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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 72 (2014) 93–114

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Dynamic stress–strain states for metal foams using a 3Dcellular model

Zhijun Zheng a,n, Changfeng Wang a, Jilin Yu a, Stephen R. Reid b,John J. Harrigan b

a CAS Key Laboratory of Mechanical Behavior and Design of Materials, University of Science and Technology of China, Hefei,Anhui 230026, PR Chinab School of Engineering, Fraser Noble Building, King's College, University of Aberdeen, Aberdeen AB24 3UE, UK

a r t i c l e i n f o

Article history:Received 4 February 2013Received in revised form5 July 2014Accepted 17 July 2014Available online 5 August 2014

Keywords:Closed-cell foamStrain fieldDynamic constitutive relationFinite element modelShock wave behaviour

x.doi.org/10.1016/j.jmps.2014.07.01396/& 2014 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ86 551 6360 3044ail address: zjzheng@ustc.edu.cn (Z. Zheng).

a b s t r a c t

Dynamic uniaxial impact behaviour of metal foams using a 3D cell-based finite elementmodel is examined. At sufficiently high loading rates, these materials respond by forming‘shock or consolidation waves’ (Tan et al., 2005a, 2005b). However, the existing dynamicexperimental methods have limitations in fully informing this behaviour, particularly forsolving boundary/initial value problems. Recently, the problem of the shock-like responseof an open-cell foam has been examined by Barnes et al. (2014) using the Hugoniot-curverepresentations. The present study is somewhat complementary to that approach andadditionally aims to provide insight into the ‘rate sensitivity’ mechanism applicable tocellular materials. To assist our understanding of the ‘loading rate sensitivity’ behaviour ofcellular materials, a virtual ‘test’ method based on the direct impact technique is explored.Following a continuum representation of the response, the strain field calculation methodis employed to determine the local strains ahead of and behind the resulting ‘shock front’.The dynamic stress–strain states in the densification stage are found to be different fromthe quasi-static ones. It is evident that the constitutive behaviour of the cellular material isdeformation-mode dependent. The nature of the ‘rate sensitivity’ revealed for cellularmaterials in this paper is different from the strain-rate sensitivity of dense metals. It isshown that the dynamic stress–strain states behind a shock front of the cellular materiallie on a unique curve and each point on the curve corresponds to a particular ‘impactvelocity’, referred as the velocity upstream of the shock in this study. The dynamic stress–strain curve is related to a layer-wise collapse mode, whilst the equivalent quasi-staticcurve is related to a random shear band collapse mode. The findings herein are aimed atimproving the experimental test techniques used to characterise the rate-sensitivitybehaviour of real cellular materials and providing data appropriate to solving dynamicloading problems in which cellular metals are utilised.

& 2014 Elsevier Ltd. All rights reserved.

; fax: þ86 551 6360 6459.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–11494

1. Introduction

1.1. General background

The dynamic mechanical behaviour of cellular materials (e.g. metal foams) has been studied extensively for many yearsand a range of useful models have been published. At sufficiently high loading rates, these materials respond by forming‘shock or consolidation waves’ (Tan et al., 2005a,2005b). Deformation localisation and strength enhancement have beenfound as two typical features of cellular metals under impact (Tan et al., 2005a,2005b; Liu et al., 2009). Accompanying these,there is an increase in the energy absorbing capacity of these cellular materials. However, it is still controversial whether,like many other continua, a cellular metal, when treated as a material per se, is strain-rate sensitive or not (Liu et al., 2009;Ma et al., 2009).

Current experimental techniques make it difficult to define the dynamic constitutive relations of cellular metals sincelocal inertia effects are not easily decoupled from the definition of the relevant parameters associated with the dynamicresponse (Liu et al., 2009). For example, the split Hopkinson pressure bar (SHPB) has been used for cellular materials toinvestigate ‘strain-rate effects’ as for solids (Kolsky, 1949). However, this method requires/assumes uniform deformationthroughout the specimen. Because of the localised nature of the dynamic deformation mechanism, this requirement isdifficult to satisfy for real cellular materials under dynamic loading using the standard SHPB technique. Fig. 3 in Deshpandeand Fleck (2000) gives a schematic diagram of an SHPB apparatus and provides high speed film observations of anactual test.

The sizes of cellular specimens and the impact velocity/loading rate are key parameters when using the SHPB techniqueor other uniaxial, dynamic-loading experimental methods. There is no standard length of a cellular specimen for such tests.Longer specimens can result in very different stresses at the two ends of the specimen. Thus, generally, the lengthof an SHPB cellular specimen should usually not be ‘too long’ but should also be chosen judiciously relative to the celldimensions.

As is well-known, under high rates of loading the deformation mechanism in cellular materials is a local one anddeformation is affected in a pseudo-discontinuous manner – ‘shock/compaction front’. This fundamental fact mitigatesagainst pseudo equilibrium. Also, the specimen needs enough cells in the radial and loading direction to avoid the influenceof boundary conditions. Thus, only for ‘low’ impact velocities, is uniform deformation a reasonable approximation.For example, Deshpande and Fleck (2000) stated that the impact velocity should be less than 50 m/s when they used theSHPB technique to investigate the dynamic constitutive behaviour of a closed-cell Alulight foam and an open-cellDuocel foam.

For high-velocity impact experiments on metal foams, inertia (stress wave) effects dominate. These lead to deformationlocalisation and to strength enhancement as observed by Reid and Peng (1997) with their direct impact tests on wood.However, it could be inaccurate to assume that the stress elevation detected in such experiments is the result of a materialstrain-rate effect (Dannemann and Lankford, 2000; Zhang et al., 2002) or a viscosity-dominated effect (Radford et al., 2005).

The general trends of increased dynamic crushing stress and energy absorption can be estimated by simple shock models(Reid and Peng, 1997) without the inclusion of strain-rate or viscosity. However, a difficulty that arises is that, as yet, thereare no experimental techniques for measuring the stress and strain and their variation on either side of a compaction wave.For this reason, quasi-static stress–strain curves were employed to model the observed dynamic enhancement in theliterature. Therefore, other methods need to be developed before a fuller understanding of the dynamic response involvingthis inertia effect can be captured experimentally and subsequently used in the solution of dynamic problems.

Recently, the problem of the shock-like response of an open-cell foam was examined by Barnes et al. (2014) andGaitanaros and Kyriakides (2014) utilising the definition and measurement of the parameters contained in a Hugoniotdescription of the high impact speed behaviour of various materials. The present study is somewhat complementary to thatapproach, but herein the aim is to elucidate an explanation and method to acquire a correct representation of the ‘ratesensitivity’ in the constitutive behaviour of a cellular material. The objective is to explore the related deformationmechanisms and to provide suggestions for improving relevant experimental test techniques.

1.2. Modelling background

Some simple shock wave models have been proposed in the literature to model the behaviour of dynamic deformationlocalisation. The simple shock model based on a rate-independent, rigid–perfectly plastic–locking (R-PP-L) idealisation wasoriginally proposed by Reid and Peng (1997) and further developed by Tan et al. (2005b) to present the first order estimationof the dynamic response. This model has been further developed by using different idealisations (Harrigan et al., 2005, 2010;Lopatnikov et al., 2003, 2004, 2007; Pattofatto et al., 2007; Zheng et al., 2012, 2013).

A rigid–linear hardening plastic–locking (R-LHP-L) idealisation was used recently by Zheng et al. (2012) to develop aShock-Mode model for high-velocity impact and a Transitional-Mode model for moderate-velocity impact. The presentpaper is directed towards providing an improvement of the Shock-Mode treatment therein. An elastic–perfectly plastic–rigid (E-PP-R) idealisation and an elastic–plastic–rigid (E-P-R) idealisation were employed by Lopatnikov et al. (2003, 2004,2007) to consider the effect of elasticity, which may affect the response of cellular materials under low-velocity impact.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114 95

The relatively simple idealisations mentioned above generally have a ‘locking/densification’ stage in the materialbehaviour formulation and it is this simplification that is the focus of this paper.

Different locking strains have been suggested in the literature (see Tan et al., 2005b; Elnasri et al., 2007; Pattofatto et al.,2007) and model predictions have been shown to be sensitive to the definition of ‘locking strain’ (Elnasri et al., 2007;Pattofatto et al., 2007). Apparently, this is because the dynamic ‘locking/densification strain’ increases with an increase ofimpact velocity as found computationally in regular honeycombs (Elnasri et al., 2007; Zou et al., 2009) and more recentlydeduced from a re-assessment of experiments on open-cell Duocel foams (Tan et al., 2012). Essentially, this is due to themeso/microstructure of cellular materials and its response under dynamic loading.

Other idealisations include a non-linear plastic hardening stage, such as the elastic–perfectly plastic–hardeningidealisation (Harrigan et al., 2010) and the rigid–power-law plastic hardening idealisations (Pattofatto et al., 2007; Zhenget al., 2013), to characterise this feature of the densification strain. However, it should be noted that all these idealisationsare based on a quasi-static stress–strain relation for cellular materials, and their corresponding shock models do not includeany intrinsic, cellular-material loading-rate effects. Whether loading-rate effects exist in cellular materials and how they areunderstood and measured is still unclear.

1.3. Outline of paper

In order to explore the dynamic material properties of cellular materials, a cell-based finite element (FE) model of adirect impact test is presented and discussed in this study. It provides a virtual ‘test’ method to explore the cell-level (meso-structural) response. This approach produces detailed information of meso-structural deformation and assists in under-standing the rate-sensitivity mechanisms appropriate to cellular materials.

The influences of the density, matrix mechanical properties, inertia and micro-inertia on the dynamic behaviour of 2DVoronoi honeycombs have been investigated previously (Tan et al., 2005b; Liu et al., 2009; Zou et al., 2009). These studieshave shown that inertia (principally the longitudinal inertia, which acts in the loading direction) is the main cause forenhancement of the plateau stress in impact tests. Micro-inertia (referred to there as lateral inertia) effects play a dominantrole in modifying the local deformation mechanism and the rate-sensitivity of the matrix materials has a relatively weakeffect in the dynamic response.

A comparison of dynamic responses between a cell-based FE model of a 2D Voronoi honeycomb and its continuum-basedFE model based on a rate-independent stress–strain relation was carried out by Ma et al. (2009). Their results in Fig. 14 of Maet al. (2009) showed that the stress at the impact (proximal) end of their cell-based FE model was a little higher than that ofthe continuum-based FE model. However, Ma et al. (2009) did not highlight this interesting difference nor discuss thepossible influential factors.

A numerical study of the dynamic “shock” compaction of cellular material is reported in this paper. Given thepredominance of the formation and propagation of discontinuities as the inherent, local deformation mechanism in cellularsolids under intense dynamic loading (impact and blast), it would appear that the relevant characterisation of thisphenomenon should be the focus of research in this area at this time. The aim of the present study is to investigate whetherthe dynamic stress–strain states for a cellular solid are different from the quasi-static stress–strain ones and to formulatehow these can be measured and used in modelling realistic boundary/initial value dynamic (impact and blast) loadingproblems.

For the investigation, the case of a prismatic metal foam specimen with an initial velocity impinging normally on a fixedrigid wall is simulated. This direct impact scenario with/without a backing mass has been implemented experimentally andanalysed by many authors (Reid and Peng, 1997; Deshpande and Fleck, 2000; Tan et al., 2005a,2005b, 2012; Zheng et al.,2012, 2013; Wang et al., 2013).

In this paper, which focuses on behaviour in which the strain-rate sensitivity of the matrix material of cellular materialsis excluded, the dynamic behaviour of cellular materials is shown to be influenced by the kinematic and kinetic features ofthe loading rate per se. To investigate the effect of loading rate, a cell-based FE model is generated by utilising a 3D Voronoi-based model for the dynamic deformation. The quasi-static nominal stress–strain relation (see the footnote on p. 68 ofZheng et al., 2012) is obtained from a virtual compression test.

The key parameters (e.g. the dynamic densification strain) may depend directly on the impact velocity or the speed of theshock front, across which the deformation is affected. It should be noted that “impact velocity” refers to the relative velocitybetween the particles behind and ahead of the shock front. In the present study, the impact velocity is the particle velocityof the portion of the cellular block that has not been brought to rest by the rigid wall, i.e. the velocity upstream of the shock.

The cellular specimen used in the present FE model has been simplified in the microstructural details deliberately. Forexample, we do not consider the influences of gas in the cells, or strain hardening and strain-rate sensitivity of cell-wallmaterial in this paper. This simplification is to enable our focus on the key deformation mechanism for this material system.The dynamic local strain and stress are clearly defined and compared with those estimated by the simple shock modelsusing the quasi-static stress–strain relation to describe the engineering constitutive behaviour of the cellular material. Thereason for the difference between the quasi-static and dynamic stress–strain relations for the cellular material is discussed,and deformation and energy absorption mechanisms are explored. Finally, some experimental procedures are proposed forinvestigating the dynamic stress–strain behaviour of cellular-type materials.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–11496

2. Cell-based finite element modelling

2.1. Voronoi structures

The 3D Voronoi technique (Okabe et al., 1992) was employed to generate closed-cell foams with a uniform cell-wallthickness. N nuclei are randomly ‘scattered’ in a given region (here, a square-section prism with volume Vfoam) based on theprinciple that the distance between any two nuclei is constrained to be larger than a given minimum distance δmin. This isdefined as

δmin ¼ ð1�kÞδ0; ð1Þwhere k is the cell irregularity, which is a parameter of value between zero and unity (Zheng et al., 2005), and δ0 is theminimum distance of any two adjacent nuclei in a tetrakaidecahedron structure, given by (Zhu and Windle, 2002)

δ0 ¼ffiffiffi3

pðVcell=4Þ1=3; ð2Þ

where Vcell¼Vfoam/N is the volume of a single tetrakaidecahedron cell.These N nuclei are copied to their surrounding neighbouring volumes by translation. All the nuclei are used to generate a

Delaunay tetrahedron configuration and its dual configuration, i.e. the Voronoi structure, see the sample in Fig. 1a, in whichevery cell consists of several polygonal surfaces. The portion of the Voronoi structure outside the given region is deleted toobtain the desired specimen of closed-cell foam. To save computational time in the FE calculation, small surfaces havingshort edges (e.g. edge lengtho0.01 mm for the sample with cell size �3 mm and cell-wall thickness �0.1 mm used in thisstudy) are eliminated.

2.2. Finite element models

The cellular specimen is constructed in a volume of 30�20�20 mm3 with 600 nuclei, as shown in Fig. 1b. The cellirregularity k and the relative density ρ are set to be 0.5 and 0.1, respectively. In the specimen, cell walls have a uniformthickness h, which is 0.094 mm. The relative density ρ of the cellular specimen is related to the cell-wall thickness h by

ρ� ρ0=ρs ¼h

V foam∑Aj; ð3Þ

where ρ0 is the density of the cellular specimen, ρs the density of its cell-wall material and Aj the area of the j-th cell-wallsurface. In the Voronoi model, a full single cell is an irregular convex polyhedron and cells are different. For simplicity, asingle cell is measured by the volume-equivalent-sphere diameter (Wadell, 1933), i.e.

di ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi6Vi=π3

p; ð4Þ

where Vi is the volume of cell i. The average cell diameter d0, which is determined by averaging all the cell diameters of thecell-based FE model, is 3.3 mm in this study. The cell-based FE model is meshed by using ABAQUS shell elements S3R (the3-node triangular general-purpose, reduced integration, hourglass controlled, finite membrane strain shell element) andS4R (the 4-node doubly curved general-purpose, reduced integration, hourglass controlled, finite membrane strain shellelement) (ABAQUS, 2010). Through a mesh sensitivity study, the characteristic length of shell element size is set to be about0.3 mm and so the model has about 170,000 shell elements including �20,000 S3R elements and �150,000 S4R elements.1

The cell-wall material is assumed here to be elastic–perfectly plastic with density ρs¼2770 kg/m3, Young's modulusEs¼69 GPa, Poisson's ratio νs¼0.3 and yield stress σys¼170 MPa. Contact between the cellular specimen and the rigid walland that between all possible surfaces in the cellular specimen are considered with a friction coefficient of 0.02, as in Zhenget al. (2005).

A direct impact scenario is considered in which the cellular specimen impinges normally with an initial velocity, V0, on astationary, fixed rigid wall, as shown in Fig. 2a. Three different initial velocities, namely V0¼150, 200 and 250 m/s, areconsidered in this paper. The dynamic simulations were carried out with the FE code ABAQUS/Explicit.

A numerical compression ‘test’ at a low constant velocity (V¼1 m/s, i.e. the corresponding nominal strain rate is �33/s)was also performed to obtain a quasi-static nominal stress–strain relation. The quasi-static cellular specimen, which has thesame structure as that used in the direct impact scenario, is sandwiched between two rigid walls to impose a constant-velocity compression condition, as shown in Fig. 2b. One wall is fixed and the other wall travels at a constant velocity of1 m/s compressing the cellular specimen. Numerically, the same specimen can be used in different simulations. This ensuresthat the microstructural randomness of the specimen does not influence our main conclusions in this paper. For a realmaterial, repeated experimental tests may be required.

The quasi-static nominal engineering stress–strain relation of the cell-based FE model is shown in Fig. 3. The nominalengineering stress at the proximal end has almost the same value as the stress at the distal end for a given nominal (global)strain, which demonstrates that the crush process is therefore essentially quasi-static. It is noted that, whilst recognising the

1 The computational time of our computer (Quad-Core CPU, 16 GB memory and Red Hat Enterprise 6.0 Linux Server) is about 25 h for the quasi-staticcompression ‘test’; about 20 h for direct impact ‘test’ with initial impact velocity 150 m/s, about 15 h for 200 m/s and about 12 h for 250 m/s.

A section

3

1

2

Fig. 1. (a) A 3D Voronoi structure with 600 nuclei in a volume of 30�20�20 mm3, (b) the corresponding cell-based FE model and (c) an image of a middlesection perpendicular to the 2nd direction.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114 97

local dependence on the specific cellular morphology as in Reid and Peng (1997), the nominal engineering stress and strainare defined as σ¼F/A and ε¼u/L, respectively. Here F is the contact reaction force on the rigid wall, A the initial nominalcontact area, L the initial length of the specimen (L¼30 mm in this paper) and u the magnitude of the compressivedisplacement of the specimen, i.e. the movement of the rigid loading wall. This approach characterises, from an engineeringperspective, the quasi-static behaviour of the cellular material, much as was done in the pioneering book by Gibson andAshby (1988).

Two idealisations are employed to characterise this nominal stress–strain relation of the cellular material under quasi-static compression (V¼1 m/s), as shown also in Fig. 3.

Rigid wall

V

Rigid wallDistal end Proximal end

V0

Rigid wall

Distal end (Free end)

Proximal end

Fig. 2. Schematic diagrams of (a) the direct impact scenario of the cell-based FE model without a backing mass and (b) the constant-velocity compressionscenario of the cell-based FE model.

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

εD

Stre

ss, σ

(MPa

)

Compression (V = 1 m/s): At the proximal end At the distal end

Idealisations: R-PP-L (ε

Strain, ε

D = 0.65) R-PH (C = 0.60 MPa)

σ0 = 5.9 MPa

Fig. 3. The quasi-static stress–strain curves obtained from the cell-based FE model, and the R-PP-L and R-PH idealisations. The plastic hardening functionfor the R-PH idealisation is defined in Eq. (5).

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–11498

One is the rate-independent, R-PP-L idealisation (Reid and Peng, 1997) with two material parameters, i.e. the initial crushstress σ0 and the ‘densification’ strain εD. These two parameters for the cellular material used are determined as 5.9 MPa and0.65, respectively. Here, the ‘densification’ strain is defined by using the maximum energy absorption efficiency function

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114 99

criterion to remove any arbitrariness, as done in Tan et al. (2005a). Although there are some other choices for determiningthe ‘densification’ strain in the literature (Gibson and Ashby, 1988; Lopatnikov et al., 2004), only one value, i.e. 0.65, is usedfor comparison purposes in the present study.

The other is the rate-independent, rigid–plastic hardening (R-PH) idealisation (Zheng et al., 2013) with an empiricalfunction for the plastic hardening stage, written as

σ ¼ σ0þCε=ð1�εÞ2; ð5Þ

where C is an empirical fitting parameter, which characterises strain hardening behaviour, and σ0 the initial crush stress asdefined above. This idealisation also has only two material parameters, however it is a more accurate representation thanthe R-PP-L idealisation, as shown in Fig. 3. The two material parameters are related to the relative density and meso-structural parameters of the cellular material. For the cellular material with ρ¼0.1 used in this study, σ0 is 5.9 MPa, as givenabove, and C is determined to be 0.60 MPa. Choosing a plastic hardening function for the R-PH idealisation permitsanalytical solutions to be obtained from earlier shock models for comparison.

3. Results and discussion

3.1. Dynamic densification strain

3.1.1. Local engineering strain field and particle velocity fieldsStrain, which is a continuum concept, has been commonly used to describe the deformation in a cellular material, despite

its having a cellular morphology. However, measuring the ‘local’ strain in cellular materials under load is almost impossibleexperimentally. A numerical ‘test’ may therefore provide some insight into this difficulty.

Zou et al. (2009) used the relative displacement between two neighbouring cross-sections to define simply the localengineering strain in a regular 2D honeycomb. They indicated that this method is more suitable for high-velocity impactthan for low-velocity impact, due to a more layer-like mode of deformation. This method also has a limitation that theresults have large data oscillations. Zheng et al. (2012) pointed out that the local strain, in a macroscopic perspective, ofcellular material should be defined statistically over the range of a few cell sizes. Accounting for this, Liao et al. (2013, 2014)developed a local engineering strain calculation method based on the optimal local deformation gradient technique (Li andShimizu, 2005). This method is also used herein, see Appendix A for the details, to define the local engineering strain fieldfrom our 3D cell-based FE simulation.

In this local engineering strain calculation method, the displacements of the nodes within a cut-off radius rc around alocal node are combined (as described in Appendix A.1) to define the local strain at this node. It is found that the local strainis sensitive to the cut-off radius rc. After a convergence analysis, see Appendix A.2, the cut-off radius rc in the local straincalculation method was set to 3.3 mm, which is equal to the average cell size d0, to ensure the accuracy of the local strainscalculated.

The strain field for ε11 (the local engineering strain in the loading direction), which is presented in the Lagrangian (un-deformed) frame of reference with coordinates (X1, X2, X3), is the focus in this paper.

As an example, a slice image at time 0.05 ms with the initial impact velocity V0¼250 m/s is shown in Fig. 4a. Hereafter,‘local strain’ always refers to the ‘local engineering strain’. The image in Fig. 4a shows that when the impact velocity is highenough, there is a compaction front, across which the strain changes significantly, approximating discontinuously. Forconsistency with the previous work, this front is termed a plastic shock front. Hereafter, the physical quantities such asstrain and stress close (in terms of the cell dimension) to and ahead of the shock front are referred as those ahead of theshock front. The physical quantities close to and behind the shock front are referred as those behind the shock front.

The 1D strain distribution (ε1 versus X1 plot) is considered along the loading direction and is obtained by averaging thelocal strain ε11 of the 3D strain field at points in the cross-sections identified by X1. Some curves of the strain distribution atdifferent times are depicted in Fig. 4b for the direct impact case of V0¼250 m/s. This figure shows that the strains ahead ofthe shock front are very small (�0.02), though not zero. Following impact, the stress behind the shock front is unloadedelastically because the ‘impact velocity’ decreases. However, the corresponding unloading elastic strain is negligible whencompared with the plastic crushing strain �0.1. By ignoring the effect of elastic unloading, the local strain behind the shockfront remains unchanged during the complete loading process.

Consider the one-dimensional velocity distribution along the 1st (loading) direction to determine the Lagrangianlocation of the shock front (namely the shock-front location hereafter), Φ(t). The particle velocity components v1 of allnodes in the middle section (see Fig. 1b) were averaged in the 2nd direction to determine the one-dimensional velocitydistribution, see Fig. 4c for the case of V0¼250 m/s, where v1 is the velocity component in the 1st direction. The one-dimensional velocity distribution in Fig. 4c shows that the velocity along the specimen has a rapid drop at a specific locationwhich can be regarded as the shock front. Therefore, the location where the velocity gradient has a maximum value is theshock-front location Φ(t) at time t. The shock-front location versus time curves for the three initial impact velocities areshown in Fig. 5.

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

Loca

l stra

in,

1

0.070.06

0.050.04

0.030.02

t = 0.01 ms

0.1 ms0.09

0.08

0 5 10 15 20 25 30

0

50

100

150

200

250

Vel

ocity

, v1 (m

/s)

Lagrangian location, X1 (mm)

Lagrangian location, X1 (mm)

0.070.06

0.050.04

0.030.02

t = 0.01 ms

0.090.08

0.1 ms

ε

Fig. 4. (a) Slice image of the 3D engineering strain field of ε11 in the cellular specimen at t¼0.05 ms, (b) one-dimensional local strain distributions and(c) one-dimensional local velocity distributions at an initial impact velocity V0¼250 m/s.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114100

3.1.2. Strains behind and ahead of the shock frontThe strain behind the shock front, εB(t), is determined by the method presented in Appendix B. Hereafter, this quantity is

referred to as the ‘shock strain’. The shock strains εB(t) versus time t curves for three different initial impact velocities areshown in Fig. 6a. The results show that the shock strain decreases during the loading process. This is because the ‘impactvelocity’ (see the definition in Section 1.3) decreases with increasing time as the kinetic energy of the cellular specimen iscontinuously transformed into internal/plastic deformation energy of the cellular specimen.

The strain ahead of the shock front can also be defined from the 1D strain distribution, see Appendix C. The curves of thestrain in the near-region ahead of the shock front versus impact time for the three initial impact velocities are shown inFig. 6b. This figure suggests that the strains ahead of the shock front almost have the same value and the average value isabout 0.02 for the three initial impact velocities, which is much less than the strain behind the shock front.

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.140

5

10

15

20

25

Shoc

k-fr

ont l

ocat

ion,

Φ (m

m)

Time, t (ms)

Direct impact:V0 = 150 m/s

V0 = 200 m/s

V0 = 250 m/s

Fig. 5. The shock-front location versus time curves for three different initial impact velocities.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114 101

3.1.3. Comparison of shock strains obtained from FE and shock modelsAccording to the continuum-based stress wave theory, the conservation relations of mass and momentum across the

shock front are given by (Wang, 2007)

vBðtÞ�vAðtÞ ¼ � _ΦðtÞðεBðtÞ�εAðtÞÞ ð6Þand

σBðtÞ�σAðtÞ ¼ �ρ0_ΦðtÞðvBðtÞ�vAðtÞÞ; ð7Þ

respectively, where _ΦðtÞ is the shock-front speed, {vA, εA, σA} are the physical quantities ahead of the shock front and {vB, εB,σB} those behind the shock front. Combining Eqs. (6) and (7) leads to

σBðtÞ ¼ σAðtÞþρ0ðvBðtÞ�vAðtÞÞ2εBðtÞ�εAðtÞ

: ð8Þ

The earlier shock models outlined in Section 1.2 are based on idealisations of the quasi-static constitutive relation. In theR-PP-L model, the shock strain is regarded as a constant and usually taken to be the densification strain (Tan et al., 2005a),which is plotted in Fig. 7b. However, it has been shown that the quasi-static densification strain underestimates the shockstrain for high-velocity impact, see Tan et al. (2012). The quasi-static densification strain may denote the onset ofdensification, but it ignores any further hardening in the curve. Therefore, an R-PP-L material model using the quasi-staticdensification strain employed by Tan et al. (2005a) will always underestimate strains associated with higher levels of stress.The removal of this assumption is the main focus of this paper.

The concept of a dynamic densification strain has been proposed by Zou et al. (2009) and Tan et al. (2012). It wasconjectured that the dynamic densification strain varies with the impact velocity, or more precisely, locally, with the shock-front speed, which can be predicted by associating the Rankine–Hugoniot relations across the shock front and the stress–strain relation with a non-linear plastic hardening stage for a cellular material. For example, assuming that the cellularmaterial is a rate-independent, R-PH idealisation with the plastic hardening function in Eq. (5), we can approximately take{vA, εA, σA}¼{v, 0, σ0} and {vB, εB, σB}¼{0, εB, σ(εB)}, where v is the (local) impact velocity that varies with time t. Thus, theshock stress in Eq. (8) can be rewritten as

σðεBÞ ¼ σ0þρ0v2=εB: ð9Þ

Applying the plastic hardening function in Eq. (5) with ε¼εB to Eq. (9) gives

σ0þCεB=ð1�εBÞ2 ¼ σ0þρ0v2=εB; ð10Þ

which leads to a very simple relation

εB ¼v

vþc1ð11Þ

with c1 being defined as

c1 ¼ffiffiffiffiffiffiffiffiffiffiffiC=ρ0

q: ð12Þ

In this study, ρ0¼277 kg/m3 and C¼0.60 MPa as given in Section 2.2, and thus c1¼46.5 m/s. The curve of shock strainversus (local) impact velocity predicted by the R-PH shock model is shown also in Fig. 7b. In the R-PH shock model, theshock strain increases with the increase of the impact velocity, which is very different to that in the R-PP-L shock model.Importantly, this approach is superseded herein.

0.00 0.02 0.04 0.06 0.08 0.100.6

0.7

0.8

0.9

1.0

Stra

in, ε

B

Time, t (ms)

Direct impact:V0 = 150 m/sV0 = 200 m/sV0 = 250 m/s

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.140.00

0.01

0.02

0.03

Stra

in, ε

A

Time, t (ms)

Direct impact:V0 = 150 m/sV0 = 200 m/sV0 = 250 m/s

Fig. 6. (a) The shock strain, i.e. strain behind the shock front, εB(t) versus time t curves and (b) the strain ahead of the shock front versus time curves forthree different initial impact velocities.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114102

The relation between the shock strain and the impact velocity controlling shock front can also be obtained from the cell-based FE model for the three initial impact velocities. The impact velocity v(t) in this example is assumed to be the averagevelocity, obtained by averaging the velocities of all the relevant nodes ahead of the shock front of the cellular specimen. The(local) impact velocities v(t) as a function of time t for the three different initial impact velocities are given in Fig. 7a. In thisexample, it shows that v(t) decreases with time. According to the relations between v(t) in Fig. 7a and εB(t) in Fig. 6a, therelation between the shock strain εB and the impact velocity v can be determined for the cell-based FE model. Thisrelationship is plotted with a short dash line in Fig. 7b. It shows that the shock strain εB increases with the increase of theimpact velocity and the curves for the three different initial impact velocities overlap. This implies that the results areindependent of the specimen length and the impact velocity as long as it is supercritical.

Compared with the results obtained from the cell-based FE models, the quasi-static densification strain used by theR-PP-L model is clearly too crude for high-velocity impact, as shown in Fig. 7b. The R-PH shock and the cell-based FE modelspredict the same trend of the εB versus v relation, but the values of the densification strain from the R-PH shock model areless than those obtained from the cell-based FE models for the impact velocities considered (75–250 m/s).

The dynamic shock strain states of the cellular material are different from those in the quasi-static stress–strain relation.From these results, it can be inferred that the constitutive behaviour is indeed loading-rate-sensitive. However, it is not inthe conventional material strain-rate sense but rather as the result of an inertia-driven mechanismwhich will be explored inSection 3.4.

3.2. Plastic crushing stress

3.2.1. Stress behind the shock frontThe stress in the small region behind the shock front, termed the shock stress hereafter, is equal to the stress at the

proximal end of the cellular specimen, because the region behind the shock front is stationary and the inertia effect in this

0

50

100

150

200

250

Vel

ocity

, v (m

/s)

Time, t (ms)

Direct impact:V0 = 150 m/sV0 = 200 m/sV0 = 250 m/s

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

50 100 150 200 2500.6

0.7

0.8

0.9

1.0

Shock models: R-PP-L (εD= 0.65) R-PH (c1= 46.5 m/s) D-R-PH (c1= 28.2 m/s)

Den

sific

atio

n St

rain

, εD

Velocity, v (m/s)

Cell-based FE model:V0 = 150 m/sV0 = 200 m/sV0 = 250 m/s

εmax = 1-ρ = 0.9

Fig. 7. (a) The impact velocity versus time t at three different initial impact velocities and (b) a comparison of shock strains obtained from the cell-based FEmodels and the shock models (The predictions of shock strains from the R-PH and D-R-PH shock models are both in Eq. (11) but with different definitionsof c1 given in Eqs. (12) and (21), respectively.).

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114 103

region can be neglected. The curve of the shock stress versus time has been generated from the cell-based FE model andshows large data oscillation.

This oscillatory behaviour is the result of both the contact algorithm of the FE analysis and the propagation of elasticstress waves. In order to exclude the influence of the data oscillation, and also to be consistent with the local strainmeasurement (see Fig. 6a), the relation between the shock stress and impact time is averaged over a certain time interval τ,which corresponds to the time when a cell with an average size is crushed to a densification state, and is determined byτðtÞ ¼ d0= _ΦðtÞ, where d0 is the average cell diameter and _ΦðtÞ the shock-front speed. The values at a particular time of thedata points for the stress are then compatible with the value of the strain field results. The average value of the shock stressand the standard deviation versus impact time are shown in Fig. 8 for different initial impact velocities. The shock stressdecreases with time.

3.2.2. Stress ahead of the shock frontThe shock models summarised in Section 1.2 are based on the quasi-static stress–strain relations. The Rankine–Hugoniot

relations across the shock front are available for considering the dynamic stress–strain relations and thus the shock modelsare not confined to rate-independent idealisations. Eq. (8) can be re-arranged to another form

σAðtÞ ¼ σBðtÞ�ρ0v

2ðtÞεBðtÞ�εAðtÞ

; ð13Þ

which gives the dynamic initial crush stress since the shock stress and the shock strain have been determined in the cell-based FE models. The stress ahead of the shock front, σA, versus impact velocity obtained from Eq. (13) is plotted in Fig. 9. Asexpected, it transpires that σA is slightly higher than the initial crush stress in the quasi-static condition, and could beregarded as constant, considering the irregular structure of the cellular material. This constant, denoted as σd

0, should be a

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.140

5

10

15

20

25

30

Stre

ss, σ

B (M

Pa)

Time, t (ms)

Direct impact:V0 = 150 m/sV0 = 200 m/sV0 = 250 m/s

Fig. 8. The stress behind the shock front versus time t at three different initial impact velocities.

50 100 150 200 2500

2

4

6

8

10

12

σd0 = 7.7 MPa

σ0 = 5.9 MPa

Stre

ss, σ

A (M

Pa)

Velocity, v (m/s)

Direct impact:V0 = 150 m/sV0 = 200 m/sV0 = 250 m/s

Fig. 9. The stress ahead of the shock front versus impact velocity at three different initial impact velocities.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114104

dynamic material parameter in the continuum description sense and the reason will be explored in Section 3.4. The value ofσd0 for the cellular material used in this study is obtained from the statistical average of the data presented in Fig. 9, which

gives 7.7 MPa with standard deviation 0.7 MPa.

3.3. Dynamic stress–strain states

The stress–strain relations behind the shock front for three different initial impact velocities can be obtained bycombining the εB versus t curves and the σB versus t curves, which are shown in Fig. 6a and in Fig. 8, respectively. Theserelations together with the quasi-static stress–strain relation are plotted in Fig. 10. The three curves corresponding to thethree initial impact velocities (V0¼150, 200 and 250 m/s) are plotted from the simulations. All the three curves overlap. Thissuggests that a unique dynamic stress–strain state exists for a particular supercritical impact velocity, irrespective of theinitial velocity. Thus, this approach explains how the dynamic behaviour can be fully described for cellular materials.Compared with the quasi-static stress–strain relation in the densification stage, the dynamic stress–strain states couldobviously be different.

The local strain in a dynamic loading case can be larger than that in a quasi-static loading case, see Section 3.4. Clearly,the stress–strain state of the cellular material (up to 250 m/s) is essentially loading-rate dependent. It should be noted thateach dynamic stress–strain state corresponds to a particular ‘impact velocity’. This will be discussed further in Section 3.4.

For convenience, a dynamic material model, denoted as the D-R-PH idealisation, is employed herein with the dynamicplastic hardening function, similar to Eq. (5), given by

σðεÞ ¼ σd0þDε=ð1�εÞ2; ð14Þ

where D is a fitting parameter and σd0 the dynamic initial crush stress. In this study, D is 0.22 MPa. This material model will

be applied in Section 3.5 to predict the energy absorption capacity of the cellular material.

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

25

30

σ0 = 5.9 MPa

Stre

ss, σ

(MPa

)

Cell-based FE model:V = 1 m/sV0 = 150 m/sV0 = 200 m/sV0 = 250 m/s

Idealisations: R-PP-L (εD = 0.65) R-PH (C = 0.60 MPa) D-R-PH (D = 0.22 MPa)

σ0d = 7.7 MPa

Strain, ε

Fig. 10. The quasi-static and dynamic stress–strain relations for the cellular material and three simple idealisations.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114 105

The local stress–strain state varies with the reducing ‘impact velocity’. It should be noticed that it is not necessary toidentify explicitly which ‘impact velocity’ corresponds to a particular stress–strain state. Since the stress–strain relation isconcave-upward, a shock wave front will form in the cellular material under high-velocity impact, and the relations acrossthe shock front associated with the dynamic stress–strain relation suitably defined through the dynamic stress–strainbehaviour within the appropriate strain range will give the corresponding relationship between the stress–strain state andthe impact velocity. An example of using the dynamic stress–strain relation to solve an initial value problem will bepresented in Section 3.5.2.

Within that strain range, Fig. 10 gives two stress–strain curves for the cellular material: one is quasi-static and the otheris dynamic as defined above. As discussed in Zheng et al. (2012), a critical velocity exists for the cellular material, having atypical value of the order of 50–70 m/s. By evaluating the balance of stresses at the two ends of specimen under constant-velocity compression, as done in Liu et al. (2009), the critical velocity is determined as Vc1¼57 m/s for the present specimen.This critical velocity is the applicable upper-limit impact velocity when using the quasi-static stress–strain relation. The useof which one of the two stress–strain curves in a particular loading scenario is based on the comparison between the impactvelocity and the critical velocity. This provides a simple guide. To exactly determine the applicable lower-limit impactvelocity when using the dynamic stress–strain relation, a new technique should be developed in further studies todetermine the stress–strain states of a cellular material when the impact velocity is moderate, corresponding to thedeformation in the Transitional Mode (Zheng et al., 2005; Liu et al., 2009).

The dynamic behaviour of stress–strain states in the densification stage has also been reported in a recent paper about2D Voronoi honeycombs by some of the authors of the present paper, see Fig. 15 in Liao et al. (2013).

Recently, two papers (Barnes et al., 2014; Gaitanaros and Kyriakides, 2014) have reported similar dynamic behaviour foran open-cell aluminium foam in an experimental and numerical investigation. Those papers characterise the dynamicshock-behaviour of metallic foam using the Hugoniot approach.

The description of the dynamic stress–strain behaviour herein is complementary to those studies but also, additionally,raises several issues, particularly regarding experimental testing to characterise the material in terms of a dynamicconstitutive representation that can be used to solve boundary/initial value problems. Suggestions of how this approach canbe applied and utilised arising from the present study are made in Sections 3.5 and 3.6. These findings herein imply that theloading-rate-dependent behaviour is common in cellular materials, whether 2D or 3D; whether closed- or open-cell.

3.4. Deformation mechanisms

3.4.1. Range of deformation mechanismsThe difference between the dynamic and quasi-static stress–strain relations for cellular materials is the result of the

different deformation modes. Deformed images of a middle section perpendicular to the 2nd direction were generated toobserve this difference.

Some deformed images are shown in Fig. 11 for quasi-static and dynamic impact conditions. Their corresponding un-deformed section image is shown in Fig. 1c. From the quasi-static deformation pattern of the section image shown inFig. 11a, it can be seen that the deformation consists of randomly distributed shear collapse bands, which are likely to beoblique to the loading direction. This deformation mode is consistent with that observed in the random 2D honeycombsunder low-velocity compression, which was termed the Quasi-static Homogeneous Mode (Zheng et al., 2005; Liu et al.,2009). When the macroscopic deformation develops further, these shear collapse bands may meet each other and theinteraction between shear collapse bands may lead to changes in the developing directions of the shear collapse bands. Theintersecting regions of the collapse bands are strengthened and further deformation is blocked until the stress in

Time: 0.032 ms Time: 0.059 ms Time: 0.075 ms

Nominal strain: 0.2 Nominal strain: 0.3 Nominal strain: 0.4

Shear collapse bands

Fig. 11. Deformation patterns of the cell-based FE model section images (a) under quasi-static compression and (b) under direct impact with an initialimpact velocity 250 m/s.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

Loca

l eng

inee

ring

stra

in,

11

Dimensionless time, t/(L/V) or t/(L/V0)

Compression (V = 1 m/s) Direct impact (V0 = 250 m/s)

Stagnant

ε

Fig. 12. The local engineering strain ε11 for the marked node in Fig. 11 versus dimensionless time.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114106

surrounding cells increases to the strength of these cells. To provide some evidence to support this strengthening effect,evolution of local strain at a typical node marked in Fig. 11 is analysed in Fig. 12, in which the strain ε11 for the quasi-staticcompression has a stagnant stage with a level of �0.3. When the impact velocity is high enough, a deformation pattern

0 5 10 15 20 25 300

20

40

60

80

100Shock models:

R-PP-L R-PH D-R-PH

( v = Vc1)

Inte

rnal

ene

rgy,

EI (J

)

Deformation, u (mm)

Cell-based FE model:V = 1 m/sV0 = 150 m/sV0 = 200 m/sV0 = 250 m/s

0 50 100 150 200 2500

5

10

15

Spec

ific

inte

rnal

ene

rgy,

U (J

/mm

3 )

Velocity, v (m/s)

Shock models: R-PP-L R-PH D-R-PH

Vc1 = 57 m/s

a

b

Fig. 13. (a) The specific internal energy absorbed by the cellular material due to the propagation of shock front versus impact velocity; (b) the total internalenergy absorbed by the cellular specimen versus axial deformation.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114 107

consisting of layer-wise collapse bands is observed, as shown in Fig. 11b. This deformation mode is similar to that observedin the random 2D honeycombs under high-velocity impact, which is termed the Shock Mode (Zheng et al., 2005; Liu et al.,2009). In this case, there is no shear collapse band in the specimen and so there is no interaction of shear collapse bands.Dynamically crushed cells are tightly stacked and thus the dynamic densification strain is larger than the onset ofdensification of the specimen under quasi-static compression.

3.4.2. The deformation mechanisms to initial crush and collapse stressesWhen the cellular specimen is under low-velocity (quasi-static) compression, relatively weak cells are first crushed and

then induce the deformation of surrounding cells to form shear collapse bands due to minimal energy-consumption. Thus,the quasi-static initial crush stress is dependent on the strength of the weakest links. However, when the loading rate ishigh, there is sufficient inertial resistance at the meso-scale level to delay the onset of the minimal energy-consuming, shearband collapse mode. A more localised layer-wise collapse mode is generated instead. In this case, the initial crush stress isdependent on the strength of each layer of cells and thus may vary with time during the impact process since randomnessand defects are inevitable in a cellular material. In a continuum sense, the influences of randomness and defects should beerased and so the dynamic initial crush stress σd

0 should be taken as a material parameter. Consequentially, the dynamicinitial crush stress is larger than the quasi-static one, i.e. σd

04σ0.During the quasi-static loading process, an increasing number of random shear collapse bands appear, and shear collapse

bands may meet each other and interact. This explains why the ‘plateau’ stage of the quasi-static stress–strain curve usuallydisplays a feature of slight plastic hardening. This shear-collapse-band interaction mechanism has not been explored in theliterature. In contrast, Barnes et al. (2014) claimed that the small increase in the stress is “caused by variation in density,anisotropy, and by some strengthening from the ends”. When the loading rate is very high, the collapsed cells do not blockthe collapse of other cells and cells can be folded at a relatively stable stress level to a large densification strain.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114108

3.4.3. Comment on the loading-rate sensitivityThe dynamic stress–strain states corresponding to different impact velocities make up a curve related to the dynamic

densification behaviour of a cellular material. The kind of loading-rate sensitivity revealed in this paper is different from thestrain-rate sensitivity of some dense metals (Meyers, 1994). For the latter, one has a set of stress–strain curvescorresponding to different strain rates, while for the former only one curve exists where each point in the densificationstage is associated with a stress–strain state behind a shock front of the cellular material under a certain impact velocity. Theuniqueness of this stress–strain state as a function of loading rate is supported by the fact that the stress and strain valuesfor virtual tests at different initial impact velocities fall on the same (i.e. unique) curve (see Figs. 7b, 9 and 10). For a cellularmaterial, the dynamic stress–strain curve is related to the layer-wise collapse mode, while the quasi-static one is related tothe random shear band collapse mode. The constitutive behaviour of the cellular material is deformation-mode dependent.

3.5. Energy absorption mechanisms

3.5.1. Specific internal energy at a particular impact velocityBased on the energy conservation relation across a shock front in stress wave theory, the specific internal energy is given

by (Wang, 2007)

U ¼ 12ðσAþσBÞðεB�εAÞ: ð15Þ

For the R-PP-L shock model, the physical quantities ahead of the shock front are {v, 0, σ0} and those behind the shockfront are {0, εD, σ0þρ0v2/εD}. Thus, the specific internal energy is obtained as (Zou et al., 2009)

U ¼ σ0εDþ12ρ0v

2: ð16Þ

For the R-PH shock model, {vA, εA, σA}¼{v, 0, σ0}, {vB, εB, σB}¼{0, εB, σ0þρ0v2/εB} with εB in Eq. (11), and so we have

U ¼ σ0vvþ

ffiffiffiffiffiffiffiffiffiffiffiC=ρ0

p þ12ρ0v

2: ð17Þ

For the D-R-PH shock model, by taking σd0 and D instead of σ0 and C, respectively, in Eq. (17), the specific internal energy

is determined as

U ¼ σd0v

vþffiffiffiffiffiffiffiffiffiffiffiD=ρ0

p þ12ρ0v

2: ð18Þ

There is the same term, i.e. ρ0v2/2, in each of the above three expressions. This term is inertial in origin, as pointed in the

literature. The predictions for U in this study are plotted in Fig. 13a. It shows that the prediction from the D-R-PH shockmodel is larger than the other two predictions when v4Vc1. It is found that there are two new energy absorptionmechanisms related to the D-R-PH shock model, namely the increase of initial crush stress ðσd

04σ0Þ and the reduction in thestrain hardening parameter (DoC), which are related to the initiation and interaction of collapse bands.

To date, predictions of energy absorption and stresses during dynamic compaction of cellular solids have relied on quasi-static properties. The key finding of this paper is that quasi-static properties, whilst providing the first-order approximationsof the stresses and the energy absorption capacity attained, do not predict accurately the stresses or energy absorption atincreasing impact velocities.

3.5.2. Application-predictions for the initial value problem in Fig. 2(a)The D-R-PH shock model for the direct impact scenario used above is first developed as follows. In this case, {vA, εA, σA}¼

{v, 0, σd0} and {vB, εB, σB}¼{0, εB, σ(εB)}. Eq. (6) gives the shock-front speed

_ΦðtÞ ¼ vðtÞ=εBðtÞ: ð19ÞThe shock stress in Eq. (8) can be rewritten as

σðεBÞ ¼ σd0þρ0v

2=εB: ð20ÞApplying the plastic hardening function in Eq. (14) with ε¼εB to Eq. (20) leads also to Eq. (11) but with c1 re-defined as

c1 ¼ffiffiffiffiffiffiffiffiffiffiffiD=ρ0

q: ð21Þ

The prediction from Eq. (11) with c1¼28.2 m/s compare well with the results of cell-based FE model, as shown in Fig. 7b.The motion of the portion behind the shock front is governed by the inertial law

dvdt

¼ � σd0

ρ0ðL�ΦðtÞÞ: ð22Þ

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114 109

By applying Eqs. (19) and (11), the left-hand side of the above expression gives

dvdt

¼ _ΦdvdΦ

¼ vεB

dvdΦ

¼ ðvþc1ÞdvdΦ

¼ 12

ddΦ

ðvþc1Þ2: ð23Þ

Combining the above two expressions and integrating leads to

v¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðV0þc1Þ2þ

2σd0

ρ0lnð1�Φ=LÞ

s�c1; ð24Þ

in which the initial conditions v(0)¼V0 and Ф(0)¼0 are used. The energy absorbed by the cellular specimen can becalculated from the change in kinetic energy, written as

EI ¼12ρ0ALV

20�

12ρ0AðL�ΦÞv2

¼ 12ρ0ALV0

2�σd0AðL�ΦÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiαþ lnð1�Φ=LÞ

q�

ffiffiffiβ

q� �2ð25Þ

where

α¼ ρ0ðV0þc1Þ2=ð2σd0Þ: ð26Þ

and

β¼ ρ0c21=ð2σd

0Þ: ð27ÞThe axial deformation of the cellular specimen is calculated as

u¼Z Φ

0εBdΦ¼

Z Φ

0

vvþc1

dΦ¼Φ�Lffiffiffiβ

q Z Φ=L

0

dξffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiαþ lnð1�ξÞ

p : ð28Þ

The predictions of the EI versus u relation from the D-R-PH shock model are plotted in Fig. 13b. They compare very wellwith the results of the cell-based FE model. In Fig. 13b, dots on the curves predicted by the D-R-PH shock model indicate thatthe impact velocity reaches Vc1.

To compare with the predictions from the rate-independent shock models, mathematical derivations for those are simplydescribed here. For the R-PH shock model, Eqs. (25) and (28) with taking σ0 and C instead of σd

0 and D, respectively, areavailable. For the R-PP-L shock model, in considering εB¼εD, it is easy to determine the impact velocity as (Zheng et al.,2012)

v¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV20þ

2σ0εDρ0

lnð1�Φ=LÞs

: ð29Þ

The axial deformation of the cellular specimen is

u¼Z Φ

0εDdΦ¼ εDΦ: ð30Þ

Thus, the energy absorbed by the cellular specimen can be explicitly expressed as

EI ¼ρ0AV

20u

2εD�σ0AðLεD�uÞln 1� u

LεD

� �: ð31Þ

The predictions from these two shock models are also presented in Fig. 13b. It shows that the R-PP-L shock model predictthe energy absorption capacity accurately only when the initial impact velocity is moderate. The deviation of the predictedvalues comes mainly from the underestimation of the densification strain at relatively high impact velocities; while theoverestimation at low velocities. The R-PH shock model always underestimates the energy absorption capacity at any initialimpact velocity. Therefore, to accurately predict the energy absorption capacity, the dynamic stress–strain curve of a cellularmaterial should be applied.

3.6. Significance to material characterisation

3.6.1. Objective of paperA basic aim of this paper is to provide a testing proposal to improve the description of the dynamic behaviour of cellular

materials. This involves providing data that is directed towards an improved treatment of the propagation of ‘shock fronts’in cellular materials under appropriate (impact velocity) conditions.

Inevitably, by recognising the first order usefulness of previous work starting with the quasi-static overview of the classof materials by Gibson and Ashby (1988), extended simply into impact loading at a high (supercritical) impact velocity byReid and Peng (1997), one of the perceived deficiencies in the latter is centred around the concept of ‘densification’, a cell-level, distinctive, meso-structural feature.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114110

Fundamentally, dynamic deformation is dominated by the effect of propagating ‘fronts’ separating un-deformed cellularstructures from heavily deformed regions in which the cells have undergone, macroscopically, very large strain changesapproaching conditions essentially governed by the consolidated cell-wall material properties. This behaviour has beeninvestigated by a 3D cellular model in this study.

Given the dominance of ‘shock’ behaviour (see footnote 4 on p. 2176 in Tan et al., 2005a) the proposals for animprovement of the cellular material description under impact conditions rest on the use of the Rankine–Hugoniotconditions which govern such modes of deformation. This is, significantly different to the assumption of uniformdeformation in a specimen required by the SHPB technique. Deformation localisation should be anticipated whenperforming relevant experimental procedures in this régime.

3.6.2. Proposed test method to generate dynamic stress–strain dataA large number of experimental investigations have been carried out on the dynamic enhancement of the crushing stress

of cellular materials and there is increasing evidence that for a given cellular material, a ‘dynamic stress–strain curve’ existsthat can be quite different from the quasi-static curve. The direct impact method described by Reid and Peng (1997) is themost common experimental technique that has been employed to inform this behaviour. In this method, only twomeasurements were taken: the initial impact velocity, V0, is recorded using interrupted light beams and the shock stress, σB,is measured using the Hopkinson bar load cell. Comparison of results with predictions based on a simplification of quasi-static data (Tan et al., 2012) suggests a dynamic material characterisation is required. However, it has not been possible todevelop a dynamic stress–strain curve from the test data that was gathered at that time.

Other studies have included the reverse impact test, in order to estimate the dynamic initial crush stress, σd0, e.g. Harrigan

et al. (2005) and Barnes et al. (2014). If required, that test could give the stress ahead of the shock front.Further experimental evidence for the differences between the quasi-static and dynamic properties was provided by

Zaretsky et al. (2012) for polyurethane foam. The Hugoniot of the foam in the stress–relative volume plane from plateimpact tests was quite different to that for quasi-static compression. Recently, Barnes et al. (2014) employed high speedphotography to take extra measurements that made it possible to construct a curve (Fig. 13 in Barnes et al., 2014) similar tothe dynamic stress–strain curve for an open-cell aluminium foam. It was shown that the “shock-induced strain” underdynamic compaction was “significantly higher than that induced quasi-statically at the same stress”.

Even in this most recent study, each test has only been used to provide a single data point on a dynamic stress–straincurve. A significant implication of the FE investigation of the 3D cellular model in this study is that one single test canprovide dynamic stress–strain data over a wide impact velocity range.

Consider a cellular material that can be modelled using the D-R-PH idealisation. By eliminating εB from Eqs. (20) and (14)with ε¼εB, the shock stress is given by

σB ¼ σd0þ

ffiffiffiffiffiffiffiffiffiDρ0

pvþρ0v

2: ð32ÞBy performing a direct impact test to produce σB(t) and v(t) (i.e. the motion of the specimen during crushing) pulses, a curveof σB versus v can be obtained, which could include as many of the data points as required. In this scenario, the shock stressσB is again measured from the output bar as stated above. The transient velocity v of the distal end of the specimen isobtained from a high-speed camera (see Fig. 14 in Elnasri et al., 2007 and Fig. 3a in Barnes et al., 2014) and image correlationsoftware. This approach is similar to that described earlier by Wu and Chang (1995).

The velocity of the cellular specimen reduces with time during impact/deformation and the traces provide ‘continuous’data, not only single data points. The material parameters σd

0 and D could be determined by fitting Eq. (32) with the σB vs. vdata. This provides an experimental method with performing only one single test enough to obtain a dynamic stress–straincurve for a given cellular material. The initial velocity of the cellular specimen is suggested to be large to obtain a large strainrange. To confirm the effectiveness of this test method, extensive experiments are encouraged, e.g. changing the initialvelocity and the length of cellular specimens.

4. Conclusions

4.1. Role of ‘virtual’ FE model

A cell-based FE model of closed-cell foam based on the 3D Voronoi technique was constructed to examine the dynamicmechanical behaviours of metal foams under direct impact. A method based on the discrete deformation gradient wasemployed to calculate the local strain field and then to determine the local strains ahead of and behind the shock front.

The results show that the dynamic densification strain increases with the increase of impact velocity and is larger thanthe quasi-static densification strain used in the R-PP-L shock model. According to shock wave theory, the stress ahead of theshock front can be estimated. This is found to be larger than the initial crush stress in the quasi-static compression.

It has been shown that the dynamic stress–strain states at each Lagrangian location in the specimen can be determinedto define a dynamic stress–strain relationship for a given cellular material. This was confirmed to be different from thequasi-static stress–strain relation. The difference is explained by analysing the deformation modes at the specimen level andthe deformation mechanisms were explored, as summarised in Table 1. The results described herein reveal the possiblemechanism explaining the loading-rate sensitivity of cellular materials.

Table 1Summary of the features of quasi-static and dynamic stress–strain relations and the corresponding deformation mechanisms. Superscripts “qs” and “d”denote the quasi-static and dynamic cases, respectively.

Features of stress–strain relations Deformation mechanisms

Quasi-static (voVc1) Dynamic (v4Vc1)

Initial crush σ0oσd0 Initiation of shear collapse band (SCB) Initiation of layer-wise collapse band (LCB)

Strain hardening C4D Interaction between random SCBs Weak interaction between LCBsDensification εqsD oεdD Strengthened cells block deformation Crushed cells are much tightly stacked

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114 111

Furthermore, this 3D cell model informs the establishment of an experimental technique to define the dynamic stress–strain for a cellular material.

As an illustration, energy absorption mechanisms, including the inertia effect as revealed by the increase of initial crushstress ðσd

04σ0Þ and the reduction of strain hardening parameter (DoC), has been explored for a cellular material underdynamic loading.

4.2. Further work

This approach could be extended in several ways to model the dynamic response of structural elements containingcellular materials. From a computational design perspective, these would require the more representative models describedabove for the cellular material in question. Inevitably, this could require more computing power and require an HPCenvironment, as anticipated in Section 4 of Tan et al. (2012). The current paper explains how the resulting data could beprocessed to provide the relevant cellular material constitutive description for dynamic applications and modelling. Aninteresting link with tomographic studies of real materials could also arise from similar approaches in the future.

Acknowledgements

The authors thank the anonymous reviewers for their valuable contribution in assisting the authors to clarify theirpresentation. The authors at USTC are grateful for the support provided by the National Natural Science Foundation of China(Projects nos. 11002140, 90916026 and 11372308). The fifth author is grateful for the support provided through the LRF(Lloyd's Register Foundation, a UK registered charity and sole shareholder of Lloyd's Register Group Ltd, invests in science,engineering and technology for public benefit, worldwide) Centre.

Appendix A. Local strain calculation method

A.1 Method description

The local engineering strain field of the cell-based FE model is obtained by the discrete deformation gradient based onthe method of least squares (Liao et al., 2013). All nodes coordinates of the cell-based FE model are taken from the FEsimulation results. In the un-deformed configuration, the coordinate of any node i is denoted as Xi and any other node witha distance from node i less than a cut-off radius rc are collected into a set Ni. The vector from node i to node j is given byDji¼Xj�Xi. In a deformed configuration, the vector from node i to node j is expressed as dji¼xj�xi, where xi is thecoordinate of node i. It is desired to find a transformation matrix Fi of node i that approximately transforms the un-deformed configuration to the deformed configuration through

dji �DjiFi: ðA:1Þ

The coordinate error is defined as

φi ¼∑jwðDijÞðDjiFi�djiÞðDjiFi�djiÞT; ðA:2Þ

where w(Dij) is a weighting function and superscript T denotes the transpose of a matrix. In this paper, the weightingfunction is defined as

wðDijÞ ¼1; jDijjorc0; jDijjZrc

(; ðA:3Þ

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114112

with rc being the cut-off radius. When the error φi meets a minimum value, the transformation matrix Fi is regarded as theoptimal deformation gradient of node i. By taking

∂Φi

∂Fi¼∑

jwðdijÞDT

ijðDjiFi�djiÞ ¼ 0; ðA:4Þ

the optimal deformation gradient Fi can be determined as

Fi ¼V�1i Wi; ðA:5Þ

where Vi ¼Σ jwðdijÞDTjiDji and Wi ¼Σ jwðdijÞDT

jidji. Based on the continuum theory, the Lagrangian strain tensor of node i isgiven by

Ei ¼12ðFiFTi �IÞ; ðA:6Þ

where I is the identity matrix. The strain component considered in this paper is the local engineering strain in the Xdirection,

ε11 ¼ 1�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ2E11

p; ðA:7Þ

which is taken positive in compression. The local engineering strain field is the interpolation of every ε11 value in the cell-based FE model.

A.2 Convergence analysis of cut-off radius

In the strain calculation method employed, the local strain of node is sensitive to the cut-off radius rc. Here, weconsidered a low-velocity compression scenario of the cellular specimen used above to determine an appropriate value ofthe cut-off radius. In the compression scenario, the nominal strain εN is defined as the compression distance divided by thelength of the specimen along the compression direction. The nominal strain should be consistent with the mean value of alllocal strains in the specimen, εmean, when the local strains are correctly defined and calculated. Some typical one-dimensional strain distributions with different rc at nominal strain 0.6 are shown in Fig. A.1a The mean value of all localstrains, εmean, determined by local strain field is given by

εmean ¼ 1L

Z L

0ε1ðXÞ dX: ðA:8Þ

By comparing with the two strain εN and εmean, their relative error, ðεN�εmeanÞ=εN , is shown in Fig. A.1b. From this figure,we find that the relative error decreases with increasing rc/d0, where d0 is the average cell diameter. When rc/d0 is largerthan 0.8, the relative error is less than 5%. In this paper, we choose rc to be equal to d0 to ensure the accuracy that the relativeerror is less than 2.5%.

Appendix B. Strain behind the shock front

The strain at the Lagrangian location Φ(t) illustrated in Fig. A.2 cannot be directly taken as the strain behind the shockfront, εB(t), because the strain at this location is obtained by using the nodes behind the shock front as well as those ahead ofthe shock front in the strain field algorithm. If we ignore the elastic deformation, the local strain behind the shock frontremains unchanged, after the shock front has passed this location. Therefore, the shock strain εB(t) can be defined in anindirect way. First, we determine the Lagrangian shock-front locationΦ(t) at a given time t and then define the strain at theLagrangian locationΦ(t) from the strain distribution at the time tend (the time corresponding to the projectile cellular modelvelocity reaching zero) as εB(t), as illustrated in Fig. A.2. The strain distribution at the time tend has a not accurate part,because of the contribution of the nodes ahead of the shock front in strain field calculation. We exclude this part and theonly the portion from point F to point C in the strain distribution at time tend is taken as the effective portion to obtain theshock strain.

0.4

0.5

0.6

0.7

0.8

Loca

l stra

in, ε

1

Lagrangian coordinate, X1 (mm)

rc = 2.0 mmrc = 2.5 mmrc = 3.0 mmrc = 3.5 mmrc = 4.0 mm

0 5 10 15 20 25 30

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.1

0.2

0.3 (εN - εmean)/εN

Rel

ativ

e er

ror

Cut-off radius, rc / d0

Fig. A.1. (a) The one-dimensional strain distribution with different cut-off radiuses and (b) the relative error between the nominal strain εN and meanstrain εmean versus rc/d0 when the nominal strain is 0.6 in the low-constant-velocity compression scenario.

0 5 10 15 20 25 300.0

0.2

0.4

0.6

0.8

1.0

Φ(t)

Shock front

εB(t)

Loca

l stra

in, ε

1

Lagrangian location, X1 (mm)

ttend

Ahead of shock frontBehind shock front

rc

F

C

A

B

DEεA(t)

rc

Fig. A.2. Schematic diagram of determining the strains across the shock front.

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114 113

Z. Zheng et al. / J. Mech. Phys. Solids 72 (2014) 93–114114

Appendix C. Strain ahead of the shock front

In the shock models mentioned in Section 1.2, the shock front is regarded as a discontinuity surface with zero width.However, the strain distribution shown in Fig. A.2 has a “certain width” because of the “averaging” effect of local strain fieldalgorithm. In order to obtain the strain ahead of the shock front, we exclude the in accurate “certain width” region (theregion close to the shock front within rc which is taken to be the average cell diameter d0 after a convergence analysis) andthe region ahead of the shock front, which ranges from D to E shown also in Fig. A.2, is considered. In order to provide ameans of comparison with these older theories, the strain ahead of the shock front for the shock theory is roughly estimatedby two steps: Step 1, linearly fitting the curve of the strain ahead of the shock front between point D and point E; Step 2,extrapolating the line to the location of the shock front, as depicted in Fig. A.2.

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