Rotational snap-through behavior of multi-stable beam-type metastructures
Zhang, Yong; Tichem, Marcel; Keulen, Fred van
DOI
10.1016/j.ijmecsci.2020.106172
Publication date
2021
Document Version
Final published version
Published in
International Journal of Mechanical Sciences
Citation (APA)
Zhang, Y., Tichem, M., & Keulen, F. V. (2021). Rotational snap-through behavior of multi-stable beam-type
metastructures. International Journal of Mechanical Sciences, 193, [106172].
https://doi.org/10.1016/j.ijmecsci.2020.106172
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ContentslistsavailableatScienceDirect
International
Journal
of
Mechanical
Sciences
journalhomepage:www.elsevier.com/locate/ijmecsci
Rotational
snap-through
behavior
of
multi-stable
beam-type
metastructures
Yong
Zhang
∗,
Marcel
Tichem
,
Fred
van
Keulen
Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, Delft, 2628 CD, The Netherlands
a
r
t
i
c
l
e
i
n
f
o
Keywords: Metastructure Snapping beam Multi-stability Rotational statea
b
s
t
r
a
c
t
Metastructuresconsistingofplanararrangementsofbi-stablesnap-throughbeamsareabletoexhibitmultiple sta-bleconfigurations.Apartfromtheexpectedtranslationalstatetransition,whenallbeamelementssnapthrough, rotationalstatesmayexistaswell.Inthispaperweexploretherotationalpropertiesofmulti-stable metastruc-turesonthebasisofbothexperimentalandtheoreticalinvestigations,anddefinetheconditionsforachieving rotationalstablestates.Resultsshowthatthemetastructureisabletorealizebothtranslationalandrotational states,whiletherotationaltransitionsrequirelessenergyascomparedtotheirtranslationalcounterparts.The influenceofgeometricparametersonrotationalstabilityisinvestigatedviaparametricstudies.Furthermore,to determinethedesigncriteriaforrotationalstability,atheoreticalinvestigationbasedonmodesuperposition principleisperformedtopredictthenonlinear-deformationofaunitcell.Thetheoreticalanalysispredictswell therotationalsnap-throughtransitionsthatareobservedinfiniteelementsimulations.Itisfoundthatthe rota-tionalstabilityisdeterminedbysettingpropervaluesforh/Landt/L(h,t,Lrepresentapexheight,thickness andspanofthebi-stablebeamstructure,respectively).Finally,weexperimentallydemonstratethattheproposed metastructurewithmultiplelayersisabletoachievelargerotationsandtranslations.
1. Introduction
Thedesignofmechanicalmetastructuresisarapidlyemergingfield, becauseofthepotentialtocreateunusualmechanicalproperties,such asultra-highstiffnessbutlightweight[1–5],negativePoisson’sratio[6– 8],andnegativethermalexpansion[9–13].Thesesuperiorfeaturesare mainlyrealizedbytherationaldesignoftheirunitstructures.For exam-ple,arrangingunitcellsinare-entrantpatterncanresultinanegative Poisson’sratioofthemacroscopicmetastructure[14].Atpresent,a va-rietyofmechanicalmetastructureshavebeenproposedandintensively studiedforvariousapplications,includingmedicalimplants[15,16], softrobots[17,18],andmicroelectromechanicalsystems[19].
Morerecently,thestrategyofutilizingbuckling,whichisgenerally avoidedinclassicalstructuraldesigns,hasbeenextensivelyreportedin metastructuraldesign[20–24].Thedesignedinstability-based mechan-icalmetastructuresallowforlargemacroscopicdeformationswith re-coverableshapechanges,andthustheyarealsoreferredtoas reconfig-urablemetastructures[25–28].Newfunctionalities,e.g.tuneable me-chanicalbehavior,canbeachievedbysuchmetastructures.However, these transformationsof geometryor reconfigurationusuallyrequire continuouspower,inthesensethatacontinuousexternalactuationis necessarytomaintainthedeformedconfiguration.Onewaytoimprove energyefficiencyistointroducebi-ormulti-stability,sothatastructure maintainsdifferentstateswithouttheneedforcontinuousenergy
sup-∗Correspondingauthor.
E-mailaddress:Y.Zhang-15@tudelft.nl(Y.Zhang).
ply.Indesigningsuchmulti-stablemetastructures,pre-shapedbeams havebeencommonlyadoptedasbasicelementssincetheyhavelarge loading-bearingcapacityandcanbeeasilymanufacturedviaadditive manufacturing[21,29].Whenanappliedloadreachesacriticallevel, thebeamwilljumpfromoneconfigurationintoanotherconfiguration, whichisreferredtoassnap-throughbehavior[30].Byarranging mul-tiplesnappingbeams,theassociatedmetastructuresareabletoexhibit multipleself-stableconfigurations.Manyexamplesofregularly assem-bledpre-shapedbeamsinoneortwodimensions(1Dor2D),referred toasmulti-stablebeam-typemetastructures(MBMs),canbe foundin literature[31–39].
In light of the stacking of the snapping beams with two stable configurations, MBMs arenormally capable of achieving a large re-versibleswitchingmotion,whichhasbeenutilizedforenergy absorp-tion[40–43].Ithasbeendemonstratedthatthemulti-stabilitycan en-hance theabilityof reducingpeakacceleration whileMBMsarestill reusable[44].Inaddition,suchMBMs,asreconfigurablestructures,can bewidelyusedfordesigningmotion-drivenmechanismsincluding de-ployablestructures,shape-changingstructures,actuators,softrobotics, andmotionsystems[45–50].Thesemotion-drivenapplicationsusually require multiple degreesof freedomandthe monolithicmulti-stable structuresmightofferacompactsolutionforsuchchallenges.For in-stance,Chenetal.[45]presentedadesignforpropulsionofsoftrobots, exploitingmotionsfromthesnappingofbeams.Santeretal.[47]
ex-https://doi.org/10.1016/j.ijmecsci.2020.106172
Received3June2020;Receivedinrevisedform10October2020;Accepted24October2020 Availableonline29October2020
ploited1Dtranslationalmotionofamulti-stablestructuretodeploya surface.Pontecorvoetal.[48]proposedtheconceptofusingbi-stable elementstoextendahelicopterrotorblade,basedonthelarge trans-lationofMBMs.Besides,Cheetal.[51]designedamulti-stable beam-typestructureusingdifferentmaterialsforeachlayertocontrolits de-formationsequences.ThedynamicbehaviorofMBMswasalso inves-tigatedtoanalyzeitsvibrationmodes[52]. However,mostproposed MBMsarelimitedin termsof allowabledirectionsandareonly pre-sentedwiththeirtranslationalmotions.Theirpotentialtorealize rota-tionalstablestateshasnotbeenrevealedandstudied,andthusthis lim-itstheirfeasibilityinmotion-drivenapplicationswhererotational move-mentsaretypicallyneeded,suchasactuators,deployablestructuresand robotics.
Toenrichmulti-stablestructure’sreconfigurationcapability,inthis workweexplorethepossibilityforMBMstoachieverotational move-ments.ThispaperaimstoanalyzetherotationalbehaviorofMBMsand studydesignconditionsforstructurestorealizeadditionalrotational states.AlthoughpreviouspapersonMBMsgiveimportantinsightsinto themechanicsofthetranslationaltransitions,thepotentialofrotational reconfigurationisrestrictedinmostpreviousdesignsandlittle informa-tionisavailableontherotationalstabilityofMBMs.Inthispaper,the rotationalbehaviorofMBMsischaracterizedviaexperimentaland nu-mericalapproaches,andthedesignrequirementsforrotationalstates areinvestigatedviaananalyticalmodel,whichcanrapidlyidentifythe designspacefortherotationalstability.Theresultingmulti-stable struc-turesexhibitbothtranslationalandrotationalmotions,whichcan fur-therenhancemulti-stablemechanisms’deploymentcapacities.For in-stance,therotationaltransitionscanbeexploitedtodesignmulti-stable bendingactuatorsinwhichrotationaldegreesoffreedomareessential, whiletheintrinsictranslationalstatecanbeharnessedtoprovide exten-sionsorcontractions.
Theremainderofthispaperisorganizedasfollows.Section2 de-scribesthestructuralgeometryofMBMsandmethodsweusein this work.Themechanicalpropertiesandresultingrotationalstablestates are characterized experimentally and numerically via a parametric study, as presented in Section 3. In Section 4, we conduct a theo-retical analysis to investigate the rotational stability. Section 5 de-scribesthemechanicalbehaviorofmulti-layermetastructures exhibit-ing large rotations and translations. Conclusions are presented in
Section6.
2. Structuraldesignandmethods
2.1. Structuralgeometryandstablestates
ThestructuraldesignofMBMsstudiedinthispaperisbasedonthe snap-through behaviorof curvedbeams,which werefirst studiedby Vangbo etal. [53]. Inthis paper,pre-shapedcurvedbeamsareused asbasicelementstoconstructmulti-stablemetastructures,asshownin
Fig.1.EachbeamelementillustratedinFig.1(b)iscomposedofacurved beam,upperandlowerframe(markedinblue).Thegeometric parame-tersoftheframesareshownintheinsettableandarefixedinthisstudy. Theoriginalshape(𝑤0 (𝑥))ofthecurvedbeamillustratedinFig.1(b)can beexpressedas: 𝑤0 (𝑥)=ℎ 2 ( 1−𝑐𝑜𝑠(2𝜋 𝑥 𝐿) ) (1)
whereℎand𝐿representinitialbeamheightandlength,respectively. Thebeam’sin-planethicknessandout-of-planethicknessaredenoted as𝑡and𝑏,respectively.Thesethreeparameters(ℎ,𝐿,and𝑡)are con-sideredasdesignparametersandtheirinfluenceonrotationalbehavior issystematicallydiscussedinthispaper.AsillustratedinFig.1(a),by seriallycombiningmultiplebeamelements,MBMscanbedesigned.
Theresultingmetastructuresandtheirstablestatesaredemonstrated inFig.1(c),wheretherepresentativeunitisreferredtoasaunitcellthat consistsoftwobeamelements:leftandrightbeamelement,asdenoted inthefigure.Theunitcellcandeformintoflatortiltedstable config-urations.Theseflatandtiltedstableconfigurationsarereferredtoas translationalandrotationalstatesrespectivelyintheremainderofthis paper.Theunitcellpossessesfourstableconfigurations:theinitialstate andthreedeformedstablestatesincludingonetranslationalandtwo rotationalstablestates.Whentheunitcellswitchesfromtheinitialto thetranslationalstate,bothbeamsundergothesamedeformationsand snaptothesecondstablestate.Therefore,thebehavioroftranslational transitionscanbeobtainedbasedonastudyforoneside,eithertheleft orrightbeamelement.However,symmetrybreaksupduringthe transi-tiontotherotationalstate,wherebothcurvedbeamsdeformwith differ-entmanners.Inparticular,theleftbeamelementsnapswhiletheright beamelementismainlyrotated.Theresultingrotationalstateisenabled byacollectiveeffectoftheleftbeamelement’ssnappingdeformations androtationaldeformationsofrightbeamelement.Consequently,
me-Fig.1. Ademonstrationofmulti-stablestructureswithtranslationalandrotationalstablestates.(a)Aschematicfortheproposedmulti-stablemetastructure.The structureisdesignedbyseriallystackingcurvedbeamswiththickframes.(b)Geometricparametersofthebeamelementaredepicted.Theout-of-planedimension isdenotedas𝑏.(c)Eachunitcell(ℎ=5mm𝑡=0.8mmand𝐿=28mm)possessesfourstablestatessuchthatitcansnapintothetranslationalstateortherotational states.
Fig.2.Characterizationofsnap-throughtransitionsoftheunitcell.(a)Stress-strainresultsforthematerialweused.(b)Experimentalsetupformeasuringrotational (left)andtranslational(right)transitions.Samplesareclampedatthebottomandconnectedtotheprintedpartatthetop.(c)Normalizedload-displacementcurves ofsnap-throughtransitions.Theblueregionsrepresentpositivestiffnessphaseswhiletheyellowareastandsforthesnappingphase,exhibitingnegativestiffness. 𝐹𝑚𝑎𝑥and𝐹𝑚𝑖𝑛representthemaximalandminimalforceofeachload-deflectioncurves,respectively.(d)Evolutionofthenormalizedstrainenergyoftheunitcell
duringtheuniaxialloading.
chanicsanddesignrequirementsfortheproposedrotationalstateare differentfromthatoftranslationaltransitions.Toestablishthedesign criteriaforreachingrotationalstates,ananalyticalstudyisperformed inSection4.
2.2. Fabricationandexperimentaltests
Tostudytheproposedmulti-stablemetastructures’behavior,aseries ofsampleswerefabricatedviaafuseddepositionprinterusingflexible thermoplasticpolyurethanes(TPU,Flex-45,RS).TPUisan environmen-talfriendlymaterialwithadvantagesofgoodwearresistanceandlarge recoverableelasticstrains.Itspropertieswerecharacterizedbystandard tensilemeasurementsaccordingtoASTMD638-14[40].Themeasured Young’smodulus(𝐸)andPoisson’sratio(𝜈)are95MPaand0.4, respec-tively.Thecorrespondingstress-straincurveisplottedinFig.2(a).
Uniaxial loadingtests were conductedto characterize metastruc-tures’snap-throughtransitionsasload-displacementcurves.Quasi-static loadingconditionswereappliedusingauniversaltestingsystem (Zwick-RoellZ005)withaloadingrateof10mm/mininadisplacement-control manner.Foreachsample,multipletestswereconductedtogetaverage load-displacementcurves.Extrapartsusedforconnectingsamplestothe testingsystemweredesignedandprintedusingpolylacticacid(PLA), asseen inFig.2(b).Toallowforunitcells’rotationsduringuniaxial testing,tworigidpartswereassembledusing screwstomimichinge connections.Loadswereappliedononesideoftheframe.For
character-izingtranslationalmotions,clampedconditionswereappliedbetween specimensandtheextraparts,asdisplayedinFig.2(b).
2.3. Numericalmethods
Finite element models were established by using software ABAQUS/Standard (2017)toexamine thestructures’mechanical be-havior, wheregeometricparametersarein accordance withthoseof experiments.A Marlowhyper-elasticmaterialmodelwas adoptedas aconstitutiverelationinfiniteelementanalysis(FEA)[54].Here,the stress-straincurveshowninFig.2(a)wasimportedtoABAQUS. Eight-nodebrickelements(C3D8)wereusedtomeshthestructuregeometry, withmeshconvergenceanalysisperformedtoensureaccuracy. Nonlin-eargeometricFEAbasedonstatic/generalprocedurewasconductedby prescribingverticaldisplacementsatspecificloadingpointsonthetop oftheunitcell,whichisconsistentwiththeexperimentalcompression testsshowninFig.2(b).Thebottomofthestructureisfullyfixedinx, y,zdirections.Thereactionforceswerecollectedineachiterationfrom nodeswhereverticaldisplacementswereapplied.
3. Mechanicalresponseforrotationaltransitions
Inthissection,wecharacterizetherotationaltransitionsofthe rep-resentativeelement,whichistheunitcelldefinedbefore.Moreover,a parametricstudyisperformedtoinvestigatetheinfluenceofprimary geometricparameters.
3.1. Characteristicsofsnap-throughbehavior
The force and vertical displacement of loading points shown in
Fig.2(b)aredenotedas𝐹and𝑑,respectively.Themeasuredforceand displacementarethennormalizedby𝐸𝑏𝑡andℎ,respectively.Fig.2(c) showsforce-displacementcurvesfortranslationalandrotational snap-throughbehavior.Itcanbeobservedthatthereisagoodquantitative agreementbetweenexperimentalandnumericalresults.Negative stiff-nessiscapturedinbothtranslationalandrotationaltransitioncurves. Theforcefirstgraduallyincreasesalongwiththedisplacementandthen instability(i.e.snapping)istriggeredwhenreachingthemaximumforce (denotedas𝐹𝑚𝑎𝑥).Underforce-controlloadingcondition,snap-through behaviortypicallyexhibits“displacementjump” afterreaching𝐹𝑚𝑎𝑥.In thisworkweapplydisplacement-controlloadingconditiontocapture thefullforce-displacementcurveshown inFig.2(c),withanegative stiffnessphase(theyellowregioninthefigure).Inthispaper,werefer tothisnegativestiffnessphaseassnap-throughtransitions.𝐹𝑚𝑖𝑛 repre-sentstheminimumforceintheforce-displacementcurve.Inthisfigure, themagnitudeof𝐹𝑚𝑎𝑥fortranslationisapproximatelytwotimeslarger thanforrotation.This isduetothefactthatin caseoftranslational transitions,twobeamsundergobendingandcompressiondeformations, whileforthecaseofrotationsmainlythebeamneartheloading posi-tiondeforms.Moreover,theminimumforce𝐹𝑚𝑖𝑛fortranslationisalso higherthanforrotationsasaresultofdeformationsoftwobeams.This impliesthatrecoveringfromthetranslationaltotheinitialstateneeds largerforceascomparedtoreversingtherotationalstate.Inaddition,we assessthesymmetryofsnap-throughtransitionbydefiningaquantity𝜂 =|𝐹𝑚𝑎𝑥/𝐹𝑚𝑖𝑛|.When𝜂 iscloseto1,thestructureintendstoexhibit sym-metrictransitions.Here,the𝜂 fortranslationaltransitionsinFig.2(c) is3.95,whereasthecounterpartofrotationalcurveis15.24.This indi-catesthattheunitcell’srotationaldeformationexhibitsamoreevident asymmetriccharacteristic ascomparedtotranslationaldeformations. Thehighlyasymmetricbehaviorof rotationsisattributedtotwo as-pects:thesnappingdeformationoftheleftbeamelementandrotational deformationsoftherightbeamelement,asdiscussedinSection2.1.
Theevolution of strain energy(𝑃) duringthe loadingprocess is showninFig.2(d).𝑃isequaltotheareaunderthecorrespondingforce displacementcurve.It indicatesthatboththetranslationaland rota-tionalstatetransitionsexhibitalocalminimumintheenergylandscape, whichcorrespondtothedeformed(translationalandrotational)stable states.Therotationalsnap-through(negativestiffness)can be under-stoodas:byapplyingprescribeddisplacementsasshowninFig.2(b), therightbeamelementismainlybent.Theleftbeamelement experi-encesbendingandcompressiondeformations,wherethecompression energyfirstincreasesandthendecreases,leadingtothedeclineofthe unitcell’sstrainenergy.Inaddition,itcanbeobservedthattheenergyof deformedstablestateislargerthanthatoftheinitialundeformedstate, whichimpliesasufficientlylargedisturbancemaycausethestructure toswitchbacktoitsinitialstatethathasalowerpotentialenergy.
3.2. Influenceofgeometricparametersonrotationalstates
Formulti-stablemetastructures,tuninggeometricparameterscan re-sultindifferentmechanicalpropertiesincludingstrength,stiffnessand stability. Rotationalbehavior of these metastructures is mainly con-trolledbythebeams’geometricparametersdefinedbefore:thebeam height(ℎ),thethickness(𝑡)andthelength(𝐿).Here,parametricstudies arecarriedoutforunitcells’rotationaltransitionsandaseriesofunit cellswithdifferentparametersistested.Moreover,wehave character-izedthetangentslopeofeachforce-displacementcurvefortheinitial androtationalstate.𝑘0 and𝑘1 arereferredtoasthetangentslopeof eachforce-displacementcurveatthepointscorrespondingtotheinitial androtationalstate,asillustratedinFig.3(a).Similarly,whenvarying ℎand𝐿,𝑘0 and𝑘1 canalsobequantifiedforthecorresponding load-displacementcurves.
3.2.1. Thicknessvariation
Resultsoftheunitcells’rotationaltransitionswithdifferent thick-ness(𝑡)arepresentedinFig.3(a)–(c).Fig.3(b)showsthechangeof𝐹𝑚𝑎𝑥 and𝐹𝑚𝑖𝑛versusthevariationofthickness.Theminordeviationsbetween experimentalandnumericalresultsaremainlyattributedto manufac-turingimperfectionsandlocaldefectsintroducedduringprinting.Itcan beseenthat𝐹𝑚𝑎𝑥isincreasingdramaticallywithrespecttothechangeof thickness.Thisisduetothefactthatbothbendingandcompression en-ergyoftheunitcellincreasemonotonicallywiththeincreasingof𝑡.The variationofminimumforce(𝐹𝑚𝑖𝑛)showsasmalldistinctionascompared tothatof𝐹𝑚𝑎𝑥.Nevertheless,itisclearthatwhenthicknessislarge,the corresponding𝐹𝑚𝑖𝑛isapositivevalue,whichmeansthatthestructure switchesbacktotheinitialstateafterremovalofload.Thisisverifiedin FEA(showninFig.3(b))aswellasexperimentaldemonstrationssuch thatunitcellswithsmall𝑡canrealizeself-stablerotationalstateswhile theunitwithathicknessof1.6mmrecoversbacktotheinitialstate uponremovingtheload(seeSupplementaryVideo).Thisshowsthat 𝑡iscrucialforrotationalstability.Moreover,resultsinFig.3(c)show thatthestiffness𝑘0 increasesmonotonicallywhen𝑡increases.However, 𝑘1 firstincreasesandthendramaticallydecreaseswhen𝑡islargerthan 1.38mminthiscase.Thisindicatesthattheunitcellpossessestunable stiffness,whichisrelatedtoitsmultiplestablestates.
3.2.2. Lengthvariation
Fig.3(d)–(f)displaytheinfluenceofthebeamlength (𝐿)on ro-tationalpropertieswhilekeepingotherparametersunchanged.Results showthatthevalueof𝐹𝑚𝑎𝑥decreaseswiththeincreaseof𝐿,while𝐹𝑚𝑖𝑛 isincreasing.Similarcharacteristicscanbecapturedforboth𝑘0 and𝑘1 thatstiffnessdecreaseswhen𝐿isincreased.Furthermore,itcanbeseen fromFig.3(e)thattheslopeofthe𝐹𝑚𝑖𝑛curvegraduallydecreases.As comparedtotheeffectof𝑡,thebeamlength𝐿hasminoreffectsonthe signof𝐹𝑚𝑖𝑛,whichisstillnegativedespiteitsmagnitudeisdecreasing. Similarly,itwasalsofoundthat𝐿hasminorinfluenceonthesignof 𝐹𝑚𝑖𝑛oftranslationaltransitions[55].Itshouldbementionedthatthis effectof𝐿isonlyapplicableforshallowandthinbeams,where𝐿is muchlargerthanℎand𝑡.
3.2.3. Heightvariation
ResultspresentedinFig.3(g)–(i)showthatthebeamheight(ℎ)has adifferentinfluenceontherotationalbehavior.Theeffectcanbe sum-marizedas:i)𝐹𝑚𝑎𝑥increaseswithanincreasingofℎ,similartotheeffect of𝑡;differently,the𝐹𝑚𝑎𝑥curvegraduallyincreasesinalinearmanner (seeFig.3(h)).ii)Whenenlargingℎ,𝐹𝑚𝑖𝑛andtheslopeof𝐹𝑚𝑖𝑛curve decrease,asshowninFig.3(h).iii)Thechangesof𝑘0 and𝑘1 exhibit different characteristics.The𝑘0 increaseswithincreasingℎin a pro-portionalmanner,whilethe𝑘1 increaseswithasmallrateofchange. Thisrelationresultsfromtheenergyvariationoftheunitcell,where theheight(ℎ)influencesbothinitialcurvatureandlengthofthebeam. Changingthegeometricparameterscancontrolthevariationofelastic energyduringdeformations.Thetunabilityofstiffness offersthe po-tentialtotunestructures’dynamicbehaviorbyswitchingintodifferent stablestates.
4. Designcriteriaforrotationalstates
From theprevious section,itis foundthatgeometricparameters (𝑡,𝐿,ℎ)affectrotationaltransitionsindifferentmanners.With chang-ingtheseparameters,therotationalresponsecanshiftfromstabilizing attherotationalstates(𝐹𝑚𝑖𝑛<0)toswitchingbacktotheinitialstate (norotationalstates,𝐹𝑚𝑖𝑛>0).Inordertoexplorethedesignspacefor realizingrotationalstablestates,wepresentananalyticalinvestigation oftherotationalbehavioronthebasisofparametersdefinedbefore.
4.1. Modelformulation
Thetheoreticalanalysisisestablishedfortheunitcell’srotational transitionsbycombiningbucklingmodesofthebeamelements.Ithas
Fig.3. Theinfluenceof𝑡,ℎ,and𝐿onrotationalresponses.(a)–(c):Theeffectof𝑡onrotationaltransitions.(a)showsexperimentallymeasuredload-displacement curvesforspecimenswithdifferentthickness.𝑘0and𝑘1aredefinedastheslopeoftangentline(orangecolor)attwostablelocations(theinitialandrotationalstate).
(b)presents𝐹𝑚𝑎𝑥and𝐹𝑚𝑖𝑛offorce-displacementcurvesasafunctionofthickness.Thepositivesignof𝐹𝑚𝑖𝑛indicatesthattherotationalstablestatedoesnotexist.(c)
representsthechangeof𝑘0and𝑘1whenvaryingthicknessinFEA.(d)–(f):Theeffectof𝐿onrotationaltransitions.(g)–(i):Theeffectofℎonrotationaltransitions.
beenreportedinliteraturethatutilizingbucklingmodescanprovidea goodestimationofload-displacementresponsesforthesnappingbeams
[35,56–58]. Here,sincetheunitcellundergoes both symmetricand asymmetricdeformations(seeFig.4(a)),dominantmodes(symmetric Mode1andasymmetricMode2)areemployedtoconstruct displace-mentfields,asdisplayedinFig.4(b).Basedontheassumed displace-mentfield,thepotentialenergyisthenformulatedtoderivethe gov-erningequations.
4.1.1. Elasticenergyforasinglebeam
Wefirstderive energyformulas foreach singleclamped-clamped pre-shapedcurvedbeamillustratedinFig.4(d),andthenformulatethe modelfortheunitcell.Thebeam’sgeometricparametersaredepictedin
Fig.4(d),with𝑋−𝑌 coordinatedefined.AccordingtoEulerBernoulli equations[59],thefirstandsecondbucklingmodesofastraightbeam thatisaxiallycompressedtobucklearegivenas:
𝑌1 =1−𝑐𝑜𝑠 ( 𝑁1 𝑥𝐿 ) with 𝑁1 =2𝜋 (2) 𝑌2=1−2𝑥 𝐿−𝑐𝑜𝑠 ( 𝑁2𝑥 𝐿 ) + 2 𝑁2 𝑠𝑖𝑛 ( 𝑁2𝑥 𝐿 ) (3)
where𝑁2 is thefirstpositivesolutionofequationtan(𝑁2 ∕2)=𝑁2 ∕2. Thedeformedbeamshape(𝑌)underagivenloadisdescribedas: 𝑌 =𝐴1 𝑌1 +𝐴2 𝑌2 and 𝑌0 =ℎ
2𝑌1 (4)
𝑑=𝑌0 −𝑌 (5)
where𝑌0 is theinitial beamshapeand𝑑 isthedisplacementof the beam.𝐴1 and𝐴2 areunknowncoefficients,which needtobe solved on thebasisoftheprinciple ofminimum energy.Theelasticenergy forthecurvedbeamisobtained,whichincludesbendingenergy(𝑈𝑏) andcompressionenergy(𝑈𝑐).Thebendingandcompressionenergyis derivedrespectivelyby:
𝑈𝑏=𝐸𝐼2 ∫ 𝐿 0 ( 𝑑2 𝑌 𝑑𝑥2 − 𝑑2 𝑌 0 𝑑𝑥2 )2 𝑑𝑥 where 𝐼= 𝑏𝑡3 12, (6) 𝑈𝑐=𝐸𝑏𝑡(Δ𝑠) 2 2𝐿 and Δ𝑠=𝑠−𝑠0 (7)
Here,𝑠is thebeam’stotallength atanyposition,while 𝑠0 is the beam’sinitiallength.Assumingsmalldeformations,𝑠canbe
approxi-Fig.4.Modelingasinglebeambasedonbucklingmodes.(a)TherotationaltransitionsobservedinbothexperimentsandFEAshowevidentasymmetricdeformation modes.(b)Thefirsttwobucklingmodes(Mode1and2)foraclamped-clampedbeam.Thesemodesareemployedasbasestoapproximatethedisplacementfield. (c)Geometryoftheunitcell.(d)Illustrationofasinglecurvedbeam.Eachbeamoftheunitcellismodeledasasinglebeam.BasedontheX-Ycoordinateinthe figure,thefirsttwobucklingmodesaredescribedbyfunction𝑌1and𝑌2,respectively.
Fig.5. Modeltherotationaldeformationoftheunitcell.(a)Unitcell’sgeometryandpredefinedparameters.𝑏istheout-of-planedimension.(b)Asobserved,the verticalbranchisbentduringrotationaltransitions.Theassociatedbendingenergyistakenintoaccountwhencalculatingthetotalpotentialenergyofthesystem.
matedas: 𝑠= ∫ 𝐿 0 √ 1+( 𝑑𝑌 𝑑𝑥 )2 𝑑𝑥≈ ∫ 𝐿 0 [ 1+1 2( 𝑑𝑌 𝑑𝑥)2 ] 𝑑𝑥, (8)
Bycombining Eqs.(7)and(8),𝑈𝑏 and𝑈𝑐can beexpressedasa functionof𝐴1 and𝐴2 ,asshowninEqs.(9)–(10).𝐶1 ,𝐶2 ,𝐷1 ,𝐷2 are con-stants,whichfollowfromintegrationsalongthe𝑥axis(seeSupporting Material). 𝑈𝑏= 𝐸𝐼 2𝐿3 [ (−ℎ𝑨𝟏+𝑨𝟐𝟏+ℎ2 4)𝐶1 +𝑨 𝟐 𝟐𝐶2 ] (9) 𝑈𝑐 = 8𝐸𝐴𝐿3 { 𝐷2 1 𝑨𝟒 𝟏+𝐷2 2 𝑨𝟒𝟐+2𝐷1 𝐷2 𝑨𝟐𝟏𝑨𝟐𝟐− ℎ2 𝐷2 1 2 𝑨 𝟐 𝟏 −ℎ 2 𝐷 1 𝐷2 2 𝑨 𝟐 𝟐+ ℎ4 𝐷2 1 16 } (10)
4.1.2. Modeltheunitcell
Basedonthederivedformulationforasinglebeam,wenowbuild themodelfortheunitcellwhichiscomposedoftwopre-shapedbeams connectedbyframes.Thegeometricparametersoftheunitcellare il-lustratedinFig.5(a),whereeachbeam’sshape(representedasleftand
rightunit)ismodeledusingtheformuladerivedinSection4.1.1. Specif-ically,eachbeam’sgeometryisdeterminedby𝑡,𝐿,andℎ,whicharethe samenotationsusedbefore.TheverticalbranchdenotedinFig.5(a) rep-resentstheverticalpartoftheupperframe,withdimensionsdepicted as𝐻 and𝑊.Asobservedinsimulation,thisverticalbranchisbent dur-ingloading,asshowninFig.5(b).Therefore,thebendingenergyofthe branchistakenintoaccountinthefollowingenergyformulation.The unitiscompressedbyaload(𝐹)atpointA,asillustratedinFig.5(a). Theangleoftheupperframeisdepictedas𝜃.
a.Kinematics
Asdiscussedintheprevioussection,theleftandrightbeam’sshape (denotedas𝑌𝐿and𝑌𝑅)canbedescribedas:
𝑌𝐿=𝐴
1 𝐿𝑌1 +𝐴2 𝐿𝑌2 and 𝑌𝑅=𝐴
1 𝑅𝑌1 +𝐴2 𝑅𝑌2 (11) where𝑌1 and𝑌2 arethefirsttwobucklingmodes,asdefined before. 𝐴1𝐿,𝐴2𝐿,𝐴1𝑅,𝐴2𝑅areunknownfactors,whichneedtobesolved.The displacementofeachbeam’smidpoint(PointBandD),denotedas𝑑𝐵 and𝑑𝐷,canbeexpressedas:
𝑑𝐵=ℎ−2𝐴
1 𝐿, 𝑑𝐷=ℎ−2𝐴1 𝑅 (12)
KinematicrelationshipbetweentheverticaldisplacementofPointA (denotedas𝑑𝐴)andC(denotedas𝑑𝐶)ispresentedinEqs.(13).The relationbetween𝐴1 𝐿and𝐴1 𝑅isderived,asshowninEqs.(14).
Fig.6.Identificationofdesignspacefor ro-tational stablestates asa function of 𝑡∕𝐿 and ℎ∕𝐿. (a) Theoretical predicted load-displacementcurvesfortheunits with dif-ferentparameters. FEAresultsare usedas references.(b)Withchanging𝑡∕𝐿andℎ∕𝐿, thedesignspaceforrotationalstablestates is identifiedbythe model. Thegrey area representsthesituation,inwhichthereare no rotationalstablestates;thecolored re-gionrepresentstherangeofparameter val-ues,forwhichthestructuredoeshave rota-tionalstablestates.Greendotsarespecific data points (denoted as 𝑡𝑏∕𝐿𝑏 andℎ𝑏∕𝐿𝑏)
alongtheboundary.(c)Thechangeof𝐹𝑚𝑎𝑥
withrespectto𝑡∕𝐿andℎ∕𝐿.(d)The𝑡𝑏∕𝐿𝑏
andℎ𝑏∕𝐿𝑏predictedbythemodelandFEA,
respectively.TheerrorbarsmarkedinFEA representthatthesimulatedℎ𝑏∕𝐿𝑏thatis
ex-actlycorrespondingto𝐹𝑚𝑖𝑛=0islocated
be-tweentheupperandlowerlimits.
𝑑𝐴=𝑑𝐶+𝐿 𝑥𝑠𝑖𝑛(𝜃) (13) 𝐴1 𝐿=𝐴1 𝑅−𝐿𝑥𝑠𝑖𝑛(𝜃) 2 ≈𝐴1 𝑅−𝐿𝑥𝜃 2 (14)
𝐿𝑥isthedistancebetweenPointAandC,whichisaconstantvalue.
b.Potentialenergy
The total potential energy for the unit under the point-force loading is formulated to obtain the four unknown coefficients (𝐴1 𝐿,𝐴2 𝐿,𝐴1 𝑅,𝐴2 𝑅).Therearefourenergycontributions,whicharethe bendingenergyofleftandrightbeam(𝑈𝑏𝐿and𝑈𝑏𝑅),thecompression energyofleft andrightbeam(𝑈𝑐𝐿 and𝑈𝑐𝑅),thebendingenergyof verticalbranch(𝑈𝑣)andpotentialenergy(𝑈𝑒).
𝑈𝑏𝐿,𝑈𝑐𝐿canbederivedbysubstituting𝐴1 𝐿,𝐴2 𝐿for𝐴1 and𝐴2 in
Eqs.(9)and(10) (seeSupportingMaterial). Similarly,bycombining
Eqs.(9),(10)and(14),wecan gettheexpressionof𝑈𝑏𝑅,𝑈𝑐𝑅,asa functionof𝐴1 𝐿,𝐴2 𝑅and𝜃.Finally,therearefourindependentunknown variables(𝐴1𝐿,𝐴2𝐿,𝜃,𝐴2𝑅)inthegoverningequations.Moreover,the bendingenergyoftheverticalbranchcanbeexpressedas:
𝑈𝑣= 𝐸𝐼𝐵 ( 𝐴𝐿 2𝐶𝑠 𝐿 +𝜃 )2 2𝐻 + 𝐸𝐼𝐵 ( 𝐴𝑅 2𝐶𝑠 𝐿 +𝜃 )2 2𝐻 (15)
where𝐸𝐼𝐵representsthebendingstiffnessoftheverticalbeamand𝐶𝑠 isaconstantvalue,asshownintheSupportingMaterial.Thepotential energy,associatedtoappliedforce,issimply:
𝑈𝑒=−𝐹𝑑𝐴 (16)
Finally,thetotalpotentialenergyis:
𝑈𝑡𝑜𝑡=𝑈𝑏𝐿+𝑈𝑐𝐿+𝑈𝑏𝑅+𝑈𝑐𝑅+𝑈𝑣+𝑈𝑒 (17) andthegoverningequationsareobtainedbytakingthederivativeof 𝑈𝑡𝑜𝑡withrespecttofourvariables(𝐴1 𝐿,𝐴2 𝐿,𝜃,𝐴2 𝑅).Theequationsare
thensolvednumericallybygivingarangeof𝐹 inputs.Consequently, thecorrespondingload-displacement(𝐹−𝑑𝐴)curvescanbeextracted. 4.2. Resultsoftheanalyticalmodel
Thecorrespondingload-displacementcurvespredictedbythe ana-lyticalmodelarepresentedinFig.6(a).Itcanbeseenthatthemodel givesagoodapproximationofrotationalsnap-throughcharacteristics thatareobservedinFEA.Specifically,thetheoreticalresultsexhibitthe sameslopeasthatofFEAinthenegativestiffnessphase.Thiscanbe explainedbythefactthatsnappingbehaviorismainlydeterminedby theasymmetricMode2,whichhasbeenusedasamodalbasisinthe analyticalformulation.Meanwhile,itisshownthatthedeformedstable positionoftherotationalstateiswellpredictedbythemodel.As com-paredtotheFEAresults,higherstiffnessareobservedinthismodel’s predictions.Thedifferencesaremainlycausedbytwofactors.First,the theoreticalmodeldoesnotconsiderhighorderbucklingmodessince using dominantmodesalready allowtocapturethecharacteristicof snap-throughdeformationatthesnappingphase,whichisourfocusin thisstudy.Second,themodelisbasedonsmalldeformationhypothesis, asreflectedintheenergyformulations.
The design criterion for the rotational stability is presented in
Fig.6(b),basedontwonon-dimensionalquantities(ℎ∕𝐿and𝑡∕𝐿).The greyregioninthefigureiscorrespondingtothecasethatthestructure cannotstabilizeatthedeformedrotationalstate,where𝐹𝑚𝑖𝑛isapositive value.Thecoloredregionmeansthattheunitisabletoachieve rota-tionalstablestatesbysettingproperℎ∕𝐿and𝑡∕𝐿.Geometrical thresh-oldsalongtheboundaryarehighlightedasgreendots,whichare de-notedas𝑡𝑏∕𝐿𝑏andℎ𝑏∕𝐿𝑏.The𝑡𝑏∕𝐿𝑏andℎ𝑏∕𝐿𝑏obtainedfromthemodel arecomparedwithFEA,asshowninFig.6(d).Itisshownthatthemodel isabletogiveareasonablepredictionoftheboundarythatdetermines therotationalstates.Therefore,itallowsarapidandfullidentification ofthedesignspaceforrotationalstabilitybeforemanufacturing.
Fig.7. 2Dmulti-layermetastructureswithserialarrangementsofunitcells.(a)Atwo-layermetastructure.(b)Athree-layermetastructure.(c)Experimentally characterizedload-displacementcurvesforthetranslationalandrotationalsnap-throughofthetwo-layermetastructure.Themetastructurecanstabilizeattheir deformedstates,asshowninthefigure.(d)Snap-throughresponsesofthethree-layermetastructures.Thetranslationalandrotationaltransitionsarecharacterized respectively.
Inaddition,fromFig.6(b),itcanbenotedthatforaspecified thick-ness,therotationalstablestatecanberealizedwhenℎ∕𝐿ofthestructure islargerthantheℎ𝑏∕𝐿𝑏.Inordertoreachtherotationalstablestate,ℎ∕𝐿 shouldbelargerthanthecriticalvaluewhile𝑡∕𝐿shouldnotexceeda specificvalue.Itcanalsobeinterpretedinthesensethatℎ∕𝑡shouldbe largeforachievingtherotationalstablestates.Besides,weexplorethe variationof𝐹𝑚𝑎𝑥withrespectto𝑡∕𝐿andℎ∕𝐿,andresultsareplottedin
Fig.6(c).Ascomparedtoℎ∕𝐿,𝑡∕𝐿hasamoresubstantialinfluenceon 𝐹𝑚𝑎𝑥.Thatis,thevalueof𝐹𝑚𝑎𝑥increasesdramaticallywiththechangeof
𝑡∕𝐿,whilethechangeofℎ∕𝐿doesnotinfluence𝐹𝑚𝑎𝑥much.Therefore,it ismoreparamounttochangetheparameter𝑡∕𝐿whenthemaximal actu-ationforceneedstobetunedtoaccommodatedifferentcircumstances.
5. Multi-layermetastructureswithrotations
Viarationallydesigningthebeamwithspecificgeometric parame-ters,wecanobtainthedesiredrotationalstatesaswellastheintrinsic translationalstatesfortheunitcell.Byassemblingsuchunitsinseries, metastructureswithmultiplelayerscanbedesigned,whichareableto exhibitlargerotationsandtranslations.Todemonstratethefeasibility ofthemulti-layermetastructurese,weexperimentallycharacterizetheir snappingtransitions.
Two-layerandthree-layermetastructuresaredisplayedinFig.7(a)– (b).Astheunitcellpossessesfourstablestates,theprintedtwo-layer andthree-layermetastructureshave42 and43 stablestates,respectively. Here,twoprimarydeformationmodesareselectedandcharacterizedby cyclicloading.Theassociatedload-displacementresponsesareplotted inFig.7(c)–(d),whereeachlayerexhibitsevidentsnap-through behav-iorsequentially.InFig.7(d), alargedifferencebetweenloadingand unloadingduringthethirdsnappingoccurs.Thishappensduetothe factthatduringuniaxialloading,thetoplayerofthethree-layer metas-tructurealsogeneratesahorizontaldisplacement,whichcannotbe han-dledbytheuniaxialtesting.Intermsofcharacterizingtherotationsof ametastructure withmorelayers,thehorizontaldisplacementof the metastructureneedstobetakenintoaccount.Themulti-stable metas-tructuresareabletostabilizeattheirdeformed(translationaland rota-tional)statesafterremovalofloads.Afterunloading,themetastructure canbefullyreversedbacktotheinitialstate,whichmeansthattheycan bereused.
More importantly, from a kinematics perspective, the proposed multi-stablestructureiscapableofnotonlyprovidingtranslationalbut alsoangularmovements,whichenrichesthestructure’sshape-morphing ability.AsshowninFig.8(a),themulti-layermetastucturecanexhibit differentcurvedshapesthatareenabledbytheunit’srotationalstates,
Fig.8.Demonstrationsofconformal-morphingonacurvedsurfacebasedontherotationalstablestates.(a)Anensembleofunitcellsstackedinseries.Bydeforming indifferenttransitionmodes,itcanswitchtocurvedshapeswithdifferentcurvatures.Theangle𝛼 isthestableangleachievedbyasingleunit’srotationaltransitions. (b)Anillustrativeexampleofconformalmorphingonaspatialfree-formsurface.Torealizeshape-matching,ourdeformedmetastructures(grey)candeforminto acurvedshapetomatchthetargetcurvedfree-formshape(red).(c)Thestableangle(𝛼)ischaracterizedexperimentallyandnumerically,asafunctionofℎfor different𝐿.
Fig.9. 3Dmulti-stablemetastructureswith rota-tionalstates.(a)Themetastructureisdesignedby arrangingunitcellsinanorthogonalpattern.(b) Thefabricatedprototypeofthe3Dmetastructure. Thestructureisabletoexhibitrotationsalongtwo directions, which result in four rotationalstable states.
asidefromthe1DextensionshowninFig.8(a).Suchdeformedcurved statescanbeusefulforshapemorphing,inwhichtheabilitytodeform intocurvedshapesisnormallyneeded.Anillustrativeexampleisshown inFig.8(b),whereconformalmorphingcanberealized.Tomatcha tar-getcurvedshape(denotedinred),itispossibletomakethe metastruc-tures(markedingrey)deformtofitintotheshapeofthetargetcurved shapeviacontrollingtherotationalstablestateofeachunitcell,while thetranslationalstatesofthepresentedstructuresshownin Fig.8(a) cannotrealizethemorphingbehaviortomatchthecurvedsurface.The largestrotationalanglecanbecalculatedas𝑛𝛼,where𝑛isthenumberof layers.Sincethebeamheight(ℎ)andlength(𝐿)havemoresubstantial effectson𝛼 thanthethickness(𝑡),weplotthechangeof𝛼 withthe vari-ationofℎand𝐿,asshowninFig.8(c).Itcanbeseenthat𝛼 increases ap-proximatelyinalinearmannerwiththeincreasingofthebeamheight. Withlargerℎand𝐿,theachievableangle𝛼 canbeenlarged.Then,
rota-tionalanglescanbeprogrammedbyadjustingthenumberoflayersand thegeometryoftheunitcells.Meanwhile,itshouldalsobenotedthata large𝛼 requiresalargedistancebetweentheupperandbottomframe. Here,weobservethatalargeℎwillleadtocontactbetweentheupper andbottomframewhenreachingtherotationalstate(seeSupporting Material).Therefore,whendesigningthestructurewithalargeℎ,the contactissueshouldbeconsidered.
Inaddition,wedemonstratetherotationsofa3Dmulti-stable metas-tructure.ThismetastructureisshowninFig.9(a),andisdesignedwith twounitsplacedinanorthogonalpattern.Itisfabricatedusingapoly-jet basedmulti-materialprinter(Objet350Connex3).FromFig.9(b),itcan beseenthatthis3Dmetastructureisabletorealizerotationsalongtwo directions,whichisanextensionof2Dstructures.Therotationalstates inFig.9(b)canbeseenasthedeformationsoftworowsof2Dunitcells arrangedinparallel.Therefore,itcanbeexpectedthatthemagnitude
ofthis3Dstructure’srotationalforce-displacementresponse,including forcethresholdsandstiffness,shouldbetwotimesaslargeasasingle 2Dunitcellwiththesamelengthscales.This3Dstructurecanbe fur-therextendedtodesignbendingactuatorsbystackingmoreunitcells. Overall,thedesignof3Dstructureherepavesthewayforforming3D deployablestructuresbypatterningstructuresinx,y,zdirections.
6. Conclusions
Inthiswork,therotationalstablestatesofmulti-stablebeam-type metastructureshavebeenpresented,apartfromtheexpected transla-tionalstates.Bothtranslationalandrotationaltransitionsexhibit neg-ativestiffness(snap-through)behaviorwhiletherotationaltransition exhibitsmoreevidentasymmetric characteristicsascomparedtothe counterpartoftranslations.Theparameters,includingthebeamheight (ℎ),thickness(𝑡)andspan(𝐿),havebeenfoundparamountfor rota-tionalstability.Moreover,thecriterion forreachingrotationalstates isestablishedthroughatheoreticalinvestigation.Theproposedmodel, validatedbyfinite elementsimulations,effectivelypredicts the rota-tionalsnap-throughtransitionsinthenegativestiffnessintervaland cap-turesthresholdsofℎ∕𝐿and𝑡∕𝐿.Itisshownthatforrealizingrotational states,ℎ∕𝐿shouldbelargerthancriticalvalueswhile𝑡∕𝐿needstobe small,and𝑡∕𝐿ismoreinfluentialthanℎ∕𝐿fortuningthemaximum forceofrotationaltransitions.Thiscriterionservesasaguidelineto tai-lorrotationalsnap-thoughbehaviorsuchascontrollingforcethresholds. Finally,wedemonstratethelargerotationsofmetastructureswith fabri-catedtwo-andthree-dimensionalmetastructures,wheretherotational stablestatescanbeprogrammedbyseriallyassemblingunitsalong mul-tipledirections.Withtheproperunitsarrangementanddesign,the pro-posedmulti-stablemetastructureswith rotationalstatescanopenup newopportunitiesfordesigningadaptivestructures.
DeclarationofCompetingInterest
Theauthorsdeclarethattheyhavenoknowncompetingfinancial interestsorpersonalrelationshipsthatcouldhaveappearedtoinfluence theworkreportedinthispaper.
CRediTauthorshipcontributionstatement
YongZhang:Conceptualization,Methodology,Validation, Investi-gation,Writing-originaldraft,Visualization.MarcelTichem:Writing -review&editing,Supervision.FredvanKeulen:Methodology,Writing -review&editing,Supervision.
Acknowlgedgments
Y.ZhangwouldliketothankChinaScholarshipCouncil(CSCNO. 201606120015)forthefinancialsupport.BradleyButatDepartmentof PrecisionandMicrosystemsEngineeringofDelftUniversityof Technol-ogyisacknowledgedforthediscussiononprinting.
AppendixA. SupportingMaterial
Supplementarydocument:thedetailedderivationof theproposed analyticalmodelandthecontactbetweentheuppperandbottomframe whenincreasingℎ.
Supplementaryvideo:Withchangingthickness,theunitcellsexhibit rotationalstatesanddonothaverotationalstates,respectively.
Supplementarymaterialassociatedwiththisarticlecanbefound,in theonlineversion,at10.1016/j.ijmecsci.2020.106172
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