International Journal of Impact Engineeringstaff.ustc.edu.cn/~jlyu/PDF/2017 Dynamic material...shock wave speed is strongly dependent on the impact velocity, but the effect of the

International Journal of Impact Engineering 99 (2016) 111�121

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International Journal of Impact Engineering

journal homepage: www.elsevier.com/locate/ijimpeng

Dynamic material parameters of closed-cell foams under high-velocity

impact

TaggedPShilong Wanga, Yuanyuan Dinga, Changfeng Wangb, Zhijun Zhenga,*, Jilin Yua

TaggedP

a CAS Key Laboratory of Mechanical Behavior and Design of Materials, University of Science and Technology of China, Hefei, Anhui 230026, PR Chinab Continuous Extrusion Research Center, Dalian Jiaotong University, Dalian, Liaoning 116028, PR China

TAGGEDPA R T I C L E I N F O

Article History:Received 10 May 2016Revised 10 August 2016Accepted 24 September 2016Available online 30 September 2016

* Corresponding author. .E-mail address: [email protected] (Z. Zheng).

http://dx.doi.org/10.1016/j.ijimpeng.2016.09.0130734-743X/© 2016 Elsevier Ltd. All rights reserved.

TAGGEDPA B S T R A C T

Cellular materials under high-velocity impact have highly localized deformation with the cells layer-wisecollapse, which is usually characterized by the propagation of shock wave. Researches have shown that theshock wave speed is strongly dependent on the impact velocity, but the effect of the meso-structural andbase-material parameters is unclear. In this study, the dynamic material parameters of closed-cell foamsare investigated with cell-based finite element models. The one-dimensional velocity distribution along theloading direction is used to capture the propagation of shock front. The shock wave speed is thus deter-mined and it exhibits a linear relationship with impact velocity when the impact velocity is high enough.The difference between the shock wave speed and the impact velocity is a dynamic material parameter ofcellular material and the effect of meso-structural and base-material parameters on this dynamic materialparameter is investigated with dimensional analysis. An expression of the dynamic material parameterwith respect to the relative density and the base-material parameters is obtained. It shows that the dynamicmaterial parameter intensively relies on the relative density and increases linearly with the relative density.The investigation of dynamic stresses with the aid of a shock model shows that the initial crushing stressincreases in a power-law tendency with the increase of relative density. As a result, a stress�strain relationinvolving the relative density of material cellular and the yield stress and density of base material isobtained for the closed-cell foams considered. The effects of hardening behaviors of cell-wall material andgas trapped within cells are also considered. It is found that the dynamic material parameter exhibits nearlylinear increase with the increase of hardening parameters of base material and initial gas pressure, whilethe dynamic initial crushing stress is independent of the strain-hardening parameter and the entrapped gaspressure but increases with the strain-rate hardening parameter linearly. These findings may be helpful forguiding the crashworthiness design of cellular materials and structures.

© 2016 Elsevier Ltd. All rights reserved.

TaggedPKeywords:

Closed-cell foamFinite element analysisVelocity fieldShock waveDimensional analysis

1. Introduction

TaggedPCellular materials are widely used as energy absorption and anti-blast materials [1,2] for their excellent capability of a wide collapsestrain with nearly unchanged load [3]. Extensive researches havebeen carried out to investigate the dynamic behaviors of cellularmaterials [4�14], which provide the foundation understandings toimprove the design of cellular materials and structures. However,there is no much knowledge about the effect of meso-structuralparameters on the dynamic mechanical properties of cellular mate-rials.

TaggedPSeveral typical features of cellular materials under quasi-staticand dynamic loadings have been found in the literature. The randomshear bands are the typical deformation feature for cellular materials

TaggedPunder quasi-static compression [15]. When the impact velocity ishigh enough, a highly localized deformation of cellular materialsappears [7,16,17]. Thus, the nominal strain is no longer applicable tocharacterize the shock-induced strain [18]. To avoid rough measure-ment of local deformation, the local strain calculation method isdeveloped by Liao et al. [19,20] to study the heterogeneous deforma-tion of honeycombs numerically. Strength enhancement, whichdescribes the increase of dynamic strength with the impact velocity[7], is another typical feature of cellular materials under impactloading. A rate-independent, rigid�perfectly plastic�locking (R-PP-L) idealization was proposed by Reid and Peng [21] to develop aone-dimensional shock model to explain the strength enhancementof wood under impact and further applied to analyze the dynamicbehavior of cellular metals [8]. Later, a series of much accurate shockmodels have been developed [6,8,10,22�24]. To further improve theaccuracy of shock models, the dynamic stress�strain behaviorshould be taken into consideration. Liao et al. [20] pointed out that

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112 S. Wang et al. / International Journal of Impact Engineering 99 (2016) 111�121

TaggedPthe difference between the shock wave speed and the impact veloc-ity commendably approaches to a constant as the impact velocity ishigh enough, and found that the dynamic stress�strain states ofirregular honeycombs are very different from those in the quasi-static stress�strain curve. Barnes et al. [11] employed a linear rela-tion to investigate the Hugoniot relation between the shock wavespeed and the impact velocity. These researches [20,11,12] haveshown that the compaction strain under dynamic impact is signifi-cantly higher than that under quasi-static loading at the same stresslevel. Zheng et al. [10] explained this phenomenon through the dif-ference in deformation mechanisms, i.e. the difference of the inter-actions between crashed bands and different status of cells beingcompacted for two types of collapse modes, namely layer-wise andrandom shear bands corresponding respectively to the dynamic andquasi-static loadings. Moreover, a dynamic rigid-plastic hardening(D-R-PH) idealization was proposed by Zheng et al. [10] with thestress�strain relation

s ɛð ÞDsd0 CDɛ

ð1¡ɛÞ2 ; ð1Þ

where sd0 is the dynamic initial crushing stress and D the dynamicstrain hardening parameter. This stress�strain relation can wellcharacterize the dynamic compression behavior of cellular materialsunder high velocity impact.

TaggedPAlthough much knowledge about the dynamic crushing behav-iors of cellular materials has been acquired, it is still unclear how themeso-structural and the base-material parameters affect the shockbehavior. Recently, the effect of relative density on the dynamicresponse and energy absorption of open-cell foams was examinedby Gaitanaros and Kyriakides [25]. Their results show that the crush-ing strength increases with the impact velocity in a concave-upwardtendency. Nevertheless, previous work was mainly focused on thequalitative understanding of the dependencies, and lack of quantita-tive investigations on the influence of the meso-structural parame-ters on the mechanical behavior of cellular materials.

TaggedPIn this study, a direct impact loading scheme is applied and thecrushing process of closed-cell foams is numerically simulated bythe 3D cell-based models modeling closed-cell foams and is demon-strated in the forms of the local strain field and mesoscopic deforma-tion pattern. Local velocity distribution is used to determine thelocation of shock front and calculate the corresponding shock wavespeed. Dimensional analysis as well as the numerical data is intro-duced to analyze the possible factors influencing the relationbetween the shock wave speed and the impact velocity. Then, thestresses behind and ahead of the shock front are obtained with the

Fig. 1. A cell-based finite element model (a) an

TaggedPaid of shock wave theory and the dynamic material parameters.Finally, the specific internal energy across the shock front is dis-cussed and further investigations are carried out to evaluate the pos-sible effect of hardening behaviors of cell-wall material andentrapped gas within closed-cells on the dynamic crushing behav-iors of cellular material.

2. Numerical method

TaggedPClosed-cell foam models with uniform cell-wall thickness areconstructed by using the 3D Voronoi technique, referring to Zhenget al. [10] for details. In this study, the Voronoi structure with 1700nuclei and cell irregularity of 0.4 is generated in a volume of 20£20£80mm3. The average cell size, d0, is about 3.3mm. The density ofVoronoi structure, r0, is changed by varying the cell-wall thickness.The relative density of Voronoi structure is expressed as rDr0/rs,where rs is the density of cell-wall material. Samples with the samerandom meso-structures but having different cell-wall thicknessesare used in this study to avoid the influence of randomness amongdifferent samples.

TaggedPThe finite element code ABAQUS/Explicit [26] is employed to per-form the numerical simulations. The cell-wall material is assumed tobe elastic-perfectly plastic with density rsD2700 kg/m3, the Young'smodulus ED69 GPa, the Possion's ratio vD0.3 and the yield stresssysD170MPa. The values of these material parameters will bechanged for necessity to explore the influence of the related variableindividually. The cell walls in the Voronoi structures are meshedwith hybrid shell elements of types S3R and S4R. According to theconvergence analysis, the element size is controlled to be »0.3mm.To save computational time, shell elements of type S3R with sharpangle are eliminated. A finite element model of closed-cell foam andthe elements meshing scheme of cell walls are shown in Fig. 1.

TaggedPThe direct impact loading scheme is applied in this study. A rigidplate is fixed and the specimen impinges onto it with an initialvelocity, V0, as illustrated in Fig. 1a. General contact is applied to allpossible contacts during crushing.

3. Results

3.1. Deformation features

TaggedPThe local strain field calculation method [19], which is based onthe optimal local deformation gradient, is employed in this study tocalculate the local strain component along the loading direction. Amiddle section parallel to the longitudinal direction is targeted to

d a cell with shell elements meshing (b).

0 10 20 30 40 50 60 70 800

50

100

150

200

250

0.0267ms

0.240ms

0.133ms

Vel

ocity

, v (m

/s)

Lagrangian location, X (mm)

Fig. 3. The velocity distribution in the specimen with a relative density of 0.1 underdynamic impact with an initial velocity V0D250m/s.

S. Wang et al. / International Journal of Impact Engineering 99 (2016) 111�121 113

TaggedPinvestigate the evolution of the strain distribution during crushing. Asequence of the strain distribution configurations in the Lagrangianframework and the deformation patterns of this section at the initialimpact velocity V0D250m/s are shown in Fig. 2. There is an appar-ent localized deformation band propagating from the proximal endto the distal end. The localized deformation band is clearly capturedby the strain field and the band localizes in a narrow width aboutone-cell size [16].

TaggedPThe cells within the collapse band are compacted tightly and thenthe crushing front spreads to the adjacent layer of cells. Such layer-wise collapse behavior is usually characterized by the shock wave[7]. During crushing, the shock front propagates forward in a nearlyplanar manner. Thus, it is reasonable to simplify the shock wavepropagation through cellular materials into one-dimensional prob-lem for convenience. The shock front divides the model into twoportions: one is compacted and the other is almost undeformed.With the compacted volume increasing, the kinetic energy is gradu-ally dissipated by the plastic collapse of cell walls.

3.2. Velocity distribution

TaggedPIt is seen that the one-dimensional strain distribution can wellcharacterize the shock wave propagation through cellular structureunder dynamic impact [10]. However, it is very time-consuming tocalculate the strain field at one moment for a large number of nodesin 3D meso-mechanical model. As a basic physical quantity, thevelocity field can also be applied to characterize the status of cellularstructures and the velocity components along the loading directioncan be extracted directly from the numerical results as well. In thispaper, one-dimensional velocity distribution in the Lagrangianframework is obtained by averaging the velocity components on thesame cross-section of the model. The one-dimensional velocity dis-tribution of cellular structure under V0D250m/s with relative den-sity rD0.1 is shown in Fig. 3. It presents an intuitive sight that there

Fig. 2. The evolutions of strain distribution of the cross-section along the loadingdirection with the initial impact velocity V0D250m/s and rD0.1.

TaggedPis an obvious shock front propagating through cellular structureunder high-velocity impact.

TaggedPThe particle velocity immediately behind the shock front is notzero but exhibits a slight velocity rebound, corresponding to thesmall peaks on the velocity distribution in Fig. 3. This may beexplained as the effect of transverse inertia, which is embedded intransverse strain for the free boundary around. The transverse straincan be calculated as the mean value of the local strains in lateraldirection perpendicular to the longitudinal axis of cellular structure.To investigate the changes in status, for example, consider theLagrangian location 5mm away from the impact end. The variationsof velocity and the transverse strain with time at this location areshown in Fig. 4. It indicates that the velocity decreases dramaticallyand the lateral expansion begins to occur when the shock front prop-agates to this location. Then, the joint movement in longitudinaldirection happens due to the effect of the plastic Poison's ratio,though the effect is nearly negligible for cellular materials [16,27].Since the kinetic energy is still very high when the shock wave prop-agates across this location, the longitudinal squeezing induced bylateral expansion results in a further compression of the compactedcells, until the transverse strain reaches a relative stable value ofabout ¡0.062 (negative sign corresponds to expansion).

TaggedPThe Lagrangian location where the velocity gradient reaches itsmaximum is defined as the shock front at current time during thepropagation of shock wave though cellular materials, as illustratedin Fig. 5. Theoretically, the shock front has no width, i.e. the physicalquantities (such as stress, strain and particle velocity) across theshock front are discontinuities and a first order singular surfaceexists [28]. However, for cellular materials, the transition practicallyhappens in a finite width about one cell size, which is the region that

Fig. 4. Variations of velocity and transverse strain with time at the Lagrangian loca-tion XD5mm.

Fig. 5. The velocity field and the corresponding distributions of velocity and velocitygradient.

Fig. 7. The shock wave speed and the impact velocity versus time.


TaggedPthe shock front spans [20,29]. From the velocity gradient, the timehistory of the Lagrangian location of shock front under different ini-tial impact velocities is obtained, see Fig. 6. The variations of theLagrangian location of shock front with time are found to be nonlin-ear. This means the speed of shock front decreases as the impactvelocity decreases during crushing.

3.3. Shock wave speed versus impact velocity

TaggedPThe shock wave speed can be determined from the shock fronthistory curve according to three-points central difference. Since theshock wave speed of one moment is deduced by the two adjacentmoments, the initial and terminal two points are abandoned. Thevelocity ahead of the shock front, i.e. the instantaneous impactvelocity, can also be determined by averaging the particle velocitywithin one cell size span immediately ahead of the shock front at acertain moment. The shock wave speed and the impact velocity ver-sus time for cellular material with relative density 0.1 under the ini-tial impact velocity 250m/s are shown in Fig. 7. During crushing,both the impact velocity and shock wave speed decrease with time,because the kinetic energy is gradually dissipated by the plastic col-lapse and densification of the cells within the shock front.

TaggedPThe relation between the shock wave speed, Vs, and the impactvelocity, v, is shown in Fig. 8. It is evidence that the shock wavespeed increases with the increase of impact velocity and it is almostin a constant intercept as the impact velocity increases to a levelhigh enough, as found in Ref. [20] for irregular honeycombs. Herein,

Fig. 6. The Lagrangian location of shock front versus time for three initial impactvelocities.

TaggedPwhen the impact velocity is high enough, the difference between theshock wave speed and the impact velocity is almost independent ofthe impact velocity, thus it can be treated as a dynamic materialparameter of cellular material, denoted as c. A similar material con-stant is also determined in solid and has the approximate value withthe bulk sound speed [30]. If using a linear function, written as VsD AC Bv with A and B being fitting coefficients [11], to fit the relationbetween the shock wave speed and the impact velocity (v>D11 X X90m/sin this study), we can find that the coefficient B is nearly indepen-dent of the relative density and approximates to 1, as shown in Fig. 9.

TaggedPThe feature that the difference between the shock wave speedand the impact velocity is asymptotic to a constant c has been cap-tured from the numerical results. For simplicity, the relationbetween the shock wave speed Vs and the impact velocity v can bewritten as

Vs D vC c: ð2ÞTaggedPThis relation can also be derived from the D-R-PH shock model [10]

and the dynamic material parameter c is related to the dynamicstrain hardening parameter D by

cDffiffiffiffiffiffiffiffiffiffiffiffiD=r0

q: ð3Þ

TaggedPSince the dynamic behaviors and the specific internal energy isrelated to the status across the shock front, acquiring knowledge ofthe dependence of the dynamic material parameter, c of cellularstructures on the meso-structural parameters and the properties ofbase material can guide the design of cellular materials for highenergy absorbing capacity. Furthermore, the dynamic behaviors of

Fig. 8. Variations of the shock wave speed with the impact velocity.

Fig. 9. Variations of the parameter Bwith different relative densities.


TaggedPcellular materials can be described conveniently by associating theD-R-PH shock model with the conservation conditions across theshock front.

4. Discussion

4.1. Dimensional analysis with numerical tests

TaggedPThe dynamic material parameter, c, may be related to themechanical properties of the base material (density rs, Young's mod-ulus E, Poisson's ratio n and yield stress sys) and the structuralparameters (cell-wall thickness h and equivalent cell size d0). For aspecific configuration of cellular material, the relative density r andthe cells number Nc of a specimen are related to the cell-wall thick-ness h and the equivalent cell size d0. Inversely, both h and d0 aredependent on r and Nc. Thus, parameter c can be expressed as

cD f ðE;n;sys;rs;r;NcÞ: ð4ÞTaggedPIn fact, the dynamic material parameter c is not very sensitive to

the cells number varying in a small range. For example, four sampleswith 1500, 1600, 1700 and 1800 cells are considered. By keepingother parameters constant, the influence of cells number on thedynamic material parameter c is evaluated, as shown in Fig. 10. Theresults show that the influence can be neglected, i.e. c is roughlyindependent of the cells number for a specified relative density ofcellular material. Thus, Eq. (4) can be simplified as

cD f ðE;n;sys;rs;rÞ: ð5Þ

Fig. 10. Variations of the dynamic material parameter cwith the cells number.

TaggedPHereafter, the number of nuclei is set to be 1700 in the extensivesimulations.

TaggedPDimensional analysis is applied to reduce the number of varia-bles. Among the five variables in Eq. (5), the relative density r andPoisson's ratio n are the dimensionless quantities. The yield stresssys and Young's modulus E have the same dimensions of ML¡1T¡2,and the dimension of density rs is ML¡3, where M, L and T are thedimensions for mass, length and time, respectively. By applying theBuckingham A-theorem [31], the dimensionless form of Eq. (5) canbe expressed as

cffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisys=rs

p DPðE=sys;n;rÞ; ð6Þwhere E/sys, n, r are dimensionless variables. From Eq. (6), we justknow the dimensionless parameter c/(sys/rs)1/2 is a function of E/sys, n and r, and cannot determine the quantitative expression with-out the aid of the measured data obtained from experimental ornumerical tests.

TaggedPFor typical coarse-grained materials or metals, which have nosignificant yield point, the yield stress sys representing the onset ofplastic flow of material is often defined as the stress at the residualstrain of 0.2%. The relevant range of E/sys from 90 to 600 practicallyfor metals is selected as the effective range in the simulations. Thevariation of the dimensionless parameter c/(sys/rs)1/2 over thischange is shown in Fig. 11. It indicates that the dimensionlessparameter c/(sys/rs)1/2 is roughly a constant when E/syD13 X X400, which is indepen-dent of the yield stress. With the increase of Young's modulus E ofbase material, the stiffness of cell-walls is strengthened and the elas-tic precursor moves much faster, which slightly reduces theresponse time for cellular material and increases the tendency totrigger the global elastic deformation. However, the dynamic crush-ing of cellular materials is mainly in the form of plastic collapse ofcell walls and the effect of elastic stage has a limited extent. Accord-ing to the numerical results and the mechanism of the dynamiccrushing behavior, the influence of the dimensionless parameter E/sy on the dynamic response is insignificant and can be regarded asnegligible in this study.

Fig. 11. Variations of the dimensionless parameter c/(sys/rs)1/2 with E/sys and Pois-son's ratio n.


TaggedPThe effect of Poisson's ratio on the dimensionless parameter c/(sys/rs)1/2 is also evaluated. For metals, the Poisson's ratio rangesfrom 0.25 to 0.45 [32]. Herein, the Poisson's ratio of the base materialranging from 0.17 to 0.35 is selected to perform the numerical simu-lations. The result shows that the dimensionless parameter c/(sys/rs)1/2 is approximately a constant over a wide range of Poisson'sratio, as shown in Fig. 11. It indicates that the dimensionless param-eter c/(sys/rs)1/2 is nearly independent of the Poisson's ratio. Thus,Eq. (6) can be simplified as


p DPðrÞ; ð7Þin which the relative density of cellular material is the only remain-ing variable.

4.2. Effect of relative density

TaggedPRelative density is one of the important parameter for cellularmaterials and it may greatly affect the mechanical behaviors, such asmodulus, collapse stress and densification strain [3]. It is possible todesign material with excellent energy absorbing capacity if the influ-ence of relative density is clear. Practically, for a kind of specific cel-lular material, the property of the base material (most of which arealuminum or polyurethane) is determinate and the relative densitymay be the only changeable parameter that could be designed effec-tively. The influence of relative density on the dimensionless param-eter c/(sys/rs)1/2 is obtained based on the cell-based finite elementmodels within a range of relative densities, as shown in Fig 12. Theresults exhibit a linear increase tendency and thus a linear fittingexpression can be estimated as


p D krC b; ð8Þwhere k and b are fitting coefficients and they are obtained as kD0.451§0.020 and bD0.0687§0.0037 for the cellular materialconsidered. As the dynamic material parameter c is related to thedynamic collapse or densification rate of the residual cells in cellularmaterials for the specific impact velocity, it naturally exhibits a pro-portional relation with the cell-wall thickness (i.e. the relative den-sity). From Eqs. (3) and (8), the dynamic strain hardening parametercan be determined as

DDsysr krC bð Þ2: ð9ÞTaggedPEq. (9) indicates that the hardening behaviors of cellular

material are highly dependent on the relative density and theyield stress of cell-wall material. It should be noted that the

Fig. 12. Variations of the dimensionless parameter c/(sys/rs)1/2 with the relative den-sity.

TaggedPfindings are restricted to the particular cell-wall material (elas-tic�perfectly plastic property) and cell-topology. The dynamichardening behaviors of cellular material may be different forother base materials (see Section 5) or cellular material with dif-ferent cell-topologies. With increasing relative density, the cell-wall thickness increases and the compressible volume becomessmall in cellular materials with negligible lateral expansion, asseen in Fig. 13. Thus, the shock front propagates much quick asthe cellular materials with a higher relative density need a com-pacted portion longer than that with a low relative density toreach the same loading displacement. More energy is needed tocollapse the relatively thick cell walls, which results in a rapiddissipation of kinetic energy. Therefore, the instantaneous impactvelocity decreases much faster with the increasing of relativedensity, as shown in Fig. 14. Since the shock wave speedpresents a linear relation with the impact velocity, as expressedin Eq. (2), the Lagrangian location of the shock front at impacttime t can be expressed as

FðtÞDZ t0vðtÞdtC ct; t< t1; ð10Þ

where t1 is the terminal time when the instantaneous impact veloc-ity, v(t) drops to a certain range that cannot trigger the shock-induced behaviors of cellular materials. The external and intrinsicfactors that may affect the shock front propagation are indicated inEq. (10), which correspond to the first term and the second term inthe equation, respectively. For the same initial impact velocity, asample with a higher relative density (i.e. thicker wall-thickness)attenuates the impact velocity faster than that with a lower relativedensity. On the other hand, a sample with a higher relative densityhas a relatively larger value of the dynamic material parameter c.These two mechanisms are competitive in the propagation of shockfront. The increase of c for samples with larger cell-wall thicknessespartly compensates for the decrease of the shock front propagationdistance due to the rapid decrease of the instantaneous impactvelocity v, resulting in the unconspicuous increasing of the locationof shock front especially during the final impact stage, as shown inFig. 14.

Fig. 13. The deformation patterns in the cross-section along the loading direction forthe cellular materials with three different relative densities under an initial impactvelocity of 250m/s.

Fig. 14. The corresponding shock front and impact velocity versus time under threerelative densities.

Fig. 16. Variation of the dynamic initial crushing stress sd0 with the relative density.


4.3. Stresses of cellular material under direct impact

TaggedPIn the D-R-PH shock model, the region ahead of the shock front istreated as a rigid body. Applying the Newton's motion law to thisregion gives

dvdt

D¡ sd0

r0�L0¡FðtÞ

� ; ð11ÞwhereF(t) is the Lagrangian location of shock front at time t, L0 andr0 the original length and density of the cellular material, respec-tively. In Section 3, the Lagrangian location of shock front and thevelocity ahead of the shock front at different moments have beenalready determined from the sectional velocity distribution, so thedynamic initial crushing stress sd D14X X0 D15X X can be calculated numericallyfrom Eq. (11). The dynamic initial crushing stresses at differentimpact velocities for three relative densities are shown in Fig. 15. Itcan be seen that the dynamic initial crushing stress of cellular mate-rials undulates slightly with the impact velocity and can be taken asa constant [10]. From Fig. 16, the dynamic initial crushing stress sd D16X X0 D17X Xvaries with the relative density r in a power-law manner

sd0=sys Darn; ð12Þwhere a and n are fitting parameters and they are obtained as aD0.726§0.016 and nD1.18§0.01 in this study. Substituting Eqs.(9) and (12) into Eq. (1), we have

s ɛð ÞDsysarn Csys rðkrC bÞ2ɛ

ð1¡ɛÞ2 ð13Þ

Fig. 15. Variations of the dynamic initial crushing stress with the impact velocity.

TaggedPfor the cellular materials considered in this study.TaggedPBased on the D-R-PH shock model [10], the shock stress, i.e. the

stress behind the shock front, can be expressed as

sB Dsd0 Cr0v

2

ɛBDsd0 Cr0vðtÞ vðtÞC cð Þ; ð14Þ

where eBD v/(vC c) is the shock strain (the strain behind theshock front). It indicates that the relation between the shockstress and the impact velocity is a quadratic polynomial and thecoefficient of the linear term is related to the dynamic materialparameter c. In direct impact tests, the instantaneous impactvelocity of cellular material and the stress at the impact end,which is equal to the shock stress, can be measured, thus thedynamic material parameters sd0 and c can be determined from Eq.(14). The relation between the impact velocity and the time can bedetermined from solving Eqs. (11) and (2) with the initial condi-tions of v(0)D V0 andF(0)D0, written as

tD r0L0sd0

Z V0v

expr02sd0

V C cð Þ2¡ V0 C cð Þ2h i( )

dV : ð15Þ

TaggedPFrom Eqs. (14) and (15), some responses (such as the shock stressand the instantaneous impact velocity) that are easily measuredexperimentally may be available to determine the dynamic stress-strain states. Therefore, the findings in this study may contribute anidea to develop new experimental techniques to study dynamicmechanical behaviors of cellular material that has highly localizeddeformation under dynamic crushing.

Fig. 17. Comparison of the shock stresses obtained from the cell-based finite elementmodel and the shock model.

Fig. 18. Effect of strain-hardening parameter h of cell-wall material on the dynamicmaterial parameters c and dynamic initial crushing stress sd0 of cellular material.


TaggedPThe dependence of mechanical response (such as the dynamicmaterial parameter c and sd0) on the properties of base material orstructural parameters studied above is limited to the initial impactvelocity not larger than 250m/s. To illustrate the reliability of mate-rial models, the shock stresses of cellular specimens at an initialimpact velocity of 300m/s obtained from the cell-based finite ele-ment model are compared with those predicted by the shock model,as shown in Fig. 17. At the early stage of impact, the impact stressobtained by numerical simulations exhibits a strong oscillation dueto the complex interactions of cell walls. With the impact eventgoing on, the kinetic energy is dissipated by the plastic collapse ofcell walls and the impact stress decreases gradually. The shock stresspredicted by the D-R-PH shock model well captures the main featureof the impact stress obtained from the cell-based finite elementmodel.

4.4. Specific internal energy under direct impact

TaggedPBy using Eqs. (8) and (12), the specific internal energy DUderived in Ref. [10] can be rewritten as a function of the relative den-sity and impact velocity

DUD sd0v

vC c C12r0v

2 D arnsys

1C krC bð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisys=rsv2

q C 12rrsv

2: ð16Þ

TaggedPThis expression illustrates that the specific internal energy is in apositive correlation with the impact velocity and relative densityand reveals the mechanism of energy dissipation across theshock front. In the process of direct impact loading, the kineticenergy of cellular material is dissipated on the propagation ofshock front (crushing band) and a decrease of impact velocitymakes the densification strain being dropped, and then theenergy absorbed of cellular material is attenuated. Nevertheless,by designing the distribution of density on the basis of theinstantaneous impact velocity, cellular material can be tailoredto keep and even exceed the original performance in energyabsorption and may also change the attenuation mechanism ofinstantaneous impact velocity. Using graded cellular materials tosatisfy the requirements of impact resistance and energy absorp-tion is an example of the crashworthiness design [33]. Underhigh-velocity impact, an increase of relative density of cellularmaterials is an effective and convenient way to improve theenergy absorbed and other mechanical properties (such as theinitial crushing stress). However, the simple idea to blindlyenlarge the cell-wall thickness of cellular material with deter-mined dimension requirement may lose the advantages to bal-ance the excellent performance and may also increase the risk ofexceeding the allowable load of the protected structures. Theexistence of complex meso-structures makes cellular materialshaving outstanding designability. The quantitative expression inEq. (16) provides a clear guideline for the crashworthiness designof cellular materials that the energy absorbed per unit volumemay be greatly improved by decreasing the coefficient k and bwhile increasing coefficient a. It should be noted that the coeffi-cients involving in Eq. (16) are determined from the same config-uration with varying cell-wall thickness (i.e. relative density) inthis study, and understanding of how the cell-topology or meso-structural design of cellular materials affects material parametersrequires further complementary investigation.

5. Further discussion

TaggedPIn the above investigations, the cell-wall material is assumed tobe elastic�perfectly plastic. Some other factors, such as hardeningbehavior of cell-wall material and gas entrapped within cells, mayalso affect the shock behaviors of cellular materials. Herein, the

TaggedPpossible effects of cell-wall material properties on the crushingbehavior of cellular material with relative density 0.1 are furtherconsidered.

5.1. Effect of strain hardening of cell-wall material

TaggedPThe cell-wall material is considered to be strain-hardening andthe corresponding true stress�strain relation in plastic stage is givenby

sDsys 1Chɛsð Þ; ð17Þwhere h is the strain-hardening parameter and es the equivalentplastic strain [34]. The dependence of dynamic material parametersof cellular material, c and sd D18X X0D19X X, on the strain-hardening parameter h isshown in Fig. 18.

TaggedPThe results indicate that the dynamic material parameter c of cel-lular material increases linearly with the strain-hardening parame-ter h of cell-wall material, but the dynamic initial crushing stress sdD20X X0 D21X Xof cellular material is independent of the strain-hardening behaviorof cell-wall material. A linear function is employed to fit the relationof c and h, given by

cD a1hC a2; ð18Þwith a1D12.3 D22 X X§ D23 X X0.8m/s and a2D29.0D24 X X§ D25 X X0.6m/s. Intuitively, thedynamic material parameter c portrays the growth rate of com-pacted region behind the shock front, as illustrated in Eq. (10). In thecompacted region, cells are in a high-strain stage and thus the defor-mation resistance of cell walls is strengthened by the strain-harden-ing behavior. Consequently, the less compaction deformation isformed than that with perfectly plastic base material, as demon-strated in Figs. 19(a) and (d), then it results in the increase of thedynamic material parameter c. Ahead of the shock front, cells remainalmost undeformed, as seen in Fig. 19(a), and the strains in thesecells are nearly zero. Accordingly, the strain-hardening behavior ofcell walls has a negligible contribution to the dynamic initial crush-ing stress sd D26X X0 D27X Xof cellular material.

5.2. Effect of strain-rate hardening of cell-wall material

TaggedPFor cell walls with strain-rate hardening property, the plasticbehavior of the base material is assumed to be

sDsys½1C rlnð _ɛs= _ɛ0Þ�; ð19Þwhere r is the strain-rate hardening parameter, and _ɛs the equivalentplastic strain rate and _ɛ0 the reference equivalent plastic strain rateof cell-wall material [34]. In this study, we take _ɛ0 D0.1 /s, as used inRef. [18], and r lies within range of 0.13�0.50. Results show thatboth of the dynamic material parameter c and the dynamic initial

Fig. 19. Deformation patterns of cellular material with rD0.1 under initial impactvelocity V0D250m/s at time tD0.16ms for different cell-wall materials: (a) linearstrain-hardening material (hD1.18), (b) strain-rate hardening material (rD0.5) andelastic�perfectly plastic material with (c) considering entrapped gas (pD 7p0) and (d)neglecting entrapped gas within cells.


TaggedPcrushing stress sd0 of cellular material exhibit a basically linearincrease with the increase of material parameter r, as depicted inFig. 20. It is also noted that the high strain-rate hardening of basematerial corresponds to a scattered dynamic material parameter c ofcellular material. This may be explained as great influence of theanisotropy issued from cells distribution and cells geometry on localcollapse of cell-walls that causes high strain-rate in intensive loadingelements [35]. Since the strain-rate in cell walls can be arisen imme-diately ahead of and behind the shock front, the strength and col-lapse resistance of base material in this two portions are enhanced.Therefore, the dynamic material parameter c and the dynamic initialcrushing stress sd0 increase with the strain-rate hardening parameter

Fig. 20. Effect of strain-rate hardening parameter r of cell-wall material on dynamicmaterial parameters c and dynamic initial crushing stress sd0.

TaggedPr. Moreover, the dependence of the two dynamic parameters c andsd0 of cellular material on the strain-rate hardening parameter r ofcell-wall material can be fitted linearly by

cD b1rCb2; ð20aÞ

sd0 D λ1rC λ2; ð20bÞwith b1D26.9D28 X X§ D29 X X4.8m/s, b2D29.3D30 X X§D31 X X1.4m/s and λ1D12.5D32 X X§ D33 X X0.9MPa, λ2D8.19D34 X X§D35 X X0.28MPa, respectively.

TaggedPBy comparing the deformation features of cellular material withsame configuration but different individual properties, an obviousdiscrepancy could be found for cellular material with strain-ratehardening of cell-wall material, as shown in Fig. 19(b). The relativelyincompact region implies that the great enhancement in compres-sion resistance happens for strain-rate hardening behaviors of cellwalls during layered collapse of cells and the quantitative descrip-tion of compaction is characterized with local strain distribution attime tD0.16ms, as depicted in Fig. 21(a). It transpires that the den-sification strain of cellular material with strain-rate hardening ofbase material behind the shock front is quite lower than that withother considered cases. Additionally, the propagation of shock frontis delayed and the width of shock front is broadened by comparingwith other considered cases at current moment. Such highlightedfeatures can be explained as the strengthened capacity of cellularmaterial to attenuate impact velocity, as shown in Fig. 21(b). For cel-lular material with strain-rate hardening of base material, the

Fig. 21. (a) Local strain distribution of cellular material with different cases at time tD0.16ms and (b) variation of instantaneous impact velocity with impact time.


TaggedPchanging compression resistance of cell walls during high-velocitycrushing makes transient velocity being attenuated in a rapid way,and then the shock-induced behavior of cellular material would beended a little early. Consequently, under a specific impact moment,the relatively long undeformed portion is retained and the impactvelocity is reduced severely for the cellular material with strain-ratehardening of base material, which greatly enhances the dynamichardening behavior of cellular material.

5.3. Effect of entrapped gas within cells

TaggedPAs basic component of realistic foams, the presence of air trappedin closed-cells has been demonstrated to affect the dynamic harden-ing behavior [36�38], but the contrary remarks conclude thatentrapped air has a negligible contribution to the strength enhance-ment [4, 39]. Considering the crushing front passing through a cell ina split second that gas cannot escape, the entrapped air may haveeffect on the shock behaviors of cellular material under high velocityimpact. To make this point clear, the trapped gas inside cells is mod-eled. For our cell-based finite element model, every closed cell isdefined as surface-based fluid cavity [26]. It is assumed that the airinside closed-cells is an ideal gas species and has the properties ofuniversal gas constant 8.314 J/(mol K), molecular weight 28.9 g/mol,molar heat capacity 28.11 J/(mol K) and the ambient atmosphericpressure p0D0.1013MPa. By maintaining initial gas temperaturewith 293.16 K, the initial gas pressure of 1»7 times atmosphericpressure inside closed-cells is considered in this study.

TaggedPThe effect of entrapped gas within cells presents that thedynamic material parameter c increases with the initial gas pressurein an approximately linear locus while the dynamic initial crushingstress sd D36X X0 D37X Xkeeps unchanged, as shown in Fig. 22. The results implythat the entrapped gas indeed has an effect on dynamic hardeningbehaviors (characterized with Eq. (3)) of cellular material. The con-tribution of gas trapped within cells to hardening phenomenon isalso observed for polypropylene foams which are compressed in afluid chamber under dynamic loading [40]. A linear relation betweendynamic material parameter c and the initial gas pressure p withincells gives

cD c1p=p0 C c2; ð21Þwith c1D0.87 D38 X X§D39 X X0.05m/s and c2D30.39D40 X X§ D41 X X0.19m/s by fitting thenumerical results. The gas pressure inside cells ahead of the shockfront remains the initial value, but behind the shock front the pres-sure sharply increases in air entrapped closed-cells due to highlylocalized deformation during dynamic crushing. The fully trapped

Fig. 22. Effect of gas pressure p in closed-cells with elastic-perfectly plastic materialof cell walls on the dynamic material parameters c and dynamic initial crushing stresssd0.

TaggedPgas in cells results in less compact deformation behind the shockfront [37], which slightly advances propagation of shock wavethrough cellular material faster than that with no entrapped air, asdemonstrated in Fig. 19(c, d). However, compared with the strain-hardening or strain-rate hardening behaviors of cell walls, the effectof entrapped gas within cells on the dynamic material parameter cof cellular material is slight, as illustrated in Fig. 21(a).

6. Conclusions

TaggedPThe dynamic deformation feature and the shock wave propaga-tion through closed-cell foams are investigated by cell-based finiteelement models. For direct impact with initial velocity 250m/s, thelocal strain field distribution clearly demonstrates the highly local-ized deformation and the nearly planar crushing front of cellularmaterials. The velocity distribution is calculated to determine thelocations of the shock front quantitatively, and the shock wave speedis then determined by numerical differential of the Lagrangian loca-tions of the shock front.

TaggedPThe difference between the shock wave speed and the impactvelocity is asymptotic to a constant, which can be regarded as adynamic material parameter, as the impact velocity exceeds a cer-tain value (say v »90m/s for the cellular materials with relative den-sity 0.1). Dimensional analysis is applied to investigate the influenceof the meso-structural parameters and the base material propertiesof the cellular materials on the dynamic material parameter c. Itshows that the relative density is the main parameter that affectsthe dynamic material parameter c of cellular materials with elas-tic�perfectly plastic base material, while the influences of the cellsize, Poisson's ratio and the ratio of elastic modulus to yield stress ofbase material in the considered ranges are almost negligible. As aresult, a linear relation of the dynamic material parameter c withrespect to the relative density is obtained.

TaggedPBased on the D-R-PH shock model and the cell-based finiteelement model, it is found that the dynamic initial crushingstress increases in a power-law form with the increase of rela-tive density. The effects of relative density and impact velocityon the shock stress were explored. The quantitative predictionbased on the shock models provides an additional method tostudy the dynamic crushing behaviors of cellular material byobtaining the instantaneous impact velocity individually. Finally,the specific internal energy across the shock front indicates thatcellular material can be tailored to acquire excellent performanceby optimizing the designing parameters a, k and b in Eq. (16) orby arranging a proper density distribution.

TaggedPEffects of strain-hardening, strain-rate hardening of cell-wallmaterial and entrapped gas within cells on the dynamic crushingbehaviors of cellular material have been analyzed. Results show thatall of the considered factors could affect dynamic material parameterc in nearly linear manners, which are ascribed to enhancement ofdeformation resistance of cell walls that results in less compactiondeformation behind the shock front. Nevertheless, the dynamic ini-tial crushing stress demonstrates negligible dependence of initialgas pressure within cells and strain-hardening behavior of cell-walls,but increases linearly with the strain-rate hardening parameter ofcell-wall material. This can be explained as reinforced deformationresistance and velocity attenuation ability of cell walls ahead of theshock front during high-velocity crushing.

Acknowledgments

TaggedPThis work is supported by the National Natural Science Founda-tion of China (Projects Nos. 11372308 and 11372307) and the Funda-mental Research Funds for the Central Universities (Grant No.WK2480000001).


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Dynamic material parameters of closed-cell foams under high-velocity impact1. Introduction2. Numerical method3. Results3.1. Deformation features3.2. Velocity distribution3.3. Shock wave speed versus impact velocity

4. Discussion4.1. Dimensional analysis with numerical tests4.2. Effect of relative density4.3. Stresses of cellular material under direct impact4.4. Specific internal energy under direct impact

5. Further discussion5.1. Effect of strain hardening of cell-wall material5.2. Effect of strain-rate hardening of cell-wall material5.3. Effect of entrapped gas within cells

6. ConclusionsAcknowledgmentsReferences

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International Journal of Impact Engineeringstaff.ustc.edu.cn/~jlyu/PDF/2017 Dynamic material...shock wave speed is strongly dependent on the impact velocity, but the effect of the