International Journal of Solids and Structures 49 (2012) 2754–2762

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International Journal of Solids and Structures

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Inertia effects on the progressive crushing of aluminium honeycombsunder impact loading

B. Hou a, H. Zhao b,⇑, S. Pattofatto b, J.G. Liu b, Y.L. Li a

a School of Aeronautics, Northwestern Polytechnical University, 710072 Xi’an, Chinab Laboratoire de Mécanique et Technologie, ENS-Cachan/CNRS-UMR8535/Université Paris 6, 61 avenue du président Wilson, 94235 Cachan cedex, France

a r t i c l e i n f o

Article history:Available online 17 May 2012

Keywords:HoneycombsImpact loadingLateral inertiaKolsky’s barNumerical simulation

0020-7683/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijsolstr.2012.05.005

⇑ Corresponding author. Tel.: +33 1 47 40 20 39; faE-mail address: zhao@lmt.ens-cachan.fr (H. Zhao).

a b s t r a c t

This paper presents the test results under quasi-static and impact loadings for a series of aluminum hon-eycombs (3003 and 5052 alloys) of different cell sizes, showing significantly different enhancements ofthe crushing pressure between 3003 honeycombs and the 5052 ones. A comprehensive numerical inves-tigation with rate insensitive constitutive laws is also performed to model the experimental results fordifferent cell size/wall thickness/base material, which suggests that honeycomb crushing pressureenhancement under impact loading is mostly due to a structural effect.

Such simulated tests provide detailed local information such as stress and strain fields (in the cell wall)during the whole crushing process of honeycombs. A larger strain (in the cell wall) under impact loadingthan for the quasi-static case before each successive folding of honeycombs is observed, because of thelateral inertia effect. Thus, differences of the ratios of the stress increase due to strain hardening overthe yield stress between 3003 and 5052 alloys lead to the different enhancements of crushing pressure.This result illustrates that the lateral inertia effect in the successive folding of honeycombs is the mainfactor responsible for the enhancement of the crushing pressure under impact loading.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Aluminium honeycombs are widely used in railway, automotiveand aircraft industries because of their excellent physical andmechanical properties such as an interesting strength/weight ratioand an outstanding capability in absorbing energy. Mechanicalbehavior for small strain under quasi-static loadings such as theelastic behavior and failure strength are well investigated for struc-tural applications (Gibson and Ashby, 1988). Elastic and fracturemodels for out-of-plane crushing (Zhang and Ashby, 1992a) andin-plane crushing (Zhang and Ashby, 1992b), as well as for trans-verse shearing (Shi and Tong, 1995), have been developed. Forthe case of larger strain, theoretical, experimental and numericalstudies have also been reported. Theoretical models can predictthe crushing pressure of honeycombs from its geometrical param-eters and wall material behavior such as the out-of-plane crushingpressure (Wierzbicki, 1983), the in-plane crushing pressure(Klintworth and Stronge, 1988), and multi-axial collapse envelope(Mohr and Doyoyo, 2004a). Other related topics such as fracturedetection using elastic waves (Thwaites and Clark, 1995), negativePoisson’s ratio honeycombs (Prall and Lakes, 1997), and foam-filledhoneycombs (Wu et al., 1995), have also been reported in the openliteratures.

ll rights reserved.

x: +33 1 47 40 22 40.

For the energy absorption applications such as protective designfor accidental collisions of high speed vehicles or the bird strike ofaircrafts, the out-of-plane behavior for large strains (up to 80%) un-der impact loading is desired. Experimental results show that thecrushing pressure of honeycombs under impact loading is higherthan that under quasi-static loading. For example, (Goldsmithand Sackman, 1992; Goldsmith and Louie, 1995) have reportedsome experimental works on out-of-plane crushing and on the bal-listic perforation of honeycombs. They have fired a rigid projectileto a target made of honeycomb and have shown that the meancrushing pressures sometimes increase up to 50% compared tothe static results. Wu and Jiang (1997), Baker et al. (1998), Harriganand Reid (1999), Zhao and Gary (1998), and Zhao et al. (2005) havealso found the similar phenomenon for metallic honeycombs withan enhancement ranging from 10% to 50%.

As the aluminium alloy is hardly rate sensitive in the range ofmoderate impact speed, there exists no plausible explanation ofthis enhancement. Indeed, the possible effect due to the eventualshock front (Reid and Peng, 1997; Pattofatto et al., 2007), airtrapped in the cell (Gibson and Ashby, 1988) cannot be appliedhere. Besides, since the testing methods used in previous worksare rather different from each other, reasonable doubt also existson the validity of those experimental results. Enhancement ornot, how much, why, are still the open questions.

This paper describes an experimental and numerical study ofthe out-of-plane compressive behavior for a series of aluminium

B. Hou et al. / International Journal of Solids and Structures 49 (2012) 2754–2762 2755

honeycombs of different size (relative densities varying from 1.78%to 4.72%) and different base material (3003 and 5052 alloys) undermoderate impact velocities. Tests in a range of impact speed from10 m/s to 28 m/s were performed using large diameter polymerSplit Hopkinson bars. The accuracy of polymeric SHPB systems isknown to be adequate to obtain reliable testing results on such softmaterials. The testing results confirmed that there does exist anenhancement from 10% to 60% for aluminum 5052 and 3003honeycombs.

A numerical model of the studied honeycombs (3003 and 5052alloys) was performed afterward using ABAQUS commercial codes.Similar enhancements were found with numerical results (>40%for 3003 honeycombs and <20% for 5052 ones). Finally, on the basisof numerical models, a comprehensive numerical study of succes-sive crushing process was performed in order to understand thereason of this enhancement as well as its dependence on the basematerials. It shows that the lateral inertia in the successive foldingof thin-wall tube structures can explain such observed enhance-ment of out-of-plane crushing pressure.

Fig. 1. Geometry of the unit cell.

2. Experimental impact rate sensitivity of 5052 and 3003aluminium honeycombs

2.1. Experimental methods and procedures

Honeycombs of various wall thickness and cell size made of5052 or 3003 alloys were tested under axial compression in theout-of-plane direction. The quasi-static experiments were per-formed using a universal tension/compression testing machine.The dynamic experiments were conducted on a SHPB (Split Hop-kinson Pressure Bar) apparatus (Hopkinson, 1914; Kolsky, 1949),commonly used as an experimental technique to study constitutivelaws of materials at high strain rates.

A typical SHPB set-up is composed of long input and output barswith a short specimen placed between them. A projectile launchedby a gas gun strikes the free end of the input bar and develops acompressive longitudinal incident wave ei(t). Once this wavereaches the bar-specimen interface, part of it er(t) is reflected,whereas the other part goes through the specimen and developsthe transmitted wave et(t) in the output bar. As the stress and par-ticle velocity of a longitudinal stress wave can be calculated fromthe strain measured by gauges, and shifted at any other place,the transmitted wave can be shifted to the output bar-specimeninterface to obtain the output force and velocity, whereas the inputforce and velocity can be determined via incident and reflectedwaves shifted to the input bar-specimen interface. The forces andparticle velocities can be then calculated as follows (Eq. (1)):

FinputðtÞ ¼ SB E ðeiðtÞ þ erðtÞÞ VinputðtÞ ¼ C0 ðeiðtÞ � erðtÞÞVoutputðtÞ ¼ C0 etðtÞ FoutputðtÞ ¼ SB E etðtÞ

ð1Þ

where Finput; Foutput;Vinput ;Voutput are forces and particle velocities atthe interfaces, SB, E and C0 are respectively the cross sectional area,Young’s modulus and the longitudinal wave speed in the pressurebars. eiðtÞ; erðtÞ; etðtÞ are the strain signals at the bar-specimeninterface.

The standard SHPB analysis (Hopkinson, 1914; Kolsky, 1949)provides an average nominal stress–strain curve, dividing the dis-placement and force respectively by the initial length and thecross-sectional area of the specimen (see Zhao and Gary, 1996).For the out-of-plane crushing tests of honeycombs, mechanicalfields are not uniform (even under quasi-static loading). Therefore,Hopkinson bars here are considered as only a loading and measur-ing system which can give accurately the force and displacementtime histories on the specimen faces without considering thedeforming characteristics (uniform or not) of the sandwiched

specimen. Instead of average stress and strain in a common SHPBtest, we use only the pressure p(t) as a function of the crush D(t)to give an overall idea of the behavior of the honeycombs. Theyare defined as follows (Eq. (2)):

pðtÞ ¼ ðFinputðtÞ þ FoutputðtÞÞ=2Ss

DðtÞ ¼Z t

0ðVoutputðsÞ � VinputðsÞÞds

ð2Þ

where Ss is the apparent area of the specimen face.It is worthwhile to notice that impact tests on such soft cellular

materials using a SHPB have two major difficulties. One is the largescatter due to the small cell/sample ratio. To overcome this diffi-culty, a large diameter pressure bar is necessary to host a largerspecimen. Another is the weak signal due to the weak strength ofhoneycombs, which leads to a low signal/noise ratio. Large diame-ter, soft, but polymeric pressure bars are used to overcome thesedifficulties. In practice, two Hopkinson bar systems were used: a60 mm PA6.6 Hopkinson bar system with input and output barsof 3 m (in LMT-Cachan), and a 30 mm diameter PMMA bar systemwith 2 m input and output bars (in the Laboratory of Dynamics andStrength, NWPU). The projectiles were made of same materials aspressure bars and their lengths are respectively 1.2 m and 0.5 m(less than half input bar length to avoid superimposition of the tailof incident impulse in viscoelastic bars, see Zhao et al., 1997). Thus,the impulse is not long enough to reach the densification point indynamic test.

Moreover, soft polymeric bars are viscoelastic materials, andthe wave dispersion effect increases greatly with the diameter ofthe bars. Consequently, as the three waves in Eq. (1) are not mea-sured at bar-specimen interfaces to avoid their superimposition,they have to be shifted from the position of the strain gauges tothe specimen faces. Kolsky’s original SHPB analysis on the basisof a one-dimensional wave propagation theory is no longer validhere. The shifting along pressure bars is performed in our experi-ments with Pochhammer and Chree’s harmonic wave propagationtheory in an infinite cylindrical bar (Davies, 1948; Follansbee andFranz, 1983), extended to viscoelastic bars (Zhao and Gary,1995). Detailed data processing procedure can be found in Zhaoet al. (1997).

2.2. Honeycomb specimens and testing results

Firstly, four 3003 alloy honeycombs referenced by the singlewall thickness h, and the minimum cell diameter S (Fig. 1) werestudied. Samples were designed as columns of 30 mm high, withhexagonal cross section containing a maximum number of cellsin a circle of 30 mm diameter (Fig. 2).

Table 1 provides a summary of honeycomb geometry parame-ters and the relative densities (see Gibson and Ashby, 1988) of fourtypes of Al3003 honeycombs.

Quasi-static tests with loading speed of 0.03 mm/s and SHPBtest between 26 and 28 m/s were performed in the out-of-planedirection (T direction in Fig. 2). For the quasi-static tests, the pres-sure (measured force divided by the nominal cross-sectional area)

Fig. 2. Hexagonal honeycomb specimens.

-2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 280

1

2

3

4

5

6

7

8

9

10

Noo3 (Al3003)

Pres

sure

(MPa

)a

Crush (mm)

Fig. 3a. Reproducibility of quasi-static experiments on honeycomb No. 3.

-1 0 1 2 3 4 5 6 70

1

2

3

4

5

6

7

8

9

10

11

Noo3 (Al3003)

Pres

sure

(MPa

)

Crush (mm)

Fig. 3b. Reproducibility of impact experiments on honeycomb No. 3.

- 3 0 3 6 9 1 2 1 5 1 8 2 1 2 4 2 7 3 00.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

No o4 (Al3003)

Dynamic Quasi-static

Pres

sure

(MPa

)

Crush (mm)

Fig. 4. Dynamic enhancement of honeycomb No. 4 pressure/crush curve.

Table 1aSummary of 3003 honeycomb parameters.

Honeycomb number Material h/S (mm/mm) Relative density (%)

1 Al3003 0.05/5.2 2.572 Al3003 0.06/4.33 3.703 Al3003 0.06/3.46 4.614 Al3003 0.04/3.46 3.08

Table 1bSummary of the parameters and experimental results of 3003 honeycombs.

Honeycombnumber

Material h/S(mm)

Relativedensity (%)

pquasi-static

(MPa)pimpact

(MPa)c(%)

1 Al3003 0.05/5.2

2.57 1.24 1.96 58.1

2 Al3003 0.06/4.33

3.70 2.79 4.06 45.5

3 Al3003 0.06/3.46

4.61 4.00 5.51 37.8

4 Al3003 0.04/3.46

3.08 1.20 1.75 45.8

2756 B. Hou et al. / International Journal of Solids and Structures 49 (2012) 2754–2762

can be depicted with respect to the measured crush displacement.For example, Fig. 3a shows the pressure/crushing displacementcurves for the three independent quasi-static tests on the specimenNo. 3 (h = 0.06 mm/S = 3.46 mm). There is a scatter on the initialbuckling peak force and the locking strain, but the average plateaulevel is very stable. The average value of the pressure between theinitial peak and locking strain is used as the quasi-static pressurepquasi-static.

For the SHPB tests, the pressure and the crush are calculatedusing Eq. (2). Fig. 3b illustrates the three repeated tests under

impact loading for the same kind of specimens No. 3 (Table 1).There exist larger oscillations probably due to impact testingimperfection. However, the scatter on the average plateau level issmall.

It is noticed that the crush reached in impact test is quite lim-ited (around 6 mm) because of limited length of impulse in theHopkinson bars system, which gives also an enlarged impressionof this oscillations. A direct comparison between dynamic and qua-si-static pressure/crush curves is shown in Fig. 4 for the case ofspecimen No. 4 (Table 1a). Quasi-static specimens undergo muchlarger crush compared to the dynamic ones. Oscillations under im-pact loading in the plateau stage in such figure are not significantlyhigher than the quasi-static case.

As in the case of quasi-static loading, an average plateau valueof the pressure pdynamic can be derived. The impact enhancementratio c is defined as follows (Eq. (3)):

c ¼ ðpdynamic � pquasi�staticÞ=pquasi�static ð3Þ

Table 1b gives the summary of testing results of all the four3003 honeycombs. From those testing results, it is clear that thestrength of 3003 alloy honeycomb under out-of-plane compressionexhibits an important impact enhancement around 40%.

Table 2Summary of the parameters and experimental results of 5052 honeycombs.

Honeycombnumber

Material h/S(mm/mm)

Relativedensity (%)

pquasi-

static

(MPa)

pimpact

(MPa)c(%)

5 Al5052 0.076/9.52

1.78 1.7 2 17.6

6 Al5052 0.076/6.35

2.76 3 3.5 16.7

7 Al5052 0.076/4.76

4.72 5.1 5.7 11.8

Fig. 5. Scheme of honeycomb cross section and the unit cell-model.

B. Hou et al. / International Journal of Solids and Structures 49 (2012) 2754–2762 2757

Secondly, three Alcore 5052 honeycombs (Duracore) were alsostudied. Samples were designed as columns of 50 mm high withhexagonal cross section containing a maximum number of cellsin a circle of 60 mm diameter. For the honeycomb of largest cells,there are still 5 cells at least in one edge of the hexagon. Quasi-sta-tic and SHPB tests (10�14 m/s) were performed in the out-of-planedirection. Table 2 gives the characteristics of different tested hon-eycombs (density, cell size and wall thickness) as well as the quasi-static and dynamic mean crushing pressure results (Zhao et al.,2005).

It is found that the enhancement for 5052 alloy honeycombs israther small (<20%), compared to that of 3003 alloy honeycombs(>40%), noting that this is a general trend obtained from differentwall thickness/cell size ratios in the same range of relative density(3 for 5052 and 4 for 3003).

2.3. Numerical simulation of honeycombs

The rate sensitivity of bulk aluminium alloy is known to besmall (<10%, see Duffy et al., 1971). Recent testing results on thin

(a)Fig. 6. Two honeycomb cell-model with diffe

aluminium sheet metals (6065 T5 in compression (Zhao, 1997),2024 T3 under shear loadings (Zhao et al., 2007)) reveal also a lim-ited rate sensitivity (<10%).

Therefore, the significant impact enhancement (>40%) observedin 3003 honeycombs cannot be directly derived from the rate sen-sitivity of the cell wall base materials. There must be a structuralreason responsible for this enhancement. For this purpose, numer-ical simulation of the above honeycomb testing results is devel-oped. Since our study focuses on the behavior of honeycombs,the modelling of the whole testing environment is not necessary.Thus, only honeycomb structures were modeled here and the load-ing environment was modeled by two rigid planes moving at theprescribed velocities corresponding to the average value of thosemeasured in the experiments. Commercial FEM code of ABAQUS/Explicit was employed for this simulated work.

In order to reduce the calculation cost with a complete honey-comb model with the same geometry as the hexagonal honeycombspecimen (Fig. 2) honeycomb specimen was simplified into a unitcell consisting of three conjoint half walls in Y-shape (Fig. 5) be-cause of its periodicity (Mohr and Doyoyo, 2004b; Hou et al.,2011; Wilbert et al., 2011). The simplified models work with sym-metric boundary conditions applied on the three non-intersectingedges of each cell wall.

It is noticed that the leg of this Y-shape cell-model is a thickwall in a real honeycomb typically made of two single-thicknessthin walls which are bonded together. In this model, we ignorethe rare delamination of the bonded interfaces and consider thestrength of the adhesive bond as infinite. Thus, the simulationsare carried out for a monolithic structure, where the thick wallsare represented by the same shell elements but with a doublethickness value.

The model is meshed with 4-node doubly curved thick shell ele-ments with a reduced integration, active stiffness hourglass control(S4R) and 5 integration points through the cell wall thickness. Inorder to determine the appropriate element size, a convergencestudy was performed among different element sizes. The elementsize is finally chosen to be 0.1 mm.

The numerical specimen is placed between two rigid planesmoving at constant velocities, which take the mean value of realtests (i.e. 27 m/s for 3003 alloy honeycomb and 15 m/s for 5052 al-loy honeycomb). In this model, general contact with frictionlesstangential behavior is defined for the whole model excluding thecontact pairs of rigid planes and tested honeycomb specimen,which are redefined by surface-to-surface rough contact to makesure that no slippage occurs.

As the real honeycomb is always far from perfect, it includes allkinds of imperfections which affect the initial peak value, but havelittle influence on the crush behavior at a large strain. These imper-fections are due to various reasons, like irregular cell geometry, un-even or pre-buckled cell walls, wall thickness variation etc. Here in

(b)rent cell-size (with initial imperfection).

Table 3Bilinear material parameters.

Material Densityq (kg/m3)

Young’smodulus E(GPa)

Poisson’sratio m

Yieldstress rs

(MPa)

Hardeningmodulus Et

(MPa)

Al3003 2700 70 0.35 70 1150Al5052 2700 70 0.35 290 500

2758 B. Hou et al. / International Journal of Solids and Structures 49 (2012) 2754–2762

this work, we generated the imperfections by preloading the per-fect specimen uniaxially by 0.1 mm. Then, the obtained displace-ment of nodes are introduced geometrically into the actualmodel used for further calculation. The value of 0.1 mm is chosento make sure that the simulated initial peak force is the same asthe one from experimental curve at uniaxial compression. Fig. 6illustrates two numerical models of different cell sizes as well asdetails of introduced imperfection. It shows that the imperfectionsintroduced by preloading is very small.

Quasi-static simulations are almost impossible to achieve withABAQUS/Standard which uses Newton’s method (or quasi New-ton’s method) as a numerical technique due to the complex nonlin-ear effects, e.g. the geometrical and material nonlinearity, thecomplex contact conditions as well as the local instability duringcrush. An alternative is to use also ABAQUS/Explicit for quasi-staticproblems. However, the explicit integration scheme of dynamicsimulation codes usually leads to very small time step which inour simulation is around ten nanoseconds for the chosen elementsize. Thus, with the loading velocity of 0.1 mm/s, the computa-tional duration for the quasi-static simulation (e.g. 130 s) will betoo large. To overcome this difficulty, automatic mass scaling tech-nique was employed to increase the time increment to 10 ls. Thequasi-static loading conditions are guaranteed by ensuring the ra-tio of the kinetic energy to the strain energy as a small value (of theorder of 10�4) with the chosen time increment. It is known thatsuch mass scaling might introduce an artificial strength enhance-ment when the introduced imperfection is small (Langseth et al.,1999). In our simulation, the imperfection obtained by 0.1 mm ax-ial crushing seems to be well-adapted because the different pre-scribed time-step do not generate significant scatter (Hou, 2011).

A bilinear elastic-plastic material model until 20% strain andperfect plastic afterwards was employed to describe the cell wallmaterial of the aluminum honeycombs. For 5052 H38 honeycomb,a yield stress of 290 MPa and a small hardening was usually admit-ted in many previous works (Papka and Kyriakides, 1994). For 3003honeycombs, parameters were identified initially with the tensiletesting results of 1 mm thick dog bone type 3003-O sheet speci-men using universal testing machine and Split Hopkinson tensilebar test (Fig. 7). One can see that its small rate sensitivity isconfirmed.

However, the hardening or thermal treatment of our 3003 hon-eycomb is unknown. The testing results of 3003-O give only a ref-erence of the real base material behavior of 3003 honeycomb. Themodel parameters of the base material such as yield stress and

−0.05 0 0.05 0.1 0.15 0.2 0.25 0.30

100

200

300

400

500

600

Strain

Stre

ss (M

Pa)

Al3003 Static Exp.Al3003 Dyn. Exp.Al3003 MoelAl5052 Model

70MPa

290MPa

500MPa

1150MPa

Fig. 7. Prescribed constitutive relation of 5052 and 3003 alloys.

hardening modulus (Table 3) were finally determined by fittingone quasi-static calculation result (i.e. No. 1 (Al3003) and No. 6(Al5052)) to the experimental one for two base materials respec-tively. These bilinear curves are also illustrated in Fig. 7.

Fig. 8 shows a comparison between experimental and simulatedpressure/crush curves for honeycomb No. 1 (Al3003) under quasi-static loading. It shows that the cell-model exhibits significant fluc-tuations at the plateau stage, which is probably due to the applica-tion of excessive symmetric boundary constraints. Actually, it iswell known that the crushing behavior of honeycombs underout-of-plane compression is regulated by the successive foldingprocedure of honeycomb cell walls. With the symmetric boundarycondition on three non-intersecting edges, the cell-model is actu-ally equivalent to a honeycomb specimen consisting of repeatedcells with identical deforming procedure, which results in a strictlysimultaneous collapse of all the honeycomb cells. Thus, in the pres-sure/crush curve, each fluctuation represents one fold formation ofthe cell wall in honeycomb structure. For the large size model, theneighboring cells interact with each other while forming the foldsand reach their local peak value at different instants, which makesthe macroscopic resulting curves smoother (Hou et al., 2011;Wilbert et al., 2011). However, the mean crushing pressure ishardly affected. We finally use this cell-model for the subsequentcalculations.

Fig. 9 shows the comparison of experimental and numericalpressure/crush curves for the honeycomb No. 1 (Al3003) underquasi-static and impact loadings. Even though the numerical pres-sure enhancement is smaller than the experimental one, whichcould be due to base material rate sensitivities or other effectsnot taken into account in this numerical model, the basic trend ispreserved.

Table 4 gives the simulated average plateau values of crushingpressure under quasi-static and impact loading for all the tested3003 and 5052 honeycombs. These numerical results suggest alsoa significant enhancement under impact for 3003 honeycombs anda small enhancement for 5052 honeycombs.

- 1 0 1 2 3 4 5 6 7 8 90.0

0.5

1.0

1.5

2.0

2.5

3.0

Noo1 (Al3003)

Quasi-static

Cell-model Experiment

Pres

sure

(MPa

)

Crush (mm)

Fig. 8. Comparison between numerical and experimental results for Al3003 cell-model under quasi-static loading.

- 1 0 1 2 3 4 5 6 7 8 9-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Noo1 (Al3003)

Static Exp. Static Cal. Dynamic Exp. Dynamic Cal.

Pres

sure

(MPa

)

Crush (mm)

Fig. 9. Comparison between the calculating and experimental results.

Table 4Simulated pressure enhancements of 3003 and 5052 honeycombs.

Honeycombnumber

Material h/S (mm/mm)

pquasi-static

(MPa)pimpact

(MPa)c (%)

1 Al3003 0.05/5.2 1.24 1.83 482 Al3003 0.06/4.33 2.27 3.57 57.13 Al3003 0.06/3.46 3.61 4.92 36.34 Al3003 0.04/3.46 1.80 2.49 38.65 Al5052 0.076/9.52 1.55 1.66 7.386 Al5052 0.076/6.35 2.99 3.58 19.87 Al5052 0.076/4.76 4.62 5.39 16.7

0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

Al5052h/S=0.076/4.72

Al5052h/S=0.076/6.35

Al5052 h/S=0.076/9.52

Al3003 h/S=0.04/3.46

Al3003 h/S=0.06/3.46

Al3003 h/S=0.05/5.2

Al3003 h/S=0.06/4.33

Dyn

amic

enh

ance

men

t (%

)

Specimen Number

Experiment Calculation

Fig. 10. Summary of the calculated and experimental dynamic enhancement forboth 3003 and 5052 alloys.

h=152 mb=1.833mm

L=0.5mm

3

1

2

v

h=152 mb=1.833mm

L=0.5mm

3

1

2

v

Fig. 11. Scheme of double-plate model.

B. Hou et al. / International Journal of Solids and Structures 49 (2012) 2754–2762 2759

Fig. 10 depicts comparison of the simulated and experimentalenhancement ratios for all the 5052 and 3003 honeycombs.

The numerical models follow well the experiments. It showsthat the numerical enhancement is around 40% for Al3003 honey-combs and less than 20% for 5052 ones. Thus, this enhancementhas its origin in the structural response and this structural re-sponse should be related to the constitutive relationship of twomaterials because the geometry and loading conditions are similarin numerical models for 3003 and 5052 honeycombs.

2.4. Lateral inertia effect on the successive folding mechanism ofhoneycombs

2.4.1. Lateral inertia effect in a simple double-plate modelThe early theoretical work on the lateral inertia effect was re-

ported by Budiansky and Hutchinson (1964). Gary (1983) showedexperimentally that the buckling of a column under compressiveimpact occurs at a larger plastic strain. Calladine and English(1984) identified velocity-sensitive type II structure and Tam andCalladine (1991) revealed that there exists, under dynamic loading,an initial phase where the compression is dominant before a sec-ond phase of bending. More sophisticated models were also re-ported (Karagiozova and Jones, 1995; Su et al., 1995). It leads tothe fact that the buckling of an elastic-plastic column under com-pressive impact occurs at a larger strain (than under quasi-staticloading) because of necessary transverse acceleration. If the elas-tic-plastic column has a strain-hardening behavior, the bucklingpeak force will also increase as shown in the numerical work ofWebb et al. (2001).

In order to illustrate this lateral inertia effect without long ana-lytical formulas and to quantitatively evaluate the magnitude ofpotential augmentation of dynamic buckling force due to inertiaeffect in the honeycombs, a double-plate numerical model withdimensions comparable to honeycomb is built using ABAQUS.The scheme of this model is shown in Fig. 11 (like most previousworks cited above), which is composed of two plates connectedwith an angle (for the initial imperfection). The size of the modelis in the same order with honeycomb cell walls with the platethickness h = 152 lm, plate width b = 1.833 mm and height ofone plate L = 0.5 mm, d is the maximum deviation of plates fromthe vertical line, which represents the magnitude of initial imper-fection of this model.

In this study, the initial imperfection employed is fixed to3.2 lm (much more exaggerated in Fig. 11) in order to avoid theundesirable elastic buckle before the plastic collapse. The double-plate model is sandwiched between two parallel rigid walls (onefixed and another moving at prescribed velocity). The loadingvelocity is 0.1 mm/s for quasi-static case and 15 m/s for dynamicloading. A surface-to-surface rough contact is defined at the inter-faces of double-plate model and rigid walls to make sure that noslippage occurs. Such simulation permits to obtain the force anddisplacement time histories, which can be converted to nominalstress and strain by being divided by the plate cross sectional areaand the initial distance between rigid walls.

Fig. 12 shows the calculated quasi-static and impact nominalstress-strain curves for both 3003 and 5052 alloys, compared withthe prescribed constitutive relations. It can be found that the col-lapse point (at which the curve begin to decrease rapidly) of thequasi-static curve coincides with the yield point. However, underimpact loading, the collapse cannot take place at yield point and

-0.025 0.000 0.025 0.050 0.075 0.100 0.125 0.1500

50

100

150

200

250

300

350

400

450

Stre

ss (M

Pa)

Strain

5052 material 5052 dynamic 5052 static 3003 material 3003 dynamic 3003 static

Fig. 12. Force divided by cross section area vs. displacement divided by length.

Fig. 14. Formation of the second fold of honeycomb cell-model.

2760 B. Hou et al. / International Journal of Solids and Structures 49 (2012) 2754–2762

the double-plate model is further compressed. Therefore, the mod-el undergoes larger plastic strains in axial direction under impactloading before the collapse occurs. As the strain hardening curveis different, the stress increase due to this larger strain is differentand especially the ratio of this stress increase over the yield stressis different.

From constitutive relation shown in Fig. 7, for the 3003 alloy,with a yield stress of 70 MPa and a hardening modulus of1150 MPa, 5% of strain enhancement induces more than 80% stressincrease. For the 5052 alloy, with a yield stress of 290 MPa and ahardening modulus of 500 MPa, 5% of strain enhancement inducesonly less than 9% of rise in stress. This is considered to be the rea-son for the pressure enhancement difference between 5052 and3003 honeycombs.

2.5. Lateral inertia effect in the honeycomb cell-models

Such a lateral inertia effect under impact loading exists also inthe successive folding of tube-like hollow structures (Langsethand Hopperstad, 1996; Zhao and Abdennadher, 2004; Karagiozovaand Alves, 2008). It has been found that in the successive crushingprocess of square tube, the corner region (intersection of two flatplates) supports most of the external loadings and the buckling

(a)

-1 0 1 2 3 4 5 6 7-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Noo1 (Al3003) Dynamic Vimpa

Pres

sure

(MPa

)

Crush (mm)

cba

C

B

fourththird foldsecond foldfirst fold

A

Dd

Fig. 13. Deformation profile (a) and pressure/c

of this corner part determines the successive peak load. For thehoneycomb structure, this corner part corresponds to the regionnear the intersectional line. In our Y-shape cell-model (as shownin Fig. 13(b) the deformed profile), the analysis should be thenfocused on the central intersectional line of three cell walls.

Fig. 13(a) shows the pressure/crush curve of a cell-model forhoneycomb No. 1 (Al3003) taken as an example. Large fluctuationwith each wave representing one fold formation is observed.We consider the formation of the second fold to illustrate the

(b)

8 9

ct=27m/s

fold

Thin wall

Thick wall

rush curve (b) of honeycomb cell-model.

B. Hou et al. / International Journal of Solids and Structures 49 (2012) 2754–2762 2761

successive folding system of honeycomb under out-of-plane com-pression. The deformed profile of cell-model at point B in Fig. 13(a)is shown in Fig. 13(b). The deformation sequence of the basic cell-model is shown in Fig. 14 (only the thick wall is displayed for sakeof illustration’s clarity).

The formation of the second fold begins from point A inFig. 13(a). At this moment, the first fold has completely collapsedand the material of the second fold begins to support loading(Fig. 14(a)). The continuous axial deformation of the second foldenables the carrying capacity of the cell-model to increase gradu-ally (Segment a in Fig. 13(a) and the deformation image inFig. 14(b)). During this process, the intersectional line (as shownin Fig. 14(a)) and its adjacent region remains straight, while theplate region has been buckled. The peak load of the second foldis reached (Point B in Fig. 13(a)) when the intersectional line andits adjacent region begin to buckle (as show in Fig. 14(c)). After thispeak point, the overall carrying capacity decreases dramatically(segment b in Fig. 13(a)) and the corresponding deformation ofthe cell-model is characterized with the bending of intersection re-gion (as shown in Fig. 14(d)). When the carrying capacity of thecell-model reaches the trough C in Fig. 13(a), the third fold initiates

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-0.04

0.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28

Noo1 (Al3003)

Equi

vale

nt s

train

Distance from the intersection line (mm)

Quasi-static Dynamic

Half width of thick wall Half width of thin wall

Fig. 15a. Strain distribution in the central line of a fold under quasi-static anddynamic loading, honeycomb No. 1 (Al3003).

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0.12

Noo6 (Al5052)

Equi

vale

nt s

train

Distance from the intersection line (mm)

Quasi-static Dynamic

Thick wall Thin wall

Fig. 15b. Strain distribution in the central line of a fold under quasi-static anddynamic loading, honeycomb No. 6 (Al5052).

and will repeat the above-mentioned process, i.e. the C-c-D-d pro-cess in Fig. 13(a).

The successive folding is controlled by the successive bucklingof region near the central intersectional line. It is important to no-tice that the intersectional line remain rather straight (Fig. 14(c))before its buckling. Thus, the initial imperfection is small enoughfor each single successive fold so that the lateral inertia effect likein the double-plate model will apply.

Figs. 15a and 15b depicts the equivalent strain profiles (stillconsidering that the strain here is the calculated wall materialequivalent strain) along the central line of one fold (seeFig. 14(d)) in both the double thickness wall and the single thick-ness wall just before the buckling (at the successive force peaks,points B, D in Fig. 13(a)). For honeycomb No. 1 (Al3003) model(Fig. 15a) as well as No. 6 (Al5052) model (Fig. 15b), one can seethat the strain reached before buckling is higher under impactloading (27 m/s for Al3003 and 15 m/s for Al5052) than the caseunder quasi-static loading. The closer is the position to the inter-sectional line; the larger is the increase in strain. The lateral inertiaeffect is then clearly seen in the successive buckling of the centralintersectional line of the Y-shape cell-model.

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50

50

100

150

200

250

300

350

Noo1 (Al3003)

Mis

es s

tress

(MPa

)

Distance from the intersection line (mm)

Quasi-static Dynamic

Fig. 16a. Stress distribution in the central line of a fold under quasi-static anddynamic loading, honeycomb No. 1 (Al3003).

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50

50

100

150

200

250

300

350

400

Noo6 (Al5052)

Mis

es s

tres

s (M

Pa)

Distance from the intersection line (mm)

Quasi-static Dynamic

Fig. 16b. Stress distribution in the central line of a fold under quasi-static anddynamic loading, honeycomb No. 6 (Al5052).

2762 B. Hou et al. / International Journal of Solids and Structures 49 (2012) 2754–2762

The stress profiles just before the buckling are also shown inFigs. 16a and 16b. On the one hand, there is a larger difference ofstress levels between impact and quasi-static loadings for 3003model which is due to the important strain hardening behaviorof prescribed 3003 constitutive law, especially the ratio betweenthe stress increase due to strain hardening and the yield stress.On the other hand, because of prescribed flat strain hardeningbehavior for 5052 model, the stress level differences for 5052 areless significant even though a noticeable strain difference due tolateral inertia is observed.

3. Conclusion

Quasi-static and impact tests with large diameter polymer SHPBwere performed on 7 kinds of aluminium (5052 or 3003) honey-comb with a relative density varying from 1.78% to 4.72%. The re-sults show that the enhancement under impact loading of thecrushing pressure depends on the base material. In fact, 4 different3003 honeycombs have an enhancement larger than 40% while the3 types of 5052 honeycombs exhibit an enhancement less than20%.

Abaqus models using a Y-shape cell-model with rate insensitiveconstitutive law were performed to simulate the studied sets ofhoneycombs under quasi-static and impact loadings. This resultconfirmed numerically this difference of enhancement.

Finally, strain and stress profiles in single Y-shape cell duringthe successive folding (especially at the force peaks) were ana-lysed. The strain reached before collapse (successive force peak)under impact loading is bigger than that under quasi-static load-ing. This is considered to be due to the lateral inertia effect of typeII structure, well studied in the past. As the 3003 alloy has a biggerratio between the stress increase due to strain hardening and theyield stress than that of 5052 alloy, the crushing pressure enhance-ment under impact loading of 3003 honeycombs is therefore big-ger than 5052 honeycombs.

Acknowledgement

The authors would like to thank 111 project of China (ContractNo.1307050) for funding the cooperation between NPU and LMT. B.Hou and Y. L. Li would also like to thank the supports of the Na-tional Science Foundation of China (Contract Nos. 10932008 and11072202).

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