Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

47, 4, pp. 737-750, Warsaw 2009

MICROMECHANICAL MODEL OF AUXETIC

CELLULAR MATERIALS

Małgorzata Janus-Michalska

Cracow University of Technology, Institute of Structural Mechanics, Cracow, Poland

e-mail: mjm@limba.wil.pk.edu.pl

An effective anisotropic continuum formulation for auxetic cellularmaterials
is the objective of this paper. A skeleton ismodelled as a plane beam elastic
structure with stiff joints. The skeleton topology, forming concave polygons,
is responsible for negative Poisson’s ratio effect. The essential macroscopic
features ofmechanical behaviour are inferred from the deformation response
of a representative volume element using the framework ofmicromechanical
analysis. The strain energy of a unit cell is calculated by adding the tensile,
shearing and bending strain energy of individual members. The equivalent
continuum is based on averaging this energy, thus formulating the basis
for computing the anisotropic stiffness matrix. The structural mechanics
methodology and ANSYS finite element code are applied to solve the beam
model of the skeleton.Graphical representationof certainmaterial constants
such as Young’s modulus, Poisson’s ratio, shear modulus and generalized
bulk modulus is given. The results of included parametric study may be
used for proper choice of geometric and material data of the skeleton for a
given structural application of the anisotropic continuum.

Key words: auxetic cellular materials, anisotropy, effective model, elasticity

1. Introduction

Auxetic materials are of interest because of enhanced material properties re-
lated to negative Poisson’s ratio. Such materials are called also dilatational
materials because they exhibit substantial volumetric changes when loaded.
First auxetic cellular structures were created as 2D silicone rubber or alu-
minum honeycombs and were extensively investigated by Lakes (1991-1993),
Lakes et al. (1988), Lakes andWitt (2002). Materials with negative Poisson’s
ratio aremore resilient than non-auxeticmaterials, and the linear strain-stress
relationship can reach up to 0.40 compressive ultimate strain compared to 0.05
strain for conventional cellular materials. Auxetic materials also exhibit lower



738 M. Janus-Michalska

stiffness and greater resistance to indentations than ordinary materials. This
propertymay beuseful in certain applications such asmatresses andwrestling
mats. Auxetic materials better redistribute strain under external loads. Ne-
gative Poisson’s ratiomay significantly influence stress distribution in contact
problems, reduce stress concentrations, substantially affect the Saint Venant
effect and yield double curvature effect in 3D bending problems. This no-
vel behaviour is important in design considerations. An increasing number of
materials and processing routines are developed and tailored to specific appli-
cations. Auxetic behaviour can be described by classical theory of elasticity
and it does not depend on scale (Lakes, 1991b). Deformation takes place on
the micro or macro level. This means that one may consider auxetic matrials
as well as auxetic structures.
Physical origin of mechanical behaviour of cellular materials is based on

structural considerations. Structural analysis of a skeleton on the micro level
explains macroscopic behaviour of such structured bodies. The overall effecti-
ve properties are determined by considerations using transition betweenmicro
andmacro scales of observation. This corresponds to effectivemodel construc-
tion (Hori andNemet-Naser, 1999;Nemat-Naser andHori, 1999). The effective
properties are then used to determine the response of structural elements on
the macro scale and emerge naturally as a consequence of micro-macro rela-
tionswithoutdependingon specificphysicalmeasurements on themacroscopic
level (Nemat-Naser andHori, 1999). Thismethod, typical formicromechanics,
has been applied to aluminum foams Janus-Michalska and Pęcherski (2003),
and a group of 3D cellular structures forming positive Poisson’s ratio cellular
materials Janus-Michalska (2005). The existing solutions concerning stiffness
matrices for auxetic cellularmaterials aremainly based on the homogenization
approach (Kumar andMcDowell, 2004). Themost recent solution for auxetic
structures is the missing ribs method by Smith et al. (2000).
The approach based onmicromechanicalmodelling presented in this paper

is advantageous because the dependence of effective properties onmicrostruc-
tural parameters is easily readable here.Thus structural topology and skeleton
material may be properly chosen in order to obtain the required properties of
the cellular material.

2. Micromechanical analysis

A micromechanical framework similar to that proposed in Janus-Michalska
and Pęcherski (2003), Janus-Michalska (2005) has been developed to deter-
mine equivalent continuum properties. The internal structure of a material is



Micromechanical model of auxetic cellular materials 739

assumed to be periodical one, and is the basis for studying overall mechanical
properties of a cellular auxetic material.

2.1. Representative unit cell

A theoretical investigation is conducted for a material exhibiting two the
dimensional reentrant hexagonal microstructure as shown in Fig.1a.

Fig. 1. A cellular structure and a representative unit cell

Formulation on the micro level begins by identifying the unit cell of the
spatially periodic array.The idea adopted here is based on the fact that the es-
sential feature of arbitraryuniformdeformation of thewhole beamstructure is
affinityofmidpointdisplacements (WarenandKraynik, 1988; Janus-Michalska
andPęcherski, 2003). It results in the same structural response of parts of the
skeleton consisting of the node and halves of beams. The representative unit
cell defined as the smallest part of the skeleton with repetitive response was
successively applied to foams (Janus-Michalska and Pęcherski, 2003), and 3D
periodic cellular structures (Janus-Michalska, 2005). The detailed description
of unit cell construction is given in Janus-Michalska (2005).

The cell corresponding to the reentrant hexagonal structure is given in
Fig.1b. The skeleton of the cell is modelled as an elastic beam structure with
a stiff joint (vertex 0). The strut midpoints are described by the position
vectors

b
0
1 =
(
0,
h

2

)
b
0
2 =
(L
2
sinγ,

L

2
cosγ

)
b
0
3 =
(
−
L

2
sinγ,

L

2
cosγ

)

where: |b01|= h/2, |b02|=L/2, |b03|=L/2. Beam lengths and γ angle should
satisfy the following conditions, which are necessary to obtain the representa-
tive volume element (RVE) shown by the dotted line in Fig.1a: Lcosγ ¬h/2
and cosγ¬L/h.



740 M. Janus-Michalska

The thickness of RVE is assumed to be H =1 in the direction perpendi-
cular to the (x,y) plane. The volume of the unit cell is

V = [(h−Lcosγ)Lsinγ]H

Three surfaces of RVE perpendicular to struts i, where i = 1,2,3 are
considered for further calculations, and they are computed as follows

A1 =
(L−hcosγ
sinγ

)
H A2 =A3 =

(h−Lcosγ
sinγ

)
H

The part of the skeleton included in RVE consists of beams of rectangular
cross-sections As = tH, where t denotes beam thickness. The skeleton vo-
lume is thus determined as: Vs = (L+ h/2)tH. The cellular material may
be characterised by its relative density: ρ∗ = ρ/ρs, where ρ and ρs denote
the cellular material and skeletonmaterial densities, respectively. The relative
density may be expressed by relation

ρ∗ =

(
L+ h

2

)
t

(h−Lcosγ)Lsinγ
The structure is specified by four geometric structural parameters: L, h, t, γ.
Three material parameters: Rse – yield stress, Es – Young’s modulus, νs –
Poisson’s ratio must be given to describe the skeleton material.
On the basis of the unit cell model, effective properties of cellular material

can be derived.

2.2. Uniform strains for linear elasticity

Uniformplain deformation of a solid with repetitivemicrostructure results
in displacement affinity (Janus-Michalska, 2005). Periodic skeleton structure
requires that the individual beams deformantisymmetrically about theirmid-
points, so the resultant forces at eachmidpoint reduce to axial and transversal
force (Janus-Michalska, 2005) (resultant bendingmoment at themidpoint va-
nishes).
Inmicromechanics, strains are defined as volumetric averages of themicro

field variables (Nemat-Naser and Hori, 1999) and read as follows

ε= 〈εs〉V =
1

V

∑

Ai

sym(ni⊗ui) dS (2.1)

where: 〈·〉V stands for the volumetric average in the skeleton s taken over V ,
ni is the outer unit normal on the boundary ni = b

0
1/|b0i |, ui – themidpoint

displacement on the surface Ai.



Micromechanical model of auxetic cellular materials 741

Three independent uniform unit deformations Kε̃, described by the follo-
wing strain vector components (Kelvin’s notation in 6-D space) are considered

1ε̃= εx =1
2ε̃= εy =1

3ε̃=
√
2εxy =1 (2.2)

This results in the following midpoint displacements:
— for uniaxial extensions

∆i(
1ε̃)= 1ε̃(b0i ·ex)ex ∆i(2ε̃)= 2ε̃(b0i ·ey)ey i=1,2,3 (2.3)

— for pure shear

∆i(
3ε̃)= 3ε̃[(b0i ·ex)ey+(b0i ·ey)ex i=1,2,3 (2.4)

where ex, ey are unit vectors parallel to the x, y axes.

2.3. Mechanical model of cellular skeleton structure

Atypicalmicrostructure skeleton consists of thickbeams,whichmaybede-
scribedby theTimoshenko beammodel. For structures consisting of long slen-
der beams, thus yielding low relative cellular material density, the Bernoulli-
Euler beam model is sufficient. Structural mechanics methods are used to
analyse the skeleton structure. Resultant forces, i.e. normal force, tangent for-
ce and bendingmoment for the sets of midpoint displacements given by Eqs.
(2.3) and (2.4), related to uniform deformation, are obtained by making use
of the ANSYS FEM code. The skeleton part within the RVE is therefore as
shown as in Fig.2.

Fig. 2. A part of the skeleton with the stiff node 0 and a local beam system of
coordinates

The following notation is used:
ξi – local coordinate axis for strut i, ξi(0)= 0, 0¬ ξi ¬ li, li =h/2

for i=1, li =L/2 for i=2,3



742 M. Janus-Michalska

ni – versor normal to the i-th beam cross-section
τi – versor tangent to the i-th beam cross-section.

Theaxial and transversal force functions are constant along thebeamaxes,
while the bendingmoment changes linearily as follows

KF̃ni(ξi)=
KF̃ni

KF̃τi(ξi)=
KF̃τi

KM̃i(ξi)=
KF̃τi(li− ξi) K =1,2,3 i=1,2,3

(2.5)

Due to linearity of the problem, superposition is possible, and thence for an
arbitrary strain state ε = (1ε,2ε,3ε), the forces are a linear combination of
solutions computed for unit strains and read as follows

Fni(ε)=
3∑

K=1

KεKF̃ni Fτi(ε)=
3∑

K=1

KεKF̃τi (2.6)

2.4. Equivalent continuum based on averaging of strain potential

The approach adopted here is based on equivalence of the strain potential
for the discrete structure and the strain potential of an effective continuum.
It refers to averaging the strain energy density (Nemat-Naser andHori, 1999)
as written below

ΦE = 〈sΦE〉V =
1

V

∫

Vs

(sΦE) dVs (2.7)

The strain potential of the beam skeletonmay be obtained using the following
formula (Piechnik, 2003)

U =

∫

Vs

(sΦE) dVs =
3∑

i=1

( li∫

0

(Fni)
2 dξi

2EsAs
+µ

li∫

0

(Fτi)
2 dξi

2GsAs
+

li∫

0

[Fτi(li− ξi)]2 dξi
2EsJ

)

(2.8)
where
Es,Gs – Young’s and shear modulus for the skeleton material
As,J – beam cross-sectional area andmoment of inertia
µ – energy cross-sectional coefficient (for rectangular cross-

-section µ=1.2).

Due to linearity of the stress-strain relationship, the strain energy density
function is represented by the following quadratic form

ΦE =
1

2
ε :S : ε (2.9)



Micromechanical model of auxetic cellular materials 743

For simplicity, it is convenient to express strains, stresses and stiffness ma-
trix in terms of 6D space (Kelvin notation), where the plane stress tensor
is represented by a vector and the stiffness tensor representation is a 3× 3
matrix.

Introducing relation (2.6) to the expression of strain potential and diffe-
rentiating with respect to strain components as follows

SIJ =
1

V

∫
Vs

∂2(sΦE)

∂(Iε)∂(Jε)
(2.10)

one obtains the following formula for stiffness matrix components

SIJ =
1

V

3∑

i=1

[ li
EsA

IF̃in
JF̃in+

( µli
GsA

+
l3i
3EsJ

)
IF̃iτ
JF̃iτ
]

I,J =1,2,3

(2.11)
It is worth to emphasise that the energy based definition of S in general does
not lead to the same quantities as those obtained through the stress-strain
relations based on averaging of stresses in the skeleton over volume of the
representative cell (Janus-Michalska, 2005; Nemat-Naser and Hori, 1999).

The stiffness matrix for the considered type of symmetry is as follows

S=



S11 S12 0
S12 S22 0
0 0 S33


 (2.12)

Its zero components result from the distribution of resultant forces KF̃ni,
KF̃τi

in the skeleton.Those forces are symmetrical for K =1,2andantisymmetrical
for K =3.

The constitutive relation for the equivalent continuum is determined by
the following linear relation

σ=S : ε (2.13)

2.5. Anisotropic linear properties

Anisotropic properties are represented by a stiffnes matrix or by a direc-
tional distribution of stiffness moduli. For plane structures, the direction is
represented by the vector n shown in Fig.3, defined through the angle θ
measured from the axis x of the cell coordinate system to the direction n.



744 M. Janus-Michalska

Fig. 3. Directions n,mwith respect to the cell coordinate system

The following stiffness moduli are considered: Young’s modulus E(n),
Poisson’s ratio ν(n,m), shear modulus G(n,m), generalized bulk modu-
lus K(n). They can be obtained using the following definitions

1

E(n)
= (n⊗n) ·C · (n⊗n)

−ν(n,m)
E(n)

= (n⊗n) ·C · (m⊗m)
1

2G(n,m)
= (n⊗m) ·C · (n⊗m

)

1

3K(n)
= I ·C · (n⊗n)

(2.14)

where
n – versor specifying the tensile direction in a tension

test
m – versor perpendicular to the direction of tension n

C=S−1 – compliance tensor
I – unit tensor
n⊗n,n⊗m – diadic tensors.

InKelvin’s notation, the diadics and tensors have the representation given
below

n⊗n=
(
cos2θ,sin2θ,

1√
2
cosθsinθ

)

n⊗m=
(
−cosθsinθ,cosθ sinθ, 1√

2
(cos2θ− sin2θ)

)
(2.15)

I=(1,1,0) C=



C11 C12 0
C12 C22 0
0 0 C33






Micromechanical model of auxetic cellular materials 745

The anisotropy parameters defined as follows

J1 =
S11
S22

J2 =
S12+2S33
S11

are introduced tomeasure the degree of anisotropy (for isotropy J1 = J2 =1).

3. Examples

3.1. Comparison with results available in literature

Theeffective elastic stiffnessmatrix S calculated according to theproposed
method is compared with the matrix S∗ available in literature (Overaker et
al., 1998) for the following geometrical data: L = 3.152mm, h = 3.152mm,
t = 0.15mm, γ = 70◦, thus yielding the material of relative density ρ/ρs =
0.1155. Skeleton material data are: Es =10GPa, νs =0.3,

sRe =100MPa.

The computed values

S∗=



73.112 −18.49 0
0 4.804 0
0 0 0.038


·kPa S=



73.122 −18.489 0
0 4.803 0
0 0 0.038


·kPa

show that application of both methods leads to similar results.

3.2. Structural modelling and obtained material properties

Geometric parameters of the skeleton structure are chosen to satisfy the
assumptions of theTimoshenko beam theory (beams are not overly thick) and
to avoid buckling (beams are not overly slender).

Calculations areperformed for thematerial dataas in theprevious example
and geometric parameters given in Table 1.
The obtained effective stiffness moduli and anisotropy coefficients for the

given structures are put together in Table 2.

The dependence of stiffness moduli on the angle θ yields plots presented
inFigs.4-7. Theplots are given for angles 0◦ ¬ θ¬ 180◦ (for 180◦ ¬ θ¬ 360◦
the plots are the same).

The analysis shows that cell shape (ratio L/h and angle γ) determines
the type of anisotropy and plots shapes. Comparison of these plots leads to
conclusion that for materials of identical cell shapes (structure (2) and (6)),
which result in similar anisotropy parameters, the curves E(θ), G(θ), K(θ)



746 M. Janus-Michalska

Table 1.Geometric parameters

Structure L h γ t
min
{
L
t
, h
t

}
max
{
L
t
, h
t

}
type [mm] [mm] [–] [mm]

(1) 1.50 1.50 80◦ 0.15 10.0 10.0

(2) 1.50 2.00 60◦ 0.15 10.0 13.33

(3) 1.50 3.00 80◦ 0.15 10.0 20.0

(4) 3.00 1.40 80◦ 0.15 9.333 20.0

(5) 1.50 3.00 60◦ 0.15 20.0 20.0

(6) 3.00 4.00 60◦ 0.15 20.0 26.67

Table 2. Stiffness moduli and anisotropy coefficients

Structure EX EY
νXY νYX max

(
G
K

)
J1 J2

type [kPa] [kPa]

(1) 46.183 1.297 −5.058 −0.142 5.52 35.556 −0.1317
(2) 6.270 1.941 −1.718 −0.532 28.9 3.230 −0.5192
(3) 20.900 2.808 −2.289 −0.307 10.1 7.442 −0.2996
(4) 20.071 0.0586 −17.708 −0.0527 14.4 342.07 −0.0495
(5) 3.6713 3.4074 −0.9823 −0.9119 108.3 1.0774 −0.8970
(6) 0.7834 0.2402 −1.786 −0.5472 34.2 3.1616 −0.5448

Fig. 4. Young’s modulus in dependence of the angle θ for different structures

are also similar and their maximum values decrease approximately with the
third power of dimension of the proportion ratio. This phenomenon may be
explained by domination of bending over tension in skeleton beams, resulting
in such a dependence (Janus-Michalska, 2005).Materials with slender skeleton
beams are very compliant.



Micromechanical model of auxetic cellular materials 747

Fig. 5. Poisson’s ratio in dependence of the angle θ for different structures

For anisotropic materials the bounds of negative Poisson’s ratio are wider
than for isotropic materials, and theoreticaly may reach infinity (for isotropic
materials in two-dimensional problems the acceptablePoisson’s ratio is limited
by −1 ¬ ν ¬ 1). The existence of directions with high auxetic behaviour in
cellular materials is bound with high anisotropy (both parameters influence
the plot shape).

Fig. 6. Generalized bulk modulus in dependence of the angle θ for different
structures

Thequantity 1/[3K(θ)] represents the relative change of volumeper tensile
unit stress in the direction given by θ. The bulk modulus is sensitive only to
the first anisotropy parameter J1.

Contrary to the bulk modulus K, shear modulus G is only sensitive to
the second anisotropy parameter J2.



748 M. Janus-Michalska

Fig. 7. Shear modulus in dependence of the angle θ for different structures

Due to lack of typical symmetry, the distribution of stiffness moduli de-
pends only on the choice of geometric andmaterial parameters, which results
in various anisotropy parameters and values of stiffness moduli.

4. Conclusions

An effective model based on the framework of micromechanical analysis is
built and applied for an auxetic cellular material to predict elastic properties
on the macroscale. The same framework can be applied for arbitrary cellular
material with open cells of a regular skeleton structure. It is especially useful
for cells representing complicated shapes of the skeleton (multiple nodes, lack
of symmetry). Various structural topologies of cellular materials and skeleton
materials result in different macroscopic properties, which can be tailored to
special demands.Theproper choice ofmicrostructural geometrical parameters
determines expected elastic properties. The sensitivity analysis of structural
parameters for the auxetic cellularmaterial allows for formulation ofmodelling
indications.

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versity Press

2. Hori M., Nemat-Nasser S., 1999, Onmicromechanics theories for determi-
ningmicro-macro relations in heterogeneous solids,Mechanics ofMaterials,31,
667-682



Micromechanical model of auxetic cellular materials 749

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cellular materials,Archives of Metallurgy and Materials, 50, 3, 596-608

4. Janus-Michalska M., Pęcherski R.B., 2003, Macroscopic properties of
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750 M. Janus-Michalska

Model mikromechaniczny materiałów komórkowych o ujemnym

współczynniku Poissona

Streszczenie

Celem pracy jest sformułowanie efektywnego anizotropowego continuum spręży-
stego dlamateriałówkomórkowycho ujemnymwspółczynnikuPoissona. Szkieletma-
teriału jest modelowany przez płaską strukturę belkową połączoną w sztywnych wę-
złach tworzącą układwielokątówwklęsłych.Kątywklęsłe w strukturzemateriału od-
powiadają za efekt ujemnego współczynnika Poissona. Poprzez zastosowaniemodelu
mikromechanicznego istotne cechy mechaniczne materiału komórkowego są wypro-
wadzone z wyników analizy komórki reprezentatywnej. Potencjał sprężysty szkieletu
komórki jest wyznaczony jako suma energii w belkach tworzących szkielet od ich
rozciągania, ścinania i zginania. Efektywne continuum jest oparte na uśrednianiu po-
tencjału sprężystego, co jest podstawą konstruowania macierzy sztywności. Metoda
analizy strukturalnej przeprowadzona za pomocą programuMES-ANSYS jest stoso-
wanadlamodelu belkowego szkieletu. Jakowynik tej analizyprzedstawionograficznie
rozkładymodułuYounga, współczynnika Poissona,modułu na ścinanie i uogólnione-
go współczynnika ściśliwości objętościowej. Studium parametryczne umożliwia prze-
śledzenie wpływu parametrów geometrycznych struktury i charakterystykmateriału
szkieletu na własności kontinuum zastępczego jako materiału o zastosowaniu struk-
turalnym.

Manuscript received November 13, 2008; accepted for print April 6, 2009