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Computational models for simulations of lithiumion battery cells
under constrained compression tests
Mohammed Yusuf Ali
a
, WeiJen Lai
b
, Jwo Pan
a
,
*
a
Department of Mechanical Engineering, The University of Michigan, Ann Arbor, MI 48109, USA
b
Department of Materials Science and Engineering, The University of Michigan, Ann Arbor, MI 48109, USA
highlights
Develop computational models for simulations of lithiumion battery cells.
Model the multiscale buckling of lithiumion battery cells.
Model the formation of kinks and shear bands in lithiumion battery cells.
Model the buckling of cover sheets and justify the length selection of cell specimens.
Model the void compaction, plastic deformation and loaddisplacement curves.
article info
Article history:
Received 15 February 2013
Received in revised form
4 May 2013
Accepted 8 May 2013
Available online 22 May 2013
Keywords:
Lithiumion battery
Representative volume element
Mechanical behavior of pouch cell battery
Kink formation
Shear band formation
Computational models
abstract
In this paper, computational models are developed for simulations of representative volume element
(RVE) specimens of lithiumion battery cells under inplane constrained compression tests. For cell
components in the ﬁnite element analyses, the effective compressive moduli are obtained from inplane
constrained compressive tests, the Poisson’s ratios are based on the rule of mixture, and the stress
eplastic strain curves are obtained from the tensile tests and the rule of mixture. The Gurson’s material
model is adopted to account for the effect of porosity in separator and electrode sheets. The computa
tional results show that the computational models can be used to examine the micro buckling of the
component sheets, the macro buckling of the cell RVE specimens, and the formation of the kinks and
shear bands observed in experiments, and to simulate the loadedisplacement curves of the cell RVE
specimens. The initial micro buckling mode of the cover sheets in general agrees with that of an
approximate elastic buckling solution. Based on the computational models, the effects of the friction on
the deformation pattern and void compaction are identiﬁed. Finally, the effects of the initial clearance
and biaxial compression on the deformation patterns of the cell RVE specimens are demonstrated.
Ó2013 Elsevier B.V. All rights reserved.
1. Introduction
Lithiumion batteries have been considered as the solution for
electric vehicles for the automotive industry due to its lightweight
and high energy density. The major design considerations of
lithiumion batteries involve electrochemistry, thermal manage
ment and mechanical performance. The electrochemistry has been
widely studied since it directly determines the battery performance
and its life cycle. Different active materials on electrodes give
different types of lithiumion batteries. However, the basic
chemical reactions of the cells are similar. For automotive appli
cations, the mechanical performance is of great importance for
crashworthiness analyses. Research works were conducted on the
safety performance of the battery cells under mechanical tests such
as nail penetration tests, round bar crush tests, and pinch tests, for
example, see Refs. [1e5]. Many research works were also conducted
to understand and model the phenomena related to intercalation
induced stresses, diffusion, debonding, cracking, and the effect of
the coatings due to reaction in the lithiumion batteries, for
example, see Refs. [6e13]. However, these research works mainly
focused on electrodes or separators and understandably do not
cover the global mechanical behavior of battery cells, modules and
packs.
Sahraei et al. [14] conducted a series of mechanical tests and
computational modeling works on commercial LiCoO
2
/graphite
*Corresponding author. Tel.: þ1 734 764 9404; fax: þ1 734 647 3170.
Email addresses: mdyusuf@umich.edu (M.Y. Ali), weijen@umich.edu (W.J. Lai),
jwopan@gmail.com,jwo@umich.edu (J. Pan).
Contents lists available at SciVerse ScienceDirect
Journal of Power Sources
journal homepage: www.elsevier.com/locate/jpowsour
03787753/$ esee front matter Ó2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.jpowsour.2013.05.022
Journal of Power Sources 242 (2013) 325e340
cells used for cell phones. The results indicate that the compressive
mechanical behavior is characterized by the buckling and densiﬁ
cation of the cell components. Other testing and modeling data
available were also conducted on commercial LiCoO
2
cylindrical or
prismatic battery cells [15,16]. However, this information is of
limited use for researchers to model the mechanical performance of
automotive highvoltage LiFePO
4
battery cells and modules for
crashworthiness analyses. Sahraei et al. [14] indicated that
computational effort is quite signiﬁcant to model local buckling
phenomenon of battery cells under inplane compression. There
fore, macro homogenized material models of the representative
volume elements (RVEs) for both the battery cells and modules
have to be developed for crashworthiness analyses with sacriﬁce of
the accuracy at the micro scale. Other than dealing with the multi
physics problem, one of the challenges of developing the compu
tational models for the battery behavior is to deal with different
models at different length scales as indicated in Ref. [14]. Therefore,
understanding the basic mechanical behavior of the lithiumion
batteries for automotive applications is very important to develop
macro homogenized material models for representative volume
elements (RVEs) of cells and modules for efﬁcient crashworthiness
analyses.
Recently, Lai et al. [17,18] investigated the mechanical behaviors
of lithiumeiron phosphate battery cells and modules by conducting
tensile tests of individual cell and module components, constrained
compression tests of RVE specimens of dry cells and modules, and a
punch test of a smallscale dry module specimen. Their results of
inplane tensile tests of the individual cell components indicate
that the active materials on electrodes have a very low tensile load
carrying capacity. For inplane constrained compression tests of cell
RVE specimens, the results indicate that the load carrying behavior
of cell RVE specimens is characterized by the buckling of cells with
a wavelength approximately in the order of the thickness of the
cells and the ﬁnal densiﬁcation of the cell components. They also
tested module RVE specimens with different heights and the results
indicate that the load carrying behavior of module RVE specimens
is also characterized by the buckling of cells with a wavelength
approximately in the order of the thickness of the cells and the ﬁnal
densiﬁcation of the module components but relatively indepen
dent of the height of the tested specimens. In addition, they
investigated the effects of adhesives between cells and foam/
aluminum heat dissipater sheets on the mechanical behavior of
module RVE specimens. The results indicate that the adhesive
Nomenclature
a
kink angle
q
shear band angle
q
i
initial shear band angle
q
f
ﬁnal shear band angle
h
s
shear band height
dkink length
wcell thickness
f
Gurson yield function for porous materials
fvoid volume fraction; it is deﬁned as the ratio of the
volume of voids to the total volume of the material
f
0
initial void volume fraction
rrelative density of a material; it is deﬁned as the ratio
of the volume of matrix material to the total volume of
the material
qeffective macroscopic Mises stress
pmacroscopic hydrostatic pressure
s
y
ﬂow stress
s
y0
yield stress
3
pl
m
average equivalent plastic strain
q
1
,q
2
and q
3
ﬁtting parameters
Sdeviatoric part of the macroscopic Cauchy stress tensor
Smacroscopic Cauchy stress tensor
n
i
Poisson’s ratio of the ith component for rule of
mixture (ROM)
V
i
volume fraction of the ith component for ROM
s
y
Gurson
adjusted ﬂow stress of the matrix
PEEQ equivalent plastic strain
RVE representative volume element
nnumber of waves
mnumber of half waves
llength of the cell
kthe spring constant of the elastic foundation on one
side of the beam and is deﬁned as the lateral force per
unit plate length per unit deﬂection of the neighbor
components in the outofplane direction
Emodulus of elasticity
bwidth of the specimen
hthickness of the neighbor components
P
potential energy of the system
Imoment of inertia
vdeﬂection of the beam in the ydirection
Pcompressive force
h
cover sheet
thickness of the cover sheet
h
anode
thickness of the anode
h
cathode
thickness of the cathode
h
separator
thickness of the separator
h
foundation
thickness of the foundation (neighbor components)
Fig. 1. A schematic view of the approaches for computational model developments.
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340326
slightly increases the compressive load carrying capacity of the
module RVE specimens. Their SEM images of the active materials
on electrodes and the results of inplane compressive and outof
plane compressive tests suggest the total volume fraction is up to
40% for the microscopic gaps between cell components and the
porosity of the separators and the active materials on electrodes.
Based on the compressive nominal stressestrain curves in the in
plane and outofplane directions, their work suggests that the
lithiumion battery cells and modules can be modeled as aniso
tropic foams or cellular materials.
The current study is focused on developing the computational
models for simulations of RVE specimens of lithiumion battery
cells under inplane constrained compression tests based on the
work of Lai et al. [17] and then comparing the computational results
with those of the tests. Fig. 1 shows a schematic view of the ap
proaches of the developments of the computational models. Two
approaches are used for the modeling of these battery cells and
modules: a detailed model (micro approach) and a less detailed
model (macro approach). This investigation will focus on the
detailed modeling of a cell RVE specimen of lithiumion batteries.
In the detailed model, the pouch cell battery is modeled as a layered
composite and the RVE material nominal stressestrain response is
obtained based on the properties of the cell components of layered
anode, cathode, separator and cover sheets. The less detailed
models were investigated in a companion study [19] to address the
length scale issue in mechanical modeling of the batteries. In those
less detailed models, a smallscale battery module was considered
as a homogenized material based on the response of the physical
testing of the module RVE specimens [18]. Both approaches are
useful to investigate the mechanical behavior of lithiumion pouch
cell batteries and modules.
The purpose of this detailed model investigation is twofold: one
is to enhance understanding of the mechanical behavior of lithium
ion battery cells used for automotive applications and the other is
to pave the groundwork for the development of user material
models to represent the battery cells and modules by homogenized
materials which are a subject of the future research. Finite element
models can be used to simulate the tensile tests for multilayered
cell and module RVE specimens. However, a simple estimation
scheme for tensile behavior is presented in Ref. [18] based on the
rule of mixture (ROM) for composite and thus the tensile behavior
of battery cells will not be addressed here. In this investigation, the
compressive behavior of cell RVE specimens under quasistatic in
plane compression tests is investigated using the ABAQUS explicit
ﬁnite element solver [20]. In this paper, the experimental results for
cell RVE specimens under inplane compression tests are ﬁrst
reviewed brieﬂy for understanding the physical deformation
pattern of the porous cell RVE specimens. Next, the Gurson’s model
for porous material is presented for characterization of the sepa
rator and the electrodes with the active materials. Then the avail
able material data are discussed and adopted for the input of the
computational model. The details of the computational model are
presented. The computational results of the deformation pattern
and nominal stressestrain behavior are then compared with the
test results. Based on the detailed computational results, the micro
buckling modes of the component sheets are identiﬁed and an
approximate elastic buckling solution of a beam with a rigid
boundary on one side and an unattached elastic foundation on the
other side is developed and used to examine the micro buckling
mode of the cover sheets for justiﬁcation of the selection of the
length of the cell RVE specimens used in the tests in Ref. [17]. Based
on the computational model, the effects of the friction between the
(a) (b)
(c)
F
F
25 mm
4.642 mm
25 mm
x
y
z
x
y
z
y
z
Cover sheet Separator
Cathode Anode Cover sheet
Fig. 2. A schematic of (a) a pouch cell and (b) a cell RVE specimen with the dimensions, and (c) a side view of a small portion of a tenunit cell RVE specimen showing the individual
cell components. The large arrows indicate the compressive direction.
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340 327
cell components and the constrained surfaces on the deformation
pattern, plastic deformation, void compaction, and the loade
displacement curve are examined. The usefulness of the compu
tational model is then presented by further exploring the effects of
the initial clearance and biaxial compression on the deformation
patterns of cell RVE specimens. Finally, some conclusions are made.
2. Experiment
A detailed description of the structure of a lithiumion battery
module used for this investigation can be found in Refs. [17,18]. Also
note that the following deﬁnitions will be used throughout the
paper. A Single unit cell represents a basic cell containing one
cathode, one anode and a separator sheet with two aluminum
cover sheets with two accompanying separator sheets. A Ten unit
cell consists of ten basic cells containing ten cathode, ten anode,
twenty one separator and two aluminum cover sheets. In this
investigation, the ten unit cell is considered as a general cell RVE
specimen that represents a typical assembled pouch cell.
Each cell consists of ﬁve major components: cover sheet, anode,
cathode, separator and electrolyte. Since the electrolyte is difﬁcult
to handle during assembly due to the safety concern, all the cell and
module RVE specimens tested in this study were made without
electrolyte at the University of Michigan. Fig. 2(a) shows a sche
matic of a pouch cell with two cover sheets and a cell RVE specimen
with the xeyezcoordinate system. A cell RVE specimen with the
dimensions is shown in Fig. 2(b). The pouch cell has a layered
structure as schematically shown in Fig. 2(b). The zcoordinate is
referred to as the outofplane coordinate whereas the xand y
coordinates are referred to as the inplane coordinates. Fig. 2(c)
shows a side view of a small portion of a tenunit cell RVE spec
imen showing the individual cell components. The assembly of the
cell components in the generic cell RVE specimen as schematically
shown in Fig. 2(c) may be slightly different from those in usual
lithiumion cells for convenience of assembly of the purchased cell
components. However, generic cell RVE specimens with slightly
different assemblies should have the similar buckling, kink and
shear band mechanisms under constrained compression as dis
cussed later due to their layered structures. Constrained compres
sion tests were conducted for cell RVE specimens with the
dimension of 25 mm 25 mm 4.642 mm. The details of the test
setup and results of the inplane constrained compression tests are
discussed in Ref. [17] and are brieﬂy reviewed in the following.
Fig. 3 shows three nominal compressive stressestrain curves of
the cell RVE specimens tested at a displacement rate of
0.5 mm min
1
. The specimens showed almost a linear behavior in
the beginning with an effective elastic modulus of 188 MPa. Note
that the effective elastic modulus obtained from the composite
ROM is 190 MPa using the effective elastic moduli obtained from
the nominal stressestrain curves of cell components under in
plane constrained compression tests. When the strain reaches
about 2%, noticeable change of the slope takes place and the curves
continue to increase gradually up to the strains of 34%. Some minor
drops were observed during the stage after the linear region due to
the development of kinks and shear bands as shown in the defor
mation patterns recorded as discussed later. The trends of all three
curves are quite consistent.
Figs. 4(a)e(d) show the deformation patterns of a cell RVE
specimen at the nominal strain of 1% in the initial linear stage, at
the nominal strain of 2% where the slope changes, and at the
nominal strains of 10% and 15%. Figs. 4(e) and (f) show the front and
back views of the tested cell RVE specimen at the nominal strain of
34%. A careful examination of the deformation pattern shown in
Fig. 4(a) indicates the initial linear stage corresponds to the
development of smooth buckling for the cell components. As the
displacement increases toward the nominal strain of 2% where the
slope starts to level off, the cell RVE specimen shows the devel
opment of kinks or plastic hinges of the cell components against
the walls, as indicated in Fig. 4(b). The presence of the kinks pro
motes the macroscopic shear band formation (strain localization in
a narrow zone), as indicated in Fig. 4(b). The shear band formation
creates a physical mechanism to accommodate efﬁciently for the
compression displacement and hence induces the load drop. As the
strain continues to increase, more kinks and shear bands form
across the cell RVE specimen as shown in Figs. 4(c) and (d).
Figs. 4(e) and (f) show the front and back views of the tested cell
RVE specimen at the nominal strain of about 34%. As shown in the
ﬁgures, the kinks are fully developed as folds and many shear bands
can be identiﬁed. After the efﬁcient compaction mechanism of
shear band formation has been completed, further compression can
only be accommodated by the micro buckling of the cell compo
nents outside the shear band regions, as marked in Figs. 4(e) and (f),
and the compression in the shear band regions.
An idealized deformation process of the cell RVE specimen un
der an inplane constrained compression test is proposed in
Ref. [17] to explain the shear band formation and is brieﬂy reviewed
here. Figs. 5(a)e(c) show schematics of a cell RVE specimen before,
during, and after the shear band formation under inplane con
strained compression, respectively. Figs. 5(d)e(f) show the detailed
schematics of the shear band formation corresponding to
Figs. 5(a)e(c), respectively. As shown in Fig. 5(a), the cell RVE
specimen (shown in gray) forms shear bands (between two parallel
dashed lines) to accommodate the volumetric reduction under
constrained compression. During the deformation, the kink angle
a
(as shown in Fig. 5(e)) keeps deceasing from 90
toward to zero
while the shear band angle
q
(as shown in Fig. 5(e)) also decreases
from the initial
q
i
to the ﬁnal
q
f
by a small amount. In the shear
band, the cell components are subjected to a compressive strain in
the z
0
direction, a signiﬁcant amount of the shear strain in the y
0
ez
0
plane and a signiﬁcant amount of rotation. Here, y
0
and z
0
represent
the local material coordinates that are ﬁxed to the material. Outside
of the shear band, the cell components are subjected to compres
sive strains in the yand zdirections. Once
a
reaches to zero, further
compressive strains are achieved by the micro buckling of the cell
components outside of the shear bands and the void reduction and
shear in the shear band as
q
continues to decrease. This is illustrated
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0
5
10
15
20
25
30
35
Test result 1
Test result 2
Test result 3
Nominal stress (MPa)
Nominal st rain
02468
0
500
1000
1500
2000
2500
3000
3500
4000
Load (N)
Displacement (mm)
Fig. 3. Nominal compressive stressestrain curves of the cell RVE specimens tested at a
displacement rate of 0.5 mm min
1
(nominal strain rate of 0.0003 s
1
).
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340328
in Figs. 5(c) and (f). It should be noted that Figs. 5(a)e(f) are
idealized with the periodic shear band structures. In tests, the shear
bands do not form at the same time and the shear band angle
q
varies due to the imperfections of the specimens.
3. Gurson’s yield function for porous materials
The anode and cathode for this investigation are graphite coated
on copper foil and LiFePO
4
coated on aluminum foil, respectively.
The copper foil has a thickness of 9
m
m and the total thickness of the
anode sheet is 0.2 mm. The aluminum foil has a thickness of 15
m
m
and the total thickness of the cathode sheet is 0.2 mm. Both the
anode and cathode sheets are doubleside coated. The separator is
made of polyethylene with the porosity ranging from 36 to 44% and
a thickness from 16 to 25
m
m according to the manufacturer
speciﬁcation.
Fig. 6 shows SEM images of the graphite and LiFePO
4
on the
anode and cathode sheets, respectively, reported in Lai et al. [17].It
should be noted that both active materials on electrodes are in a
powder form held together by the binder and therefore possess a
high degree of porosity as seen in the SEM images. It is not the
intent of this paper to characterize the composition and
morphology of the active materials on the electrodes. For example,
the porosities of the active materials on the current collectors as
shown in Figs. 6(a) and (b) are difﬁcult to measure and characterize.
However, the electrodes with the porous active materials can be
computationally treated as homogenized porous sheets. The results
of the tensile tests of the cell component sheets will be used to
determine approximately the plastic parts of the stressestrain
curves of the component sheets as homogenized materials, and the
results of the constrained compression tests of the cell component
specimens will be used to determine approximately the compres
sive moduli of the component sheets as homogenized materials as
detailed later in this paper and in Lai et al. [17].
It should be mentioned that the electrode sheets are idealized as
homogenized porous sheets in the micro computational models
Fig. 4. Deformation patterns of a cell RVE specimen during a compression test at the displacement rate of 0.5 mm min
1
: (a) at the nominal strain of 1% in the initial linear stage, (b)
at the nominal strain of 2% where the slope changes, (c) at the nominal strain of 10%, (d) at the nominal strain of 15%, (e) at the nominal strain of 34% af ter the test (front view), and
(f) at the nominal strain of 34% after the test (back view).
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340 329
used for the cell RVE specimens in this paper and the computational
effort is quite extensive. It is possible toconsider the active materials
on the current collectors as particles for the electrode sheets in
computational models at smaller scales. However, this paper is
focused on development of computational models at the scales of
homogenized cell component sheets, and development of compu
tational models at smaller scales is outof the scope of this paper. The
Gurson’s model for porous materials is adopted to model the elec
trode sheets with the active materials as homogenized materials in
the ﬁnite element analyses. Also, the separator sheets used in the
cell RVE specimens are manufactured with a high degree of porosity
to hold electrolyte. Therefore, the Gurson’s model for porous
materials is also adopted to model the separator sheets as homog
enized materials in the ﬁnite element analyses. A brief description of
the Gurson’s model [20] is presented in the following.
Gurson [21] proposed a yield function
f
for porous materials
containing a small volume fraction of voids. In porous materials, the
void volume fraction fis deﬁned as the ratio of the volume of voids
Shear band
y
z
d
w
i
d
f
y'
z' y'
z' z'
y'
hs
y
z
(a) (b) (c)
(d) (e) (f)
Fig. 5. Schematics of a cell RVE specimen (a) before, (b) during, and (c) after inplane
constrained compression. (d)e(f) are detailed schematics showing the shear band
formation corresponding to (a)e(c), respectively. The yand zcoordinates are the global
coordinates and the y0and z0coordinates in (d)e(f) are the local material coordinates
rotating with the cell components.
Fig. 6. SEM images of (a) graphite and (b) LiFePO
4
on the anode and cathode sheets, respectively.
f
= 0 (Mises)
f
= 0.1
f
= 0.2
f
= 0.4
y
p
y
q
= 0 (Mises)
compression ( )
tension ( )
y
y

0
f
0
f
0
f
(a)
(b)
Fig. 7. (a). A schematic of the Gurson’s yield contour in the normalized hydrostatic
pressure (p)eMises stress (q) plane [20]. (b). A schematic of uniaxial behavior of a
porous material with a perfectly plastic matrix material and the initial void volume
fraction f
0
[20].
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340330
to the total volume of the material. The relative density of a ma
terial, r,deﬁned as the ratio of the volume of matrix material to the
total volume of the material, can also be used. Note that f¼1r.
The Gurson’s yield function
f
was later modiﬁed by Tvergaard [22]
to the form
f
¼q
s
y
2
þ2q
1
fcos hq
2
3p
2
s
y
1þq
3
f
2
¼0 (1)
where q¼ð3S:S=2Þ
1
.
2
represents the effective macroscopic
Mises stress, p(¼S:I/3) represents the macroscopic hydrostatic
pressure,
s
y
represents the ﬂow stress of the matrix material, which
is expressed as a function of the average equivalent plastic strain
3
pl
m
of the matrix for strain hardening materials, and q
1
,q
2
and q
3
are
the ﬁtting parameters. Here, Srepresents the deviatoric part of the
macroscopic Cauchy stress tensor S. The macroscopic Cauchy stress
Sis based on the current conﬁguration of a material element with
voids. For f¼0(r¼1), the material is fully dense, and the Gurson’s
yield function reduces to the Mises yield function. The model
generally gives physically reasonable results only for f<0.1
(r>0.9). Tvergaard [22] introduced the ﬁtting constants q
1
,q
2
and
q
3
to ﬁt the numerical results of shear band instability in square
arrays of cylindrical holes and axisymmetric spherical voids. One
can recover the original Gurson’s yield function by setting up
q
1
¼q
2
¼q
3
¼1. In the current investigation, q
1
¼1.5, q
2
¼1, and
q
3
¼q
1
2
¼2.25 [20].
Fig. 7(a) shows a schematic of the Gurson’s yield contour in the
normalized hydrostatic pressure (p)eMises stress (q) plane for
porous materials in comparison with that of the Mises material
model. The porous material model reduces to the Mises material
model as the void volume fraction freduces to zero. Fig. 7(b) shows
a schematic of the uniaxial behavior of a porous material with a
perfectly plastic matrix material and the initial void volume frac
tion f
0
. Here the yield stress is denoted as
s
y0
. The porous material
softens in tension and hardens in compression. The porous material
hardens in compression due to the reduction of the void volume
fraction. Phenomenological hyperfoam and crushable foam mate
rial models are available in ABAQUS. However, more material input
data are needed for these foam models and additional material data
for the cell components are not available. Therefore, the Gurson’s
material model is adopted here for modeling the separator and the
electrodes with the active materials.
4. Available material data for cell components
Tensile tests were conducted for the individual cell compo
nents such as anode, cathode, separator and cover sheets, and the
test results were discussed in detail in Lai et al. [18]. Inplane
constrained compression tests were also conducted for the
anode, cathode, separator, and cover sheets to estimate the
compressive elastic moduli, and the test results were discussed in
detail in Lai et al. [17]. Although these tests are constrained
compression tests, only the apparent elastic part of the stresse
strain responses appear to be useful to obtain the effective
compressive elastic moduli for individual components. The effec
tive compressive elastic moduli may account for the local micro
buckling that occurs at a very small load level for each component
sheet and has indistinguishable impacts to the measurable
macroscopic response. Due to the compressive loading of the cell
RVE specimens, the effective compressive elastic moduli are thus
used for the electrodes, separator and cover sheets in the ﬁnite
element analyses of the cell RVE specimens under constrained
compression tests.
For elast iceplastic materials, the plasticstrain hardening behavior
is essential forthe input of elasticeplastic ﬁnite element analyses. For
the current investigation, the elasticeplastic tensile stressestrain
data for the components obtained in Ref. [18] are used to deﬁne the
strain hardening behavior due to the difﬁculties to obtain such data
under uniaxial ‘unconstrained’compression tests. For the ABAQUS
solver, the tensile tests data must be converted to the true stress and
true strain format for elasticeplastic ﬁnite element analyses. There is
no simple way to convert the engineering stressestrain curves of the
anode, cathode and separator sheets with high porosity. The con
version to the true stressestrain curve is based on the usual
assumption of plastic incompressibility for metal plasticity for lack of
the detailed information on the detailed microstructure of the anode,
cathode and separator. With the composite rule of mixture for the
void and matrix and the assumption of the constant total volume of
the void and matrix, the engineering stressestrain curve is converted
to the true stressestrain curves. The anode and cathode fail at very
low strains. The separator is very thin and is expected not to
contribute signiﬁcantly to the overall load carrying capacity of cells
and modules. Therefore, the conversion to the true stressestrain
curve with the plastic incompressibility seems to be a reasonable
option for lack of further information.
The tensile and effective compressive moduli, tensile yieldstress
and Poisson’s ratio of the cell components are listed in Table 1. For
the linear part of each stressestrain curve, the modulus is calcu
lated based on each data point with respect to the origin of the
stressestrain curve. A stable average value for a range of the strain
of the apparent linear behavior is selected as the tensile modulus
for that speciﬁc material. The yield stresses for the materials of the
cell components are selected where the stresses deviate from the
apparent linear ranges. Poisson’s ratios of 0.33 for copper, 0.33 for
aluminum, 0.45 for polymer and 0.2 for the active layers are used to
obtain the effective Poisson’s ratio for the separator, anode, cath
ode, and cover sheets using the composite rule of mixture (ROM).
The effect of the void volume fraction on the Poisson’s ratio is
estimated by treating the void as a component with zero Poisson’s
ratio using the ROM as
n
ROM
¼X
n
i¼1
n
i
V
i
(2)
where
n
i
and V
i
are the Poisson’s ratio and the volume fraction of
the ith component, respectively. Table 1 lists the Poisson’s ratios
Table 1
Material properties used in the ﬁnite element analyses.
Tensile
modulus
(MPa)
Effective compressive
modulus (MPa)
Tensile yield
stress (MPa)
Poisson’s
ratio
Anode,
graphite/Cu
4700 83 2.11 0.21 (f¼0%)
0.17 (f¼20%)
0.13 (f¼40%)
Cathode,
LiFePO
4
/Al
5100 275 1.48 0.21 (f¼0%)
0.17 (f¼20%)
0.14 (f¼40%)
Separator 500 90 10.53 0.25 (f¼44%)
Cover sheet 5600 575 9.74 0.41
Table 2
Thicknesses and densities of the battery cell components.
Thickness, mm Density, kg m
3
Anode, graphite/Cu 0.2 934
Cathode, LieFePO
4
/Al 0.2 1712
Separator 0.02 795
Cover sheet 0.111 1338
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340 331
for the cell components for different values of the initial void vol
ume fraction f. The Poisson’s ratios listed for anode and cathode in
Table 1 are corresponding to the assumed void volume fraction f
listed in the parenthesis. It should be noted that the void volume
fractions of the active materials on electrodes are difﬁcult to
measure due to the fact that the graphite and lithium iron
phosphate particles are loosely bonded together by a weak binder.
The void volume fraction 44% of the separator provided by the
manufacturer is adopted here. The thicknesses and the densities of
the cell components are also listed in Table 2.
Figs. 8(a) and (b) show the representative tensile nominal and
true stressestrain data of the cell components, respectively, and
Fig. 8(c) shows the stresseplastic strain curves of the cell compo
nents used in the ﬁnite element analyses. For the electrodes, the
stresseplastic strain curves are provided up to the strain of failure
in the tests. In ABAQUS, the stresses are kept constant outside the
input strain range. In this case, this represents a perfectly plastic
extension of the stresseplastic strain curves of the electrodes since
they are apparently ﬂat near the failure strains. This automatic
extension of the input material data is used to avoid numerical is
sues that may arise due to error tolerance check used in regular
izing the userdeﬁned data in ABAQUS/explicit code.
As mentioned earlier for the micro and macro buckling analyses,
the compressive elastic moduli of the cell components are used and
are listed in Table 1. The cover sheet is modeled as the Mises ma
terial with the isotropic hardening rule of ABAQUS and the stresse
plastic strain curve shown in Fig. 8(c) is used. For the separator
sheets, the initial void volume fraction fis set at 0.44 based on the
manufacturer speciﬁcation. The plastic behaviors for the cell
components are provided based on the stresseplastic strain curves
shown in Fig. 8(c). The strain hardening behavior for the matrix
material is obtained by scaling the tensile stresseplastic strain
curves using the ROM as
s
y
Gurson
¼
s
y
1f(3)
where
s
y
and
s
y
Gurson
represent the ﬂow stress and adjusted ﬂow
stress of the matrix, respectively. The stresseplastic strain curve
based on equation (3) will be referred as ‘adjusted strain hardening
curve’. It should be mentioned again that the stresseplastic strain
curves are estimations based on the available information. Many
assumptions are made to obtain these curves. One goal of this
investigation is to understand the physical mechanisms of these
RVE specimens under constrained compressive tests. The details of
the various aspects of compression tests simulations for cell RVE
specimens are described in the following sections.
5. Computational models
The results of the compression tests of cell RVE specimens show
that the layers of the RVE specimens are deformed by multiscale
buckling phenomenon eboth layer micro buckling and global
macro buckling. Ideally, in a conﬁned space with no clearance and
only with the presence of the porosity in the separatorand the active
materials onthe electrodes, the dense parts of the layers(copper foils
in anodes, aluminum foils in cathodes, and aluminum and polymers
in cover sheets)would get the room for buckling only by compressing
the relatively softer and porous active materials on electrodes and
separator layers laterally or in the outofplane direction. However,
the initial microscopic gaps between the cell components also allow
some rooms for buckling. The following approach is adopted and
tested in developing the buckling model presented here. As the load
increases,the local buckling of the individual sheetsof the specimens
develops and then the kinks of the cell components start to form
laterally adjacent to the wall of the ﬁxture. The kinematics of
development of kinks and shear bands is an efﬁcient way to compact
these porous sheets with plastically incompressible inner copper or
aluminum foils. Based on the experimentalobservations [17], the cell
global buckling or shear bands come from the plastic hinges or sharp
bending due tothe rigid constraint of theﬁxture wall and the module
Nominal strain
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Nominal stress (MPa)
0
25
50
75
100
125
Separator
Cover sheet
Anode (Graphite/Cu)
Cathode (LiFePO4/Al)
True strain
0.00 0.05 0.10 0.15 0.20 0.25 0.30
True stress (MPa)
0
25
50
75
100
125
Separator
Cover sheet
Anode (Graphite/Cu)
Cathode (LiFePO /Al)
(a)
(b)
Plastic strain
0.00 0.05 0.10 0.15 0.20 0.25 0.30
Stress (MPa)
0
25
50
75
100
125
Separator
Cover sheet
Anode (Graphite/Cu)
Cathode (LiFePO4/Al)
(c)
Fig. 8. (a) The representative tensile nominal stressestrain data of the cell compo
nents obtained from Ref. [18], (b) estimated true stressestrain data based on (a), and
(c) the stresseplastic strain curves of the cell components used in the ﬁnite element
analysis.
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340332
global buckling comes from the smooth bending due to a more relax
environment from the soft foam padding. The formation of these
kinks and shear bands in computational simulations is the key to
properly simulate the buckling behavior of the cell RVE specimens
under constrained compression tests.
The simulation of the cell RVE (ten units) specimens under
compression tests will be presented here. The similar approach can
be used for module RVE specimens under compressive loading to
estimate the nominal stressestrain response that can be used as an
input for a less detailed modeling [19].Fig. 2(b) shows a schematic
of a cell RVE specimen with the dimensions. It should be noted that
the stackup of the cell components gives a total thickness of
4.642 mm. However, the constrained compression test ﬁxture has a
conﬁnement dimension of 25 mm 25 mm 5 mm. In the
beginning of the compression test, a clearance of 0.358 mm was
present in the lateral or thickness direction, and is modeled
accordingly in the ﬁnite element analyses.
Fig. 9(a) shows the ﬁnite element model setup for a cell RVE
specimen compression test using the ABAQUS/Explicit commercial
ﬁnite element code. The xeyezcoordinate system is also shown.
The explicit ﬁnite element solver is used for this simulation for a
better contact stability among all the thin sheets during the buck
ling and under large deformation. The vertical length 25 mm and
the thickness 4.642 mm of the ﬁnite element model are similar to
the cell RVE specimen. For computational efﬁciency, only a half of
the cell RVE specimen width of 25 mm is used in the ﬁnite element
model. However, it should be noted that the symmetric boundary
condition was not applied in the model due to the nature of the
problem. The nominal stress vs. nominal strain curves of the
computational simulations will be compared with those of the
tests.
The compression test ﬁxture is made of steel that has a very high
stiffness compared to the cell components. Therefore, the
conﬁnement surfaces are assumed to be rigid and modeled by
planar rigid surfaces. In the ﬁnite element model setup, the spec
imen mesh is surrounded by six rigid surfaces. The rigid surfaces
contacting with the edges of the cell RVE component sheets with
the normals in the xand ydirections have zero clearance. The rigid
surfaces contacting the cover sheets with the normal in the zdi
rection are 5 mm apart and provide a total of 0.358 mm initial
lateral clearance with 0.179 mm on each side. The reference nodes
of all the rigid surfaces except the top one have six degrees of
freedom constrained. The top rigid surface can only move in the
vertical direction and is given a velocity boundary condition. The
general contact algorithm of ABAQUS/Explicit is used to model the
contact interaction between the surfaces of the cell components
that contact with one another and with the rigid surfaces. All the
contact surfaces are assumed to be in friction contact with each
other and an appropriate value of the coefﬁcient of friction is used
in the simulations as a ﬁtting parameter. Fig. 9(b) shows detailed
views of the meshes of each layer. The anode, cathode and cover
sheets are modeled by linear hexahedral full integration solid ele
ments (C3D8 of ABAQUS). Only a single layer of elements are used
to model each layer and a mesh size of
D
x¼0.25 mm and
D
y¼0.25 mm is used. For the cathode and anode sheets,
D
z¼0.2 mm. For the cover sheets,
D
z¼0.111 mm. The thin sepa
rator is modeled by linear quadrilateral reduced integration shell
elements (S4R of ABAQUS) with a thickness of 0.02 mm for
convergence and computational efﬁciency.
The compression test speed of 0.008 mm s
1
is considered as a
quasistatic condition. Using the explicit dynamics solver to model
a quasistatic event requires some special considerations. It is
computationally impractical to model the process by a time step to
satisfy the CouranteFredricheLevy condition of numerical stability.
A solution is typically obtained either by artiﬁcially increasing the
loading rate or the speed of the process in the simulation, or
increasing the mass of the system, or both. A general recommen
dation is to limit the impact velocity to less than 1% of the wave
speed of the specimen, and a mass scaling of 5e10% is typical to
achieve a desirable stable time increment. Also the kinetic energy
of the deforming specimen should not exceed a small fraction (1e
5%) of the internal energy throughout the quasistatic analysis. The
densities of the cell components are very low and the mesh size in
this simulation is ﬁne enough to capture the micro and macro
buckling behaviors. Therefore, for a reasonable computational time,
the ﬁnite element analysis is conducted at a speed of 200 mm s
1
and with a uniform mass scaling of 100 times of the actual mass.
The deformation speed and the kinetic energy are very low and
meet the recommendations of the quasistatic analysis for the
explicit solver even though a higher mass scaling is used for
computational efﬁciency. Different percentages of mass scaling
were examined. The results showed some impact on the initial part
of the stressestrain response up to a strain of about 1.5% and the
results are generally comparable.
(a) (b)
x
y
z
Velocity boundary
condition is applied Anode
Cathode
Separator
Cover sheet
Cell RVE
Fig. 9. (a) The ﬁnite element model setup for a cell RVE specimen under constrained compression, and (b) detailed views of the meshes where the anode, cathode and cover sheets
are modeled by linear hexahedral solid elements and the separator sheet is modeled by linear quadrilateral reduced integration shell elements.
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340 333
6. Computational results
Fig. 10 shows the initial and deformed shape of the battery cell
model under quasistatic inplane compression and the corre
sponding experimental results. Initially, the cell RVE model is
conﬁned by six rigid surfaces as described earlier as shown in
Fig. 10(a). The top rigid surface is moved downward in the ydi
rection with a velocity boundary condition. Fig. 10(b) shows the
deformed shape of the model after the compressive displacement
boundary condition is applied and held with the initial void volume
fraction of f¼0.2 for the electrodes with the active materials at the
nominal strain of 34%. In this case, the initial void volume fraction
of f¼0.2 is used in estimating the Poisson’s ratios and adjusting the
strain hardening curves for the anode and cathode sheets with the
active materials. In the model, a coefﬁcient of friction of 0.1 is
adopted for all the contacting surfaces as a general value. Many
simulations with multiple combinations of the parameters such as
the void volume fraction and coefﬁcient of friction were conducted.
Only the combinations of parameters giving reasonable results will
be presented here. Fig. 10(c) is a zoomin view of (b). Fig. 10(d)
shows a deformed cell RVE test specimen after the inplane
compression test. The cell RVE specimen after the compression
test shown in 10(d) is one of the three cell RVE specimens tested.
Note that a different tested specimen is shown in Fig. 4. The spec
imen showed here has a fairly regular buckling pattern and was
selected for comparison of the buckling pattern with that of the
computational results. Regular buckling patterns are obtained from
the computational models since these computational models do
not have signiﬁcant irregularities or imperfections. The buckling
patterns of the deformed ﬁnite element model are found similar
and comparable to that of the test specimen.
Figs. 11(i)e(vi) show successive snapshots of the deformation of
the cell RVE specimen during the buckling simulation. Figs. 12(i)e
(vi) show the successive snapshots of the equivalent plastic strain
(PEEQ) of the cell RVE specimen during the buckling simulation. In
Fig. 11(ii) for the strain of 1.7%, the cover sheets on both sides of the
cell specimen appear to buckle independently but the buckling is
restricted by the rigid walls. The computational results not shown
here indicate that the ﬁve buckles shown in Fig. 11(ii) develop
successively one by one from the top to the bottom and the softer
neighbor separator, anode and cathode sheets buckle with the
stiffer cover sheets. In Fig. 11(iii) for the strain of 3.4%, the buckling
peaks or valleys appear to adjust and synchronize with the macro
buckling mode of the cell RVE specimen as a homogenized beam or
plate. The absolute value of the nominal stress starts to drop at the
strain of about 2% with the increasing compressive nominal strain
based on the computational results and this appears to be related to
the starting of the macro buckling of the cell RVE specimen as a
homogenized beam or plate.
Fig. 11(iv) for the strain of 11.9% shows that kinks start to form
adjacent to the cover sheets and shear bands are formed between
the opposite pairs of kinks. As shown in Figs. 11(v) and (vi) for the
strains of 22.1% and 34%, respectively, the kinks become folds and
the folds have different depths. The spacing between the folds in
Fig. 10. (a) A cell RVE half model is conﬁned by six rigid surfaces, (b) the deformed shape of the model after the compressive displacement is applied (with the effective elastic
compressive modulus and f¼0.2 for the electrode sheets with the active materials at the nominal strain of 34%), (c) a zoomin view of (b), and (d) a deformed cell RVE specimen
after the inplane compression test.
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340334
fact is not the same due to the friction and imperfections. The
plastic hinges or bends of the cell components are found smoother
in the computational models due to the large size of the elements in
the ﬁnite element analyses compared to those of the tests where
the bends are sharper with almost rectangular corners as shown in
Figs. 4(d) and 10(d) for two different tested cell RVE specimens.
The shear bands are formed in the sheets between the two
opposite kinks as schematically shown in Fig. 11(iv). As the defor
mation progresses, the shear bands in the computational models
become slightly wider in the middle of the specimen compared to
those of the tests due to the smoothing of the bends coming from
the large element size in the ﬁnite element analysis. It should be
noted that in order to capture the local bending more accurately,
more layers of linear elements would have been appropriate to
model each sheet. However, for computational efﬁciency and for a
very high length to thickness ratio of each sheet, only a single layer
of element is used for modeling each sheet to sufﬁciently capture
the micro and macro deformation patterns.
The compaction of the voids in the components and microscopic
gaps between the components, along with the initial clearances,
allows room for further compression in a conﬁned space. The kinks
grow up to certain depths and the surfaces collapse with further
compressive loading as shown in Fig. 11(v). On the other hand, the
stack of sheets that apparently are vertical between the two kinks
on the same side are carrying loads by further deformations as
shown in Figs. 11(iv)e(vi) and 12(iii)e(vi). The shape of this vertical
zone across the thickness direction appears to be triangularly
shaped whose apex is at the tip of the kink on the opposite side.
Fig. 12(vi) shows that at the end of the compression, the values of
the PEEQ are higher near the top of the specimen compared to
those near the bottom. This can be attributed to the friction effect
on the top portion due to the progress of compressive deformation.
Figs. 13(i) and (ii) show the distributions of the void volume
fraction at the nominal strains of 8.5% and 34% of the constrained
compression simulation, respectively. Only the void volume frac
tions of the anode and cathode sheets of the cell RVE specimen are
displayed in these plots. Fig. 13(i) shows that during the deforma
tion at the nominal strain of 8.5%, the voids along the outer
boundaries of the shear bands where large bending occurs are
consumed. Fig. 13(ii) shows that at the end of the compression at
the nominal strain of 34%, the void volume fraction decreases more
near the top of the specimen compared to that near the bottom.
This can be attributed to the friction effect on the top portion due to
the progressive nature of the compressive deformation.
Fig. 14 shows a comparison of the nominal stressestrain curve
from the ﬁnite element analysis with that of the test results ob
tained from Lai et al. [17]. As mentioned earlier, the nominal stresse
strain curve of the ﬁnite element analysis is based on the Gurson’s
material model with the initial void volume fraction f¼0.2 for the
electrodes with the active materials and a friction coefﬁcient of 0.1
based on a parametric study. The results of the parametric study
show that the formation of the kinks and shear bands affects the
stress where the slope of the nominal stressestrain curve of the
computational results changes whereas a higher coefﬁcient of
friction raises the nominal stress to a higher value at a large strain.
The results of the parametric study will not be reported here for
brevity. The SAE 60 class ﬁlter has been used to postprocess the
computational stressestrain responses to ﬁlter the computational
noise if present and for consistency in comparing the curves from
computations [20]. The computational results show that the
stresses drop slightly after the ﬁrst noticeable global buckling at a
strain of about 2% and this is in agreement with the experimental
results. After reaching a strain of about 2%, the global buckling for
the cell RVE specimen as a homogeneous beam begins. The stress
then gradually increases as the densiﬁcation or compaction con
tinues as the strain increases. The results are compared fairly well
with the test results in general. However, the computational
response drops slightly after the strain of 25%.
The importance of the establishment and validation of the
detailed computational model in this investigation can be
demonstrated by exploring two example cases in order to visualize
the effect of clearance and biaxial compression on the deformation
patterns of cell RVE specimens under constrained compression.
Only the deformation patterns of the two example cases are brieﬂy
presented here for demonstrating the usefulness of the computa
tional model to understand the underlying physics of the cell RVE
specimens under constrained compression.
The effect of the initial clearance on the shear band formation of
a cell RVE specimen is demonstrated by using three initial clear
ances of zero, 0.358 mm (for the current model) and 0.716 mm in
the ﬁnite element analyses. Figs. 15(a)e(c) show the deformation
patterns of the cell model under quasistatic inplane compression
for the three clearance cases at the nominal strain of 34%. As the
initial clearance in the ﬁnite element models decreases from
0.716 mm to 0.358 mm and then to zero, the number of kinks in
creases, the kink depth decreases, and the number of shear bands
(i) (ii) (iii)
(iv) (v) (vi)
Kink
Shear
band
Triangular
vertical zone
during initial
buckling
y
z
y
z
y
Fig. 11. The successive snapshots of the deformation of the cell RVE specimen during
the buckling simulation at the nominal strains of (i) 0%, (ii) 1.7%, (iii) 3.4%, (iv) 11.9%, (v)
22.1% and (vi) 34%.
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340 335
increases from 8, 10 to 15, respectively. The details of the compu
tational results will be reported with the corresponding experi
mental results in the future.
Fig. 16 shows the deformation pattern of a cell RVE specimen
under equal biaxial constrained compression based on the model
shown in Fig. 10(a). Here, the top and front rigid surfaces are moved
downward in the ydirection and horizontally in the xdirection,
respectively, with the velocity boundary conditions such that
compressive nominal strains are equal in both xand ydirections.
The kinks and shear bands are formed inclined to both xand y
Fig. 12. The successive snapshots of the equivalent plastic strain (PEEQ) of the cell RVE specimen during the buckling simulation at the nominal strains of (i) 1.7%, (ii) 3.4%, (iii) 6.8%,
(iv) 13.6%, (v) 23.8% and (vi) 34%.
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340336
directions in Fig. 16. The number of kinks and shear bands on side S
is about half compared to that of the side L due to the length ratio of
one half for sides S and L in the computational model. Fig. 16 also
shows the pattern of interactions of the shear bands initiated from
sides L and S. The details of the computational results will be re
ported with the experimental results for biaxial compression in the
future.
7. Discussions
Based on the experimental observations of the cell RVE speci
mens under inplane constrained compression, the physical
Fig. 13. The distributions of the void volume fraction at the nominal strains of (i) 8.5%
and (ii) 34% of the constrained compression simulation. Only the void volume fractions
of the anode and cathode sheets of the cell RVE specimen are shown.
Nominal strain
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
Nominal stress (MPa)
0
5
10
15
20
25
30
35
40
Gurson's model
f
= 0.2
Test result 1
Test result 2
Test result 3
Fig. 14. A comparison of the nominal stressestrain curve from the ﬁnite element
analysis using the Gurson’s material model with those of the test results. In the ﬁnite
element analyses, all the contact surfaces are assumed to be in friction contact with a
friction coefﬁcient of 0.1.
Fig. 15. The deformation patterns of a battery cell under quasistatic inplane
compression for three initial clearances of (a) zero, (b) 0.358 mm of the current model
and (c) 0.716 mm at the nominal strain of 34%. The ﬁnite element models are similar to
the model described in Fig. 10(a) with different initial clearances.
Fig. 16. The deformation pattern of the cell RVE specimen under equal biaxial con
strained compression based on the model shown in Fig. 10(a) at the nominal strain of
22% in the xand ydirections.
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340 337
mechanism to accommodate the compression starts with the
elastic buckling of the cell components. When a cell RVE specimen
is under inplane constrained compression, the component sheets
buckle independently with the lateral constraints from the
neighbor component sheets as indicated in Ref. [17]. Since the
component sheets were only packed together, each component
sheet can be treated as an individual sheet or thin plate under in
plane compression with the lateral constraints which can be
treated as unattached elastic foundations.
For the anode, cathode and separator sheets in the middle
portion of the cell RVE specimens, the buckling mode will be
dominated by the constraints on both sides of the sheets. Lai et al.
[17] presented the buckling load solutions for the cell RVE speci
mens by treating the cell component as a uniform straight beam
supported by two equal unattached elastic foundations under end
loads with both ends hinged based on the solution listed in Refs.
[23,24]. The cover sheets have only one unattached elastic foun
dation and are free to buckle to the unconstrained side due to the
small clearances in the die for the cell RVE specimen during the
test. However, the small clearances will limit the cover sheets to
fully develop a lower order buckling mode. For the neighbor anode,
cathode and separator sheets near the cover sheets, they can start
to buckle in a lower order mode with the cover sheets but will
also be constrained by the rigid walls through the cover sheets.
The detailed results of the ﬁnite element analyses indicate that
the cover sheets and the neighbor anode, cathode and separator
sheets are constrained by the rigid walls and buckle in a high order
mode.
Therefore, treating the cover sheet as a beam with one unat
tached elastic foundation on one side and a small or zero clearance
to a rigid wall on the other side appears to be a reasonable approach
to gain insight on the buckling behavior and the number of the
waves or half waves for the cell RVE specimens. The details of an
elastic RayleigheRitz buckling analysis are presented in Appendix
A. The results of the elastic buckling analysis indicate that the
number of waves, n, is proportional to the length lof the cell. In
other words, the wavelength of the buckling is independent of the
specimen length l. Since the cell RVE specimens buckle with mul
tiple half waves, the selection of the length for the cell RVE speci
mens to represent lithiumion battery cells with a full length under
inplane constrained compressive loading conditions in Lai et al.
[17] appears to be reasonable.
8. Conclusions
In this paper, computational models are developed for simula
tions of representative volume element (RVE) specimens of
lithiumion battery cells under inplane constrained compression
tests. First, the loadedisplacement data and deformation patterns
for cell RVE specimens under inplane constrained compression
tests are brieﬂy reviewed. For the corresponding ﬁnite element
analyses based on ABAQUS, the effective compressive moduli for
cell components are obtained from inplane constrained
compressive tests, the Poisson’s ratios for cell components are
based on the rule of mixture, and the stresseplastic strain curves of
the cell components are obtained from the tensile tests and the rule
of mixture. The Gurson’s material model is adopted to account for
the effect of porosity in separators and in the active layers of anodes
and cathodes. The computational results show that the computa
tional models can be used to examine the micro buckling of the
component sheets, the macro buckling of the cell RVE specimens,
and the formation of the kinks and shear bands observed in ex
periments, and to simulate the loadedisplacement curves of the
cell RVE specimens. The computational results also suggest the
micro buckling of the component sheets controls the macro
buckling of the cell RVE specimens and then the formation of the
kinks and shear bands. The initial micro buckling mode of the cover
sheets in general agrees with that of the approximate elastic
buckling solution of a beam with a rigid boundary on one side and
an unattached elastic foundation on the other side. The elastic
buckling solution indicates that the buckling wavelength is a
function of the elastic bending rigidity and the outofplane elastic
modulus of the cell RVE specimens. The results further suggest that
the length of the cell RVE specimens is appropriately selected and
the constrained compressive behavior of the cell RVE specimens
can represent that of battery cells with a full length. Based on the
computational models, the effects of the friction between the cell
components and the constrained surfaces on the deformation
pattern, plastic deformation, void compaction, and the loade
displacement curve are identiﬁed. Finally, the usefulness of the
computational model is demonstrated by further exploring the
effects of the initial clearance and biaxial compression on the
deformation patterns of cell RVE specimens.
Acknowledgments
Helpful discussions with Yibing Shi, Guy Nusholtz, and Ronald
Elder of Chrysler, Saeed Barbat, Bill Stanko, Mark Mehall and Tau
Tyan of Ford, JenneTai Wang, Ravi Nayak, Kris Yalamanchili and
Stephen Harris of GM, Christopher Orendorff of Sandia National
Laboratory, SeungHoon Hong of University of Michigan, and
Natalie Olds of USCAR are greatly appreciated.
Appendix A. Buckling of a beam on an elastic foundation and
rigid boundary
Fig. A1 shows a uniform straight beam under end loads and
with one unattached elastic foundation on one side and a rigid
boundary on the other side. Both ends are hinged and the beam
is supported by the elastic foundation through the lateral pres
sure proportional to the deﬂection in the ydirection. Here, k
represents the spring constant of the elastic foundation on one
side of the beam and is deﬁned as the lateral force per unit plate
length per unit deﬂection of the neighbor components in the
outofplane direction. The spring constant kcan be expressed in
terms of the outofplane elastic modulus Eof the cell RVE
specimens as
k¼Eb
h(A1)
where brepresents the width of the specimen, and hrepresents the
thickness of the neighbor components.
In calculating the critical value of the compressive force for a
beam with an unattached elastic foundation on one side and a rigid
boundary on the other side, the energy method can be used to
develop an approximate solution [24]. The potential energy func
tion for the beam can be expressed in terms of the strain energy of
beam bending, the strain energy of the elastic foundation, and the
work done by the compressive force as
P
¼EI
2Z
l
0
d
2
v
dx
2
!
2
dxþk
2Z
l
0
v
2
dxP
2Z
l
0
dv
dx
2
dx(A2)
where
P
is the potential energy of the system, Eis the modulus
of elasticity of the beam, Iis the moment of inertia of the beam, v
represents the deﬂection of the beam in the ydirection, lis the
length of the beam, kis the spring constant for the elastic un
attached foundation, and Pis the compressive force. Note that
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340338
the friction effects are not considered in this simple beam
analysis.
According to the computational results for the cell RVE speci
mens with a small or zero tolerance in the die, the buckling mode
appears to be periodic. The deﬂection vof the beam must be pos
itive due to the rigid boundary on the left side as shown in Fig. A1.
The deﬂection vis assumed in a form as
v¼a
n
sin
2
n
p
x
l(A3)
where nis an integer and a
n
is a coefﬁcient. Substituting the
deﬂection vin equation (A3) into equation (A2) and evaluating the
integrals, the potential energy becomes
P
¼a
2
n
EI n
4
p
4
l
3
þ3
16 kl Pn
2
p
2
4l(A4)
For the minimum potential energy, the critical buckling load is
determined at v
P
/va
n
¼0as
P¼4EI
p
2
l
2
n
2
þ3
16
kl
4
n
2
p
4
EI(A5)
where the integer nrepresents the number of waves as indicated in
equation (A3). It should be noted that in the buckling analysis in Lai
et al. [17], the value of mcorresponds the number of half waves for
the buckling of the entire cell RVE specimens or the number of half
waves of the anode, cathode and separator sheets in the middle
portion of the cell RVE specimens.
Following the argument presented in Ref. [24] with consider
ation of nas an integer, the value of nat which the number of waves
changes from nto nþ1 giving the same value of Pcan be obtained
from equation (A5) as
n
2
ðnþ1Þ
2
¼3
16
kl
4
p
4
EI (A6)
The solution for equation (A6) is expressed as
n¼1
20
@ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1þ4l
2
p
2
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
3
16
k
EI
r
s11
A(A7)
Equation (A7) is similar to the solution for the number of half
waves for the buckling of a beam supported by an attached or
unattached elastic foundation as listed in Timoshenko [24] and
used in Lai et al. [17].
Equation (A5) is plotted in Fig. A2 to demonstrate the change of
the compressive load Pwith nusing the values of k,l,Eand Ithat are
listed in Table A1 based on the experimental results presented in Lai
et al. [17]. Since nis an integer, equation (A5) is discrete in nature.
Fig. A2 shows that the solution of nof 12 in equation (A7) corre
sponds to the minimum or critical buckling load for the beam.
However, when nis treated as a real number, equation (A5) be
comes differentiable and the value of nat which the minimum or
critical compressive load Pcan be determined. Therefore, consid
ering nas a real number, vP/vn¼0 gives
n¼3k
16EI
1
4
l
p
(A8)
It should be noted that when 1 is neglected on the left hand side
of equation (A6) for large n’s, equation (A6) becomes equation (A8).
For the value of nin equation (A8), the minimum or critical buckling
load P
c
can be determined as
P
c
¼2ﬃﬃﬃ
3
pðkEIÞ
1
2
(A9)
Equations (A7) and (A8) give the values of n11.62 and 12.11,
respectively, based on the values of k,l,Eand Ithat are listed in
Table A1 and are obtained from the experimental results pre
sented in Ref. [17]. Both equations give the same number of waves
of 12 for the cover sheets for the minimum or critical buckling
load. From the computational results for the cell specimen with
the zero clearance, the number the waves for the initial buckling
mode of the cover sheets is 8 which is comparable to the
approximate solution obtained from the RayleigheRitz method
presented above. Equation (A8) appears to be a reasonable esti
mation for the current investigation and simple enough to show
that the number of waves, n, is proportional to the length lof the
cell.
Equation (A8) can be rewritten as
l
n¼16EI
3k
1
4
p
(A10)
As indicated in equation (A10), the wavelength, l/n, is inde
pendent of the specimen length. For the cell RVE specimens with
the small clearance of 0.358 mm, the number of half waves for the
buckling for the entire cell specimen is 7 and 10 from experimental
and computational results, respectively. The computational results
also show that as the clearance increases, the number of half waves
decreases. Since the cell RVE specimens buckle with multiple half
waves, the selection of the length for the cell RVE specimens in the
experimental investigation of Lai et al. [17] appears to be reason
able based on equation (A10).
Fig. A1. A schematic of a uniform straight beam with a rigid boundary on one side and
an unattached elastic foundation on the other side under end loads. Both ends are
hinged and the beam is supported by the elastic foundation through the lateral
pressure proportional to the deﬂection in the ydirection.
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340 339
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Table A1
The values of the parameters for the cell RVE specimens used in the elastic buckling
solution for calculation of the critical buckling load and the number of waves.
Parameters Value
h
cover sheet
0.111 mm
h
anode
0.200 mm
h
cathode
0.200 mm
h
separator
0.020 mm
h
foundation
¼10h
anode
þ10h
cathode
þ21h
separator
þ1h
cover sheet
4.531 mm
E(¼E
cover sheet
) 575 MPa
Ið¼ I
cover sheet
Þ¼
bh
3
cover sheet
12
2.85E03 mm
4
b(¼b
cell
)25mm
l(¼l
cell
)25mm
E
cell
(from outofplane cell RVE compression tests) 8.5 MPa
Equation (A1):k¼E
cell
b
h
foundation
46.899 MPa
Fig. A2. The compressive load Pas a function of the number of waves, n.
M.Y. Ali et al. / Journal of Power Sources 242 (2013) 325e340340