Post-bifurcation and stability of a finitely strained hexagonal honeycomb subjected to equi-biaxial in-plane loading

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Post-bifurcation and stability of a finitely strained hexagonal honeycomb subjected to equi-biaxial in-plane

loading

Christelle Combescure, Pierre Henry, Ryan Elliott

To cite this version:

Christelle Combescure, Pierre Henry, Ryan Elliott. Post-bifurcation and stability of a finitely strained

hexagonal honeycomb subjected to equi-biaxial in-plane loading. International Journal of Solids and

Structures, Elsevier, 2016, 8889 (24), pp.296 - 318. �10.1016/j.ijsolstr.2016.02.016�. �hal-01647871�

ContentslistsavailableatScienceDirect

International Journal of Solids and Structures

journalhomepage:www.elsevier.com/locate/ijsolstr

Post-bifurcation and stability of a finitely strained hexagonal honeycomb subjected to equi-biaxial in-plane loading

Christelle Combescure

^a,b

, Pierre Henry

^a,b

, Ryan S. Elliott

^a,∗

aDepartment of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA

bLaboratoire de Mecanique des Solides (CNRS UMR 7649), Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, France

a rt i c l e i nf o

Article history:

Received 27 October 2015 Revised 9 December 2015 Available online 10 March 2016 Keywords:

Bifurcation

Honeycomb structures Post-buckling Stability

a b s t ra c t

Thebucklingandcrushingmechanicsofcellularhoneycombmaterialsisanimportantengineeringprob- lem. Motivated bythe pioneeringexperimentaland numerical studies ofPapkaand Kyriakides (1994, 1999a,b), wereviewthe literature onfinitely strained honeycombssubjectedto in-planeloading and identifytwoopenquestions:(i) Howdoesthemechanical responseofthehoneycombdependonthe applied loadingdevice?and (ii) What canthe Blochwaverepresentationof allboundedperturbations contributetoourunderstandingofthestabilityofpost-bifurcatedequilibriumconfigurations?Toaddress theseissueswemodelthehoneycombasatwo-dimensionalinfiniteperfectperiodicmedium.Weuse analyticalgrouptheorymethods(asopposedtothemorecommon,butlessrobust,imperfectionmethod) tostudythehoneycomb’sbifurcationbehaviorunderthreedifferentfar-fieldloadingsthatproduce(ini- tially)thesameequi-biaxialcontractivedilatation.UsinganFEMdiscretizationofthehoneycombwalls (struts),wesolve theequilibrium equationsto findtheprincipal andbifurcatedequilibrium pathsfor eachofthethreeloadingcases.Weevaluatethestructure’sstabilityusingtwocriteria:rank-oneconvex- ityofthehomogenized continuum(longwavelengthperturbations)andBloch wavestability(bounded perturbationsof arbitrarywavelength).We findthat thepost-bifurcation behavior isextremely sensi- tivetotheappliedloadingdevice,inspiteofacommonprincipalsolution.Weconfirmthattheflower modeisalwaysunstable,aspreviouslyreported.However,our(firstever)Blochwavestabilityanalysis ofthepost-bifurcatedequilibriumpaths showsthattheflowermodeisstablefor allsufficientlyshort wavelengthperturbations. Thisnewresultprovidesarealisticexplanationforwhythismodehasbeen observedinthefinitesizespecimenexperimentsofPapkaandKyriakides(1999a).

© 2016ElsevierLtd.Allrightsreserved.

1. Introduction

The buckling and crushing mechanics of cellular honeycomb materials subjectedto in-plane loading hasbeen a long-standing problem of practical importance to the engineering community, primarily dueto their wide use inshock absorption andmitiga- tion.The theoreticalinterest in thisproblemstems fromthefact that it represents a prototypical example of nonlinear mechani- cal systemsexhibiting instabilities that lead to strain localization andquasi-staticpropagatingshear-bands.Theexcellentmonograph by Gibsonand Ashby(1997),now inits second edition,provides athoroughpresentation,fromthematerials scienceviewpoint,of whatisknownaboutthebehaviorofsuchmaterials.

From the wide literature on the subject, thereview of which isbeyondthescopeofthiswork,attentionisfocusedhereonthe

∗ Corresponding author. Tel.: +1 6126242376.

E-mail address: [email protected] (R.S. Elliott).

two-dimensionalproblemofin-planecrushingofhoneycombs.De- tailedstudieshaveappearedonthistopicinthemechanicslitera- ture,bothexperimental(PapkaandKyriakides,1994,1998,1999a;

1999b,amongothers)andtheoretical(Ohnoetal.,2002;Okumura et al., 2002; Saiki etal., 2005, 2002; Triantafyllidis and Schraad, 1998).However,manyopen questionsremainabouttheparticular mechanisms by which honeycomb materials progressively buckle inasequencethatultimatelymakesup theirhighlycomplexme- chanicalresponsetoin-planeloading.Moreover,thereseemstobe someambiguityintheliteratureregardingtheapplicationandef- fect ofdifferent mechanicalloading devicesas well asambiguity inregardtothestabilityofthebifurcatedequilibriumpaths.

Inthispaper,wefirstreviewtherelevantliteraturewithafocus on the onsetof instability,initial post-bifurcation andthe stabil- ityoftheresultingbifurcatedequilibriumpathsinfinitelystrained cellular solidswithhexagonal symmetry underdifferent in-plane loading conditions. Next, we investigate the behavior of an infi- nite, perfect,two-dimensionalhexagonal honeycombsubjected to equi-biaxial compression from multiple loading devices. We use http://dx.doi.org/10.1016/j.ijsolstr.2016.02.016

0020-7683/© 2016 Elsevier Ltd. All rights reserved.

analytical symmetry group theory methods (as opposed to the morecommon,butlessrobust,imperfectionmethod)tostudythe honeycomb’s bifurcation behavior under three different far-field loading conditions that produce (initially) the same equi-biaxial contractive dilatation: displacementcontrol, deadloadBiot stress control,andliveloadpressurecontrol.Usingtheprimitiveunitcell (smallest possible periodic domain) and an FEM discretizationof thehoneycombwalls(struts),wesolvetheassociatedequilibrium equations,identifytheprincipal pathandbifurcationpointsalong thispath.Eachbifurcationpoint correspondsto alossofsymme- try (either rotational,translational, orboth)along the bifurcating equilibrium paths and group theory methods identify an appro- priate primitive unit cell foreach such path.Using the appropri- ate unit cell and the corresponding FEM mesh, we solve the as- sociated equilibrium equations andcompute the post-bifurcation behavior for the honeycomb in each of the three loading cases.

Forallobtainedequilibriumconfigurations,weevaluatethestruc- ture’sstabilityusingtwocriteria:rank-oneconvexityofthehomog- enized continuum (for long wavelength perturbations)and Bloch wavestability(forboundedperturbationsofarbitrarywavelength).

The presentation is organized as follows: in Section 2 we review the recent literature on the theoretical and numerical modeling of instability andbifurcation of honeycombs subjected to in-plane loading.In Section 3we describe our theoretical and numericalmodelofaperfect,infinite,two-dimensionalhexagonal honeycomb structure andthevarious loading devicesconsidered.

In Section 4we presentthe equilibriumandstabilityproblemof interest and discuss the symmetry-based branch-following and bifurcation techniques employed in the numerical (FEM based) solution of the nonlinear equilibrium equations.In Section 5 we report the results obtained from our simulations, and finally in Section 6we concludewithacriticaldiscussion andperspectives ontheobtainedresults.

2. Previousinvestigationsofthein-planeloading ofhoneycombs

Thestudyofcellularsolidshasbeenamajortopicofresearch inmechanics over thepastfew decades.However theanalysisof the stability andpost-bifurcation behavior of regular honeycomb structures hasonlybeenaddressedrelativelyrecently.Thefollow- ingisashortbibliographicreviewofpaperswhicharerelevantto thecurrentstudy.

2.1. Experimentalcharacterization

TheinvestigationbyPapkaandKyriakides(1994)ontheuniax- ial in-planecrushingofhexagonal aluminumhoneycombwasthe first in-depthstudyon thistopicfromthemechanics standpoint.

These authors established the link between a bifurcation on the principal solution and the onset of localization due to the sub- criticalnatureofthepost-bifurcatedequilibriumpath(i.e.areduc- tion inloadcarryingcapacityof thestructurewithincreasing bi- furcationamplitude).Theyalsoshowedhowthe repetitionofthe abovemechanismexplainsthesequentialcrushingofthecellrows, leadingtotheplateaubehaviorintheload–deformationresponse, whichisthemostcharacteristic(andtechnologicallyuseful)aspect ofthenonlinearbehaviorofthesematerials.

The subsequent work of Papka and Kyriakides (1999a) is an experimental studyofthe biaxialcrushingof circularcellhoney- combs andprovidesone ofthe bestavailable accountsofthe bi- axial,in-planeloadingbehaviorofhexagonalhoneycombmaterials.

The authors usea custom madetestingfacility tocrush polycar- bonatehoneycombspecimensofsize18×21unitcells,investigat- ing severalbi-axialityratios. Theexperiments aimtoapply affine

displacement-controlboundaryconditionstothespecimen¹:

x=FX,

∀

^X∈

∂

, ⁽¹⁾

where^{isthebodyofthespecimen,}∂^{itsboundary,and} F=

_F

xx 0

0 Fyy

, (2)

istheformof thedeformation gradient(with respecttoa Carte- siancoordinatesystemalignedwiththeframeofthetestingfacil- ity)usedtodescribetheaffineboundaryconditionsthatthefacil- ityiscapable ofprescribing. Figs. 1and2 show thedeformation sequenceofthehoneycomb materialunderequi-biaxial compres- sion(Fxx=Fyy<1).Theinitialresponseofthehoneycombiselastic up to theonset of instability wherea flower-like mode develops.

Much ofthe specimenbehaves like thecluster ofcells shownin Fig.2,witharather undeformedcentral cellandsixsurrounding highlydeformedcells.Thisregularpatternisbrokenupinseveral places,duetoacombinationofthreefactors:edgeeffects,friction attheslidingboundaryconditionsandincompatibilityofthemode ofdeformationwithafinitespecimen.

Figs.3and4showthedeformationsequenceofthehoneycomb material withFyy/Fxx=2. The behavior ofthe honeycombs under theseloadingconditionsissimilartotheequi-biaxialcompression.

At the onset of instability, an almost flower-like mode develops.

Thismodeissimilartotheequi-biaxialflower-likemodebutnow thehexagonalC₆symmetryofthemodeisbrokencausingtheen- tireflower-like patternto experiencea uniformdeviationaligned withtheprincipal axesoftheappliedaffineboundaryconditions.

Thepatternis notasuniformthroughoutthe specimenasinthe caseofequi-biaxialcompression,andthefinalcollapseofthespec- imenspreadsfromtheselocalizedhigh-deformationregions.

Fig.5showsthelocalizeddeformationsequenceofthehoney- combmaterialwithFyy/Fxx=3.Herethebehaviorissimilartothat whenF_yy/F_xx=2,exceptthatnowacompetingrectangularmodeis observedinsomeregionsofthespecimen.

The experiments of Papka and Kyriakides (1999a) show that the in-plane biaxial crushing of circular cell honeycomb materi- als is a complex mechanical process involving an initial loss of stabilityassociated witha flower-like orrectangular deformation mode followed by strain localization and propagation and fin- ishing with densification of the material.These experimental re- sults(aswellastheirprevious work,PapkaandKyriakides, 1994;

1998) have inspired a number oftheoretical studies that aim to explain the observed behavior associated with the onset of in- stabilityinsuch materials. Thesetheoretical studiesare reviewed next.

2.2.Characterizationoftheonsetoffailure

The paper by Triantafyllidis and Schraad (1998) performs a semi-analytical study of the onset of failure for infinite two- dimensional hexagonal cell honeycomb materials subjected to a

1In this paper we use the direct tensor notation of Tadmor et al. (2012) . In this notation x = FX has the indicial form x i= F iJX J(where summation on repeated in- dices is implied) and represents the contracted multiplication operation. A tensor product of a second-order tensor and a first-order tensor (vector) is denoted T ⊗a with indicial form T ija k. A centered dot between two vectors, as in a ·b , represents the inner product. Thus, the fully contracted product of a second-order tensor is written a ·( Tb ) with indicial form T ija ib j. A double dot represents the inner prod- uct on second-order tensors, as in T : S and has the indicial form T ijS ij. This notation is also extended to the fully contracted product of a fourth-order tensor T : K : S with indicial form T ijK ijklS kl.

Fig. 1. Sequence of deformed configurations of honeycomb material subjected to equi-biaxial ( Fxx= Fyy) loading conditions. From Papka and Kyriakides (1999a) with permis- sion.

Fig. 2. Deformation patterns of a cluster of cells subjected to equi-biaxial ( F xx= F yy) loading conditions. From Papka and Kyriakides (1999a) with permission.

Fig. 3. Sequence of deformed configurations of the honeycomb material subjected to biaxial Fyy/Fxx= 2 loading conditions. From Papka and Kyriakides (1999a) with permis- sion.

Fig. 4. Deformation patterns of a cluster of cells subjected to biaxial F yy/F xx= 2 loading conditions. From Papka and Kyriakides (1999a) with permission.

Fig. 5. Deformation patterns of a cluster of cells subjected to biaxial F yy/F xx= 3 loading conditions. From Papka and Kyriakides (1999a) with permission.

far-fieldcompressivefirstPiola–Kirchhoff stressoftheform

= 1

A1A2

_cos

θ

−sin

θ

sin

θ

^cos

θ

_cos

φ

⁰

0 ^sin

φ

×

_cos

θ

^sin

θ

−sin

θ

^cos

θ

, ⁽³⁾

given with respect to a Cartesian coordinate frame aligned with the symmetry axes of the honeycomb consistent with that used byPapkaandKyriakides(1999a).Here,A₁ andA₂ aretheunitcell side lengths (as given in Fig. 6), ^{is the amplitudeof loading,} θ ^{is the}orientation angleof the principal stress axes andφ ^de-

scribesthebiaxialnatureoftheapplieddeadloading.Proportional loading(fixedvaluesofφ ^andθ^{)paths are}considered,forwhich anon-dimensionaldisplacement^{/h(wherehisthe}out-of-plane thicknessofthehoneycomb),work-conjugatetotheapplied load- ing,canbedefined. Inordertobettermodelrealhoneycombma- terialstheverticalhexagonalcellwallshavetwicethethicknessof theothers.Thehoneycombmaterialitselfismodeledusingafully nonlinear(largedisplacementandlargerotation)beammodeland a linearly-elastic/linearly-plastic constitutive model. The unit cell chosentorepresenttheinfiniteperiodicsystemisshowninFig.6, andperiodicboundaryconditionsareenforced.

Using this model of the infinite honeycomb material the au- thorsinvestigatetheonsetoffailure(definedasthefirstoccurrence

oflossoflocaluniquenessoftheprincipalequilibriumpath)along everypossible proportionalloading path.Theonset offailurecan occur dueto the occurrence of a bifurcation point, a maximum- loadpoint, or the onset of plasticityalong the principal equilib- rium path. Note that this definition of onset of failure does not necessarilycoincidewiththeprincipalequilibriumpath’sonset of instability.

The authors apply a semi-analytical technique combining the theory ofBloch wavesanda finite-element model ofthe honey- combunit cell, to establish the theoretical failure surface of the perfectinfinitehoneycomb.² Theyalsoinvestigategeometricscale effectsandimperfectioneffectsonthefailuresurfaces.

2The stability of Bloch waves has been identified as an important measure of stability for microstructured materials since, at least, the pioneering 1993 work of Geymonat et al. (1993) . Indeed, these ideas have been extensively employed ever since by Triantafyllidis and co-workers ( Dobson et al., 20 07, 20 06; Elliott, 20 07; El- liott et al., 2006a, 2006b, 2011; Gong et al., 2005; Jusuf, 2013; Michailidis et al., 20 09; Michel et al., 20 07; Sorkin et al., 2014; Triantafyllidis and Schraad, 1998 ) and they are starting to propagate into the current applied literature as well ( Laroussi et al., 2002; Liu and Bertoldi, 2015 ). There is also a history of recognition that bifur- cating equilibrium branches that are described by Bloch waves exist in many phys- ical problems with spatial periodicity. For example, in the nonlinear Schrödinger equation where work on these ideas has existed since at least the early 1990s ( Heinz et al., 1992 ).

Fig. 6. Planar section of a perfect, infinite honeycomb, and chosen unit cell. From Triantafyllidis and Schraad (1998) with permission.

Fig.7ashowstheonsetoffailuresurfaceforvaryingloadpaths, withθ=0ofparticularinterestinthecurrentstudy.Theseresults showthatformostloadingpaths,theonsetofplasticdeformation, due to bending of the honeycomb cell sides, is the first failure modetobeencountered.However, thereexistsaregimeofnearly equi-biaxial loading forwhich the onset of failure occurs dueto bifurcation.Fig.7bshowsthatwithinthisloadingregimethebifur- cationmodeisonethathasaperiodicityequaltothatofthesim- ulationunitcell.Thisisincontrasttothefirstbifurcationforload- ingpathswith|1||2|,correspondingtoφ≈3π^/2,forwhich

thebifurcationmodeisalong-wavelength,strain-like,deformation ofthe honeycomb. Ofprimary interest to the current study,the results of Triantafyllidis and Schraad (1998) show that under equi-biaxial loading honeycomb structures undergo initial failure due to elastic bifurcation and that this bifurcation is associated witha buckling mode that is periodic witha spatial wavelength that is of the same order of magnitude as the honeycomb cell size.

In a related study, Ohno et al. (2002) considered a three- dimensionalratebased finiteelementformulationwithanelastic constitutiverelationtoexaminethebehaviorofhexagonalhoney- combstructures withall cell sidesof equal thickness. Using ob- servedbuckling patterns,a 2 ×2 fourhexagon unit cell wasse- lectedforsimulationwithinahomogenizationframeworkandthe onsetof instability behavior was investigatedfor various propor- tional loading conditions. In this study the proportional loading paths correspond to specification of the principal Cauchy stress components.(This is in contrast to the previous investigation of Triantafyllidis and Schraad (1998), which takes the more phys- ically realizable approach of controlling the components of the firstPiola–Kirchhoff stress.)Theauthorsfoundnolossofellipticity (longwavelength instability, also calledrank-one convexity), but instead found short wavelength bifurcation for all loading paths.

DependingontheparticularratioofprincipalCauchystresses,one, two,orthreecriticaldeformationmodeswereidentifiedattheon- setofinstability.Thecaseofthreecriticalmodescorrespondingto equi-biaxial compression.Based onthe three criticaldeformation modesobtained numerically, Ohno etal. (2002) used ad-hoc ar- gumentstoconjecturethatthreeequilibriumdeformationpatterns werepossibleforthepost-bifurcationbehaviorofthehoneycomb.

These modes are depictedin Fig. 8. The single criticalmode, ModeI,wasfoundtobethefirstinstabilityassociatedwithload-

ingpathswherecompressiveloadingisprimarilyalignedalongthe directionparalleltoonehexagoncellside.Whentheloadingispri- marily alignedalong the direction perpendicular to one hexagon cellside,thentwosimultaneouscriticalmodesoccur attheonset ofinstabilityandModeII(alinearcombinationoftwovariantsof Mode I) waspredicted. Finally,when the loading is equi-biaxial, threecriticalmodesoccurattheonsetofinstability andMode III waspredicted.

2.3. Characterizationofpost-bifurcationbehavior

Okumuraetal.(2002)extendedtheworkofOhnoetal.(2002) to numerically study the post-bifurcation dynamics of the elas- tichoneycombstructureunderCauchystresscontrolanddisplace- mentcontrolloadingconditions.The2×2fourhexagon unitcell wasused andartificialdisplacement constraints were applied,as needed,toensurethatonlyoneinstabilitymodeoccurredforeach loadingcondition. Ofparticular interestforthe currentwork, are the results forequi-biaxial loading.Okumura etal. (2002) found that for equi-biaxial displacement control Mode III was stable.

ModeIandModeIIwereunstableandfoundtodynamicallytran- sitiontoModeIII.Incontrast,underCauchy stresscontrolModeI and Mode II were found to be stable with Mode I having the lowest internal energy. Mode III was unstable and found to dy- namically transitionto Mode II. This result showsthat thepost- bifurcation behaviorofthe honeycombis sensitiveto theloading conditions.However, duetothesomewhatunusualchoiceofcon- trolling theCauchy stresscomponents,the physicalinterpretation ofthesedynamicssimulationsissomewhatproblematic.Okumura etal. (2002)also report that theinitial post-bifurcation behavior ofModeIIandModeIIIshowslongwavelengthinstability(lossof rank-one convexity).Unfortunately,dueto anapparent misunder- standingofthetheoryanditsimplications,Okumuraetal.(2002) incorrectly dismiss this loss ofellipticity based on a faultyargu- mentofincompatibilitywiththeirimposedboundaryconditions.

The finalwork that we wouldliketo highlighthereisthat of Saiki et al.(2005), who apply group theoretic methods to inves- tigate the post-bifurcation behavior of the honeycomb structure.

Based ontheprevious works,describedabove,Saiki etal.(2005) make the “sound engineering choice” to study the 2 × 2 four hexagon cell subjected to periodic boundary conditions. The au- thorsfirst performa purely theoretical bifurcation analysisbased

Fig. 7. (a) Onset of plasticity, initial bifurcation and maximum load surfaces in macroscopic stress space. (b) Regions of the initial bifurcation surface corresponding to two different bifurcation mode shapes. Here, iare the principal components of the first Piola–Kirchhoff stress. From Triantafyllidis and Schraad (1998) with permission.

Fig. 8. Onset of instability buckling modes as suggested by Ohno et al. (2002) . (a) Mode I (anti-rolls). (b) Mode II (rectangles). (c) Mode III (flower-mode). From Ohno et al.

(2002) with permission.

onthenonlinearequilibriumequationsforthe2×2periodichon- eycombsubjectedtoisotropicbiaxialloading.Thisprovidesasetof equilibriumequationsthatisequivariant (symmetric)withrespect tothesymmetrygroupofthe2×2unitcell,includingD₆(six-fold hexagonal rotations and mirror) symmetries, as well as internal- translations. From this analytical asymptotic bifurcation analysis, Saikietal.(2005)reviewthewell-knownbifurcationresultsasso- ciatedwithbreakingofD₆ symmetryandthenderivenewresults for bifurcations associated with breaking of internal-translation symmetry. Theyshow analytically that the post-bifurcating equi- librium paths that emerge from such a bifurcation point corre- spondpreciselytotheModeI(anti-rolls),ModeII(rectangles),and Mode III (flower-mode) configurations that were previously pro- posed,usinganad-hoctechnique,byOkumuraetal.(2002).

In the second half of their work, Saiki et al. (2005) perform anumerical post-bifurcationanalysis ofthe2 × 2four cell peri- odic honeycomb problem using a corotational large rotation and smallstrainBernoulli–Eulerbeamformulationandalinearlyelas- ticstress-strainmodel.Theauthorsapplyall-arounddisplacement controlboundary conditionsto theperiodic modelcorresponding to a macroscopically uniform dilatational contraction. Using this numericalmodel,the authorsapply their grouptheory based bi- furcation results in order to compute the initial post-bifurcation behavior associated witha seriesof internal-translation-breaking and D₆-symmetry-breaking bifurcations. These results provide a completeandrigorousinvestigationofthepost-bifurcationbehav- ior for the 2 × 2 four cell honeycomb subjected to equi-biaxial compressionunderdisplacementcontrol.However,theauthorsdo notaddress, inany manner,the stabilityofthese post-bifurcated solutions.

From theliterature review presentedhere, we haveidentified twosignificant open theoreticalquestionsassociatedwiththe in- plane loading of honeycomb structures. First, the dependence of thestructuralresponse onthe mechanicalloadingconditions has notbeensufficientlyexploredanddiscussed,andsecond, thesta- bility,withrespecttoperturbationsofallwavelengths,ofthepost- bifurcatedequilibriumsolutionshasnotbeeninvestigated.Inorder toaddress thesetwoissues, inthepresentwork, we explore the stabilityofthehoneycomb’spost-bifurcatedequilibriumsolutions, using the Bloch wave method, and discuss how these solutions andtheirstabilitydependonthreeloadingconditions:(i)displace- mentcontrol,(ii)deadloadBiotstress³ control,and(iii)liveload two-dimensionalhydrostaticpressurecontrol. Eachoftheseload- ingconditions hasa clearphysicalinterpretation andprovide in- terestinginsightintothebucklingbehaviorofthehoneycomb.

3. Modeling

As withmanyoftheresearchesdiscussed intheprevioussec- tion,we modelthe hexagon cell honeycomb structure asa two- dimensionalperfectperiodicbodyofinfiniteextent.Thesolidsides (struts) of the hexagons are idealized as one-dimensional frame memberscapableofundergoinglargedisplacementsandrotations.

Thevertices where threehexagon sidescome together are taken tobehaveasweldedjoints,whichmeansthatthethree120degree anglesbetweenthethreesidesmustremainthesameduringany deformation.However, thejoint mayundergoan overall rotation.

The constitutive model for extension, bending, and shear of the framemembersistakentobelinearlyelastic.Basedontheresults ofTriantafyllidis andSchraad (1998),the (initial)post-bifurcation equilibriumandstabilitybehaviorpredictedbythismodelisappli- cableto all realhexagonal geometry honeycomb structures com-

3The Biot stress loading is closely related to dead load first Piola–Kirchhoff stress loading. This is discussed more extensively in Elliott et al. (2011) .

Table 1

Geometry and elastic constant data used with the FEAP FEM program to model the honeycomb unit cell.

Hexagon side length (cm) 0.5 Strut thickness (cm) 0.05 Strut cross sectional area (cm ²) 0.05 Strut moment of inertia (cm ⁴) 1.041667 ×10 ⁻⁵ Young’s modulus (MPa) 69

Poisson’s ratio 0.3

posed of rate-independent solids subjected to equi-biaxial com- pressiveloadings.

3.1. Finiteelementdiscretizationandnumericalimplementation

For any given calculation, an isolated unit cell of the hon- eycomb structure is selected and meshed with a finite element method (FEM) program. This provides the energy, forces, and stiffness as a function of the nodal degrees of freedom (DOFs) from the FEM code. Letting the FEM DOFs be given by u= (^u1,u₂,u₃,...,u_N),thesequantitiesmayberepresentedas

w(^u), (4)

f(^u)^{≡ −}

∂

^w

∂

^{u, (5)}

K(^u)≡

∂

^2w

∂

^u

∂

^{u, (6)}

whereNisequaltothenumberofDOFspernodetimesthenum- ber ofnodes inthe unit cellFEM mesh,w isthe unit cell strain energy, fisthe residual force, andK the tangentstiffness ofthe isolated unitcell forthearbitrarydeformation ufromthe stress- free perfecthexagonal honeycombreferenceconfiguration.Exam- pleunitcellFEMmeshesasusedinthisworkareshowninFig.9. InthisworktheFEMcodeFEAP(Taylor,2011)isused.Thetwo- dimensionallargedisplacementlargerotationframeelementwith shear deformation included aspart of the standard FEAP imple- mentationisusedwithgeometricparametersandelasticconstants asgiveninTable1. Thegeometric valuesare consistentwiththe small scale honeycomb specimens used by Papka and Kyriakides (1999a)andthematerialconstantsareconsistentwithtypicalval- uesforthelinearlyelasticbehaviorofrubbers.

The nodal DOFs implemented by FEAP for this element are the nodal displacements in two directions and a nodal counter- clockwiserotationintheplane.Thus,fornodeithenodalDOFvec- tor isu⁽ⁱ⁾=(^u(xⁱ⁾,u⁽_yⁱ⁾,θ⁽ⁱ⁾)^{. TheFEM meshis} constructedin such a way that nodes at the boundary of the isolated unit cell are matched intopairs corresponding to translation periodicity.Then the remaining interior nodes are numbered. Thisspecial number- ing facilitates the applicationof periodic boundary conditions as describedbelow.

In order to allow for the application of uniform deformation, periodicboundary conditions,andfar-fieldloading,the FEMcode iscoupledwithacustomcodethatappliesdisplacementandrota- tion constraints inorder to ensure that half theboundary nodes have displacements corresponding to an affine deformation and thattheremaining (periodic)boundarynodeshavedisplacements androtationsthatsatisfyappropriateperiodicityconditions.Inpar- ticular,thecustomcodeimposeswhatwecallLagrangianCauchy–

BornKinematicssuchthat thedeformedpositionx⁽ⁱ⁾ ofeach node isgivenby

x⁽ⁱ⁾=F(^X(i)+S⁽ⁱ⁾), ⁽⁷⁾ whereFistheuniformdeformationgradienttensor(withcompo- nentsFxx,Fxy,Fyx,andFyy),X⁽ⁱ⁾ isthereferencepositionvectorof

x y

x y

x y

x

y x

y

Fig. 9. 1 ×1, 1 ×2, and 2 ×2 unit cell FEM mesh examples.

node i, and S⁽ⁱ⁾ is the reference shift vector of node i. This kine- matic choice, where the reference shift vector is acted upon by theuniformdeformation mapping,allows forthenaturaldescrip- tionofmacroscopicallyuniformdeformation oftheinfinitestruc- ture.Thus,thecustomcodedefinesaDOFvectorconsistingofthe components ofthedeformation gradient tensor,shiftvectors and rotations forhalf theunit cell boundarynodes⁴ andshift vectors androtationsforeach interiornode.Inparticular,theDOF vector istakentobe

v=(^F,v¯)=(^Fxx,Fxy,Fyx,Fyy,S⁽_x¹⁾,S_y⁽¹⁾,

θ

^(1),...,S(x^Nb/2),S⁽_y^Nb/2),

θ

^(Nb/2),S(x^Nb+1),S⁽_y^Nb+1),

θ

^(Nb+1),...,Sx^(Nn),S_y^(Nn),

θ

^(Nn)), (8) whereN_b isthenumberofboundarynodesintheunit cellmesh, andNn isthetotalnumberofnodesintheisolated unitcellFEM mesh. The vector v¯ will be called the internal nodal degrees of freedomvector.

Thekinematics ofEq.(7)implicitlydefinea mappingbetween thecustomcodeDOFvectorvandtheFEMDOFvectoru,i.e.,u(v).

Thismappingisused,alongwiththechain-rulefordifferentiation, todefinethestrainenergyperreferenceprimitiveunitcellandits derivativesfortheinfinitehoneycombstructure:

W(^v)≡1

nw(^u(^v)), ⁽⁹⁾

∂

^W

∂

^{v ≡1n}

∂

^w

∂

^u

∂

^u

∂

^{v, (10)}

∂

^2W

∂

^v

∂

^{v ≡}

1 n

∂

^2w

∂

^u

∂

^u

∂

^u

∂

^v

∂

^u

∂

^v+

∂

^w

∂

^u

∂

^2u

∂

^v

∂

^v

, (11)

4The periodic boundary conditions imply that only half of the boundary nodal DOFs are independent.

wherenisthenumberofprimitiveunitcellscontainedwithinthe chosenFEMunitcell.

Finally, in order to study the uniform equilibrium configura- tionsofthehoneycombmaterialwhensubjectedtothethreedis- tinctfar-fieldloadingconditionsconsideredinthisworkwedefine thefollowingthreetotalenergy(perreferenceprimitiveunitcell) functions:

For dilatational displacement control: This loading condition correspondstoprescribingthedeformationgradienttensortohave theform F=(¹−/2)^I, where ^{is the} non-dimensional scalar loadingparameter (with > 0corresponding to decreasingunit cellarea)andIisthetwo-dimensionalidentity tensor.Wedefine thetotalenergyfunctionfordisplacementcontrolas

E¯(^v¯;)≡W((¹−/2,0,0,1−/2,v¯))^{. (12)}

Forequi-biaxialBiotstresscontrol:Thisloadingconditionadds thepotentialenergyofadeadloadBiotstress P_B=−λ^Iconjugate

to thestrain measureU−I. Here,to avoid thecomplications as- sociated with rigid-bodyrotations,⁵ we restrict attentionto pure stretchesandintroducethereducedDOFvector

q=(^U,v¯)=(^Uxx,Uyy,Uxy,v¯). ⁽¹³⁾ GivenareducedDOFvector,q=(^Uxx,Uyy,Uxy,v¯)^wecanobtain the corresponding full DOF vector v=(^Uxx,Uxy,Uxy,Uyy,v¯) ^which maythenbeusedastheargumenttothehoneycombstrainenergy functionW(v).Thus,we obtainthetotalenergyfunctionfordead loadcontrolas

Eˆ(^q;

λ

)≡W((^Uxx,Uxy,Uxy,Uyy,v¯))^−tr(^PB(^U−I))^Acell

=W((^Uxx,Uxy,Uxy,Uyy,v¯))⁺

λ

^tr(^U−I)^Acell, ⁽¹⁴⁾

5Note that rigid-body rotations are eliminated by the use of the symmetric right- stretch tensor in the definition of the reduced degree of freedom vector q . Further, the use of the right-stretch tensor instead of the full deformation gradient elimi- nates the highly complicated bifurcation behavior associated with rigid-body rota- tions encountered at zero load. For an in-depth discussion of this bifurcation prob- lem in a general setting, see Chillingworth et al. (1982 , 1983) .

whereλ ^{is the scalar loading parameter having physical dimen-}

sionsofforce per unit length(λ> 0corresponds tocompressive loading),andA_cellisthereferenceareaoftheprimitiveunitcell.

Finally for live two-dimensional hydrostatic pressure control:

Thisloading conditionaddsthe potential energyof alive hydro- staticloadp,havingphysicaldimensionsofforceper unitlength, whichisconjugatetotherelativedecreaseinareaoftheprimitive unitcell.Again,we restrictattentiontopurestretchesandobtain thetotalenergyfunctionforliveloadcontrolas

E˘(^q;p)≡W((^Uxx,Uxy,Uxy,Uyy,v¯))^+p[det(^U)−1]A_cell, ⁽¹⁵⁾

wheredet(^U)=a_cell/A_cell givestheratioofdeformedtoreference unitcellarea.

4. Equilibriumandstability

Inthissectionwestatetheequilibriumequationsforeachload- ing condition, discuss the elimination of rigid-body translations fromtheequilibriumequationsolutionsets,anddiscusstheeval- uationofstabilityfortheobtainedequilibriumsolutions.Thenwe identifythenumericalpath-followingstrategythatisusedtomap outtheequilibriumsolutions asa functionoftheloadingparam- eterforeach loadingcondition, andcommentonthe handlingof bifurcationpointswheretwoormoreequilibriumpathsintersect.

4.1.Equilibriumconfigurations

We apply the principle ofstationarypotential energy andob- tainequilibriumequations,whichimplicitlydefinetheequilibrium configurations, by setting the first derivative of the total energy equaltozero

∂

^E¯(^v¯;)

∂

^{v¯ =0,}

∂

^Eˆ(^q;

λ

)

∂

^{q =0,}

∂

^E˘(^q;p)

∂

^{q =0. (16)}

With v¯∈R^M (as defined in (8)) and q∈R^M+3 (as defined in (13)),thesecorrespondtosystemsofnonlinearalgebraicequations ofsizeM,M+3,andM+3,respectively.Eachsystemofequations alsohasasingleloadingparameter(^,λ^,andp,respectively).

4.1.1. Eliminationofrigid-bodytranslations

AsdefinedinSection3,thetotalenergyfunctionsareinvariant torigid-bodytranslationsinthetwospatialdirections,correspond- ingtoDOFvectorswhereallinternalshiftvectorsareequal.Con- figurationsthat differonlybyarigid-body translationare consid- eredphysically equivalent.This impliesthatthe spaceofall con- figurations can be partitioned into equivalence classes consisting ofconfigurations that differ only by a rigid-body translation.An additional consequence of the energy invariance with respect to rigid-bodydisplacements isthat theequilibriumEqs. (16)are in- sensitivetosuch changesinconfiguration.Inparticular, itiseasy toshow that,forall t=(⁰,0,0,¯t)∈T⊂R^M whereTisthetrans- lationsubspaceofconfigurations,

∂

^E¯(^v¯;)

∂

^{v¯ ·t¯=0,}

∂

^Eˆ(^q;

λ

)

∂

^{q ·t=0,}

∂

^E˘(^q;p)

∂

^{q ·t=0, (17)}

forall (^v¯;),(q;λ^{) and(q;p),}respectively.Thus, ifaconfigura- tionq₀ satisfiesEq.(16),thensodoesq₀+tforeveryt∈T.

Itisdesirabletomodifytheenergyfunctionsinsuchawaythat onlyonerepresentativefromeachequivalenceclassisidentifiedas anequilibriumconfiguration.⁶Weachievethisgoalbyaddingaso-

6See Footnote 5 .

calledPhantomenergy(Jusuf,2010)termtoobtain

E¯(^v¯;)≡E¯(^v¯;)+1 2

_N

b/2

i=1

S_x⁽ⁱ⁾+

Nn

i=Nb+1

S_x⁽ⁱ⁾

2

+1 2

_N

b/2

i=1

S⁽_yⁱ⁾+

Nn

i=Nb+1

S⁽_yⁱ⁾

2

, (18)

Eˆ(^q;

λ

)≡Eˆ(^q;

λ

)+1 2

_N

b/2

i=1

S⁽_xⁱ⁾+

Nn

i=Nb+1

S⁽_xⁱ⁾

2

+1 2

_N

b/2

i=1

S⁽_yⁱ⁾+

Nn

i=Nb+1

S_y⁽ⁱ⁾

2

, ⁽¹⁹⁾

E˘(^q;p)≡E˘(^q;p)+1 2

_N

b/2

i=1

S⁽_xⁱ⁾+

Nn

i=Nb+1

S⁽_xⁱ⁾

2

+1 2

_N

b/2

i=1

S⁽_yⁱ⁾+

Nn

i=Nb+1

S⁽_yⁱ⁾

2

, (20)

where the additional terms are positive semi-definite and have value zero only for configurations withzero rigid-body displace- ment.Thisensures thatonlyzerorigid-bodydisplacementconfig- urationsaresolutions tothecorresponding equilibriumequations,

∂

E^¯

∂

^{v¯ =0,}

∂

E^ˆ

∂

^{q =0,}

∂

E^˘

∂

^{q =0. (21)}

4.1.2. Stabilityofanequilibriumconfiguration

The identified equilibrium configurations are infinite periodic structures withvarious far-field loading conditions applied. Prac- ticalstabilitydefinitionsforsuchinfiniteproblemsarehardtofor- mulateandjustify.Amorereasonableapproachistoformulatethe definitionsandideasofstabilityintermsofboundaryvalueprob- lems(BVPs)forfinitebodies.Thismaybeachievedinareasonably straight-forward fashion; for example, see Truesdell and Toupin (1960)andComoandGrimaldi(1995).Insuchaframework,there isa naturaldefinition forthe stabilityofan equilibriumconfigu- ration for a BVP. Engineers like to think of a loss of stability as beingeither a structural instability ora material instability. Struc- turalinstability,suchascolumnbuckling,istypicallydrivenbyge- ometric nonlinearities. Material instability,such as shear-banding or phase transformation, is typically thought of as anything that isnota structuralinstability.Unfortunately,theseengineeringcon- ceptscannotbepreciselydefined.Inpractice,oneidentifiesvarious materialstabilitycriteriawhicharepoint-wiseconditions(specified intermsofthepropertiesofamaterial’sconstitutiverelation)that are necessary for stability of an equilibrium configuration for all possible BVPs. In particular, herewe will think of the strain en- ergyfunction W(v)presented inSection 3asamaterial constitu- tivelawandconsidertheclassicstabilityconditionoflocal⁷ rank- one convexity (Como and Grimaldi, 1995; Truesdell et al., 2004).

Thisconditionconcernsthe elasticmoduli (secondderivatives)of thehomogenizedcontinuum (see Elliottetal., 2006b)energyden- sityW^∼(^F)^{.Thesemoduliareobtainedbystaticallycondensingthe}

7Here local refers to the fact that we only consider the criterion evaluated for deformation gradient F 0associated with the uniform equilibrium configuration of interest.

internalnodaldegreesoffreedomfromW(v)andaregivenby⁸:

∂

^2W∼

∂

^F

∂

^F=

∂

^2W

∂

^F

∂

^F−

∂

^2W

∂

^F

∂

^v¯

∂

^2W

∂

^v¯

∂

^v¯

−1

∂

^2W

∂

^v¯

∂

^{F. (22)}

In terms of these homogenized continuum moduli, the local rank-one convexity (RK1) stability condition says that in order for the uniform equilibrium configuration characterized by v0= (^F0,v¯₀)to be stable, it is necessary that

(^a⊗k)^:

∂

^2W∼

∂

^F

∂

^F

F=F0

:(^a⊗k)>0, ⁽²³⁾

forall non-zerovectors aandk.This conditionisalsoknown as thestrongellipticitycriterion(Truesdelletal.,2004).

Foraperiodicmicrostructuredmaterialmodelsuchastheone studied here,the RK1conditionshouldbe generalizedbyconsid- eringperiodic perturbations onall possible unit cells.This corre- spondstothePhononorBlochwavestability criterionasdiscussed in depth by Elliott etal. (2006b). The Bloch wave criterion con- sidersall boundedperturbationsofthe equilibriumconfiguration, andisasetofnecessaryconditionsfortheconfigurationtobean energy minimizing state. The criterion is equivalent to requiring all positive eigenvalues, associated witheigenvectors of theform

δ^v=(⁰,δ^v¯),forthe Hessianofthe unit cellenergy forallpossi- bleunit cells.The Blochtheorem(Geymonatetal.,1993) ensures thattheseeigenvectorsoftheHessianoftheunitcellenergymay be written as follows.Starting witha primitive unit cell forthe equilibriumconfiguration,whichhaslatticevectorsG₁andG₂ and FEM nodes numbered j=1,2,...,m, we can label each node in theextendednon-primitiveunitcell intermsofitsprimitiveunit cellimageindices(¹,²,j),with¹ and² integers.Thus,theref- erencepositionofnode(¹,²,j)isgivenintermsofthereference positionofnode(0,0,j)as

X(¹,²,j)≡¹G1+²G2+X(⁰,0,j).

Now, the Bloch theorem says that eigen-perturbations (also calledBloch waves)fortheequilibriumconfigurationcan bewrit- tenas

δ

^u(^1,2,j)⁼

δ

^u(^0,0,j)

k

exp

k·X(^1,2,j)

, (24)

where k=2πⁱ(^k1G¹+k₂G²) ^{is the} eigen-perturbation wave vec- tor written withrespect to the primitive unit cell reciprocallat- tice vectors G¹ and G² and scaled by a factor of 2π^{i, with i}

the unit imaginary number. Due to the periodicity of the Bloch wave expression,only k vectorsfromtheFirst Brillouin Zone(BZ) needtobeconsidered,asdescribedbyDove(1993).Fig.10shows the lattice vectors and reciprocal lattice vectors, along with the fullysymmetricfirstBZ,fortheundeformedhexagonalhoneycomb structure.

Theprimitiveunitcellperturbationδ^{u(0,0,j)inEq.(24)isnot}

determined bythe Blochtheorem,butingeneraldependsonthe wave vectork.Inparticular,itisdetermined bysolvingan eigen- value problemon the primitive unit cell (which has3m degrees offreedomcorrespondingtotwospatialdisplacementcomponents andone rotationforeach FEM node).Thus,foreach wave vector k there will be 3m distinct eigenvectors δ^{u(0, 0, j; k, h), where}

h∈{¹,2,...,3m}^{is a label}distinguishing each eigenvector.Asso- ciated witheach eigenvector is an eigenvalue ^{(k, h). Thus, the} Bloch wave stability conditionsays that inorder fortheuniform

8Note, some care must be taken to eliminate the zero-eigenvalues of ^∂_∂v²_¯^W_∂v¯ as- sociated with rigid-body translations when computing the inversion indicated in Eq. (22) .

Fig. 10. Lattice vectors, reciprocal lattice vectors, and fully symmetric First Brillouin Zone for the hexagonal honeycomb lattice.

equilibriumconfigurationtobestable,itisnecessarythat⁹

mink,h (^k,h)>0. ⁽²⁵⁾

The different Bloch wave eigenvalues for each wave vector k can,forwave vectorsin a neighborhoodof theso-called^-point k=0, be usefully split into two categories: acoustic eigenvalues that are associated with Bloch wave deformation patterns that haveaslowspatialvariation(i.e.,longwavelengthmodes),andop- tic eigenvalues that are associated with Bloch wave deformation patternsthat havea rapid spatialvariation (i.e., shortwavelength modes).

In principle, Eq. (25) involves an infinite set of wave vectors takenfromthefirstBZoverwhichtheBloch wavecriterion must beevaluated.Thisis,ofcourse,notpossibleandthereforeafinite gridof wave vectorsisselected forevaluationof theBloch wave criterion.However,duetotheexistenceoftheunitcellenergy,and thusthesymmetryofitsHessian,theBlochwaveeigenvaluessat- isfy the identity (−k,h)=(^k,h)^.Further, considering the ro- tational symmetry of the problem one finds that only k vectors fromthe irreducible part ofthe Brillouinzone provide indepen- dentBloch wave eigenvaluesthat mustbe consideredinorder to evaluate the Bloch wave stabilitycondition. Fig. 11shows a plot ofBloch wave eigenvaluecontours for the hexagonalhoneycomb structure.

AlthoughtheirreducibleBZprovidesthesmallestdomainofin- dependentBlochwaveeigenvalues,thisdomainchangessizeonce thestructureisdeformedandthehexagonalsymmetryisbroken.

ToavoidsuchcomplexitiesinourBlochwavealgorithms,we pre- fertouseakvectorgridoverafullone-halfofthefirstBZ.Thatis, weconsiderwavevectorcomponentsthatsatisfy−1/2<k₁≤1/2, and0 ≤ k₂ ≤ 1/2, asindicated by the computedzone labeled in Fig.11.Usingthisregiontakesadvantageonlyofthelatticeinver- sionpropertyofthematerial’s translationalsymmetry groupand, thus,guaranteesthatouralgorithmwillworkcorrectlyforallde- formedconfigurationsofthehoneycomb.Additionaldetailsrelated

9It must be noted that when k = 0 there will be two zero eigenvalues corre- sponding to rigid-body translation. We do not wish to consider such rigid-body motions as instabilities, and therefore, (implicitly) exclude these eigenvalues from consideration in Eq. (25) .