InternationalJournalofImpactEngineering28(2003)161182In-planedynamiccrushingofhoneycombsaniteelementstudyD.Ruana,G.Lua,*,B.Wangb,T.X.YucaSchoolofEngineeringandScience,SwinburneUniversityofTechnology,Hawthorn,Victoria3122,AustraliabDepartmentofMechanicalEngineering,BrunelUniversity,Uxbridge,MiddlesexUB83PH,UKcDepartment
of Mechanical Engineering, Hong Kong University of Science &
Technology, Clear Water Bay, Hong
KongReceived25October2001;receivedinrevisedform24June2002AbstractThe
in-plane dynamic behaviour of hexagonal aluminium honeycombs has
been studied in this paper
bymeansofniteelementsimulationusingABAQUS.Theinuencesofhoneycombcellwallthicknessandimpact
velocity on the mode of localised deformation and the plateau
stress are investigated. Deformationmodes inthe X1directionchange
signicantlywithdifferent values of cell wall thickness
andimpactvelocity. Threedeformationmodesareobserved.
Theyaresummarisedinamodeclassicationmapandtwoempirical formulae are
obtainedfor critical velocities at whichdeformationmode changes.
Twodeformationmodes are observedinthe X2direction. The
plateaustresses are relatedtothe cell wallthickness by a power law
for a given velocity and they are proportional to the square of
velocity for a highvelocity. An empirical formula for plateau
stress at high impact velocities is derived in terms of the cell
wallthicknessandvelocity.r
2002ElsevierScienceLtd.Allrightsreserved.1. IntroductionHoneycombis
atype of cellular material withatwo-dimensional arrayof hexagonal
cells(Fig. 1a). It is used in a variety of applications:
frompolymer and metal honeycombs
foradvancedaerospacecomponentstoceramichoneycombsforhigh-temperatureprocessing[1].Muchtheoretical
andexperimental workhas beencarriedout onhoneycombs under
staticloading [1,2].One particularfeatureof honeycombsis softening
withineach celland localisationofdeformation(Fig.
1b).Inrecentyears,theniteelementmethodhasbeenusedtoanalysethestaticcrushingbehaviourofhoneycombs.NumericalsimulationsbyPapkaandKyriakides[3,4]*Correspondingauthor.Fax:+61-3-9214-8264.E-mailaddress:[email protected](G.Lu).0734-743X/02/$
- seefrontmatter r 2002ElsevierScienceLtd.Allrightsreserved.PII:S 0
7 3 4 - 7 4 3 X( 0 2 ) 0 0 0 5 6 - 8provide a good insight into the
crushing process of honeycombs. Silva and Gibson [5] studied
theeffect of randomly removed individual cell edges on the Youngs
modulus and the plateau stress inregular hexagonal honeycombs.
GuoandGibson[6] investigatedthebehaviour of intact
anddamagedhoneycombsusingABAQUS. Theyanalysedtheeffect of
isolateddefectsof varioussizes and that of the separation distance
between two defects on the elastic and plastic behaviourof
honeycomb. It was impossible to conduct such investigations
basedmainly ona
micro-mechanicalanalysisofasinglecell.Honeycombsbehavedifferentlyunderdynamicloading,
mainlyintheirdeformationmodesand stress levels. Stronge et al. [79]
proposed one-dimensional models to account forthe strain-softening
within each cell and strain rate effect. Dynamic tests were
performed by Baker et al. [10],andWuandJiang[11]
byusingagasguntostudytheout-of-planepropertiesof
aluminiumhoneycombs. ZhaoandGary[12]
studiedthein-planeandout-of-planecrushingbehaviourofaluminiumhoneycombs
using the split Hopkinsonpressure bar (SHPB). However, in
thesestudies,onlyafter-testspecimenswereobservedandthewholecrushingprocessofthecellsandspecimens
was not studied. No theoretical model has been proposed for
two-dimensionalspecimens.NomenclatureA aparameter(Eq. (11))B
aparameter(Eq. (11))h honeycombcellwallthickness(Fig.
1)L0originallengthofhoneycombsampleintheX1directionl
lengthofhoneycombcelledge(Fig. 1)p exponent(Eq. (11))s
honeycombcellsize(Fig. 1)t timeX1horizontalloadingdirection(Fig.
2)X2verticalloadingdirection(Fig. 2)Greeklettersd
displacementedlocking(ordensication)strainofhoneycombv
impactvelocityvc1criticalvelocityatwhichdeformationmodechangesfromXmodetoVmodevc2criticalvelocityatwhichdeformationmodechangesfromVmodetoImoder
densityofhoneycombrsdensityofhoneycombcellwallmaterials
plateaustressofhoneycombs0staticplateaustressofhoneycombsysyieldstressofhoneycombcellwallmaterialD.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
162Inthispaper,byusingtheniteelementmethod,westudythein-plane(X1andX2direction)deformationofhexagonal
aluminiumhoneycombsunderimpactloading. Theinuencesofcellwall
thickness and impact velocity on the deformation mode and plateau
stress are discussed.
Theresultsaresummarisedwithamodeclassicationmapandempirical
formulaefortheplateaustress.2. FiniteelementmodelTheFEmodel
usedisshowninFig. 2. It
has16cellsintheX1directionand15intheX2direction. The cell size is s
4:7 mm and l 2:7 mm (corresponding to b 1201) for all cases andthe
cell wall thickness h varies from 0.08 to 0.5 mm. ABAQUS/EXPLICIT
[14] was employed forthedynamicanalysis. Thecell wall material
wasassumedtobeelastic, perfectlyplasticwithaYoungs modulus of 69
GPa and a yield stress of 76 MPa. Each edge of the cell wall was
modelledwith threeshell elements (type S4R). The model consisted of
a total of2280 shell
elements.Eachhexagonalcellwasdenedasasingleself-contactsurface.Self-contactwasalsodenedbetweenthe
outside faces of the cell which might contact other cells during
crushing. Through aFig. 1.
Aluminiumhoneycomb:(a)Undeformedhoneycomb;(b)Deformedhoneycombinaquasi-staticcompressiontest.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
163convergencestudy, it wasdeterminedthat
veintegrationpointsthroughthethicknessof
theelementsprovidedsufcientlyaccurateresults.A constant velocity
was applied to a rigidplate, which crushed the honeycomb. The
velocityvwas variedfrom3.5to280 m/sinorder tostudy theeffectofthe
loadingrate.When crushinginthe X1 direction, all degrees of freedom
of the left edge of the specimen were xed and the top andbottom
edges were free. A horizontal constant velocity (X1 direction) was
applied to the right faceof theplate. Similarly, forimpact
inthevertical (X2) direction, all degreesof freedomof
thebottomedgewerexedandtheleftandrightedgeswerefree. Avertical
constantvelocitywasappliedtothetopfaceofthestrikingrigidplate.3.
Resultsandanalysis3.1. Deformationmodes3.1.1.
DeformationmodesintheX1directionFigs. 35 show deformation modes in
theX1direction under various impact velocities for caseh 0:2 mm.
For Fig. 3, v 3:5 m/s. Initial localisation occurs when the
displacement of the rightedge is small (Fig. 3a) and this produces
an X shaped band starting from the stricken end.
Withtheincreaseofthedisplacement, asecondlocalisedX
bandisdevelopedfromthexededgeFig. 2.
FEmodelandarigidimpactplateusedinthesimulation.Thevelocityisconstantduringcrushing.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
164anditintersectswiththerstlocalisationbandtoformarhombusatthecentreofthespecimen(Fig.
3b and c). As the crushing proceeds, more localised bands occur
with one more layer of cellscrushedalongtheX bands (Fig. 3d). After
that, localisationtakes placewithinthecentralrhombus (Fig. 3e and
f). Finally, when deformation within the rhombus exhausts, more
localisedFig. 3. Crushingof ahoneycombinthe X1direction, h 0:2 mm,
v=3.5 m/s. Deformations arelocalisedinitiallywithin an X shaped
band. (a) d 5:6 mm (t 1:6 ms); (b) d 11:9 mm (t 3:4 ms); (c) d 18:2
mm (t 5:2 ms); (d)d 24:5 mm (t 7:0 ms); (e) d 34:0 mm (t 9:7 ms);
(f) d 40:3 mm (t 11:5 ms); (g) d 46:6 mm (t 13:3 ms); (h)d 56:0
mm(t 16:0
ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
165bands occur near the loading edge (Fig. 3g) until the honeycomb
is completely crushed(Fig. 3h). The X shaped localisation band was
also observed in our quasi-static experiments, seeFig. 1(b).At a
higher impact velocity of 14 m/s (Fig. 4), a number of cells near
the right edge of the modelisslightlycrushedwithinaV
shapedblockandnoobviouslocaliseddeformationbandareobservedat
thebeginningof deformation(Fig. 4a). Afterwards,
alocaliseddeformationbandFig. 4. Same conditions as Fig. 3, except
v=14m/s: (a) d 5:6 mm(t 0:4 ms); (b) d 11:9 mm(t 0:85 ms); (c)d
18:2 mm(t 1:3 ms);(d) d 24:5 mm(t 1:75 ms);(e) d 34:0 mm(t 2:43
ms);(f) d 40:3 mm(t 2:88 ms);(g) d 46:6 mm(t 3:33 ms);(h) d 56:0
mm(t 4:0
ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
166occursneartheloadingedge(Fig. 4b). ThebandisoftheshapeofX
butisslightlyslimmerthanthat underalowerspeedimpact (asinFig. 3).
Withfurtherdeformation, morelocalisedbands develop progressively
(Figs. 4ce). When the displacement increases to about 40 mm, a
newobliquelocalisedbandoccurs,fromthexededgeandgrowstowardsalongtheexistingonesFig.
5. SameconditionsasFig. 3, except v=70m/s: (a)d 5:6 mm(t 0:08 ms);
(b)d 11:9 mm(t 0:17 ms); (c)d 18:2 mm (t 0:26 ms); (d) d 24:5 mm (t
0:35 ms); (e) d 34:0 mm (t 0:49 ms); (f) d 40:3 mm (t 0:58 ms);(g)
d 46:6 mm(t 0:67 ms);(h) d 56:0 mm(t 0:8
ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
167which seem to have stopped developing (Figs. 4fh). Then the
bands interact with each other
andthesampleistotallycrushed.NoobviouslocalisedX orV
shapedbandhasbeenfoundthroughthewholecrushingprocess whenthe impact
velocityis higher, for example, at 70 m/s (Fig. 5).
Onlyalocalisedtransversebandperpendiculartotheimpactisobservedattheloadingedgeanditcontinuestopropagate,
layer bylayer, tothexededge. This deformationis inamanner of plane
wavepropagation[13].Figs. 6 and7 showthe deformationmodes for cell
wall thicknesses of 0.5 and0.08 mm,respectively. Impact
velocitiesarethesameasinFig. 4(14 m/s). ThelocalisedbandshowninFig.
6(a) is almost the same as that obtained in our experiment (Fig.
1b) and by Guo and Gibson[6] for the static case. Fig. 6(a) shows a
localised band starting fromthe impact edge forh 0:5
mm.However,forthesamevalueofdeformation,noobviouslocalisedbandisobservedforh
0:2 mminFig. 4(a). InFig. 6(b),
thecellsinthelocalisedbandarecrushedmorethanthoseinFig. 4(b). Fig.
6(e)showsthatthelocalisedbandfromthexededgeoccurswhenthecompresseddisplacementoftherightedgeis34
mm,whileforh 0:2 mm,thebandoccursatadisplacement of about 40 mm, as
inFig. 4(f). All these indicate that increasingthe cell
wallthicknessof thehoneycombhasthesameeffect asdecreasingtheimpact
velocityintermsofdeformationmode. Further evidence for this is that
for thinwall cells nolocalisedbandisdevelopedfromthexededge(Fig.
7),
withthedeformationmodebeingsimilartothecaseofthickerwallcells(Fig.
5).3.1.2. ModeclassicationmapThe above deformation characteristics
may be summarised as follows. Oblique localised bandsare present
when the impact velocity is low (3.5 and 14 m/s). The lower the
velocity, the wider theX shaped band (Figs. 3 and 4). With the
increase of the impact velocity, localised bands tend tobe
transverse (Fig. 5); also a localised band starting from the xed
edge occurs later in the process.For example, for an impact
velocity of v 3:5 m/s, the localised band which starts from the
xededge is observed when crush deformation reaches d 11:9 mm (t 3:4
ms) (Fig. 3b). However forv 14 m/s, it canonlybeobservedat d 40:3
mm(t 2:88 ms) (Fig. 4f).
Nolocalisedbandappearsfromthexededgewhenthevelocityishighenough(Fig.
5).Atthisstage,wemaybroadlyclassifyalltheobserveddeformationmodesintheX1directioninto
three types. Type one is of X shaped deformation mode, as shown in
Fig. 3(a). The featureof this modeis that X
shapedlocalisedbandscanbeobservedclearlywhenhoneycombiscrushedbyadeformationassmallas5.6
mm(d=L0 7%).Type three is I mode, as shown in Fig. 5. In this mode,
there is no obvious oblique localisedband through the whole
crushing process and only vertical bands normal to the loading
directionarefound. Typetwoisatransitional modebetweentheX andI
modes, suchasshowninFig.
4(a),andisnamedtheVmode.Thelocalisedbandsareoblique,buttheydonotformacompleteX
shapewhenahoneycombiscrushedbyadisplacementof5.6 mm(d=L0 7%).Fig. 8
is a sketch of these three modes. The I and X modes are distinctive
and the V
modeislessso.Basedonthisconvention,weplot,logarithmically,thedeformationmodesintheX1directionof
honeycombswithdifferent cell wall thicknessinFig. 9.
Deformationmodesareall of Xmode at low velocities and I mode at
high velocities. The critical velocity at which
deformationD.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
168mode switches from one type to another depends on the cell wall
thickness. For cell wall thicknessbetween 0.08 and 0.4 mm, the
deformation is of X mode when the velocity is lower than 7
m/s.However, for cell wall thickness of 0.5 mm, X mode occurs when
velocities are as high as 14 m/s.Fig. 6. Same conditions as Fig. 3,
except h 0:5 mm, v 14 m/s: (a) d 5:6 mm(t 0:4 ms); (b) d 11:9 mm(t
0:85 ms); (c) d 18:2 mm (t 1:3 ms); (d) d 24:5 mm (t 1:75 ms); (e)
d 34:0 mm (t 2:43 ms); (f)d 40:3 mm(t 2:88 ms);(g) d 46:6 mm(t 3:33
ms);(h) d 56:0 mm(t 4:0
ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
169The lowest velocities for an I mode are 70 m/s for honeycombs
with a cell wall thickness of 0.1and 0.2 mm, 100 m/s for a cell
wall thickness of 0.3 mm, and 140 m/s for a cell wall thickness of
0.4and0.5 mm,respectively.Fig. 7. Same conditions as Fig. 3, except
h 0:08 mm, v 14 m/s: (a) d 5:6 mm(t 0:4 ms); (b) d 11:9 mm(t 0:85
ms); (c) d 18:2 mm (t 1:3 ms); (d) d 24:5 mm (t 1:75 ms); (e) d
34:0 mm (t 2:43 ms); (f)d 40:3 mm(t 2:88 ms);(g) d 46:6 m(t 3:33
ms);(h) d 56:0 mm(t 4:0
ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
170From a dimensional analysis, a nondimensional critical velocity
v=s=r1=2is often adopted fordynamic response of solids. We follow a
similar argument for the whole honeycomb specimen andhere that this
dimensional group is only dependent upon another group, h=l;
assuming the size ofa honeycombspecimenis immaterial. But since
sph=l2andrph=l [1], we wouldexpectvph=lp; which corresponds to a
straight line in Fig. 9. From Fig. 9 vc1is almost independent
ofh=l;butvc2ph=lp:Theempiricalequationsforthetwocriticalvelocitiesare,respectively,vc1
14 m=s 1andvc2 277h=lpm=s: 2Fig. 8.
Sketchofthethreetypesofthedeformationmodes.(m/s)lh
c220.03"I" mode"V" mode(transitional mode)"X" mode
c1 0.2
(m/s)h/l0.110100= 277= 14 (m/s)Fig. 9. Deformation mode map in
terms ofh=land v (() denotes deformation mode type oneX mode, with
Xshaped localised bands, (D denotes deformation mode type threeI
mode, with the localised bands beingperpendicular
totheloadingdirection, (J) denotes deformationmodetypetwoV mode,
whichis
in-betweenmodesoneandthree).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
1713.1.3. DeformationmodesintheX2directionThe deformation modes in
the X2 direction for honeycombs with a cell wall thickness of 0.2
mmare shown in Figs. 10 and 11. The impact velocities are 7 and 70
m/s, respectively. When v 7 m/s,lightlycrushedcellsformanarch at
thelowerpart of themodel (Figs. 10aandb). Thisissimilar to the V
mode for the X1 direction (Because the cells that form this V
shaped band areless severely crushed than those in the X1direction,
the V mode observed in the X2direction isless visible than that in
the X1direction.). Localised bands are composed of heavily crushed
cells,perpendicular to the loading direction, at the upper middle
position of the model (Fig. 10b). ThisissimilartotheI
modeintheX1direction. However,
whentheimpactvelocityisrelativelyhigh, say 70 m/s, the upper
loading edge is crushed rst and it forms a localised band
horizontally(Fig. 11). When v 70 m/s, there is no V mode and only I
mode is observed. As the
crushingcontinues,morelocalisedbandsareformedlayerbylayer,asshowninbothFigs.
10(c)(f)andFigs. 11(b)(f).Fig. 10. Crushing of a honeycomb in the
X2 direction, h 0:2 mm, v 7 m/s: d 5:6 mm (t 0:8 ms); (b) d 11:9
mm(t 1:7 ms);(c) d 18:2 mm(t 2:6 ms);(d) d 34:0 mm(t 4:9 ms);(e) d
40:3 mm(t 5:8 ms);(f) d 46:6 mm(t 6:7
ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
172Fig. 12 shows the deformation of a honeycomb with a cell wall
thickness of 0.5 mm. The impactvelocity is 7 m/s. The V mode can be
observed rst (Fig. 12a). Then heavily crushed cells
formhorizontallylocatedbands(Fig.
12bf).However,therstnoticeablelocalisedbandoccursataposition that
is one cell-layer lower than that in Fig. 10. This indicates that
the position of the rstlocalised band occurs further from the
loading edge with the increasing of the cell wall
thicknessearlierlayersarestrongerbecauseoftheirlateralinertia,whichisprobablythesameastheeffectoflateralinertiainstrutproblem.3.2.
Plateaustress3.2.1. X1directionIn the simulations, the force
between the honeycomb and the striking rigid plate are calculated.A
typical force-displacement curve is shown in Fig. 13 for v 14 m/s
and h 0:2 mm, and Fig. 14demonstrates the effect of the impact
velocity. In general, there are initial, and for high
velocities,subsequent peaks. However, because our main interest is
in the plateau stress, which is importantFig. 11.
SameconditionsasFig. 10, exceptv 70 m/s: (a) d 5:6 mm(t 0:8 ms);
(b) d 11:9 mm(t 1:7 ms); (c)d 18:2 mm(t 2:6 ms);(d) d 34:0 mm(t 4:9
ms);(e) d 40:3 mm(t 0:58 ms);(f) d 46:6 mm(t 0:67
ms).D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
173forenergyabsorption, andthevaluesof
thepeakpointsareheavilyinuencedbythecut-offfrequencyofawavelterintheFEcalculation,
thesepeakforcesarenotdiscussedhere.
Therelationshipbetweentheforceandthedisplacementforothercell wall
thicknessissimilar,
butwithdifferentvaluesforpeakandplateauforces.The values of the
dynamic plateau stresses, which are calculated as the ratio of the
plateau forcetoloadingsectional area, arelistedinTable1.
Theoretical staticplateaustressesarecalculatedFig. 12. Same
conditions as Fig. 10, except h 0:5 mm, v 7 m/s: (a) d 5:6 mm(t 0:8
ms); (b) d 11:9 mm(t 1:7 ms);(c) d 18:2 mm(t 2:6 ms);(d) d 34:0
mm(t 4:9 ms).0 10 20 30 40 50 60050100150200Initial peak force
Plateau forceForce (N)Displacement (mm)Fig. 13.
Dynamicforce-displacementcurveofahoneycombintheX1direction,h 0:2
mm,v 14
m/s.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
1740 10 20 30 40 50 600255075100=35m/s=14m/s=7m/s=3.5m/sTheoretical
static forceForce (N)Displacement (mm)Fig. 14.
Dynamicforce-displacement curveintheX1directionof
honeycombswiththesamecell wall thicknessof0.2
mmatdifferentimpactvelocities.Table1Plateaustressforhoneycombsunderimpactloadingv(m/s)
h(mm)0.08 0.2 0.3 0.4 0.5s0a(MPa) Static 0.0506 0.317 0.712 1.266
1.978Calculateddynamicplateaustress s(MPa)X1direction 7 0.0513
0.332 0.770 1.349 2.25214 0.061 0.365 0.829 1.478 2.38535 0.134
0.550 1.106 1.899 3.13470 0.556 1.600 2.491 3.777 5.772100 1.061
2.966 4.677 6.802 10.268140 1.959 5.378 8.784 12.500 16.983200
3.731 10.286 16.700 23.700 31.698280 7.546 20.492 33.500 47.009
60.403X2direction 14 0.069 0.453 1.009 1.775 2.58635 0.193 0.771
1.636 70 0.501 1.549 2.960 100 0.953 3.014 4.734 140 1.815 5.260
8.575 ValuesofparameterAinEq. (3) 96.7 255.3 405.5 560.5
736.5aObtainedfromEq.
(4.26b)inRef.[1]multipliedbythefactorof1.15.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
175followingGibsonandAshby[1], whichiss0 23sysh=l2:
(Wemultiplytheir formulafor
theplateaustressbyafactorof1.15toaccountfortheplanestrainconditionofthecellwalls.)Forthe
same cell wall thickness, dynamic plateau stresses are higher than
the theoretical static values.When the impact velocity is
sufciently high, the deformation mode is a propagation of a
planeplasticwave.Globally,thestaticstressstraincurvesofhoneycombsexhibitahardeningwithalocking
strain, though at a micro-level for each cell, there is a
softening. This suggests that a shockwave
theorymaybeapplicableinthiscase.Thesimplestformforthedynamicstressisthen[13]s
s0rv2ed s0Av2; 3where s is the dynamic plateau stress, s0is the
static plateau stress as listed in Table 1 for varioushoneycombs, r
is the density of the honeycomb, v is the impact velocity, edis the
locking strain ofhoneycombsunderstaticloading,
andAisaparameterwhichequalstor=ed:
Thedensitiesofthesehoneycombscanbecalculatedbyusingthefollowingequation[1]:r
rs23phl:
4Thecalculatedvaluesofdensityforhoneycombswithcellwallthicknessof0.08,0.2,0.3,0.4and
0.5 mm are 91.9, 229.8, 344.7, 459.6 and 574.5 kg/m3, respectively.
The static locking strain ofhoneycombsdecreaseswithcell wall
thickness. However, itsactual valueisnot available. Weassume that
the static locking strain for honeycombs with wall thickness of
0.08, 0.2, 0.3, 0.4 and0.5 mmare0.95,0.9,0.85,0.82and0.78,sothatEq.
(3)besttstheFEresultsintheregionofhigh velocity. The values of
parameter A ( r=ed) in Eq. (3) for h 0:08; 0.2, 0.3, 0.4 and 0.5
mmarethencalculatedandlistedinTable1.Figs. 15(a)(e) show the change
of plateau stress s s0 with impact velocity, where
calculateddynamicplateaustressesfromFEareplottedasdiamondsymbols.Forallthevaluesofthecellwall
thickness, theplateaustressincreaseswithimpactvelocity.
Whenthevelocitiesarehigherthanacertainvalue,plateaustresses s
s0showagoodcorrelationtovelocitiesbyasquarelaw, corresponding to a
slope of 2 in the double logarithmic plot. The slope is lower than
2, whenhisrelativesmall, say0.08 mm, andlargerthan2,
whenhisrelativelarge, 0.30.5
mminourcalculation.ThevalueofparameterAinEq.
(3)isrelatedtothecellwallthicknessorh=l;fromtheleast-squarettingasA
4742hl 23115hl 0:75 5andisplottedinFig.
16.Thestaticplateaustressforvarioush=l canbedescribedby[1]s0 1:15
23syshl 2: 6ThusEq. (3)canbere-writtenass 0:8syshl 24742hl 23115hl
0:75"
#v27D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
1761 10 1001E-30.010.1110 FE data-0=96.72-0 (MPa) (m/s)1 10
1000.010.1110FE data -0=255.32-0 (MPa) (m/s)1 10 1000.010.1110FE
data-0=405.52-0 (MPa) (m/s)1 10 1000.1110FE data
(m/s)-0=560.52(a)(b)(d) (c)-0 (MPa)1 10 1000.1110FE-0=736.52 -0
(MPa) (m/s) (e)Fig. 15.
ChangeofplateaustressesintheX1directionwiththeimpactvelocity:(a)h
0:08 mm;(b)h 0:2 mm;(c)h 0:3 mm;(d)h 0:4 mm;(e)h 0:5
mm.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
177or,notingthat sys 76 MPainourFEcalculationssys 0:8hl 262hl 241hl
0:01"
#106v28ThisistheequationforthedynamicplateaustresswhenthevelocityishighforanImode.Forexample,whenv
100 m/sssys 1:39hl 20:41hl 0:0001: 9Forh=l oftherangeof0.030.18,Eq.
(9)canalsobeapproximatedbyapowerlawssys 0:9hl 1:2: 10Equations for
some higher velocities are listed in Table 2, and the power laws
have an exponentsmallerthan2.Fig.
17showstherelationshipbetweentheplateaustressandthecellwallthicknesswhentheimpact
velocityis kept constant. Withinthe impact velocityrange of 7280
m/s, the
plateaustressesshowageneralgoodcorrelationtothecellwallthicknessortheratioofh=l
byapowerlaw,whichisssys Bhl p; 11Here, B is a constant and p is an
exponent. Both B and p depend on the impact velocity, as shownin
Fig. 17. p is equals to 2 at low velocities, which is the same as
static cases. At high velocities, p isless than 2. Both Band p
obtained here fromcurve tting equal or approximate to the0.02 0.04
0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.200100200300400500600700800
values of parameter A A=4742(h/l)2+3115(h/l)+0.75Ah/lFig. 16.
ParameterAinEq. (3)versush=l
withthebestttedcurve.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
178correspondingvaluesinTable2. Inthiscase, Eq. (8)
canbeusedasanempirical
formulafordynamicplateaustressathighimpactvelocity.3.2.2.
X2directionThe force-displacement curve in the X2 direction is
similar to that in the X1 direction, but differsin peak forces and
plateau forces. The plateau stresses in the X2direction are given
in Table 1 and0.01 0.1 11E-41E-30.010.1=280m/s
/ysh/l/ys=5.4(h/l)1.1/ys=2.9(h/l)1.2/ys=1.5(h/l)1.2/ys=0.9(h/l)1.2/ys=0.6(h/l)1.2/ys=0.7(h/l)1.7/ys=0.9(h/l)2.0/ys=0.9(h/l)2.1=200m/s=140m/s=100m/s=70m/s=35m/s=14m/s=7m/sFig.
17. Plotof
s=sysintheX1directionversush=l:Table2ExpressionsofplateaustressderivedfromEq.
(8)forhighvelocitiesVelocity(n)(m/s) sintheformofEq. (8)
sintheformofpowerlaw140ssys 2:0hl 20:8hl 0:0002ssys 1:5hl
1:2200ssys 3:28hl 21:64hl 0:0004ssys 2:7hl 1:1280ssys 5:66hl
23:21hl 0:0008ssys 5:4hl
1:2D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
179Figs. 18(a)(c). Eq. (3) is assumed to be able to describe the
plateau stress in the X2 direction whenimpact velocity is high
enough. The values of density r and locking strain edare the same
for thehoneycomb in both X1and X2directions when the cell wall
thickness is the same. Forh 0:08 mm (h=l 0:03) shown in Fig. 18(a),
FE results are on a straight line when the velocity
isbetween14and140 m/s. Inthiscase,
itcanbesaidthattheplateaustressesarerelatedtotheimpactvelocitybyasquarepowerlaw.InFigs.
18(b)and(c),circlesymbolsareonthestraightlineswhenvelocitiesarelargerthan70
m/s, forhoneycombwithcell wall thicknessof 0.2 mm(h=l 0:07) and 0.3
mm(h=l 0:11), respectively. This indicates againthat plateau
stressesincreasewiththeimpact
velocityfollowingasquarelawwhenthevelocitiesarelargerthanacriticalvelocity,andatarateslowerthanasquarelawwhenthevelocitiesaresmallerthanthiscriticalvelocity.Similarly,
when the impact velocity is constant, plateau stresses in the X2
direction also increasewith the cell wall thickness by power laws,
as shown in Fig. 19. The exponent is 2 at low
velocities,whichisthesameasthestaticcases.Itislessthan2athighvelocities.Bandparethesamefor10
1000.010.11FE data (m/s)10 1000.1110FE data (m/s)10 1000.1110FE
data (m/s)(a) (b)(c)_0=96.7v2-0 (MPa)_0=255.32 -0 (MPa)_0=405.52-0
(MPa)Fig. 18.
ChangeofplateaustressesintheX2directionwiththeimpactvelocity:(a)h
0:08 mm;(b)h 0:2 mm;(c)h 0:3
mm.D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
180loadinginbothX1andX2directionsatv 100and140 m/s.Eq.
(8)isalsoapplicableforhigh-speedimpactintheX2direction.4.
ConclusionsBothdeformationmodesandplateaustressesofhoneycombsunderin-planedynamicloadingarerelatedtothecellwallthicknessandimpactvelocity.Three
different types of deformation modes are observed when the
honeycomb is loaded in theX1direction.
LocaliseddeformationcausesobliqueX shapedbands(X mode) at
thelowimpactvelocity, butproducesvertical localisedbands(I
mode)perpendiculartotheloadingdirection at high impact velocity.
Atransitional V mode is present when the velocity ismoderate. The
localised bands occur closer to the loading edge when the impact
velocity increases.Reducingthethickness of thecell wall has
thesameeffect as increasingtheimpact
velocity,leadingtoformationoflocalisedbandsclosertotheloadingedge.V
modeisalsoobservedintheX2directionwhencrushingatlowvelocity.
Thecellsthatcomposethe Vshaped bandsare
lessheavilycrushedthanthoseintheX1direction.OnlyImodeisobservedathighimpactvelocity.ThisImodelocalisedbandcontinuestogrowintheadjacentcelllayers.ThepositionoftherstlocalisedbandintheImodedifferswiththeimpactvelocityandcell
wall thickness.
Thehighertheimpactvelocityandthethinnerthewallthickness,thenearertherstlocalisedbandoftheImodetotheloadingedge.Arelationshiphasbeenestablishedbetweentheplateaustresses,
thecell wall thicknessandvelocity (Eq. (8)). Plateau stresses show
a good correlation to the cell wall thickness or the ratio
of1E-30.010.1/ys0.01
0.1h/l/ys=1.5(h/l)1.2=140m/s/ys=0.9(h/l)1.2/ys=0.7(h/l)1.3/ys=0.7(h/l)1.6/ys=1.3(h/l)2.0=100m/s=70m/s=35m/s=14m/sFig.
19. Plotof
s=sysintheX2directionversush=l:D.Ruanetal./InternationalJournalofImpactEngineering28(2003)161182
181h=l by a power law for a given impact velocity. The exponent
equals to 2 at low velocities, which
isthesameasthestaticcase.However,athighvelocities,theexponentislessthan2.The
plateau stresses in both X1and X2directions increase with the
impact velocity by a
squarelawwhenthevelocityissufcientlyhigh,whichisinagreementwiththeReidPengequation.AcknowledgementsThe
authors wish to thank the Australia Research Council for the
nancial support and WorleyFEAfor an academic license of ABAQUS. The
fourth author (T.X. Yu) would like toacknowledge the support
fromthe HongKongResearchGrant Council (RGC) under
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