Hyperbolic origami-inspired folding of triply periodic minimal surface structures

Delft University of Technology

Hyperbolic origami-inspired folding of triply periodic minimal surface structures

Callens, Sebastien J.P.; Tümer, Nazlı; Zadpoor, Amir A.

DOI

10.1016/j.apmt.2019.03.007

Publication date

2019

Document Version

Final published version

Published in

Applied Materials Today

Citation (APA)

Callens, S. J. P., Tümer, N., & Zadpoor, A. A. (2019). Hyperbolic origami-inspired folding of triply periodic

minimal surface structures. Applied Materials Today, 15, 453-461.

https://doi.org/10.1016/j.apmt.2019.03.007

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ContentslistsavailableatScienceDirect

Applied

Materials

Today

jo u r n al hom e p a g e :w w w . e l s e v i e r . c o m / l o c a t e / a p m t

Hyperbolic

origami-inspired

folding

of

triply

periodic

minimal

surface

structures

Sebastien

J.P.

Callens

,

Nazlı

Tümer,

Amir

A.

Zadpoor

DepartmentofBiomechanicalEngineering,DelftUniversityofTechnology(TUDelft),Mekelweg2,2628CDDelft,TheNetherlands

a

r

t

i

c

l

e

i

n

f

o

Received30January2019

Receivedinrevisedform26February2019 Accepted15March2019 Keywords: Minimalsurfaces Shape-shifting Foldingkinematics Curvature Geometry Architectedmaterials

a

b

s

t

r

a

c

t

Origami-inspiredfoldingmethodspresentnovelpathwaystofabricatethree-dimensional(3D) struc-turesfrom2Dsheets.Akeyadvantageofthisapproachisthatplanarprintingandpatterningprocesses couldbeusedpriortofolding,affordingenhancedsurfacefunctionalitytothefoldedstructures.Thisis particularlyusefulfor3Dlattices,possessingverylargeinternalsurfaceareas.Whilefoldingpolyhedral strut-basedlatticeshasalreadybeendemonstrated,morecomplex,curvedsheet-basedlatticeshavenot yetbeenfoldedduetoinherentdevelopabilityconstraintsofconventionalorigami.Here,anovel fold-ingstrategyispresentedtofoldflatsheetsintotopologicallycomplexcellularmaterialsbasedontriply periodicminimalsurfaces(TPMS),whichareattractivegeometriesformanyapplications.Theapproach differsfromtraditionalorigamibyemployingmaterialstretchingtoaccommodatenon-developability. OurmethodleveragestheinherenthyperbolicsymmetriesofTPMStoassemblecomplex3Dstructures fromanetofself-foldablepatches.Wealsodemonstratethatattaching3D-printedfoldableframesto pre-strainedelastomersheetsenablesself-foldingandself-guidedminimalsurfaceshapeadaptionupon releaseofthepre-strain.ThisapproacheffectivelybridgestheEuclideannatureoforigamiwiththe hyperbolicnatureofTPMS,offeringnovelavenuesinthe2D-to-3Dfabricationparadigmandthedesign ofarchitectedmaterialswithenhancedfunctionality.

©2019TheAuthors.PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

1. Introduction

Stochasticsheet-based micro-architecturesare ubiquitousin engineeringandnaturalmaterialsandappearintheformoffoams, sponges,bonetissue,orattheinterfaceofphase-separated materi-als[1].Theirperiodiccounterparts,beingmoretractabletostudy, have received widespread attention too, especially geometries basedontriplyperiodicminimalsurfaces(TPMS).Minimalsurfaces aresurfacesthatlocallyminimizeareaandaredefinedtohave van-ishingmeancurvatureeverywhere(H=0),whichgivesrisetotheir saddle-shapedappearance(withGaussiancurvatureK≤0).Triply periodicminimalsurfacesformaspecialclassofminimalsurfaces thatarebicontinuousandperiodicinthreedirections,hencethey extendinfinitelyanddividespaceintotwocontinuous,intertwined labyrinths[1].WhilemathematicianswerethefirsttostudyTPMS followingtheseminalworkofSchwarz[2],thefrequent observa-tionsofTPMSmorphologiesin awiderange ofnaturalsystems

[3],rangingfromself-assembledlipids[4]tobutterflywingscales

[5],hassparkedtheinterestofotherscientistsaswell.Indeed,the uniquestructure-propertyrelationshipsofferedbyTPMShave

con-∗ Correspondingauthor.

E-mailaddress:s.j.p.callens@tudelft.nl(S.J.P.Callens).

tributedtowarddevelopmentofhighlyefficientcellularsolids.For example,TPMS-basedstructureshavebeenshowntocombinehigh yieldstress,lowelasticmodulus,exceptionallyhighfatigue resis-tance,andbone-mimickingtransportproperties,makingtheman idealgroupofbonesubstitutes[6–8]Otherexamplesinclude pho-tonicmetamaterials[9],architectedmaterials[10,11],orporous membranestructures[12].Thefunctionalityoflatticestructures in general,and TPMS-basedsolidsin particular,couldbevastly augmented withplanarsurface-functionalization processes. For instance,preciselycontrolledsurfacenanopatternscouldenhance the optical [13], wetting [14], osteogenic [15], and antimicro-bial[16]propertiesofsurfaces,whileplanarprinting/imprinting techniquesenableintegrationofembeddedelectronicsinto mate-rials[17]. Theincorporation ofsurface-relatedfunctionalities is particularlyattractiveforTPMSstructures,giventheirverylarge surface-to-volume ratios. However, most 3D lattice structures, especially those based on TPMS, can currently only be manu-factured using 3D printing techniques, which are incompatible withtheplanarfunctionality-inducingprocesses.Tocircumvent thisincompatibilitybetween3Dprintingandplanarprocesses,an origami approachhasrecentlybeenproposed[18],where peri-odicbeam-basedlatticeswereshown tobefoldablefroma flat startingstate,therebyenablingsurfacefunctionalizationpriorto folding.However,giventhehyperbolic,i.e.non-developable,nature

https://doi.org/10.1016/j.apmt.2019.03.007

454 S.J.P.Callensetal./AppliedMaterialsToday15(2019)453–461

ofTPMS,suchconventionalorigamitechniquesareinherently ill-equippedtotackletheproblemoffoldingTPMSmorphologiesfrom aflatstate[19].Therefore,weintroduceafundamentally differ-entapproachthatcircumventsthedevelopabilityconstraintand enablesthefoldingofhyperbolicminimalsurfacemorphologies, byleveragingsheetstretching.Duetotherequirementofsheet stretching,onecoulddescribethisfoldingmethodas“origomu”, signifyingthefoldingofstretchablerubber-likesheets(“ori”means folding,“gomu”meansrubber),asopposedtoorigami,signifying thefoldingofnon-stretchablepaper-likematerials.While compu-tationaltoolsanddifferentialgrowth-basedfabricationmethods forthegenerationofnon-developablegeometriesfromflatsurfaces havebeendevelopedinrecentyears[20–23],theseapproaches typ-icallyrequirecomplicatedmaterialprogrammingandtheresulting shapesareoftenrestrictedtotopologicaldisks.Thefoldingmethod thatweintroduceinthispaperenablesthefoldingoftopologically complexporousstructures withminimalsurface morphologies, whilerequiringverylimitedmaterialprogramming.Therationale behindourapproachconsistsofrealizingcurvedminimalsurface patchesfromaflatstate,bycombiningrigidfoldableframeswith pre-strainedelastomersheets.Multipleofthesefoldablepatches couldthenbeconnectedtogetherin anetand usedasbuilding blockstofoldamyriadof3DTPMS-basedarchitectures,ranging fromsingleunitcellstolargerassembliesconsistingofmultiple unitcellsand3Dstackableminimalsurfacelayers.

2. Results

2.1. HyperbolicgeometryofTPMS

Triply periodic minimal surfaces belong to the realm of

hyperbolicgeometryandarisefromsymmetryoperationson fun-damentalpatches.ThisideaofconstructingaTPMSstructureas a 3D puzzle using a single, saddle-shaped puzzlepiece that is repeatedthroughoutthestructureiscentraltoourapproach.As ademonstrativeexample,thetranslationalunitcellofthe well-knownSchwarzPsurface(Fig.1a),canbetiledbyafundamental asymmetricpatch(Flächenstück) throughtwo symmetry opera-tions:mirrorreflections abouttheplanelinesofcurvature, and two-fold()rotationsaboutthestraightlines.Theresulting tri-angulartiling,withangles/2,/4,and/6,isnotcompatiblewith theEuclideanplaneE2_{,butisatilingofthehyperbolicplaneH}2_(it

isthe*246tilinginorbifoldnotation[24]),asseenintheconformal Poincarédiskmodel(Fig.1a).Thisillustratestheinteresting fea-turethataportionofH2_{canbeembeddedin3DEuclideanspaceE}3

bywrappingitontotheperiodicminimalsurface(albeitwithsome curvaturedistortion),analogoustoembeddingE2_inE3_bywrapping

itontoacylinder[24,25].Withinthecontextofthispaper,however, thisintrinsicconnectionbetweenH2_andE3_{underpinstheinherent}

complexityoftryingtounwrapTPMStoaflatstate,i.e.E2_.Thesame

minimalsurfacecouldbetiledwithdifferentpatches(Fig.1b),all constructedfromsomesymmetryoperationsontheelementary asymmetricpatch.Withinthewealthofknown,intersection-free TPMSandtheirrespectivesurfacepatches,ourapproachcovers thosesurfacesthatcouldbetiledbystraight-edgedskewpolygonal patches(homeomorphictoadisk).Anecessary(butnotsufficient) conditionthereforeistheexistenceofembeddedstraightlinesin theTPMS,whichareaxesoftwo-foldrotationandformthe“linear skeletalnet”ofthesurface[26].TPMSwithembeddedstraightlines weretermedbyFisherandKochas“spanningminimalsurfaces”

[27],andtheyarenecessarilyalsominimalbalancesurfaces(i.e. thetwolabyrinthsonbothsidesofthesurfacearecongruent)[28]. Ourfoldingapproachappliestothosespanningminimalsurfaces forwhichthegeneratingpatchisasurfacespanningaskew poly-gon.ThisexcludescertainTPMSsuchastheHsurface,whichdoes

containstraightlinesbutcannotbetiledbyskewpolygonalpatches

[27],andthewell-knownGyroidsurface,whichdoesnotcontain embeddedstraightlinesatall[28](seeSI).Thepresentedapproach does,however,coverarangeofotherwidelystudiedTPMS,fourof whichareincludedinthispaperasexamples(seeFig.1c,tiledby skewpolygonpatches):thePsurface,theDsurface(adjointtoP), theCLPsurface,andtheC(P)surface(complementarytoP).

2.2. Foldableminimalsurfacepatches

Thekeytoourorigamiapproachistherationaldesignofskew polygonal patches that could be flattened. We achieve this by addinghingesatsomeoftheverticesoftheboundaryframewhile keepingtheedgelengthsconstant,enablingacontinuousfoldingof theframefromaskewpolygontoa(simple)flatpolygon.Foraskew n-gon(n≥44),thisapproachrequires2≤k≤n/2hingesatthe ver-tices,whiletheotherverticesarekeptfixed.Sincetheinternalangle sumoftheskewpolygonsissmallerthanthatofsimpleflat poly-gons,i.e.



must increase during the unfolding motion(Fig.1d–g). Apply-ingtheGauss-Bonnettheoremtotheskewpolygonalboundary frameindicatesthatthefoldedframemustenclosenegative Gaus-siancurvature(seeSI)[29].Consequently,thefolding/unfoldingof thepolygonalboundaryframenecessitatesachangeinthe Gaus-siancurvatureofthesurfacespanningtheframethat,according toGauss’Theorema Egregium,canonlybeaccommodatedbyan areadistortion[30].Indeed,flatteningthesaddleshapedpatches requiresthesurfacespanningtheframetostretch(seeSection4). Inotherwords,themetricofthesurfacehastotransitionbetween aEuclidean(flat)metricandanon-Euclidean(saddle-shaped) met-ric.Withconventionalorigamifolding,thiswouldnotbepossible andthefoldedstructureswouldremainintrinsicallyflat(exceptfor somediscretepointsofnon-zeroGaussiancurvatureinsome tech-niques[31]).Inourfoldingapproach,therequiredareadistortion duringfoldingisachievedbyattachingrigid,foldableframestoa bi-axiallypre-strainedelastomersheet.Thepre-strainintheflatsheet entailstwokeybenefits:releasingthepre-strainedsheetdrives self-foldingoftheattachedframefromtheflatstatetothefolded state,andtheremainingpre-straininthesheetforcesittoadopta minimalsurfaceshape,bythevirtueofenergy(orarea) minimiza-tion.Thisisanalogoustothefamousdemonstrationsofminimal surfaceformationthatareobtainedwhendipping3Dwireframes inasoapsolution:thesoapfilmadoptsaminimum-energy mini-malsurfaceshape.Thesameprinciplehasbeenemployedtocreate physicalmodelsofminimalsurfaces(beforetheadventof3D print-ing)usingstretchedfabricsorpolymer sheets[32,33],and asa meanstoactuatecertainorigamitessellations[34].

Thefourpatchesconsideredhereandtheirfoldingkinematics are shown in Fig. 1d–g. The skewhexagonal frames for the P and Dsurfaces(Fig.1d,e) areequilateraland equiangular, and arethePetriepolygonsoftheregularoctahedron andthecube respectively.TheP-patchisflattenedtoanequilateraltriangle,and theD-patchtoanequilateralhexagonwithanglesof/2and5/6. Thissimplefolding/unfoldingkinematicsentailsarotationofangle aroundthree“creases”thatconnectthehingevertices(thedotted lines).InthecaseofthehexagonalpatchoftheCLPsurface(Fig.1f), onlytwohingesarerequired,and,thus,one“crease”.Thepatchcan thenbeflattenedtoarectanglewithsidesland2l.Finally,thepatch fortheC(P)surfaceisaskewoctagonwithalternatinganglesof/2 and/3,whichisflattenedtoabow-tieshapewithangles,/3, and4/3.Contrarytothethreeotherpatcheswherethelocation ofthehingepointsremainsfixed,theflatteningoftheC(P)patch requires in-planesliding ofthe hingepointsduring thefolding motion(seeSection4).TheC(P)skewoctagonalpatchcouldbe flattenedintodifferentshapesthatdonotrequireslidinghinges,

Fig.1. GeometryofTPMSandpatchfolding.(a)AtranslationalPunitcelldecoratedwiththehyperbolic*246tilingofthefundamentalasymmetricalpatch.(b)Alternative patchestotilethePsurface,showntogetherwiththeconventionalunitcell.(c)ThefourTPMSconsideredhere.Fromlefttoright:P,D,CLP,andC(P)surface.(d)Folding kinematicsforthestraight-edgedskewpolygonalpatchesoftheP,D,CLP,andC(P)surfaces,respectively.

butthesealternativeshapesareunfitforbuildingan“overlap-free” 2Dnetthatcouldbefoldedinto3DTPMSmorphologies(seeSI). 2.3. Patchconnectionsandunitcellfolding

Nowthatasuitablefolding/unfoldingapproachforthe mini-malsurfacepatches,i.e.thepiecesofthe3DTPMSpuzzle,hasbeen obtained,thenextkeystepistoconnectpatchestogethertobuild largerportionsoftheminimalsurfaces.Assuch,wedevelopa fold-able2D“net” thatresultsina 3DportionoftheTPMSonceall patcheshavebeenfolded.We identifytwo possibleattachment strategies, namelyedge-connections and vertex-connections.In thefirsttype (Fig.2a),two patchesareconnected bymeansof a-rotationaroundtheircommonedgeu,whichisaninherent propertyofthestraightlinesembeddedinminimalsurfacesand justifiesourfocus onstraight-edged skewpolygonalpatches.A consequenceofthis-rotationisthatthecommonedgeremains coplanarwithtwoadjacentedges

v

v

duringtheentirefoldingmotion(Fig.2a).Theconnectionbetween bothpatchesistherefore“rigid”andthereisnoneedtoactuatethe foldingofonepatchrelativetotheother.Thevertex-connection type attachestwopatchesat a vertexthat isnot ahinge point (Fig.2b).Avertex-connectionisestablishedasa2rotationabout anaxisnthatisnormaltotheedgesu1and

v

where=cos−1



||u1||||v1||



.Infact,thistypeofconnectionisthe resultoftwoconsecutiveedge-connections,i.e.a-rotationover

v

edge-connections,theedgesu1,

v

v

thatthevertex-connectionisalsorigidandcanbephysically real-izedwithouthavingtoaccountfortherelativemotionsbetween bothpatchesattheconnectingvertex.Experimentingwithpatch connections quickly reveals the most crucial challenge in our

foldingstrategy:avoidingoverlapsinthe2Dnet.Thischallenge arisesasaconsequenceoftryingtoconfinethehyperbolictilingof TPMSpatchestotheEuclideanplane.Forexample,thetilingwith skewhexagonsoftheDsurface(Fig.2c),isahyperbolic(6,4)tiling wherefourhexagonsmeetateveryvertex.Attemptingtoachieve thiswiththeflattenedhexagonalpatchesleadstoanoverlapping 2Dnet,whilethefoldedconfigurationisfreeofoverlaps(Fig.2c). ThisshowsthatitisnottrivialtounwrapTPMSmorphologiesinto 2Doverlap-freenets.Ourrationalapproachtoovercomethis chal-lengeconsistsoffirstcreatingfoldable,overlap-freenetsforthe TPMSunitcells,andusingthoseunitcellnetsasprototilesinthe constructionofoverlap-freenetsforlargerstructures.Thus,instead ofassemblinglargermorphologiespatchbypatch,weproposeto firstdefinethenetforasingletranslationalunitcelland conse-quentlyconnectthoseunitcellnetstogethertobuildlarger struc-tures.AsshowninFig.2d–g,thetranslationalunitcellsfortheP,D, CLPandC(P)surfacesrespectivelycouldallbefoldedfrom overlap-free2Dnetsconsistingentirelyofvertexconnections.Incaseofthe P,DandCLPsurfaces,theunitcellconsistsof4patcheswhilethe C(P)unitcellisconstructedusing6patches.Aconsequence(and advantageduringphysicalrealization)ofusingvertex-connections isthatallpatchesfoldinthesamedirection,whichisnotthecase foredge-connections,causingtheunitcellnettocloseinonitself, analogoustothefoldingofapapercube.ExceptfortheCLPunit cellnet,thevertex-connectedunitcellnetspresentedinFig.2are notunique,i.e.differentarrangementsofvertex-connectedpatches couldbegeneratedthatfoldintothesametranslationalunitcell. Examplesof differentnetsthat foldinto thesametranslational unitcellareshowninFig.S8(seealsoSI).Ingeneral,thenetfor aunitcellconsistingofnpatcheswouldneedatmostn−1 vertex-connectionsconnectingtwopatchestogether.However,anetwith fewerconnections couldbedesignedif morethan twopatches

456 S.J.P.Callensetal./AppliedMaterialsToday15(2019)453–461

Fig.2.Connectingpatches.(a)Theedge-connectionoftwoPpatches.(b)Thevertex-connectionoftwoDpatches.Atransparentpatchindicatesapatchthatfitsinbetween twovertex-connectedpatches.(c)Whentryingtoconformthehyperbolic(6,4)tilingoftheDsurfacetotheflatplane,onefrequentlyencountersoverlapsinthe2Dnet. (d–g)ThefoldingofTPMSunitcellsconsistingofvertex-connectedpatches.

couldbeattachedtogetheratthesamevertex(e.g.inthecaseofthe PandC(P)surfaces,seealsoSI).Thesamerationaletodesignnets forlargerTPMSstructuresapplies,independentofthechoiceofunit cellnet.However,certainunitcellnetsmightbepreferableinorder toconstructlargerassemblies,asexplainedinthenextsection. 2.4. Multiple-unitcellassemblies

ToestablishfoldingoflargerTPMSassembliesconsistingof mul-tipleunitcells,weconnectthe2Dnetsofseveralunitcellstogether. Thisrequirestheuseofedge-connections betweenthedifferent unitcells,sincefurtheruseofvertex connectionswouldleadto overlapsin3D,aconsequenceofthe“closing”oftheunitcells.To avoidoverlapsinthe2Dnets,however,notalledgesareavailable foredge-connections.Theadmissibleedgesforthefourunitcell netsconsideredherearehighlightedinFig.3a.Onlyalongthese admissibleedges,twounitcellscanbeconnectedwithoutcausing

overlapsinthe2Dnet.Theadmissibleedgesofagivenunitcellnet arethoseedgesthatlieontheedgesoftheconvexpolygonthat definestheconvexhullofthepatchvertices(seeFig.S8).Thusan edgewhoseendpointsareverticesoftheconvexhullwouldbean admissibleedgeforedge-connections.Alledgesthatarecontained entirelywithintheconvexhull(andnotontheboundary)are inad-missibleforedge-connections.Asimpleexampleoftwoconnected PunitcellsisprovidedinFig.3b,illustratingtheoppositefolding directionsofbothunitcells.Furthermore,Fig.3bshowshow con-nectingtwounitsalongoneedge,e.g.u1,mayprohibitaconnection

alonganotheredge,e.g.v1,asthiswouldotherwiseleadto

over-lapsin2D.Thenumberofadmissibleedgesvariesdependingon thechosenunitcellnet.Forexample,someunitcellnetsofthe C(P)surfaceallowforonlyasingleedge-connection,makingthem unsuitabletoextendthe2Dnetbeyondtwounitcells(seeFig.S8 andSI).ThenetthatisdepictedinFigs.2gand3a,ontheother hand,hassixadmissibleedges(appearinginthreepairs).Choosing

Fig.3.Foldingmultiple-unitassemblies.(a)Theunitcelledgesthatareavailableforedge-connections(highlightedinred)withoutcausingoverlapsinthe2Dnet.(b)Folding oftwoedge-connectedPunitcells.(c)Foldingofa10-unit-cellnetofthePsurfacewithoutimplementingsequentialfolding(toprow),givingrisetocollisions,andwith sequentialfolding(bottomrow),toavoidcollisions.Theyellowpatchintheleftpanestartsfoldingwithadelayrelativetotheotherpatches(fullyfoldedconfigurationat t=1).(d)Foldingofa14-unit-cellnetoftheDsurfacewithoutrequiringsequentialfolding.(e)Thesequentialfoldingofa7-unit-cellnetoftheC(P)surface,withafolding delayappliedtothecentralpatch(yellowintheleftpane).(f)Sequentialfoldingofa10-unit-cellassemblyoftheCLPsurface,containingthreeseparatefoldingstarting timestoavoidcollisions.SeeSIforadditionalmorphologiesandSIMovie1throughSIMovie6.(Forinterpretationofthereferencestocolorinthisfigurelegend,thereader isreferredtothewebversionofthisarticle.)

thelatterunitcellnetasaprototilewouldthusoffermore free-domtoconstructalargernet,sinceitcouldconnecttothreeother unitcells.ToexplorethefoldingoflargerandmoregeneralTPMS morphologies,weconstructedacomputationaltoolthatcalculates thefolding andtheresulting3Dconfigurationofauser-defined input2Dnet.Startingfromasingleunit cell,weextendthe2D netbyaddingmoreunitcells,withoutcausingoverlaps,and ver-ifytheresultingfoldingmotionandthefinal3Dmorphology.The underlyingfoldingkinematicsofourapproachissurprisingly sim-ple,sinceallfoldinginformationiscapturedinthekinematicsof asinglepatch(assumingallpatchesfoldsimultaneously)andin thewaythepatchesareconnectedtogetherthroughvertex-and edge-connections(seeSection4).Alargevarietyof3DTPMS-based structurescouldbeobtainedwithouthavingtodeterminea sepa-ratefoldingstrategyforeachsimplybyvaryingthe2Darrangement ofthepatches(seeSI).Whilestilltractableforsmallerstructures, therelationshipbetweenagiven2Dnetandtheresulting3D struc-turebecomesincreasinglycomplexforlargerstructureswithmany patchconnections,involvingintricatefoldingmotionsand poten-tiallyoverlappingpatchesin3Dthatdonotoverlapin2D,whichare

detectedinthetoolbycheckingforduplicatesetsofvertex coordi-nates.Usingthisexplorativetool,amultitudeof2Dnetscouldbe designedtofoldawiderangeofcomplexminimalsurface struc-tures,someofwhichareshowninFig.3andmorecanbefoundin theSI(e.g.minimalsurfacestring-likemorphologiesorstackable layers).InFig.3c–f,thefoldingof multipleconnected unitcells fortheP,D,C(P),andCLPsurfacesisillustrated(seeSIMovie1to SIMovie6).Duetothecomplexfoldingmotionsarisingforsuch largestructures,collisionsduringfoldingcouldoccur,asshownin thetoprowofFig.3cforthefoldingoftenPunitcells.We demon-stratethatsequentialfolding,i.e.temporalcontroloverthefolding motion,couldalleviatethisproblem.Asasimpleexample,aslight delayinthefoldinginitiationofacentrallylocatedunitcellinthe netsoftheP(Fig.3c)andC(P)(Fig.3e)morphologiesissufficient toenablecollision-freefolding,bymaintainingadequate separa-tionbetweentheoutwardsextendingarms.FortheCLPexamplein

Fig.3f,threedifferentstartingmomentsareimplementedtoavoid collisions,aconsequenceofthehighaspectratiooftheunitcell net,whiletheassemblyof14Dunitcells(Fig.3d)didnotrequire sequentialfolding.ThefoldingsequencesshowninFig.3c–fare

458 S.J.P.Callensetal./AppliedMaterialsToday15(2019)453–461

examplesthatdemonstratehowsmallchangesinthefolding initi-ationofcertainunitcellscouldpreventcollisions.However,many differentfoldingsequencescouldbeemployedtoachievethesame result,e.g.sequencesinvolvingvariationsinthefoldingspeedand thestartingtimeoftheunitcells,individualpatches,oreven indi-vidualhinges,therebyofferinggreaterdesignfreedomtoensure collision-freefolding.Sequentialfoldingcouldbephysically real-izedindifferentways,e.g.usinglocalizedexternaltriggers[35]or built-indesignfeatures[36].

2.5. Self-foldingexperiments

Wephysicallyrealizedourself-foldingminimalsurface struc-turesbyattachingstretchedelastomersheetsto3Dprintedfoldable frames (see Section 4). Upon release, the strain energy in the sheetscausestheflatpolygonalframetoself-foldintothedesired skewpolygonalconfiguration,andthesheetspanningtheframe adoptsanenergy-minimizingsaddle-shapedgeometry, approxi-matingtheminimalsurface(Fig.4a).Beingacombinationof(semi-) rigidbeamsandflexiblesheets,ourstructuresrepresentaspecial caseofKirchhoff–Plateausurfaces[37],inwhichvirtuallyallframe deformationisconcentratedatthehinges.Thedirectionoffolding oftheframeiscontrolledbytheeccentricpositionofthesheetwith respecttothehingelocation:thepre-stretchedsheetisattached ononesideoftheframe,whilethehingelayer(seeSection4and Fig.S4)issituated ontheotherside,ensuringpreferential fold-inginthedirectionofthesidetowhichthesheetisattached.The levelofpre-strainintheelastomersheetsshouldatleastbehigh enoughtoaccommodatetherelativeamountofareashrinkagethat occursinthesheetduringthefoldingmotion,whichvariesbetween approximately10% and 30% depending onthe patchtype (Fig. S1b).Moreover,thestrainenergystoredinthepre-stretchedsheets shouldbehighenoughtodrivethefoldingmotionoftheframe,i.e. toovercomethebendingresistanceatthehingesandthe gravita-tionalforcesactingontheframe,andalsotokeeptheframeinthe foldedconfigurationafterwards.Duringtheexperimentspresented here,thesheetswerebi-axiallystrainedby50%inbothdirections (seeSection4),whichenabledtherapidself-foldingoftheframe andresultedinampleresidualtensioninthesheettomaintainthe frameinitsfoldedconfiguration.Weassessedthemean(H) cur-vatureprofileofthesheetsurfaceonthebasisofmicro-computed tomographyscansoftheself-foldedpatches(seeSection4), find-ingthatHisclosetozeroeverywhere(Fig.4b).Thisdemonstrates thattheself-foldedpatchesadoptashapeveryclosetotheideal minimalsurface,asminimalsurfacesaremathematicallydefined ashavingH=0everywhere.Whiledeviationsfromtheideal mini-malsurfaceshapeariseasaconsequenceof,e.g.,sheetwrinklingat thehinges,non-uniformsheetstraining,andcompetitionbetween thebendingandstretchingenergiesofthefinitethicknesssheet

[38],wedemonstratethatarelativelysimplecombinationofrigid andflexiblecomponentsenablesthefoldingofcomplex,hyperbolic shapesthatareincompatiblewithtraditionalorigamimethods.In additiontoindividualpatches,wealso3Dprintedunitcellnets ofthefourTPMS,whichwereself-folded tothefinal configura-tionafterattachingandreleasingthepre-strainedsheetmaterial (Fig.4c).Sinceallpatcheswithinthesameunitcellfoldinthesame direction,aconsequenceofusingvertex-connections,thesheetis attachedtothesamesideforallpatcheswithintheunitcell,which isconvenientduringfabrication.Todemonstratetheself-folding capabilityofthisapproach,atimesequenceoftheself-foldingof theCLPunitcellisshown inFig.4d(seealsoSIMovie8).Upon release,thepre-stretchedlatexsheetrapidlycausestheframeto self-fold,andthebuilt-instoppingmechanismscauseittostopat thedesiredconfiguration(seeSection4).Moreover,weshowthat attachingunitcellnetstogetherusingedge-connectionsenables theself-foldingoflargerassemblies(Fig.4e).Asademonstrative

example,fourconnectedunitcellsfortheD-surfaceareshownin

Fig.4e,butthisapproachisalsoapplicabletolargermorphologies aslongascollisionsduringfoldingareavoided.

3. Discussion

Whereaspreviousorigami-baseddesignshavebeenrestricted toprimarilydevelopablegeometries,suchaspolyhedralstructures andclassicalorigamitessellations,thepresentedapproachrealizes theself-foldingofpreviouslyunfoldable,non-developable,TPMS structuresthroughtherationaldesignoffoldablesurfacepatches andtheirconnections.WhilewefocusedhereonfourTPMStypes, otherspanningminimalsurfacescouldalsobeconstructed,ifa suit-ableflatteningoftheskewpolygonalpatchisfound.Byelucidating thefolding kinematics ofthe fourtypes of generatingpatches, andbyconnectingmultiplepatchesusingeithervertex-or edge-connections,alargevarietyoffoldable3Dmorphologiescouldbe generatedthatareallaportionoftheinfiniteminimalsurface.Our focushasbeenongenerating2Dnetsbyfirstconstructingunitcells, usingvertex-connections,andconnectingunitcellstogetherusing edge-connections.However,manydifferentfoldable2Dnetscould begenerated,e.g.tofoldperiodiclayersofTPMSunitcellsthatcould bestackedtoassemblearbitrarilylargeportionsoftheTPMS(see SI).Therelativelysimplefolding“rules”ofourapproach,capable ofdescribingcomplexfoldingmotions,couldpotentiallybenefit fromefficientoptimizationalgorithmstouncoverfoldablenetsfor specificTPMSmorphologies.DuetothecomplexityoftheTPMS morphologies,akeychallengeinthefurtherdevelopmentofthe presentedorigamiapproachistheabilitytoaccuratelycontrolthe foldingmotion,i.e.notonlythetemporalaspectbutalsothefinal configuration,aswellasfindingwaystolockthestructureonce folded.Thepresentedapproachoffersnewandexciting perspec-tivesinthedevelopmentofmetamaterials,duetoregainedaccess totheflatstartingsurface.Weenvisionnotonlyorigami biosys-temapplications[39],e.g.biomimetictissueengineeringscaffolds withosteogenicand bactericidalsurfacenano-patterns,butalso bi-continuousmembranesforfluidtransferwithtailored wettabil-ity(e.g.,self-cleaningmembranes)orTPMS-basedstructureswith embeddedelectroniccomponents. In this work,we focused on sheet-basedstructures,butbeam-basedlatticesderivedfromthe boundaryframescouldalsobefolded.Finally,ourapproachisnot strictlyboundbyaspecificlengthscale,meaningthatitcouldalso inspiretheself-foldingofarchitectural-scaletensilestructures,nor isitlimitedtospecificconstituentmaterials,aslongasasufficient areadistortionofthesheetsurfacesandtherigidityoftheboundary framescanbeobtained.

4. Materialsandmethods

4.1. Patchkinematics

Thefoldingkinematicsofthestraight-edgedTPMSpatcheswere implementedinMatlab(Mathworks,USA)bycalculatingthe Carte-siancoordinatesoftheverticesasafunctionofthefoldangle,from zerountilthefinalfolded configuration.Thevertexcoordinates inthefoldedconfigurationswereobtainedfromFisherandKoch

[27].Bycalculatingthedifferencebetweentheinternalanglesum inthefolded,



i,andflat,



i˛fi,configurations,therequired

amountof angularchangeforunfoldingwasobtainedforevery hingevertex.Fornhingedvertices,theangularchangethatneeds tobeaccommodatedforeveryhingebythefolding/unfoldingis thereforegivenas:

˛n

(





Fig.4. Self-foldedphysicalmodels.(a)3D-printedfoldableframesforthefourpatchtypesinflat(toprow)andfolded(bottomrow)configurationsafterthestretched latexsheetshavebeenattached(seeSection4).(b)Themeancurvatureestimatedusingthe3Dreconstructionsofthefourpatchtypesobtainedfrommicro-computed tomographydata(seeSection4andSIMovie7formoreinformation).(c)3D-printedfoldableTPMSunitcellsintheflat(toprow)andfolded(bottomrow)configurations. (d)Theself-foldingoftheCLPunitcellthroughthepre-tensionpresentinthelatexsheet(seeSIMovie8).(e)AnassemblyoffourDunitcellsintheflat(left)andfolded (right)configurations.Allscalebarsare20mm.

Thefinalfoldanglewascalculatedusingthetrigonometric rela-tionsthatwereobtainedforthefoldingmotionandbyconsidering theorthogonalprojectionoftheskewpolygonalpatch.As men-tioned in themain text,the bowtie patch for theC(P) surface requiredthein-planeslidingofthehingevertices.Thisisillustrated intheFig.S1a,where_ı1andı2representtheamountofin-plane

slidingofthetwopairsofoppositehingeverticestoaccommodate thefolding.Thesedistancesaregivenby:

ı1=l



































4.2. Minimalsurfacegeneration

Thewidely-used(andfreelyavailable)SurfaceEvolversoftware

[40] was used to find the minimal surface spanning a given

boundary frame. The software numerically finds the minimal

surfacebyminimizingthesurfaceenergyusingagradientdescent method.UsingMatlab,theinputfilesforeverydesiredfoldangle weregenerated, containingthevertexcoordinatesand theedge numbersof theboundaryframe. Thesurface wasthenevolved

using two consecutive gradient descent and mesh refinement

steps,supplemented withequiangulation and vertexaveraging,

until area convergence was achieved. The resulting minimal

surfacemorphologieswerethenexportedas.objfilesand were renderedinKeyShot5(Luxion,USA).Asdescribedinthemaintext, thefoldingmotionoftheframeentailsanareadistortionofthe surfacespanningtheframe.ThisisillustratedinFig.S1b,showing theevolutionofthenormalizedsurfaceareaA/A0duringfolding,

obtainedbyfindingtheminimalsurfacespanningtheframesat everyfoldingstepusingtheSurfaceEvolversoftware.

4.3. Foldingkinematicstool

AMatlab toolwasdeveloped toexplore thefoldingof user-defined2Dnetsconsistingofflatpatchesconnectedtogetherusing eithervertex-oredge-connections.Byemployingthesymmetry properties that both types of connections entail(see themain text),thevertex coordinatesofeverypatchinthenetcouldbe determinedforeveryfoldanglefromtheinitialpatch,byusing appropriaterotationmatricesandtranslationvectors.For exam-ple,copyingtheinitialpatchalongoneofitsedgesrequiresrotating thecoordinatesbyanglearoundthatedge(Fig.S2).Thus,forboth edge-connectedpatchesinFig.S2,thecoordinatesofavertexinthe copiedpatchqiarerelatedtothecoordinatesofthecorresponding

vertexintheoriginalpatch_piby:

460 S.J.P.Callensetal./AppliedMaterialsToday15(2019)453–461

whereR()istherotationmatrixforarotationofangleaboutthe unitvector ˆr= r ||r||andisgivenas(32): R()=

⎛

⎜

⎝

x(1−cos()) ˆrxˆry(1−cos())− ˆrzsin() ˆrxrˆz(1−cos())− ˆrysin() ˆ

rxˆry(1−cos())− ˆrzsin() cos()+ ˆr2y(1−cos()) ˆryrˆz(1−cos())− ˆrzsin()

ˆ

rxˆrz(1−cos())− ˆrysin() ˆryrˆz(1−cos())− ˆrxsin() cos()+ ˆr2z(1−cos())

⎞

⎟

⎠

Thevectortrepresentsatranslationtoensurethattheoriginal patchanditsrotatedcopyareconnectedatthedesiredvertex.For example,foranedge-connectionabouttheedgedefinedbypoints p1andp2(Fig.S2),thetranslationvectorisgivenby:

t=p2−R()·p2

We verifiedthe absenceof coinciding patchesin the folded configurationsbycheckingforduplicatecoordinatesinthetotal coordinatematrix.Thefoldingsequencesformultiple-unit assem-blieswerevisuallyexaminedforpatchcollisionsaftertheminimal surfaceswereaddedtotheboundaryframesinSurfaceEvolver.To avoidcollisions,sequentialfoldingwasimplementedbyusing mul-tiplefoldanglevariablesithatincreaseatthesameratebutstart

atdifferenttimes.Fig.S3showstwoedge-connectedPunitcells thatwerefoldedaccordingtodifferent,showingthatoneunitis essentiallya“time-shifted”copyoftheother,i.e.

a=b−ıt

whereıtrepresentsthetimeshiftandcanbepositiveornegative.

4.4. Designoffoldableframes

Foldable patchand unit cellframes were designed in Solid-Works2016(DassaultSystèmes,France).Theframeedges,which aresupposedtobehaverigidly,weregivenasquare2mm×2mm cross-section,whilethehingesweredesignedasa0.3mmlayer connecting the rigid edges and facilitating the folding motion throughlocalbending.Thishingedesignenabledefficientfolding yetsimplefabrication.Atthelocationofthehinges,theframeedges weregivenachamfersuchthatthefoldingoftheframewouldbe haltedatthedesiredfoldangle(Fig.S4).

4.5. 3Dprinting

Theframeswere3DprintedonanUltimaker2+FDMprinter (Ultimaker,TheNetherlands)usingpoly-lacticacid(PLA)filaments witha0.25mmdiameternozzleandalayerthicknessof0.6mm. InthecaseoftheC(P)surface,aflat‘star’patch(seeSItext)was printed,whichwasmanuallydeformedintothebow-tie config-urationafterprintingtoenablethein-planeslidingofthehinge verticesuponfolding.Latexsheets(150␮mthick,TheraBand,USA) werebi-axiallystretched(ε1=ε2≈0.5)andfixedtoacuttingboard.

Theflat frames were adhesively bonded tothe stretched latex sheetsusingacyanoacrylateadhesive(Bison,TheNetherlands)and werecuredatroomtemperature.Next,thelatexsheetwascut alongtheoutsideboundaryoftheframe,therebyonlyretainingthe stretchedlatexspanningtheframe.Theframewasthenreleased fromthecuttingboardandwasallowedtoself-foldintothefinal configuration.

4.6. Micro-computedtomographyandcurvatureestimation Micro-computedtomography(␮-CT)imagesoffour3Dprinted self-folded patches (one from each type of the minimal sur-facesconsideredhere)wereacquiredusingaPhoenix Nanotom scanner (General Electric, USA). Tomographic reconstructions weremadewitha slice increment of22.5␮m,and a matrix of 2284×2284 pixels. The voxel size of the volumetric data was

22.5␮m×22.5␮m×22.5␮m. Asthe resultingsize of the volu-metricdatasetswastoolarge(over20GB)tobeopenedbyany imageanalysissoftwarepackage(e.g.,3Dslicer,Fiji) ona desk-topcomputerwithIntel(R)Xen(R)E5-2687W(2cores)at3.40GHz and64.0GBRAM,each3Dvolumetricintensityimagewasresized byafactorof0.4usingthefunction“imresize3”availablein Mat-lab(Mathworks,USA).Subsequently,eachvolumetricdatasetwith avoxelsizeof56␮m×56␮m×56␮mwaspost-processedusing Mimics(version14.01,Materialise,Belgium).Usingthissoftware, allpatchesweresegmentedand3Dmodelswerereconstructed basedonthesegmentationresults.Duringthereconstruction,the smoothingfunctionavailableinMimicswasappliedwitha smooth-ingfactorof1.0.Toensureproperdefinitionofthecontourofthe patches,smoothingeffectswerevisuallyexamined.Usingthesame software,3DmodelswereexportedasSTLfiles.Themeancurvature ofeachtriangulatedpatchsurface(.STL)wasestimatedusingthe “vtkCurvatures”classofTheVisualisationTookit(VTK)inPython

[41].

Dataavailability

Alldatausedtogeneratetheseresultsisavailableinthemain textorsupplementarymaterial.Thecodescouldbeobtainedfrom thecorrespondingauthoruponrequest.

Acknowledgements

TheauthorsarethankfultoArjanThijssenfromtheMicrolab atDelft Universityof Technologyforexecuting the␮-CTscans. ThefirstauthoristhankfultoRyanC.Haywardforintroducingthe term“origomu”duringapresentationatIMECE2018andtoGerd Schröder-TurkforteachingonthegeometryofTPMSduringthe summerschoolon“Geometryandtopologyincontemporary mate-rialsscience”.Theresearchleadingtotheseresultshasreceived fundingfromtheEuropeanResearchCouncilundertheERCgrant agreementno.[677575].

AppendixA. Supplementarydata

Supplementarydataassociatedwiththisarticlecanbefound,in theonlineversion,atdoi:10.1016/j.apmt.2019.03.007.

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