature that different types of materials exhibitdifferent maximum volumetric compression ratio values.For example, Lakes stated that for the open-cell polymericfoam he employed the optimal volumetric compression ratiowas between 1.4 and 4 [3]. More recently, Choi and Lakesreported that in order to have a successful auxetic conversionthe volumetric compression ratio had to be between 2 and 5[32, 36]. Although optimum compression ratio ranges havebeen suggested, it has to be remembered that the optimumpermanent compression to achieve the best negativePoisson’s ratio depends on the initial foam density andappears to have a purely geometrical origin, as the samenegative Poisson’s ratio value was observed for metallic andpolymeric foams with the same relative density [36]. Themajority of Rc values reported in the literature fell withinthis range [2, 7, 10, 27, 28, 41]; however, experiments withgreater volumetric compression ratios have also beenreported but only with respect to mechanical testing andnot volumetric effects [31].

The compression ratio has also been described in termsof volumetric percentage change [10, 42], which differs fromthe volumetric compression ratio, as no optimum percentagerange has been outlined. However, from estimates givenin the literature, the optimal percentage range may beapproximated as being between 30 and 94% of the originalvolume; converting these values into volumetric com-pression ratios, it can be found that the optimal range isbetween 1.43 and 16.1. While this range seems too wide to beconsidered as a useful optimal range, it is in agreement withthe commonly accepted 2–5 range previously defined byLakes [32, 36], which as a percentage equates to 50–80% ofthe original volume.

The volumetric compression ratio is the most importantand influential parameter in the auxetic manufacture [4, 5,28, 32], in fact it has been observed to influence both theauxetic microstructure and the mechanical propertiesof the foam. However, the volumetric compression ratiois also susceptible to the effects of other variables, suchas temperature and pore size. For example, Choi and Lakesreported that when a large volumetric compression ratio isapplied together with time and temperature values outsidethe optimum conditions, the ribs in the foam become stucktogether [29]. It has been observed that applying smallervolumetric compression ratios produces samples with a verylow negative or almost positive Poisson’s ratio values [27].However, Wang et al. [32] noted that, within the optimumvolumetric compression range of 2–5, specimens with lowervolumetric compression ratios were found to exhibit greaternegative Poisson’s ratios. This behaviour was once again

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observed by Bianchi et al. [28], where it was further noticedthat lower volumetric compression ratios produced greatertangent modulus values.

Metallic foams are also affected by the volumetriccompression ratio applied during the conversion process.During a study of the non-linear properties of a copper-basedfoam, Choi and Lakes reported an optimal volumetriccompression ratio of 3.6 [36]. This value was very similar tothose reported for polymeric foams with comparable density.Interestingly, Choi and Lakes found that the metallic foamswere subjected to the same problems reported for polymericfoams, e.g. when a volumetric compression ratio above theoptimum range is applied the auxetic behaviour becameobstruct because the ribs come into contact hinderingthe unfolding mechanism [36]. It should be noted that tothe author’s knowledge little to no work has been publishedon how the porosity of metallic foams affects the overallvolumetric compression ratio.

4 Foam structure and deformation mechanismThe cellular structure of conventional polymeric foams iswell-known; during the auxetic conversion process thecellular structure completely changes and therefore it isfundamental to understand the deformation mechanismsinvolved. In order to understand the counterintuitivebehaviour of auxetic foams, numerous studies have analysedhow the cell ribs and the foam structure changes undercompression and tensile loading and proposed numeroustheories that will be summarised below.

4.1 Bow-tie or re-entrant cell model In the ear-liest studies reported by Lakes the permanent inwardprotrusion of the cell ribs during the conversion processforming a ‘bow-tie’ structure was highlighted; this structureis also known as re-entrant cellular structure [3]. Furtherwork by Lakes confirmed that the performance of the auxeticfoams is a result of the rib behaviour [43]. Following theseobservations, the re-entrant cell model shown in Fig. 8 wasestablished; according to this model, the application of atensile load results in deformation of the cell ribs by bending,which in turn leads to unfolding of the cells to their originalstructure. This is further demonstrated by following the ribmovement in the scheme reported in Fig. 8. Applying atensile load, points A and B expand further apart attempting



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to achieve the original hexagonal geometry of the cell, whilepoints C and D maintain a fixed position; the expansion ofA-B produces an increase in volume. Lakes also observed thatthe bending was not crucial, with the same effect generated byemploying stretchable spring elements, with no rotationalconstrains, that are allowed to pivot freely. Rothenburg et al.[44] further proposed a similar approach in which eachelastic element within a randomly oriented network can berepresented as shock absorbing unit. By assuming that all theorientations of the elastic elements have equal probabilityof occurring, deformation can be said to be similar to that ofisotropic elastic materials; therefore the Poisson’s ratio isshown to be determined by the ratio between normal andtangential stiffness (l), where a negative Poisson’s ratio isachieved when l> 1.

Figuundbow

� 20

v ¼ 1� l

3þ l: (13)

Within this arrangement, should three elastic elementsform a triangular structure, a compressive or tensile load inone direction will cause the structure to either compress orexpand. Interestingly, by introducing a structural kink intoan elastic element, the axial stiffness of the collapsed bar isgreatly reduced, where the kink angle determines the stiffnessreduction. It is on this principle that the re-entrant structureproposed by Lakes achieves a negative Poisson’s ratio.

4.2 Honeycomb model In 2000, Evans and Alder-son presented a two-dimensional (2-D) model that describedthe different structures of conventional and auxetic foamsunder tensile loading [11]. This model represented theconventional foams with a repeatable honeycomb structuremade of hexagonal geometries (as shown in Fig. 9a) and

re 9 2-D representation of conventional and auxetic foamser loading: (a)conventional honeycombstructureand (b)auxetic-tie structure (adapted from Ref. [11]).


the auxetic foam with a repeatable bow-tie structure (asshown in Fig. 9b), as also proposed by Lakes [43]. In orderfor this model to be valid it was assumed that the foampresented a state of cellular symmetry, since it is known thatexperimental foams are asymmetric and still convertible toauxetic it could be concluded that the cellular symmetry isnot required for auxetic conversion to occur [5, 45]. Whentensile loading is applied in the y-direction, the hexagonalcells in the conventional foam become elongated along thestretch path and contract in the x-axis producing a positivePoisson’s ratio (see Fig. 9a). By applying the same tensileloading along the y-axis, the bow-tie cell in the auxetic foamelongate in both the direction of stretch and the perpendiculardirection. However, this behaviour is anisotropic, as loadingin the y-axis will not yield the same results as loading in thex-axis [37]. On the other hand, Lakes stated that althoughsome materials experienced anisotropic behaviour not allauxetic materials are anisotropic [3, 37]. Moreover, it hasbeen suggested that auxetic foams produced via an isotropiccompression ratio have a tendency to exhibit superiorproperties compared to their anisotropic counterparts [5].

The 2-D honeycomb model explained above has beenwidely accepted by the research community for under-standing the cellular structure and deformation mechanismof 2-D foams [18, 25, 27, 46]. However, alternative modelshave been proposed to explain the auxetic behaviour, most ofwhich work in conjunction with the bow-tie model.

4.3 Missing rib model Smith et al. [25] proposed themissing rib foam model because, at the time of publication,they believed that the models proposed by other groups weresatisfactory in describing the stress–strain behaviour ofconventional foams but were limited in describing theauxetic behaviour. In this model the cellular internal anglesare not subjected to angular changes, but a fraction of the cellribs are removed, as shown in Fig. 10. This model is bestemployed in the calculation of strain-dependent Poisson’sratio function, opposed to small-strain Poisson’s ratios, asdefined by Berthelot and Reinf [47]. The advantage of thismodel is that both auxetic and conventional materials canbe described using realistic geometries and stress–strainbehaviours could be predicted slightly more accurately. Thedisadvantages of this model result from both the assumptionthat needs to be made (real world foams contain alreadybroken ribs which are randomly distributed) and its inabilityto accurately describe the compression/heating effects onfabrication. From this model Smith et al. [25] suggested thatduring several scenarios of sample fabrication, it is plausibleto presume that there are multiple mechanisms causingthe auxetic conversion, i.e. concaved re-entrant cells andmissing ribs.

4.4 Rigid triangle model In 2005 Grima et al. [42]proposed a further model to explain the 2-D behaviour of theauxetic foams. This model assumed that the microstructuralchanges induced during the compression/heating stage of thefabrication preserved the cell joint geometry, whilst the rib

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Figure 10 Schematic of the missing rib foam model showing (a) anintact conventional foam structure and the cut version with cell unitsshaded, (b) more detailed version of selected cells for both intact andcut versions alongside their geometry parameters (adapted fromSmith et al. [25]).

Figure 11 Steps in the rigid triangle model: (a) hexagonal honey-combmodel foraconventional foam, (b) rotationofrigidunitsmodelfor auxetic foams, (c) ideal model for the rotating rib model wherejoints are shownasperfect rigidlyequilateral triangles (adapted fromGrima et al. [42]).

lengths were subjected to the main deformations, resulting inbuckling. Furthermore, this model assumed that due to theincreased thickness at the cell vertices it is possible to treatthe joints as rigid, thus allowing the behaviour to bedescribed in terms of beam mechanics, rotation and perfectlyrigid equilateral triangles, although in reality the ribs, jointsor microstructure are not perfect. When the foam is loaded,the model predicts that the triangular joints rotate to return totheir original arrangement causing the rib to twist and unfoldproducing a volumetric expansion, as shown in Fig. 11.Based on this method the auxeticity depends mainly on thejoint rigidity and the flexibility of the connection ribs, withother factors being less influent. Grima et al. [42] believe thatthis model presents a dominant mechanism responsible forthe auxetic behaviour found after fabrication.

4.5 Three-dimensional (3-D) model Althoughnumerous models in 2-D have been suggested and accepted,a 3-D model has yet to be agreed. Several authors haveattempted to produce a 3-D model starting from the 2-Dmodel and applying to it an extra axis utilising dodecahe-drons and tetrakaidecahedrons [42, 45, 48–52]. Thisapproach is acceptable for a simplistic analysis, such asthose performed by a Finite Element Analysis, as the overallauxetic foam structure and behaviour are reverted to abasic level, where little to no variation to the structure isconsidered.

In order to determine the true mechanisms of the auxeticfoams, some researchers have begun to study cell inter-actions in 3-D analysing a small area and applying theresults to the whole specimen body. One of such studieswas undertaken by Chan and Evans [37], that employed

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microscopy to examine the microstructure and the defor-mation mechanisms of conventional and auxetic materials.By applying a compressive or tensile load the cell ribs inboth conventional and auxetic foams undergo a rangeof deformation. The dominant deformation mechanismdepended on the loading type and the foam cell structure.Conventional foams in compression deform primarily byflexure and buckling when high strains are applied. Duringtensile loading, the ribs perpendicular to the load alsodeform by flexure. When a tensile load is applied, the cell ribsof conventional foams deform through a combination ofhinging, stretching and flexing mechanisms (Fig. 12). Failurefinally occurs as a result of tensile fracture. In auxetic foams,it was found that compressive and tensile loading producethe same deformation mechanisms observed in conventionalfoams, with the addition of rib rotations. Based on theseobservations, Chan and Evans suggested that, as a generalrule, auxetic and conventional foams under compression aredominated by flexure and that under tensile loading they arecontrolled by flexure and stretching [37].

Although each of the currently presented models providevaluable if not limited insight, these models are toosimplistic, often requiring the presence of perfect micro-structure geometry. However, should various aspects ofthe previously suggested models be considered alongside



hys

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Figure 12 A finite element sample (2.2 mm3) of an auxetic foam of 45 PPI, demonstrating the response to tensile loading conditions,undertaken at the University of Southampton.

imaging techniques such as computed tomography (CT), amore accurate model could be produced. Unlike opticalmicroscopy and SEM, CT is capable of producing true 3-Ddata, not only of the surface but throughout the volume of abody. For auxetic foams this is extremely important, as theauxetic ribs deform in three-dimensions. CT is capable ofanalysing large samples, but the achievable resolution isaffected by both the area to be scanned and the type ofequipment employed. CT is a non-destructive technique andtherefore allows studying the changes in microstructureunder various experimental conditions. CT can also beutilised in the application of finite element analysis, an areararely studied in current literature. An example of this workcan be seen in Fig. 12, where a real world auxetic foamsample of 45 PPI at a volumetric compression ratio of4.88 was scanned, meshed and analysed in an attempt tounderstand localised rib movement under tensile loadingconditions, while investigating how a simulated responsecorresponds to the model demonstrated in Fig. 13. Someprelininary work using the CT technique has been reportedby Elliott et al. [53] and Gaspar et al. [54].

4.6 Models for metallic foams While the majorityof the research has been carried out on polymeric foams,


some authors have also studied or speculated the possibledeformation mechanisms in metallic foams. Friis et al. [2]suggested, based on the ideal foam theory, that the maindeformation mechanism in copper foams would be plastichinging. Microscopy studies of re-entrant foam structuresdemonstrated that plastic rib buckling was also present.Successively, Choi and Lakes suggested that the behaviourof polymeric and metallic foams is attributed to the ribs of thematerial, with both materials experiencing re-entrant cellularstructures as a result of the auxetic conversion, resulting ina 3-D auxetic behaviour [1, 3, 7]. It was further found forcopper foams that the ribs could form plastic hinges byyielding at a relatively small strain, unlike polymeric foamwhich are elastomer [36].

4.7 Granular mechanics Rothenburg et al. [44]studied the behaviour of negative Poisson’s ratio byassuming materials to be granular, for instance, consistingof randomly packed smooth stiff spheres connected byimaginary normal and tangential springs. Within this system,the interactions occur through frictional interfaces, resist-ance to compression (stiffness) and force transmission alongthe axis of the connecting particles centres and tangential tothe plane of contact, as per real regular granular materials. By

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Figure 13 Diagram showing the key deformation mechanisms in conventional foam under compression and tension, (A) conventionalfoam vertex and ribs, (B) forces acting on the ribs under compression, (C) deformation mechanism for each of the ribs during compression,(D) forces acting on the ribs under tension, (E) deformation mechanism for each of the ribs during tension, where blue arrows representforces upon the ribs and dotted red lines represent previous positions of the ribs before force.

considering the material as a 2-D isotropic plane, thePoisson’s ratio [Eq. (14)] is once more shown to becontrolled by the ratio between normal and tangentialstiffness,

www

v ¼ 1� l

4þ l: (14)

Shufrin et al. [55] later explored the effects of negativePoisson’s ratios in planar isotropic structures throughreference units (Fig. 14), each comprising of an infinitelyrigid hexagonal core surrounded by six linear elastic circularframe arches of conventional Poisson’s ratio. By tailoringthe units through various frame and core shapes, internalstructures and bonding types, shear stiffness is once more

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able to be greater than the normal stiffness resulting innegative Poisson’s ratio.

Pasternak and Dyskin expanded the granular mechanicsapproach to produce a 3-D granular system with the inclusionof imaginary rotational elastic springs, where the ratio betweennormal stiffness (kn) [N m�1] to shear stiffness (ks) [N m�1]was again shown to be controlling parameter in determiningPoisson’s ratio [Eq. (15)] [56]. The inclusion of rotationalfreedom failed to add any additional controlling parameters.

v ¼ kn � ks

4kn þ ks

: (15)

Following this relationship, Pasternak and Dyskenproceeded to present two isotropic structures (ball linkmodel and thin shell model) capable of achieving a



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Figure 14 Isotropic structural units: (a) clamped connection, (b)semi-hinge connection and (c) hinge connection, adapted fromShufrin et al. [55].

Figure 15 Ball link unit model where (a) unloaded and (b) tensileloaded, adapted from Pasternak and Dysken [56].

theoretical negative Poisson’s ratios of �1 (Figs. 15 and 16,respectively) [56]. When tested the ball link model wasfound to exhibit a Poisson’s ratio close to�1, whilst the thinshells model only exhibited a Poisson’s ratio of �0.27. Atheoretical model was also suggested, in which sliding rodsare inserted into a crack to allow a sliding but restrict sheardisplacement. This system however is difficult to implementand thus has not been physically tested, although atheoretical maximum negative Poisson’s ratio of �0.33was suggested.

Figure 16 Thin shell model consisting of seven hollow balls gluedfrom Pasternak and Dysken [56].


Interestingly Pasternak and Dyskin showed that as thePoisson’s ratio of an isotropic material becomes closer to�1,other mechanical properties will change depending upon ifthe shear modulus or the Young’s modulus is fixed. In thecase of a fixed Young’s modulus, the tensile strength of amaterial with a crack will increase indefinitely, whilst theshear modulus will tend to infinity. Alternatively, fixingthe shear modulus will cause the Young’s modulus to tend tozero and will decrease the tensile strength [56].

5 Mechanical properties of auxetic foams Under-standing the mechanical properties of a material is funda-mental to highlight its strengths and weaknesses and toevaluate any possible application. It has been reported thatthe conversion process of conventional polymeric andmetallic foams into auxetics enhances the mechanicalproperties compared to the preliminary material. Many ofthe observed improvements in mechanical properties aredirectly linked to the change in the cellular structure andthe increase in density of the material due to the volumetriccompression ratio applied [1, 7, 37, 43]. The increase inmechanical properties has been observed in both isotropicand anisotropic materials and has been noticed to affect theshear resistance [7, 8, 11, 24, 43, 44, 55–57] the indentation

together: (a) before loading and (b) bi-axial compression, adapted

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resistance [1, 2, 7, 11, 24, 26, 58–60], the fracture toughness[1, 7, 9, 11, 24, 43, 61], the compression [2, 24], the shearmodulus [24, 43, 44, 55, 56], the stiffness [9, 10, 44, 55, 56],acoustic dampening [1, 7, 43, 62, 63], the dynamicperformance [24], the optical passive turning of structuralvibration [64] and the viscoelastic loss [27, 65–67].

The improvements in mechanical properties observedafter the auxetic conversion have led numerous researchers toconsider many possible applications for auxetic foamsranging from smart filtration systems to protective sportsequipment [1, 11]. The mechanical enhancements observedare due to the changes to the four elastic constraints of thematerial: Young’s modulus (E) [Pa], shear modulus (G) [Pa],bulk modulus (K) [Pa] and Poisson’s ratio, which respectivelymeasure stiffness, rigidity, compressibility and volumetricchange under strain [11]. For these constraints there is not adiscernible characteristic length scale, which implies that amicrostructural dimensions smaller than 1 mm could exhibita negative Poisson’s ratio [3]. The following equations showhow the elastic constraints are related to each other and whyaltering a constraint will affect the others [68]:

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E ¼ 9KG

3K þ G; (16)

E
G ¼
2 1þ vð Þ ; (17)

E
K ¼
3 1� 2vð Þ ; (18)

1 3K � 2G� �

v ¼2 3K þ G

: (19)

These equations also provide understanding of theproperty enhancements exhibited by auxetic materials. Thetwo main mechanical properties of auxetic foams studied inthe literature have been hardness (indentation) and toughness(energy absorption), both properties are well-established asthe primary applications of the conventional foams arepackaging and cushioning [29].

5.1 Indentation of auxetic foams Lakes firststudied the effects of indentation on an auxetic foam byassuming a localised pressure distribution, where the pressureis proportional to (1� n2)/E, and from this expression it isevident that for a material with a Poisson’s ratio close to �1any indentation is very difficult [3, 69]. At the same time thismaterial becomes extremely compressible because the shearmodulus exceeds the bulk modulus. This behaviour is due tothe connection between the bulk modulus, the shear modulusand the Poisson’s ratio through the equation:

K¼ 2G 1þvð Þ1� 2vð Þ : (20)

Conversely, a material with a Poisson’s ratio approach-ing 0.5, i.e. rubber, is incompressible because the bulk

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modulus is greater than the shear modulus. In addition, Lakesstudied the effects of indention on a wrestling mat madefrom elastomeric foams (n� 1/3) [8]. He assumed that theindentation could be described as indentation rigidity:

P

w¼ E

2a1� v2� �

; (21)

where a is the radius of a circular localised pressure [mm], Pthe localised pressure [Pa] and w is the indentation depth[mm], as described by Timoshenko and Goodier [69]. Theindentations experienced by the mat were further analysedfor small and large impacts [8]. For small impacts (narrow)it was assumed that the circular pressure distribution couldbe taken as an elastic half space yielding:

F

u

� �narrow

¼ Gan

1� vð Þ ; (22)

where F is the indentation force [N], u and an are themaximum displacement [mm] and the radius for narrowindentation [mm], respectively. For impacts greater than themat thickness (wide), it is assumed that both compressionand force are uniformly distributed, with the Poisson’s ratioeffect controlled giving:

F

u

� wide

¼Ga2

w

2h 1þvð Þ1� 2vð Þ ; (23)

where h is the mat thickness measured in mm and aw is theradius for wide indentation [mm].

From this study, Lakes suggested that the continuumtheory of elasticity was probably not the best approach todescribe the auxetic behaviour and proposed using theCosserat theory of elasticity that takes into account boththe maximum stresses and a natural length scale to allowmicrostructural size to be incorporated into the prediction offailure [8, 70, 71].

Subsequently, Evans and Alderson confirmed the theorypresented by Lakes treating the indentation as an effect ofuniform pressure distribution that for an isotropic materialcan be described as proportional to [(1 – n2)/E]� 1 (Fig. 17)[11]. By applying the classic theory of elasticity, which statesthat a material can have a Poisson’s ratio varying from�1 toþ0.5, it can be seen that the indentation resistance increasestowards infinity for a negative Poisson’s ratio.

Furthermore, Evans and Alderson presented a relation-ship to describe how the elastic constraints affect one anotherwith respect to the shear modulus (G) [11].

G¼ 3K 1� 2vð Þ2 1þvð Þ : (24)

In order to keep the same representation employed byLakes [Eq. (20)], this equation can be re-written in terms ofthe bulk modulus (K) to give [3]:

K¼ 2G 1þvð Þ3 1� 2vð Þ : (25)



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Figure 17 Reaction of conventional and auxetic materials to loading under Hertzian indentation, adapted from Ref. [11].

In direct response to Lakes’ original paper [3], Burnspresented a relationship for the elastic constrains in term ofthe bulk modulus [72], similar to that already given by Evansand Alderson [Eq. (25)] [11]; however, the discrepanciesbetween the original work of Lakes [Eq. (20)] and the laterworks published by other authors [Eq. (25)] should be noted[11, 72]. Even if all the three sources employed the sametheory and treat the indentation as a uniform pressuredistribution, the equations obtained are different. Whenthe original literature is compared to the work of Evans,Alderson and Burns, it can be seen that the uniform pressuredistribution only accounts for a negative Poisson’s ratio,unlike in the later works, Poisson’s ratio could be eitherpositive or negative [3, 11, 72]. Furthermore, the equationpresented in the later literature is set to the power of �1, avalue unaccounted for in the original work [3, 11]. Thediscrepancies continue when looking at the relationshipbetween the elastic constraints; with the work presented byLakes lacking of a 3 in the denominator, compared to theequations reported by Burns and Evans and Alderson [3, 11,72]. It is unknown why these discrepancies occurred, it ispossible that they could be the result of either an error withinthe text or a failure to apply the correct concepts. Because ofthese discrepancies found within the original literature thework by Burns, Evans and Alderson and the equations thatthey present should be considered to be correct.

5.2 Toughness Toughness is an important mechan-ical property for a polymeric porous material since itdetermines the maximum energy absorption of the foamper unit, and can be determined by taking the integral of astress–strain curve, and applying the following equation:

� 20

Energy

Volume¼Z ef

0

sde; (26)


where e and ef are the strain and the strain upon failure and sis the stress [Pa].

In 1987 Lakes observed that the Poisson’s ratio affectsthe toughness of the auxetic materials and that as thePoisson’s ratio approaches �1, the material is expected tobecome extremely tough [3]. Using the critical tensile stress:

s¼ pET

2r 1� v2ð Þ ; (27)

where T is the surface tension [N m�1] and r the circularcrack radius [mm], this behaviour can be explained [3, 8,73]. Further to elastic constraints, the material toughnessand the Young’s modulus are also affected by both its non-linear properties and structural aspects [43]. This initialwork was continued by Lakes and Choi who outlined thedifferences between conventional and auxetic foams interms of material toughness; they observed that re-entrantfoams have an initial lower stiffness (Young’s modulus)compared to conventional ones, but as a result of non-linearbehaviour during a large deformation, the energy density ishigher [29]. Furthermore, the energy absorption can beincreased by using conventional foams with higherdensities; this however is not always desirable as it resultsin a higher stiffness [26, 29]. In order to understand the fulleffect of energy absorption for auxetic and conventionalfoams, Lakes and Choi studied the behaviour of foamssubjected to a range of volumetric compression ratios undercompressive and tensile stresses [29]. They found that thematerial behaviour was substantially different for each ofthe re-entrant specimens with toughness factors (1.7, 2.1,2.3, 2.6 and 3.2) increasing with increasing volumetriccompression ratio (2.0, 2.6, 3.2, 3.7 and 4.2) [29].

Likewise, Choi and Lakes analysed the materialtoughness by applying a methodology similar to that

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described in to conventional and auxetic copper foams withand without annealing.[36] For un-annealed auxetic foamswith volumetric compression ratios of 2.0, 2.5 and 3.0 thematerial toughness increased by factors of 1.4, 1.5 and 1.7compared to conventional foams [29]. It was found that theapplication of annealing could be used to influence theoverall toughness of the specimens however not alwayspositively. At a volumetric compression ratio of 2.0annealing was found to improve the mechanical toughnessby 33%, whist annealed samples compressed to a factor of2.5 gained a minimal increase in toughness. Once thevolumetric compression ratio reached a factor of 3.0 thetoughness decreased, a phenomenon that was not observed inpolymeric auxetic foams [36].

Alternatively to the single quasi-static loading methodundertaken by Choi and Lakes [29], Bezazi and Scarpastudied the material toughness under quasi-static cyclicloading via a fatigue test [9]. The auxetic foams exhibitedsignificantly higher stiffness degradation, energy absorptionand lower rigidity loss over a large number of cycles whencompared to their conventional counterparts. For N numberof cycles, the energy dissipated per unit volume (Ed) isgiven by:

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Ed ¼Z emax

emin

sdx; (28)

where emin and emax are the minimum and maximum strain.The graphical data reported in Fig. 18, where loading

levels (r), are defined as [74]:

Figure 18 Load vs. displacement for different levels of loading for(a) auxetic and (b) conventional foams, adapted from Ref. [9].
r¼Umax
Ur

; (29)

where Umax and Ur are the maximum displacement at aparticular level and at failure (measured in mm), showgreater energy absorption for the hysteresis loops of theauxetic foams, which with support of the stress–straincurves of static testing demonstrates the increase inresistance to failure and mechanical resilience. Thehysteresis cycle areas for both auxetic and conventionalfoam increased with the load applied. On average auxeticfoams dissipate 2.4 times the amount of energy dissipated byconventional foams for the first cycle under differentloadings. Furthermore, Bezazi and Scarpa stated that therewas different stiffness degradation behaviour for tension–tension and compression–compression loading, although itwas found that the difference between each was not directlylinked to the energy absorption [9].

Bianchi et al. [4] continued to study the energydissipation during quasi-static and cyclic loading undertension. They observed that when the negative Poisson’sratio approached zero the energy dissipation decreased.Furthermore, this study demonstrated that there was nosignificant statistical correlation between the various auxeticsample batches analysed. Interestingly, Bianchi et al.reported that during quasi-static cyclic tests, conventional

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foams exhibit energy dissipation greater than that of auxeticfoams and deduced that the overall compression is thecontrolling factor in the final Poisson’s ratio.

To further understand the mechanical properties ofauxetic foams, Bianchi et al. [5] studied how the energydissipation during both tension and compression is affectedwhen the foam is subjected to a four phase fabricationprocess (as described in Section 2.1.1), as shown in Fig. 19.An interesting observation was that the energy dissipation ison average greater for second phase auxetics than for firstphase auxetics, although first phase auxetic experienced farless scattering of the results. It was further observed, for bothtensional and compressive loading, that the conventionalphase demonstrate on average smaller energy dissipationcompared to the auxetic counterparts. However, theconventional phases exhibited mechanical stiffness of upto an order of magnitude higher when compared to theauxetic phases [25, 27, 37, 60, 75].

In addition to quasi-static cyclic loading, Scarpa et al.[10] have studied the energy absorption of conventional,auxetic and iso-volumetric materials at different strain ratesunder tensile loading. Strain rates of 8, 10 and 12 s�1 and



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Figure 19 Energy dissipation under (a) compression and (b) ten-sion for as received (~), returned (�), first auxetic (^) and secondauxetic (&) phases for polyurethane foam, adapted from Ref. [5].

instrument head displacements of 120, 150 and 180 mm s�1,respectively were applied to 19 mm diameter, 15 mm longsamples; the small sample size was employed to achieve thedesired strain rate. From the tests two main characteristicswere observed: (i) by increasing the tensile strain, thePoisson’s ratio approaches zero and before becomingpositive; (ii) that enhancement to mechanical propertiesunder compression was not only the resultant of the increasein density employed during fabrication of auxetic foams.They also reported that for the auxetic samples tested themechanical properties, e.g. mechanical strength and stiff-ness, were affected by the fabrication method and that theauxetic foams had an order of magnitude increase comparedto the conventional foams.

Similar work was undertaken by Scarpa et al. [27] whichstudied the energy absorption under higher strain rates,where values of 15 and 38 s�1 were employed comparable tovelocities of 1.5 and 3.8 m s�1, respectively. They noticedthat during dynamic crushing the auxetic foams showeda clear time-load history while the conventional foamsfailed to demonstrate any form of loading resilience. This


behaviour can be associated with micro-inertia (an extensionof the classic continuum mechanics theory that includesspatial gradients of acceleration to be considered into motionequations) [76], localisation effects and strain rates affect thedynamic crushing behaviour within the foams [2, 26, 27, 38].Furthermore, the auxetic specimens were found to have a lowsensitivity to strains induced by stresses lower than 0.54 MPaand once this value was exceeded the specimens exhibitedstiffening effects during the densification process.

6 Auxetic foam applications The enhancements inmechanical properties offered by auxetic foams compared toconventional foams have led several researchers to considerpossible applications for this new class of materials.Currently conventional foams are utilised in applicationssuch as packaging, cushioning, air filtration, shock absorp-tion and sound insulation [11, 26]. It could be anticipated thatin the future conventional foams will be replaced by auxeticfoams in many of these applications as a result of theirenhanced mechanical properties.

Further novel applications for the auxetic foam technol-ogy have been suggested. Smart filtration systems allowthrough the pores only particles with a certain size that couldbe varied by varying the pore size of the foam by applyingdifferent loads [11]. This application could be taken a stepfurther and be applied for smart drug delivery to patients.In this case the amount of drugs delivered could becontrolled by whether or not the pores were opened orclosed and this could be influenced by the swelling of thebody and the consequent pressure exerted on the foam [15].Another application of auxetic foam could result from theircharacteristic double curvature reported by Lakes [3]. Thedome-like shape when the foam is bent could be applied tomattresses to give support to the doubly curved human bodyform [11].

Other notable uses for auxetic foams stem from militaryapplications where auxetic foams and other auxetic materialsare being studied for use in ballistic protection, for instanceMitsubishi has recently patented a bullet design whereone component is made of auxetic material, in an attempt tocreate an overall Poisson’s ratio of zero. By achieving anegative Poisson’s ratio, lateral expansion would be reducedwhen travelling down the gun barrel [11].

Another application proposed by Choi and Lakes is apress fit fastener [7], which would work by tangentiallycontracting when placed into a socket, and expanding whenattempted to be removed. An additional suggestion by Lakesis linked to a wrestling mat that experiences both smalllocalised impacts (knees) and large distributed impacts(human torso) during its use, which can be calculatedthrough Eqs. (19) and (20) [8]. Currently, wresting matscontain elastomeric foams with Poisson’s ratio of approxi-mately 0.3; however, Lakes proposes that due to theenhanced indentation and energy distribution mechanismsof negative Poisson’s ratio materials, auxetic foam wouldoffer the greatest performance, as opposed to rubber which isconsidered to be not as effective.

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Other possible applications include structural integritystructures [6, 10], sandwich components [1, 7, 10, 13, 64],smart structures [11, 27], biomedical components [18, 77],dynamic and multi-physics applications [26, 27, 78–81],knee pads [1], tear resistant sponges [1] and sound absorbingmaterials [1].

7 Summary

(i) C

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onventional polyurethane foam cells can be repre-sented as pentagonal and hexagonal structures.However, in reality the pores are slightly elongatedand not perfectly symmetrical.

(ii) A
uxetic polyurethane foam cells can be represented aseither re-entrant cells or random rib structures.
(iii) B
oth 2-D and 3-D models have been proposed toexplain the auxetic behaviour. These models include:the rigid triangle, honeycomb, missing rib and bow-tiemodels. Although each of the models account for someof the behaviours observed, no one of these modelsalone can account for all behaviours observed andtherefore they should be considered together.
(iv) T
he auxetic fabrication requires a temperature equal orgreater than the softening temperature of the polymericmaterial.
(v) A
literature review indicates that an optimum heatingtime for the process has not been identified yet and thatheating time ranging from 6 to 50 min has beenreported. This is likely due to the authors utilisingdifferent equipment, materials and different sizes.
(vi) F
ailure to apply a correct volumetric compression ratiowill result in transitional samples, where both conven-tional and auxetic cell units are present but withoutpresenting auxetic behaviour.
(vii) T
he optimum volumetric compression ratio is between2 and 5 or between 50 and 80% of the original size.
(viii) P
ore size has been found to be a secondary factorinfluencing the auxetic behaviour whilst the cellgeometry is believed to be of primary influence.
(ix) A
uxetic materials have been found to demonstrateenhanced properties including mechanical hardness,toughness, stiffness and dampening.
(x) T
here are a numerous potential applications for auxeticsamples including smart filters, biomedical apparatus,protective sports clothing, packaging and numerousother smart systems.
Acknowledgements The authors gratefully acknowledgethe financial support of EPSRC and Dstl (EP/G042195/1).

References

[1] R. Lakes, Nature 358, 713 (1992).[2] E. A. Friis, R. Lakes, and J. B. Park, J. Mater. 23, 4406

(1988).[3] R. Lakes, Science 235, 1038 (1986).

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[4] M. Bianchi, F. L. Scarpa, and C. W. Smith, J. Mater. Sci. 43,5851 (2008).

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