渐变折纸结构 83页

ＴＩＡＮＪＩＮＵＮＩＶＥＲＳＩＴＹ中国第—臓代大＾ＦＯＵＮＤＥＤＩＮｉ８９５全日制工程硕士学位论文＿＿＿＿＿＿＿＿？戀■？１匪机械工程领域：作者姓名：谢瑞康天津大学研究生院＿２０１５年丨１２月．．．：？．．． 独创性声明本人声明所呈交的学位论文是本人在导师指导下进行的研究工作和取得的研宄成果，除了文中特别加以标注和致谢之处外，论文中不包含其他人己经发表或撰写过的研究成果，也不包含为获得天津大学或其他教育机构的学位或证书而使用过的材料一。与我同工作的同志对本研究所做的任何贡献均己在论文中作了明确的说明并表示了谢意。学位论文作者签名：謝鋒Ｉ签字日期：年以月以日学位论文版权使用授权书本学位论文作者完全了解天津大学有关保留、使用学位论文的规定。特授权天津大学可以将学位论文的全部或部分内容编入有关数据库讲行柃索，并采用影印、缩印或扫描等复制手段保存、汇编以供查阅和借阅。同意学校向国家有关部门或机构送交论文的复印件和磁盘。（保密的学位论文在解密后适用本授权说明）学位论文作者签名：导师签名：〇ａ：２ｉＪｒ签字日期年Ａ月Ｍ日签字日期：年月乂Ｒ 天津大学硕士学位论文渐变折纸结构TheGradedOrigamiStructures学科专业：机械工程作者姓名：谢瑞康指导教师：陈焱教授企业导师：宋宝敬高工天津大学机械工程学院二零一五年十二月 摘要渐变结构在自然界中普遍存在，诸多生物通过自身组织的渐变来优化整体结构性能，以适应其生活环境下的特定载荷条件。本文将折纸图案和渐变结构结合起来，形成渐变折纸结构，以探究折纸更广泛的应用。本文首先探究了形成渐变折纸结构的方法。基于刚性折纸条件，提出了改变单元长度，改变单元角度和改变单元数三个基本方法。选取了目前研究最广泛的一种折纸图案Miura-ori作为研究对象，形成了多种多层叠加的渐变结构，并研究了它们的几何参数和特性。针对折纸管状结构，选取Arc-Miura和Arcpattern两种图案作为研究对象，应用上述三种基本方法及其混合后形成的新方法对图案进行渐变，形成了刚性折纸管状结构。其次，在得到的渐变折纸结构的基础上，对单层渐变Miura-ori作为夹芯的三明治结构在三点弯和渐变Arcpattern在轴向冲击下的反应进行了研究。结果表明在三点弯这种载荷条件下，渐变Miura-ori作为三明治结构夹芯相对于正常的Miura-ori夹芯，能量吸收和材料利用率都得到了较大提升。最后，对折纸结构的一种加工方法进行了介绍，应用3D打印机进行折纸结构的加工，其中折痕和刚性平面分别用软材料和硬材料打印。结果表明这种方法对于折纸结构是适用并省时的。关键词:折纸图案，刚性折纸结构，Miura-ori图案,超材料。I ABSTRACTTherearemanyexcellentgradedstructuresexistinginnaturetooptimizethemechanicalpropertiesinvariousloadsituationsbyadjustingthedistributionofmaterials.Inthisthesis,rigidorigamiandgradedstructureconceptarecombinedtogethertoformthegradedorigamimetamaterials.Thefirstpartofthethesisfocusesonthemethodstoformgradedrigidorigamipatterns.Threeindependentmethodsareproposed,includingchangingthelengthofcreaselines,changingthesectorangle,andchangingthenumberofunits.Oneofthemostfamousrigidorigamipatterns,Miura-ori,ischosentogeneratethegradedorigamistructures,andthegeometricparametersandpropertiesarestudiedbriefly.Inthesecondpart,derivativeMiurapatterns,Arc-MiuraandArcpatternareselectedtoachievethegradedorigamitubes.Thegeometricparametersandcorrespondingrelationshipsareexplored.Thethirdpartisdevotedtotheengineeringapplicationsofthesestructures.Numericalsimulationsbasedonthefiniteelementmethodareemployedtoassessthemechanicalproperties,inwhichquasi-staticthree-pointbendingresponseofsandwichbeamswithgradedMiura-oricoreandquasi-staticimpactresponseofgradedorigamitubesareexplored.TheinvestigationrevealsthatsandwichbeamwithgradedMiura-oricorehavepreferableenergyabsorptioncapabilityinthree-pointbendingcomparedwiththenormalMiura-oricore.Manufacturingoforigamipatternsformsthefinalpartofthethesis.Prototypesofrigidandnon-rigidorigamipatternaregeneratedusing3Dprinter.Theircreaselinesandpanelsareprintedbydifferentmaterials,oneisthesoftmaterialandtheotherishardmaterial.Thismanufacturingmethodisapplicabletovariousorigamipatterns,whichshowstimesaving.Keywords:Origamipattern,gradedrigidorigamistructure,Miura-ori,metamaterial.II CONTENTS摘要................................................................................................................................IABSTRACT.................................................................................................................IICONTENTS...............................................................................................................IIILISTOFFIGURES....................................................................................................VLISTOFTABLES.................................................................................................VIIINOTATION...............................................................................................................IXChapter1INTRODUCTION.....................................................................................11.1GradedStructures.....................................................................................................11.2Origami....................................................................................................................11.3Aim,ScopeandOutline...........................................................................................2Chapter2LITERATUREREVIEW..........................................................................42.1CurrentResearchonGradedStructures...................................................................42.1.1GradedStructuresinNature..........................................................................42.1.2ApplicationofGradedStructures..................................................................62.2CurrentResearchonOrigamiPatterns....................................................................72.2.1OrigamiandRigidOrigami..........................................................................72.2.2RelationshipbetweenOrigamiandSpherical4RLinkage...........................92.3OrigamiGeometry.................................................................................................10Chapter3GEOMETRYOFGRADEDORIGMAIMETAMATERIALS...........153.1ConditionofRigidOrigamiPatterns.....................................................................153.2GradedOrigamiMetamaterialsBasedonMiura-ori.............................................223.2.1GradedMiura-oriPatternBasedonIndependentMethods........................223.2.2GradedMiura-oriPatternBasedonCombinedMethodsandClassification..............................................................................................................................273.3GradedOrigamiMetamaterialsBasedonTapered-Miura.....................................313.4Poisson’sRatioofGradedOrigamiMetamaterials...............................................333.4.1In-planeEndtoEndPoisson’sRatioofGradedSingle-layerOrigamiMetamaterial........................................................................................................33III 3.4.2Out-of-planePoisson’sRatioofGradedMulti-layerOrigamiMetamaterial..............................................................................................................................363.5Summary................................................................................................................42Chapter4GEOMETRYOFGRADEDORIGAMITUBES.................................434.1GradedArc-MiuraPattern.....................................................................................434.1.1GradedArc-MiuraPatternBasedonIndependentMethods.......................434.1.2GradedArc-MiuraPatternBasedonCombinedmethods...........................464.2GradedArcPattern................................................................................................484.2.1GradedArcPatternBasedonIndependentMethods..................................484.2.2GradedArcPatternBasedonCombinedMethods.....................................50Chapter5MECHANICALBEHAVIOUROFGRADEDORIGAMISTRUCTURES...........................................................................................................525.1Quasi-staticImpactResponseofGradedOrigamiTubes......................................525.2Three-pointBendingResponseofSandwichStructureswithGradedMiura-oriCore..............................................................................................................................575.2.1Three-pointBendingofSandwichBeamwithGradedMiura-oriCoreBasedonChangingtheNumberofUnitsinyDirection.....................................575.2.2Three-pointBendingofSandwichBeamwithGradedMiura-oriCoreBasedontheCombinationofChangingtheLengthofCreaseLinesandSectorAngleinxDirection.............................................................................................58Chapter6THEMANUFACTUREOFORIGAMISTRUCTURES....................61Chapter7FINALREMARKS..................................................................................647.1MainAchievements...............................................................................................647.2FutureWorks.........................................................................................................64REFERENCES...........................................................................................................66发表论文和参加科研情况说明.................................................................................68ACKNOWLEDGEMENTS......................................................................................69IV LISTOFFIGURES1.1Designoftheorigamitent……………………………………………………………………22.1Microstructureofbamboo……………………………………………………………………42.2Microstructureofstems……………………………………………………………………...52.3Schematicattachmentoftendonstomusclesandbones……………………………….…….52.4Schematicattachmentofasubmarinemusseltoastoneviathemusselbyssus......................62.5SchematicofafunctionallygradedbeamthatgivesPoisson-curvingbehavior…………......72.6Origamimodels………………………………………………………………………………72.7VertexwithNcreaselines.……………………………………………………...……………82.8StripewithNfoldedlines……………………….……………………………...……………82.9Definitionoforigamipatternanditscorrespondingsphericallinkage……………...……..92.10Assemblyoffourspherical4Rlinkages………………………………………...………..102.11Miura-oripatterngeometry…………………….……………………...…………...……..112.12ThefoldingprocessofaMiura-basedmetamaterial……….…….….…………...……..112.13Arc-Miurapatterngeometry………………..………………………………...…………....122.14TessellationofArc-Miurapattern………………………………….…..……...…………..132.15Arcpatterngeometry………………………………………………………...………...…..142.16TessellationofArcpattern……………………………………….…..……...………...…...143.1Coordinatesystemandlinkagegeometricparametersinlinks(i-1)iandi(i+1)connectedbyrevolutejointi………………………………………………………………………….163.2Definitionofasinglespherical4Rlinkage………………………………......……………..173.3Thecreasepatternofcase1…………………………………………..…….……………..183.4Thecreasepatternofcase2……………………………………………..…...……………..203.5Thecreasepatternofcase3………………..………………………………...……………..213.6GradedMiura-oripatternbasedonchangingthelengthofcreaselinesinxdirection……..233.7GradedMiura-oripatternbasedonchangingthesectorangleinxdirection…………..…..233.8GradedMiura-oripatternbasedonchangingthenumberofunitsinxdirection…………..243.9GradedMiura-oripatternbasedonchangingthelengthofcreaselinesinydirection……..253.10GradedMiura-oripatternbasedonchangingthenumberofunitsinydirection…..……..253.11GradedMiura-oripatternbasedonchangingthesectorangleinzdirection……….……..263.12GradedMiura-oripatternbasedonchangingthenumberofunitsinzdirection…..……..273.13Thegradedmulti-layerMiura-oristructurebasedonthecombinationofchangingthelengthV ofcreaselinesandsectorangleinxdirection,changingthenumberinydirection,andchangingthesectorangleandnumberofunitsinzdirection………………………………..293.14Thegradedmulti-layerMiura-oristructurebasedonthecombinationofchangingthelengthofcreaselinesandsectorangleinxdirectionandchangingthesectorangleinzdirection…………………………………………………………………..………….…..…...293.15Thegradedmulti-layerorigamistructurebasedonthecombinationofchangingthelengthofcreaselines,sectorangleandnumberofunitsinxdirection,changingthelengthofcreaselinesandnumberofunitsinydirection,andchangingthenumberofunitsinzdirection…………………………………………………………………………….…..…...303.16Thegradedmulti-layerMiura-oristructurebasedonthecombinationofchangingthelengthofcreaselines,sectorangleandnumberofunitsinxdirection,changingnumberofunitsinydirection………………………………………………….……………………….....………..303.17Tapered-Miurapatterngeometry…….……………………………………….…..………..313.18Twotapered-Miuraunitswithdifferentsectoranglesthatstackedtogether…….….…..323.19Thegradedmulti-layertapered-Miurastructurebasedonthecombinationofchangingthelengthofcreaselinesandsectorangleincirculardirection,andchangingthesectorangleinzdirection.……………………..………...…………………....………………………………..323.20Thegradedmulti-layertapered-Miurastructurebasedonthecombinationofchangingthelengthofcreaselines,sectorangleandnumberofunitsincirculardirection,andchangingthelengthofcreaselinesinradialdirection.………….………………………..………...……..333.21ThegradedMiura-oripatternbasedonchangingthelengthofcreaselinesandsectorangleinxdirection……………………………………………………….………............…..………..343.22TheendtoendPoisson’sratioofgradedMiura-oripattern.…….………………………..353.23TheinfluenceofbtotheendtoendPoisson’sratio………………..…….……………..363.24Threecasesofgradedmulti-layerorigamimetamaterials………….……………....……..363.25Theinfluenceof3totheout-of-planePoisson’sratio(cases1and2)……...………....…..393.26Theinfluenceof4totheout-of-planePoisson’sratio(cases1and2).……..…..……..….393.27Theinfluenceof3totheout-of-planePoisson’sratio(case3)……………..………..…...413.28Theinfluenceof4totheout-of-planePoisson’sratio(case3)……….…………..……....424.1GradedArc-Miurapatternbasedonchangingthelengthofcreaselines…….....………..…..434.2GradedArc-Miurapatternbasedonchangingthesectorangle………......……..……….…..444.3GradedArc-Miurapatternbasedonchangingthenumberofunits………..……….………..454.4GradedArc-Miurapatternbasedonthecombinationofchangingthelengthofcreaselinesandchangingthesectorangle……..……………………...………………….….......…...…..464.5GradedArc-Miurapatternbasedonthecombinationofchangingthelengthofcreaselinesandchangingthenumberofunits……..………………………………….……..…….....…..474.6GradedArc-MiurapatternbasedonthecombinationofchangingthesectorangleandVI changingthenumberofunits……….……………………..…………..…………..…...…..474.7GradedArc-Miurapatternbasedonthecombinationofchangingthelengthofcreaselines,changingthesectorangleandchangingthenumberofunits………….……….………..…..484.8GradedArcpatternbasedonchangingthelengthofcreaselines……..…..……………..…..494.9GradedArcpatternbasedonchangingthesectorangle……………...………...………..…..494.10GradedArcpatternbasedonchangingthenumberofunits……....…..………..………..…..504.11GradedArcpatternbasedonthecombinationofchangingthelengthofcreaselinesandchangingthesectorangle……………...……...………………………..…………....…...…..504.12GradedArcpatternbasedonthecombinationofchangingthelengthofcreaselinesandchangingthenumberofunits……..………….………………………………….…….....…..514.13GradedArcpatternbasedonthecombinationofchangingthesectorangleandchangingthenumberofunits……….…………………………..……...………..…….…………..…...…..514.14GradedArcpatternbasedonthecombinationofchangingthelengthofcreaselines,changingthesectorangleandchangingthenumberofunits…...…………………….……..……..…..515.1AE/IEvsdisplacementcurvesofthreemeshdensities……..……..……………………..…..535.2KE/IEvsdisplacementcurvesofthreeanalysistimes……..……..………………….…..…..545.3Thedeformationprocess:(a)GA-3-(170-170-170);(b)GA-3-(178-178-178)……..….…..555.4Thedeformationprocess:(a)GA-3-(174-174-174);(b)GA-3-(174-174-170)……..…......565.5Thesandwichstructure…………………….……………………………………………..…..575.6Thequasi-staticthree-pointbendingresponseofsandwichbeamwithMiura-oricoreandgradedMiura-oricorebasedonchangingthenumberofunitsinydirection………….…..585.7Deformationprocessofsandwichbeam:(a)Miura-oricore;(b)GradedMiura-oricore…...585.8GradedMiura-oricorebasedonthecombinationofchangingthelengthofcreaselinesandchangingthesectorangleinxdirection.…………………..………….……………....…..595.9Thequasi-staticthree-pointbendingresponseofsandwichbeamwithMiura-oricoreandfourcasesofgradedMiura-oricore……..………………………………………….......……..…..595.10Deformationprocessofsandwichbeam:(a)Miura-oricore;(b)GradedMiura-oricore(Graded-6)……………………………..……………………………....……..…………...….606.1Differentmountain-valleyfolddistributionsofsquare-twistpattern………………......…..616.2Thedesignoftype1inpart-foldedconfiguration.…………………………………….....…..626.3Thedesignoftype1inflat-deployedconfiguration.…………..………………….……..…..636.4Thedesignedprototypeoftype2..…………………..……………………………….......…..63VII LISTOFTABLES5.1Meshsizeconvergencetestresults...........................................................................................535.2Analysistimeconvergencetestresults.....................................................................................545.3FEAresultsofnormalorigamitubes.......................................................................................555.4FEAresultsofgradedorigamitubeswithcase1.....................................................................565.5FEAresultsofgradedorigamitubeswithcase2.....................................................................565.6NumericalsimulationresultsofsandwichstructureswithMiura-oricoreandgradedMiura-oricorebasedonchangingthenumberofunitsinydirection..………….…………585.7NumericalsimulationresultsofsandwichstructureswithMiuracoreandfourcasesofgradedMiura-oricore………………………...………………….…………………………………60VIII NOTATIONaPatternconstant:straightsidelength.bPatternconstant:zigzagsidelength.wUnitwidth.lUnitlength.hUnitheight.RPatternvariable:foldedradius.WWidthofwholestructure.HLengthofwholestructure.DDiameterofcylinder.lDistancebetweentwosupportedcylindersinthree-pointbending.mPatternconstants:numberofunitsinxdirection.xmPatternconstants:numberofunitsinydirection.ymPatternconstants:numberofunitsinzdirection.znNumberofmethodsinxdirection.xnNumberofmethodsinydirection.ynNumberofmethodsinzdirection.zPatternconstants:sectorangle.Patternvariable:longitudinaldihedralangle.APatternvariable:lateraldihedralangle.BPatternvariable:longitudinaledgeangle.APatternvariable:lateraledgeangle.BWLeeEndtoendPoisson’sratio.OutofplanePoisson’sratio.WHDensity.EYoung’smodulus.IX Yieldstress.yuUltimatestress.FmaxInitialpeakforce.FmAverageforce.ZCoordinateaxisalongtherevoluteaxisofjointi;iXCoordinateaxiscommonlynormaltoZandZ,XZZ;ii1iiii1TherotationanglefromXtoX,positiveaboutZ(meettheii1iiright-handrole);dThecommonnormaldistancebetweenXandX;iii1aii(1)ThecommonnormaldistancebetweenZiandZi1;TherotationanglefromZtoZ,positiveaboutaxisXii1i1ii(1)(meettheright-handrole).TTransformationmatrixfromcoordinateframejtoi.ijIUnitmatrix.RZiHomogeneousmatricesofrotationiaboutaxisZi.TZdiHomogeneousmatricesoftranslationdialongaxisZi.TX(aii1)Homogeneousmatricesoftranslationaii(1)alongaxisXi1.RX(ii1)Homogeneousmatricesofrotationii(1)aboutaxisXi1.1,2,3,4,5Subscriptdenotingdifferentpatternparameters.M,VSubscriptdenotingmountainandvalleycreaseparameters.s,pSubscriptdenotingstraightandpolarcreaseparameters.X Chapter1INTRODUCTIONChapter1INTRODUCTION1.1GradedStructuresTherearemanysmartgradedstructuresinnature,suchasbamboo,barelyand[1]corn.Thegradeddistributionofcellnumberorcellsizesinplantsoranimalscorrespondstothestressdistributioncausedbyexternalorinternalenvironment,achievingstrongbutlight-weightstructures.Oneofearlyresearchofgradedmaterialwascarriedoutin1984byNiinoetal.toobtaintheheat-resistantcompositematerialsthatareusableinspacestructuresand[2]fusionreactors.Sincethen,functionallygradedmaterialswerepopularizedandextendedtoseveraldifferentareas:aerospace,medicine,defense,energyconversion[3]devicesandoptoelectronics.Forexample,Houetal.studiedtheflatwisecompressionandedgewiseloadingofsandwichpanelswithgraded[4]conventional/auxetichoneycombcores,Diasetal.usedthegradedconceptinfibercementasameanofoptimizingthefiberquantitydistribution,whichreducesthefiber[5]totalcontentswithoutanobviousreductiononmodulusofrupture.1.2OrigamiOrigamiisanancientartoffoldingpaperintointricateshapes.Inthepastseveraldecades,ithasinterestedmoreandmoreresearchersinawiderangeoffieldsandobtainedmanyuniqueengineeringapplications.Forexample,inthefieldofspacetechnology,RobertJ.Langusedtheorigamitechniquetoallowthegianttelescope,[6]knownasEyeglass,foldedtofitintotherocket.Andorigamiactasthecoreofsandwichstructureswasunderdevelopedtoreplacetheexistingrocketorairplane[7]hall.Inbiomedicalfield,byintroducingtheorigamipatternintotubes,atypeof[8]deployablestentformedicalandsurgeryapplicationswasexplored,asshowninFig.1.1.Rothemundexploredthe“DNA-Origami”techniquetofoldthelong,single-standardDNAmoleculesintoarbitrarytwodimensionalshapes,thuscreating[9]thehighcomplexitynanostructureshapes.Amongorigamipatterns,rigidorigamiisasetoforigamipatternsthateachsurfacesurroundedwithcreaselinesisnotstretchingortwistingduringfoldingmotion,andallcreaselinesactasrotationalhinges.Hence,itcanbeeasilymanufacturedfromrigidengineeringplates,suchasthinpanelsandhardsheets,and[10]themachining-inducedresidualstressissmall.Rigidorigamiisalsousefulforthedesignofvariousindustrialproductsandmathematicalmodeling.Manyrigidorigamipatternshavebeenproposedforengineeringapplications.MaandYouimprovedtheenergyabsorptioncapabilityandreducingthepeakforceofvehiclecrushboxby[11]introducingtheorigamipattern.Nojimaetal.exploredsomestrongandfunctional1 Chapter1INTRODUCTIONultra-lightweightcoresbasedonorigamitechnique:athinflatwithperiodicallyslitsorpunchedoutportionsisfoldinazigzagwaytoproduceathree-dimensional[12]structure.Thesestructuresmaybeabletoactasgoodimpact-resistantstructures,acousticabsorbers,noiseinsulatorsorheatretainers.WangandCheninvestigated[13]methodstogeneratepatternedcylindersfromflatpaper,whichextendedthetypesoforigamitubes.[8]Fig.1.1Designoftheorigamitent.1.3Aim,ScopeandOutlineTheaimofthisthesisistodesignneworigamistructuresthatmoresuitableforengineeringapplication,calledgradedorigamistructures.Itisachievedbycombiningtheorigamitechniqueandgradedconcepttogether.ThefocusofthisthesisismainlyonMiura-oripattern,asmostoftheorigamiapplicationprojectpossessesabasicMiura-origeometry.Thethesisconsistsofthefollowingsixchapters.Chapter2isaliteraturereviewcoveringtheresearchfiledofgradedstructuresandorigamipatterns.Inthefirstpart,particularattentionisgiventothegradedstructuresexistinginnatureandtheresearchtrend.Inthesecondpart,theorigamipatternsanditsapplicationareintroduced,thegeometryofMiuraandMiura-derivativepatternsaregiven.Chapter3explorestheconditionofrigidorigamipatternsandproposesthebasicmethodstoconstructgradedMiura-oripattern.Thesebasicmethodsarechangingthelengthofcreaselines,changingthesectorangleandchangingthenumberofunits.Byapplyingthesemethods,gradedorigamimetamaterialsaregeneratedfromMiura-oriandtapered-Miurapattern.AndPoisson’sratioofgradedsingle-layerandmulti-layerMiura-oristructuresarestudied.Chapter4usesthemethodsdevelopedinchapter3toformthegradedorigamitubes.Thegeometricparametersandcharacteristicsofeachpatternarestudied.Specificvertexwithfivecreaselinesisintroducedtoformgradedstructuresthoughchangingthenumberofunits.Chapter5coverstwotopicsrelatedtothemechanicalpropertiesofgraded2 Chapter1INTRODUCTIONorigamistructures.Oneisthethree-pointbendingresponseofsandwichbeamwithgradedMiura-oricore.Theotherisquasi-staticaxialimpactresponseofgradedorigamitubes.Chapter6givesamanufacturingmethodoforigamipatterns.3Dprinterischosentogeneratetheorigamistructures.Twobasicmaterials,hardmaterialverowhiteandsoftmaterialtangoblack,areused.Anddifferentmixedmaterialsaregeneratedfromthesetwomaterials.Inthedesign,comparativelysoftmaterialsactasthecreaselineandcomparativelyhardmaterialsactasthethinpanel.Thisdesignmethodisusefulformostoftheorigamipatterns,andthemanufacturingprocessistimesaving,Conclusionsanddiscussionsonfutureworksaregiveninsection7,whichendsthepaper.3 Chapter2LITERATUREREVIEWChapter2LITERATUREREVIEW2.1CurrentResearchonGradedStructuresContinuouschangesinthemicrostructure,densityorporositycanresultingradinmechanicalandthermalpropertyofthematerials.TheresearchonFGMs,knownasfunctionallygradedmaterials,hasfocusedonseveraldifferentareas:processing,[14]mechanicalproperties,failurebehavior,biomaterialandthermalproperties.2.1.1GradedStructuresinNatureGradedmaterialsarenotahumaninventionandplayanimportantroleinnature.Therearemanysmartgradedstructuresexistedinplantsandanimals.Thegradedstructuresofbamboo,asshowninFig.2.1,havebeenexploredby[15-16][17][18]manyresearchers,suchasAmada,LieseandMilwich.Takingthewholestructureofbamboointoconsideration,theincreaseddensityofnodes,diameterandthevolumefractionoffibers(alignedwithvascularbundlesdistribution)fromtoptobottomcorrespondtothebendingmomentdistributioncausedbywindloads.What’smore,thevolumefractionoffibersinthecross-sectionofbamboo’sculmisdecreasedfromtheoutersurfacetoinnersurface,whichconformstothestressdistributioncausedbythebendingmoment.[18]Fig.2.1Microstructureofbamboo.Thebarely,wheat,pea,cornandgrassstemsalsohavethesimilargraded[1,19-21]structureinthebasicparenchyma.Figure2.2showsthegradeddistributionofthecellnumberandcellsizesfromoutsidetoinsideofthestem,whichcorrespondstothestressdistributioncausedbywind.Thegradedstructuresofplantsmentionedaboveareadjustingthestructuresandmaterials(mainlyadjustthetopologyofthecellmicrostructure)tooptimizetheoverallmechanicalproperties,whichpossessahigh-stresspropertiesinthehigh-stressregionandalow-stresspropertiesinthelow-stressregion,thusreducingthewhole4 Chapter2LITERATUREREVIEWmassandadaptingthemselvestotheenvironment.(a)(b)(c)(d)(e)[1][1][19][20]Fig.2.2Microstructureofstems:(a)Barely;(b)Corn;(c)Wheat;(d)Grass;(e)[21]Pea.[22]Fig.2.3Schematicattachmentoftendonstomusclesandbones.Asforanimals,thetendonsofhumanbodyandmusselbyssusofsubmarinemusselaresmartgradedmaterials,whichareaveryeffectivesolutiontoattachacompliant,softtissuestructure(muscle)toastiff,hardtissuestructure(boneorstone),[22]asshowninFig.2.3andFig.2.4.Thegradedstiffnessoftendontissuesservestotransferloadsacrossjointsbyavoidingthestressconcentrationatanypoint.Ontheonehand,tendonstransmittensileforcesgeneratedbymusclecells.Ontheotherhand,theyaresubjectedtocompressionandshearattheboneinterface.Forthemussel5 Chapter2LITERATUREREVIEWbyssus,theproximalpartofthethread(connectedtothesofttissueofthemussel)iselastic,butthedistalpartisstiffandensuresastrongattachmenttothesurface,e.g.,rocks.ThisstructureshowsanexcellentcombinationofmechanicalpropertiesduetocontinuousalterationofE-modulusalongthefiber.[22]Fig.2.4Schematicattachmentofasubmarinemusseltoastoneviathemusselbyssus.2.1.2ApplicationofGradedStructuresManyresearchersdevotethemselvesontheapplicationofgradedconceptinimprovingthemechanicalpropertiesandmaterialutilization.Condeetal.exploredtheweightsavingpotentialofgradedmetalfoamcoresandwichbeamsunderagraded[23]appliedmomentinyield-limiteddesign.Zhangetal.focusedonimprovingthe[24]loadcapacityofceramicsbyusingagradedstructure.LimexploredaPoisson-curvingstructurethatencounterssignificantchangein[25]itsthicknessasaresultofaprescribedcurvature.ThemicrostructureschematicforsuchstructureisshowninFig.2.5.Thehexagonalstructureandre-entrantstructurelieaboveandbelowtheneutralaxis,respectively,andthePoisson’sratiochangesfromthetoptobottomaccordingtothespecifiedrequirement.AsaresultofthedifferentPoisson’sratiobetweenhexagonalandre-entrantstructure,bendingindirectionAleadingtothesignificantthickening,whilebendingindirectionBleadingtothethinningofthebeam.Thegradedstructureusedinthesandwichstructureforimprovingtheenergy-absorptioncapabilityoffoamcoreshasalsobeenproposed.Zhouetal.demonstratedthatagradedfoamcore,whichisafoamcorewithgradeddensitythroughthethicknessofthepanel,haveexcellentimpactresistance,comparingwith[26]thenormalconstantdensitycore.6 Chapter2LITERATUREREVIEW[25]Fig.2.5SchematicofafunctionallygradedbeamthatgivesPoisson-curvingbehavior.2.2CurrentResearchonOrigamiPatterns2.2.1OrigamiandRigidOrigamiOrigamiisanancientartoffoldingpaperorcardsinto2Dor3Dintricateshapes.Inartisticfield,thereisagreatnumberoforigamipatterns.Figure2.6showsanumberofdifferentorigamimodels.Thecreaselinesofeveryorigamipatternaredividedintotwogroups:mountainlineandvalleyline.Thebasictechniqueoforigamiisthearrangementandselectionofcreaselines.Manyorigamipatternsarederivedfromnature,suchasleaves,wingsofinsectsandtheconfigurationofwarms.Butamongthesepatterns,onlyafewofthembelongstorigidorigami.Nojimaexploredacomprehensivelistofrigidorigamipatterns.Thesepatternsaredividedintotwoparts,theplanefoldthatpatternswerefoldedinto3Dplaneshapesandthe[27]cylindricalfoldthatpatternswerefoldedintocylindricalorspiralconfigurations.Fig.2.6Origamimodels.[28-31]Therearesometheoremsforrigidorigamipattern.Iftherigidorigamipatternisflat-foldableandflat-deployable,thefollowingconditionsarenecessary,7 Chapter2LITERATUREREVIEWThenumberofmountainlinesminusthenumberofvalleylinesatasinglevertexis2,whichcanbeexpressedasMV2(Maekawa’stheorem).Thesumofalternateanglesaroundasinglevertexis(Kawasaki’stheorem1).Forconsecutiveangles,,atasinglevertex,ifisthesmallest12233423angle,thentwocreaselinesformingmustbetheoppositeline,itistosay,oneis23themountainlineandtheotherisvalleyline(Kawasaki’stheorem2).Thesumofsectoranglesaroundasinglevertexis2,thatistosay:1223...nn1n12(2.1)Fig.2.7VertexwithNcreaselines.Nojimagavethegeometricconditionsforcylindricalfolding,whichjustifiedthe[27]closingconditioninthecircumferentialdirectionwhenthepatternwasfullyfolded.ItconsidersthefoldingofastripconsistingofN-foldlines(Niseven),asshowninFig.2.8.Afterthestripefoldedalongcreases①,②…,theaxischangesfromX0toX,X….Itisassumedthattheparalleltopandbottomedgeisparalleltothe12axisX,and,areinclinedangles.Whenthecompletelyfoldedstripe012Nareclosed,thefollowingrelationshipissatisfied,NN22123(2.2)[27]Fig.2.8StripewithNfoldedlines.8 Chapter2LITERATUREREVIEW2.2.2RelationshipbetweenOrigamiandSpherical4RLinkageIfalinkagehasfourrevolutejointswithaxesintersectingatasinglepoint,thenthelinksmoveonasphere,andtheassemblyiscalledaspherical4Rlinkage.Recently,WangandChenusedtheassemblyofspherical4Rlinkagetostudyrigid[13]origamipattern.Theypointedoutthatcreaselinescanbeseenasrevolutejoints.Whenseveralcreaselinesintersectatasinglevertex,theorigamipatternbecomesasphericallinkage.Ifthenumberofcreaselinesis4,theorigamipatternisaspherical4Rlinkage,whichhasonlyonedegreeoffreedom.(a)(b)Fig.2.9Definitionoforigamipatternanditscorrespondingsphericallinkage:(a)Origamipattern;(b)Correspondingspherical4Rlinkage.Forasingleclose-loopinalinkage,DenavitandHartenbergpointoutthatthe[32]productofthetransformmatricesequalstheunitmatrix,i.e.,TTTT21324314I(2.3)Tisthetransformationmatrixfromcoordinateframejtoi.ijFortheassemblyofspherical4Rlinkage,asshowninFig.2.10,spherical4Rlinkagesareconnectedtogetherthroughthesharedaxis,whichcanbeseenasacommonjoint,thusformingaclosedloop.9 Chapter2LITERATUREREVIEWFig.2.10Assemblyoffourspherical4Rlinkages.2.3OrigamiGeometryMiura-oriPatternMiura-ori,seeFig.2.11,isacommonrigidorigamipattern.Itisconstructedoffourparallelogramplanes,andownsmanyusefulcharacteristicsincludingflat-foldability,flat-deployability,onedegree-of-freedom,negativein-planePoisson’sratioandtessellation.Itpossessesrepetitiveunitgeometrygeneratedfromrepetitiveparallelogramsizes.SameordifferentMiurapatterncanstacktogethertoformthe[33]stackedconfiguration,seeFig.2.12.AunitofMiura-oriisshowninFig.2.11(c)and(d),whereaandbaresidelengths,isthesectorangle,andareABdihedralangles,andareedgeangles,w,handlareunitwidth,height,andAB[34]length,respectively.Thefollowingrelationshipshavebeenestablished:21cosBA1cos4cos(2.4)22cossincoscos(2.5)AA22cossincoscos(2.6)BBwb2sin/2B(2.7)hacosA2(2.8)la2sinA2(2.9)10 Chapter2LITERATUREREVIEWInordertodefinethenumberoflayersinx,yandzdirections,weintroducem,xmandm.Intotal,therearethirteenparametershavebeendefined:a,b,,m,yzxm,m,,,,,w,handl.Thefirstsixareconstantparameters,astheyyzABABremainunchangedduringthefoldingprocess,andthefollowingparametersarevariables.Amongthesevariables,sixindependentrelationshavebeenestablished,thusonegivenvariableparametercandefineotherparameters.(a)(b)(c)(d)Fig.2.11Miura-oripatterngeometry:(a)Creasepattern;(b)Part-foldedconfiguration;(c)Basicunitandconstants;(d)Configurationvariables.[33]Fig.2.12ThefoldingprocessofaMiura-basedmetamaterial.11 Chapter2LITERATUREREVIEWArc-MiuraPatternArc-MiurapatterniscreatedbychangingthesectoranglealternativelyinlongitudinaldirectionoftheMiura-oripattern,asshowninFig.2.13.Thispatternhasfoursidelines,a,a,bandb,andtwosectorangles,and.Butfourofthemareindependentduetothefollowingrelationships,bbsinsin(2.10a)abcosabcos(2.10b)(a)(b)(c)Fig.2.13Arc-Miurapatterngeometry:(a)Basicunitandconstants;(b)Configurationvariables;(c)Frontprojection.Equations2.4-2.6canbereformulatedatM-verticesandV-vertices,whereM-verticesimpliestheintersectionpointsofthreemountainlinesandonevalleyline,andV-verticesimpliestheintersectionpointsofthreevalleylinesandonemountainline.ThesubstriptMorVdenotesthevariablesonM-verticesorV-vertices,givingtwodihedralanglesand,andfouredgeangles,,,and.MAVAMBVBMAVAM-verticesandV-verticesliealongconcentriccylinders,calledoutercylinder(radiusisdefinedasR1)andinnercylinder(radiusisdefinedasR2),respectively.Andthe12 Chapter2LITERATUREREVIEW[34]followingrelationshipshavebeenestablished:22Raaa12caosVA21cosVAMA(2.11)22Raaa2caos21cos(2.12)2MAVAMAwb2sinVB22sinbMB2(2.13)(a)(b)Fig.2.14TessellationofArc-Miurapattern:(a)Creasepattern;(b)Part-foldedconfiguration.ArcPatternAlteringthecreaseorientationonaMiurapatternformstheArcpattern.ThebasicunfoldedunitoftheArcpattern,asshowninFig.2.15,consistsoffouridenticalparallelogramplates.Threepatternconstantsaredefined:a,b,andsectorangle.Wecanderiveonemoreusefulconstants:sidelengthaab2cos.AsdefinedintheMiurapattern,fourconfigurationparameters,,,ABAcanbeverified.ItisconvenienttodefineadditionalfoldedparameterswithwandBR:wb2sinB/2(2.14)22Raaa2caos21cos2(2.15)AA13 Chapter2LITERATUREREVIEW(a)(b)(c)Fig.2.15Arcpatterngeometry:(a)Basicunitandconstants;(b)Configurationvariables;(c)Frontprojection.(a)(b)Fig.2.16TessellationofArcpattern:(a)Creasepattern;(b)Part-foldedconfiguration.14 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSChapter3GEOMETRYOFGRADEDORIGMAIMETAMATERIALS3.1ConditionofRigidOrigamiPatternsTherigidityoforigamipatternbasedonthreecasesisdiscussedinthissection.Theyarecase1tocase3,inwhichcase2andcase3arethecombinationoforigamiandkirigamitechniques.Inthefollowingdiscussion,wefirstintroducethekinematicsofspherical4Rlinkage.Thentheassemblyofspherical4Rlinkage(case1tocase2)isexplored,andtheconditiontoformingrigidorigamipatternisgiven.AsshowninFig.3.1,thecoordinateframesforthelinksandjointsarecreated[32]byfollowingtheD-Hparameters,inwhich,ZCoordinateaxisalongtherevoluteaxisofjointi;iXCoordinateaxiscommonlynormaltoZandZ,XZZ;ii1iiii1TherotationanglefromXtoX,positiveaboutZ(meettheii1iiright-handrole);dThecommonnormaldistancebetweenXandX;iii1aii(1)ThecommonnormaldistancebetweenZiandZi1;TherotationanglefromZtoZ,positiveaboutaxisX(meettheii1i1ii(1)right-handrole).ThecoordinatetransformationfromframeFi:XiYiZitoframeFi+1:Xi+1Yi+1Zi+1canberepresentedasRTTZZX()()(iiida(i1))(RXi(i1))FFii1(3.1)inwhich,cossin00iisincos00R()ii(3.2a)Zi00100001isthehomogeneousmatricesofrotationaboutaxisZ,ii15 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSFig.3.1Coordinatesystemandlinkagegeometricparametersinlinks(i-1)iandi(i+1)connectedbyrevolutejointi.10000100TZ()di(3.2b)001di0001isthetranslationdalongaxisZ,ii100aii(1)0100TX(()aii1)(3.2c)00100001isthetranslationaalongaxisX,andii(1)i110000cossin0R()ii(1)ii(1)(3.2d)X(ii1)0sincos0ii(1)ii(1)0001istherotationaboutaxisX.ii(1)i1Forspherical4Rlinkage,allrevoluteaxesintersectatonevertex,seeFig.3.2.Therefore,thefollowingrelationshipissatisfied,dddd01234(3.3)aaaa01223344116 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSFig.3.2Definitionofasinglespherical4Rlinkage.LiuandChenhaveexploredtherelationshipsbetweenrotationofeachjointi[35]andthelinki(i+1)"stwist,whichcanbeexpressedasfollows,ii(1)cossinsincoscossinsincosii1211iiiiiiii111iiii21cossinsincoscossinsinsinsinii11ii21iiii11ii121iiiicoscoscoscos0ii23ii1ii12ii1(3.4)coscossinsincoscoscosi1111iiiiiiiii2i11ii2(3.5)sinsincos0ii21ii12i2Case1Inthenetworkofspherical4Rlinkages,seeFig.3.3,adjacentrotationaljointsofabbctwoconnectedspherical4Rlinkageskeepaligned.Thatistosay,,,4141cd.Therefore,therotationofjointJa1cantransferthroughthealignjoints,end41atthejointJd4.Toensurethenetworkismobile,therotationanglesofjointsJa1andJd4shouldbeequal.Sothekinematiccompatibilityconditionofthenetworkisasfollows,aabbccdd14141414(3.6)ad14TherangeofeachrotationalangledependsonthedistributionofiMountain-Valleycreases.InFig.3.3,thevalleylineisrepresentedbydashline,andthemountainlineisrepresentedbysolidline.Ifthepatternisflat-foldableandflat-deployable,therangeofmountainfoldline’srotationandvalleyfoldline’sM17 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSrotationis,V0M,V0(3.7)Itisassumedthatlink’stwistofeachlinkageisasfollows,abcd12a12b12c12dabcd23a23b23c23dA:B:C:D:(3.8)abcd34a34b34c34dabcd41a41b41c41dBycombiningEqns.3.4and3.6-3.8together,wecanobtainthefollowingrelationshiptoensuretherigidityofthispattern,tantantantancoscosabcdad,,(3.9)tantantantancoscosabcdbcFig.3.3Thecreasepatternofcase1.Case2Basedoncase1,westudytherigidityoforigamipatternwithpunched-outpanels,seeFig.3.4.Theblackportionsarepunchedoutofthesheet,twopartsareconnectedtogethertoformthenewpattern,fromFig.3.4(a)toFig.3.4(b).Itisassumedthatforeachlinkage,asshowninFig.3.4(a),thelink’stwistis,18 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSabcd12a12b12c12dabcd23a23b23c23dA:B:C:D:abcd34a34b34c34dabcd41a41b41c41defgh12f12g12e12hefgh23e23f23g23hE:F:G:H:efgh34e34f34g34hefgh41e41f41g41h(3.10)ijkl12j12i12k12ljkli23i23j23k23lI:J:K:L:ijkl34i34j34k34lijkl41i41j41k41lmn12m12nmn23m23nM:N:mn34m34nmn41m41nAmongtheseangles,thefollowingrelationshipisassumedtobesatisfied,gfe,lmnK(3.11)Similartocase1,forFig.3.4(a),basedonEqns.3.9-3.11,whentanbcdjtantancoscosabcostantantancoscoscosbcdikc,(3.12)tantantancoscoscosjihilctantantancoscoscosjihhgdallloopssatisfythecorrespondingcompatiblecondition.Asinthefoldingprocess,thedistancebetweenCandKshouldbeequaltothedistancebetweenCandI.Thenthefollowingrelationshipcanbededuced,4KsinL4LLLIKCKcosLLLLILLMMNNCcos(3.13)2sinK2Inwhich.meansthedihedralangleoftwofacesclosetoJ.44LK4K4KCombinedwiththegeometry,seeFig.3.4(b),thefollowingrelationshipissatisfied:LLLLLLIKILMN,KCNCLM(3.14)19 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSTherefore,Eqns.3.10-3.12and3.14canbecombinedtogethertoensuretherigidity.(a)(b)Fig.3.4Thecreasepatternofcase2:(a)Twoindividualparts;(b)Theconnectedorigamipattern.20 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSCase3Fig.3.5Thecreasepatternofcase3.AsshowninFig.3.5,M1M2iscutoff.Itisassumedlink’stwistofeachlinkageis,abcd12a12b12c12dabcd23a23b23c23dA:B:C:D:abcd34a34b34c34dabcd41a41b41c41dfgeh12f12g12e12hefgh23e23f23g23hE:F:G:H:(3.15)efgh34e34f34g34hefgh41e41f41g41hji12j12iij23i23jI:J:ij34i34jij41i41j21 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSThecorrespondingcompatibleconditionis,aabbccddeeffgghhiijj13141314131413131314aj14(3.16)BycombiningEqns.3.4-3.5,3.7and3.15-3.16together,wecanobtainthefollowingrelationshiptoguaranteetherigidity,jihgfaebcd,,(3.17)jihgfaebcdItshouldbenotedthatEqn.3.17canguaranteethedistancebetweenAandEfortheleftpartandrightpartisthesame,whichissimilartocase2.3.2GradedOrigamiMetamaterialsBasedonMiura-ori3.2.1GradedMiura-oriPatternBasedonIndependentMethodsBasedonthediscussionofrigidorigamipatternsinsection3.1,wecanobtainthegradedrigidorigamistructures.Ascase1containstheconditionthatthesectoranglesindifferentverticesarenotthesame,itcanbeseenasamethodtoachievegradedstructuresbychangingthesectorangle.Forcases2and3,theycanbeseenasamethodtoachievegradedorigamistructuresbychangingthenumberofunitsandsectorangle.Thenwecanderivethreeindependentways:changingthelengthofcreaselines,changingthesectorangleandchangingthenumberofunits.ByapplyingthesemethodstoMiura-ori,thegradedMiura-oristructurescanbeachieved.Andthedesignparametersare:a,b,,mx,myandmz.xdirectionThreebasicmethodsthatforminggradedMiura-oripatterninxdirectionareshowninFigs.3.6-3.8.Forchangingthelengthofcreaselines,allunitshavethesamesectoranglebutdifferentlengthofcreaselines(sidelengthb),soothergeometricparametersremainunchangedexceptunitwidthw,seeFig.3.6.AsshowninFig.3.7(a)and(b),bychangingthesectorangle,adjacentunitswithdifferentsectoranglesand()areconnectedtogetherthroughthesameedgeangleandsidelengtha.FromtherelationshipdeducedfromEqns.2.4-2.6,A21cosA2cosBcot1(3.18)1cosAitcanbeseenthatwhenpartIisfullyfolded,partIIisinstillpart-foldedconfiguration.Thatistosay,thispatternisnotflat-foldable.Butitisrigid-foldable,correspondingtotherigiditydiscussionofcase1insection3.1.22 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALS(a)(b)(c)Fig.3.6GradedMiura-oripatternbasedonchangingthelengthofcreaselinesinxdirection:(a)Creasepattern;(b)Part-foldedconfiguration;(c)Foldingsequenceofapapercardprototype.(a)(b)(c)Fig.3.7GradedMiura-oripatternbasedonchangingthesectorangleinxdirection:(a)Creasepattern;(b)Part-foldedconfiguration;(c)Foldingsequenceofapapercardprototype.23 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSGradedMiura-oripatternformedthroughchangingthenumberofunitsisreflectedinFig.3.8,whichcombinesthekirigamiandorigamitechniquestogether.ThelineCDiscutofffromthesheet.Allunitshavethesamedihedralanglesandedgeangles,andthisgradedorigamistructureisflat-foldable,rigid-foldable(sametotherigiditydiscussionofcase3insection3.1)andone-DOF.TheheighthofpartIishalfofpartII.(a)(b)(c)Fig.3.8GradedMiura-oripatternbasedonchangingthenumberofunitsinxdirection:(a)Creasepattern;(b)Part-foldedconfiguration;(c)Foldingsequenceofapapercardprototype.ydirectionInthischapter,itisassumedthatduringthefoldingprocess,themountainzigzaglinesorvalleyzigzaglineslieonaplane,causingthatthestructurecanbestackedinzdirectiontoformmulti-layerstructures.Thentwoindependentmethodsareleftinydirection.Theyarechangingthelengthofstraightcreaselinesandchangingthenumberofunits,asshowninFig.3.9andFig.3.10.Theunitheighthandlengthlarechangedwiththevariationofstraightcreaselines,seeFig.3.9.Asanglevariablesarenotchanged,itisrigid.InFig.3.10,blackportionsarepunchedoutofthesheet,causingtheunitnumberofpartⅣishalfofpartⅢ.Thesetwotypesareflat-foldableandflat-deployable.ForthepatternreflectedinFig.3.10,itisalsorigid,whichbelongstotherigiditydiscussionofcase2insection3.1.24 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALS(a)(b)(c)Fig.3.9GradedMiura-oripatternbasedonchangingthelengthofcreaselinesinydirection:(a)Creasepattern;(b)Part-foldedconfiguration;(c)Foldingsequenceofapapercardprototype.(a)(b)(c)Fig.3.10GradedMiura-oripatternbasedonchangingthenumberofunitsinydirection:(a)Creasepattern;(b)Part-foldedconfiguration;(c)Foldingsequenceofapapercardprototype.25 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSzdirectionGradedmulti-layerorigamimetamaterialscanbeformedthroughstackingdifferentMiura-oripatternstogether.Inotherwords,weapplytheindependentsmethodsinzdirection,generatinggradedmulti-layerorigamimetamaterials.Asthesectorangleandlengthofstraightcreaselinearerelevanttoeachotherforstackedlayers,twoindependentmethodsareleft,changingthesectorangleandchangingthenumberofunitsofdifferentlayers.AsshowninFig.3.11,Miura-oriwithdifferentsectoranglesarestackedtogether,forminggradedmulti-layerfoldablemetamaterials.[33]Andthefollowingthreerelationshipsaresatisfied,acos21(3.19)acos12bb21(3.20)Duetothedifferentsectorangleofdifferentparts,thisstructureisflat-foldablebutnotflat-deployable,seeFig.3.11(c).(a)(b)(c)Fig.3.11GradedMiura-oripatternbasedonchangingthesectorangleinzdirection:(a)Creasepattern;(b)Part-foldedconfiguration;(c)Foldingsequenceofapapercardprototype.Ifwechangethelengthofzigzagcreaselinebofdifferentlayersandstackthemtogether,gradedmulti-layerorigamimetamaterialbasedonchangingthenumberofunitsisobtained,asshowninFig.3.12,where26 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSbn21b(3.21)Itisassumedn2,andnmeansthevariationindex.(a)(b)(c)Fig.3.12GradedMiura-oripatternbasedonchangingthenumberofunitsinzdirection:(a)Creasepattern;(b)Part-foldedconfiguration;(c)Foldingsequenceofapapercardprototype.3.2.2GradedMiura-oriPatternBasedonCombinedMethodsandClassificationThemethodsthatforminggradedorigamistructuresinx,yandzdirectionscanbecombinedtogethertogivemoreinterestinggradedstructures.Hereweclassifythesemethods(includingindependentmethods)andgivesomeexamples.Thereareintotal127methods(includingindependentmethods),andthenumber127iscalculatedbynnnnnnnnnnnnxyzxyxzyzxyz(3.22)wherenisthenumberofmethodsinxdirection,includingthecombinedmethods,xforexample,thecombinedmethodthatchangingthelengthofcreaselinesand123changingthesectorangleatthesametime,andnccc;nisthenumberx333y12ofmethodsinydirection(ncc);nisthenumberofmethodsinzdirectiony22z12(ncc).z22Hereweassumethatallthegradedstructureshavethefeaturethatm3,manyzsituationsareeliminated,leaving79methods.Thesemethodsaredefinedintotwo27 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSgroups,oneisthatthevarydirectionsarethesameforeverylayers(group1),theotheristhatthevarydirectionsarenotalwaysthesame(group2).Group1Inthisgroup,thereare30methodsintotal.Themostcomplexcircumstancesisasfollows,xyzmethodlength+anglenumberangle+numberThestructureofeachlayerandthestackedconfigurationarepresentedinFig.3.13.Foreachlayer,thepatternisdividedintotwoparts,onewithsectorangle,1andtheotherwithsectorangle,bbcoscosissatisfiedtoensurethatthe21122outercontourisrectangle.Thisstructureisflat-foldablebutnotflat-deployableasthesectorangleischanged.Bysplittingthiscombinedmethod,other29gradedmulti-layerorigamistructuresareobtained.OneofthesestructuresisreflectedinFig.3.14,whichiscreatedbychangingthelengthofcreaselinesandsectorangleinxdirectionandchangingthesectorangleinzdirection.Group2Themethodsthatincludingchangingthenumberofnumberofunitsinxdirectionorchangingthelengthofcreaselinesinydirectionareclassedintogroup2,asnotalllayershavethesamevarydirection.ThemostcomplexexampleisreflectedinFig.3.15,andthemethodisasfollows,xyzmethodlength+angle+numberlength+numbernumberWecanseethatthebottomadjacenttwolayershaveoppositevariationdirections:thelengthofcreaselinesinydirectionandthenumberofunitsinxdirection.Thedifferentunitconfigurationmodeofthebottomtwoadjacentlayersisforcontinuousstackingprocess.Thiscombinedmethodcanalsobespiltup,resultinginother48methodsthatbelongtogroup2.AnexampleisshowninFig.3.16,whichisachievedbyusingthecombinedmethodthatchangingthelengthofcreaselines,sectorangleandnumberofunitsinxdirection,changingthenumberofunitsinydirection.Duetothepropertythatadjacenttwolayershavesomeoppositevariationdirectionsofthispattern,itisassumedthatzdirectionisnotunchangedifthesectorangleandtotalnumberofunitsarenotchanged,whichisapplicableforothercircumstancesofgroup2.28 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSFig.3.13Thegradedmulti-layerMiura-oristructurebasedonthecombinationofchangingthelengthofcreaselinesandsectorangleinxdirection,changingthenumberofunitsinydirection,andchangingthesectorangleandnumberofunitsinzdirection.Fig.3.14Thegradedmulti-layerMiura-oristructurebasedonthecombinationofchangingthelengthofcreaselinesandsectorangleinxdirection,andchangingthesectorangleinzdirection.29 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSFig.3.15Thegradedmulti-layerorigamistructurebasedonthecombinationofchangingthelengthofcreaselines,sectorangleandnumberofunitsinxdirection,changingthelengthofcreaselinesandnumberofunitsinydirection,andchangingthenumberofunitsinzdirection.Fig.3.16Thegradedmulti-layerMiura-oristructurebasedonthecombinationofchangingthelengthofcreaselines,sectorangleandnumberofunitsinxdirection,changingthenumberofunitsinydirection.30 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALS3.3GradedOrigamiMetamaterialsBasedonTapered-MiuraByincliningthestraightcreaselinesofaMiurapattern,tapered-Miurapatternis[34]obtained,asshowninFig.3.17.Threedirectionsaredefinedforthispattern,circulardirection,radialdirectionandzdirection.Wecanobtainthegradedcurved-Miuramatematerialbyvaryingtheparametersoftapered-Miurapattern,similartogradedMiurapattern.Inthecirculardirection,therearethreeindependentmethods,changingthesectorangleandchangingthelengthofcreaselinesandchangingthenumberofunits.Inradialdirection,onemethodleft,changingthelengthofcreaselines.Forzdirection,changingthesectorangleistheonlyway.Fortwolayerswithdifferentsectoranglethatconnecttogether,asshowninsFig.3.18,thefollowingrelationshipsofconstantparametersshouldbesatisfied:acos2s1(3.23)acos12ssinps22sinps22cos(3.24)sinps11sinps11cosTheoverallmethodscanalsobedividedintotwogroups,thesamewithgradedmulti-layerMiura-basedmetamaterial.Themostcomplexstructuresofgroup1andgroup2areshowninFig.3.19andFig.3.20.Andthesetwostructurescanbedividedintoother6and11gradedorigamistructuresthatbelongtogroup1andgroup2,respectively.(a)(b)Fig.3.17Tapered-Miurapatterngeometry:(a)Creasepattern;(b)Part-foldedconfiguration.31 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSFig.3.18Twotapered-Miuraunitswithdifferentsectoranglethatstackedtogether.sFig.3.19Thegradedmulti-layertapered-Miurastructurebasedonthecombinationofchangingthelengthofcreaselinesandsectorangleincirculardirection,andchangingthesectorangleinzdirection.32 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSFig.3.20Thegradedmulti-layertapered-Miurastructurebasedonthecombinationofchangingthelengthofcreaselines,sectorangleandnumberofunitsincirculardirection,andchangingthelengthofcreaselinesinradialdirection.3.4Poisson’sRatioofGradedOrigamiMetamaterials3.4.1In-planeEndtoEndPoisson’sRatioofGradedSingle-layerOrigamiMetamaterialAsshowninFig.3.21,weexplorethein-planeendtoendPoisson’sratioofgradedMiura-oripatternbasedonchangingthelengthofcreaselinesandsectorangleinxdirection.ThelengthandwidthareBBWwwb2sin2bsin(3.25)22cosBLa2cbosB2(3.26)cos233 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSFig.3.21ThegradedMiura-oripatternbasedonchangingthelengthofcreaselinesandsectorangleinxdirection.Fromthefollowingrelationshipcos222coscosB2sinB21cosBA1cos4cos2cos21cosBA1cos4cosAcos(3.27)sin2Bcos2Anotherexpressionofwidthcanbededuced,2BBcos2Wb2sin21bcos(3.28)22cos2BycombiningEqns.3.25and3.28together,thePoisson’sratioisobtained,WdLWLeeLdWcos2bbsinBB1cos222cos2BB2B2abcossincossin1222(3.29)2B2caboscos222BBcosbcossin22B22cosbcos2cos21cos2Bcos22Inwhich,34 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSB1B2arcsinsin1sin(3.30)2WithEqns.3.29and3.30,againestcurveisplottedwiththeWLeeBassumptionthat60,changesfrom60to50with5internal,seeFig.3.22.Wecanseethatbyintroducingthegradedconcept,canbedesignedtoWLeebeclosetoastablevaluewithincertainlimits.Forexample,bychanginginto55,thevalueofhaslittlechangeswhenchangesfrom70to110.WLeeBTheinfluenceofbtoPoisson’sratioisalsostudied.Wechoose60,55,bb2,bb8,2band0.5b,respectively.ThevalueofWLeeisshowninFig.3.23,fromwhichwecanseethatthelengthofbhaslittleinfluencewhenchangesfrombeginning(around70)to110,andbecomesBWLeebiggerwiththedecreaseofb.Fig.3.22TheendtoendPoisson’sratioofgradedMiura-oripattern.35 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSFig.3.23TheinfluenceofbtotheendtoendPoisson’sratio.FromFig.3.22,Fig.3.23andthediscussionsabove,wecanconcludethatbyadjustingthegradedgeometricparameters,thein-planeendtoendPoisson’sratiocanbedesignedtonear0inagivenrangeofdihedralangle.PartIcanbedesignedtobethemainpartofthewholestructure.BychangingthegeometricparameterofpartII,wecanadjustthePoisson’sratiocloseto0inagivenrangeof.B3.4.2Out-of-planePoisson’sRatioofGradedMulti-layerOrigamiMetamaterialThreecasesaretakenintoconsiderationtoexploretheout-of-planePoisson’sratioofgradedmulti-layerorigamimetamaterialsbasedonchangingthesectorangle,asshowninFig.3.24.Andcase3isthecombinationofcase1andcase2.(a)(b)(c)Fig.3.24Threecasesofgradedmulti-layerorigamimetamaterial:(a)Case1;(b)Case2;(c)Case3.36 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSCase1Thewidth:BWb2sin(3.31)12Theheightoffourunits:AA12A3A4Hacosacosacosacos(3.32)12342222WithEqns.2.4,3.19,3.31and3.32,thePoisson’sratioisdeduced,WdHWHHdW2Bcos222Bcoscos122B2coscos122Bcoscos12cos122cosBcos222Bcos22coscoscos222B2tan22Bcos1cos2cos122Bcoscos322Bcos32cos33coscos22Bcoscos12cos122cosBcos422Bcos42coscoscos44222BBcos122coscos12coscos2cos22cos122Bcos122Bcoscoscoscos34cos2cos234(3.33)Case2Theheightoffourunits:AA24A3Hacosacosacos(3.34)234222BycombiningEqns.2.4,3.19,3.31and3.34together,wecangettheout-of-plane37 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSPoisson’sratioofcase2,WdHWHHdW2Bcos1cos2cos122Bcoscos222Bcos22coscoscos2222Bcoscos212cos122tanBcosBcos32ccoscos22Bcosos323322Bcos1cos2cos122Bcoscos422Bcos42cos44coscos2cos1122BBcos22coscoscoscos23cos2cos223cos122Bcoscos(3.35)4cos42Forcase1andcase2,itisassumedthatthebottomtwolayershavefixedsectorangles(1245,70)and31.ThroughEqns.3.30,3.33and3.35,weplotthePoisson’sratioagainstdihedralanglecurveofthesetwocases.Figure3.25B1showsthatwhenisfixed,theinfluenceoftothePoisson’sratio.Figure3.2643reflectstheinfluenceoftoPoisson’sratiowhenisgiven.Fromthesetwo43figures,itcanbeseenthatthePoisson’sratioofcase1becomessmallerwiththeincreasingofor.Butforcase2,thePoisson’sratiobecomessmallerwiththe34increasingof4orthedecreasingof3.When355,145and2470(case2),asshowninFig.3.25,thePoisson’sratioisnear0withlittlevariation.38 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSFig.3.25Theinfluenceoftotheout-of-planePoisson’sratio(cases1and2).3Fig.3.26Theinfluenceoftotheout-of-planePoisson’sratio(cases1and2).4Case3Theheightoffourunits:AA12A3A4Hacosacoscosacoscosacoscos(3.36)12342222WithEqns.2.4,3.19,3.31and3.36,theout-ofplanePoisson’sratioofcase3isestablished,39 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSWdHWHHdW2Bcos222Bcoscos122B2coscos122Bcoscos12cos122cosBcos222Bcos22coscoscos222B2tan22Bcos1cos2cos122Bcoscos322Bcos32cos33coscos22Bcoscos12cos122cosBcos422Bcos42coscoscos44222BBcos122coscos12coscos2cos23cos122Bcos122Bcoscoscoscos34cos2cos234(3.37)Case3isthemixedtypeofcase1andcase2.Itisalsoassumedthatthebottomtwolayershavefixedsectorangles(45,70).InFig.3.27(a),itisassumed12.ButinFig.3.27(b),.Figures3.27and3.28showtheinfluenceof31313andtothePoisson’sratio,respectively.WecanseethatthePoisson’sratio4becomessmallerwiththeincreasingoforthedecreasingof.AsshowninFig.433.27(b),thePoisson’sratioiscloseto0when32.5,andthefluctuationissmall.3Fromthediscussionsmentionedabove,itcanbeseenthatbyadjustingthegeometricparameter,sectorangle,canbedesignedtobecloseto0forcasesWH2and3.40 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALS(a)(b)Fig.3.27Theinfluenceoftotheout-of-planePoisson’sratio(case3):(a);(b)331.3141 Chapter3GEOMETRYOFGRADEDORIGAMIMETAMATERIALSFig.3.28Theinfluenceoftotheout-of-planePoisson’sratio(case3).43.5SummaryBasedontheconditionofrigidorigamipatternthathasexplored,afamilyofgradedrigidorigamipatternsbasedonMiurapatternhasbeenproposed.Threebasicmethodsthatforminggradedorigamistructuresarediscussed,andthecombinedmethodsarealsotakenintoconsideration.Thegeometricrelationshipsandgeometricfeaturesofeachpatternarestudied.AstherearemanycasesthatcanformgradedMiura-basedmetamaterial,aclassificationisgivenforclearunderstanding.Themechanicalproperty,Poisson’sratio,isalsoexplored.ThePoisson’sratioofgradedsingle-layerMiurapatternbasedonchangingthelengthofcreaselinesandsectorangle,multi-layergradedMiurapatternbasedonchangingthesectoranglearediscussed,respectively.Forgradedmulti-layerorigamimetamaterial,threecasesareproposed,inwhichtheconnectedwaysaredifferent.Theresultsshowthatbyintroducinggradedconceptintosingle-layerMiurapattern,thePoisson’sratiocanbedesignedtoaroundzeroinalimitedrangeofdihedralangleofthemainpart.Andformulti-layerstructures,theout-of-planePoisson’sratioofcases2and3canbedesignedtoaroundzeroduringthefoldingprocessbyadjustingthesectorangles.42 Chapter4GEOMETRYOFGRADEDORIGAMITUBESChapter4GEOMETRYOFGRADEDORIGAMITUBESTwoorigamipatterns,Arc-MiuraandArcpatternareselectedtogenerategradedorigamitubes.ThreebasicmethodsthatgenerategradedMiura-oripatternaretakenintoconsideration.Andwefocuesontheaxialdirectionofthetubes.Thegeometricrelationshipofeachstructureisdiscussedbriefly.Theconditiontoformrigidpatternisalsogiven.4.1GradedArc-MiuraPattern4.1.1GradedArc-MiuraPatternBasedonIndependentMethodsForgradedArc-Miurapatternbasedonchangingthelengthofcreaselines(sidelengthb),seeFig.4.1,unitwidthisthegradedparameter.ThefoldingprocessisshwoninFig.4.1(c).Astheangleparametersremainunchanged,thisstructureisrigid-foldable,flat-foldableandflat-deployable.(a)(b)Fig.4.1GradedArc-Miurapatternbasedonchangingthelengthofcreaselines:(a)Creasepatern;(b)Foldingsequenceofapapercardprototype.AsshowninFig.4.2,alterationofthesectorangleorgivesagradedArc-Miurapattern.Oneofsixinterrelatedconfigurationvariables,,,,w,BMAVAand,canbeseenasthegradedparameter.FortwounitswithdifferentsectorMBVBanglesthatconnectedtogether,edgeanglesandshouldbethesame.FromVAMAthefollowingequationsdeducedfromEqns.2.4-2.6:4cos1VA1224tan112tan(4.1)2221cosMA11tantan143 Chapter4GEOMETRYOFGRADEDORIGAMITUBES4cos1VA2224tan222tan(4.2)2221cosMA22tantan2therelationshipofsectoranglescanbeestablished:tantan12(4.3)tantan12Thatistosay,when,andaregiven,canbeestablishedtoensurethe1122rigidity.SimilartogradedMiura-oripattern,thispatternisnotflat-foldable.(a)(b)(c)Fig.4.2GradedArc-Miurapatternbasedonchangingthesectorangle:(a)Creasepattern;(b)Part-foldedconfiguration;(c)Foldingsequenceofapapercardprototype.ThestructureofgradedArc-MiurapatternthatcontainsdifferentnumberofunitsfordifferentsectionsisreflectedinFig.4.3.ThenumberofcreaselinesinvertexAandvertexBisfour,butthenumberofcreaselinesinvertexCisfive.Byintroducingsuchvertex,twounitscanbecombinedintooneunit,thusgeneratingthisgradedArc-Miurapattern.44 Chapter4GEOMETRYOFGRADEDORIGAMITUBES(a)(b)(c)(d)Fig.4.3GradedArc-Miurapatternbasedonchangingthenumberofunits:(a)Creasepattern;(b)Part-foldedconfiguration;(c)Frontprojectionoftwoparts;(d)Foldingsequenceofapapercardprototype.AsshowninFig.4.3(a)and(b),itisassumedthatthetoppart(Ⅱ)isArc-Miuralikepattern.Iftoppartisnotconsidered,therelationshipofedgeanglesofthebottompart(Ⅰ)canbededuced:.Forthetoppart,thisrelationshipshouldbealsoAA12satisfiedtoensuretherigidfoldingprocess.FromrelationshipdeducedfromEqns.2.4-2.6,24coscosA122(4.4)1cossincosBitcanbeseenthatforthetoppart(whereofallunitsarethesame),isequalBA1towhenor.ThecasethatisexcludedbecauseA212121245 Chapter4GEOMETRYOFGRADEDORIGAMITUBESthedistributionofisdifferentfromthebottompart,asshowninFig.4.3(c).AWhen,thecreaselinesformingperlayerareliealongparallelplanesfor12Athetoppartandthebottompart.Inthiscase,thetoppartandbottompart,whichareallrigidorigamipatterns,canbeconnectedtogetherthroughthesameedgeangleAandlengthofcreaselinestoformthegradedrigidArc-Miurapattern.Butitisnotflat-foldable.Forthispattern,itisassumedthatissatisfied.Thesection12betweenthetoppartandbottompartisdefinedasthetransitionsection.Thedensedottedlineischangingfromvalleylinetomountainlineinthefoldingprocess.4.1.2GradedArc-MiuraPatternBasedonCombinedmethodsBycombingthebasicindependentmethods,fournewgradedorigamitubesaregenerated.AsshowninFig.4.4,anewstructureisachievedthroughchangingthelengthofcreaselinesandchangingthesectorangle.Twogradedparameterscanbeselectedinthiscircumstance,unitwidthwandoneofthefollowingfivevariables,,,,and.Forthispattern,M-verticesandV-verticescanbeBMAVAMBVBdesignedtoliealongconcentriccylindersbysatisfyingthefollowingrelationships,bb1122coscos(4.5a)bb1122coscos(4.5b)(a)(b)Fig.4.4GradedArc-Miurapatternbasedonthecombinationofchangingthelengthofcreaselinesandchangingthesectorangle:(a)Creasepattern;(b)Foldingsequenceofapapercardprototype.Bycombiningthefollowingtwomethodstogether,changingthelengthofcreaselinesandchangingthenumberofunits,thefeatureofthesetwogradedorigamistructurescanbeintegratedtoconstructanewone.AsshowninFig.4.5,thenumber46 Chapter4GEOMETRYOFGRADEDORIGAMITUBESofunitsandunitwidtharetwogradedparameters.ThesequenceshowninFig.4.6isgeneratedbycombiningchangingthesectorangleandchangingthenumberofunitstogether.Thenumberofunitsandoneconfigurationvariablearetwogradedparameters.(a)(b)Fig.4.5GradedArc-Miurapatternbasedonthecombinationofchangingthelengthofcreaselinesandchangingthenumberofunits:(a)Creasepattern;(b)Foldingsequenceofapapercardprototype.(a)(b)Fig.4.6GradedArc-Miurapatternbasedonthecombinationofchangingthesectorangleandchangingthenumberofunits:(a)Creasepattern;(b)Foldingsequenceofapapercardprototype.ThecircumstancethatgradedArc-Miurapatternbasedonthecombinationofchangingthelengthofcreaselines,changingthesectorangleandchangingthenumberofunitsisshowninFig.4.7.Theunitwidthw,oneconfigurationangle47 Chapter4GEOMETRYOFGRADEDORIGAMITUBESvariableandthenumberofunitscanbechosenasgradedparametersatthesametime.(a)(b)Fig.4.7GradedArc-Miurapatternbasedonthecombinationofchangingthelengthofcreaselines,changingthesectorangleandchangingthenumberofunits:(a)Creasepattern;(b)Foldingsequenceofapapercardprototype.4.2GradedArcPattern4.2.1GradedArcPatternBasedonIndependentMethodsAsshowninFig.4.8andFig.4.9,thegradedstructuresaregeneratedbychangingthelengthofcreaselinesandsectorangle,respectively.Thephysicalmodelsofthesestructuresarealsogiven.AndthegradedparametersarethesamewithgradedArc-Miurapattersapplyingsamemethods.Bychangingthenumberofcreaselinesinspecificvertex,agradedorigamitubecanbeconstructed,asshowninFig.4.10.Thefollowingrelationshipcanbeobtainedfromthegeometries,22cosAA11sincos11cos(4.6)22cosAA22sincos22cos(4.7)AA212(4.8)4cosA11cosB11(4.9)cosA111cos2144cosA21cosB21(4.10)cosA221cos214FromtheEqns.4.6-4.8,thefollowingequationcanbefound,48 Chapter4GEOMETRYOFGRADEDORIGAMITUBES42422sin11cosAA2cos1sin2cos112cos2cot1(4.11)A222sin2Eqns.4.9-4.11canbecombinedtogethertodeterminetherelationshipbetweensectorangles,anddihedralangles,,,.Thatistosay,whenoneof12B1B2A1A2configurationvariablesandconstantanglesaregiven,otherparameterscanbeestablished.ThesectorangleandthelengthofcreaselinesofthetoppartandthebottompartshouldbethesameforgradedArcpatternbasedonthisindependentmethod.Itistosay,andbb.Sectorangeldoesn’taffecttherelationshipbetweenthe2121toppartandbottompart.(a)(b)Fig.4.8GradedArcpatternbasedonchangingthelengthofcreaselines:(a)Creasepattern;(b)Physicalmodel.(a)(b)Fig.4.9GradedArcpatternbasedonchangingthesectorangle:(a)Creasepattern;(b)Physicalmodel.49 Chapter4GEOMETRYOFGRADEDORIGAMITUBES(a)(b)(c)Fig.4.10GradedArcpatternbasedonchangingthenumberofunits:(a)Creasepattern;(b)Part-foldedconfiguration;(c)Physicalmodel.4.2.2GradedArcPatternBasedonCombinedMethodsThegradedparametersofthefollowinggradedArcpatternsthatarereflectedinFigs.4.11-4.14basedonthecombinedmethodsaresimilartogradedArc-Miurapatternusingthesamemethod.Itisassumedthattherelationshipbbcoscos1122issatisfiedtoensureverticesallliealongacylinderinpart-foldedconfigurationforthesequenceshowninFig.4.11(b).(a)(b)Fig.4.11GradedArcpatternbasedonthecombinationofchangingthelengthofcreaselinesandchangingthesectorangle:(a)Creasepattern;(b)Physicalmodel.50 Chapter4GEOMETRYOFGRADEDORIGAMITUBES(a)(b)Fig.4.12GradedArcpatternbasedonthecombinationofchangingthelengthofcreaselinesandchangingthenumberofunits:(a)Creasepattern;(b)Physicalmodel.(a)(b)Fig.4.13GradedArcpatternbasedonthecombinationofchangingthesectorangleandchangingthenumberofunits:(a)Creasepattern;(b)Physicalmodel.(a)(b)Fig.4.14GradedArcpatternbasedonthecombinationofchangingthelengthofcreaselines,changingthesectorangleandchangingthenumberofunits:(a)Creasepattern;(b)Physicalmodel.51 Chapter5MECHANICALBEHAVIOUROFGRADEDORIGAMISTRUCTURESChapter5MECHANICALBEHAVIOUROFGRADEDORIGAMISTRUCTURES5.1Quasi-staticImpactResponseofGradedOrigamiTubesAnumberofnumericalsimulationsofthequasi-staticaxialcrushingofgradedArcpatternwereconductedtoseewhetherthegradedstructureinorigamitubehadanyinfluences.Andthecrosssectionoftubewasquadrangle.[36]CommercialFEAsoftwareAbaquswasappliedtostudytheaxialcrushingprocess.Thedetailedmodelingtechniquewasgivenbelow.First,quasi-staticaxialloadingwasconductedtoignoretheinfluenceofdynamiceffect.Themainpointofthismethodwastochooseapropertimethatshouldbelongenoughtoensurecrushingprocessquasi-staticbutalsoasshortaspossibletosavetheanalysistime.RecommendedbyAbaqusdocument,theratioofkinematicenergytoinitialenergyshouldbelow5%duringmostoftheanalysistimetoignorethedynamiceffects.Second,four-nodeshellelementS4Rwasadoptedtomeshthesidewallsoftubes,withfiveintegrationpointsthroughthethickness.Itallowedfinitemembranestrainsandarbitrarylargerotations,andwascheaptosolveproblemswithlargedeformationandcomplicatedcontacts,buthastheproblemofhourglassing.Toignorethehourglassingeffect,theratioofartificialenergytoinitialenergyshouldbebelow5%,recommendedbyAbaqusdocument.Generalcontactwasemployedtomodelthecontact,withafrictioncoefficientof0.25.Foreachtubes,itstopandbottomendwerefixedonthecorrespondingrigidpanelbyacoupleconstraint.What’smore,noinitialimperfectionswereintroducedtothetubemodel.Anannealedmildsteelwasselectedfortheanalysis,withdensity37332.3kgm/,Young’smodulusE=190.5Gpa,Poisson’sratio0.3,yieldstress287.9MPa,ultimatestress506.9MPaandisotropichardening.uuTheconvergencetestsofmeshdensityandanalysistimewereconducted.GA-3-(170-170-170)wasselectedforthetestsandtworecommendsinAbaqusdocumentationwerechecked.ItwasassumedthatforGA-m-(--),GAmeantthegradedArcpattern,mmeantthetotalnumberoflayer,,andrepresentedthedihedralangleofeachlayerfrombottomtotop,respectively.ForBmeshdensity,threetypeswereused,coarsewith13608elements,mediumwith27600elements,andfinewith57420elements,andanalyzedunderanidenticaltimeof0.02s.ThemeancrushingforceFofeachelementdensitywascalculatedandnormalizedmagainstthatofthefinemeshdensity.AsshowninTab.5.1,thedateshowedacleartrendofconvergence,itistosaythedifferencebetweenFformiddlemeshdensitym52 Chapter5MECHANICALBEHAVIOUROFGRADEDORIGAMISTRUCTURESandfinemeshdensitywasbelow5%.InFig.5.1,theratiooftheartificialenergytointernalenergywasplottedagainstdisplacement.Anditcanbeseenthattheratiowasbelow5%forbothmiddleandfinemeshdensity.Thereforethemediummeshdensitywasselected,wheretheelementsizewas1mm.Foranalysistime,tubewithmeshsize1mmwasanalyzedunderthreeanalysistimes,i.e.,shortwith0.01s,middlewith0.02sandlongwith0.04s.AcleartrendofconvergencewasobservedfromthedatereflectedinTab.5.2.ThedifferencebetweenFmformiddleanalysistimeandlonganalysistimewasbelow5%.Theratioofkinematicenergy(KE)tointernalenergy(IE)wasbelow5%forallofthethreeanalysistimesduringmostofthecrushingprocess,seeFig.5.2.Thereasonwhythisratiowashighatthebeginningcamefromthefactthattheinternalenergywassmallduetolittlematerialdeformation.Thereforetheanalysistime0.02swasselected.Table5.1Meshsizeconvergencetestresults.MeshdensityMeshsize(mm)NumberofelementsNormalizedFmCoarse1.4136081.096Medium1276001.040Fine0.7574201Fig.5.1AE/IEvsdisplacementcurvesofthreemeshdensities.53 Chapter5MECHANICALBEHAVIOUROFGRADEDORIGAMISTRUCTURESTable5.2Analysistimeconvergencetestresults.AnalysistimeDuration(s)NormalizedFmShort0.011.023Middle0.021.004Long0.041Fig.5.2KE/IEvsdisplacementcurvesofthreeanalysistimes.Theorigamitubeswithdifferentdihedralangleswerestudiedfirst,fivedihedralangleswereselected,from170to178with2interval.Andthelayerofeverymodelwas3.ThenumericalsimulationresultsareshowninTab.5.3.Itcanbeseenthatfororigamitubeswithsmalldihedralangels,thedeformationprocessfollowsthecreasepattern,butwhenthedihedralangelisbiggerthan172,thedeformationprocessdon’tfollowingthepattern.Withtheincreaseofthedihedralangles,theinitialpeakforceFincreasescorrespondingly.Whenthecrushingprocessofmodelmaxfollowsthepattern,themeaningcrushingforcebecomesbiggerwiththeincreaseofdihedralangle.ThedeformationprocessoftwomodelsfollowingpatternandnotisshowninFig.5.3.54 Chapter5MECHANICALBEHAVIOUROFGRADEDORIGAMISTRUCTURESTable5.3FEAresultsofnormalorigamitubes.ModelFmax(kN)Fm(kN)FollowingpatternGA-3-(170-170-170)24.4513.94YGA-3-(172-172-172)27.2814.06YGA-3-(174-174-174)31.1316.19NGA-3-(176-176-176)36.6716.22NGA-3-(178-178-178)44.5115.67N(a)(b)Fig.5.3Thedeformationprocess:(a)GA-3-(170-170-170);(b)GA-3-(178-178-178).Byintroducingthegradedorigamiconceptintothisorigamitube,theinfluenceofgradedparameterwasstudied.Twocaseswereadopted,asshowninTab.5.4andTab.5.5.Thebasicsmallestdihedralangle170isnotchangedfortwocases.Fromtheresults,itcanbeseenthatbyintroducingonelayerwithsmalldihedralangle(170),themodelGA-3-174changestoGA-3-(174-170-170)orGA-3-(174-174-170),andthedeformationprocessbecomesfollowingthepattern,seeFig.5.4.Butwhenoneofthedihedralanglelargerthan174,thepatternisnotfollowingthepattern.Thatistosay,theintroducedgradedstructurehasalimitedinfluencetothedeformationprocess.55 Chapter5MECHANICALBEHAVIOUROFGRADEDORIGAMISTRUCTURESTable5.4FEAresultsofgradedorigamitubeswithcase1.ModelFmax(kN)Fm(kN)FollowingpatternGA-3-(174-170-170)24.9014.18YGA-3-(176-170-170)36.6716.20NGA-3-(178-170-170)25.0215.87NTable5.5FEAresultsofgradedorigamitubeswithcase2.ModelFmax(kN)Fm(kN)FollowingpatternGA-3-(174-174-170)24.6714.47YGA-3-(176-176-170)32.2416.12NGA-3-(178-178-170)44.6615.27N(a)(b)Fig.5.4Thedeformationprocess:(a)GA-3-(174-174-174);(b)GA-3-(174-174-170).56 Chapter5MECHANICALBEHAVIOUROFGRADEDORIGAMISTRUCTURES5.2Three-pointBendingResponseofSandwichStructureswithGradedMiura-oriCoreQuasi-staticthree-pointbendingresponseofsandwichbeamwithgradedMiura-oricorewasstudied.Four-nodeshellelementS4Rwasadoptedtomeshthebeam,withfiveintegrationpointsthroughthethicknessandglobalelementsize1mm.Thebeamconsistsoftwofacesheetsandonecore.Inthesimulation,debondingbetweencoreandfacesheetswasnotconsidered,thereforethefacesheetsandcoreweretiedtogether.Thebeamwasrestedontwofixedsupportedcolumns.Thetopcolumn,initiallycontactingwiththetopfacesheet,moveddownwardstocrushthebeam.Acrushtime0.03swasselected,whichlimitedthekineticenergytointernalenergyratiotobelessthan5%.Generalcontactwasemployedtomodelthecontact,withafrictioncoefficientof0.25.Thematerialthicknessofthefacesheetandcorewere1mmand0.5mm,respectively.Crushdistancewas25mm.SEAmeanttheenergyabsorptionperunitmass.5.2.1Three-pointBendingofSandwichBeamwithGradedMiura-oriCoreBasedonChangingtheNumberofUnitsinyDirectionThelengthofsandwichbeamwasL=180.5mmandthedistancebetweensupportedcolumnswasl=140mm,asshowninFig.5.5(b).Thewidthandheightofthebeamwere40mmand10mm,respectively.DiameterofthreecolumnswereallD=5mm.TheparametersofMiura-oricore,refertoFig.2.11,werea=20mm,b=10mm,64.3.(a)(b)Fig.5.5Thesandwichstructure:(a)GradedMiura-oricore;(b)Sandwichbeamunderthree-pointbending.ThesimulationresultisshowninFig.5.6andTab.5.6,anditcanbeseenthatbyintroducingthegradedconceptintosandwichbeamwithMiura-oricore,themassofthewholesandwichbeamcanbereducedwithoutchangingtheenergyabsorptionobviously.Whenconsiderthemass,theenergyabsorptioncapability(SEA)increases18%comparedwiththesandwichbeamwithMiura-oricore.57 Chapter5MECHANICALBEHAVIOUROFGRADEDORIGAMISTRUCTURESFig.5.6Thequasi-staticthree-pointbendingresponseofsandwichbeamwithMiura-oricoreandgradedMiura-oricorebasedonchangingthenumberofunitsinydirection.Table5.6NumericalsimulationresultsofsandwichstructureswithMiura-oricoreandgradedMiura-oricorebasedonchangingthenumberofunitsinydirection.EnergySEAFmaxFmax/MassTypeMass(kg)absorption(kJ/kg)(kN)(kN/kg)(kJ)Miura-ori0.200.100.507.3636.8GradedMiura-ori0.170.100.597.2442.59(a)(b)Fig.5.7Deformationprocessofsandwichbeam:(a)Miura-oricore;(b)GradedMiura-oricore.5.2.2Three-pointBendingofSandwichBeamwithGradedMiura-oriCoreBasedontheCombinationofChangingtheLengthofCreaseLinesandSectorAngleinxDirectionLength,widthandheightofthesandwichbeamwere90mm,28.7mmand10mm,58 Chapter5MECHANICALBEHAVIOUROFGRADEDORIGAMISTRUCTURESrespectively.ThedistancebetweensupportedcolumnswasL=70mm.Eachsandwichcorecontainednineunits,asshowninFig.5.8.ParametersofnormalMiura-oricorewerethesamewithprevioussection.ForgradedMiura-oricores,lengthofcreaselinebandedgeanglewerechangedandbbcoscoswassatisfiedtoensureB11nnthatthecontourisrectangle.ofGraded-impliedthegradedparameter,edgeangle,wheren1.BBnB1BFromFig.5.9andTab.5.7,itcanbeseenthattheenergyabsorptionofsandwichbeamswithgradedMiura-oricoresareallincreasedexceptgraded-6,Takingthemassintoconsideration,forgraded-18,SEAincreases28.5%,whichhavethebiggestimprovementcomparedwiththesandwichbeamwithnormalMiura-oricore.Fig.5.8GradedMiura-oricorebasedonthecombinationofchangingthelengthofcreaselinesandchangingthesectorangleinxdirection.Fig.5.9Thequasi-staticthree-pointbendingresponseofsandwichbeamwithMiura-oricoreandfourcasesofgradedMiura-oricore.59 Chapter5MECHANICALBEHAVIOUROFGRADEDORIGAMISTRUCTURESTable5.7NumericalsimulationresultsofsandwichbeamswithMiura-oriandfourcasesofgradedMiura-oricore.EnergySEAFmax/MassTypeMass(g)Fmax(N)absorption(J)(kJ/kg)(kN/kg)Miura-ori61.61135.982.215413.9587.87Graded-661.68131.412.136043.7597.99Graded-1261.87148.052.396574.27106.26Graded-1862.20176.702.847356.04118.26(a)(b)Fig.5.10Deformationprocessofsandwichbeam:(a)Miura-oricore;(b)GradedMiura-oricore(Graded-6).FromthenumericalsimulationresultsofsandwichbeamswithMiura-oricoreandtwotypesofgradedMiura-oricore,itcanbeconcludedthesandwichbeamwithgradedMiura-oricorecouldhavebetterenergyabsorptionperformancecomparedwithMiura-oricoreunderthree-pointbendingbyadjustingthegradedparameter.ForthesandwichbeamwithMiura-oricore,thefailuremodeistheformationoflocalizedplastichingesofthefacesheetalongthebeam,asshowninFig.5.7(a),orthecombinationoflocalizedplastichingesofthefacesheetandbucklingofcorenearthemiddleofbeam,asshowninFig.5.10(a).Largeplasticdeformationoccursneartheseplastichinges,butonlysmalldeformationappearsonotherpartsofthebeam.Mostofthematerialsarewastedinthiscircumstance.ByusingthegradedMiura-oricore,thematerialcanbeeffectivelyused,givingthesandwichbeamahighstresspropertyinthehighstressregion,andlowstresspropertyinthelowstressregion.60 Chapter6THEMANUFACTUREOFORIGAMISTRUCTURESChapter6THEMANUFACTUREOFORIGAMISTRUCTURESInthischapter,weproposeadesignmethodfororigamiprototypes.A3Dprinter,ObjectConnex350isadoptedtoprintthedesignmodel.Twobasicmaterialsareverowhiteandtangoblack,whichareallUV-curableresinmaterials,thatistosay,theywillbesolidifiedwhenexposedtoUV.Adesignstandardisthatthesoftmaterialactasthejoints(creaselines),hardmaterialactastherigidpanel.Weuseanorigamipattern,square-twistpattern,togenerateitscorrespondingprototype.Thereasonwhywechoosethispatternisbecausefourdifferenttypescan[37]begeneratedwhenthedistributionofmountainlineandvalleylinearechanged.Twoofthesefourtypesarerigid-foldable,butothersarenon-rigid.Inthiscircumstance,wecandesigntherigidprototypeandnon-rigidprototypefromthesamepattern.Thatistosay,throughthedesignoftwotypesofsquare-twistpattern,thedesignmethodoftwokindsoforigamistructures,rigidandnon-rigid,canbederived.Basedthemethodsexploredinthischapter,thegradedorigamistructuresmentionedinchapters3and4canbeachieved.(a)(b)Fig.6.1Differentmountain-valleyfolddistributionsofsquare-twistpattern:(a)type1;(b)type2.AsshowninFig.6.1,twotypesofsquare-twistpatternaregiven,type1isrigid-foldable,buttype2isnon-rigid.First,wefocusontherigidsquare-twistpattern,type1.Asitisrigid,thefacesbetweencreaselinesarenotdeformingduringthefoldingprocess,thematerialverowhiteisselectedtomodelthefaces.Forcreaselines,astheyneedtorotateduringfoldingmotion,tangoblackisselectedtomodelthecreaselines(joints).Twodesignedconfigurationsarechosen,oneispart-foldedconfiguration,andtheotherisflat-deployedconfiguration.Forpart-foldedconfiguration,asshowninFig.6.2,thecreaselinesandrigidfacesaredividedintotwoparts,whichwillconvergeintoonepartinthesoftware61 Chapter6THEMANUFACTUREOFORIGAMISTRUCTURESObjectStudio.Thefacesaremodeledwiththinrigidpanel,whosethicknessis0.8mm.Butforcreaselines,itis0.5mm.Thereasonthathingeisthinthanpanelcomesfromthefactthatthethicknessisenoughtoensurethefoldingprocessandcanreducetheinterference.Asforthewidthofjoints,2.5mmisturnedouttobeapropervaluetoensurethesmoothfoldingprocessandavoidlargererror.Theoffsetofhingesrefertoeveryfaceofadjacentpanelsis0.15mm.Forflat-deployedconfiguration,seeFig.6.3,italsocontainstwoparts.Differentfromthedesigninpart-foldedconfiguration,theoffsetofhingesinfluencethedistributionofmountainlineandvalleyline.Thejustificationofmountainlineandvalleylineisbasedonthedifferentoffsetdistance.Formountainline,theoffsetdistanceiszeroreferringtothecorrespondingdatum,butforvalleyline,itis0.3mm.(a)(b)Fig.6.2Thedesignoftype1inpart-foldedconfiguration:(a)Designprocess;(b)Prototype.62 Chapter6THEMANUFACTUREOFORIGAMISTRUCTURES(a)(b)Fig.6.3Thedesignoftype1inflat-deployedconfiguration:(a)Designprocess;(b)Prototype.Fornon-rigidsquare-twistpattern,type2,wecan’tdeterminethepart-foldedconfigurationprecisely.Therefore,flat-deployedconfigurationischosentogeneratethisstructure.Assomefacesaredistortedduringfoldingprocess,notallthefacescanbemodeledbyverowhite(hardmaterial).Thus,apropermixedmaterialthatcannotonlydeformduringfoldingmotion,butalsostrongenoughtoensurethefoldingprocess,isneeded.Wechoosesixdifferentmixedtypesforexperiment:DM_9840_shore40,DM_9850_shore50,DM_9860_shore60,DM_9870_shore70,DM_9885_shore85andDM_9895_shore95.Theresultsshowthatwhenthemixedmaterialisrubber-likematerial,DM_987_shore70,whoseelongationatbreakis75%,thefoldingprocessisrelativelygood.Asthehingesarealsodistortedinotherdirections,thematerialtangoblackisnotapplicable.Thenwetestthematerialofhinges,usingthementionedsixmixedmaterials.Asaresult,DM_987_shore70isdemonstratedaproperone.Fig.6.4Thedesignedprototypeoftype2.63 Chapter7FINALREMARKSChapter7FINALREMARKSTheaimofthisthesisistoexplorewaysofgeneratinggradedorigamistructuresbycombininggradedconceptwithorigamistructures.Inthischapter,themainachievementsinthedesignofgradedrigidorigamistructuresandthemechanicalpropertiesarepresented,followedbyanoverviewoffuturework.7.1MainAchievementsFirstly,threecasesofrigidorigamistructuresareexplored.Basedonthesecases,threebasicmethodsareproposedtogenerategradedorigamistructures,includingchangingthelengthofcreaselines,changingthesectorangleandchangingthenumberofunits.Andcombinedmethodsareobtainedfromthepermutationandcombinationofbasicmethods.Oneofthemostfamousorigamipatterns,Miura-ori,ischosentoformmulti-layergradedorigamimetamaterials.79differentstructuresbasedonMiura-oriand19differentstructuresbasedontapered-Miuraareachieved.TheinfluencesofgradedparameterstoPoisson’srationarealsostudied.Secondly,derivativeMiurapattern,Arc-MiuraandArcpatternareselectedtoachievethegradedorigamitubesusingthesimilarmethod.Inthedesignofgradedorigamitubesbaseonchangingthenumberofunits,weintroducepointwithfivecreaselines,whichcanbeseenasspherical5Rlinkage.Thirdly,numericalsimulationsbasedonthefiniteelementmethodareemployedtoassessthemechanicalpropertiesofgradedorigamistructures,inwhichquasi-staticthree-pointbendingresponseofsandwichbeamswithgradedMiura-oricoreandquasi-staticaxialcrushingresponseofgradedorigamitubesareexplored.TheinvestigationrevealsthatsandwichbeamswithgradedMiura-oricorehavepreferableenergyabsorptioncapabilityinthree-pointbendingcomparedwiththenormalMiura-oricore,butthegradedstructureinorigamitubeshavesmallinfluencetothedeformationofwholestructure.Finally,amanufacturemethodoforigamipatternsisintroduced.Weusethe3Dprintertoprinttheorigamistructures.Twobasicmaterials,verowhiteandtangoblack,areusedtomodeltherigidfacesandcreaselines,respectively.Fornon-rigidpatterns,amixedmaterial,DM_9870_shore70ischosentomodelthefaceswithlargedeformation.Andtheresultshowsthatthismethodistimesavingandcanbeusedforvariousorigamistructures.7.2FutureWorksThisthesishasestablishedthemethodstoformgradedorigamistructures.The64 Chapter7FINALREMARKSmechanicalpropertiesofsandwichstructureswithgradedorigamipatternandgradedorigamitubesareexplored.Manydetailedanalysisaboutgradedorigamistructurecanbeconductednext.First,anexperimentisneededtoverifythenumericalsimulationresults.Second,metamaterialwithgradedstiffnesscanbeexploredunderthefoundationofgradedorigamistructures.Third,thisthesisgivesaresearchopportunitytotheconnectionofdifferentorigamistructures,thusformingmoreusefulorigamimetamaterials.65 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发表论文和参加科研情况说明发表论文和参加科研情况说明发表论文：[1]Xie,R.,Chen,Y.andGattas,J.M.,Parameterisationandapplicationofcubeandeggbox-typefoldedgeometries.InternationalJournalofSpaceStructures,2015,30(2),99-110.[2]Xie,R.,Li,J.andChen,Y.,Thegradedorigamistructures,ProceedingsoftheASME2015InternationalDesignEngineeringTechnicalConferences,2015,DETC2015-46081.68 ACKNOWLEDGEMENTSACKNOWLEDGEMENTSThisthesiswouldnotbepossiblewithouttheongoinghelpandsupportfrommyfriends,familyandcolleagues.ThefirstandmostimportantpersonIwouldliketothankismysupervisor,Prof.YanChen,whoseguidanceandadvicesleadmetotheworldoforigami.Heruniqueviewstodifferentproblemsandsparklingimaginationsbenefitedmeenormouslyduringeverydiscussion.Thehighstandardsthatshesetsforthescientificresearchalsodramaticallyinfluencemefrombeginningtoend.WithintheMotionStructureLaboratory,IwouldliketothankRuiPeng,DegaoHouandFufuYang,whocontributealotinvariouswaystomystudyinTianjinUniversity.Theiradvicesandhelpstomyresearchandinconstructingprototypesinspiremeandpromoteformationofthisthesis.IwouldalsoliketothankProf.ZhongYouforhisadvicesandhelpstomystudy,Dr.JiayaoMaforhishelponthenumericalsimulationandDr.Gattasforhishelponmakingphysicalmodelsandtakingphotos.Finally,Iwouldliketothankmyparents,brotherandsisterfortheirtremendousencouragement,loveandsupport.69 天津大学全曰制工程硕士学位论文＊：腿圓