Theory of strains in auxetic materials
aa r X i v : . [ c ond - m a t . m t r l - s c i ] A p r Theory of strains in auxetic materials
Raphael Blumenfeld , , Sam F. Edwards
1. ISP and ESE, Imperial College, London SW7 2AZ, UK2. Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK (Dated: October 22, 2018)This paper is dedicated to Prof. Jacques Friedel, an inspirational scientist and a great man.His excellence and clear vision led to significant advances in theoretical physics, which spilled intomaterial science and technological applications. His fundamental theoretical work on commonplacematerials has become classic. We can think of no better tribute to Friedel than to apply a funda-mental analysis in his spirit to a peculiar class of materials - auxetic materials. Auxetic materials, ornegative-Poissons-ratio materials, are important technologically and fascinating theoretically. Whenloaded by external stresses, their internal strains are governed by correlated motion of internal struc-tural degrees of freedom. The modelling of such materials is mainly based on ordered structures,despite existence of auxetic behaviour in disordered structures and the advantage in manufacturingdisordered structures for most applications. We describe here a first-principles expression for strainsin disordered such materials, based on insight from a family of ‘iso-auxetic’ structures. These arestructures, consisting of internal structural elements, which we name ‘auxetons’, whose inter-elementforces can be computed from statics alone. Iso-auxetic structures make it possible not only to iden-tify the mechanisms that give rise to auxeticity, but also to write down the explicit dependenceof the strain rate on the local structure, which is valid to all auxetic materials. It is argued thatstresses give rise to strains via two mechanisms: auxeton rotations and auxeton expansion / con-traction. The former depends on the stress via a local fabric tensor, which we define explicitly for2D systems. The latter depends on the stress via an expansion tensor. Whether a material exhibitsauxetic behaviour or not depends on the interplay between these two fields. This description hastwo major advantages: it applies to any auxeton-based system, however disordered, and it goesbeyond conventional elasticity theory, providing an explicit expression for general auxetic strainsand outlining the relevant equations.
I. Introduction by Sam F. Edwards
At the end of the war, which had isolated Francefrom the English speaking world, several French scien-tists moved to UK universities, in particular to studysolid state theory. The outstanding person in the UK atthe time was Nevill Mott in Bristol and Jacques Friedelmoved to Bristol to work in Mott’s group. At the time,the field of theoretical physics was moving into the useof field theory to elucidate elementary particle theory, adirection favoured in Cambridge University and in Lon-don. The Bristol group, however, specialised in electronicstudies, an area that Friedel preferred. I remember hispapers at that time, which had a wonderful clarity anddiscussed down-to-earth type of problems. It was re-freshingly in stark contrast to the renormalisation the-ory, which was the fashion in quantum field theory atthe time.Sometime later, Mott moved to Cambridge andJacques returned to Paris. This reminds me my firstconference in Paris, where I gave my first paper. It wasnonsense, I regret to say, for it tried to separate greenfunctions for the real and imaginary parts of the wavefunction. Fortunately, none of the attendants in thatconference exposed it.Anyway, I recall Friedel giving wonderful lectures incambridge, where his work was held at very high esteem.Years later Cambridge University awarded him an Hon-orary Doctor of Science and I had the pleasant task ofarranging a dinner for him. Friedel was also involved in setting up the European physical society, where I was ac-tive, and I recall him giving valuable advice on its struc-ture.A central sociological problem in theoretical physics isto choose the problem to work, for there are many bril-liant people working at the forefront of the field. Think-ing of Friedel’s work on electronics in parts of systems, itoccurred to me that one should be able to do statisticalmechanics on continuous systems in contrast to particu-late systems. With my coauthor here, Raphael Blumen-feld, I have developed this idea by studying the entropyof particulate systems in the continuum. For example,in conventional thermal systems the entropy S is a func-tion of pressure, volume, energy and number of particles, S ( E, P, V, N ) and one of the most useful concepts it leadsto is the temperature T = ∂E/∂S . We have applied theseto granular systems where the entropy is due to config-urational disorder and the volume takes the role of theenergy. Consequently, the analogue of temperature is the‘Compactivity’ X = ∂V /∂S . There are other quantitiesthat dictate the states of granular matter, the simplestbeing the response of stresses to the entropy, X = ∂σ/∂S ,which we called the Angoricity (note that the Angoric-ity is in fact a tensor). An even richer and more general’thermodynamics’ is required when we study mixtures.This paper is dedicated to Jacques Friedel and, in thespirit of the close relations of his theoretical works withreal materials, we can think of no better tribute to himthan to present a fundamental theory that aims to under-stand the physics of a peculiar class of materials - auxeticmaterials.‘ II. General introduction
Auxetic materials, i.e. materials with negative Pois-son’s ratio, expand when stretched and contract whencompressed, in contrast to most conventional materials.This is due to correlated degrees of freedom in the inter-nal elements that theses materials are made of. Theseelements are reversibly foldable and, in effect, can beregarded as the basic constituents of cellular solids. Inthe following, we call these foldable elements ‘auxetons’.Macroscopic auxetic structures can be manufactured ofpolymers[1] or metals[1, 2]. They can exist on a range oflength-scales and, in particular, can be constructed outof molecular building blocks[3–5]. Auxetic materials areuseful in applications requiring high shear to bulk modulior compactification on impact, e.g. for energy absorbingmaterials and bullet-proof armours.Both natural[6–9] and man-made[10, 11] auxetic ma-terials have been discovered, made and studied. Muchof the theoretical analysis, however, is carried out on or-dered models, such as two-dimensional inverted cell hon-eycombs. Although models of the auxeticity phenomenonin ordered structures is convenient for analysis purposes,the ubiquity of disordered such materials and the littleexisting understanding of deformations in the presence ofdisorder require a more general theory. Here we describesuch a theory, based on a recent suggestion made in [19].The aims of this paper are the following. First, wedescribe a new family of disordered auxetic structures,called iso-auxetic (IA) structures, for which it is possibleto identify clearly the basic strain mechanisms. Second,we show that elasticity theory is not necessary for thedescription of auxeticity, implying that using negativePoisson’s ratio as a descriptor has a limited utility. Third,we present an explicit expression for the auxetic strain interms of local expansive and rotational fields. In thisexpression, the fields are coupled to the stress throughwell-defined tensors, which we discuss. Fourth, we showthat auxeton rotations are essential to the understandingof the global behaviour and that the rotational field canbe modelled without resorting to non-symmetric stresses.This obviates models based on Cosserat theory[13].The paper is structured as follows. We first intro-duce the new family of IA structures. These are struc-tures whose inter-auxeton forces can be determined fromstatics alone. This property distinguishes IA from moreconventional auxetic structures, which we term elasto-auxetic (EA). Specifically, the stress field equations ofisostaticity theory differ significantly from those of con-ventional elasticity in that they are based on local stress-structure relations, as opposed to the usual stress-strainrelations[14–16]. We next describe an extension of a re-cent result for yield of granular systems to IA structuresand write down explicitly the IA strain equation in termsof two local fields: an expansive and a rotational. It isthen argued that the mechanism for auxeticity depends only on these two fields and is therefore independent ofthe particular way that the structure transmits stresses,whether isostatically or elastically. Hence, the auxeticexpression for the strain is valid for all auxetic materials.This, in turn, implies that general auxeticity needs tobe described by a theory that goes beyond elasticity. Aparticular implication of this conclusion is that negativePoisson’s ratio in auxetic materials should be regardedonly as a descriptor of the ratio of strains in perpendic-ular directions, not as a ratio of elastic moduli. We alsoargue that, although the way the form of the strain ex-pression is the same for all auxetic materials, the strainsdeveloping in IA structures differ markedly from thosedeveloping in EA structures under the same loading con-ditions. We conclude with a discussion of the results.
II. Iso-auxetic structures
In the following discussion, we consider planar auxeticmaterials made of 2D elementary units that connect totheir neighbours at exactly three points. We call these el-ements ‘auxetons’. Aiming at a theory of disordered ma-terials, we do not require that the auxetons be identical,nor that the system possess any type of symmetry, trans-lational or otherwise. Rather, we consider systems whoseauxetons comprise a mixture irregular sizes, shapes andorientations. A wide variety of such structures can beconstructed, some of which are illustrated in figure 1.
FIG. 1.
Examples of auxetons made of three-contact buildingblocks. Each auxeton can expand and rotate when forces areapplied to its ends, termed ‘contacts’ in the text.
We constrain our auxetons to have three ‘contacts’with their neighbours and connect these contacts byimaginary straight lines into triangles (the dashed bluelines in figure 2). This construction results in a planargraph of triangles, connected at their vertices. The tri-angles enclose polygons, which we call in the followingcells.When loaded by external forces, the auxetons trans-mit those to one another through ‘inter-auxeton’ forces.The contacts between auxetons may or may not be free-jointed. One expects the latter to be more common, inwhich case a contact can support a certain threshold oftorque moment without yielding. This gives rise to afinite overall stress threshold for straining the material.Consider then a structure, made of N ( ≫
1) auxetons,stressed below the yield threshold by a set of externalforces. Below the yield threshold, the system is in me-chanical equilibrium and all the inter-auxeton forces and
FIG. 2.
A section of a disordered auxetic structure, made ofjoining auxetons at their contacts. The contacts are joined bystraight lines (blue dashed) into a triangle. These trianglesare then used to characterise the contact network in a well-defined manner. torques are balanced. Since every auxeton has three con-tacts then the number of contacts is 3 N/ O ( √ N ),where the latter term is a boundary correction, whichcan be neglected for N ≫
1. Since each contact trans-fers one force vectors, there are overall 3 N force compo-nents. These can be determined uniquely by the threebalance equations for every auxeton - one of torque andtwo of force components. It follows that this structureis statically determinate, or isostatic . Hence the nameiso-auxetic. A familiar textbook statically determinatesystem is that of a ladder on a frictional floor leaningagainst a frictionless wall. The forces that the wall andthe floor apply to the ladder can be determined uniquelyfrom its three balance equations. It is important to notethat, as in the case of the ladder problem, the deter-mination of the discrete inter-auxeton forces requires noknowledge whatever of the elastic properties of neitherthe auxetons nor the contacts. Since the stress field isnothing but a continuos representation of the large num-ber inter-auxeton forces, it must reflect the nature of thediscrete solution and therefore be independent of localelastic moduli. It follows that elasticity theory, whichdoes rely on knowledge of the elastic moduli, is inappli-cable for IA structures.For later discussion, it is useful to recall the continuum2D stress equations of isostaticity theory - the theory ofstresses in isostatic structures, ∂σ ij ∂x i = g j (1) σ ij = σ ji (2) Q ij σ ij = 0 (3)Eqs. (1) and (2) represent force and torque balances, re-spectively, with σ the stress tensor and g external forces. Eq. (3) is a constitutive relation between the static stressand the local structure, which is characterised by a sym-metric fabric tensor Q [17 ? , 18]. This replaces the stress-strain relations in conventional elasticity and is indeedindependent of the elastic moduli of the material.In most known isostatic systems these equations arehyperbolic, leading to solutions that ‘propagate’ alongcharacteristic paths in the material. This means that theresponse to a localised force source in 2D is generically apair of force chains. In contrast, EA materials respond tolocalised sources by ‘dispersing’ the stress field in all di-rections, subject to local stress-strain relations. The dif-fernce between the two types of solutions stems from thenature of the stress field equations - while the equations ofelasticity theory are elliptic , eqs. (ForceBal)-(ConstEq)of isostaticity theory are hyperbolic . The different stresstransmission is bound to affect macroscopic behaviour,as will be discussed below.The global auxetic behaviour is the result of local fold-ing and unfolding of auxetons when stressed. This localresponse is independent of whether the rest of the struc-ture is isostatic or not, it is only dependent on the localexpansion and rotation of the auxetons. This leads tothe conclusion that the strain can be written in termsof local expansion and rotation fields regardless of theisostatic or elastic nature of the material. This conclu-sion is significant for two reasons. One is that, in IA, thestress is independent of elastic moduli. The other is, thatin IA we can write the strain explicitly in terms of theexpansion and rotation of auxetons, which means thatthe same expression holds for EA materials. This givesinsight into the description of auxeticity in general andin particular into the coupling between the local strainand the local structure. Additionally, this suggests thatelastic constants need not play as major a role as in con-ventional materials. Another important implication isthat the negative Poisson’s ratio, which such materialsexhibit, is only a descriptor of the ratio of perpendicularstrains and is of little use in terms of describing bulk elas-tic moduli because these cannot be obtained simple ho-mogenisation of small-scale regions. It is also worthwhileto note, before we continue, that this description shouldapply not only to all auxetic materials made of foldableauxetons, but also to those made of rigid ones[20].Before we proceed, we must comment on a much de-bated issue: whether or not auxeticity theory necessitatesresorting to Cosserat stress theory[13], which allows forexistence of a non-symmetric stress tensor. This is nota question of formalism, but rather of the underlyingphysics. A symmetric stress tensor means that residualtorque moments vanish on the continuum length-scales.Differently put, it means that there are no external couplemoments on the system that require balancing mechani-cally by the mechanical stress field. By letting the stresstensor be non-symmetric on macroscopic scales, Cosserattheory implies that there exist external couples that thestress must balance. Thus, a theory that invokes onlysymmetric stresses does not resort to such additional in-put and must be preferable for modelling of large-scaleauxetic behaviour. For this reason we prefer the aboveformulation, which includes (eq. (2).Nevertheless, it is important to point out that a sym-metric stress tensor can still allow existence of local ro-tational fields of the material upon straining. In otherwords, although the stress field must be symmetric un-der no external couples, the strain field may have non-symmetric contributions. Indeed, local such contributionarise from rotation of auxetons and it is at the core ofauxetic behaviour, as will be discussed in the next sec-tion. III. Auxetic strain and field equations
To relate the strain to the local structure one has tohave first a quantitative description of the structure, how-ever disordered. Such a descriptor is the fabric tensor Q ij of eq. (3). This tensor plays a key role in modelling aux-etic strains, as will be seen below. Consider a disorderedstructure of auxetons, comprising an arbitrary mixtureof elements, such as those shown in figure 1. The modelto be described below has been discussed initially in [19]and it is general in that it applies to any arbitrary struc-ture of the above auxetons. Specifically, the disorder caninvolve both auxetons of different sizes and of differentshapes. Connecting the three contact points around eachauxeton by straight lines, as described above, the planeis tiled into a network of triangles of different sizes andshapes, all interconnecting at their vertices - the contactpoints. The triangles enclose polygons, which we will callin the following cells. According to Euler relation[21], asystem of N ≫ ∼ / √ N and therefore negligible. All triangleedge are assigned directions, making them into vectors r that circulate the triangles in the clockwise direction(figure 3).Every triangle is assigned a centroid, defined as the meanposition vector of its three vertices. Similarly, every cell isassigned a centroid, defined as the mean position vectorof the contacts (triangle vertices) that surround it. Inmechanical equilibrium, the cell polygons must be convexto be stable. This means that, a vector R cg extendingfrom the centroid of triangle g to the centroid of one ofits neighbour cells c , intersects one of the triangle edgevectors, which we can index r cg (figure 3).The vectors R cg and r cg can be regarded as the di-agonals of a quadrilateral, called ‘quadron’, which playsa significant role in granular and cellular physics[22–24].Each quadron is associated uniquely with a pair cg (seefigure 4). This construction allows us to quantify thelocal structure by quantifying the shape of every cg -quadron tensorially as the outer product C cgij = r cgi R cgj (4) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) cg R r cg gc FIG. 3.
Characterisation of the auxeton structure in 2D. Wemake the edges of the representative triangle g into vectors, r cg , by assigning the edges a direction such that they circulatearound the triangle in the anti-clockwise direction. The vector R cg extends from the centroid of the grain contacts to thecentroid of an adjacent cell c . (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) r R cgcg Quadron q
FIG. 4.
Quadron tessellation in 2D . The vectors r cg and R cg make the diagonals of the cg -quadron. The quadrons are theelementary units that tessellate the system. The quadronshape is quantified by a local structure tensor, C cgij = r cgi R cgj . The tensor appearing in the isostaticity stress equationsis the symmetric part of C cg , summed over the cellsaround triangle gQ g = 12 ǫ − · X c around g C cg + ( C cg ) T · ǫ (5)where ǫ is the π/ C T is the transpose of C .Armed with a quantitative description of the localstructure, it is possible now to relate it to the strain.Suppose that the structure is in mechanical equilibriumunder a set of external forces and these forces are in-creased. Eventually, the system crosses what is known asthe yield surface and it starts deforming. As will becomeclearer below, whether the deformation is auxetic or notdepends on the structure of the auxetons, their configu-ration and the magnitude of the local stresses. The aimof the following is to describe the equations that governthe strain, given the local structure and the local stress.Central to the model is the observation that only aux-eton rotation and expansion (or shrinking, which canbe regarded as negative expansion) can give rise to dis-placement. The expansion corresponds to pure folding/ unfolding of auxetons. Thus, the local strain e , dueto changes in shape and volumes of the triangles, canbe written as a superposition of a rigid triangle rota-tion, e rot , and triangle (non-uniform) expansion, e ex . Forexample, auxetic materials composed of rigid auxetons,such as those studied in [20], can be described by e rot alone. In the following we consider only the symmetrisedstrain, but there is no reason why the treatment shouldnot apply to non-symmetric strains equally well. Notethat the dependence of the strain on the stress is onlythrough the responses of both these modes of motion tolocal stresses. This is, in fact, the main difference be-tween this theory and elasticity-based descriptions thatrelate directly the strain to the stress.The (symmetrised) strain due to rotation at the centroidof auxeton g is given directly by the tensor Q g [25], e rot,gij = Q gijkl θ gkl (6)where θ gkl is its angle of rotation, which depends on thelocal stress. Eq. (6) is written so that it holds both in2D and in 3D. In 3D, this expression is symmetric underexchange of the indices i and j , but anti-symmetric underexchange of k and l . This is due to the anti-symmetricnature of the description of the axes of rotations kl . In2D, there is only one axis of rotation, perpendicular tothe plane, and the indices kl are redundant, which re-duces Q ijkl to the tensor Q ij of eq. (5). This expressionhas been derived first in [25] for granular media, whereit gives rise to dilation. It comprises the only relevantcontribution to the strain when the auxetons are rigidand, as such, should also describe well the systems dis-cussed in [20]. For what follows, it is important to notethe observation in [17, 18] that Q is a measure of thelocal rotational (or chiral) deviation of the auxeton froma global zero average. This rotation is best quantified bythe sign of Tr { Q } .When elements can also fold and unfold, their expan-sions depend on the local stress. Significantly, there is noreason to expect that auxeton expansions be isotropic;depending on the choice of shape and the local structurearound them, auxetons may expand differently in differ-ent directions. The expansive strain rate can be relateddirectly to the local stress via e exp,gij = E gijkl σ gkl (7)where the non-isotropic expansion can be modelled into E g and different auxeton shapes would be described by different such matrices. Limiting the description to sym-metric strains imposes some constraints on the local ex-pansion tensor E , making its properties similar to thoseof the conventional compliance matrix in linear elastic-ity. However, such similarity would not exist for non-symmetric strains. For example, for such strains, E neednot be symmetric under exchange of i and j . It is im-portant to note that, whilst the strain may have non-symmetric components, for example to describe large-scale vorticity, the stress cannot if there are no externalcouples to balance the excessive torque. This is one ofthe reasons that the following theory cannot be mappedreadily to elasticity theory, nor to Cosserat theory.The total strain can be written then as e gij = E gijkl σ gkl + Q gijkl θ kl ( σ g ) (8)This relation is reminiscent of the yield equations in gran-ular systems[25], but for two important differences. Oneis that, in granular systems, the rotating elements (thegrains) can also slide relative to neighbours, a mechanismthat auxetons do not possess, which gives rise to an ad-ditional, plasticity-like term. The other difference is thatauxetons can fold and unfold (the E -dependent term),which rigid grains cannot.Relation (8) makes good sense on the auxeton level.However, to be of use to materials that contain manyauxetons, it must be coarse-grained (homogenised) to thecontinuum. To this end, one must average it over smallvolumes, containing sufficiently many auxetons. The ex-pansive term on the right hand side of (8) gives no prob-lems - one can average E and σ independently to obtaina continuum-scale contribution. This is no different thanthe practice in conventional elasticity and plasticity mod-els.In contrast, the rotational term requires a careful con-sideration. Coarse-graining over the rotation field of aregion of volume V , h θ i = (1 /V ) P g θ g , can be carriedout by replacing the volume average by a surface sum(or integral, for large enough regions), using Stokes the-orem. This leads immediately to the observation thatthe contribution to such an average comes only from theboundary of the region. Hence, if the system does not ro-tate globally, then the rotation per auxeton decays fastas the averaging volume increases and the macroscopicrotation has a zero average. It turns out that the tensor Q possesses exactly the same property. Since this tensormeasure the local chiral fluctuation of an element, an av-erage over a region decays to the global zero average atexactly the same rate as h θ i .On the face of it, these two observations may seem toimply that the rotational contribution to the strain van-ishes on large scales. This, however, is not the case. Thereason is that both θ and Q possess the same local anti-correlations: when one auxeton rotates in one direction,elements in contact with it are more likely to rotate inthe opposite, rather than in the same, direction. Simi-larly, if the tensor Q g measures the rotation of an auxetonat a given direction, nearest-neighbours of g are morelikely than not to have Q ’s whose trace has the oppo-site sign. This anti-correlation means that, while each ofthese terms averages to zero independently over increas-ing regions, their product h Q ijkl θ kl i adds constructively over nearest neighbours, leading to a finite large-scale av-erage. It is exactly this average that leads to measurablebulk strain due to rotations of rigid particles in granularsystems (dilation). We therefore conclude that eq. (8)has a well defined homogenised large-scale version e ij = E ijkl σ kl + Q ijkl θ kl ( σ ) (9)The only remaining question is how to derive local con-tinuous expressions for the rotational term. This can bedone, using the renormalisation approach taken in [26].A word of caution: the existence of a macro-scale con-tinuous theory does not imply that the strain is auxetic.Relation (9) gives the correct dependence of the strainon the local fields, but whether the ratio between per-pendicular strains (Poisson’s ratio) is negative or positivedepends on the relative contribution of the two terms onthe right hand side of this relation.The advantage of relation (9) is that it identifies theprecise role that the local structure plays in the couplingto the strain and the stress. As such, it is an importantingredient in the field equations of auxeticity theory. Tocomplete the theory for IA structures, one still needs thelocal rotational response to the local stress, θ kl ( σ ). Thisrelation is still missing and work to derive it is ongoing.Thus, the full set of auxeticity field equations in d dimen-sions consist of:(i) d ( d + 1) / d ( d − / d ( d − / θ kl ( σ ).As in any theory, constitutive information is required.For the theory described here, this comprises the consti-tutive tensors E and Q , which could be obtained eitherphenomenologically or modelled theoretically for specificstructures.The solution for quasi-static deformation then pro-ceeds as follows. First, one solves for the stress fieldfrom eqs. (1)-(3). From this solution one finds the localrotational field θ kl ( σ ), using the local rotation - stressrelation. Substitution of the rotational field, the consti-tutive fabric tensor Q and the expansion tensor E intoeq. (9) one then derives the total local strain.Since the dependence of the strain on local rotationand expansion of elements is valid regardless of the stress state, then all this theory, but for the stress field equa-tions, applies to any auxetic material. In particular,it applies to EA materials, where the stress equationsshould then be replaced by those of elasticity theory. Inother words only the closure relation (3) is replaced bySaint tenant compatibility conditions[27], supplementedwith phenomenological or modelled expression for thestress-strain relations. IV. Discussion and conclusions
To conclude, we have described a theory for strains inauxetic materials. The theory’s main contribution is theexplicit relation between the local auxetic strain and thelocal rotation and expansion of auxetons - the elementarybuilding blocks of auxetic materials. This is a refinementof the currently existing elasticity theory which lumpsthese two contributions together into a stress-strain rela-tion. The identification of these strain mechanisms makesit possible to eventually arrive such a relation, since thelocal magnitudes of auxeton rotations and expansions dodepend on the local stress. However, the explicit decom-position to rotation and expansion give insight into thecorrect symmetryies and details of such a stress-strainrelation.Furthermore, using elasticity theory for IA would leadto erroneous results, which originate from two sources.Firstly, the stress state cannot be derived from elasticitytheory and is likely to exhibit non-uniform force-chain-like fields. Secondly, the rotational and expansion re-sponses to the stress are of completely different nature.For example, the averaging properties of Q and E arecompletely different - while the latter has a well-definedmacroscopic homogenised value, the latter does not. Thisis despite both terms having homogenised large-scale con-tributions.A significant implication of the above is that all aux-etic, whether IA or EA, must follow the universal straineq. (9). However, the stress state, which determines thelocal rotation and expansion of auxetons, depends on thecorrect stress description and this may vary between dif-ferent families of materials - isostaticity theory for IA andelasticity theory for EA. This then leads to the intrigu-ing conclusion that the auxetic behaviour of IA and EAmaterials should be markedly different, with the formerexhibiting more non-uniform local auxetic behaviour.It is emphasised that eq. (9) does not ensure auxeticity,but rather it describes correctly the strain as a functionof the local rotational and expansive fields. Whether thematerial exhibits a bulk negative Poisson’s ratio dependson the different contributions of the two terms in thestrain relation. [1] R.S. Lakes, Foam structures with a negative Poissons ra-tio , Science , 1038 (1987).[2] E.A. Friis, R.S. Lakes, J.B. Park,
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