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1 Copyright © 2009 by ASME
Proceedings of the ASME International Mechanical Engineering Congress and Exposition IMECE 2009
November 13-19, 2009, Lake Buena Vista, Florida, USA
IMECE 2009-12654
NONLINEAR ELASTIC CONSTITUTIVE RELATIONS OF AUXETIC HONEYCOMBS
Jaehyung Ju Research Associate
John Ziegert Timken Chair Professor of Design
Joshua D. Summers Associate Professor
Georges Fadel Exxon-Mobil Employees Chair Professor of Engineering
Clemson Engineering Design Application and Research Group (CEDAR) Department of Mechanical Engineering
Clemson University Clemson, SC 29634-0921
ABSTRACT When designing a flexible structure consisting of cellular
materials, it is important to find the maximum effective strain
of the cellular material resulting from the deformed cellular
geometry and not leading to local cell wall failure. In this
paper, a finite in-plane shear deformation of auxtic honeycombs
having effective negative Poisson’s ratio is investigated over
the base material’s elastic range. An analytical model of the in-
plane plastic failure of the cell walls is refined with finite
element (FE) micromechanical analysis using periodic
boundary conditions. A nonlinear constitutive relation of
honeycombs is obtained from the FE micromechanics
simulation and is used to define the coefficients of a
hyperelastic strain energy function. Auxetic honeycombs show
high shear flexibility without a severe geometric nonlinearity
when compared to their regular counterparts.
1. INTRODUCTION
Cellular materials have been used in the aerospace and
automobile applications such as impact resistance, light weight
design, thermal management, and multifunctional properties
(e.g., [1, 2]). Hexagonal honeycomb structures are the most
commonly used two-dimensional cellular solids. The
mechanical properties of honeycombs have been studied for
more than three decades [3-10]. The earlier models were
developed for the evaluation of their effective properties and
considered cell wall collapse strengths of regular honeycombs
over a linear range as a function of cell geometries [3-11].
Gibson and Ashby extended the previous models to establish a
clear linear constitutive relation of honeycombs, frequently
called Cellular Material Theory (CMT) [6]. Masters and Evans
refined the CMT by extending the flexure deformation of
beams to stretching and hinging [7]. Recently, Park and Gao
used the coupled stress theory and micromechanical technique
to obtain the effective properties of a hexagonal honeycomb
[10].
The practical applications using cellular solids have been
focused primarily on i) high energy impact energy absorption
applications such as military armors and helmets and ii) ultra
light weight design associated with high strength and low
density such as light weight components used in automobile
and aerospace structures. Thus, the research to date has
focused on tailoring the geometries of cells to maintain required
effective bending, tensile, and compressive stiffness and/or
local impact resistance. A notable absence in the literature on
design and characterization of cellular materials is an
exploration of performance under shear loading conditions –
Cellular materials’ practical design approach has not been
explored well by tailoring in-plane properties related to the
flexible structural design.
The authors’ previous work on tailoring honeycombs having a
relatively low effective shear stiffness (4 to 6MPa) and high
effective shear strain (~15%) shows that auxetic cellular
materials having effective negative Poisson’s ratio may be a
candidate for the design of flexible structures in shear [11].
The unique properties of the negative Poisson’s ratio structures
appear to avoid high local cell wall stress, resulting in high
2 Copyright © 2009 by ASME
effective shear elongation without local cell wall failure (e.g.,
[13-15].
Generally, if cellular materials are under a high strain condition,
they undergo both geometric and material nonlinearities
associated with elastic buckling and plastic deformation,
respectively. In order to explain the large deformation of
cellular solids, nonlinear constitutive relations must be
developed. The nonlinear constitutive relations of regular
honeycombs whose cell angle is 30o have been studied over
high compressive, tensile, and shear strains [16, 17]. Zhu and
Mills investigated the in-plane uni-axial behavior of regular
honeycombs under compression considering material and
geometric nonlinearities using an analytical study and
experimental observations [16]. Lan and Fu extended the work
of Zhu and Mills to in-plane tensile and shear behaviors of
regular honeycombs using an analytical method [17].
Recently, the high effective elastic strains of cellular materials
have been used for a hyperelastic model; Hohe et al. developed
a hyperelastic model of cellular foams using a strain energy
based homogenization procedure [18, 19]. In this study, a FE
simulation of cellular materials under simple shear is used to
generate effective stress-strain curves and to find proper strain
energy potentials related to their constitutive relations. A
detailed workflow is shown in Figure 1.
Figure 1. Workflow of a FE hyperelastic modeling of cellular
meta-materials
2. THEORETICAL BACKGROUND
2.1. Brief Review on Homogenization of Cellular Solids
Micromechanics approaches have been utilized to obtain the
effective engineering properties and local stress distributions of
multiple phase materials or non-homogeneous solids [20-22].
Structures with heterogeneities at the micro-structural scale are
conventionally analyzed with equivalent homogenized (or
effective) properties [20-24]. The homogenized properties are
obtained from volume averaging of the response of a
representative volume element (RVE). The length scale of the
RVE, y, must be small enough as compared to the length scale
of the macrostructure, X, in order for it to be treated as a typical
point in the structure under study. The homogenized scales, f,
are obtained by volume averaging of the variables in the RVE,
following the definition:
1( )
Xf f y dV
V (1)
According to the Hill-Mandel hypothesis [22], the condition for
macroscopic homogeneity assumes equivalence of strain
energy between the actual heterogeneous media and the
equivalent homogenized ones:
: : (2)
The microscopic stress σ and strain ε fields satisfying the
homogeneity condition can be obtained by solving boundary
value problems for the RVE with one of the following three
boundary conditions [24, 25]:
i. Uniform traction:
ˆ ˆ( ) ( )T n y n y on V (3)
ii. Uniform strain:
u y on V (4)
iii. D-periodicity:
u y u y y u y kD on V (5)
where 𝑘 is a 3 × 3 array of integers, 𝑛 (𝑦) the surface normal
vectors. D is the period of the periodic displacement function,
( )u y , interpreted as local perturbations to macroscopic strain
based displacement fields.
2.2. Elastic Properties and Large Deformation
Simulation of Cellular Solids with Displacement Boundary Conditions
Numerical tests with ABAQUS 6.8 for effective properties (E11,
E22, and G12) of auxetic honeycombs are conducted. A shear
deformable beam element (B22 in ABAQUS) is used for the FE
simulation. The B22 elements follow Timoshenko’s beam
theory and have 3 element-nodes; 1 internal and 2 end nodes,
each of them having 3 degrees of freedom; horizontal
displacement, vertical displacement, and rotation. Single cell
and multi cell results confirmed that the single cell results
capture the behavior of a multi cell auxetic honeycomb
structure. Large deformation simulations for simple shear tests
on auxetic honeycombs are carried out to obtain stress-strain
curves for each displacement loading step by calculating
effective stresses and effective strains using Equations (6),(7),
and (8).
3 Copyright © 2009 by ASME
* 1 111
1 2
R R
A wL , * 1
11
1L
(6)
* 2 222
2 1
R R
A wL , * 2
22
2L
(7)
* 1 112
2 1
Top TopR R
A wL , * 1
12
2
Top
L
(8)
where R1 and R2 are the total reaction forces on the prescribed
boundary along the x1 and x2 directions, respectively. δ1 and δ2
are the displacements along the x1 and x2 directions,
respectively (Figure 2). A1 and A2 are the cross-sectional areas
perpendicular to the x1 and x2 directions, respectively. L1 and
L2 are the length of the honeycomb cell structures along the x1
and x2 directions. w is the width of honeycombs. The effective
Poisson’s ratios are
* 212
11 2L
(9)
* 121
22 1L
(10)
The displacement boundary conditions are shown in Figure 2.
The more details on the boundary conditions can be found in
the references [25, 26].
Figure 2. Schematics of the displacement boundary conditions
for effective moduli; E11, E22, and G12
2.3. Brief Review of Cellular Material Theory
The basic parameters for hexagonal geometries are shown in
Figure 3 indicating cell angle, θ, cell height, h, cell length, l,
and cell wall thicknesses, t.
Figure 3. Unit cell geometries for (a) auxetic and (b)
conventional honeycombs
In-plane linear constitutive relations which are modified from
those provided in [4] can be expressed as
3
* *
11 112
cos
sin sin
tE
hll
(11)
3
* *
22 223
sin
cos
ht l
El
(12)
3
* *
12 122
sin
1 2 cos
ht l
El h h
l l
(13)
where σ*11, σ*22, and τ*12 are the effective stresses of cellular
materials in the x1, x2, and the shear directions, respectively.
ε*11, ε
*22, and γ*
12 are the effective strains of cellular materials
in the x1, x2, and the shear directions, respectively. E is the
Modulus of a base material.
3. HYPERELASTIC MODELING
The uni-axial and simple shear deformations of auxetic
honeycombs are simulated with finite strains (up to 30%) using
a material’s nonlinear elasto-plastic information. Local and
global stresses of cellular solids are checked for each loading
step with increasing global shear strains. In this study,
polycarbonate is used as a base material since polycarbonate
honeycomb coupons can be easily prepared using a 3D printing
technique. Experimental validation is planned for a future
study. A uni-axial stress-strain curve of a continuous
polycarbonate is obtained from the literature as shown in Figure
4 [27]. The modulus and Poisson’s ratio of the polycarbonate
are 2.7GPa and 0.42, respectively. Yield strength and yield
strain in the true stress and true strain values of the
polycarbonate are 81MPa and 3.75%, respectively.
4 Copyright © 2009 by ASME
Figure 4. Uni-axial stress-strain curve of a continuous
polycarbonate [27]
Even though the incompressible property may not apply to the
uni-axial tension of auxetic cellular materials, because their
effective Poisson’s ratios are not 0.5, it appears that
incompressibility can be applied to the simple shear of auxetic
honeycombs under an assumption of a reasonably small volume
change. Therefore, incompressible deformation is assumed for
the simple shear deformation of auxetic honeycombs in this
study.
3.1. Simple Shear of a Auxetic Honeycomb
A shear strain of 0.3 is applied to an auxetic honeycomb (θ=-
30o, h=4.2mm, l=2.1mm, and t=0.42mm). The boundary
conditions and the deformed geometry are shown in Figure 5.
The deformed geometry of the auxetic honeycomb does not
appear to have a severe buckling.
Figure 5. Simple shear deformation of an auxetic honeycomb
Figure 6 shows the effective stress-strain curve of the auxetic
honeycomb under simple shear loading. Local maximum von
Mises stresses are also indicated for the global loading step.
Figure 6. Effective stress-strain curve of a auxetic honeycomb
under the simple shear loading
Considering polycarbonate’s yield strength in the true stress
value, 81MPa, the polycarbonate auxetic cellular material
shows an effective elastic range up to ~16% shear elongation
without cell wall yielding.
The current empirical model based on the finite element
simulation represents a finite shear deformation of an auxetic
honeycomb including a base material’s nonlinearity. In Figure
7, the current model is compared with the CMT model
represented by Equation (13). As can be shown in Figure 7, a
nonlinear effect is observed up to 0.3 shear strain of the auxetic
honeycomb. Considering the material’s nonlinear data in Figure
4 and local cell geometry in Figure 6, the global nonlinearity
appears to be from the polycarbonate’s nonlinear elasto-plastic
properties. Lan and Fu also showed the nonlinear effect of the
regular honeycombs when the shear loading is applied [16].
They showed that the nonlinear analytical model coincides with
Gibson’s results for small strains, but the nonlinear effect is
getting higher with an increasing effective strain of cellular
materials.
In order to develop a nonlinear constitutive relation of the shear
deformation, the deformation mapping, x X with
undeformed (Xi) and deformed (xi) configurations for the
simple shear is given by [28]
*
1 2 1 2 2 3 3ˆ ˆ ˆX X X e X e X e
(14)
The deformation gradient tensor, ij i jF x X with X and x
the coordinates of a honeycomb in the original un-deformed
and deformed configurations, respectively, is represented as
0
20
40
60
80
100
120
140
160
180
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Tru
e s
tre
ss (
MP
a)
True strain
5 Copyright © 2009 by ASME
*
*
1 0
0 1 0
0 0 1
F
(15)
where γ* denotes the amount of effective shear of honeycomb,
The determinant of F is 1, implying that there is no effective
volume change for the in-plane simple shear of the
honeycombs.
The corresponding right Cauchy-Green deformation tensor is
*
* * * * *2
1 0
1 0
0 0 1
TC F F
(16)
and
*2
1 3I tr C (17)
2
2 *2 *2 *4 *2
2 1
*2
1 13 4 3
2 2
3
I I tr C
(18)
The left Cauchy-Green deformation tensor is given by
*2 *
* * * *
1 0
1 0
0 0 1
TB F F
(19)
The Cauchy stress tensor for the in-plane simple shear is given
by
*
12
* *
12
0 0
0 0
0 0 0
(20)
From the relation between the Cauchy stress and the strain
energy potential function for incompressible materials, which is
given by [28]
* * * *
1
1 2 2
2 2W W W
pI I B B BI I I
(21)
where p is the hydrostatic pressure and W(I1, I2) is the strain
energy potential function. Therefore, the effective shear
stresses for the simple shear deformation are expressed as
* * *3 *
12 1
1 2 2
*
1 2
2 2 2
2
W W WI
I I I
W W
I I
(22)
Reduced polynomial models are used in this study. The models
are known to predict behavior for complex deformation states
when test data are available for only a single deformation state.
A general form of the reduced polynomial models is given by
[28]
2
1
1 1
13 1
N Ni i
i el
i i i
W C I JD
(23)
where Ci and Di , constants obtained from test data, control the
shear behavior and the bulk (hydrostatic) compressibility.
Assuming little hydrostatic effect and using Equations (22) and
(23) gives one Ci values which are shown in Table 1.
Table 1. Cis of the reduced polynomial models for N=1, 2, and
3(auxetic honeycomb).
C1 (MPa) C
2 (MPa) C
3 (MPa)
N=1 (Neo-Hookean) 0.554 N/A N/A
N=2 0.465 0.785 N/A
N=3 (Yeoh) 0.441 1.369 -3.707
Corresponding curves are shown in Figure 7. Both N=2 and
N=3 (Yeoh model) show a good fit for the simple shear of the
auxetic honeycomb.
Figure 7. Plots of the hyperelastic models - auxetic honeycomb
3.2. Comparison with the Shear Deformation of a Regular Honeycomb
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0 0.05 0.1 0.15 0.2
eff
ect
ive
sh
ear
str
ess
(M
Pa)
effective shear strain
virtual test
N=1
N=2
N=3
6 Copyright © 2009 by ASME
In order to compare the shear deformations of the auxetic
honeycomb with the regular counterpart, the FE simulation is
carried out with a regular honeycomb. A shear strain of 0.3 is
applied to a regular honeycomb (θ=30o, h=4.2mm, l=4.2mm,
and t=0.42mm). The boundary conditions and the deformed
geometry are shown in Figure 8. As can be seen in the Figure
7, severe buckling is observed from the deformed geometry of
the regular honeycomb.
Figure 8. Simple shear deformation of a regular honeycomb
Figure 9 shows the stress strain curve of the regular honeycomb
under the simple shear loading condition with an indication of
local maximum von Mises stresses for several loading steps.
The polycarbonate regular honeycomb shows an effective
elastic range up to a shear strain of 0.075 without cell wall
yielding.
Figure 9. Effective stress-strain curve of a regular honeycomb
under simple shear loading
A nonlinear effect for the shear loading is also indicated by
comparing the linear CMT model developed by Gibson et al.
[3]. Compared to the deformation of the auxetic honeycomb,
the regular honeycomb shows higher geometric nonlinearity at
a strain lever of 0.2 or higher. Lan and Fu also indicated that
the geometric nonlinearity is more severe with an increasing
cell angle in the regular honeycombs associated with a higher
buckling condition [16]. A material nonlinear effect appears to
be dominant compared to the geometric counterpart up to a
shear strain of 0.15 in the regular honeycomb.
For a shear flexible design, auxetic honeycombs appear to be
preferable geometries associated with low geometric
nonlinearity up to a high effective shear strain.
The shear constants of the reduced polynomial models are
obtained from the shear stress-strain curves for a material’s
elastic range and shown in Table 2. Corresponding plots are
shown in Figure 10. N=2 and N=3 show a good fit with the
virtual shear test data.
Table 2. Cis of the reduced polynomial models for N=1, 2, and
3 (regular honeycomb)
C1 (MPa) C2 (MPa) C3 (MPa)
N=1 (Neo-Hookean) 0.909 N/A N/A
N=2 0.513 3.492 N/A
N=3 (Yeoh) 0.543 2.759 4.652
Figure 10. Plots of the hyperelastic models – regular honeycomb
SUMMARY AND CONCLUDING REMARKS
A FE based homogenization technique was used to find the
effective stress-strain of an auxetic honeycomb structure under
simple shear loading. Using a hyperelastic assumption for the
shear loading of an auxetic honeycomb, a strain energy
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 0.02 0.04 0.06 0.08
eff
ect
ive
sh
ear
str
ess
(M
Pa)
effective shear strain
virtual test
N=1
N=2
N=3
7 Copyright © 2009 by ASME
potential is developed for empirical constitutive relations to
describe an auxetic honeycomb’s shear behavior.
Based on the finite shear strain of auxetic and regular
honeycombs, the following conclusions are reached:
Auxetic honeycombs show lower geometric nonlinearity
than regular honeycomb counterparts when subjected to a
high effective shear strain, resulting in higher shear
flexibility.
Auxetic honeycombs show about two times higher
effective shear strain than regular honeycombs do.
The linear CMT model fit reasonably well the nonlinear
model up to an effective shear strain of 0.03.
ACKNOWLEDGMENTS This work is supported by two grants: South Carolina NASA –
EPSCoR and NIST – ATP. The work presented in this study
does not necessarily represent the views of our sponsors whose
support we gratefully acknowledge.
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