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: zjzheng@ustc.edu.cn (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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t

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

d

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

i

i

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

m

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

l

l

s

m

a

t

i

c

t

s

l

o

p

i

t

c

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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