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Mechanics of Materials 100 (2016) 219–231
Contents lists available at ScienceDirect
Mechanics of Materials
journal homepage: www.elsevier.com/locate/mechmat
Research papaer
Dynamic crushing of cellular materials: A unique dynamic
stress–strain state curve
Yuanyuan Ding
a , Shilong Wang
a , Zhijun Zheng
a , ∗, Liming Yang
b , Jilin Yu
a
a CAS Key Laboratory of Mechanical Behavior and Design of Materials, University of Science and Technology of China, Hefei 230026, PR China b Mechanics and Materials Science Research Center, Ningbo University, Ningbo 315211, PR China
a r t i c l e i n f o
Article history:
Received 31 January 2016
Revised 19 May 2016
Available online 8 July 2016
Keywords:
Cellular material
Wave propagation
Finite element method
Dynamic stress–strain state
Local stress–strain history curve
a b s t r a c t
Cellular materials under high loading rates have typical features of deformation localization and stress en-
hancement, which have been well characterized by one-dimensional shock wave models. However, under
moderate loading rates, the local stress–strain curves and dynamic response of cellular materials are still
unclear. In this paper, the dynamic stress–strain response of cellular materials is investigated by using the
wave propagation technique, of which the main advantage is that no pre-assumed constitutive relation-
ship is required. Based on virtual Taylor tests, a series of local dynamic stress–strain history curves under
different loading rates are obtained by Lagrangian analysis method. The plastic stage of local stress-strain
history curve under a moderate loading rate presents a crooked evolution process, which demonstrates
the dynamic behavior of cellular materials under moderate loading rates cannot be characterized by a
shock model. A unique dynamic stress–strain state curve of the cellular material is summarized by ex-
tracting the critical stress–strain points just before the unloading stage on the local dynamic stress–strain
history curves. The result shows that the dynamic stress–strain states of cellular materials are indepen-
dent of the initial loading velocity but deformation-mode dependent. The dynamic stress–strain states
present an obvious nonlinear plastic hardening effect and they are quite different from those under quasi-
static compression. Finally, the loading-rate and strain-rate effects of cellular materials are investigated. It
is concluded that the initial crushing stress is mainly controlled by the strain-rate effect, but the dynamic
densification behavior is velocity-dependent.
© 2016 Elsevier Ltd. All rights reserved.
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. Introduction
Cellular materials have been extensively used as core materials
f anti-blast sacrificial claddings ( Hassen et al., 2002; Liao et al.,
013 b) and impact energy absorbers for their lightweight and su-
erior energy absorption capability. Studying the dynamic mechan-
cal behavior of cellular materials has become an important re-
earch direction in the field of impact dynamics. However, two
oupled dynamic effects, namely inertia effect and strain-rate ef-
ect, should be taken into consideration when the dynamic me-
hanical behavior of materials is involved ( Wang, 2005 ). The split
opkinson pressure bar (SHPB) technique ( Kolsky, 1949 ) has been
eveloped to uncouple these two dynamic effects and the dynamic
ehaviors of many solid materials have been determined by this
echnique. Nevertheless, due to the localized deformation nature
f cellular material ( Deshpande and Fleck, 20 0 0 ), the assumption
f uniform deformation along the specimen is no longer satisfied
∗ Corresponding author. Fax: + 86 551 6360 6459.
E-mail address: [email protected] (Z. Zheng).
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ttp://dx.doi.org/10.1016/j.mechmat.2016.07.001
167-6636/© 2016 Elsevier Ltd. All rights reserved.
or cellular materials under impact loading. Therefore, the applica-
ion of SHPB for cellular materials under dynamic loading is still a
ontentious issue.
The inertia effect, which leads to stress enhancement and
eformation localization as observed by Reid and Peng (1997 ),
ominates the dynamic behavior of cellular materials under high
elocity loading. According to the particular dynamic deforma-
ion features, some shock models were proposed to character-
ze the dynamic behavior of cellular materials. Based on a rate-
ndependent, rigid–perfectly plastic–locking (R-PP-L) idealization,
shock model was first proposed to model the impact response
f wood ( Reid and Peng, 1997 ) and further applied to character-
ze the dynamic crushing behavior of metallic foams under im-
act/blast loading ( Hassen et al., 2002; Main and Gazonas, 2008 ).
first-order approximation for engineering designs of cellular ma-
erials could be estimated by the R-PP-L shock model ( Harrigan
t al., 1999; Tan et al., 2005 ). A rate-independent, rigid–linear
ardening plastic–locking (R-LHP-L) idealization was employed by
heng et al. (2012 ) to investigate the dynamic behavior of cellu-
ar materials deformed in the shock mode and in the transitional
ode. A rate-dependent, rigid–linear hardening plastic–locking
220 Y. Ding et al. / Mechanics of Materials 100 (2016) 219–231
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(D-R-LHP-L) idealization was developed by Wang et al. (2013 b) to
study the energy conservation and critical velocities of cellular ma-
terial. In order to avoid the oversimplified approximation of "lock-
ing stage" used in the above models, a rigid–power-law harden-
ing idealization ( Pattofatto et al., 2007; Zheng et al., 2013 ) and
an elastic–perfectly plastic–hardening idealization ( Harrigan et al.,
2010 ) were further proposed. However, most of above works did
not consider the loading-rate sensitivity of cellular materials. Re-
cently, Zheng et al. (2014 ) proposed a rate-independent, rigid–
plastic hardening (R-PH) idealization and a dynamic one (D-R-PH)
to characterize the quasi-static stress–strain curve and the dynamic
stress–strain states of cellular materials, respectively. Barnes et al.
(2014 ) and Gaitanaros and Kyriakides (2014 ) carried out dynamic
experiments and simulations of open-cell aluminum foams and in-
vestigated the Hugoniot relation of shock wave speed and par-
ticle velocity. The nonlinear plastic hardening behavior and the
loading-rate effect of cellular materials under high velocity impact
are much clear, but there are some different opinions in the lit-
erature ( Zheng et al., 2014; Barnes et al., 2014; Gaitanaros and
Kyriakides, 2014 ). For example, Zheng et al. (2014 ) reported the
quasi-static and dynamic initial crushing stresses of cellular ma-
terials are different due to different deformation mechanisms, but
Barnes et al. (2014 ) regarded that the stress ahead of the shock
front is at the same level as the first local stress maximum of the
quasi-static stress-strain curve. These investigations are based on
the assumption of the shock-like deformation patterns, which may
be improper for some impact cases, and the shock models are only
suitable for the cases under high velocity loading. Thus, the dy-
namic behaviors of cellular materials have not been comprehen-
sively understood, especially for the case under moderate loading
rates.
Wave propagation techniques, which contain no constitutive as-
sumption, can be used to study the dynamic behavior of mate-
rials ( Wang et al., 2013 a). The application potential is that the
dynamic constitutive relation can be deduced directly from a se-
ries of physical quantity measurements regardless of the two cou-
pled dynamic effects, because the interaction of the inertia ef-
fect and strain-rate effect is naturally and implicitly considered
in the wave propagation technique. As a wave propagation tech-
nique, Lagrangian analysis method ( Fowles and Williams, 1970;
Cowperthwaite and Williams, 1971; Grady, 1973 ) gets the favor
of researchers. However, the traditional Lagrangian analysis should
consider a boundary condition, because it involves integral opera-
tions. In other words, a combination of boundary stress and par-
ticle velocity or a combination of boundary strain and particle
velocity should be measured simultaneously, which requires two
gauges at one position. A method combining the Lagrangian anal-
ysis and the Hopkinson pressure bar technique was proposed by
Wang et al. (2011 ) to overcome this difficulty, and the physical
quantities (stress, particle velocity, etc.) at the interface between
the specimen and the pressure bar can be obtained simultaneously.
Based on this technique, the "1 sv + n v " and "1 s ε + n ε " inverse anal-
ysis methods were developed according to the measured particle
velocity field or strain field ( Wang et al., 2011 ). However, these
methods are not suitable for soft materials, because the bound-
ary data cannot well match with the measured velocity data in a
specimen under impact experiments. Wang et al. (2013 a) proposed
a much convenient method of Lagrangian analysis using the pre-
known zero initial condition, but only investigated the dynamic
constitutive behavior of aluminum foam under moderate velocity
impact. When this Lagrangian analysis method (called "n v + T 0 ")
with the Taylor-Hopkinson bar experimental device is applied, a
very high impact velocity, say v > 200 m/s, may hardly be realized.
Some other limitations, such as the accuracy of digital image cor-
relation, also restrict the applicability of the "n v + T 0 " Lagrangian
analysis in experiment for cellular materials.
Fortunately, the finite element simulation based on cell-based
odels can make up the deficiencies in the experimental study,
nd it can offer sufficient data for theoretical analysis. Cellular
aterials can be well simulated by the 3D Voronoi technique
Zheng et al., 2014 ). By applying virtual tests, detailed and accu-
ate data of boundary stress, nodal displacement and velocity can
e obtained easily, which may hardly be measured in real experi-
ents.
In this paper, the dynamic behaviors of cellular materials are
nvestigated by using the Lagrangian analysis method. A brief in-
roduction of Lagrangian analysis method is presented in Section 2 .
he local stress–strain response of cellular materials is determined
y the Lagrangian analysis method based on the virtual Taylor test
n Sections 3 and 4 . The discussion on stress–strain states of cellu-
ar materials obtained by the Lagrangian analysis method is carried
ut in Section 5 , followed by conclusions in Section 6 .
. Lagrangian analysis method
In the case of one-dimensional wave propagation, when ig-
oring the influences of heat conduction, body force and internal
ower source, mass and momentum conservation equations in La-
rangian coordinates are given by
∂v ∂X
∣∣∣∣t
= − ∂ε
∂t
∣∣∣∣X
(1)
nd
0 ∂v ∂t
∣∣∣∣X
= − ∂σ
∂X
∣∣∣∣t
, (2)
espectively, where σ , ε, v are stress, strain and particle velocity,
espectively; X and t are Lagrangian coordinate and time, respec-
ively; ρ0 is the initial density of specimen. Here, the stress and
train are positive for compressive case, and negative for tensile
ase.
The mass conservation equation ( Eq. (1) ) establishes a rela-
ion between strain ε and particle velocity v , while the momen-
um conservation equation ( Eq. (2) ) provides a relation between
tress σ and particle velocity v . Therefore, the relationship of strain
nd stress can be built with the aid of velocity field. Neverthe-
ess, those quantities are connected by their first order derivatives,
hich means initial or boundary conditions should be provided to
olve this problem.
Consider the case that the particle velocity profiles v ( X i , t ) at
osition X i ( i = 1, 2, …) have previously been measured from nu-
erical or experimental tests. The first order partial derivatives
v / ∂ X at time t j ( j = 1, 2, …) and ∂ v / ∂ t at position X i can be numer-
cally calculated. Hence, ∂ ε/ ∂ t at position X i and ∂ σ / ∂ X at time t jan be indirectly obtained from Eqs. (1) and (2) , respectively. Since
he initial strain at t = 0 is usually known, the strain field ε( X i , t ) at
osition X i ( i = 1, 2, …) can then be determined by numerical in-
egral operation, and the stress field σ ( X, t ) can be determined in
he same way if the boundary stress is measured simultaneously.
However, based on the experimental study, the Lagrangian anal-
sis methods should be combined with the path-line method,
hich was first introduced by Grady (1973 ) in order to aid the
erivative computation of Lagrangian analysis. In other words, due
o the incompleteness of experimental technique, the distance of
wo adjacent Lagrangian positions is not small enough to obtain
ccurate partial derivatives ( ∂ σ / ∂ X and ∂ v / ∂ X ), and the path-line
ethod switches the first order derivatives containing variable X to
he partial derivatives containing variable t by the total differenti-
tion along the path-line. Using the path-line method, researchers
ust need to know velocity profiles no less than 3 positions, and
he relationship of stress and strain can be calculated by the La-
rangian analysis. The stress wave propagation characteristics in a
Y. Ding et al. / Mechanics of Materials 100 (2016) 219–231 221
(a) (b)
XRigid wallFree end
V0
Fig. 1. A cell-based finite element model (a) and its Taylor impact scenario (b).
Fig. 2. Deformed configurations in the Taylor test obtained from the cell-based finite element model.
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pecimen should be identified to apply the path-line method, but
his may bring a big error.
In fact, the path-line method is not necessary if there is suffi-
ient data obtained from a test. The virtual experiment (e.g. cell-
ased finite element method) can offer a detailed particle ve-
ocity field, which may not be measured completely in experi-
ents. Thus the strain field and the stress field can be determined
traightly from Eqs. (1) and (2) . In order to facilitate determination,
qs. (1) and (2) can be converted to the difference equations
i, j+1 − ε i, j = − ∂ v i, j
∂X
∣∣∣∣t j
(t j+1 − t j
)(3)
nd
i + 1 , j − σi, j = −ρ0
∂ v i, j
∂t
∣∣∣∣X i
( X i +1 − X i ) , (4)
here ∂ v i , j / ∂ X and ∂ v i , j / ∂ t can be obtained by central difference
∂ v i, j
∂t =
1
2
(v i, j+1 − v i, j
t j+1 − t j +
v i, j − v i, j−1
t j − t j−1
),
∂ v i, j
∂X
=
1
2
(v i +1 , j − v i, j
X i +1 − X i
+
v i, j − v i −1 , j
X i − X i −1
). (5)
Based on the virtual Taylor test, the stress, strain and velocity
rofiles at free end can be acquired simultaneously, and the veloc-
ty profiles at all element nodes can be easily extracted from finite
lement simulations. Hence, the dynamic strain–stress curve can
e obtained by using the Lagrangian analysis method.
. Finite element modeling and virtual Taylor test
Closed-cell foam models with a uniform cell-wall thickness
re generated by employing the 3D Voronoi technique (see Ref.
heng et al. (2014 ) for details). The cell-wall material of the
oronoi structure is assumed to be elastic, perfectly plastic with
= 69 GPa, ν = 0.3, Y = 170 MPa and ρs = 2700 kg/m
3 , where E, ν ,
and ρs are the Young’s modulus, Poisson’s ratio, yield stress
nd density, respectively. The relative density of the Voronoi struc-
ure used in the numerical simulations is set as ρ0 / ρs =0.1, where
0 is the initial density of the Voronoi structure. The cell ir-
egularity is 0.4. The macroscopic properties can be well simu-
ated by using Voronoi structures with at least five cells along the
hortest length direction, as pointed in Andrews et al. (2001 ). So
he cellular specimen used in this paper is constructed in a vol-
me of 30 ×20 ×20 mm
3 with 600 nuclei, and the average cell
ize, d , is about 3.34 mm, as illustrated in Fig. 1 (a). The numeri-
al simulations are performed by the explicit finite element code
ABAQUS/Explicit), and the cell walls of the Voronoi structure are
odeled with S3R and S4R shell elements.
A conventional Taylor impact scenario is considered as a dy-
amic virtual test in this study, and the X coordinate is established
t the free end, as shown in Fig. 1 (b). During the test, the specimen
mpinges normally with an initial velocity of 250 m/s onto a fixed
igid target, and it deforms as a 1D shock front propagating from
he striking end to the free end, as shown in Fig. 2 . The deforma-
ion patterns of the cellular specimen change from the shock mode
o the transitional mode, as the velocity of the uncompressed part
222 Y. Ding et al. / Mechanics of Materials 100 (2016) 219–231
Fig. 3. The quasi-static stress–strain curve (b) obtained from the cell-based finite element model under the constant-velocity compression scenario (a).
Fig. 4. Time history of particle velocity in the cellular specimen under an initial impact velocity of 250 m/s.
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of the specimen decreases gradually and finally becomes zero. The
kinetic energy of the specimen vanishes gradually and is trans-
formed into the internal energy.
A virtual compression test, in which the specimen of cellular
material is fixed at one end and loaded at the other end with a
low constant velocity (say V = 1 m/s), was performed to obtain the
quasi-static nominal stress–strain curve, as depicted in Fig. 3 . The
same specimen is used in order to ensure that the results are not
influenced by micro-structural randomness among different speci-
mens.
4. Results
4.1. Particle velocity of the numerical model
The Lagrangian analysis method is based on the continuum me-
chanics and thus it cannot be directly applied to porous/cellular
materials. Some averaging procedure should be carried out to ana-
lyze the data in a cellular material. In this paper, in order to elimi-
nate the influence of meso-structures in cellular material, the local
velocity profiles are substituted by averaging the velocity profiles
in a scale of one-cell. For instance, the average velocity profile at
the Lagrangian position X is calculated by averaging all nodal ve-
ocities from X − d /2 to X + d /2 in the cellular Voronoi structure,
here d is the average cell size.
The particle velocity profiles v ( X i , t ) have three distinct stages
f motion. An example with an initial impact velocity of 250 m/s
s shown in Fig. 4 . Initially, the time history of particle velocity de-
reases sharply from the initial velocity 250 m/s at the beginning,
nd the velocity falling point is corresponding to the arrival of the
oading elastic wave front. Thus, the elastic wave speed of this
ellular structure can be estimated to be 40 0 0 m/s. Subsequently,
here is a short period of transition in the velocity profiles until the
rrival of shock wave front. In the second stage, the velocity curve
ecreases rapidly and approaches to zero when the shock wave
ropagates in the cell strip of the corresponding Lagrangian po-
ition. However, this phenomenon of rapid velocity change would
isappear at the position away from the impact end, which can
e explained as the result of the unloading effect induced by the
nloading waves reflected from the free end of the structure. By
oughly analyzing the shock wave arrival time at different posi-
ions, it is estimated that the shock wave speed is about 270 m/s
ear the impact end, and it decreases gradually and finally van-
shes with the action of the unloading elastic waves. At the last
tage, the velocity profile almost equals to zero, which indicates
hat the shock wave has propagated through this position and the
aterial at this position is in a stationary state.
Y. Ding et al. / Mechanics of Materials 100 (2016) 219–231 223
Fig. 5. The dynamic strain history curves in the cellular specimen impacted at an
initial velocity of 250 m/s.
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.2. Dynamic strain and stress history curves
Once the particle velocity profiles of the specimen under virtual
aylor test have been established, the dynamic strain profiles ε( X i ,
) and the dynamic stress profiles σ ( X i , t ) can be directly obtained
y applying the Lagrangian analysis method through Eqs. (3) and
4) with the zero-boundary and initial conditions ( σ (0, t ) = 0 and
( X , 0) = 0), as shown in Figs. 5 and 6 , respectively.
It can be seen from Fig. 5 that when the shock wave arrives, the
ynamic strain increases rapidly and approaches to a local locking
train, which reflects the degree of local densification due to the
hock wave propagation in the specimen. As expected, the local
ocking strain decreases with the distance away from the impact
nd, and becomes very small at locations X = 0 ∼ 7 mm, where the
hock wave almost vanishes and disappears.
As shown in Fig. 6 , the stress near the impact end increases
rom zero to an initial crushing stress rapidly at the beginning,
nd then gradually increases to a densification stress, the value of
hich decreases with the distance away from the impact bound-
ry. This phenomenon is probably due to the decreasing velocity
head of the plastic wave in the specimen. At last, the stress de-
Fig. 6. The dynamic stress-time curves and the boundary stress curve in
reases from the densification stress to zero, and this is related to
he unloading behavior since the corresponding strain has already
pproached to the locking strain in this stage. When noticing the
tress history curve of the position close to the free end, we find
hat the second upward trend disappears. This phenomenon indi-
ates that no plastic wave exists in this region.
The oscillatory phenomenon at the beginning stage of curves in
ig. 6 is a result of the interaction of loading and unloading elas-
ic waves. To make the issue clearly understood, the stress curves
nd the corresponding strain curves at the same position are plot-
ed simultaneously in Fig. 7 . It clearly shows that the strain corre-
ponding to the initial oscillatory part of the stress is very small
nd stays within the elastic strain limit of cellular materials.
.3. Verification of the stress and strain obtained by Lagrangian
nalysis method
To quantitatively demonstrate the rationality of the stress and
train fields determined by the Lagrangian analysis method, the
ccuracy of the stress and strain results should be verified. Some
tress and strain indexes are introduced to confirm the correctness
f the stress and strain fields.
For the stress field, since the stress data of the compressive
egion in the virtual experiment can hardly be extracted directly
rom the cell-based FE model, the boundary stress-time curve (the
ed broken line in Fig. 6 ) is chosen as a stress index. The bound-
ry stress curve coincides with the upper envelope curve of the
tress field determined by the Lagrangian analysis method. This is
ecause the region behind the shock front is stationary and the in-
rtia effect can be neglected, which means that the stress in this
egion is identical. Thus, the stress-time curves obtained by the La-
rangian analysis can be verified indirectly.
For the strain field, the nominal strain, ε N , is selected to verify
he strain field. The nominal strain can be expressed as
N = �L/L, (6)
here �L is the total deformation of the specimen, and L is its
riginal length. In order to compare with the nominal strain, the
verage strain, ε avg , is introduced by averaging the strain field
long the loading direction, given by
avg =
1
L
∫ L
0
ε(X )d X . (7)
the cellular specimen under an initial impact velocity of 250 m/s.
224 Y. Ding et al. / Mechanics of Materials 100 (2016) 219–231
Fig. 7. The strain and stress curves at several Lagrangian positions.
Fig. 8. Comparisons of the nominal strain and average strain under initial impact velocities of 180 m/s and 250 m/s.
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To ensure the strain-time profiles are correctly calculated at dif-
ferent loading rates, comparisons of the nominal strain and the av-
erage strain under initial impact velocities of 180 m/s and 250 m/s
are carried out. The results show that the strain field obtained by
the Lagrangian analysis method can well estimate the deformation
of cellular materials, as shown in Fig. 8.
However, the comparison of the nominal strain and the average
strain based on the Lagrangian analysis is an indirect evaluation at
the macroscopic level, which ignores the local strain distribution.
In order to verify the local strain, the local strain field calculation
method ( Liao et al., 2014; Liao et al., 2013 a) based on the opti-
mal local deformation gradient technique is employed. The local
strain distributions at three different times, namely 0.02, 0.05 and
0.08 ms, obtained from the local strain calculation method and the
Lagrangian analysis method are presented in Fig. 9 . As can be seen,
the results of strain distribution obtained by the two methods are
in satisfactory agreement.
Tiny differences of two strain measures (the nominal strain
and the local strain) obtained by Lagrangian analysis method and
other strain calculation methods indicate the correctness and va-
lidity of the application of Lagrangian analysis method for cellular
materials. t
.4. Local dynamic stress–strain history curves
The local dynamic stress–strain history curves at Lagrangian
osition X i can be directly acquired by eliminating the time t
rom strain profile ε( X i , t ) and stress profile σ ( X i , t ), as shown in
ig. 10 . The stress–strain curves for X < 8 mm have not been taken
nto consideration since there is no plastic deformation. For X ≥ mm, the local stress–strain history curves can be obviously di-
ided into an elastic stage, a plastic deformation stage and an un-
oading stage. The elastic and unloading stages are controlled by
he Young’s modulus of cellular material, which is insensitive to
he loading rate. The most critical stage should to be taken into
onsideration is the plastic deformation stage, which is extremely
ependent on the loading velocity. Thus, the local dynamic stress–
train history curves can be classified into two categories, corre-
ponding to the transitional mode and the shock mode. A general
wareness of the shock mode for cellular materials is that it is con-
rolled by a structural shock wave and the plastic stage manifests
s a linear Rayleigh chord, of which a similar phenomenon can be
ound in the local stress–strain history curves at Lagrangian posi-
ions X > 14 mm. However, the understanding of plastic deforma-
ion under transitional mode is unclear, and it is thought as an
Y. Ding et al. / Mechanics of Materials 100 (2016) 219–231 225
Fig. 9. The local strain distributions obtained by the local strain calculation method and the Lagrangian analysis method at different times.
Fig. 10. The dynamic local stress–strain history curves obtained by the Lagrangian
analysis method under an initial impact velocity of 250 m/s.
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ntermediate state between the homogeneous mode and the shock
ode and its deformation configuration contains random shear
ollapse bands and layer-wise collapse bands, corresponding to the
eformation characteristic of the homogeneous and shock modes.
convex plastic stage, which is found in the local stress–strain his-
ory curves at Lagrangian positions 8 ≤ X ≤ 14 mm, may character-
ze this complex deformation process under the transitional mode.
. Discussion
.1. Typical local stress–strain history curves
Two typical local stress–strain history curves, corresponding to
he shock mode and the transitional mode, of the cellular mate-
ial at an initial impact velocity of 250 m/s are investigated and
nalyzed in this section. Here, the Lagrangian positions X = 25 mm
nd 10 mm are taken as examples to illustrate the evolution of the
tress–strain state under the shock mode and transitional mode,
espectively.
At position X = 25 mm, the stress initially increases to an initial
rushing stress σ c rapidly. In the second stage, the stress–strain
urve increases linearly with a slope of the chord connecting the
nitial crushing stress σ c and the critical stress–strain state, which
s the critical point just before unloading. The corresponding ve-
ocity ahead of shock front is at a high level, which is much larger
han the second critical velocity of cellular materials ( Li et al.,
014 a), as shown in Fig. 11 . In the last stage, with the action
f the unloading wave reflected from the free end of the cel-
ular specimen, the curve decreases dramatically and approaches
o zero with a linear path paralleling to that in the elastic stage
pproximately.
Theoretically, when a plastic shock wave propagates along a
pecimen bar, there exists a first-order singular interface, across
hich a series of physical quantities (such as stress, strain
nd velocity) jump from the pre-shock states to the post-shock
tates. As illustrated in the book "Foundations of Stress Waves"
Wang, 2005 ), the speed of the plastic shock wave is determined
y the slope of Rayleigh chord linking the two states ahead of and
ehind the shock front. In the Taylor test, the material behind the
hock front is in a stationary state, so the shock wave speed can be
etermined by the loading velocity ahead of the shock front. The
ayleigh chord is a virtual chord denoting the discontinuous jump
f physical quantities across the shock front. However, in consider-
ng the mesoscopic inhomogeneous deformation of cellular mate-
ials, an average velocity in the scale of one cell is used to do the
agrangian analysis in this study, so the local stress–strain curve
ay characterize the average stress and strain within a cell strip.
his conclusion can be illustrated by a sequence of sectional de-
ormation patterns of a cell-width strip at X = 25 mm, as depicted
n Fig. 12 , where the pattern number corresponds to the number
arked in Fig. 11 . From Nos. 1 to 9, the stress and strain both
ncrease along a similar Rayleigh chord in the local stress–strain
istory curve, while a progressive plastic collapse of the strip is
ound in Fig. 12 . From the Rayleigh chord and the density of cellu-
ar material, the shock wave speed can be estimated as 252.5 m/s.
he velocity ahead of the shock front in the plastic stage of lo-
al stress–strain curve at X = 25 mm is not much changed (about
25 m/s), thus the difference between the shock wave speed and
he impact velocity is about 27.5 m/s. The linear stress–strain be-
avior in the plastic stage can be explained as a linearly-growing
ompressed part in one cell, as shown in Fig. 12 . The stress–
train points in the local stress–strain history curve represent the
226 Y. Ding et al. / Mechanics of Materials 100 (2016) 219–231
Fig. 11. The local stress–strain and velocity-strain history curves at Lagrangian position X = 25 mm.
Fig. 12. Sequence of deformation patterns corresponding to the stress–strain points marked in Fig. 11.
5
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e
t
d
c
t
l
i
t
e
s
a
a
i
a
s
t
t
o
g
average mechanical response within a cell strip at Lagrangian po-
sition X = 25 mm.
At position X = 10 mm, the stress–strain history curve exhibits
the same trends in the elastic and unloading stages, but a crooked
curve is appearing in the plastic stage, as shown in Fig. 13 . The
corresponding velocity in this stage almost locates in the zone
below 120 m/s, at which rough but not apparent layer-wise col-
lapse bands are observed in the deformation patterns, as shown
in Fig. 14 . Under a moderate velocity, the plastic shock wave is
weakened and the complex interactions of elastic wave and plas-
tic wave dominate this stage. It can be roughly inferred that the
crooked plastic stage is in correlation with the local inertia effects
of cellular materials. As the plastic deformation is a relatively long
process in the transitional mode, the large velocity variation leads
to a more apparent local inertia effect. At first the velocity ahead
of plastic wave is at a relatively high level (about 100 m/s) and the
compressed part in the current cell-width strip increases with the
deformation. As the deformation continues, the velocity decreases
to a low level and the compressed part increases slowly. Thus, the
average stress displays a downward trend after the first rise in the
plastic stage and it also can be concluded that the stress–strain
points in the plastic stage is highly dependent on the deformation
process.
.2. Stresses behind and ahead of the shock front
A compressive discontinuity interface (also called shock front)
oes exist in a cellular material when the impact velocity is high
nough. Significant changes of stress and strain take place across
he shock front. The local stress distributions at different times un-
er the initial impact velocity of 250 m/s are shown in Fig. 15 . They
apture fairly well the propagation behavior of the shock wave in
he Voronoi structure. The region behind the shock front in cellu-
ar material is compressed tightly and stationary, thus the stress
n this region almost maintains a constant value, which is equal
o the boundary stress. The region ahead of the shock front is an
lastic stage, in which the stress distributes with a linear slope. The
tress between the elastic region and compressed region is also in
close linear transition distribution for the average effect of stress
s mentioned above. Two important stress quantities, namely the
nitial crushing stress and the shock stress (the critical stresses
head of and behind the shock front), can be obtained from the
tress distribution. The shock stress is associated with the inflec-
ion point from the shock wave region to the platform densifica-
ion stress within 5% error, and the initial crushing stress is the
ther inflection point from the elastic region to the shock wave re-
ion, obtained by the intersection of the lines of two regions.
Y. Ding et al. / Mechanics of Materials 100 (2016) 219–231 227
Fig. 13. The local stress–strain and velocity-strain history curves at Lagrangian position X = 10 mm.
Fig. 14. Sequence of deformation patterns corresponding to the stress–strain points marked in Fig. 13.
Fig. 15. The local stress distribution in the cellular specimen under the initial im-
pact velocity of 250 m/s.
c
s
s
o
i
v
a
l
t
c
f
s
t
e
i
r
w
H
2
t
l
Two zones of the initial crushing stress, distinguished by the
ritical position of mode transformation between transitional and
hock modes (the green solid point), are shown in Fig. 15 to de-
cribe the dynamic initial crushing behavior. With the increasing
f velocity ahead of the shock front in the virtual Taylor test, the
nitial crushing stress also increases, and this feature is much ob-
ious in the transitional mode. As discussed in Zheng et al. (2014 )
nd Wang et al. (2013 a), the initial crushing stress under dynamic
oading is higher than that under quasi-static compression. Hence,
he initial crushing stress is bound to increase from a quasi-static
rushing stress to a dynamic one as the loading velocity change
rom the first critical velocity to the second critical velocity (tran-
itional mode), and a similar phenomenon is found here. However,
he variation of initial crushing stress under shock mode is differ-
nt from that reported in Zheng et al. (2014 ). It shows a slightly
ncreasing trend with the increase of loading velocity (local strain
ate, as discussed below).
The shock stress increases with the increasing loading rate,
hich is known as the stress enhancement ( Reid and Peng, 1997;
arrigan et al., 1999; Tan et al., 2005; Liu et al., 2009; Li et al.,
014 b). A similar phenomenon that the shock stress decreases with
he shock front propagating away from the impact end in the Tay-
or test is found and depicted by the red points in Fig. 15 , and the
228 Y. Ding et al. / Mechanics of Materials 100 (2016) 219–231
Fig. 16. The dynamic stress–strain state curve and quasi-static stress–strain curve
for cellular materials.
c
d
s
s
s
s
t
m
s
c
l
c
s
q
(
u
m
y
c
q
e
s
5
p
l
s
(
h
t
t
t
(
s
F
σ
f
σ
w
h
t
Z
R
(
ε
w
t
l
F
F
i
fi
t
c
v
D
9
m
variation is consistent with the theoretical prediction of the D-R-
PH shock model proposed in Zheng et al. (2014 ).
5.3. Dynamic stress–strain states ahead of and behind the shock front
The local stress–strain points represent the average quantities
among one cell-width strip and depend on the deformation pro-
cess. The strain–stress states corresponding to critical points just
before unloading in the dynamic local strain–stress history curve
can characterize the dynamic stress–strain states when the plas-
tic wave propagates through the corresponding cell. According to
the statistic mechanics, two series of dynamic stress–strain states
and their standard deviations are shown in Fig. 16 with the initial
impact velocities of 180 m/s and 250 m/s. Two overlapping stress–
strain state curves indicate that a unique curve of dynamic stress–
strain states exists to characterize the dynamic constitutive be-
havior of cellular materials. Three distinct phases of stress–strain
states can be found and defined as quasi-static, transitional and
shock phases. For comparison, the quasi-static stress–strain curve
is also plotted in Fig. 16 . Significant differences between the dy-
namic stress–strain states and the quasi-static stress–strain curve
indicate that the deformation of cellular materials is sensitive to
the loading rates ( Zheng et al., 2014 ).
Due to the low loading velocity in the quasi-static phase, the
stress–strain states are consistent with those in the quasi-static
stress–strain curve. In the shock phase, the noticeable feature is
that the dynamic densification strain is larger than the strain under
quasi-static compression at the same stress level, which can be ex-
plained by the difference in deformation mechanisms ( Zheng et al.,
2014 ). Under quasi-static loading rates, the deformation of cellu-
lar materials consists of a series of random shear collapse bands.
According to the principle of minimum energy, the weakest shear
collapse bands at every moment compose the quasi-static defor-
mation. Under high impact velocities, inertia effect dominates the
deformation process, and the crushed cells deform layer by layer
and are stacked compactly, as depicted in Fig. 12 . The stress in
the shock phase increases with the increasing loading velocity and
it is known as the stress enhancement. According to the shock
wave theory, the stress enhancement can be expressed as ρ0 ν2 / ε B ,
where ν and ε B is the current loading velocity and shock strain,
respectively. Thus, the shock phase of the dynamic stress–strain
states can be explained as a result of deformation localization and
stress enhancement. In the transitional phase, the deformation of
ellular materials is not a onefold mode and contains both ran-
om shear collapse band and layer-wise collapse band. Thus, the
tress states in this phase are higher than those in the quasi-static
tress–strain curve.
It should be emphasized that the stress–strain states behind the
hock front are not enough to characterize the 1D dynamic stress–
train curve of cellular materials. The stress–strain states ahead of
he shock front (i.e. the initial crushing states) need to be supple-
ented, as shown in Fig. 16 . Under high loading rates, the stress–
train state first reaches to the initial crushing state, and then in-
reases to the one in the dynamic stress–strain state curve along a
inear chord. Under moderate loading rates, the stress–strain state
hanges from the initial crushing state to the one in the dynamic
tress–strain state curve along a crooked chord.
The stress–strain behaviors under high velocity impact and
uasi-static compression have been discussed in Zheng et al.
2014 ), but the issue of the stress–strain states of cellular materials
nder moderate velocity impact (corresponding to the transitional
ode) is still open. In this paper, by applying the Lagrangian anal-
sis method, the stress–strain states under the transitional mode
an be calculated and they are quite different from those in the
uasi-static stress–strain curve. This means the moderate inertia
ffect cannot be neglected and the physical mechanism under tran-
itional mode needs to be further investigated.
.4. Comparison with the R-PP-L and D-R-PH shock models
In the shock models, a shock front propagates from the im-
act end to the free end and the physical quantities (particle ve-
ocity, strain and stress) jump across this shock front. The R-PP-L
hock model ( Reid and Peng, 1997 ) and the D-R-PH shock model
Zheng et al., 2014 ) are employed to characterize the dynamic be-
aviors of the cellular material considered. The R-PP-L idealiza-
ion ( Reid and Peng, 1997 ) has two material parameters, namely
he plateau stress and the locking strain, which are usually de-
ermined by applying the maximum energy absorption efficiency
Tan et al., 2005; Avalle et al., 2001 ). In this study, using the quasi-
tatic stress–strain curve of the cellular material as presented in
ig. 16 , the plateau stress and the locking strain are determined as
pl = 6.48 MPa and ε L = 0.64, respectively. The stress–strain relation
or the D-R-PH idealization ( Zheng et al., 2014 ) is written as
( ε ) = σ d 0 + Dε/ (1 − ε) 2 , (8)
here σ d 0
is the dynamic initial crushing stress and D the strain
ardening parameter. Only two material parameters are involved in
he D-R-PH idealization and their values used here are taken from
heng et al. (2014 ), i.e. σ d 0
= 7.7 MPa and D = 0.22 MPa. For the D-
-PH shock model, the strain behind the shock front is given by
Zheng et al., 2014 )
B =
v v + c
, (9)
here c = ( D / ρ0 ) 1/2 .
The results obtained by Lagrangian analysis method show that
he densification strain is highly dependent on the impact ve-
ocity and increases with the increasing of impact velocity, see
ig. 17 . The variation of the densification strain has two stages.
or the high impact velocity ( v > 100 m/s), the densification strain
ncreases slowly with the increase of impact velocity, and it veri-
es the prediction obtained from the D-R-PH shock model. Under
he moderate impact velocity ( v < 100 m/s), the change of densifi-
ation strain shows a rapidly descent with the decrease of impact
elocity, and it is completely different from the predictions of the
-R-PH shock model. Only when the impact velocity is close to
0 m/s, the densification strain obtained by the Lagrangian analysis
ethod is consistent with the locking strain of the R-PP-L model,
Y. Ding et al. / Mechanics of Materials 100 (2016) 219–231 229
Fig. 17. Comparison of densification strains obtained by the Lagrangian analysis
method and the shock models.
o
e
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e
v
m
R
V
(
V
t
D
t
5
t
e
i
m
(
u
u
I
e
s
T
i
p
S
s
Z
t
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w
e
r
b
s
s
i
e
n
t
d
s
b
d
M
d
s
c
s
i
therwise they are quite different for the cases under high or mod-
rate velocity impact.
The shock front position denoting the compressive discontinu-
ty interface can be determined by the maximum of absolute veloc-
ty gradient, | ∂ v / ∂ X | max . According to the three-point central differ-
nce, the relation between the shock wave speed and the impact
elocity can be obtained, as shown in Fig. 18 . For the R-PP-L shock
odel, the shock wave speed can be expressed as ( Wang, 2005;
eid and Peng, 1997 )
s = v / ε L . (10)
For the D-R-PH shock model, the shock wave speed is given by
Zheng et al., 2014 )
s = v + c. (11)
The results show that the R-PP-L shock model fails to predict
he shock wave speed in a wide range of impact velocity, but the
-R-PH shock model can well predict the shock wave speed when
he impact velocity is high enough.
Fig. 18. Comparison of shock wave speeds obtained by the
.5. Strain-rate and loading-rate sensitivity
The strain-rate effect of cellular materials were often inves-
igated by using SHPB tests ( Deshpande and Fleck, 20 0 0; Zhao
t al., 2005; Yu et al., 2006 ), but the conclusions showed conflict-
ng strain-rate sensitivity for both open-cell and closed-cell alu-
inum foams, as reported by Liu et al. (2009 ) and Zhao et al.
2005 ). Under high-velocity impact, the SHPB technique may be
nsuitable for cellular materials, because the basic assumption of
niform stress distribution along the specimen cannot be satisfied.
n this section, the local strain rate is introduced to investigate its
ffect on the dynamic response of cellular materials.
It has been concluded that the dynamic densification strain
hould be dependent on the impact velocity ( Zou et al., 2009;
an et al., 2012; Zheng et al., 2014 ) and the dynamic behav-
ors of cellular materials under high-velocity impact can be well
redicted by the D-R-PH shock model ( Zheng et al., 2014 ), see
ection 5.4 . Thus, we confirm that the shock stress and the shock
train are mainly dependent on the impact velocity as found by
heng et al. (2014 ). Thus, the loading-rate effect is considered as
he leading factor of the dynamic densification behavior of cellular
aterials.
The initial crushing stress was considered to be independent
ith the loading rates in the literature ( Zheng et al., 2014; Barnes
t al., 2014; Gaitanaros and Kyriakides, 2014 ). Barnes et al. (2014 )
egarded that the initial crushing stresses are the same value for
oth dynamic and quasi-static cases, but Zheng et al. (2014 ) con-
idered that the dynamic initial crushing stress was another con-
tant value, which is higher than that under quasi-static load-
ng when considering the difference in deformation modes. How-
ver, in this study, we find that the initial crushing stress is
ot a constant, as shown in Fig. 15 . It should be noted that
he initial crushing stress as a material parameter could not be
irectly dependent on the loading velocity. The initial crushing
tress should be dependent on the local strain rate, as discussed
elow.
The local strain rate distribution is related to the velocity gra-
ient, which can be obtained directly by using Eq. (1) , see Fig. 19 .
ountain-like regions with one-cell width in the local strain rate
istribution show that a local layer-wise collapse band exists in the
pecimen and the strains at positions behind and ahead of this
ollapse band are all in a steady state. With the time goes, the
train rate peak moves from the impact end to the free end and
ts value becomes smaller gradually. It indicates the collapse band
Lagrangian analysis method and the shock models.
230 Y. Ding et al. / Mechanics of Materials 100 (2016) 219–231
Fig. 19. The local strain rate distribution of cellular materials under initial impact velocity of 250 m/s.
Fig. 20. The initial crushing stress versus strain-rate in the Taylor test.
6
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l
n
propagation and the process of kinetic energy dissipation by the
cellular material.
The initial crushing stress, σ c , has been determined in Fig. 15 ,
and the corresponding local strain rate can be extracted from
Fig. 19 , according to the Lagrangian position where the local initial
crushing behavior happens, as presented in Fig. 15 . Thus, a relation
between the initial crushing stress and the strain rate can be es-
tablished, which indicates that the initial crushing stress increases
with the increasing of strain rate, as shown in Fig. 20 . We perform
a power-law fitting procedure with
σc /σq 0
= ( ̇ ε / ̇ ε 0 ) n , (12)
where σ q 0
is the quasi-static initial crushing stress, ˙ ε the local
strain rate, n the power-law index, and ˙ ε 0 a reference strain rate.
For the studied cellular materials, the quasi-static initial crushing
stress is obtained from Fig. 3 (b) as σ q 0
= 5.99 MPa, and the values
of other parameters are determined by applying the least squares
fitting method as ˙ ε 0 = 1390 s −1 and n = 0.141.
Thus, the initial crushing stress is mainly controlled by the
strain-rate effect, but the strain-rate effect can almost be neglected
compared with the loading-rate effect for the dynamic densifica-
tion behavior of cellular materials.
. Conclusions
In this study, the Lagrangian analysis method with virtual Tay-
or tests is employed to investigate the dynamic behavior of cel-
ular materials. The averaging operation of particle velocity in a
cale of one cell is carried out to make the Lagrangian analysis
ethod feasible and credible for cellular materials. The local strain
nd stress profiles of cellular materials are obtained by applying
he Lagrangian analysis method and their accuracy is verified by
ntroducing three indexes including the engineering strain, the lo-
al strain distribution and the boundary stress.
The local dynamic stress–strain history curves of cellular ma-
erials for all Lagrangian positions are presented and they demon-
trate the stress–strain evolution process under a local impact ve-
ocity. Under high and moderate loading rates, the plastic stage
f local stress–strain curve presents linear and crooked evolution
rocesses, respectively. For the shock mode, the linear increas-
ng stress–strain points are caused by the shock wave propaga-
ion with an almost constant velocity in the cell strip of a spe-
ific Lagrangian position. For the transitional mode, rough but not
ayer-wise collapse bands are observed, but its physical mechanism
eeds further investigations.
Y. Ding et al. / Mechanics of Materials 100 (2016) 219–231 231
c
t
t
t
s
s
e
v
a
l
t
f
f
s
r
s
n
d
(
m
d
q
T
t
t
l
A
d
F
W
R
A
A
B
C
D
F
G
G
H
H
H
K
L
L
L
L
L
L
M
P
R
T
T
W
W
W
W
Y
Z
Z
Z
Z
Z
The stress distributions at different time are investigated to
apture the shock wave propagation behavior. The results reveal
hat the shock stress is sensitive to the local loading rate and
he initial crushing stress under dynamic loading is higher than
hat under quasi-static loading. The comparison between the re-
ults obtained by Lagrangian analysis method and the shock model
hows that the D-R-PH shock model ( Zheng et al., 2014 ) can well
xplain the dynamic behavior of cellular materials under high-
elocity loading, but it is not suitable for the case under moder-
te loading rates. The loading-rate and strain-rate effects of cel-
ular materials are further investigated, and the results reveal that
he initial crushing stress is mainly controlled by the strain-rate ef-
ect with a power law, while the loading-rate effect is the leading
actor of the dynamic densification behavior of cellular materials.
A unique curve composed by a series of dynamic stress–strain
tates is presented and it is independent of the initial loading
ate. Under high-velocity impact, the dynamic stress–strain states
how the effect of the nonlinear plastic hardening, where the dy-
amic densification strain is larger than the quasi-static strain un-
er the same stress level. This confirms the findings in Zheng et al.
2014 ). However, the stress–strain states of cellular materials under
oderate-velocity impact, which were not comprehensively ad-
ressed, are explored in this study and it is found that they are
uite different from those in the quasi-static stress–strain curve.
he significant differences among the dynamic stress–strain his-
ory curves and the quasi-static stress–strain curve indicate that
he deformation mechanism of cellular materials is sensitive to the
oading rates.
cknowledgments
This work is supported by the National Natural Science Foun-
ation of China (Projects Nos. 11372308 and 11372307 ) and the
undamental Research Funds for the Central Universities (Grant No.
K2480 0 0 0 0 01 ).
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