dynamic behavior of graded honeycombs...

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Dynamic Behavior of Graded Honeycombs — — A Finite Element Study

C.J. Shen, G. Lu, T.X. Yu

PII: S0263-8223(12)00540-5

DOI: http://dx.doi.org/10.1016/j.compstruct.2012.11.002

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Please cite this article as: Shen, C.J., Lu, G., Yu, T.X., Dynamic Behavior of Graded Honeycombs — — A Finite

Element Study, Composite Structures (2012), doi: http://dx.doi.org/10.1016/j.compstruct.2012.11.002

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Composite Structures xxx (2012) xxx–xxx

COST 4901 No. of Pages 13, Model 5G

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Contents lists available at SciVerse ScienceDirect

Composite Structures

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

Dynamic behavior of graded honeycombs – A finite element study

C.J. Shen a, G. Lu a,⇑, T.X. Yu a,b

a School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singaporeb Mechanical and Material Science Research Center, Ningbo University, Ningbo 315211, China

a r t i c l e i n f o a b s t r a c t

1819202122232425

Article history:Available online xxxx

Keywords:Graded honeycombDynamic stressDeformation modeDensification velocity

2627282930313233

0263-8223/$ - see front matter � 2012 Published byhttp://dx.doi.org/10.1016/j.compstruct.2012.11.002

⇑ Corresponding author.E-mail address: [email protected] (G. Lu).

Please cite this article in press as: Shen CJ et al. D10.1016/j.compstruct.2012.11.002

This paper aims to investigate the in-plane dynamic mechanical behavior of the functionally graded hon-eycomb. Finite element simulations were carried out using ABAQUS/EXPLICIT. In each case, a constantvelocity was applied to the impact plate which then crushed the honeycomb. The values of the velocitywere from 5 to 100 m/s. From the observation of the deformation profiles, three deformation modes havebeen identified when the strongest layer was placed at the impact end while two deformation modeshave been found when the weaker layer was placed at the impact end. The critical velocities, at whichthe deformation modes transits were determined with a classification map. Dynamic stress and quasi-static plateau stress were obtained from the loading and the supporting plate, respectively. Thestress–displacement curves were compared with the analytical prediction based on the shock wave the-ory. Then, the absorbed energy under constant velocity loading was investigated. To further investigatethe influence of the stress gradient on the behavior of the graded honeycomb, different initial velocitiesversus the impinging mass were imposed on the impact plate. By considering a critical case, in which thevelocity reduces to zero when the honeycomb structure is just fully crushed, the densification velocitywas obtained. The results confirm that the positive gradient will enhance the energy absorption capacityof honeycombs.

� 2012 Published by Elsevier Ltd.

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1. Introduction

As a classic species of cellular materials, honeycombs have at-tracted a great deal of attention owing to their outstanding proper-ties in energy absorption, high relative stiffness and strength, heatinsulation and so on. Pioneering work on the mechanical proper-ties of the honeycomb structures dates back to 1963 by McFarland[1]. In this early work, a semi-empirical formula was proposed topredict the mean crushing stress for honeycomb under axial com-pression. Afterwards, a great number of investigations have beenconducted analytically [2–8] and experimentally [9–16] on thein-plane and out-of-plane properties of the honeycomb structuresunder quasi-static and dynamic loading. Compared to the theoret-ical predictions, Wu and Jiang [9] experimentally studied thequasi-static behavior of the honeycombs. Baker et al. [13] designeda gas gun to study the out-of-plane properties of aluminum honey-combs. Zhao and Gary [14] studied the in-plane and out-of-planecrushing behavior of honeycombs by using a split Hopkinson pres-sure bar. The in-plane and out-of-plane properties of the honey-combs have been well summarized by Gibson and Ashby [17].Finite element modeling (FEM) is widely used in the research ofthe honeycombs [18–22], especially in dynamic cases.

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ynamic behavior of graded hon

Under quasi-static in-plane loading, collapse always occurs atthe weakest row or band of the cells, which is dominated by thedistribution of the quasi-static plateau stress and initial imperfec-tions [6,10]. Honeycombs behave very differently under dynamicin-plane loading in terms of the deformation mode and plateaustress. The inertial effect significantly influences the crushingbehavior. Three different deformation modes in uniaxial impactedhoneycombs were identified based on numerical simulations[20,21], namely ‘‘X’’, ‘‘V’’ and ‘‘I’’ patterns. The critical velocities,at which the deformation modes change, were obtained empiri-cally based on the numerical simulations. The ‘‘X’’ mode usuallytakes place under low impact velocities while the ‘‘I’’ mode occursunder high impact velocities and the ‘‘V’’ mode is a transitionalmode between these two.

The above studies mainly focused on the honeycomb withhomogenous materials and uniform cells. However, in nature, cel-lular structures vary in cell wall thickness, size and shape. For cel-lular solids, a gradual increase in the cell size or in materialstrength can influence many properties such as mechanical shockresistance and thermal insulation. The understanding of mechani-cal properties of functionally graded cellular structures is of impor-tance for the proper application and utilization of these materials.

In general, the stress–strain curve of cellular material possessesthe feature sketched by the fine line in Fig. 1 and it is characterizedby three stages, viz. a short elastic stage, a long plateau stage and

eycombs – A finite element study. Compos Struct (2012), http://dx.doi.org/


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Behavior

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Graded Honeycombs ——

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¦Å¦Åd

¦ Ò

¦ Òp

plateau stage

R-P-P-L model

densification

elastic stage

Fig. 1. Stress–strain relationship for the cellular material and the R–P-P–L materialmodel.

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28 November 2012

finally a densification stage [17]. Reid and Peng [23] firstly pro-posed a rigid–perfectly-plastic–locking (R–P-P–L) model (Fig. 1)for the cellular material and derived the equation for the dynami-cally-enhanced stress of the cellular material under dynamic load-ing based on shock wave theory.

rd ¼ rp þq0

edV2 ð1Þ

where q0 is the density, ed is the locking (densification) strain, andrp is the quasi-static plateau stress of the cellular material. From Eq.(1), it is noted that two kinds of gradients can be introduced to ahoneycomb block, i.e. the gradient in the density q0, and that inthe quasi-static plateau stress rp.

By using the large deformation bending theory, Gibson and Ash-by [17] theoretically derived the quasi-static plateau stress rp, thedensity of the undeformed cell q0 and the locking strain ed, as,

rp ¼23

tl

� �2

rys ð2Þ

q0 ¼2ffiffiffi3p t

l

� �qs ð3Þ

ed ¼ 1� kq0

qsð4Þ

where t and l are the thickness and the length of the cell edge,respectively; rys and qs are the yielding stress and density of theparent material, respectively; and k is a constant, which is ideallyequal to 1. It should be noted the cell of the honeycomb block can-not be wholly densified, and so k is always greater than 1. Hu andYu [24] determined the constant k = 4/3 from the analysis of theevolution of the honeycomb shape after uniaxial compression.

Combining Eqs. (1)–(4) gives the dynamic stress,

rd ¼23

tl

� �2

rys þqsffiffi

3p

2lt

� �� k

V2 ð5Þ

Although the above equations are determined for homogenous hon-eycombs, they provide good insight into the graded honeycombstructure (GHS). Eq. (5) shows that the gradient can be introducedinto the structure by merely changing the yielding stress or the den-sity of the parent material; the thickness ratio t/l not only influencesthe quasi-static plateau stress but also the density of the cellularmaterial.

There are limited studies on the mechanical properties ofgraded cellular structures. Ali et al. [25] investigated the gradedstructure with hexagonal cells with a low impact velocity of upto 20 m/s. Different wall-thickness of the cell was assigned to eachlayer. The main contribution of this work is that the plastic hinge

Please cite this article in press as: Shen CJ et al. Dynamic behavior of graded hon10.1016/j.compstruct.2012.11.002

length within the cell was incorporated in the equations derivedby Gibson and Ashby [17]. Ajdari et al. [26] investigated the com-pressive uniaxial and biaxial behavior of functionally graded Voro-noi structures, hence with gradient in density and stress. In theirlater work [27], the regular hexagonal honeycombs with gradeddensities under dynamic loading were also investigated by usingfinite element method. Results showed that decreasing the relativedensity in the direction of crushing enhances the energy absorp-tion of honeycombs at early stages of crushing. A failure mode cri-terion for piece-wise functionally graded sandwich compositeswas proposed by Avila [28]. Most recently, Zeng et al. [29] investi-gated the influence of the density gradient on the mechanical re-sponse of graded polymeric hollow sphere agglomerates underimpact loading. Focused on the early stage, the force at the distalend was found much higher when the strongest layers were placednear the impinged end.

In this paper, the response of graded honeycomb structures un-der in-plane impact loading is investigated. The deformationmodes and the energy absorption capacity are systematically stud-ied. We start this problem as the simplest case of gradient cellularmaterials, in an attempt to decouple the two most dominant fac-tors in the dynamic crushing of cellular materials (i.e. quasi-staticplateau stress and density). In reality, changing thickness would bethe easiest way to introduce the gradient. However, it would in-volve gradient both in the quasi-static plateau stress and density(see Eq. (5)), which would make it difficult to see the effect ofchanging one of the two alone. Hence, an artificial case is inventedfor academic investigation, in which a gradient only in the materi-als’ yielding stress is introduced to the honeycomb structure, in or-der to reveal any new phenomena and to define some parametersfirst, such as gradient and other dimensionless parameters.

As a first step, gradient in stress is introduced by assigning dif-ferent values of yielding stress to the parent material of each layerof a regular hexagonal honeycomb. The deformation modes are ob-served under a constant impact velocity. Deformation profiles areobtained for each deformation mode. The critical velocities, atwhich the deformation mode changes, are obtained empiricallyin the terms of the impact velocity, stress gradient and the thick-ness ratio of cells. To further investigate the influence of the gradi-ent on energy absorption, the densification velocity of a givenimpact mass is obtained, at which the instant velocity of theimpinging plate reduces to zero when the honeycomb block is justfully crushed.

2. Finite element model

The Finite Element (FE) model is similar to that previously pro-posed by Ruan et al. [18], but a gradient in the material’s yieldingstress is introduced to represent the graded hexagonal honeycombstructure (GHS) (Fig. 2a). The structure is composed of 16 cells inthe X1 direction and 15 in the X2 direction. The material propertyof all the layers is the same except that the yielding stress varieslayer by layer. It is noted that the neighboring layers have cellssharing the same side. Within a cell, five edges are assigned thesame value of yielding stress while the right edge has a differentvalue which is for the next cell (Fig. 2b). The parent material is as-sumed to be elastic, perfectly plastic with E = 69 GPa and v = 0.3,where E and v are Young’s modulus and Poisson ratio of the mate-rial, respectively. The cell is regular with the same value of edgelength, which is l = 2.7 mm, and the corresponding angle betweenthe edges is 120�. The wall-thickness of the cells t and the densityof the parent material qs are 0.2 mm and 2700 kg/m3, respectively.Hence, the density of the honeycomb is q0 ¼ 2=

ffiffiffi3pðt=lÞqs ¼

230 kg=m3. The out-of-plane width of the honeycomb isb = 0.9 mm. Hourglass controlled, four nodes, reduced integration




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A

V0

B

Layer 1 2 3 4 5 .....

X1

X2

(a)

Material 1

Material 2

(b)Fig. 2. (a) Finite element model with 16 hexagonal cells in the X1 direction and 15cells in the X2 direction. (b) Within a cell, five edges have the same value of yieldingstress while the right one is assigned with different value of the next cell. Plate A isfixed and plate B is the impinging plate.

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shell elements (type S4R) are used to mesh the structure. Each cellwall is meshed into eight shell elements with one-layer elementsalong the out-of-plane direction. Both plates A and B are definedas rigid body. As shown in Fig. 2, plate A is fixed by constrainingall the degrees of freedom at the reference point; plate B onlycan move along the X1 direction. Interaction properties are im-posed using general contact conditions and surface to surface kine-matic contact conditions between the honeycomb block and thetwo rigid plates without considering friction. Furthermore, theout-of-plane displacement of nodes within the X1–X2 plane is con-strained so as to prevent the honeycomb block from out-of-planebulking. Five integration points through the thickness of the shellelements are adopted.

For clarity, it is defined that the stress gradient is negative whenthe strongest layer is placed at the impact end while it is positivewhen the weakest layer is placed at the impact end. The stress gra-dient of GHS is defined as follows,

g ¼ Drys

ravellayerð6Þ

where Drys is the difference of the material’s yielding stress be-tween two layers; rave is the average stress of the materials’ yield-ing stress of the honeycomb block and llayer is the length of each cellin X1 direction. Non-dimensional parameters are selected to inves-


tigate the dynamic behavior of the honeycomb under impact load-ing. In our study, the average materials’ yielding stress of the parentmaterial is rave = 77 MPa. Using Eq. (2), the average plateau stress ofthe cellular material is obtained as r0 = 0.28 MPa.

Non-dimensionalization of the parameters is implemented incurrent study. The displacement of the impinging plate is non-dimensionalized respected to the initial length of the GHS,

l � u=l0 ð7Þ

where l0 is the total length of the graded honeycomb block.A characteristic velocity of the GHS is defined as,

cp �ffiffiffiffiffiffiffiffiffiffiffiffiffiffir0=q0

qð8Þ

Substituting Eqs. (2) and (3) into Eq. (8) gives,

cp �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ffiffiffi3p t

l

� �rave

qs

sð9Þ

Substituting the values of the physical parameters, the charac-teristic velocity of the GHS is cp = 34.92 m/s.

Hence, the non-dimensional velocity can be determined asfollows,

m � V=cp � V=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ffiffiffi3p t

l

� �rave

qs

sð10Þ

The non-dimensional gradient is defined as follows,

H � gl0 ¼ nDrys

raveð11Þ

where n is the number of the layers. The non-dimensional gradientcan be regarded as the ratio of the difference in the yielding stressbetween the impinged and distal ends and the average yieldingstress.

3. Finite element results

3.1. Deformation modes

Constant velocities are applied to the impinging plate (i.e. plateB). The deformation profiles are obtained to investigate the defor-mation modes of the GHS under dynamic loading.

3.1.1. Deformation modes – positive gradientWhen the gradient g is positive, the weakest layer is placed at

the impact end. Under low velocity impact, the deformation ismainly dominated by the distribution of the quasi-static plateaustress of the layers. Since the weakest layer is placed at the im-pinged end, the initial collapse occurs at the impinging end regard-less of the impact velocity. Figs. 3–5 show deformation modes of aGHS with the gradient H = 0.42 under various impact velocities. Asshown in Fig. 3, when V = 5 m/s deformation localization occursinitially at the loading end. It is found that some cells in the otherlayers with stronger material collapse while the cells in the 15thlayer remains undeformed, which produces a ‘V’ shaped band(Fig. 3a. Afterwards, the ‘‘V’’ shaped band continues to propagateto the supporting end (Fig. 3b and c). With the increase of the dis-placement of the impinging plate, the cells ahead of the ‘‘V’’ shapedband begin to collapse (Fig. 3d). At a higher impact velocity of20 m/s (Fig. 4), the ‘‘V’’ shaped band appears later than the case de-picted in Fig. 3. Besides, no deformation is found near the support-ing end during the whole crushing. When the impact velocityincreases to V = 35 m/s (Fig. 5), only a localized transverse bandin the shape of ‘‘I’’ is observed near the loading end and it continuesto propagate, layer by layer to the supporting end, which is like theshock wave propagation.




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Fig. 3. Deformation profile of the graded honeycomb in the case of H = 0.42, V = 5 m/s. (a) u = 7 mm; (b) u = 21 mm; (c) u = 35 mm; (d) u = 49 mm. u is the displacement of theimpinging plate.

Fig. 4. Deformation profile of the graded honeycomb in the case of H = 0.42, V = 20 m/s. (a) u = 7 mm; (b) u = 21 mm; (c) u = 35 mm; (d) u = 49 mm. u is the displacement ofthe impinging plate.




28 November 2012

The deformation profile for the GHS with a higher stress gradi-ent of H = 1.66 is shown in Fig. 6. The impact velocity is the sameas for Fig. 4 (i.e. V = 20 m/s). Comparison between Figs. 4 and 6shows that the ‘‘V’’ shaped band appears in the honeycomb withlower stress gradient (Fig. 4) while only ‘‘I’’ shaped band appearsduring the whole crushing process in the cases of higher stress gra-dients (Fig. 6). This indicates that the stress gradient influences thedeformation mode under the dynamic loading.


3.1.2. Deformation modes – negative gradientWhen the gradient is negative, the strongest layer is placed at

the impact end while the weakest layer is at the supporting end.The deformation modes are more complex than the cases with po-sitive gradient. Figs. 7–9 show the deformation modes for the hon-eycomb with negative gradient of H = �0.42 under various impactvelocities. Fig. 7 depicts the response under the velocity V = 5 m/s.The collapse of cells initiates from the supporting end while no




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Fig. 7. Deformation profile of the graded honeycomb in the case of H = �0.42, V = 5 m/s. (a) u = 7 mm; (b) u = 21 mm; (c) u = 35 mm; (d) u = 49 mm. u is the displacement ofthe impinging plate.




28 November 2012

deformation occurs at the loading end. Afterwards, a ‘‘V’’ shapedband appears near the supporting end and it becomes moreobvious with the cells’ collapse near the shaped band. Near theend of the crushing, some cells adjacent to the impact plate (i.e.plate B) begin to collapse. The deformation process seems a rever-sal of that shown in Fig. 3, as expected. Under low impact loadingcondition, the deformation process is determined by the distribu-tion of the quasi-static plateau stress no matter where the load isapplied—the weakest part deforms first.


The deformation profile of the honeycomb under a higher im-pact velocity of V = 20 m/s is given in Fig. 8. The initial collapseoccurs at the loading end (though stronger) while no obviousdeformation occurs at the supporting end. Afterwards, some cellsnear the impinged end remain undeformed while others in themiddle of the specimen begin to collapse, producing a ‘‘V’’ shapedband. Subsequently, the cells near the supporting end begin to col-lapse, forming a new ‘‘V’’ shaped band (Fig. 8c). With the increaseof the displacement, the ‘‘V’’ shaped band near the impinged end




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almost remains unchanged and further deformation isconcentrated near the supporting end. Finally, the bands interactwith each other until the whole structure is crushed.

For V = 50 m/s the deformation profile is depicted in Fig. 9. Sim-ilar to the cases with positive gradient under high impact veloci-ties, only a localized transverse band is observed at the loadingend, which then propagates to the supporting end. For the cellsadjacent to the supporting end, some small deformation still canbe found.

The deformation modes of the honeycomb with larger magni-tudes of the stress gradient, �1.02 and �1.66, are shown in Figs.10 and 11, respectively. The impact velocity is the same as forFig. 7 (i.e. V = 5 m/s). The initial deformation at the supportingend in Fig. 7 is also present in Figs. 10 and 11 while no deformationoccurs at the loading end. Comparing these figures, it is noted thatthe ‘‘V’’ shaped band is more obvious in the honeycomb with alower magnitude of the stress gradient, indicating that the defor-mation mode is significantly influenced by the local quasi-staticplateau stress under low velocity impact.

3.1.3. Mode classification mapThe above deformation characteristics can be summarized as

follows. Two deformation modes are found for the GHS with posi-tive gradient while three deformation modes in the GHS with neg-ative gradient.

When the gradient is positive, the first deformation mode is ‘‘V’’mode, which takes place when the impact velocity is low (Fig. 3)


and is characterized by oblique localized bands initiating fromthe loading end. The ‘‘V’’ shaped band occurs at the loading endand then extends to the other layers. With the increase of the im-pact velocity, localized bands tend to be orientated in the transver-sal direction (Figs. 4 and 5), which is the second, known as ‘‘I’’mode.

When the gradient is negative, the deformation processes arequite complex and may involve a change in the mode. Hence, themodes are classified according to the initial deformation mode.Three deformation modes exist: two involve a ‘‘V’’ mode, either ini-tiating from the supporting end or from the impacting side and onefor ‘‘I’’ mode. The first is ‘‘VS’’ mode, as shown in Fig. 7. In thismode, deformation occurs from the ‘‘supporting’’ end, resultingin a ‘‘V’’ shaped band. With the increase of the impact velocity,the inertial effect gradually influences the deformation mode sothat more and more deformation takes places at the loading end.The second is ‘‘VL’’ mode, in which the initial deformation occursat the ‘‘loading’’ end, resulting in a ‘‘V’’ shaped band. ‘‘VL’’ modeis a mixed mode, in which further deformation may be concen-trated at the supporting end in the later stage of deformation(Fig. 8). The third is ‘‘I’’ mode, which appears under high velocityimpact (Fig. 9). This ‘‘I’’ mode is similar to the case for graded hon-eycomb with positive gradient under high velocity impact. Com-pared with the deformation modes for the homogenoushoneycomb under impact loading [20], the ‘‘VS’’ mode is the qua-si-static deformation mode while the ‘‘I’’ mode represents thedeformation mode under high velocity impact.




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-2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 1.6 2.00.0

0.5

1.0

1.5

2.0

2.5

"I" mode

"VL" mode

"VS" mode

Gradient (Θ)

Crit

ical

vel

ocity

(υ cr

)

"V" mode

"I" mode

Θ 0

Vcr2

Vcr1

Θ 0

Fig. 12. Non-dimensional critical velocities versus the gradient.



28 November 2012

Fig. 12 plots the non-dimensional critical velocities for the GHSagainst the stress gradient, based on FE simulations of a number ofcases. In terms of the stress gradient H, the thickness ratio t/l, theaverage quasi-static plateau stress rave and the density of the par-ent material qs, two empirical equations are proposed for the crit-ical velocity as,

Vcr1 ¼ �0:27Hþ 0:43ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ffiffiffi3p t

l

� �rave

qs

sð12Þ

Vcr2 ¼ �0:51Hþ 1:31ð Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1ffiffiffi3p t

l

� �rave

qs

sð13Þ

It should be noted that when the gradient is positive (i.e. H > 0),the first critical velocity does not exist.

3.2. Dynamic stresses in the graded honeycombs

In this section, the dynamic stress in graded honeycomb isinvestigated by analyzing the reaction stresses of the two plates(i.e. plates A and B). For the low velocity impact, the stresses ofthe two plates are nearly the same as the quasi-static plateaustress, neglecting the inertia effect. On the other hand, when theimpact velocity is sufficiently high, the deformation mode is apropagation of a plane plastic wave (i.e. ‘‘I’’ mode) regardless ofthe stress gradient, indicating that the shock wave theory may be


applicable in this case. Eqs. (1) and (2) theoretically give the dy-namic stress and plateau stress for the honeycomb under dynamicand quasi-static loading, respectively. The locking strain ed is as-sumed to be a constant at ed = 0.82, which is determined fromthe FE simulations.

Fig. 13 shows the reaction stresses versus the non-dimensionaldisplacement of the impact plate (i.e. plate B) when the gradient ispositive. Under the low velocity impact, the stresses at the sup-porting and loading ends are almost the same. Compared withthe analytical prediction some deviations are found at the finishingstage of crushing. The stresses are much smaller than the analyticalprediction. The reason can be found from the deformation profiles.For example, it is found that the cells ahead of the compactionband are collapsed in Fig. 3d. Under high velocity impact, the dy-namic stress is obtained from the loading end while the stress onthe supporting end is equal to the quasi-static plateau stress atthe compaction front. The analytical prediction shows a goodagreement with the FE results, indicating that the shock wave the-ory can be employed for the GHS with positive gradient.

Fig. 14 represents the reaction stresses versus the non-dimen-sional displacement of the impinging plate when the gradient isnegative and the deformation mode is either ‘‘VS’’ or ‘‘I’’. For thelow velocity impact, similar to the cases in Fig. 13, the reactionstresses of the loading and supporting plates are almost the same,which can be regarded as quasi-static behavior. Under the highvelocity impact, the stress at the loading plate is much higher thanthat at the supporting plate due to the inertial effect. The stressesdecrease with the increase of the displacement because the col-lapse occurs from the loading end with strong material to the sup-porting end with weak material. However, it is found that there is asudden drop for the stress after the initial peak force and the valueis much smaller than those predicted by the shock wave theory.The reason is that the collapse of the cells is not layer by layer asassumed analytically. For example, in Fig. 9b the cells behind thecompaction front are not fully densified while some cells in theother layers are already densified, leading to the deviation fromthe analytical prediction.

Under an intermediate velocity impact, the transitional defor-mation mode takes places in the GHS with negative gradient. Thereseem two compaction fronts appearing in this mode and it is diffi-cult to give a good analytical prediction by using shock wave the-ory. Fig. 15a represents the stress–displacement relationship ofsuch a case of V = 20 m/s and H = �1.02. The corresponding defor-mation profiles are shown in Fig. 15b. The whole crushing processcan be divided into two phases. In Phase I, the stress of the loadingend is higher than that at the supporting end, indicating that mostdeformation occurs adjacent to the loading end; the compaction




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Stre

ss (M

Pa)

V=30m/s

Displacement (μ)

Displacement (μ)Displacement (μ)

Displacement (μ) Displacement (μ)0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Fig. 13. The stress–displacement relationship for the GHS with positive gradient. (a) H = 0.42, V = 5 m/s and V = 35 m/s; (b) H = 1.04, V = 5 m/s and V = 35 m/s; (c) H = 1.66,V = 5 m/s and V = 30 m/s. Note: the displacement is the non-dimensional one respect to the initial length.



28 November 2012

band adjacent to the loading end becomes more obvious with thecollapse of the cells ahead of the compaction band. At thebeginning of Phase II, there is a sudden increase in the stress atthe supporting end. The reason is that no further deformation oc-curs at the loading end at that moment, but at the supportingend instead, forming a new compaction band. Afterwards, thestress at the supporting plate increases due to the inertial effect.Meanwhile, the stress at the loading end suddenly drops to a verylow value because the compaction band adjacent to the loading


plate stops and the stress becomes close to the quasi-staticplateau stress at the new compaction band adjacent to the sup-porting end.

3.3. Energy absorption capacity of the graded honeycomb

In this section, the energy absorbed by the GHS under the dy-namic loading conditions is investigated, including the constantvelocity loading and mass impact with initial velocity.




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Displacement (μ) Displacement (μ)

Displacement (μ)Displacement (μ)

Fig. 14. the stress–displacement relationship for the GHS with negative gradient. (a) H = �0.42, V = 5 m/s and V = 50 m/s; (b) H = �1.04, V = 5 m/s and V = 80 m/s; (c)H = �1.66, V = 5 m/s and V = 100 m/s. Note: the displacement is the non-dimensional one respect to the initial length.



28 November 2012

3.3.1. Energy absorbed under constant impact velocityFig. 16 depicts the absorbed energy by the GHS versus the dis-

placement of the impinging plate. For the low velocity impact(Fig. 16a), at the early stage the energy absorbed by the homoge-nous honeycomb is larger than that for GHS but they approachnearly the same value when the displacement reaches the maxi-mum. The reason is that the inertia effect can be neglected so thatthe deformation always initiates from the weakest layer and thentransmits to the stronger layers regardless the loading direction.


With the increase of the impact velocity, the inertia influencesthe deformation modes. For the GHS with negative gradient, thedeformation may initiate from the loading end (i.e. strongestlayer), resulting in slightly larger amount of energy absorbedthan the homogenous structure at the early stage (Fig. 16b–d).For the GHS with positive gradient, the deformation always initi-ates from the loading end (i.e. weakest layer), thus the absorbedenergy is always smaller than the homogenous structure at theearly stage.




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Phase IIPhase I

(a)

(b)

Fig. 15. (a) The stress–displacement relationship for the transitional mode ‘‘VL’’ inthe GHS with negative gradient. H = �1.04, V = 20 m/s; (b) the correspondingdeformation profile at the instants l = 0.078, l = 0.267, l = 0.457 and l = 0.647.

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Displacement (μ)

Θ=1.66Θ=1.66

Θ=0Θ=1.66Θ=1.66

Fig. 16. Absorbed energy versus the displacement of the impinging pla



28 November 2012


It is found that the energy absorbed by the GHS with positivegradient approaches to nearly the same value as the homogenoushoneycomb block regardless of the velocity. The reason is that onlyone compaction band propagates during the whole crushing, sothat the average dynamic stress is the same as that of the homog-enous honeycomb. On the other hand, under high velocity impact,the energy absorbed by the GHS with a negative gradient is muchsmaller (Fig. 16c and d). This is because the deformation mode forthe GHS with negative gradient is very different from that of thehomogenous honeycomb; that is there are two compaction bandsfor the GHS with a negative gradient under an intermediate impactvelocity. Even under higher velocity impact, small deformationscan still be found at the cells adjacent to the supporting end.

3.3.2. Energy absorbed under a mass impact with initial velocityWhen the honeycomb is subjected to impact of a mass with ini-

tial velocity, the velocity of the impinging plate cannot remain con-stant and it decreases with the displacement of the impingingplate. Consequently, the deformation modes may change fromone to another during a response process. Hence, the absorbed en-ergy under a constant velocity loading cannot truly represent theenergy-absorbing capacity of the GHS. In this section, an initialvelocity is applied to the impinging plate (i.e. plate B). Dependingon the magnitude of the initial input energy from the impingingmass, there are two possible outcomes. When the initial kinetic en-ergy is sufficiently high, the structure will be fully crushed with aresidual velocity of the impinging plate. On the other hand, if theinitial kinetic energy is relatively low the honeycomb block willnot be fully densified when the velocity of the plate reaches zero.

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Θ=0Θ=1.66Θ=1.66

te. (a) V0 = 5 m/s; (b) V0 = 20 m/s; (c) V0 = 60 m/s; (d) V0 = 100 m/s.




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28 November 2012

Hence, a special initial velocity, called densification velocity md, canbe defined by considering a critical case, in which the GHS is fullycrushed when the input kinetic energy is just exhausted. In thiscase, when the displacement of the rigid plate reaches the maxi-mum, the velocity decreases to zero. Since there is no heat transferor energy loss in the whole crushing process, the energy absorbedby the GHS can be determined by the conservation of the energy.

Under the quasi-static loading the honeycomb collapses layerby layer and the energy absorption capacity of the GHS is calcu-lated as,

Eq ¼Z

A0rpdu ¼ A0edl0r0 ¼r0

q0med ð14Þ

where rp is the plateau stress, r0 is the average plateau stress, andm = q0l0A0 is the total mass of the GHS. Since the average plateaustress is defined as a constant, the quasi-static energy-absorbingcapacity of graded honeycomb blocks with different gradient isthe same.

The conservation of the energy requires the energy-absorbed bythe graded honeycomb under a mass impact loading as,

Ed ¼12

Cr0

q0mm2

d ð15Þ

where C = G/m, G is the mass of the impinging plate. Hence, the dy-namic enhancement of the energy absorption can be defined as,

w � Ed

Eq¼ Cm2

d

2edð16Þ

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sific

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ty (

υ d )

Θ = 1.66

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.21.0

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Mass ratio (Γ= G/m)

Ener

gy ra

tio (

Ψ)

(b)

Mass ratio (Γ= G/m)

(a)Θ = 0Θ =-1.66

Θ = 1.66 Θ = 0Θ =-1.66

Fig. 17. (a) Densification velocity versus the mass ratio; (b) energy ratio versus themass ratio.


where md is the non-dimensional densification velocity.Harriagan et al. [30] theoretically gave the densification velocity

for the homogenous cellular rod as,

md ¼1C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2Cþ 1Þed

pð17Þ

Eq. (17) shows that the densification velocity is only related to themass ratio C.

Fig. 17a depicts the densification velocity md against the massratio C from FE simulation. It shows that the densification velocitymd decreases with the increase of the mass ratio C. Compared withthe homogenous honeycomb, it is found that the densificationvelocity is higher when the gradient is positive while it is lowerwhen the gradient is negative. Similar to the homogenous honey-combs, the densification velocity is significantly influenced bymass ratio C. Using Eq. (16), the corresponding energy ratio w isplotted against the mass ratio C in Fig. 17b. The positive gradientenhances the energy ratio while the negative gradient weakensthe energy ratio.

Most recently, the authors [31] conducted theoretical deriva-tion of the one-dimensional graded cellular rod under impact byusing R–P-P–L material model. Assuming only a single shock com-paction front, the densification velocity md is given in a closed form,

md ¼1C

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Cþ 1þ 1

6H

� �ed

sð18Þ

Compared with Eq. (17), the gradient H is now incorporatedinto the equation. Combing with Eq. (16), the energy ratio for thegraded cellular rod is,

w ¼ 1þ 12Cþ 1

12CH ð19Þ

It is easily seen that placing the weakest material at the im-pinged end leads to higher energy absorption, which is consistentwith the FE results.

4. Conclusion

In this paper, the dynamic of crushing band and wave trappingof the graded honeycomb structure under in-plane dynamic load-ing are analyzed by using FE simulation. A gradient in the mate-rial’s yielding stress has been introduced to a regular honeycombstructure. Constant velocity loading was used to analyze the defor-mation modes, critical velocities and the dynamic stress. To inves-tigate the energy absorbing capacity of the GHS, both the cases ofconstant velocity loading and a mass impact with initial velocityare employed.

Two types of deformation modes have been observed when thegradient is positive while three types of deformation modes existwhen the gradient is negative. The deformation modes are nearlythe same when the velocity is either small or sufficiently high,whether the gradient is positive or negative. ‘‘V’’ shaped band ini-tiates from the weakest layer at the low velocity impact while ‘‘I’’shaped band occurs from the loading end at the high velocity im-pact. A transitional mode is present when the velocity is interme-diate and the gradient is negative. More deformation has beenfound at the loading end with the increase of the impact velocity.A map has been constructed showing the critical velocities.

The analysis of the stresses indicates that the shock wave theoryis applicable only when one compaction fronts appear. The reac-tion stress at the supporting end is equal to the quasi-static plateaustress of the layer under collapse; the stress at the loading end isthe dynamic stress at the compaction front.

Densification velocity of the GHS is not only related to the massratio but also the stress gradient. The investigation of the energy




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absorption shows that the positive gradient enhances the energyabsorbing capacity of the GHS.

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[12] Papka SD, Kyriakides S. Biaxial crushing of honeycombs – Part II: analysis. Int JSolids Struct 1999;36(29):4397–423.

[13] Baker WE, Togami TC, Weydert JC. Static and dynamic properties ofhighdensity metal honeycombs. Int J Impact Eng 1998;21(3):149–63.

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[21] Zou Z, Reid SR, Tan PJ, Li S, Harrigan JJ. Dynamic crushing of honeycombs andfeatures of shock fronts. Int J Impact Eng 2009;36:165–76.

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[23] Reid SR, Peng C. Dynamic uniaxial crushing of wood. Int J Impact Eng1997;19:531–70.

[24] Hu LL, Yu TX. Dynamic crushing strength of hexagonal honeycombs. Int JImpact Eng 2010;37:467–74.

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[26] Ajdari A, Canavan P, Nayeb-Hashemi H, Warner G. Mechanical properties offunctionally graded 2-D cellular structures: a finite element simulation. MaterSci Eng A Struct 2009;499:434–9.

[27] Ajdari A, Nayeb-Hashemi H, Vaziri A. Dynamic crushing and energy absorptionof regular, irregular and functionally graded cellular structures. Int J SolidsStruct 2011;48:506–16.

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Zhao H, Gary G. Crushing behaviour of aluminium honeycombs under impact loading. Int J Impact Eng 1998, 21: 827–836.

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Acknowledgement The supports from NSFC (No. 11028206) and the State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology) (Project KFJJ11-1M) are acknowledged.

dynamic behavior of graded honeycombs...

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