## Abstract

We systematically analyse the mechanical deformation behaviour, in particular Poisson's ratio, of floppy bar-and-joint frameworks based on periodic tessellations of the plane. For frameworks with more than one deformation mode, crystallographic symmetry constraints or minimization of an angular vertex energy functional are used to lift this ambiguity. Our analysis allows for systematic searches for auxetic mechanisms in archives of tessellations; applied to the class of one- or two-uniform tessellations by regular or star polygons, we find two auxetic structures of hexagonal symmetry and demonstrate that several other tessellations become auxetic when retaining symmetries during the deformation, in some cases with large negative Poisson ratios *ν*<−1 for a specific lattice direction. We often find a transition to negative Poisson ratios at finite deformations for several tessellations, even if the undeformed tessellation is infinitesimally non-auxetic. Our numerical scheme is based on a solution of the quadratic equations enforcing constant edge lengths by a Newton method, with periodicity enforced by boundary conditions.

## 1. Introduction

Materials with negative Poisson ratios, termed *auxetic* by Evans *et al.* [1], were once believed a rarity but have recently been found in amazing variety. Poisson's ratio *ν* can be expressed as
with *ϵ*_{∥} the imposed strain in a given direction and *ϵ*_{⊥} the resulting strain in the perpendicular direction. If, for a given elongation along one direction, the material expands in the perpendicular direction, Poisson's ratio becomes negative. Such auxetic behaviour has been reported in polymeric and metal foams [2], carbon ‘buckypaper’ nanotube sheets [3], coulombic crystals in ion plasmas [4], elastic strut frameworks [5], tetrahedral framework silicates [6], micro-porous polymers [1], *α*-cristobalites [7], cubic metals [8,9] and self-avoiding membranes [10]. Locally auxetic behaviour has been observed in semicrystalline polymer films [11]. Related phenomena are the negative normal stress in bio-polymer networks [12] and the dilatancy of granular media [13]. Inspired by these findings, technological applications, such as enhanced shock absorption [14], self-cleaning filters [15], tunable photonic crystal devices [4] and molecular-scale strain amplifiers [16], have been proposed. Complex physical behaviour beyond the mechanical properties results, e.g. phonon dispersion [17] and wave propagation or attenuation [18,19]. For a broader discussion of Poisson's ratio in the context of modern materials, see the review article by Greaves *et al.* [20].

## 2. Periodic bar-and-joint frameworks as models for auxetic structures

The widespread appearance of auxetic behaviour results from generic features of complex micro-structure that is common to these materials, rather than from specific interactions. The universal appearance of auxetic behaviour cannot be based on some particular homogeneous material property, but must be related to general features of a complex micro-structure below a certain length scale. To first order, that micro-structure is often approximated by a *periodic bar-and-joint framework* of rods (usually stiff), freely pivoting at mutual joints of two or more rods.

A frequent geometric element of auxetic structures are *re-entrant* elements (non-convex polygons; e.g. see the inverted honeycomb pattern), but other mechanisms based on rotating or stretching motifs have also been proposed [21].

It seems timely to search in a systematic way for auxetic structures and their building principles. Here, we describe (i) the methodology for systematic numerical analyses of the deformation of symmetric structures and (ii) as results of this analysis, several novel auxetic frameworks and new (deformation) mechanisms with transitions from non-auxetic to auxetic behaviour at finite strains.

We focus on bar-and-joint frameworks, henceforth referred to as *frameworks*, consisting of stiff rods of constant length that pivot freely at mutual joints (figure 1). Specifically, we will focus on periodic and symmetric bar-and-joint frameworks, which can be interpreted as tessellations (or tilings) of the plane by polygons with straight edges [22].

Mathematically speaking, *S*=(*K*,*E*) is an embedded graph consisting of a set of nodes, *K*, and a set of edges, *E*. Every node *i*∈*K* corresponds to a joint, with coordinates **p**_{i}={*x*_{i},*y*_{i}}. Every edge *e*={*i*,*j*}∈*E* (with *i*,*j*∈*K*) corresponds to a rigid bar of length *l*_{{ij}} that defines the distance constraints
2.1

The solutions of this system of quadratic equations are permissible configurations compatible with the bar length equations.

The geometric object given by the polynomial equations is called an *affine variety* [23]. A *deformation* is a continuous one-dimensional hyperpath **P**(*δ*)={**p**_{i}(*δ*)} through the configuration space that fulfils equations (2.1) for all *δ* with **P**(0) the initial configuration.

Our results are based on the geometric exploration of the full affine variety for finite values of *δ* and not only for the limit of infinitesimally small values of *δ* usually considered in infinitesimal rigidity theory [24].

We study the deformation of portions of the frameworks that correspond to a single unit cell or multiple translational unit cells and apply periodic boundary conditions. Because it is possible that an initially periodic infinite framework does not retain its symmetry during an imposed deformation, the restriction to one or a few translational unit cells represents a restrictive assumption. Note the discussion by Borcea & Streinu [25] on how periodic boundary conditions can induce non-genuine infinitesimal mechanisms.

## 3. Methodology

In general, the deformation behaviour of a framework need not be unique, that is, the affine variety of equation (2.1) can have dimensions greater than one. A continuum of deformation modes is possible, that is, multi-dimensional solutions of equations (2.1). However, Poisson's ratio is well defined and a purely *geometric* property only with respect to a single unique deformation path. If the deformation mode is not unique, Poisson's ratio can only be defined by identifying one deformation path from the continuum of solutions, and defining Poisson's ratio with respect to that unique mode. In this article, two approaches are used to reduce multi-dimensional continua of deformations to single deformation modes, namely by constraining symmetry or by requiring minimization of an angular energy functional.

### (a) Deformation with symmetry constraints

Most of the results of this article are obtained by enforcing that a framework retains some or all of its symmetries throughout the deformation (in addition to periodicity).

A periodic bar-and-joint framework can be built from a translational unit cell by appending copies translated by all possible integer multiples of two linearly independent lattice vectors **a**^{0} and **b**^{0} (figure 2). For periodic bar-and-joint frameworks, we assume that the deformation mode retains the periodicity of the structure, i.e. an extended or infinite fraction of the structure responds to an applied strain in the same way as a single translational unit cell (with lattice vectors **a**^{0} and **b**^{0}, figure 2).

A symmetric bar-and-joint framework *S* is mapped onto itself under the action *g*(*S*) for all elements *g*∈*G* of the symmetry group^{1} *G* of the tessellation/framework, *g*(*S*)=*S*.

If the solution space of a symmetric bar-and-joint framework is two- or more-dimensional, it can often be reduced to a unique deformation mode by enforcing that the deformed configurations *S*_{δ} maintain all or some of the symmetries of the original, i.e. *g*′(*S*_{δ})=*S*_{δ} for elements *g*′∈*G*′ of a subgroup *G*′ of *G*. Often, highly symmetric bar-and-joint frameworks are rigid when constraining most or all of the symmetries, have one or more subgroups with a unique deformation and have ambiguous deformation modes if too many of the symmetry constraints are relaxed [27].

Symmetry constraints may impose immediate constraints on Poisson's ratio. For example, if the framework retains hexagonal or square symmetry during the deformation, Poisson's ratio is *ν*(*δ*)=−1. Importantly, the existence of symmetries in the undeformed initial framework alone (without constraining the symmetries during the deformation) is not sufficient to determine the values of Poisson's ratio; see for example, the study of a system with cubic symmetry by Norris [28]. Similarly, the limits −1≤*ν*≤0.5 only apply to isotropic and homogeneous materials, excluding the frameworks studied here. For frameworks, an isotropic Poisson ratio of *ν*=−1 for all directions is possible (and realized e.g. in figure 1), but values *ν*<−1 or *ν*>1 can only be achieved for a specific direction and if the network is anisotropic (i.e. not of hexagonal or square symmetry).

We note that the question of uniqueness of the deformation and of its determinacy is related to rigidity theory [29], Laman's theorem for the rigidity of finite graphs [30], and to the generalization of Maxwell's rule [31] for the determinacy of periodic structures [32]. Note also the discussion of periodic auxetic deformations of unimode metamaterials constructed from rigid bars and pivots by Milton [33].

### (b) Minimization of harmonic angular spring energy functionals

Similar to the Kirkwood–Keating model described, for example, by Sahimi & Arbabi [34], we introduce a harmonic energy functional penalizing deviations of vertex edge angles from their value in the initial undeformed configuration
3.1with *P*(*δ*) the configuration when the imposed strain is *δ* and *P*(0) the initial configuration. 〈*jik*〉 represents a pair of edges {*j*,*i*} and {*i*,*k*} that are adjacent (part of the same polygon) at node *i*, and the sum over all such edge pairs of the considered unit cell; *α*_{〈jik〉}(*P*) is the angle formed by the edges {*j*,*i*} and {*i*,*k*} at vertex *i* in the configuration *P*. This potential is inspired by early work of Kirkwood [35] and Keating [36]. This energy functional is applied in §4*a* below.

### (c) Poisson's ratio of periodic bar-and-joint frameworks

Poisson's ratio characterizes the contraction or extension of a material in the horizontal direction to an applied uni-axial vertical deformation.^{2} For a rectangular sample of size *h*_{0}×*l*_{0}, Poisson's ratio *ν* is defined as the ratio of Cauchy strains *ν*=−((*h*−*h*_{0})/*h*_{0})/((*l*−*l*_{0})/*l*_{0}), where *l*_{0}×*h*_{0} is the size of the undeformed sample. For bar-and-joint frameworks with a rectangular translational unit cell that remains rectangular under strain *δ*, this definition is valid and yields the same result as the general definition given below.

For general periodic bar-and-joint frameworks, we define the horizontal strain by the construction in figure 2, of relevance to this study since frameworks with hexagonal unit cells are also considered, e.g. figure 1: **e**_{∥} is the direction of the given horizontal strain *ϵ*_{∥}, now called *δ*, here chosen in the direction of the lattice vector **a**, i.e. **e**_{∥}=**a**^{0}/|**a**^{0}|. Note, however, that the direction can be arbitrary and is not limited to lattice directions. Given the strain *δ* along **e**_{∥}, the deformations of the translational unit cell are and , where and are the projections of the initial vectors **a**^{0} and **b**^{0} onto **e**_{∥}; and are the projections of the finite deformations. The projections onto the perpendicular direction are implicit functions *a*_{⊥}(*δ*) and *b*_{⊥}(*δ*) of *δ* that result from equations (2.1). This leads to the following definition of Poisson's ratio:
3.2This is motivated by an experimental set-up with the structure fixed at the top and bottom layer and then stretched or compressed. Importantly, for rectangular translational unit cells, this definition constrains *b*_{∥}(*δ*)=0 (and hence prevents pure rotations), but allows for changes in the angle between **a**(*δ*) and **b**(*δ*), i.e. shear. For inhomogeneous or non-isotropic structures, *ν*(*δ*) depends on the applied strain direction **e**_{∥}. Our method allows for arbitrary **e**_{∥} that are not lattice vectors.

Given a strain *δ*, Poisson's ratio defined by equation (3.2) gives the ratio of lateral to orthogonal deformations with respect to the initial structure with *δ*=0. Commonly, this definition is used for infinitesimal strains , but it also applies to finite values *δ* (figures 1 and 3). We define the *instantaneous Poisson ratio* *ν*_{inst}(*δ*) as the Poisson ratio of the bar-and-joint framework already deformed by *δ* when a further infinitesimal strain *dδ* is applied (figures 3 and 4).

### (d) Numerical solution by the Newton scheme with singular value decomposition

Analytic solutions of equations (2.1) are, in general, not known, but roots of these equations, i.e. the node coordinates and lattice parameters *a*_{⊥}(*δ*) and *b*_{⊥}(*δ*), can be found numerically by iterative Newton methods [37], with an affine deformation as initial, non-permissible configuration. The symmetry constraints *g*(*S*)=*S* are easily integrated into this scheme, and each symmetry appears as an additional linear equation. Structures that have a multi-dimensional solution space (or infinitesimal phantom mechanisms) imply an under-determined Jacobian matrix *J* that cannot be inverted. Such degeneracy is dealt with by a singular value decomposition method that identifies the solution with smallest displacement of the coordinate values [27]. Within tolerances, it is numerically straightforward to decide whether the structure is rigid, has a unique deformation or if the solution space is multi-dimensional.

Solutions for finite *δ* are obtained by computing successive intermediate permissible configurations for incremental steps that sum to *δ*, with random perturbations added to the initial and intermediate non-permissible configurations. It is possible that, starting from the affine deformation of the framework by a factor (1+*δ*) in the strain direction, the Newton scheme does not converge to a solution of the edge equations—even if a solution exists and is unique. Bearing in mind that the goal of our study are continuous deformation paths from deformation 0 to a finite value *δ*, we achieve a finite strain *δ* by a number *n* of smaller increments *Δδ*. In each increment, the strain is increased by *Δδ* until *δ* is reached.

Numerically, the deformation mode that minimizes *E*[*P*(*δ*)] for a given value of *δ* is determined by random sampling using a Monte Carlo approach. As described above, the deformation with finite *δ* is obtained by small increments of size *Δδ*, starting at *δ*=0. A number *m* of possible solutions *P*_{i}(*δ*′+*Δδ*) of the edge equations (equation (2.1)) for strain *δ*′+*Δδ* are computed by adding small evenly distributed random numbers to all vertex coordinates of one of the *n* solutions *P*_{j}(*δ*′) with *j*=1,…,*n*, before application of the Newton scheme that evolves the vertex positions to fulfil the edge equations; for the first step *n*=1 with *P*_{1} equal to the initial undeformed configuration. For each solution, *P*_{i}(*δ*′+*Δδ*), the value of *E*[*P*_{i}] is computed. Out of the *m* solutions *P*_{i}(*δ*′+*Δδ*), those *s* solutions that have minimal energy value are kept as initial configurations for the next increment. Typical values used for the parameters in this article are *s*/*m*≈0.1, *m*≈250 and *Δδ*≈*δ*/100.

## 4. Results

We have applied the analysis described above to the 35 one- and two-uniform tessellations of the plane by regular or star polygons. These are tessellations with straight edges where all corners are vertices. These tessellations are enumerated by Grünbaum & Shephard [22], whose nomenclature is used below. One of the key results is that among these, there are two novel auxetic mechanisms, without enforcing any symmetry constraints (in one case, not even periodicity).^{3} These are the two-uniform tessellations (3^{6};3^{2}.4.3.4), here called *triangle–square wheels* or *TS wheels*, and (3^{6};3^{2}.6^{2}), here called *hexagonal wheels* or *H wheels* (figure 3). Both only allow compressions and their unique deformations to retain periodicity and hexagonal symmetry, owing to the edge equations only and without symmetry constraints, yielding *ν*=−1 for Poisson's ratio in any direction.

The deformation behaviour of the TS wheels, with maximal symmetry *p*6*mm*, is shown in figure 1*a*. We refer to it as TS wheels as it consists of a triangulated hexagonal wheel that rotates during the deformation, surrounded by a layer of alternating triangles and squares. The structure does not have any re-entrant elements. The deformation of the hexagonal wheels (H wheels) is shown in figure 1*b* with the name motivated by a rotating triangulated hexagon surrounded by a ring of hexagons that deform and develop re-entrant angles during the deformation.

The TS wheel structure has only one degree of freedom; the corresponding deformation is a shearing deformation of all squares, with all other parts rigid. This deformation mode is unique, apparently by virtue of the edge equations only without constraining neither symmetry nor periodicity.^{4} In contrast to the TS wheels, the deformation of the H wheels is only unique when the primitive unit cell symmetry is retained; larger unit cells show finite mechanisms and the minimization of angular energies results in much weaker auxetic behaviour.

Lakes [39] classifies auxetic materials with respect to three different features of the microstructure: rotational degrees of freedom, non-affine deformation kinematics or anisotropic structure. The two presented ones belong to the class of auxetic structures that rely on the chirality^{5} for the auxetic property and can be assigned to the ones with rotational degrees of freedom. The H wheels are similar to the proposed chiral honeycomb by Prall & Lakes [40], when the triangulated hexagons are replaced by circles. Also the geometry described by Milton [41] shares a feature with both the H wheels and the TS wheels, namely the universal property of this auxetic class that the lattice points are decorated by rigid objects that rotate.

The 35 tessellations also contain another known auxetic mechanism in *p*1 (i.e. without constraint symmetry), namely the (3.6.3.6) called the *trihexagonal tessellation* or *kagome structure*, already discussed in earlier studies[42–46]. Kapko *et al.* [43] have noted that the number of collapse mechanisms grows with the size of the unit cell and have also considered crystallographic symmetry constraints for the deformation of this tessellation.

The relevance of these results for engineered realizations made of homogeneous linear elastic material has been demonstrated by observation of *ν*<0 in specimens produced by selective electron beam melting [47] of the TS wheel structure, and corroborated by finite-element calculations [38]. The approximate agreement, in terms of Poisson's ratio, between the framework with stiff rods and flexible joints on the one hand and the linear-elastic homogeneous solid structure with rigid joints is somewhat surprising, but points towards the importance of geometric principles for the deformation behaviour of auxetic structures. Published research on bending- versus stretching-dominated behaviour of cellular materials supports the idea that for bending-dominated structures, an approximation by bar-and-joint frameworks is valid [48–50]. An akin approximation of truss structures by pin-jointed frameworks has been carried out by Wicks & Guest [51], who discuss actuation of a single bar in periodic square, triangular and kagome lattices.

The second principal result of our systematic exploration is the occurrence of auxetic behaviour upon finite deformations. There are a number of bar-and-joint frameworks that are not auxetic for small deformations, but become auxetic at finite values of *δ*. In these cases, the behaviour is clearly anisotropic as the structures are, in all cases, neither square nor hexagonal for the value of *δ* where *ν* or *ν*_{inst} vanish. As an example, the deformation of the two-uniform tessellation (3^{3}.4^{2};3^{2}.4.3.4)_{1} is shown in figure 3. Further uniform tessellations that are unique in *p*1 (i.e. without symmetry constraints) and become auxetic upon finite deformation are (3^{2}.4.3.4) [22], called the *snub square tessellation*, and (3^{3}.4^{2};3^{2}.4.3.4)_{2}, discussed by Grima *et al.* [21].

Finally, our third main result is the occurrence of large negative Poisson ratios when symmetry constraints are imposed. A collection of examples is compiled in figure 4, which includes the ‘kites’ structure [52] and the *inverted honeycombs* [53,54]. Among the investigated tessellations, there are two with negative Poisson ratios at *δ*=0, in non-hexagonal and non-square symmetry groups. These are the two-uniform (3^{2}.4.3.4;3.4.6.4) tessellation, with a constant Poisson ratio of −1 in both the primitive *cm* and the *p*2 space group (interestingly, the deformation modes in these two groups are different), and the two-uniform (3.4^{2}.6;3.6.3.6)_{2} tessellation with a Poisson ratio smaller than −1 in lattice direction [01] (figure 4*c*); this tessellation can only be compressed but not stretched. The deformation behaviour depends strongly on the direction **e**_{∥} of the applied strain and can change from non-auxetic to auxetic only at finite strain, and finally to *ν*<−1, even at *δ*=0. Without the complete analysis of the algebraic variety defined by equation (2.1) presented in this study, such complex deformation behaviour is not detectable.

We frequently observe that tessellations adopt a unique deformation mode *and* are auxetic if their symmetries are constrained to preserve glide planes, a symmetry element of plane group *cm* given in the International tables of crystallography [26]. If such constraints are imposed, the following tessellations become auxetic when a strain is applied in the direction of the primitive lattice vectors: (3.4.6.4), (4.8^{2}), (3^{6};4^{4}.4.12), (3^{3}.4^{2};3.4.6.4), (3.4^{2}.6;3.4.6.4), (3.12^{2}) and (6^{3}). Upon large enough deformations, the last one of these, (6^{3}), is congruent to the inverted honeycomb pattern (cf. figure 4*b*). Some of these constrained bar-and-joint frameworks have large negative Poisson ratios *ν*_{inst}<−1, making them promising candidates for applications as strain amplifiers [16].

Finally, figures 5 and 6 illustrate the obvious observation that a tessellation with ambiguous deformation modes for *p*1 (no constraint symmetries) may have unique deformation modes when subgroups of the full symmetry are used as constraints and that these deformation modes may be different for different subgroups.

### (a) Symmetry constraints versus energy minimization

From a physics perspective, a symmetry constraint may be expected as a secondary effect resulting from minimization of an energy functional, somewhat similar to the molecular bonds model of polyphenylacetylene described by Grima & Evans [42]. This motivates our second approach to the reduction of ambiguous deformation continua to single unique deformation modes, namely by picking the deformation path that minimizes a given energy functional, see also §3*b*. The energy functional could be the harmonic bond angle energy *E*[*P*] penalizing average deviations [*α*(*δ*)−*α*(*δ*=0)]^{2} of vertex angles *α* from the initial value in equation (3.1), or more complicated energy functionals.

A detailed study of the relationship between the deformations that minimize energy functionals such as *E*[*P*] and those that result by constraining the symmetry group is beyond the scope of this article. However, figures 7 and 8 elucidate this relationship, and its subtleties, for the deformation of the elongated kagome structure for two directions of applied strain.

Figure 7 shows the deformation modes and the corresponding Poisson ratio for strain perpendicular to [01]. With only translational symmetry constraints (either *p*1 or *c*1), the deformation is ambiguous, whereas for both subgroups *c*11*m* (including glide planes) and *c*2 (without glide planes), the deformation is unique; however, the modes for *c*11*m* and *c*2 are clearly different, as evidenced by the configurations as well as the Poisson ratio *ν*_{inst}(*δ*). It is then interesting to note that the deformation that minimizes *E*[*P*] without any symmetry constraints, except for pure translation *c*1,^{6} corresponds to one of these groups, namely *c*11*m*. This is an interesting observation considering that we have identified several tessellations with *ν*_{inst}<−1 if glide plane symmetries are constrained.

Interestingly, when the strain is applied in the orthogonal direction (i.e. perpendicular to [10]; figure 8), we again observe two distinct and unique deformation modes for *c*11*m* and *c*2. However, for that strain direction, the unconstrained energy-minimizing mode is the same as the *c*2 mode, in contrast to the situation above.

## 5. Discussion

Bar-and-joint frameworks provide possibly the simplest model to study deformations of cellular materials, as applied strain causes only geometric deformations without any resulting forces. Such models are not suitable to answer questions regarding forces such as ‘How stiff is the cellular structure?’, but can be used to address the simpler question ‘Can a cellular structure be deformed?’ and, if so, ‘What is the geometric deformation mode?’.

Because of the simplicity of the bar-and-joint frameworks and of the edge equations that govern their deformation, a systematic exploration of the deformations of framework geometries is possible, for example, by investigation of the large classes of periodic tessellations. A bar-and-joint framework can adopt one of three states: rigid (zero degrees of freedom), floppy with a unique deformation mode (one degree of freedom) or underdetermined (with two or more degrees of freedom). For the deformation modes of cellular matter, those frameworks with a single degree of freedom are most relevant.

When exploring the vast class of tessellations as models for models of auxetic frameworks, we have here shown that symmetry constraints are a useful method to reduce the degrees of freedom of a framework. In many cases, a subgroup of the full symmetry group of the undeformed tessellation can be found such that the deformation becomes unique. This paper may represent the first instance where the dependence of framework deformation on symmetry is systematically investigated. However, implicit assumptions about the symmetry preserved under strain are not uncommon; even the standard inverted honeycomb structure with its re-entrant elements, often depicted as the archetypal auxetic model, does not have a unique deformation mode, unless one assumes that rectangular lattice vectors (or equivalently glide plane symmetries) are maintained during the deformation.

While symmetry constraints were here largely used as a means to an end, namely to obtain structures with a single degree of freedom, it appears likely that preserved symmetries could also emerge as the result of physical forces. Figures 7 and 8 demonstrate some of the subtleties of this approach that require more in-depth investigation. Of particular interest, both theoretically and for applications such as strain amplification, would be an energy functional that, when minimized, leads to the preservation of glide plane symmetries; as has been demonstrated here, glide plane symmetries are preserved, as we have here demonstrated that glide plane symmetry constraints lead to particularly large negative values of Poisson's ratio, below −1, evidently in anisotropic structures such as those shown in figure 4.

It is an interesting question to what degree the force-less deformations of a bar-and-joint framework and those of a linear-elastic body relate to one another. Clearly, the structure of the equations underlying the two processes are very different [55,56]. Nevertheless, if the deformation mode of a bar-and-joint framework is unique, one may expect that geometry is the fundamental determinant of the deformation behaviour and that the resulting deformation modes are robust to changes of the microscopic physics. One may then expect that the deformation modes of the framework are similar to those of a linear-elastic realization of the same structure (without freely jointed hinges, and with edges replaced by linear-elastic beams). The analysis by Mitschke *et al.* [38] of Poisson's ratio for the TS wheel structure appears to conform to that expectation; for both a realization of this structure as a linear-elastic cellular solid by selective ion beam melting (*ν*≈−0.75) and for the bar-and-joint framework (*ν*=−1), negative Poisson ratios of similar magnitude were found. Note also the relationship given by Blumenfeld & Edwards [55] between local structural objects (so-called *auxetons*) and global auxetic deformations.

Knowledge of the symmetry-constrained deformation modes, presented here for both infinitesimal and for finite strains, may be useful structural data for the development of rigidity criteria for periodic structures [32], similar to the celebrated Maxwell counting rule for an isostaticity condition [31]. Furthermore, the auxetic framework configurations identified here can be usefully applied for topological optimization algorithms that optimize a given property (here e.g. Poisson's ratio) under given constraints [57].

The analysis described here specifically addresses planar and symmetric structures. It is however of relevance to two obvious generalizations, namely to planar disordered structures and to three-dimensional symmetric structures. Auxetic behaviour of disordered planar structures has been discussed theoretically by Blumenfeld & Edwards [55], and has also been observed experimentally, at least locally, by Franke & Magerle [11] in elastomeric polypropylene films. Franke & Magerle [11] have identified the mechanism that leads to the locally auxetic behaviour in these semi-crystalline films as an angular constraint between crystalline lamellae that, in a rough approximation, correspond to load-bearing bars of a bar-and-joint framework (but are, however, not of fixed length). The fixed angles between lamellae can be loosely interpreted as a local symmetry constraint, hence relating to the work presented in this article.

The generalization to three-dimensional tessellations and networks is numerically straightforward (in particular, the complexity is polynomial in the number of joint coordinates and constraining equations). Our systematic approach of finding auxetic mechanisms by searching existing structure archives has proved fruitful in two dimensions. The identification of a large number of inherently three-dimensional auxetic mechanisms, without planar equivalent, is in our opinion more likely to be achieved by a systematic search of the large number of spatial tessellation and network structures than by conceptual generalization of planar models.

## Acknowledgements

We acknowledge financial support by the German Science Foundation (DFG) through the Engineering of Advanced Materials Cluster of Excellence (EAM) at the Friedrich-Alexander University Erlangen-Nürnberg (FAU) and the research group Geometry and Physics of Spatial Random Systems under grant nos. SCHR1148/3-1 and ME1361/12-1.

## Footnotes

↵1 Symbols for the different

*plane groups*are taken from the*International tables of crystallography, volume A1*[26]. Note that for certain subgroups, non-conventional settings, e.g.*c*2,*c*11*m*, are used [26] with the corresponding cell transformation.↵2 Note that this deformation does

*not*necessarily correspond to a uni-axial strain. In contrast to the situation typically studied here, the term uni-axial strain means a uni-axially stressed material exhibiting no vertical strain, identical to a vanishing Poisson ratio with respect to the axial direction.↵3 A preliminary account of one of these, TS wheels, has been given by Mitschke

*et al.*[38].↵4 We have studied the deformation behaviour for systems of

*N*×*N*translational unit cells with*N*=1,3,6, with periodic boundary conditions. From the observation that even the deformation for*N*=6 maintains the internal (unconstrained) periodicity, we conclude that the edge equations alone constrain periodicity. It is noteworthy that this tiling is the only auxetic one within the 31 one- and two-uniform tilings that possess this feature.↵5 In the initial state

*δ*=0, both the TS wheels and the H wheels are in a degenerate singular rigid position, i.e. being on a singular point in configuration space leading to rigidity that can immediately be removed by any small perturbation breaking the mirror symmetry resulting in chiral symmetry.↵6 The symmetry group

*p*1 with lattice vectors**a**and**b**has no symmetries except for periodicity. This applies equally to the ‘centred’ group*c*1 with lattice vectors**a**and**b**; however,*c*1 is periodic under translations by (**a**−**b**)/2 and (**a**+**b**)/2.

- Received September 5, 2012.
- Accepted October 11, 2012.

- © 2012 The Author(s) Published by the Royal Society. All rights reserved.