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Finite auxetic deformations of
plane tessellations
Holger Mitschke1 , Vanessa Robins2 , Klaus Mecke1
rspa.royalsocietypublishing.org
and Gerd E. Schröder-Turk1
1 Theoretische Physik, Friedrich-Alexander Universität
Research
Cite this article: Mitschke H, Robins V, Mecke
K, Schröder-Turk GE. 2013 Finite auxetic
deformations of plane tessellations. Proc R
Soc A 469: 20120465.
http://dx.doi.org/10.1098/rspa.2012.0465
Received: 5 September 2012
Accepted: 11 October 2012
Subject Areas:
structural engineering, geometry, mechanics
Keywords:
Poisson’s ratio, strain amplification, tilings and
tessellations, isostaticity, skeletal frameworks,
cellular structures
Author for correspondence:
Gerd E. Schröder-Turk
e-mail: [email protected]
Erlangen-Nürnberg, Staudtstrasse 7B, 91058 Erlangen, Germany
2 Applied Maths, Research School of Physical Sciences and
Engineering, The Australian National University, Canberra,
0200 Australian Capital Territory, Australia
We systematically analyse the mechanical deformation
behaviour, in particular Poisson’s ratio, of floppy barand-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
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rspa.2012.0465 or
via http://rspa.royalsocietypublishing.org.
ν =−
⊥
,
with the imposed strain in a given direction and
⊥ the resulting strain in the perpendicular direction.
c 2012 The Author(s) Published by the Royal Society. All rights reserved.
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The widespread appearance of auxetic behaviour results from generic features of complex microstructure 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 pi = {xi , yi }. 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)
|pi − pj |2 − l2{ij} = 0 ∀ {i, j} ∈ E.
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) = {pi (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.
..................................................
2. Periodic bar-and-joint frameworks as models for auxetic structures
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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].
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(a)
d=0
Figure 1. Novel auxetic frameworks: (a) triangle–square wheels and (b) hexagonal wheels, which include both rotating and
re-entrant elements. For both structures, the edge equations alone ensure unique deformations that retain the hexagonal
symmetry and hence yield ν(δ) = −1. Two-sided arrows (orange) represent the imposed deformation along e , here chosen
to be in the direction of a0 . The one-sided, horizontal arrows (grey) are the lattice vector b0 . (Online version in colour.)
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
a0 and b0 (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
a0 and b0 , figure 2).
..................................................
d = –0.15
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(b) d = 0
d = –0.1
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(a)
4
b||
e^
0
b (d)
0
l
e||
l¢
e^
e||
a||
a0
a (d)
a^
b^0
(b)
b^
(c)
e^
0
b0
e ||
b0||
e^
0
b (d )
e ||
a||
a0
a (d)
a^
b0
e^
0
e||
b||
0
e||
b (d)
e^
a (d)
a0
b^0
b^
Figure 2. Set-up for the deformation of the translational unit cell used for the definition of Poisson’s ratio in equation (3.2).
The strain δ is applied in the e direction (here the vertical direction). (a) General case: the crystal system is not retained, the
unit cell may be or may become oblique, (b) sheared rectangular deformation also with loss of crystal system (rectangular to
oblique), and (c) crystal system retaining deformation of a hexagonal unit cell.
A symmetric bar-and-joint framework S is mapped onto itself under the action g(S) for all
elements g ∈ G of the symmetry group1 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 Sδ maintain all or some of the symmetries of the original, i.e. g (Sδ ) = Sδ 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
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. c2, c11m, are used [26] with the corresponding cell transformation.
..................................................
b0
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b0||
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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
E[P(δ)] =
[αjik
(P(0)) − αjik
(P(δ))]2 ,
(3.1)
jik
with 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
jik
the sum over all such edge pairs of the considered unit cell; αjik
(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 §4a 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 h0 × l0 , Poisson’s
ratio ν is defined as the ratio of Cauchy strains ν = −((h − h0 )/h0 )/((l − l0 )/l0 ), where l0 × h0 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 , now called δ, here
chosen in the direction of the lattice vector a, i.e. e = a0 /|a0 |. 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 a(δ) = a0 (1 + δ)e + a⊥ (δ)e⊥ and b(δ) = b0 (1 + δ)e + b⊥ (δ)e⊥ ,
where a0 = a0 , e = |a0 | and b0 = b0 , e are the projections of the initial vectors a0 and b0
onto e ; a (δ) = (1 + δ)a0 and b (δ) = (1 + δ)b0 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:
ν(δ) = −
|a⊥ (δ)|+|b⊥ (δ)|−|a0⊥ |−|b0⊥ |
|a0⊥ |+|b0⊥ |
|a (δ)|+|b (δ)|−|a0 |−|b0 |
|a0 |+|b0 |
=
1−
|a⊥ (δ)|+|b⊥ (δ)|
|a0⊥ |+|b0⊥ |
δ
.
(3.2)
This 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
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.
..................................................
(b) Minimization of harmonic angular spring energy functionals
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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].
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(a) 1.0
6
0.8
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n (d )
0.6
0.4
n(
0.2
d)
0
n
–0.2
inst
–0.4
0
(b)
–0.1
(c)
d=0
d ª –0.103
(d )
–0.2
–0.3
d
Poisson’s ratio
(d)
–0.4
d ª –0.268
–0.5
(e)
d ª –0.42
Figure 3. Transition to auxetic behaviour (ν < 0) at finite strains of the two-uniform tessellation (33 .42 ; 32 .4.3.4)1 , without
symmetry constraint. Both finite (grey) and instantaneous (black) Poisson ratios are given as defined in §3c. The four thick
symbols in (a) correspond to the configurations (b–e) representing the compression pathway that is unique in p1. The strain δ
is applied in vertical direction (along the downwards pointing lattice vector a) and the maximal compression (when
√ collapse of
0
− 1 ≈ −0.482. (b) initial configuration with square unit cell; (c) δ = 3/a 0 − 1 ≈
adjacent rods occurs) is δ = 1/a√
−0.103 with νinst = 0; (d) δ = 2/a 0 − 1 ≈ −0.268 with ν = 0, i.e. horizontal extension equal to its initial value;
(e) δ = −0.42 with almost collapsed edges. (Online version in colour.)
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 δ → 0, but it also applies to finite values δ (figures 1 and 3). We define
the instantaneous Poisson ratio νinst (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.
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(a)
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(c)
(b)
(d)
kites
0
νinst(d)
–1
–2
inverted honeycomb
–3
elongated kagome
–4
–0.5
0
0.5
1.0
d
Figure 4. Auxetic structures with νinst < −1 as candidates for strain amplification. Note that the deformation of the
elongated kagome structure are studied in detail. (a) Kites (p1), (b) inverted honeycomb (cm), (c) elongated kagome
(3.42 .6; 3.6.3.6) (cm)—the retained symmetry groups are given in parentheses, (d) instantaneous Poisson ratio. Note the
anisotropy of these structures; the value of γ refers to vertical applied strain. The light grey (green) solid and dashed
lines represent symmetries, the vertical arrows the direction e and the horizontal arrows the direction e⊥ . (Online version
in colour.)
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 δ. In each increment,
the strain is increased by δ 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 δ, starting at δ = 0. A number m of possible
solutions Pi (δ + δ) of the edge equations (equation (2.1)) for strain δ + δ are computed
by adding small evenly distributed random numbers to all vertex coordinates of one of the n
solutions Pj (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 P1 equal to the initial undeformed
configuration. For each solution, Pi (δ + δ), the value of E[Pi ] is computed. Out of the m solutions
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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 (36 ; 32 .4.3.4), here called triangle–square wheels or TS wheels, and
(36 ; 32 .62 ), 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 p6mm, is shown in
figure 1a. 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 1b 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
chirality5 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 p1 (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 finiteelement calculations [38]. The approximate agreement, in terms of Poisson’s ratio, between the
3
A preliminary account of one of these, TS wheels, has been given by Mitschke et al. [38].
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.
4
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4. Results
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Pi (δ + δ), those s solutions that have minimal energy value Emin := minm
i=1 E[Pi (δ + δ)] 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 δ ≈ δ/100.
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(a)
Figure 5. Unique deformations of the truncated square tessellation (4.82 ) in subgroups of the full symmetry group p4mm. Note
that the deformation in subgroup p2mm in (c) is identical also to the deformation mode in subgroup pm (not shown). See the
corresponding figure 5 in the electronic supplementary material for an animation of the deformation modes. Deformation mode
of (4.82 ) in subgroup (a) p4, (b) c2mm and (c) p2mm. (Online version in colour.)
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 νinst vanish. As an example, the deformation of the two-uniform tessellation (33 .42 ; 32 .4.3.4)1
is shown in figure 3. Further uniform tessellations that are unique in p1 (i.e. without symmetry
constraints) and become auxetic upon finite deformation are (32 .4.3.4) [22], called the snub square
tessellation, and (33 .42 ; 32 .4.3.4)2 , discussed by Grima et al. [21].
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(b)
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(a)
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 (32 .4.3.4; 3.4.6.4) tessellation, with a constant Poisson ratio of
−1 in both the primitive cm and the p2 space group (interestingly, the deformation modes in these
two groups are different), and the two-uniform (3.42 .6; 3.6.3.6)2 tessellation with a Poisson ratio
smaller than −1 in lattice direction [01] (figure 4c); 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), (4.82 ), (36 ; 44 .4.12), (33 .42 ; 3.4.6.4), (3.42 .6; 3.4.6.4), (3.122 ) and (63 ).
Upon large enough deformations, the last one of these, (63 ), is congruent to the inverted
honeycomb pattern (cf. figure 4b). Some of these constrained bar-and-joint frameworks have large
negative Poisson ratios νinst < −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 p1 (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.
..................................................
Figure 6. Unique deformations of the great rhombitrihexagonal tessellation (4.6.12) in two different subgroup embeddings of
the full symmetry group p6mm. See the corresponding figure 6 in the online supplementary material for an animation of the
deformation modes: (a) subgroup p6 and (b) subgroup p31m. (Online version in colour.)
rspa.royalsocietypublishing.org Proc R Soc A 469: 20120465
(b)
10
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(a)
60
–1.0
d)
–1.5
E(
40
–2.0
20
–2.5
0
energy, E (d )
Poisson's ratio, ninst (d )
d = –0.05
d = –0.2
80
2
80
1
ninst (d)
0
60
d)
E(
40
–1.0
energy, E (d )
d=0
d = –0.05
Poisson's ratio, ninst (d )
100
(b)
20
–2.0
0
d = –0.2
100
0
ninst (d)
–0.5
80
–1.0
60
–1.5
d
E(
–2.0
–2.5
0
)
40
energy, E (d)
Poisson's ratio, ninst (d)
d = –0.05
d=0
(c)
20
0
–0.1 –0.2 –0.3 –0.4 –0.5
d
Figure 7. Deformation mechanisms of the elongated kagome tiling by imposing a strain in a direction perpendicular to [01]
for three different deformations (the [01] direction is the horizontal direction in this figure): (a) constraint to symmetry group
c11m including a glide plane, (b) constraint to symmetry group c2 and (c) minimizing E[P] from equation (3.1) with only purely
translational symmetry c1 with translation vectors (1, 0) and (1/2, 1/2). The tessellation is the same as in figure 8c, however
with strain applied in the orthogonal direction to figure 8c. (Online version in colour.)
(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
..................................................
ninst (d)
–0.5
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d=0
11
100
0
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(d )
100
2.0
80
1.5
60
d)
E(
1.0
40
energy, E (d)
Poisson's ratio, ninst (d)
0
(d )
20
0.5
0
2.5
100
2.0
80
n inst
(d )
Poisson's ratio, ninst (d)
20
2.5
1.5
60
d)
E(
1.0
40
20
0.5
0
energy, E (d )
40
inst
–2.0
80
energy, E (d )
Poisson's ratio, ninst (d )
d=0
d = –0.2
d=0
n
12
60
n inst
d = –0.1
–1.5
)
0
(c)
d=0
E
–1.0
(d
–2.5
(b)
d = –0.1
–0.5
100
0
–0.1
–0.2
d
–0.3
0
–0.4
Figure 8. Deformation of the same tessellation as in figure 7 for the orthogonal strain direction (i.e. strain perpendicular to
[10]) (in this figure, the [10] direction is the vertical direction): (a) constraint: c11m, (b) constraint: c2 and (c) minimal energy
deformation amongst all configurations with C1 periodicity. (Online version in colour.)
modes, namely by picking the deformation path that minimizes a given energy functional, see
also §3b. 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.
..................................................
d = –0.05
0
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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 barand-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
6
The symmetry group p1 with lattice vectors a and b has no symmetries except for periodicity. This applies equally to the
‘centred’ group c1 with lattice vectors a and b; however, c1 is periodic under translations by (a − b)/2 and (a + b)/2.
..................................................
5. Discussion
13
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Figure 7 shows the deformation modes and the corresponding Poisson ratio for strain
perpendicular to [01]. With only translational symmetry constraints (either p1 or c1), the
deformation is ambiguous, whereas for both subgroups c11m (including glide planes) and c2
(without glide planes), the deformation is unique; however, the modes for c11m and c2 are
clearly different, as evidenced by the configurations as well as the Poisson ratio νinst (δ). It is then
interesting to note that the deformation that minimizes E[P] without any symmetry constraints,
except for pure translation c1,6 corresponds to one of these groups, namely c11m. This is an
interesting observation considering that we have identified several tessellations with νinst < −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 c11m and c2. However,
for that strain direction, the unconstrained energy-minimizing mode is the same as the c2 mode,
in contrast to the situation above.
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SCHR1148/3-1 and ME1361/12-1.
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We acknowledge financial support by the German Science Foundation (DFG) through the Engineering
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14
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