The Mechanics of Ribbons and Möbius Bands

350 Pages·2015·9.124 MB·English

by  Roger Fosdick, Eliot Fried

Roger Fosdick
Editors
Eliot Fried
The Mechanics
of Ribbons and
Möbius Bands
The Mechanics of Ribbons and Möbius Bands
Roger Fosdick (cid:2) Eliot Fried
Editors
The Mechanics of Ribbons
and Möbius Bands
Previously published in Journal of Elasticity Volume 119,
Issues 1–2, 2015
Editors
RogerFosdick EliotFried
AeropsaceEngineeringandMechanics MathematicalSoftMatterUnit
UniversityofMinnesota OkinawaInstituteofScienceandTechnology
Minneapolis,Minnesota,USA Okinawa,Japan
ISBN978-94-017-7299-0 ISBN978-94-017-7300-3(eBook)
DOI10.1007/978-94-017-7300-3
SpringerDordrechtHeidelbergNewYorkLondon
©SpringerInternationalPublishingSwitzerland2016
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Contents
(cid:3)
Foreword SpecialInvitedCollectionontheMechanicsofRibbons
andMöbiusBands
(cid:3)
R.Fosdick E.Fried 1
TranslationofMichaelSadowsky’sPaper“AnElementaryProoffortheExistenceof
aDevelopableMÖBIUSBandandtheAttributionoftheGeometricProblemtoa
VariationalProblem”
(cid:3)
D.F.Hinz E.Fried 3
TranslationandInterpretationofMichaelSadowsky’sPaper“TheoryofElastically
BendableInextensibleBandswithApplicationstotheMÖBIUSBand”
(cid:3)
D.F.Hinz E.Fried 7
TranslationofMichaelSadowsky’sPaper“TheDifferentialEquationsofthe
MÖBIUSBand”
(cid:3)
D.F.Hinz E.Fried 19
TranslationofW.Wunderlich’s“OnaDevelopableMöbiusBand”
R.E.Todres 23
Gamma-LimitofaModelfortheElasticEnergyofanInextensibleRibbon
(cid:3)
N.O.Kirby E.Fried 35
“Wunderlich,MeetKirchhoff”:AGeneralandUnifiedDescriptionofElastic
RibbonsandThinRods
(cid:3)
M.A.Dias B.Audoly 49
EquilibriumShapeswithStressLocalisationforInextensibleElasticMöbiusand
OtherStrips
(cid:3)
E.L.Starostin G.H.M.vanderHeijden 67
BendingPaperandtheMöbiusStrip
(cid:3)
S.Bartels P.Hornung 113
RoadmaptotheMorphologicalInstabilitiesofaStretchedTwistedRibbon
(cid:3) (cid:3)
J.Chopin V.Démery B.Davidovitch 137
TheShrinkingFigureEightandOtherSolitonsfortheCurveDiffusionFlow
(cid:3) (cid:3) (cid:3) (cid:3)
M.Edwards A.Gerhardt-Bourke J.McCoy G.Wheeler V.-M.Wheeler 191
KinematicalAspectsofLevi-CivitaTransportofVectorsandTensorsAlongaSurface
Curve
J.Casey 213
(cid:3)
Non-EuclideanRibbons GeneralizedSadowskyFunctionalforResidually-Stressed
ThinandNarrowBodies
E.Efrati 251
TheSecond-OrderL2-FlowofInextensibleElasticCurveswithHingedEndsinthe
Plane
(cid:3) (cid:3)
C.-C.Lin Y.-K.Lue H.R.Schwetlick 263
BucklingofNaturallyCurvedElasticStrips:TheRibbonModelMakesaDifference
(cid:3)
B.Audoly K.A.Seffen 293
ResidualStressesandPoisson’sEffectDriveShapeFormationandTransitionof
HelicalStructures
(cid:3) (cid:3)
Z.Chen X.Han H.Zheng 321
RepresentationforaSmoothIsometricMappingfromaConnectedPlanarDomain
toaSurface
(cid:3) (cid:3)
Y.-C.Chen R.Fosdick E.Fried 335
Erratumto:FourPapersPublishedintheJournalofElasticity(2015)119(1–2)
(cid:3)
R.Fosdick E.Fried 351
DOI10.1007/978-94-017-7300-3_1
ReprintedfromJournalofElasticityJournal,DOI10.1007/s10659-015-9516-7
Foreword
SpecialInvitedCollectionontheMechanicsofRibbons
andMöbiusBands
RogerFosdick1·EliotFried2
©SpringerScience+BusinessMediaDordrecht2015
ThisvolumeoftheJournalofElasticitycontainsacollectionofpapersdedicatedtothehis-
toricaldevelopmentofandcurrentresearchinterestsinthemechanicsoftheMöbiusband.
Itcontainsfourtranslationsoflandmarkpapers,originallywritteninGerman,thatplayed
amajorroleinthedevelopmentofthisandrelatedtopicsinmechanics:threepublishedby
Sadowskyfromthe1930’sandonepublishedbyWunderlichin1962.Inaddition,thereare
twelvecurrentresearchpapersandreviewsthatprovideinsightintotheintricatemechanics
ofstretchableandunstretchableelasticbands,theirpreferredequilibriumshapesaswellas
thegeometryofsurfacesandtherepresentationofisometricmappings.AMöbiusbandneed
notbearuledsurface,butitmaybeadevelopablesurface,whichisakindofruledsurface,
withtheadditionalpropertythatitmaybecontinuouslyflattenedintoaplanarformwhile
preservingitsintrinsiclengthsandangles,i.e.,theresultofanisometricmappingofaflat
domainintoasurface.Thepapersinthiscollectionaddressmathematicalandcomputational
issuescoveringthiswiderangeofpossibilities.
TheMöbiusbandwasformallyidentifiedasanobjectofmathematicalinterestinthemid-
nineteenthcentury.Thefirstpublicationstoincludediscussionsofitstopologicalproperties
werethoseofListingin1862andMöbiusin1865.Ithas,though,beenreportedthatboth
ListingandMöbiusrecognizedtheimportanceoftheMöbiusbandalittleearlierin1858and
that in his1847 studyof topologyListing even madepassing remarks concerning twisted
ribbon-likesurfaces.
B R.Fosdick
[email protected]
E.Fried
[email protected]
1 DepartmentofAerospaceEngineeringandMechanics,UniversityofMinnesota,Minneapolis,
MN55455-0153,USA
2 MathematicalSoftMatterUnit,OkinawaInstituteofScienceandTechnology,Onna,
Okinawa904-0495,Japan
1 Reprintedfromthejournal
R.Fosdick,E.Fried
TheinfluenceoftheMöbiusbandnowextendswellbeyondmathematicstoencompass
multiplebranchesofscienceandengineering,architecture,philosophy,psychology,andthe
musical, visual, literary, and performing arts. New directions for exploiting its intriguing
topologicalpropertiesinscienceandengineeringhaveemergedinresponsetorecentbreak-
throughs in the ability to fabricate objects with molecular-scale precision. Novel ideas of
inductionlessresistorsandsuperconductorswithhightransitiontemperature,molecularen-
gines,andhelicalmagnetismhavebeenproposed.Itistheintriguingone-sided,one-edged,
nonorientablenatureofMöbiusbandsthatisdrivingmuchofthemodernworktowarddis-
coveriesandapplicationsofnanotechnologicalimportance.
Thisvolumeisintendedtoenhancegrowthin,andprovideinsightfor,theadvancement
offundamentalresearchanddiscoveryinmechanicsrelatedtoribbonsandtheMöbiusband.
Thecontributionsarewideinscopeandtheyillustratetheimportantrolethatmathematical
modelingandcomputationplayinthisnovelareaofresearch.
Reprintedfromthejournal 2
DOI10.1007/978-94-017-7300-3_2
ReprintedfromJournalofElasticityJournal,DOI10.1007/s10659-014-9490-5
Translation of Michael Sadowsky’s Paper
“An Elementary Proof for the Existence of a Developable
MÖBIUS Band and the Attribution of the Geometric
Problem to a Variational Problem”
DenisF.Hinz·EliotFried
Received:8August2014/Publishedonline:12September2014
©SpringerScience+BusinessMediaDordrecht2014
Abstract This article is a translation of Michael Sadowsky’s original paper “Ein ele-
mentarer Beweis für die Existenz eines abwickelbaren MÖBIUSschen Bandes und die
ZurückführungdesgeometrischenProblemsaufeinVariationsproblem.”whichappearedin
Sitzungsberichte der Preussischen Akademie der Wissenschaften, physikalisch-mathemati-
scheKlasse,17.Juli1930.–Mitteilungvom26.Juni,412–415.PublishedonSeptember12,
1930.
Keywords Möbiusband·Energyfunctional·Bendingenergy
MathematicsSubjectClassification 74K20·74K10·53A04·74G55·01A75
TranslationoftheOriginalPaper
MÖBIUS[1]illustratedthebandbearinghisnamebydescribinghowonemaybeconstructed
bybendingarectangularsheetofpaper.Subsequently,ithasbeenaskedwhetherthiscon-
structioncanbeachievedsolelyasaconsequenceofthecomplianceofthesheetinbending
or whether stretching is also required. In other words—the question has been raised as to
whethertheMÖBIUSbandisdevelopableinthestrictsense.
MÖBIUS himselfdidnotaddressthisquestion,sinceitwasirrelevantforhispurposes.
The developability of his band has been challenged by many, including, it is rumored,
H.A. SCHWARZ. In the present work, the existence of a developable band is established
onelementarygeometricgrounds.
CitationsofthistranslationshouldreferalsotoSadowsky’soriginalpaper,ascitedintheAbstract.
D.F.Hinz·E.Fried(B)
MathematicalSoftMatterUnit,OkinawaInstituteofScienceandTechnology,Okinawa904-0495,
Japan
e-mail:[email protected]
D.F.Hinz
e-mail:[email protected]
3 Reprintedfromthejournal
D.F.Hinz,E.Fried
Fig.1 AdaptationofFig.1from
theoriginalversionofthepaper
Fig.2 AdaptationofFig.2from
theoriginalversionofthepaper
Imagine an elongated rectangle of flexible but completely inextensible paper. Further,
imagine a rigid cylindrical rod with circular cross-section. Construct two parallel planes
tangent to the rod. A bendable but inextensible paper strip may then be positioned to lie
withinoneoftheseplanes,wraphalfwayaroundtherod,andliewithintheremainingplane.
SeeFig.1foradepictionofthedescribedarrangement.
Considernowthreecylindricalrodswithcircularcross-sections,oneofwhichhasdiam-
eterequaltothesumofthediametersoftheremainingtwo.Therectangularstripofpaper
maythenbethreadedbetweentherodstoformaMÖBIUSband.Theresultingsurfacecon-
sists of three planar sections and three semicylindrical sections and, thus, is developable.
Anillustrationoftheconstructionforrodsofdiametersd,d,and2d isprovidedinFig.2.
Theelementaryconnectionsbetweentheparticularanglesandlengthsneededtoformsuch
abandarenotofinteresthere,sinceonlytheexistenceofthebandmatters.
TheforegoingdescriptiveconstructionofadevelopableMÖBIUSbandadmitsanequiv-
alent analytical construction. If attention is restricted to the previously considered simple
caseinwhichthediametersoftherodsared,d,and2d,theprojectionsontothedrawing
planeofthemidlinesoftherectangularportionsofthebandlieontheedgesofanequilateral
trianglewithsidelengthL,andtheaxesofallthreerodsareparalleltothedrawingplane.
Let A, B, C, D, E, and F be points on the midline of the MÖBIUS band and let A(cid:2), B(cid:2),
C(cid:2),D(cid:2),E(cid:2),andF(cid:2)betheirprojectionsontothedrawingplane.LetAB,BC,etc.denotethe
distancesmeasuredalongthemid-linefrompointAtoB,B toC,etc.andletA(cid:2)B(cid:2),B(cid:2)C(cid:2),
etc.denotethelengthsoftherectilinearconnectionsbetweenA(cid:2)andB(cid:2),B(cid:2)toC(cid:2),etc.inthe
drawingplane.WhereasthesectionsAF,BC,andDEofthemidlineremainrectilinear,the
sectionsAB,CD,andEF aretwistedalonghelicalcurves.Thesehelicalcurvesintersect
Reprintedfromthejournal 4