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Strain rate effects on sandwich core materials: An experimental and analytical investigation

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Acta Materialia 51 (2003) 1469–1479www.actamat-journals.com
Strain rate effects on sandwich core materials:An experimental and analytical investigation
U. Chakravarty, H. Mahfuz
∗
, M. Saha, S. Jeelani
Tuskegee University, Department of Mechanical Engineering, Room 103, James Center, Tuskegee 36088, USA
Received 25 July 2002; received in revised form 8 November 2002; accepted 18 November 2002
Abstract
Poly-vinyl chloride (PVC) based closed-cell foams were tested at different strain rate under compression loadingranging from 130–1750 s
1
using a modiﬁed Split Hopkinson Pressure Bar (SHPB) apparatus, consisting of polycarbon-ate bars. Foams with different density and microstructure were examined. The attainment of stress equilibrium withinthe specimen at various strain rates was examined. It was found that the stress equilibrium was reached early at lowerstrain rate as compared to higher strain rate. Both the peak stress and absorbed energy were found to be dependent onfoam density and strain rate, although foam density was found to be a more dominating factor. A model based on unitcell geometry of the closed-cell foam was also developed to predict the absorbed energy at high strain rate. Theproposed model is found to be promising in predicting the energy absorption during high strain rate loading.
©
2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.
Keywords:
PVC foam; High strain rate; Split Hopkinson Pressure Bar (SHPB); Closed-cell foam modeling
1. Introduction
Cellular foams are increasingly being used ascore materials in conjunction with high strengthskins, to produce strong, stiff and lightweight sand-wich structures for aerospace and marine appli-cations. Due to their higher impact resistance andenergy absorbing capability, cellular foams arebeing also used extensively in automobile appli-cations. The foam can be subjected to high strainrates, which are typically a couple of orders of
∗
Corresponding author. Tel.:
+
1-334-727-8985; fax:
+
1-334-724-4399.
E-mail address:
ememah@tusk.edu (H. Mahfuz).
1359-6454/03/$30.00
©
2003 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved.doi:10.1016/S1359-6454(02)00541-4
magnitude higher than those of quasi-static rates.Again, the mechanical behavior of foam dependson the structure of the cell and the density of foam.Cellular foam can be found as closed-cell or open-cell conﬁguration. Closed-cell foams can absorbmore energy than open-cell foams because of theentrapped gas within the cell, which acts as amedium of energy absorption. During compressionloading, the entrapped gas inside the cell can com-press either isothermally or adiabaticallydepending upon the strain rate. Thus, the mechan-ical properties of cellular foams in the context of these strain rates will be essential to the designers.Extensive studies were reported previously forboth metallic and polymeric foams at quasi-staticand lower strain rate by many authors [1–3]. High
1470
U. Chakravarty et al. / Acta Materialia 51 (2003) 1469–1479
strain rate data for metallic foams have beenreported in the references [4–10]. Hall and Guden[4] investigated the effect of density and strain rateon compressive failure stress and energy absorp-tion capacity of closed-cell aluminum foam up to2000 s
1
. Despande [5] and Paul [8] have alsocharacterized metallic foam under compressiveloading at different strain rates. Recently, a fewresearchers investigated polymeric foams atmedium and high strain rate [11–15]. Daniel andRao [11] and Zhao [12] examined different typesof polymeric foams under compressive loading upto 1000 s
1
and 250 s
1
, respectively. Thomas andMahfuz [13] have investigated PVC based foamsusing modiﬁed SHPB up to strain rate of 2000 s
1
.Thomas and Mahfuz [14] have also examinedstrain rate effect and temperature sensitivity of PVC based foams at high strain rate up to 1400s
1
.Traditionally the energy absorption at higherstrain rate has been calculated by using the conven-tional stress–strain relationship derived fromSHPB, which was based on the assumption that thestress equilibrium exists within the specimen dur-ing the test, and is independent of the strain rate.However, it is reported that the attainment of stressequilibrium is a function of strain rate and speci-men length [11]. Thus, the conventional stress–strain approach may not yield accurate estimationof energy absorption at high strain rate. Recently,Park and Zhou [16] have estimated the energyabsorption at high strain rate using the forces andparticle velocity histories and found it to be moreaccurate. In the present study, an analytical formu-lation based on unit cell geometry for closed-cellfoam has been developed to determine the energyabsorption at various strain rates. The absorbedenergy was also calculated using both force andparticle velocity approach and the conventionalstress–strain method to assess the validity of theproposed unit cell model. Time required to reachequilibrium for different strain rates was alsoinvestigated.
2. Materials and test method
Three different grades of PVC based closed-cellfoams of densities 75, 130, and 300 kg/m
3
weretested in compression using a modiﬁed Split Hop-kinson Pressure Bar (SHPB) setup, having all poly-carbonate striker, incidence and transmitter bars.Two different grades of foams with density of 130kg/m
3
but with different microstructure were alsotested. The specimens were cut from 12.5-mmthick panel with 12.5-mm square cross-sectionalarea, and tested in the thickness direction.
3. Theoretical consideration
3.1. Split Hopkinson Pressure Bar (SHPB)
Conventional SHPB setup uses a long incidenceand transmitter steel bar, while a relatively shortspecimen is placed in between the bars. Schematicdiagram of the bar-specimen assembly used inSHPB setup is shown in Fig. 1. Strain gages areplaced in the incidence and transmitter bar to rec-ord strains as a function of time. The impact of thestriker bar on incidence bar results in a compress-ive incidence pulse,
e
I
(
t
); a portion of this pulsegets reﬂected as a tensile pulse,
e
R
(
t
) at the inter-face of the incidence bar and specimen, and therest of the incidence pulse is transmitted as a com-pressive pulse,
e
T
(
t
) through the transmitter barafter traveling [
e
I
(
t
)
+ ε
R
(
t
)] pulse within the speci-men [12,17–20].The classical assumption of homogeneous stressand strain ﬁelds in the specimen yields an averagestress and strain relationship as follows [20,21].
s
S
(
t
)
A
b
A
S
E
b
e
T
(
t
) (1)
e
S
(
t
)
2
C
b
L
S
t
0
e
R
(
t
)
dt
(2)where
A
b
, E
b
, C
b
are the cross-sectional area, themodulus, and the wave speed of the bar material,and
A
s
, L
s
are cross-sectional area and the lengthof the specimen.
3.2. Impedance ratio and the use of polycarbonate bars
The stress–strain relationships, shown in Eqs.(1) and (2), are only applicable when the homogen-
1471
U. Chakravarty et al. / Acta Materialia 51 (2003) 1469–1479
Fig. 1. Schematic diagram of bar-specimen assembly used in typical SHPB Set-up.
eity of stress and strain ﬁelds exists within thespecimen. Estimation of energy absorption basedon non-homogenous stress–strain relationship willnot be accurate. It is reported that the measurementof the input and output forces at two ends of thespecimen can be used as a tool to check the stresshomogeneity. The forces and particle velocities atboth ends of the specimen can be calculated asdetailed in reference [20]. Daniel and Rao [11]have shown through wave propagation analysisthat the steel–foam–steel conﬁguration takes aboutﬁve times higher than polycarbonate–foam–poly-carbonate conﬁguration to reach stress equilibrium.This phenomenon can be visualized through theconcept of impedance ratio,
r
, as explained in ref-erences [11,12]. The impedance ratio is muchlower for steel–foam–steel conﬁguration whencompared to polycarbonate–foam–polycarbonateconﬁguration. A setup with low impedance ratioyields weak transmitted wave [21–24]. Table 1shows the impedance ratio and the ratio of trans-mitted to incidence signal for both the steel–foam–steel and polycarbonate–foam–polycarbonate sys-tem. It is seen that the impedance ratio for thepolycarbonate system is almost two order magni-tude higher than the steel system, which conse-
Table 1Comparison of the steel–foam–steel and polycarbonate–foam–polycarbonate systemFoam Steel–foam–steel Polycarbonate–foam–polycarbonateImpedance
T
/
I
Impedance
T
/
I
ratio,
r
ratio,
r
R75 7.52x10
4
0.0030 0.0190 0.0732H130 2.12x10
3
0.0085 0.0533 0.1922R300 4.49x10
3
0.0178 0.1125 0.3636
quently improves the transmitted signal by about20 to 25 times for the polycarbonate system. Thus,the polycarbonate bars are far better than the tra-ditional steel system for testing foam specimens.
3.3. Energy absorption
A conventional stress–strain approach can beused to calculate the energy absorption accuratelywhen the difference between the two forces is verysmall. Otherwise, the force and particle velocityhistories at two bar-specimen interfaces need to beused [16].
3.3.1. Force–particle velocity approach
The work carried out by the input bar to thespecimen can be written as
W
in
(
t
)
t
0
F
in
(
t
)
u
in
(
t
)
dt
A
b
E
b
t
0
[
e
I
(
t
) (3)
e
R
(
t
)]
u
in
(
t
)
dt
On the other hand, the work done by the specimento the output bar can be written as
W
out
(
t
)
t
0
F
out
(
t
)
u
out
(
t
)
dt
(4)
A
b
E
b
t
0
e
T
(
t
)
u
out
(
t
)
dt
Thus, the total energy absorbed by the specimen isthe difference between
W
in
(
t)
and
W
ou
t
(t)
as follows
E
absorbed
(
t
)
W
in
(
t
)
W
out
(
t
)
A
b
E
b
t
0
[
e
I
(
t
) (5)
1472
U. Chakravarty et al. / Acta Materialia 51 (2003) 1469–1479
e
R
(
t
)]
u
in
(
t
)
dt
A
b
E
b
t c
e
T
(
t
)
u
out
(
t
)
dt 3.3.2. Considering foam cell geometry
An optical micrograph of a typical closed cellPVC foam is shown in Fig. 2a. The ﬁgure showscell edges, cell faces and their relative dimensions,as they are viewed perpendicular to cell faces. Anidealized cubic model of the closed cell foam asshown in Fig. 2b can be extracted from thisgeometry for theoretical consideration. The cubicmodel has cell edge thickness of
t
e
, cell face thick-ness of
t
f
, and edge length of
l
. The total energyabsorbed can be calculated based on the defor-mation (elastic and plastic) of the cell walls andthe compression of the ﬂuid within the cell as thefoam deforms [1].
3.3.2.1. Deformation of cell walls
The defor-mation of the cell wall consists of linear elastic andplastic regimes. Only a very small amount of energy is absorbed in the linear regime, whereasmost of the energy is absorbed due to cell collapseby buckling, yielding or crushing at near-con-stant load.
3.3.2.1.1. Energy absorption due to elasticdeformation
The total energy absorbed up to yield point canbe determined from cell edge bending and facestretching work as follows [1].
Fig. 2. (a) Optical micrograph of a typical cubic cell closed cell H130 foam. (b) Schematic view of a model for closed-cell foam.
[
E
absorbed
]
y
1
l
3
C
1
12
S
d
2
C
2
12
E
s
e
2
y
V
f
(6)where
S,
d
, E
s
are the bending stiffness, deﬂectionat yield point, and modulus of foam material,
V
f
isthe volume of the cell face,
e
y
is yield strain and
C
1
,
C
2
are constants. After substitution,
e
y
= δ
l
,S
E
s
I l
3
, V
f
=
l
2
t
f
, and
I
t
4
e
, Eq. (6) can be reduced to[
E
absorbed
]
y
C
5
E
s
t
e
l
4
d
l
2
C
4
E
s
d
l
2
t
f
l
(7)where
C
4
,
C
5
are constants. Again, the edge andface thickness are related to the relative foam den-sity and the cell length as follows:
t
e
l
a
1
f
0.5
r
∗
r
s
0.5
(8a)
t
f
l
a
2
(1
f
)
r
∗
r
s
(8b)where
f
is the volume fraction of the solid in thecell edge,
r
∗
and
r
s
are density of foam and solidmaterial, and
a
1
,
a
2
are constants. Finally, theexpression for the absorbed energy due to elasticdeformation can be written as:[
E
absorbed
]
y
a
E
s
f
2
r
∗
r
s
2
e
2
y
b
E
s
(1 (9)
f
)
r
∗
r
s
e
2
y
1473
U. Chakravarty et al. / Acta Materialia 51 (2003) 1469–1479
where
a
, and
b
are constants containing
a
1
, a
2
, C
4
,and
C
5
.3.3.2.1.2. Energy due to plastic deformation
During plastic deformation, the cell edge buck-ling and face buckling occurs. However, the contri-bution of the face buckling on total energy is verysmall and can be neglected. Thus, the total energyabsorbed due to edge buckling can be written as:[
E
absorbed
]
y
D
e
D
e
y
s
crit
d
e
(10)where
D
is the densiﬁcation strain. Using therelationships,
s
crit
=
F
crit
A
face
,
F
crit
=
n
2
p
2
E
s
I l
2
,
I
t
4
e
,
A
face
=
l
2
, and
r
∗
r
s
t
e
l
2
, the expression forabsorbed energy can be written as:[
E
absorbed
]
y
D
y
E
s
r
∗
r
s
2
(
e
D
e
y
) (11)where
y
is a constant containing
n,
p
etc.
3.3.2.2. Energy absorption due to compression of entrapped gas
The contribution of the gas inside the cell onenergy absorption cannot be neglected. Thetrapped gas has a profound effect on energyabsorption capacity for closed-cell foams. Let usconsider a closed-cell foam with initial volume,
V
0
,relative density,
r
∗
r
s
, and initial gas pressure,
P
0
.The volume will decrease from
V
0
to
V
due to theapplication of axial compressive strain, accordingto the following equation:
V V
0
1
e
(1
2
n
∗
) (12)where
n
∗
is Poisson’s ratio. The volume of the gasinside the cell will also decrease from
V
0
g
to
Vg
due to the application of compressive strain as fol-lows,
V
g
V
0
g
1
e
(1
2
n
∗
)
r
∗
r
s
1
r
∗
r
s
(13)If the pressures of gas before and after com-pression are
P
0
and
P
, respectively, then from Boy-le’s law assuming isothermal compression process,we can write
PV
g
P
0
V
0
g
(14)Using Eqs. (13) and (14) the change of pressure,
P
, within the cell can be written as
P
P
0
e
(1
2
n
∗
)1
e
(1
2
n
∗
)
r
∗
r
s
(15)Energy absorption due to isothermal com-pression of gas inside the cell occurs at two differ-ent stages similar to cell walls; elastic and plasticdeformation. The expression of energy absorptionduring elastic deformation of the cell walls can bederived after integrating the change of pressure of inside gas with respect to strain as follows [1]:[
E
absorbed
]
y
e
y
0
P
/
d
e
P
0
(16)
e
y
1
r
∗
r
s
1
2
n
∗
ln1
r
∗
r
s
1
e
y
(1
2
n
∗
)
r
∗
r
s
During plastic deformation, foam specimendeform plastically with constant load, thus, one canneglect the effect of Poisson’s ratio. Thus, theexpression for absorbed energy can be written as:[
E
absorbed
]
y
D
e
D
e
y
P
/
d
e
e
D
e
y
P
0
e
1
e
r
∗
r
s
d
e
(17)
P
0
(
e
D
e
y
)
(1
r
∗
r
s
)ln1
e
D
r
∗
r
s
1
e
y
r
∗
r
s
3.3.2.3. Total absorbed energy
For isothermal elastic and plastic deformation,

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