EFFECT OF RELATIVE DENSITY ON THE DYNAMIC IMPACT BEHAVIORS OF CLOSED-CELL FOAM
Shilong Wang CAS Key Laboratory of Mechanical Behavior and
Design of Materials, University of Science and Technology of China
Hefei, Anhui 230026, PR China
Yuanyuan Ding CAS Key Laboratory of Mechanical Behavior and
Design of Materials, University of Science and Technology of China
Hefei, Anhui 230026, PR China
Changfeng Wang Continuous Extrusion Research
Center, Dalian Jiaotong University Dalian, Liaoning
116028, PR China
Zhijun Zheng* CAS Key Laboratory of
Mechanical Behavior and Design of Materials, University of
Science and Technology of China
Hefei, Anhui 230026, PR China
Jilin Yu CAS Key Laboratory of
Mechanical Behavior and Design of Materials, University of
Science and Technology of China
Hefei, Anhui 230026, PR China
* Corresponding author. Email: [email protected] (Z.J. Zheng)
ABSTRACT
Dynamic behaviors of closed-cell foam are investigated
with cell-based finite element models based on the 3D Voronoi
technique. The typical deformation feature of cellular structures
under high-velocity impact is layer by layer collapse, like a
shock wave propagating in the specimen. The one-dimensional
velocity distribution of the structure is calculated to characterize
the propagation of shock front and thus the shock wave speed is
determined quantitatively. It is found that the shock wave speed
has intense dependence on the impact velocity for a specific
relative density. The difference between the shock wave speed
and the impact velocity is asymptotic to a constant as the impact
velocity increases. This constant can be therefore regarded as a
dynamic material parameter. The influence of relative density
on this dynamic material parameter is investigated. The results
show that the shock wave speed at a specific impact velocity
increases with the increase of the relative density of cellular
structure in a certain extent. An expression of the shock wave
speed with respect to the impact velocity and relative density is
obtained. The dynamic strain hardening parameter is lower than
that in the quasi-static one, which indicates different
mechanisms of the deformation under high-velocity and quasi-
static loadings.
INTRODUCTION
Cellular materials especially metallic foams have been
applied to many fields involving impact and blast protection,
such as military, automotive, and packaging applications [1], for
their excellent mechanical properties. The existence of pores
makes cellular materials have high specific energy absorption.
The nearly stable load with long compression displacement of
metallic foams [2] provides an opportunity to be an ideal
material that is widely used where structures need to be
protected or to attenuate unfavorable energy [3]. Adequate
knowledge of metallic foams especially the dynamic responses
is the premise to guide and design materials that satisfy special
demands.
The layer-wise collapse and strength enhancement are the
two typical features of cellular materials under high-velocity
impact [4]. A simple one-dimensional shock model was
proposed by Reid and Peng [5] to explain the strength
enhancement of wood under impact loading and a rate-
independent, rigid−perfectly plastic−locking (R-P-P-L)
idealization was further proposed to provide a first order
understanding of its dynamic response. Subsequently, several
shock models [6-9] have been developed to characterize the
dynamic responses of cellular materials more accurately.
Proceedings of the ASME 2016 35th International Conference on Ocean, Offshore and Arctic Engineering OMAE2016
June 19-24, 2016, Busan, South Korea
OMAE2016-54562
1 Copyright © 2016 by ASME
However, quasi-static compressive stress-strain curves are
frequently used in these simple shock models. It has been
shown that under high-velocity impact, metallic foams exhibit a
new kind of rate-sensitivity [10], so using the quasi-static stress-
strain relations loses the rationality. Consequently, experimental
tests [11] or cell-based numerical simulations [10, 12-13] have
been demonstrated to understand the features of the shock wave
propagation in cellular materials. Under dynamic impact, the
densification strain of the metal foams could reach to a higher
value than that predicted theoretically using quasi-static stress-
strain curves and the micromechanical features should be taken
into consideration to acquire the more realistic results [14]. For
such reason, a method to obtain the local strain field was
developed by Liao et al. [12] to characterize the deformation
feature of irregular honeycombs quantitatively. The shock wave
speed of the honeycomb under impact was calculated and the
results show that the difference between shock wave speed and
the corresponding impact velocity is asymptotic to a constant as
the impact velocity increase to a critical value. Barnes et al. [11]
and Gaitanaros et al. [13] adopted a linear Hugoniot relation to
fit the relationship between impact velocity and shock wave
speed. Based on the dynamic features and the conservation
relations across the shock front, Zheng et al. [10] proposed a
rate-dependent, rigid–plastic hardening (D-R-PH) shock model
with the dynamic stress-strain relation of cellular materials
written as:
d
0 2( )
(1 )
D
, (1)
where D and σd
0 are the dynamic hardening parameter and the
dynamic initial crushing stress, respectively. The mechanism of
the impact velocity sensitivity of cellular materials was
explained as the layer-wise collapse deformation and its
interactions. These findings strengthen the understanding of the
features of the shock wave propagation through cellular
material under dynamic loading, but it is still unclear how the
material parameters (e.g. the relative density) affect the values
of shock wave speed and the dynamic behaviors of cellular
structures.
In this paper, a 3D Voronoi structure is used to study the
deformation features and velocity distributions of cellular
materials. The shock wave speed is then calculated. The effect
of relative density on the shock wave speed of metallic foams
under constant-velocity impact is investigated. A quantitative
relation of the shock wave speed with respect to the impact
velocity and the relative density is established. Finally, the
strain hardening behavior and the strain behind shock front of
the cellular structures are investigated.
NUMERICAL MODELS
The meso-mechanical model of cellular material is
constructed by using 3D Voronoi technique [15]. Since the
principle of 3D Voronoi configuration has the same physical
nature with the foaming process of metal foams, Voronoi
structures could well characterize the meso-mechanical
structures of foams. And also, based on cell-based finite
element models, the dependence of one parameter on the
concerned response of cellular structures could be investigated
individually. The main procedure to obtain the geometric
configuration of a Voronoi structure follows roughly these steps:
generating random nuclei, constructing convex hulls and
forming a Voronoi structure, see Ref. [10] for details. In this
study, the specimen with a Voronoi structure is generated in a
volume of 20 × 20 × 30 mm3 with 600 nuclei, and its cell
irregularity [16] is set to be 0.4, as shown in Fig. 1. The average
cell size, d0, is about 3.3 mm. The cell-wall thickness is
identical. The density of Voronoi structures, ρ0, can be changed
by varying the cell-wall thickness. The relative density of
Voronoi structures is expressed as ρ = ρ0/ρs, where ρs is the
density of base material. The base material is assumed to be
elastic-perfectly plastic with density ρs = 2700 kg/m3, Young‟s
modulus E = 69 GPa, Possion‟s ratio ν = 0.3 and yield stress σys
= 170 MPa. The finite element code ABAQUS/Explicit [17] is
used to perform the numerical simulations. The cell walls of
Voronoi structures are meshed with shell elements of types S3R
and S4R.
Uniaxial constant-velocity compression is considered as the
loading scheme in this study. The specimen is sandwiched
between two rigid plates: one is fully constrained and the other
moves along the longitudinal direction of the specimen with a
constant velocity, as shown in Fig. 2. A linear multi-point
constraint is applied to the surfaces at the loading direction to
avoid the rollover of the specimen under high velocity impact.
The general contact is applied to satisfy the complex contact
behavior among the cell walls with a friction coefficient of 0.02,
as used in Ref. [16].
(a) (b)
Figure 1. (a) A finite element model of closed-cell foam and (b)
a longitudinal section.
Figure 2. A schematic diagram of the constant-velocity
compression of a cellular specimen.
2 Copyright © 2016 by ASME
RESULTS AND DISCUSSION
Quasi-static Responses and the R-PH Idealization
During the compression process, the cell walls of cellular
structure experience initially elastic deformation and then
collapse as the load increases to the yield limit of the cell wall
material. Under the quasi-static compressive load, random shear
bands appear in the region that has the relatively weak strength
at the early stage of the loading, following the minimal energy-
consumption principle. With the crushing process continuing,
the collapse spread to the neighboring cells and new collapse
bands may appear for the strengthening effect of collapse bands
[10], as shown in Figs. 3②~③.
Figure 3. The quasi-static deformation sequence of cellular
structure with relative density ρ = 0.1 under V = 10 m/s.
0.00 0.15 0.30 0.45 0.60 0.75 0.90
0
5
10
15
20
25
④
③
③
③
②
Nom
inal
str
ess
(MP
a)
Nominal strain
= 0.10
R-PH idealization
(a)
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
0
2
4
6
8
10
12
Quasi-static data of C
Power-law fit
Init
ial
cru
shin
g s
tres
s,
0 (
MP
a)
Relative density,
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Har
den
ing
par
amet
er,
C (
MP
a)
Quasi-static data of 0
(b)
Figure 4. Quasi-static stress-strain curve of the cellular structure
with ρ = 0.1 (a) and variation of the parameters in the R-PH
idealization with the relative density (b).
A typical nominal stress-strain relation of the cellular
structure under quasi-static compression is shown in Fig. 4a.
The stress-strain curve consists of three typical stages: a linear-
elastic stage, a long plastic plateau stage and a rapid
compaction stage. The deformation configurations in Fig. 3 are
corresponding to the status on the stress-strain curve with
sequence numbers ②~④. The rate-independent, rigid−plastic
hardening (R-PH) idealization [10], expressed as σ(ε) =
σ0+Cε/(1-ε)2 with σ0 and C respectively being the initial crush
stress and the strain hardening parameter, could characterize the
plastic deformation and hardening behavior of the cellular
structures well, as shown in Fig. 4a. The dependence of the two
material parameters, σ0 and C, on the relative density of the
specimen is investigated by fitting the quasi-static stress-strain
curves for a range of relative density. The results show that both
of them increase with the increase of relative density in power-
law forms, as shown in Fig. 4b, and the relations could be
written as
1.37
0 ys
1.50
ys
/ 0.885
/ 0.115C
, (2)
which indicate that the quasi-static response of cellular material
is strongly dependent on the relative density.
Velocity Field and Its One-dimensional Distribution
As a basic physical quantity, the velocity field could be
applied to characterize the status of cellular structures. The
velocity components along the loading direction could be
extracted directly from the numerical results, thus the velocity
field could be calculated conveniently. For simplicity, the
velocity field distribution of the longitudinal section parallel to
the loading direction is used to represent the status of the whole
model at a specific moment as the uniaxial impact scenario
satisfies the assumptions of one-dimensional stress distribution
and the uniform distribution of stress-strain states along the
longitudinal direction of specimen [18]. The contours of
velocity distribution normalized with the impact velocity V, i.e.
v/V, and the corresponding deformation configurations of
cellular structures with relative density ρ = 0.1 and nominal
strain εN = 0.35 under different impact velocities are shown in
Fig. 5.
When the impact velocity is beyond a critical value,
corresponding to the transition between the Transitional mode
and the Shock mode [19], a highly localized shock-like
deformation appears and a nearly planar shock front is
developed. The shock wave, which is used to characterize the
layer-wise collapse [4], propagates from the loading end to the
support end in cellular materials under dynamic impact. The
one-dimensional velocity distribution is calculated by averaging
the velocity component in the loading direction over the cross-
sections. The results are shown in Fig. 6. The shock-like
deformation could be captured well from the velocity
distribution once the impact velocity reaches a critical value,
whilst there is no appearance of “shock” forming at low loading
velocities.
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Figure 5. Dimensionless velocity distribution contours of
cellular structures and the corresponding deformation
configurations.
0 5 10 15 20 25 30
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
Lagrangian location(mm)
Dim
ensi
on
less
vel
oci
ty,
v/V
= 0.1
N = 0.35
V = 10 m/s
V = 60 m/s
V = 120 m/s
V = 200 m/s
Figure 6. One-dimensional velocity distributions of the cellular
structure.
Shock Wave Speed
The velocity distribution under constant-velocity impact of
200 m/s shows that the velocity along the impact direction
drops rapidly at a specific location, which is considered as the
location of the shock front, as shown in Fig. 7a. The velocity
gradient is calculated and used to determine the accurate
location of the shock front at a specific time, as demonstrated in
Fig. 7b. The velocity gradient curve also shows that the shock
front has a finite width of about one cell diameter. Theoretically,
the physical quantities (such as stress, strain and velocity)
across the shock front are discontinuities for a solid material.
However, for cellular materials with an intrinsic characteristic
length, these physical quantities are meaningful only in the
average sense with a scale at least equal to or larger than the
characteristic length. Actually, it always takes time to activate
the response for cellular structures under dynamic impact [12,
20].
The Lagrangian location of the shock front is defined as the
absolute maximum value of the velocity gradient distribution, as
demonstrated in Fig. 7b. The shock locations at different times
for different impact velocities are shown in Fig. 8. It transpires
that the shock front moves with time in a nearly linear way
during crushing. The shock wave speed is determined by fitting
the data linearly. The result shows that the shock wave steadily
propagates through the cellular material under constant-velocity
compression, and the shock wave speed increases with the
increase of impact velocity. The conservation of mass across the
shock front [18] is expressed as:
[ ] [ ]sv V . (3)
If the velocity jump, [v] = VB−VA, across the shock front can be
determined, the physical quantities that associate with the shock
wave speed could be described quantitatively by combining
with the conservation relationships across the shock front.
0 5 10 15 20 25 30
-200
-150
-100
-50
0
0.108ms0.081ms
0.054ms
Vel
oci
ty(m
/s)
Lagrangian location(mm)
V = 200 m/s
0.027ms
(a)
0 5 10 15 20 25 30
-225
-180
-135
-90
-45
0
d0/2
Vel
oci
ty(m
/s)
Lagrangian location(mm)
N = 0.5
V = 200 m/s
= 0.1
Velocity
d0
-125
-100
-75
-50
-25
0
Gradient
Shock frontV
elo
city
gra
die
nt(
ms-1
)
(b)
Figure 7. (a) The velocity distribution and (b) the velocity
gradient of cellular structure with ρ = 0.1 under constant-
velocity impact compression of 200 m/s.
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0.00 0.05 0.10 0.15 0.20 0.25 0.30
0
5
10
15
20
25
30
= 0.1
V = 90 m/s
V = 120 m/s
V = 200 m/s
Linear fitting
Sh
ock
fro
nt(
mm
)
Impact time(ms)
Figure 8. Variations of shock front location with time for three
different impact velocities.
Velocity Ahead of the Shock Front
The velocity ahead of the shock front during crushing is
non-zero but has a slight graded distribution over the
Lagrangian location, as founded in Fig. 7. The material status at
a specific location changes from a pre-existing non-zero value
to the final velocity when the shock front passes through. Thus,
the region ahead of the shock front but away from the support
end could be served as a precursor of the incoming shock wave
during crushing. The behavior of the velocity transition in this
region is probably explained as the dispersion of the elastic
bending wave in cell walls [21]. Quantitatively, the region
around the Lagrangian location Φ(t) ranging from Φ(t) + d0/2 to
Φ(t) + 3d0/2 is defined as the effective length on the specimen
to calculate the velocity ahead of the shock front at time t, as
depicted in Fig. 7b. By averaging the cross-sectional velocities
over the region, the variations of velocity ahead of the shock
front with the dimensionless time t/tend for different constant
impact velocities are obtained, as shown in Fig. 9, where tend is
the end time corresponding to the time when the shock front
reaches the support end.
0.0 0.2 0.4 0.6 0.8 1.0
0
10
20
30
40
50
60
Vel
oci
ty,
VA (
m/s
)
Dimensionless time, t/tend
V = 150 m/s
V = 200 m/s
V = 250 m/s
V = 300 m/s
Average velocity, VA
Figure 9. The velocity ahead of the shock front versus
dimensionless time.
It is found that the velocity ahead of the shock front is
roughly independent of the impact velocity and keeps at a level
with some fluctuation, which may be due to the interaction of
the elastic wave in cell walls. Fig. 9 also indicates that the
velocity at the initial and terminal stages, when the shock front
locates within one cell size from the ends of the specimen, are
relatively low due to the boundary effect. Thus, the velocities at
these two stages are not used to calculate the average particle
velocity ahead of the shock front, VA, as schematically
illustrated in Fig. 9.
Effect of Relative Density on Shock Wave Speed
The variations of the shock wave speed with the impact
velocity for specimens with different relative densities are
shown in Fig. 10. It is found that the shock wave speed
increases with the increase of impact velocity as well as the
increase of the relative density. When the impact velocity is not
high enough but in a shock-formed range, denoted in the figure
by the red dash dot line-enclosed ellipse, the difference between
the shock wave speed and the impact velocity has a trend to
increase with the decreasing of impact velocity. However, the
difference is no longer dependent on the impact velocity and
tends asymptotically to a constant value as the impact velocity
is high enough. Therefore, the difference could be regarded as a
dynamic material parameter. This feature has been captured in
honeycombs [12] and in open-cell aluminum foams [13-14].
Here, the influence of the relative density on this dynamic
material parameter is further investigated quantitatively.
Since the dynamic material parameter is velocity-
independent as the impact velocity beyond a critical value (say
V ~120 m/s in this study), a linear expression between the shock
wave speed and the impact velocity with coefficients associated
with the relative density is adopted to fit the relation as
s ( ) ( )V k V c , (4)
where k(ρ) and c(ρ) are material parameters.
For a given cellular structure with relative density ρ, the
coefficients, k and c, are determined by fitting the relation of the
shock wave speed Vs and the impact velocity V. By changing
the relative density, a series of numerical data of k and c are
obtained, as shown in Fig. 11. The parameter k has an
approximated value of 1 and the maximum relative error is less
than 5%, thus k is considered to be independent of the relative
density of cellular material and its value could be taken as 1 for
convenience. The parameter c exhibits a nearly linear tendency
with the relative density and a linear fitting expression is
estimated as
( ) 23.2(1 5.40 ) m/sc . (5)
Thus, Eq. (4) could be written as
sV V c , (6)
where V is the impact velocity and c is the dynamic material
parameter. For constant-velocity impact, the impact velocity V
should be revised by considering the velocity ahead of the
shock front but for direct impact scenario, it is the instantaneous
impact velocity.
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60 120 180 240 300
90
135
180
225
270
315
Impact velocity
Sh
ock
wav
e sp
eed
, V
s (m
/s)
Impact velocity, V (m/s)
Figure. 10. Variations of the shock wave speed with the impact
velocity for specimens with different relative densities
0.000 0.075 0.150 0.225 0.300
0
15
30
45
60
Numerical data
Linear fitting
Par
amet
er,
c(m
/s)
Relative density,
0.0
0.5
1.0
1.5
2.0
c = 125.28+23.20
Numerical data Par
amet
er,
k
Figure 11. Variations of parameters k(ρ) and c(ρ) with the
relative densities.
Strain Hardening Behavior
According to the D-R-PH model proposed by Zheng et al.
[10], the dynamic material parameter could be expressed as c =
(D/ρ0)1/2
, where D is the dynamic strain hardening parameter.
By applying Eq. (5), the strain hardening parameter D can be
expressed as a function of the relative density as
2 2
0 1.45 (1 5.40 ) MPaD c . (7)
A comparison of the dynamic strain hardening parameter D with
the corresponding quasi-static strain hardening parameter C
determined in Eq. (2) is shown in Fig. 12.
It is seen that both the strain hardening parameters increase
with the increase of relative density, which could be physically
understood as the fact that increase of the relative density may
strengthen the flexure resistance of cell walls, leading to the
increase of the strain hardening parameter in cellular materials.
For a given relative density, the stain hardening parameter under
quasi-static loading is larger than that under dynamic impact,
which is related to the difference in the deformation
mechanisms [10]. Under dynamic crushing, the cells behind
shock front are much tightly stacked and could reach a higher
strain compared with the quasi-static behavior. Hence, the R-PH
idealization underestimates the dynamic response of cellular
materials in the densification stage under high enough impact
loading.
Prediction of the Strain behind the Shock Front, εB
The strain behind the shock front εB is further evaluated
based on Eq. (3) as the strain ahead of the shock front is nearly
zero and the strain jump could be simplified as [ε] = εB. The
dynamic material parameter c is obtained from the numerical
results with the explicit expression of Eq. (5), thus variations of
the strain behind the shock front, εB, with the impact velocity
can be determined, as shown in Fig. 13. It transpires that the
strain behind the shock front increases with the impact velocity
and is asymptotically approaches the theoretical value of εmax =
1−ρ. For comparison, it is assumed that the R-PH idealization
could characterize the dynamic behavior of cellular material
reasonably, then a similar definition of the dynamic material
parameter could be written as cqs = (C/ρ0)1/2
, where the
hardening parameter C has already determined as a function of
the relative density in Eq. (2) based on the quasi-static stress–
strain relation.
0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16
0.0
0.2
0.4
0.6
0.8
1.0
1.2
C (Quasi-static loading)
D (Dynamic impact)
Power-law fitting
Har
den
ing p
aram
eter
(M
Pa)
Relative density,
Figure 12. Comparison of the strain hardening parameter D with
the corresponding quasi-static strain hardening parameter C.
50 100 150 200 250 300
0.5
0.6
0.7
0.8
0.9
Dynamic results R-PH prediction
= 0.05 = 0.05
= 0.10 = 0.10
= 0.15 = 0.15
Str
ain
beh
ind
sh
ock
fro
nt,
Impact velocity(m/s)
Figure 13. Variation of strain εB with the impact velocity for
cellular structures with different relative densities.
The prediction of εB based on the R-PH idealization has the
same trend with the dynamic results. However, for a specific
relative density of cellular structure, the strain εB obtained by
6 Copyright © 2016 by ASME
the dynamic finite element simulation is much higher than that
obtained using the quasi-static compressive stress-strain curve.
The difference is attributed to the difference of deformation
modes under different loading rates [10]. So the D-R-PH shock
model is an improvement of the R-PH shock model for cellular
materials under dynamic impact.
CONCLUSIONS
The cell-based finite element model constructed using 3D
Voronoi technique is employed to investigate the effect of
relative density on the shock wave speed of closed-cell foam
under constant-velocity compression. One-dimensional velocity
distributions are calculated to characterize the propagation of
shock wave in cellular materials. It provides that there exists a
nearly planar shock front propagating through cellular materials
under high velocity.
The shock wave speed is determined based on the velocity
distribution along the impact direction. The results reveal that
the difference between the shock wave speed and the impact
velocity is asymptotically tending to be a constant as the
increase of the impact velocity, and the shock wave speed
increases linearly with the relative density. Consequently, an
explicit expression of shock wave speed with respect to the
impact velocity and the relative density is determined.
The influence of relative density on the strain hardening
parameter is investigated and analyzed. The result shows that
the strain hardening parameter increases with the relative
density. For a specific relative density, the strain hardening
parameter under quasi-static loading is higher than that under
dynamic impact, which indicates different mechanisms of the
deformation under high-velocity and quasi-static loadings.
Additionally, the strain behind shock front εB determined from
the dynamic finite element results is much higher than that
obtained using the quasi-static stress-strain curve.
ACKNOWLEDGMENTS
This work is supported by the National Natural Science
Foundation of China (Project No. 11372308) and the
Fundamental Research Funds for the Central Universities
(Grant No. WK2480000001).
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8 Copyright © 2016 by ASME
