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Hierarchical honeycomb material design and optimization: Beyond linearized behavior
Christelle Combescure, Ryan Elliott
To cite this version:
Christelle Combescure, Ryan Elliott. Hierarchical honeycomb material design and optimization: Be- yond linearized behavior. International Journal of Solids and Structures, Elsevier, 2017, 115-116, pp.161 - 169. �10.1016/j.ijsolstr.2017.03.011�. �hal-01648159�
ContentslistsavailableatScienceDirect
International Journal of Solids and Structures
journalhomepage:www.elsevier.com/locate/ijsolstr
Hierarchical honeycomb material design and optimization: Beyond linearize d b ehavior
Christelle Combescurea,b,∗, Ryan S. Elliotta
aDepartment of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
bLaboratoire de Mécanique des Solides (CNRS UMR 7649), Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, France
a rt i c l e i n f o
Article history:
Received 13 December 2016 Revised 18 February 2017 Available online 15 March 2017 Keywords:
Stability Bifurcation
Hierarchical honeycomb Resilience
Non-linear material properties Buckling
a b s t r a c t
Thispaperexplorestheimportanceofnonlinearmaterialpropertiesinthedesignofhierarchicalhoney- combmaterials.Therecentliteratureonthedesignandoptimizationoflinearmaterialpropertiesforhi- erarchicalhoneycombsisreviewed.Thenafullnonlinearpost-bifurcationnumericalanalysisisperformed forfiverepresentativehierarchicalhoneycombstructures.Particularattentionispaidtothefollowingfour nonlinearmaterialproperties:thecriticalloadλcatwhichthestructurefirstexperiencesaninstability;
theplasticcriticalloadλpatwhichtheonsetofplasticitywouldoccur(ifnoelasticinstabilityoccurred);
thestablepost-bifurcatedstructureofthehoneycomb;andthepurelyelasticresilienceofthenonlinear material.Itisfoundthatalthoughthe honeycomb’slinearYoung’smodulus isoptimallymaximizedat ahierarchyratioofγ1 ≈30%,thecritical loadisreducedbyafactoroftwo(relative tothestandard honeycomb)atthisratio.Further,thecriticalloaddisplaysamonotonedecreasingtrendwithincreasing hierarchyratio.Asimilartrendisfoundfortheplasticcriticalload.Anon-monotonetrendforthere- silienceisdiscoveredandexplainedbyaqualitativechangeinthestablepost-bifurcatedstructureforthe hierarchicalhoneycombswhichoccursasthehierarchyratioisincreased.Theobservedlossofstrength (decreasedcriticalload)issignificantandmaynegateanyadvantagesoftheincreasedYoung’smodulus.
Thisresultdemonstratestheimportanceofconsideringnonlinearpropertiesandtheirimplicationsinthe designandoptimizationofhierarchicalmaterials.
© 2017ElsevierLtd.Allrightsreserved.
1. Introduction
Hierarchyisobservedinmanyorganicmaterialsandbiological systems,asnotedbyAjdarietal.(2012).Itisalsocommonlyem- ployed in engineeredsystems, microstructured materials, and ar- chitecturaldesigns(Lakes,1993).Indeed,aself-similarhierarchical microstructurecanenhancemechanicalbehaviorandprovidesthe opportunityto tailoramaterial’sproperties,suchasmassdensity andYoung’s modulus, foritsspecific intended use.As afirst and basic applicationof this principle ofhierarchy, honeycomb sand- wich structures have proven their efficiency andare now widely usedinindustry(GibsonandAshby,1999).
1.1. Designoflinearmaterialpropertiesthroughhierarchy
Incorporating additional levels of hierarchy in honeycomb structures may lead to even more interesting materials. Indeed, Ajdarietal.(2012)proposedanewdesignthatincludeshierarchy
∗ Corresponding author at: Safran Research and Technology, Rue des Jeunes Bois - Châteaufort 91128 Magny-les-Hameaux, France.
E-mail address: [email protected] (C. Combescure).
inhoneycombstructuresbyreplacingeachvertexofanhexagonal honeycombstructurebyasmallerhexagon(seeFig.1).Todescribe thegeometryofa1storderhierarchicalhoneycombstructure,the ratioofthesmall hexagon’sedgelength b tothe largehexagon’s edgelengtha isdefined:γ1 ≡ b/a.Further,adimensionlessrela- tive density(i.e.,areafraction) isdefinedin termsofγ1 andthe thicknessofthecellwalls(alsocalledstruts)t:ρ≡√23(1+2γ1)at. Usinganapproximateanalyticalapproach,Ajdari etal.(2012)de- termined the effect of γ1 on the hierarchal honeycomb’s lin- earized elastic properties: Young’s modulus and Poisson’s ratio.
Theyshowedthat theYoung’s moduluscanbe enhanced,relative totheγ1=0standardhoneycomb,byafactoroftwowhilekeep- ingthedensityρconstant.Similarly,thePoisson’sratiocanbere-
ducedtoone-thirdofitsstandard(γ1=0)value.Thesetrendscan beobservedinFig.2.
Asγ1 increases thehexagon strut thickness mustdecrease in ordertomaintainaconstantdensityρ.Asaconsequence,theas- pect ratio t/a of the hexagon walls decreases, leading to an in- creasedsusceptibilitytoinstabilities.
http://dx.doi.org/10.1016/j.ijsolstr.2017.03.011 0020-7683/© 2017 Elsevier Ltd. All rights reserved.
162 C. Combescure, R.S. Elliott / International Journal of Solids and Structures 115–116 (2017) 161–169
Fig. 1. Hierarchical honeycomb structures. (A) Unit cell of hierarchical honeycombs with regular structure and 1st order hierarchy. (B) Images of honeycombs fabricated using 3D printing. From Ajdari et al. (2012) .
Fig. 2. (A) Normalized stiffness and for hierarchical honeycombs with 1st order hierarchy versus γ1. The finite element results are shown for honeycombs with three different densities. Experimental results for structures with different γ1values are also shown (black circles). (B) Poisson’s ratio versus γ1. From Ajdari et al. (2012) .
1.2.Designofnonlinearmaterialpropertiesthroughhierarchy
Indeed, it is well known that standard honeycomb struc- tures (γ1=0) exhibit complex nonlinear elastic bifurcation behavior when subjected to in-plane loading. For example, Combescureetal.(2016)haverecentlyshownthatwhensubjected to an isotropic in-plane dead-load, a honeycomb made of a lin- earelasticrubber-likematerialhasaninitiallystiff linearresponse.
However,uponreachingacriticalload,λc(atabout12.25kN/m,in
Fig.3)the structureundergoes amultiple bifurcationresulting in acomplexpost-bifurcationbehaviorthatinvolvesthreecompeting equilibriumstructures. Thisbehaviorisillustrated inFig.3where thethreebifurcatingequilibriumstructures (ModeI,Mode II,and
Mode III) are shown emerging, with significantly reduced stiff- ness (slope), from the bifurcation point on the principal loading path.Asshown,thebuckledequilibriumpathsterminatewhenthe hexagonstrutsfirstmakecontactwitheachother.Thiscontactcan beviewedastheonsetofirreversibilityinthedeformationhistory ofthehoneycombstructure.Additionally inFig.3,thestabilityof the bifurcating equilibrium structures is indicated, withMode III beingthemoststablestructureinthiscase.
Theseresultsreveala numberofimportantnonlinearmaterial properties.First,thecriticalbifurcationloadλc.Thismaterialprop- ertyisimportantbecauseitrepresentstheloadatwhichthema- terialexperiencesasuddenanddramaticlossofstiffness. Second, thecriticalplasticloadλp.Itisimportanttoknowiftheonsetof
Fig. 3. Dead load control bifurcation diagram and deformed configurations associated with the triple Bloch-wave bifurcation of the hexagonal honeycomb structure. The diagram shows Mode I (orange), Mode II (red), and Mode III (blue) bifurcated equilibrium paths. The deformed equilibrium configuration at three loading parameters ( = 0 . 05 , 0.3, and 0.55) is shown for Mode I (first row), Mode II (second row), and Mode III (third row). Adapted from Combescure et al. (2016) . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
instabilityforhoneycombstructuresoccursduetoelasticbuckling, asinFig.3,orduetoplasticity.Forthehoneycombmaterialstud- ied by Combescure et al. (2016), plasticity is initiated when the strain ina hexagon strut reachesa value of 1.52% corresponding totheaverageyield stressvalueof35MPaforthistype ofmate- rial.Thiscorrespondstoacriticalplasticloadofλp≈6.5λc thatis
significantly higherthan theelastic criticalload. Third,the stable post-bifurcatedequilibriumstructureforthefullyelasticmaterial.
Thismaterialpropertycanhaveadramaticeffectontheobserved post-buckling behavior of the material. Different structures have significantly differentcharacteristicsandknowingwhichstructure isstablepredicts thepropertiesthatwill beobservedin practice.
Finally,theelasticmaterial’sresilience.Amaterial’sresilienceisthe amountofenergythatcanbestoredelasticallyandrecoveredupon removalofloads(Callister,2000).Forthenonlinearbehaviorofthe standard honeycomb structure showninFig. 3,thematerial’s re- silience corresponds tothe area under thestable portions ofthe unbuckledandbuckledconfigurations.Itiseasy toseethatama-
jorityofthehoneycomb’sresilienceisdueto thepost-bifurcation regime andthat thisvalue dependssignificantly on which bifur- catedconfigurationisstable.
Thedesignofhierarchicalhoneycombmaterialswithrespectto nonlinearproperties, such asthosejust mentionedabove, is cur- rentlyan active area ofresearch andisthe mainarea of interest inthispaper.Existingstudiesalong theselineshavefocused only ontheonsetofinstability(elasticand/or plastic).Inparticular,re- cent studies of plastic collapse under uniaxial and biaxial stress (Haghpanahetal.,2014b,2013)andelasticinstabilitycollapseun- dergeneralbiaxialstress(Haghpanahetal.,2014a)havebeenper- formedforhierarchicalhoneycombstructures.Thesestudiesderive approximateresultsinanalyticalclosedform(validatedbynumer- icalresults)thatprovideacompletemapoftheonsetofplasticity
λp as well as elastic buckling loads λc in the three-dimensional generalstressspace.
Although these studies are the first to look at the nonlin- ear onset of elastic and plastic instability of hierarchical honey-
164 C. Combescure, R.S. Elliott / International Journal of Solids and Structures 115–116 (2017) 161–169
Fig. 4. Bifurcation modes (A) I and (B) II for hierarchical honeycomb structures identified from finite element simulations. From Haghpanah et al. (2014a ).
Fig. 5. Bifurcation modes (A) I, (B) II and (C) III for hierarchical honeycomb struc- tures identified from finite element simulations. From Mousanezhad et al. (2015b ).
combmaterials, their methods introduce a number ofsignificant approximations.For example,elastic bucklingbehavior wasstud- iedusinganalytical methodsbasedon thebeam-columnsolution (TimoshenkoandGere,2009).Inorderforsuchananalyticalstudy tobe possible, theauthors’ hadto introduce predefined andfixed bucklingmodes (motivated by eitherthe results offinite element simulations or buckling modes found in the existing literature).
Onlythen is it possible to develop closed-formrelations forthe bucklingstrength ofthehierarchicalhoneycombs. Inparticular,it wasassumedthat thesmall hexagons ofthe hierarchical honey- combstructuredeformasrigidbodies.Thisassumptionresultsin asignificant restrictionoftherangeofγ1 forwhichtheir results can be considered valid. Moreover, only two bifurcation modes wereidentifiedfromthefiniteelement computations(seeFig.4), whereasMousanezhadetal.(2015b)identified threemodes, sim- ilar tothe three modesof standard hexagonal honeycomb struc- tures(seeFigs.3and5).
Inordertomorefullyinvestigatethesenonlinearmaterialprop- ertiesandto look,forthefirsttime, atthestablepost-bifurcated equilibriumstructureandtheresilienceofhierarchicalhoneycomb structures, the presentstudy uses a predictive methodbased on group representation theory (Healey, 1988; Zingoni, 2002; Saiki etal.,2005;Okumuraetal.,2002)andbranch-followingandbifur- cationtechniques (Elliott etal., 2002; Elliott, 2007) to determine howthesenonlinear properties varyover thewhole range ofγ1
forequibiaxial dead-load conditions. One aim ofthis studyis to complementthe resultspresentedinMousanezhadetal.(2015a), whostudytheinfluenceofhierarchyratioonthestablebifurcated modes and Poisson’s ratio of hierarchical honeycombs subjected touniaxialcompression.Inparticular,Mousanezhadetal.(2015a) showthat,underuniaxialcompressionalongeitherthexoryaxes, thestablebifurcation modechanges whenthehierarchyratioex- ceeds37.5%.
Intheremainderofthispaper,webrieflydescribethemodeling andsimulation techniquesusedinSection2.These methodspro- vide accurate load–deformation paths andrigorouslycharacterize
thestabilityofeach(unbuckledandbuckled)honeycombequilib- riumconfiguration.Wethenpresent,inSection3,detailedresults for the behavior of five hierarchical honeycomb structures with values ofγ1 spanningthe entirephysically relevant range.From these results,we highlight the variation of four nonlinear mate- rial properties:the critical bifurcation loadλc,the criticalplastic
loadλp,the stablepost-bifurcated equilibriumstructure, andthe resilience.
2. Methodsandmaterialconstitutiveparameters
We follow the approach recently developed in Combescure et al. (2016), where the finite element method (FEM) code FEAP (Taylor, 1987) is used to discretize the honey- comb hexagon struts using two-dimensional large displacement large rotation frame elements with shear deformation. Thus, hexagon vertices (where three struts join) are modeled as rigid welded joints such that the angle at the joint between any two strutsisalways120°.However,duringloading,ajointmaydeform as a rigid-body. That is, it may translate in two-dimensions and rotate about the out-of-plane axis. The FEAP frame elements are coupled with a linear elastic material constitutive law using parameter values of E=2.3 GPa, ν=0.3 for Young’s modulus andPoisson’s ratio,respectively.In orderto investigatethe onset of plasticity along the primary equilibrium path, a criterion on the plasticyield strain isset up atp=0.0152.A referencestrut length of a=2.0 cm and thickness of t=0.175 cm provide a reference relative density of ρ=0.101 for the standard (γ1=0) honeycombstructure.Thesevaluesareconsistentwiththoseused by Ajdari et al. (2012) in their study of hierarchical honeycomb structures.
AsinCombescureetal.(2016),theFEMcodeiscoupledwitha customcomputationalcodethatimplementsuniformdeformation with periodic displacement oscillations (i.e., Lagrangian Cauchy–
Born Kinematics) of a hierarchical honeycomb unit cell to obtain the honeycomb material’s effective strain energy (per unit cell) W(U,v¯),whereUisthesymmetricright-stretchtensordescribing thematerial’suniformdeformationandv¯isavectorofdegrees-of- freedom(DOFs)representingtheperiodicdisplacementoscillations correspondingtotheFEMmeshnodalDOFs.Thecustomcodealso implementsatwo-dimensionalfar-fieldisotropicdead-loadingde- vicesuchthatthetotalenergy(perunitcell)fortheinfiniteperfect hierarchicalhoneycombmaterialisgivenby
Eˆ(q;λ)=Eˆ(U,v¯;λ)≡W(U,vˆ)+λtr(U−I)Acell, (1) whereq≡(U,vˆ) isanarraycontainingthecompletesetofDOFs, λis the scalarloading amplitude,Iis the two-dimensional second- orderidentitytensor,andAcell isthereferenceareaofaprimitive hierarchical honeycomb unit cell. The deformation parameter as- sociatedwithλis=tr(I−U)whichrepresentsthecompressive Biotstrain.
The hierarchical honeycomb material’s equilibrium configura- tionsasafunctionoftheloadingarethengivenbythederivative ofitsenergy
∂Eˆ
∂q= ∂Eˆ
∂U
∂Eˆ
∂vˆ
=0. (2)
Theseequationsaresolvedusingbranch-followingandbifurcation methods(asdescribedinCombescureetal.,2016)toobtainbifur- cationdiagramssuchastheoneshowninFig.3.
Inaddition tothegeneration ofbifurcation diagrams,the cus- tom numerical code performs a detailed and rigorous stability analysisofeachequilibriumconfiguration.Thisincludesevaluation oftherank-oneconvexityconditionaswell astheBloch wavesta- bilitycondition.Together,thesetwocriteriaproviderigorousneces-
Fig. 6. A. Finite element meshes for the 1 × 1 cell of hierarchical honeycombs with γ1 = 0 % (classical), 5%, 10%, 30% and 45%, from left to right. Similar symbols on boundary nodes indicate pairs of periodic boundary nodes. B. Example of periodic arrangement of hierarchical honeycombs with γ1= 30 %. The 1 ×1 cell is represented in black surrounded by dashed lines.
saryconditionsfortheequilibriumconfigurationtobestable(with respecttoallboundedperturbations)intheLyapunovsense.
3. Resultsanddiscussion
In this section, we present results for five representative hi- erarchical honeycombstructures with valuesofγ1=0%,5%, 10%, 30%, and 45%, holding the density at1 ρ=0.101. We selected
γ1=0% since thiscorresponds to the well known standard hon- eycombstructure. Thevalues ofγ1=5% and10% representsmall and medium perturbations ofthe standard honeycomb structure.
AccordingtotheresultsofAjdarietal.(2012)avalueofγ1=30% provides a nearly optimal Young’s modulus for the hierarchical honeycomb structure.Finally,γ1=45%represents astructure ap- proaching theupperlimit ofγ1=50%, wherethe entirehexagon strutoflengthaisreplacedbytwohexagonsofsidelengtha/2.
Fig. 6 shows the FEM meshes2 we used for the (1 × 1 unit cell) principal equilibrium path computations. As discussed inCombescureetal.(2016),periodicboundaryconditionsondis- placements and rotations are applied to corresponding pairs of boundary nodes (shown withsimilar symbols in Fig.6). Amesh convergence study (not presented here) determined that meshes with3 elementsper hierarchical hexagon strutprovide sufficient precisionwithacceptablecomputationalexpense.
3.1. γ1=0%— standardhoneycombstructure
ConsistentwiththeresultsofCombescureetal.(2016) shown in Fig. 3, the classical honeycomb structure subjected to equi- biaxial compressive loading hasa linear load versus deformation principal equilibrium path as shown in Fig. 7. The Cascading- Cauchy-Born algorithm (Sorkin et al., 2014) identifies a 2 × 2 unitcellthatmustbeusedtocomputethebifurcatingequilibrium
1For constant density, the strut thickness ratio follows the relation t/a = (1 + 2 γ1)t 0/a, where t 0/a = 0 . 0875 is the thickness for zero hierarchy.
2Even though, for the cases of γ1= 30% and 45%, the small hexagon nodes fall outside the unit cell outline, all nodes and finite elements shown can be unambigu- ously assigned to a single unit cell.
Fig. 7. Bifurcation diagram for the hierarchical honeycomb with γ1= 0 %. The prin- cipal equilibrium path is stable (solid line segment) for λ< λc ≈55 kN/m and unstable (dashed line segment) for larger load values. At the critical load λcthree energetically distinct equilibrium paths bifurcate from the principal path. An open circle labeled with a number on the path corresponds to a Hessian bifurcation point and indicates the number of eigenvalues that become zero simultaneously.
Fig. 8. Finite element mesh for the 2 × 2 honeycomb unit cell. Similar symbols on boundary nodes indicate periodic boundary nodes.
pathsforthehoneycombstructure.ThisunitcellisshowninFig.8. Theidentificationofthe2× 2unit cellisbasedontheunstable wave-vectorsresultingfromaBloch-wavestabilityanalysisonthe principal equilibrium path. Along the principal equilibrium path,