Geometry: the Leading Parameter for the Poisson's Ratio of Bending-Dominated Cellular Solids

Accepted Manuscript

Geometry: The leading parameter for the Poisson’s ratio of bending-dominated cellular solids

Holger Mitschke, Fabian Schury, Klaus Mecke, Fabian Wein, Michael Stingl, Gerd E. Schroder-Turk¨

PII: S0020-7683(16)30142-1 DOI: 10.1016/j.ijsolstr.2016.06.027 Reference: SAS 9206

To appear in: International Journal of Solids and Structures

Received date: 14 December 2015 Accepted date: 21 June 2016

Please cite this article as: Holger Mitschke, Fabian Schury, Klaus Mecke, Fabian Wein, Michael Stingl, Gerd E. Schroder-Turk,¨ Geometry: The leading parameter for the Poisson’s ratio of bending-dominated cellular solids, International Journal of Solids and Structures (2016), doi: 10.1016/j.ijsolstr.2016.06.027

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Highlights • Broad comparison of the deformation of cellular solids with mechanisms of frameworks.

• Eﬀective Poisson’s ratios agree within the regime of bending-dominated cellular solids.

• Geometric framework model emphasizes the importance of morphology for elastic properties.

• Determination of inﬁnitesimal framework mechanisms oﬀers versatile microstructure design tool.

ACCEPTED MANUSCRIPT

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Geometry: The leading parameter for the Poisson’s ratio of bending-dominated cellular solids

Holger Mitschkea, Fabian Schuryb, Klaus Meckea, Fabian Weinb, Michael Stinglb, Gerd E. Schröder-Turkc aTheoretische Physik I, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7B, 91058 Erlangen, Germany bDepartment Mathematik, Applied Mathematics 2, Friedrich-Alexander-Universität Erlangen-Nürnberg cMurdoch University, School of Engineering & Information Technology, Mathematics & Statistics, Murdoch WA 6150, Australia

Abstract Control over the deformation behaviour that a cellular structure shows in response to imposed external forces is a requirement for the effective design of mechanical metamaterials, in particular those with negative Poisson’s ratio. This article sheds light on the old question of the relationship between geometric microstructure and mechanical response, by comparison of the deformation properties of bar-and-joint-frameworks with those of their realisation as a cellular solid made from linear-elastic material. For ordered planar tessellation models, we find a classification in terms of the number of degrees of freedom of the framework model: first, in cases where the geometry uniquely prescribes a single deformation mode of the framework model, the mechanical deformation and Poisson’s ratio of the linearly-elastic cellular solid closely follow those of the unique deformation mode; the result is a bending- dominated deformation with negligible dependence of the effective Poisson’s ratio on the underlying material’s Poisson’s ratio and small values of the effective Young’s modulus. Second, in the case of rigid structures or when geometric degeneracy prevents the bending- dominated deformation mode, the effective Poisson’s ratio is material-dependent and the ˜ Young’s modulus Ecs large. All analysed structures of this type have positive values of ˜ the Poisson’s ratio and large values of Ecs. Third, in the case, where the framework has multiple deformation modes, geometry alone does not suffice to determine the mechanical deformation. These results clarify the relationship between mechanical properties of a linear- elastic cellular solid and its corresponding bar-and-joint framework abstraction. They also raise the question if, in essence, auxetic behaviour is restricted to the geometry-guided class of bending-dominated structures corresponding to unique mechanisms, with inherently low values of the Young’s modulus. Keywords:ACCEPTEDcellular structures, homogenisation, MANUSCRIPT floppy frameworks, auxeticity, mechanical metamaterial

Email address: [email protected] (Holger Mitschke) Preprint submitted to Int. J. Solids Struct. 22nd June 2016 ACCEPTED MANUSCRIPT

1. Introduction Geometry, topology and spatial structure play a crucial role for the deformation behaviour and mechanical properties of cellular materials. This observation underlies the very concept of mechanical meta-materials (Zheng et al., 2014; Florijn et al., 2014; Bückmann et al., 2014; Kadic et al., 2012; Bertoldi et al., 2010; Overvelde et al., 2012), has been recognised by classic work summarised in the book of Gibson and Ashby (1997) or Gibson et al. (2010) and is relevant for the design of actuating or buckling structures (Wicks and Guest, 2004; Guiducci et al., 2015). Cellular solids are two-phase materials, consisting of a solid phase occupying a volume fraction φ of the total space and its complement, the void phase (φ is synonymous for relative density ρ˜ ρs). Many incarnations of cellular materials are possible, including closed-cell foams where the void phase consists of individual hollow cavities, open-cell foams where the solid phase resembles~ a spatial network, etc. Typically, one distinguishes cellular solids from other porous materials by the volume fraction, which is often considered φ 30% (Gibson and Ashby, 1997). While many realisations of cellular solids have a disordered spatial structure, cellular solids with an ordered structure also occur, e.g. honeycombs. The≲ mechanical response of cellular solids to imposed strains depends on a multitude of parameters, including those describing geometric aspects of the spatial structure but also those describing properties of the material constituting the solid phase. This concept is formalised by so-called effective mechanical properties, that is, the mechanical properties of samples of the material with dimension much larger than the microstructure (Torquato, 2002; Milton, 2002). The significant dependence on spatial structure and geometry is recognised in the scaling laws of mechanical properties with volume fraction φ, and their dependence on the type and topology of the spatial structure, see the books by Gibson and Ashby (1997); Gibson et al. (2010); Ashby et al. (2000) or the publication Ashby (2006); note the cross-over behaviour from these scaling laws to the predictions of effective medium theory for low-porosity porous materials (Nachtrab et al., 2011). Similarly, the essential distinction of ‘bending-dominated’ or ‘stretching- dominated’ behaviour reflects differences in spatial structure of the material (Ashby, 2006). Other examples for the dependence on geometry are the relevance on details of the structure, e.g. for low-density open-cell foams the cross-sectional shape (Jang et al., 2008). Despite the significant role of geometry for the effective mechanical properties, the properties of the constituent material also affect the effective mechanical response. We here denote the effective ˜ Poisson’s ratio as ν˜cs and the effective Young’s modulus as Ecs whereas the linear-elastic solid material properties of the solid phase are denoted as νs and Es. The indices ‘s’ and ‘cs’ are the initials of solid and cellular solid. The constituent solid material is considered isotropic and homogeneous and hence, characterised by these two material constants. The effective properties need not to be isotropic with the possibility of different in-plane values 12 ACCEPTED21 1 2 MANUSCRIPT ˜ ˜ 1 ν˜cs , ν˜cs , Ecs , and Ecs for different directions of the applied strain. In general, the ( ) ( ) ( ) ( ) ˜ response of the effective properties ν˜cs and Ecs to changes of the material properties νs and

1Note that for orthotropic symmetry, e.g. honeycomb family in Fig. 3, the number of independent in-plane 12 ˜ 1 21 ˜ 2 moduli is three with the reciprocal relation ν˜cs Ecs ν˜cs Ecs (Gibson and Ashby, 1997). ( ) ( ) ( ) ( ) 3= ACCEPTED MANUSCRIPT

Es is not a priori a simple relationship — as we explore below. In particular for biological cellular materials, an additional materials dependence may arise if the constituent material is not isotropic which, for example, is known for wood (Cave, 1968). The structure of cellular materials where the solid phase resembles a network-like structure with edges that are joined at common vertices can be represented by bar-and-joint framework models, also known as bar/strut framework models (Torquato and Stillinger, 2010, Connelly and Whiteley, 1996; therein considered as a subclass of tensegrity frameworks without cables and extendable strut members), pin-jointed frameworks (Pellegrino and Calladine, 1986), (pin-jointed) trusses (Hutchinson and Fleck, 2005) or skeletal structures (Laman, 1970). In this publication bar-and-joint framework models are simply referred to as frameworks. These are composed of stiff edges of fixed length that pivot freely at common vertices, see Fig. 1 (a). All vertex displacements that preserve all edge lengths are allowed deformation modes of these structures. Various aspects of these models have been studied, in particular for periodic and/or symmetric systems (Kangwai et al., 1999; Guest and Hutchinson, 2003; Connelly et al., 2009; Ross et al., 2011; Treacy et al., 2014), and disordered systems (Plischke, 2007; Moukarzel, 2012; Thorpe et al., 2002). Constraint counting rules such as the famous Maxwell (1864) counting rule can give conditions for either stiff or floppy modes, with extensions that take self-stresses into account (Calladine, 1978) or periodic and symmetric systems (Borcea and Streinu, 2010; Ross et al., 2011; Guest and Fowler, 2014; Borcea and Streinu, 2013; Mitschke et al., 2013b). The rigidity, i.e., lack or presence of deformation modes, also relates to questions of infinitesimal rigidity (Roth, 1981; Calladine and Pellegrino, 1991; Salerno, 1992; Connelly and Servatius, 1994; Graver, 2001; Garcea et al., 2005). Infinitesimal rigidity is a sufficient but not necessary condition for rigidity. Rigid frameworks can have, in contrast to floppy frameworks, infinitesimal flexes. For the prediction of finite mechanisms symmetry analysis is useful (Kangwai and Guest, 1999; Guest and Fowler, 2007; Ross et al., 2011). The stiffness of disordered variants of the bar-and-joint models relates to questions of rigidity percolation (Obukhov, 1995; Chubynsky and Thorpe, 2007; Mao et al., 2013; Lubensky et al., 2015). The relationship between deformations of bar-and-joint models and those of cellular solids is the focus of this article. As we show below, a close relationship between unique deformations of the bar-and-joint models and the effective deformation of the corresponding cellular solid exists, particularly w.r.t. Poisson’s ratios. However, a priori, there are notable differences between the two models: first, for bar-and-joint models no notion of forces exists; unless rigid, the system reacts to an imposed strain with a force-free deformation (i.e. vanishing Young’s modulus). Second, in contrast to linear-elastic continuum theory, the vertex displacements are in general non-affine transformations, imposed the spatial structure and topology. (In this context note that the mathematical structure of the equations for effective deformation behaviourACCEPTED is different to linear-elastic theory MANUSCRIPT of homogeneous solids (Blumenfeld and Edwards, 2012).) Beam models are in some sense intermediate to the two models discussed above, the cellular solid and the bar-and-joint framework. In beam models — that may be loosely viewed as force-loaded versions of framework models — edges or struts are no longer rigid, but allow for bending, stretching and in 3D possibly torsion. The beams can be modelled by 4 ACCEPTED MANUSCRIPT

(a) Force-free beam hinging in a ﬂoppy framework ε

θ = 120◦ θ = 150◦ (b) Linear-elastic beam bending in a cellular solid σ

θ = 120◦ θ = 120◦

Figure 1: Bar-and-joint framework models and cellular solids: (a) A bar-and-joint framework consists of stiff struts that can pivot freely at the vertices and angles of adjacent struts can change, e.g. here θ; all deformations are force-free mechanisms, in other words with an effective Young’s modulus E˜ 0; the deformation can be invoked by an applied strain ε at the boundaries; a deformation is only possible, if the framework is floppy, otherwise it is rigid and no mechanism exists. (b) A cellular solid is a two-phase material,= consisting of a solid phase with given linear-elastic material properties, and a hollow phase. The deformations are prescribed by linear-elastic theory; imposed stresses σ result in bending (as here depicted), shearing, twisting, stretching or other deformation modes of structural elements which determine the magnitude of the ˜ effective Young’s modulus Ecs.

Bernoulli beams or Timoshenko beams leading to a commonly used finite element approach to calculate mechanical properties (ABAQUS, 2011; Jang et al., 2010). These models provide intuitive insight into ‘bending-dominated’ and ‘stretching-dominated’ deformation regimes. The beam model corresponding to a stiff framework can still deform by stretching (length extension) of the beams; this is called the stretching-dominated deformation — of great importance for structural engineering, e.g. utilised in trusses (Dewdney, 1991; Lewandoski, 2004; Rinke and Kotnik, 2010), tensile-only structures (Berger, 2005), tensegrities (Connelly and Back, 1998; Guest, 2011) and as well in micro-structured material designs (Cheung and Gershenfeld, 2013; Hutchinson and Fleck, 2006). The beam model corresponding to a flexible bar-and-joint framework may be rigid due to enforced constant angles at the vertices; however, by beam bending (Fig. 1 (b)) a motion reminiscent of the force-free displacement mode of the bar-and-joint framework model exists (bending-dominated). As we show in this article this intuitive relation between rigid structures and stretching-dominated modes on the one hand and flexible structures and bending-dominated modes on the other hand, is also observed for the deformation of linear-elastic cellular solids. In the context of this article, it is important to distinguish between two different numerical finite elementACCEPTED approaches, those for beam models MANUSCRIPT and those for two-phase cellular solids. Beam models represent the structure as a graph or network of edges connected at vertices; mechanical stiffness then results from using deformable edge models with mechanical stiffness, such as Bernoulli or Timoshenko beams. Also — often less prominently discussed — assumptions are required for the mechanical properties of the vertices, usually but not always enforcing

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constant angles between emanating edges (Jang et al., 2008; Liu and Quek, 2013). By contrast, finite element calculations for cellular solids contain no notion of edges or vertices. Rather the solid phase and the void phase are spatially tessellated (by e.g. by tetrahedral elements (Strek et al., 2008; Zienkiewicz and Taylor, 2005; Hughes, 2012) or a cubic voxel grid (Kapfer et al., 2011; Watanabe et al., 2012)), with corresponding values of the material coefficients assigned to these elements. Especially, in the low-density limit φ 0 of open-cell cellular solids, there is a common expectation that the two approaches are related. The findings of this article indirectly support this expectation, of a relationship→ between beam models or bar-and-joint frameworks on the one hand and the cellular solid on the other. We here compare the force-free deformation modes (also called mechanisms) of bar-and-joint framework models with the deformation modes of the corresponding cellular solids, in terms of Poisson’s ratios. The particular class of framework geometries are based on regular tiling models of the plane. Our key finding is a criterion to determine if the properties of the cellular solid reflect those of the bar-and-joint framework model. We find good agreement when the bar-and-joint framework model has a unique mechanism and is not geometrically degenerate (see below). For all cases discussed, this leads to bending dominated behaviour with low stiffness. These geometry-related questions are particularly relevant for auxetic materials (Greaves G. N. et al., 2011; Milton, 1992; Grima and E., 2000; Grima et al., 2005; Chetcuti et al., 2014; Bertoldi et al., 2010; Franke and Magerle, 2011; Dirrenberger et al., 2011). For these materials, characterised by negative values of the Poisson’s ratio, the essential role of microstructural geometry has been recognised. The results of this article further support this intimate relationship between geometry and mechanics. Our findings suggest the conjecture that auxetic deformations are only possible in bending-dominated cellular solids whose behaviour is largely reflecting a force-free unique bar-and-joint framework deformation mode. If this was generally true, it would imply that auxetic deformations always correspond to soft materials with low values of the Young’s modulus. A proposal by Rothenburg et al. (1991) suggests to use pistons as beams which can be more easily extended than bended (i.e. bending compliance of the “beams” is lower than their stretching compliance) lead to auxetic behaviour for rigid frameworks (see also Warren and Kraynik, 1987).

2. Bar-and-joint framework models and linear-elastic cellular solids Bar-and-joint frameworks. A bar-and-joint framework (here planar) consists of a graph embedded in the Euclidean plane, that is, a set of nodes K (or junctions or vertices) connected by a set of edges E. Henceforth the term ‘bar-and-joint’ is for brevity omitted. Every node i corresponds to a joint, with coordinates pi xi, yi . Every edge e i, j corresponds to a rigid bar of ﬁxed length l i,j . The edges are considered rigid unbendable bars ACCEPTED MANUSCRIPT= ( ) = { } of ﬁxed length that pivot freely and force-free{ } at the vertices. Any vertex displacement that leaves all edge lengths unchanged is permissible, and corresponds to a force-free deformation of the framework model. These permissible vertex displacements are characterised by being solutions to the edge equations 2 2 d pi, pj, a, b, γ l ij 0 i, j E, (1) 6 ( ) − { } = ∀{ } ∈ ACCEPTED MANUSCRIPT

where d , , a, b, γ is the distance function between two points, potentially taking periodicity into account (given by the lattice parameters a, b and γ). A one-dimensional hyperpath P δ (p⋅ i⋅ δ is) called a deformation or mechanism of the framework, if the edge equations are fulfilled for every value of the control parameter δ (this control parameter could be e.g. the( ) applied= { ( strain)} in a given direction or, as in Fig. 3, an angle). We here specifically consider periodic framework models, that is, edges and nodes are specified for a translational unit cell; the infinite framework is then obtained by repeated application of translations corresponding to the crystallographic lattice vectors. The periodicity is characterised by specifying the length of two lattice vectors, a and b, and the angle, γ, between them. While we require that the framework remains periodic throughout any deformation, we allow the two lattice parameters a and b and the angle γ to change. These three variables are hence additional variables in the edge equations. Analysis of the edge equations can reveal how many different mechanisms a framework possesses. We here choose the Newton-Raphson scheme described in Mitschke et al. (2013a) to identify mechanisms. This scheme finds the roots of the left-hand side of Eq. (1) by a Newton-Raphson algorithm and uses singular value decomposition to treat underdetermined matrices and determine their rank. This rank corresponds to the number of degrees of freedom of the framework, providing a simple way to characterise the deformation behaviour of the system. We here distinguish three types of frameworks: First, a framework may not allow for any deformations, i.e., be rigid. Second, the edge equations of a framework may have a single unique deformation mode. The system only has a single way to respond to an applied strain or otherwise imposed deformation. Third, the edge equations may permit several deformation modes. The response of the network to an applied strain is not uniquely defined by the edge equations. The focus of this article is on frameworks with a single unique mechanism. Poisson’s ratio is, for frameworks, only defined when there is a single unique mechanism. Specifically, for periodic networks considered here, we compute Poisson’s ratio by imposing a strain in the direction of a given lattice parameter, i.e. changing e.g. a 1 δ a. The Poisson’s ratio then is defined using the resulting contraction in the direction perpendicular to a. For all structures analysed here, the unique mechanisms are either of hexagonal→ ( + symmetry,) in which case the Poisson’s ratio is ν˜ff 1 (see also the discussion in Mitschke et al. (2013b)), or of rectangular symmetry (which means that γ π 2 does not change throughout the course of the deformation). In these latter= − cases, the vector perpendicular to lattice vector a is the second lattice vector b, and Poisson’s ratio is= defined~ as b δ b δ 1 ν˜ 12 δ , (2) ff a′ δ a δ 21 ( ) ν˜ff δ ′( )~ ( ) ( ) ( ) = − = 2 where theACCEPTED index ‘ff’ is the abbreviation of floppy (MANUSCRIPT)~ ( framework) .( )The derivatives a δ and b δ are determined numerically by first-order difference scheme using the result of′ the computed′ changes of the lattice vectors by the Newton-Raphson solution of the edge equations.( ) ( )

2 Indices used by authors in previous publications νinst and νSS are consolidated into ν˜ﬀ and νLE and νFEM into ν˜cs (Mitschke et al., 2011, 2013a). 7 ACCEPTED MANUSCRIPT

(a) (b) (c) γ ~a

θ = 30 θ = 0 θ = 30◦ ◦ ◦ −

Figure 2: Geometric degeneracy of bar-and-joint framework mechanisms: Snapshots of the unique deformation behaviour P θ of the periodic honeycomb family (under the assumption of rectangular symmetry, i.e. constant γ 90°). When horizontal compression is applied to members of this family with θ 0°, as illustrated in (a) and (c),( the) bar-and-joint framework’s response is unambiguous, following the deformation P θ with θ increasing.= The response of the member with θ 0°, to applied horizontal compressive strain≠ is ambiguous (∆θ 0 or ∆θ 0), somewhat akin to an unstable saddle point, see (b). ( ) S S = > < The deformation behaviour of cellular solids can become complicated by ‘geometrically degenerate’ situations, of relevance to the results of this article. They are cases where despite the presence of a unique deformation mechanism of the framework, the deformation of the cellular solid differs from that of the framework. Fig. 2 illustrates the problem, for a family of deformations of the honeycomb lattice. The figure shows three snap-shots of a mechanism of the honeycomb lattice, that can be parameterised e.g. through the angle θ, see also Fig. 3. (Note, technically speaking we have forced the framework to maintain rectangular symmetry; otherwise the mechanism is not unique.) Even though the mechanism represents a unique solution P θ , there is a degenerate situation where the application of horizontal strain does not lead to the unique deformation described by the mechanism. For θ 0 (conventional honeycomb,( Fig.) 2(a)) and re-entrant honeycomb (Fig. 2(c)), an application of strain directly leads to the expected mechanism, with θ increasing upon horizontal compression.> The situation in Fig. 2(b) (θ 0) is different. Application of horizontal strain leads to an ambiguity in the direction into which the vertices alongS S the horizontal straight lines move. In the context of linear-elastic cellular= materials, these degenerate cases lead to cases where the mechanical behaviour is solely determined by the stretching contribution of the horizontal straight bars. Throughout this article, the term ‘geometry uniquely prescribes the deformation behaviour’ means that the framework (i) only has a single unique deformation mode and (ii) is not in a degenerate situation. This situation relates to bifurcation points of kinematic paths, as discussed for finite frameworks in Kumar and Pellegrino (2000) and Yuan et al. (2012). More severe cases of geometric degeneracy when joints are in identical position (e.g. in Fig. 2 forACCEPTEDθ 90° and θ 90°) occur as limiting MANUSCRIPT frameworks for the below given examples. In these cases typically pore spaces vanish which result in a different topology of the structure. More details= are given= for− each discussed example.

Cellular solids. Cellular solids are, as stated above, two-phase models where an indicator function χ p speciﬁes whether a point p in the translational unit cell belongs to the solid

( ) 8 ACCEPTED MANUSCRIPT

(a) θ(δ) (b) Beam hinging in the honeycomb framework

90◦ 30◦ 0 30 90 − − ◦ ◦ ◦ ~a Beam theory with 4 bending only Floppy framework l θ 2 h ) FEM of cellular solid δ

( θ = 15◦ θ = 30◦ θ =45◦ 0 (12) ˜ ν Beam theory with 2 − bending, shearing & stretching 4 ~ − 1 b 2 (c) Beam bending in the honeycomb cellular solid at δ = 0

t 15 l 10 θ ) δ ( 5 h

(21) θ = 30◦ θ = 30◦ θ =30◦ ˜

ν 0 5 − 10 − 0.6 0.4 0.2 0 0.2 − − − Imposed change δ of ~b(θ = 30◦)

Figure 3: (Colour online) Floppy frameworks and linear-elastic cellular solids as two perspectives for mechanical deformations of cellular materials: the two plots in (a) show the Poisson’s ratios, ν˜ 12 and ν˜ 21 , for strain applied in the two lattice directions a and b, for a cellular material whose structure( is) given( by) the periodic honeycomb family generated by varying the angle θ from 90° to 90° as illustrated ⃗ in (b). The parameter δ denotes the length change of the⃗ lattice vector b w.r.t. the starting configuration, here θ 30°; δ relates to the angle θ with θ δ arcsin δ hδ 1 ; the conventional− (convex) honeycombs 2 l 2 ⃗ correspond to θ 0°, and the re-entrant honeycombs to θ 0°. The solid curves are analytic predictions from beam theory= taking either only bending into( account) = (red, + Gibson+ and Ashby (1997, eqs. (4.13) and (4.14))) or bending, shearing> and stretching (blue, Gibson and Ashby< (1997, eqs. (4B.8) and (4B.17))). Finite element method (FEM) data ν˜cs ( ) for the linear-elastic cellular solid is well described by the latter function whereas the bar-and-joint framework estimates ( ) coincide with the former. For δ not too close to the degenerate case δ θ 0° 0.2, geometry prescribes the mechanical deformation leading to similar values of ν˜cs and ν˜ff. Material parameters for FEM are νs 0.2, t l 0.1, and h l 2, which gives for φ θ 0° 0.2 using eq. (4.1a)( = of) Gibson= − and Ashby (1997). = ~ = ~ = ( = ) = phase (ACCEPTEDχ p 1) or if it belongs to the pore MANUSCRIPT domain (χ p 0). Given a framework, the corresponding cellular solid is constructed by inflating the edges to cylindrical( ) = (in 3D, or rectangular in 2D) elements of a( given) = radius (in 3D, or width in 2D). Mathematically, χ p is 1 if the distance from p to the nearest edge (taking the periodicity into account) is less than the specified radius. The solid volume fraction of this cellular solid ( ) 9 ACCEPTED MANUSCRIPT

1 is φ V χ p dp where the integral extends over the whole translational unit cell and V is the area of the unit cell. Effective= ∫ ( linear) elastic properties of the cellular solids are determined by a standard finite element method (FEM) where we model the structure by a so-called ersatz material approach (Bendsøe, 1989) on a regular mesh. In the core of this approach we multiply the local material properties for each finite element by a local pseudo density value. When a strut of the structure intersects the finite element we assign one to the pseudo density (solid), otherwise the local pseudo density variable is set to 10 6 to scale down the material to void. This allows to evaluate the elastic behavior of arbitrary− complex structures on a sufficiently fine regular mesh. Under the assumption of an infinitely often repeated periodic structure the effective macroscopic structural properties can be expressed by a linear elastic stiffness tensor found by asymptotic homogenization (see Allaire (2002); Bendsøe and Sigmund (2003) and references therein). Numerically the finite element analysis is performed for a rectangular base cell with periodic boundary conditions for the displacement. The system is solved for three test H strains and with the solutions all coefficients of the homogenized stiffness tensor Cij can be directly computed. Under the assumption of an orthotropic homogenized tensor, the ˜ effective Poisson’s ratios ν˜cs and Young’s moduli Ecs can be directly computed out of the tensor coefficients (Altenbach and Altenbach, 1994). See also Sigmund (1994).

3. Results A priori the Poisson’s ratio, and other deformation properties, of bar-and-joint frameworks are different to those of cellular solids. This section provides data demonstrating the conditions under which the two models give similar deformation behaviour. This is the case when the deformation mode is dictated by geometry (that is the bar-and-joint framework model has a single unique solution). In those cases, the ‘linear-elastic’ Poisson’s ratio of the linear-elastic cellular solid closely follows the ‘geometric’ Poisson’s ratio of the bar-and-joint framework. To avoid confusion, we clarify the notation in Figs. 3 to 6 . The Poisson’s ratio ν˜ff of a bar-and-joint framework is the infinitesimal/instantaneous Poisson’s ratio, that is, the ratio of relative lateral to perpendicular extension when a small infinitesimal strain in perpendicular direction is applied. The Poisson’s ratio ν˜cs of the cellular solid is the linear-elastic Poisson’s ratio. It is obtained for a cellular solid which is in equilibrium (i.e., exhibits no stresses) in the configuration corresponding to the bar-and-joint framework deformation with value δ, by means of mathematical homogenization. We first consider the deformations of the well-known honeycomb family (Gibson et al., 1982) comprising both the conventional (convex) honeycomb and the re-entrant (non-convex) honeycombACCEPTED with auxetic behaviour (Fig. 3).MANUSCRIPT As a bar-and-joint framework this deformation corresponds to a unique deformation mode described by the angle θ or strain δ as parameter (this deformation is unique when constraining spatial periodicity and a rectangular unit cell). Given rectangular symmetry, two Poisson’s ratios ν˜ 12 and ν˜ 21 are defined corresponding to strain imposed along the two lattice directions 10( )and 01( ,) respectively. For the bar- 21 and-joint framework model an applied vertical strain leads to a Poisson’s ratio ν˜ff which [ ] [ ] ( ) 10 ACCEPTED MANUSCRIPT

is a smooth function of θ or δ exhibiting the expected transition from negative to positive values at θ 0. A strain in the orthogonal (i.e. horizontal) direction leads to a more interesting behaviour with a divergence at θ 0; these divergences correspond to degenerate situation where= the non-vertical edges form infinite straight, horizontal lines. This situation is degenerate, in the sense that the response of= the material to an applied horizontal strain is ambiguous. As a concrete example illustrating the key result of our study, Fig. 3 shows that the 12 21 corresponding Poisson’s ratios ν˜cs δ and ν˜cs δ of the linear-elastic cellular solid closely ( ) ( ) 12 21 follows the floppy bar-and-joint framework values ν˜ff δ and ν˜ff δ , except near the degenerate case at δ 0.2. Put differently,( ) the( ) mechanical( ) response( ) of the force-loaded system (cellular solid) reflects the bar-and-joint framework( model) in all( situations) where the deformation is determined= − by the geometry. The deformation in Fig. 3 induced by a strain δ can be conveniently parameterised by the tilt angle θ defined in subfigure (b) and (c), and used by the analytic work of Gibson and Ashby (1997). Fig. 3 shows finite element method (FEM) data for cellular solid realisation (for numerical detail see Section 2) as well as the analytic predictions by Gibson and Ashby (1997) for beam theory with considering only bending (red lines), and one incorporating bending, shearing and stretching (blue lines). FEM is everywhere well described by the full 12 21 beam theory, even near the degenerate case, both for ν˜cs and ν˜cs . Fig. 3 also shows data for the bar-and-joint framework model (red symbols), essentially( ) ( representing) geometry only, which agrees well with the FEM data and the analytic models for θ 15° (sufficiently far from the degenerate case). Note further, the close agreement with beam theory when only bending contributions are considered, even for θ 15°. ≳ Fig. 4 provides further clarification under what circumstances the strain-induced deformation of a linear-elastic solid is well describedS S ≲ by the bar-and-joint framework model, and hence, determined by geometry alone. The figure shows data for the so-called TS-wheel structure family denoted in Grünbaum and Shephard (1987) as (36; 32.4.3.4), for different values of the free parameter δ, different values of the material Poisson’s ratio νs of the constituent material, and as function of the solid volume fraction φ. As a bar-and-joint framework model this structure is auxetic with ν˜ff 1 independent of δ. Clearly, for the two degenerate cases (δ 0 where the bar-and-joint framework deformation mode is ambiguous, and δ δmin where the structure= − represents a stretching-dominated cellular solid) the linear-elastic cellular solid= Poisson’s ratio is very different, ν˜cs 0, from ν˜ff 1, see the blue functions= in Fig. 4; for φ 0 the Gurtner and Durand (2009) value of 1 3 is recovered, for the homogeneous solid at φ 1 the Poisson’s ratio corresponds> to the= materials− Poisson’s ratio, with a continuous= transition between these two limits. The effective~ Poisson’s ratio ν˜cs shows a notable dependence= on νs for all values of φ. A similar observationACCEPTED has been reported in Day et al. MANUSCRIPT (1992, Fig. 5). By contrast, for intermediate values of δ, a transition is observed as function of volume fraction φ. For low volume fractions, in this case φ 0.3, we find ν˜cs to be clearly negative, close to the value ν˜ff 1. Further only marginal dependence of the constituent materials Poisson’s ratio νs is observed. Essentially, the deformation≲ of the linear-elastic cellular solid = − 11 ACCEPTED MANUSCRIPT

(a) 0.5 νs =0.5 (c) 0.4 δ = 0 0.4

1/3 stretchin = 0 g-d δ δmin = 0.268 omi − nate 0.2 5 d 0.2 2 5 2 0 . 0. 0 2

. 05 . − 0 − = 0 δ

= − − ) 0 νs =0 δ

φ = 1 =

( . δ 0 δ cs

ν = − δ 0.2 0.2

− − 1 . 0 − Eﬀective = 0.4 media theory 0.4 − − δ Poisson’s ratio ˜ 2 0.6 d 0.6 . − e − 0

t − a in = δ 0.8 om 0.8 − d − g- in

d 225 en . 1 b 1 0 − 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 − − =

Volume fraction φ δ (b) 1 . 268 . 0 0 − − = = δ min δ φ = 0.286 φ = 0.5 φ = 0.815 φ = 0.1

Figure 4: (Colour online) Deformation of a TS-Wheel cellular solid, illustrating the geometry- guided and material dominated limits: (a) Poisson’s ratio ν˜cs as function of the solid volume fraction φ (illustrated in (b)) for six cellular solid realisations of the TS-Wheel structure each at a different folding state δ, see the ‘filmstrip’ in (c), and two different material parameters (νs 0 and νs 0.5 corresponding to the lower and upper bounding curves of each band), determined by FEM. When the morphology prescribes the deformation behaviour (all cases except for δ 0 and δ δmin and for low= volume= fractions φ 0.3) the behaviour of the cellular solid is close to that of the bar-and-joint framework, which has ν˜ff 1 for all δ ( ); note the independence of the material’s parameter= in agreement= with results of Day et al. (1992), reproduced< in Milton (2002, chap. 4). For δ 0 and δ δmin (blue bands), where morphology does not= − prescribe the∎ deformation behaviour, ν˜cs φ differs greatly for all φ from ν˜ff 1 ( ), and looses its dependence on the material parameter νs slower with=φ 0 (where= the expected result of 1 3 ( ) is recovered, e.g. Christensen (1995)). The symbol is the( ) experimental value for δ 0.01=of− Mitschke∎ et al. (2011) where the FEM simulation data point lies within the→ measurement error bars. ~ ▲ ACCEPTED∎ MANUSCRIPT= − corresponds to that of the bar-and-joint framework model; material deformation by geometry. For larger volume fractions φ 0.3, ν˜cs continuously approaches the limit of the homogeneous solid, ν˜cs νs. For φ 0.3, increasingly strong dependence of ν˜cs on νs is observed. Both indicates that the deformation> modes of the cellular solid and the bar-and-joint framework = > 12 ACCEPTED MANUSCRIPT

(a) 0.5 0.4 0.3 0.2 0.1 0 (c) Kagome (d) H-Wheels (e) TS3-Wheels 1 2 1 3 − − − − − ~ ~0 -Wheels ) 3 δ

H-Wheels δ 0 ( ) δ 0 ( ) δ 0 ( ) ( TS cs Kagome TS-Wheels ˜ ν = = = 1 2 − ~ 1 δ 0.1 δ 0.1 δ 0.1 (b) 1 ν˜ﬀ δ − = − = − = − )

δ ( )

( 1

2 -Wh. TS-Wh. cs 3 H-Wh. Kagome ˆ E ~ TS 0 δ 0.5 ( ) δ 0.42 ( ) δ 0.29 ( ) 0.5 0.4 0.3 0.2 0.1 0 min min min Folding parameter δ = − ≈ − ≈ − − − − − − Figure 5: Auxetic cellular solids with 3- or 6-fold symmetry: Frameworks with a unique mechanism that naturally retain a hexagonal symmetry (and hence have isotropic Poisson’s ratio ν˜ff 1) represent a class of bending-dominated isotropic auxetic cellular solids, for all values of the free parameter δ of the mechanism (Mitschke et al., 2013b). (a) The effective Poisson’s ratio of the cellular solid= ν˜−cs is close to 1 except for the degenerate situation for δ 0 and when a topological transition to the rigid triangular framework occurs realised through vanishing of a pore by overlapping of bars for δ δmin. (b) The ˆ normalised− effective Young’s modulus Ecs resembles→ the behaviour of ν˜cs pointing out structures with ˆ ˆ ˜ stretching-dominated (high Ecs) and those with bending-dominated behaviour (low Ecs);→Ecs is normalised by the upper Hashin-Shtrikman bound (from Torquato et al. (1998)) indicating which configuration is close ˜ to the structure with the highest possible value of Ecs for a given φ. (c) to (e) Selected graphical data for Kagome, H-Wheels and TS3-Wheels; for TS-Wheels see Fig. 4(c); Note mechanisms of TS-Wheels and TS3-Wheels are unique for periodic boundary conditions containing several lattice points in contrast to Kagome and H-Wheels which become ambiguous. Material parameters for FEM are νs 0.2 and width to length ratio of the bars t l 0.1. = ~ = are clearly distinct, with geometry alone not sufficient to determine the mechanical response. Fig. 5 demonstrates the broader validity of the relationship between geometry and deformation properties suggested by Fig. 4. While Fig. 4 showed for a specific structure (TS-Wheels) that the effective Poisson’s ratio of the cellular solid for φ 0.3 is in agreement with that of the geometric mechanism illustrated for four points of the mechanism path (δ 0.05, 0.1, 0.2, 0.225), Fig. 5 substantiates this observation by≲ an detailed analysis of the full mechanism path for fixed bar thickness to length ratio t l of 0.1 for TS-Wheels and= − additionalACCEPTED− three− − planar cellular structures. MANUSCRIPT For the Kagome lattice, the H-Wheels, the TS-Wheels and the TS3-Wheels tessellation, we find a close relationship~ between ν˜ff and ν˜cs for all values of the deformation parameter δ for which the mechanism is not degenerate or effectively stiff. As a framework all of these have ν˜ff 1, as the unique mechanism retains hexagonal symmetry. For intermediate values of δ, the Poisson’s ratio ν˜cs adopts clearly negative values ν˜cs 0.75 similar to ν˜ff. Only in the= limiting− cases (δ 0 or δ δmin)

< − 13 = = ACCEPTED MANUSCRIPT

θ(δ) (e) Snub (f) Chiral elon. (g) Elongated 0° 30° 60° 90° 120° square snub square snub square (a) 4

) (e) δ ( 2 (f) (12) cs ˜ ν (g) 0 δmax (θ = 120◦) δmax (θ = 120◦) δmax (θ = 120◦) ν(12) (b) 1 ˜ﬀ

) (e) ~a ~a ~a δ ( 1/2 (f) (1) cs θ θ

ˆ θ E (g) (g) ~b ~b ~b 0 (c) δ = 0 δ = 0 δ = 0 10 (e) (e)

) (f) (f) δ ( 0 (g) (21) cs ˜ ν (21) 10 ν˜ﬀ − δ = 0.1 δ = 0.1 δ = 0.102 − − − (d) 1 (e) (e)

) (f) (f) δ ( 1/2 (g) (2) cs ˆ E

0 δmin (θ = 0◦) δmin (θ = 0◦) δmin (θ = 0◦) δ 0.4 0.3 0.2 0.1 0 δ min − − − − max Imposed change δ of ~a(θ = 90°)

Figure 6: (Colour online) Anisotropic cellular solids based on a family of periodic tilings with 2- or 4-fold symmetry with positive and negative Poisson’s ratio. The frameworks of the three tilings (e) snub square, (f) chiral elongated snub square and (g) (uni-) elongated snub square consist of rigid (i.e. triangulated) parallelograms or rigid trapezoids and floppy squares which only allow a unique mechanism. In contrast to Fig. 5, the initial frameworks (δ 0) are not maximally unfolded, elongation or contraction of lattice vector a is possible with δmax θ 120° 0.04 and δmin θ 0° 0.48. Initial frameworks are shown in the second row of (e) to (g) with lattice= vectors and the considered angle θ. References for the frameworks and the⃗ relation between δ and( =θ are) given≈ in Section 3.( (a)= and) ≈ − (c) Effective Poisson’s ratios 12 21 ν˜ and ν˜ agree for both models of this article, namely ν˜ff of the mechanism and ν˜cs of the cellular solid determined( ) ( by) FEM. Similarly to Figs. 3 to 5 , exceptions are on the hand degenerate configurations, here at δmax for horizontal strains in (a) and at δ θ 60° 0.1 for vertical strains in (c), and on the other hand when topology changes, here by full collaps of the squares at δmin θ 0° in (a) and (c). Note ν˜ff δ of the different frameworks are identical and( hence= a) ≈ single− data line is plotted using here the result given in Eq. (3) (red curves). Differences in the cellular solid values are due( to→ different) width to length ratios( ) of the bars: (e) t l 1 44, (f) t l 1 41, and (g) t l 1 22. For instance, these variations result in different ˆ 1 ˆ 2 δ’s when the square pores vanish. (b) and (d) Normalised effective Youngs moduli Ecs and Ecs reveal ˆ ( ) ˆ( ) again the two regimes~ = ~ of bending-dominated~ = ~ (low~ E=cs)~ and stretching-dominated behaviour (high Ecs). The latter one dominates the deformations for δ δmax in the horizontal direction (b) and for δ 0.1 θ 60° in the vertical direction (d) which are degenerate configurations w.r.t. the denoted directions. For the tiling in (f) both deformation behaviour contribute= at δ 0.1 θ 60° due to tilted aligned bars≈ − by 45( °=w.r.t.) a (see third row of (f)). When the topological transition to the triangular framework at δmin θ 0° occurs, stretching dominates in all directions and elastic properties≈ − ( = become) isotropic (last row of (e) to (g)). Note⃗ ˜ 1 ˜ 2 ( → ) Ecs and Ecs are normalised by isotropic Hashin-Shtrikman upper bound (see Fig. 5 for details) and νs 0.2. ( ) ACCEPTED( ) MANUSCRIPT = the Poisson’s ratio ν˜cs does not reflect the deformation mode of the geometric mechanism, rather adopting the positive values between 1 3 and 1 2 typical for stretching-dominated cellular solids. At δ 0 the configurations are degenerate (similarly to Fig. 2), but not rigid, ~ ~ = 14 ACCEPTED MANUSCRIPT

with geometry not uniquely determining the deformation mode; the mechanical compression as a cellular solid does not lead to a deformation representing the geometric mechanism, but rather involves stretching of some of the bonds, hence the value ν˜cs and also relatively large magnitudes of the Young’s modulus. At δ δmin (the minimal permissible value of delta specific to each mechanism) all three tessellation models adopt the same geometry, the triangular lattice (with superposed double edges);= this structure has the expected value ν˜cs 1 3 (Day et al. 1992; Christensen 1995, 2000; Torquato 2002, chap. 16.2.5; Milton 2002, chap. 4.6; Gurtner and Durand 2009). =Fig.~ 6 broadens this relationship further, by demonstrating a corresponding result to Fig. 5 for framework models where the Poisson’s ratio ν˜ff is not constant 1, but varies with δ. These are tilings, illustrated by the examples in Fig. 6 (e-g), where the deformation pathway (as δ is varied) does not retain hexagonal or square symmetry; rather, the− unique mechanism resulting from the edge equations retains a lower symmetry and hence allows variations of ν˜ff with δ. These lower-symmetry framework models are the snub square tiling (Fig. 6 (e), also known as 32.4.3.4 with symmetry p4gm), a chiral elongated snub square tiling (Fig. 6 (f), 3 2 2 also known as 3 .4 ; 3 .4.3.4 1 with symmetry p4gm) and a (uni-) elongated snub square ( ) 3 2 2 tiling (Fig. 6 (g), also known as 3 .4 ; 3 .4.3.4 2 with symmetry p2gg). The vertex signatures in parentheses( of the tessellations) are described in Grünbaum and Shephard (1987). As bar-and-joint frameworks, these( have unique mechanisms) that retain rectangular symmetry; hence, the deformation is characterised by two Poisson’s ratios and two Young’s moduli for different directions of applied strain. In contrast to deformation pathways that retain 12 21 hexagonal or square symmetry (which necessarily implies ν˜ff δ 1) both ν˜ and ν˜ are functions of δ. Poissons ratios ν˜ff of snub square and elongated snub square tilings( ) are already( ) published in Grima et al. (2008), denoted as ‘Type α rhombi’( and) = − ‘Type II α parallelograms”, respectively. Poissons ratios ν˜ff of chiral elongated snub square tiling has been shown in Mitschke et al. (2013a). Following the naming scheme of Grima et al. (2008) chiral elongated snub square tiling can be denoted as ‘Type II α (isosceles) trapezoids’. Poissons ratios ν˜ff have been determined analytically giving the same result for all three tilings 1 1 ν˜ 12 δ 1 ff 21 cos θ δ 1 ( ) ν˜ff δ 2 ( ) = ( ) = − = 1 ( ( )) + 1, (3) ( ) 1 π 1 cos 2 arcsin 2 3 2 δ 3 2 = » √ − a θ a θ π 2 2 sin θ π 3 2 + + − + with δ θ 1, its inverse θ δ accordingly, a θ 2l1 sin θ a θ π 2 2 3 ( )− ( = ~ ) (( + ~ )~ ) π 3 2 and b θ 2l2 sin θ »π 3√ 2 where for snub square l1 l2 l and for uni-elongated ( ) = ( = ~ ) = + − ( ) ( ) = (( + snub squareACCEPTEDl1 2l2 2l with l as the only MANUSCRIPT bar length (Grima et al., 2008). An analogous derviation~ )~ ) of (a )θ= and b((θ +for~ the)~ chiral) elongated snub square= = tiling gives an identical = = 12 21 result except with l1 l2 3l. The comparison of the Poisson’s ratios ν˜ff and ν˜ff of the ( ) ( ) 12 21 ( ) ( ) bar-and-joint frameworks with√ the FEM data for ν˜cs and ν˜cs of the cellular solid confirms the key observation= of this= article: except in the( vicinity) of( degenerate) or stiff limits (see caption of Fig. 6), we find very good agreement between ν˜cs and ν˜ff. This finding implies that 15 ACCEPTED MANUSCRIPT

(a) (b) 2× 2×

Figure 7: Simulation of compression testing of the TS-Wheels structure: A sample of finite size (5 5 unit cells) is compressed by a sequence of quasi-static FEM simulations. The upper inserts show close-ups of the cellular solid, whereas the lower inserts show the unique mechanism of the corresponding bar-and-joint× framework model. The overall deformation behaviour of the cellular solid resembles closely the unique mechanism of the bar-and-joint framework, with deviations close to the pistons or to the open boundaries. Parameters of this setup: volume fraction φ 0.13, bar width to length ratio t l 1 25, νs 0.2, initial configuration in (a) is the TS-Wheels framework contracted by δ 0.08; the lower insert in (b) shows the TS-Wheels framework contracted by δ 0.16. = ~ ≈ ~ = = − = − the agreement between the Poisson’s ratios of the cellular solid and those of the bar-and-joint model does not rely on high structural symmetry. In addition, we have analysed the deformation behaviour of several tessellations with lower symmetry, both in terms of the geometric mechanism and the mechanical response, confirming the result of Fig. 5 for non-hexagonal symmetry. Because of the lower symmetry these structures have two different Poisson’s ratios, for different directions of strain, related 12 ˜ 1 21 ˜ 2 by ν˜cs Ecs ν˜cs Ecs , and the Poisson’s ratios vary with δ. Except in the vicinity of the ( ) ( ) ( ) ( ) degenerate or stiff limits, we find very good agreement between ν˜cs and ν˜ff, even for those = 12 configurations with large ν˜ff 4. Interestingly, we find different( ) types of deformations of the cellular solid can be distin- guished by the magnitude of the≈ Young’s modulus, that is, the forces needed to drive the deformation. For all network solids analysed here, we find that those deformations that follow a geometric mechanism have substantially lower values of Young’s modulus than those corresponding to the stiff or degenerate limits. Consider for example the intermediate ranges ˆ of the deformation shown in Fig. 5 for which Ecs 1.

≪ 4. Summary This article has reaffirmed an intimate relationship between a geometric micro-structure and mechanical deformation, specifically Poisson’s ratio. Namely, when geometry clearly prescribes a unique mode in which the micro-structure can respond to an imposed strain, the Poisson’s ratio of the cellular solid corresponds to a good approximation that of a simple bar-and-jointACCEPTED framework model, hence it isMANUSCRIPT determined by geometry alone. In particular this implies the independence (to first order) of Poisson’s ratio of the constituent material properties and the volume fraction, in the low-density limit. The effective stiffness, quantified by Young’s modulus, in these geometry dominated regimes is very small, compared to the values observed for situations where the response to

16 ACCEPTED MANUSCRIPT

an imposed strain involves stretching. This corroborates the relationship to the deformation ˜ modes of floppy bar-and-joint framework models which are force-free (Eff 0). The deformation behaviour of a finite sample of the TS-Wheels structure (Mitschke et al., 2011) in a compression test setup supports the practical relevance of these= results. Fig. 7 shows a finite sample of this structure realised as a cellular solid. The compression test is modelled as a quasi-static sequence of FEM calculations, that is, the deformed structure (on the right) is obtained from the undeformed structure (on the left) by repetition of the following steps. FEM is used to calculate a stress field of the structure, and a small displacement corresponding to this stress is calculated. We then assume that this new (stressed) configuration represents an equilibrium (unstressed) structure and repeat the steps. Importantly, in line with the key result of this article, the mechanical deformation of this cellular solid follows the unique mechanism of the underlying bar-and-joint framework closely, except near the interfaces with the piston or with the open boundaries. These findings emphasise an intriguing aspect about the origin of a particularly interesting mechanical response, auxetic deformations with negative Poisson’s ratio. In all auxetic materials to date, the negative Poisson’s ratio results as a geometric effect, akin to the identification of a single relevant deformation mode of a floppy bar-and-joint framework model which are force-free. Auxetic behaviour has never been observed in a stiff structure, where the deformation necessarily involves stretching and hence large values of Young’s modulus. Note the possibility of auxetic deformations in materials where bending is artificially suppressed, e.g. by spring-piston-models, Rothenburg et al. (1991). An implication would be that the ˜ Young’s modulus Ecs of auxetic structures cannot reach the large values of cellular materials corresponding to stiff bar-and-joint frameworks (assuming equal volume fractions). For the sake of completeness we want to mention homogenization based optimization approaches to find auxetic structures (see for example Sigmund (1995) which is based on a truss model which shares some properties with the bar-and-joint-frameworks) and Sigmund (2000)). The ˆ potential usefulness of stiff (high Ecs) and simultaneously auxetic materials motivates further investigation of this observation and a search for exceptions.

Acknowledgements The authors would like to acknowledge the funding of the Deutsche Forschungsgemeinsch- aft (DFG) through the Cluster of Excellence Engineering of Advanced Materials.

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