The shape of a Möbius strip
The Möbius strip, obtained by taking a rectangular strip of plastic or paper, twisting one end through 180 ∘ , and then joining the ends, is the canonical example of a one-sided surface. Finding its characteristic developable shape has been an open problem ever since its first formulation in refs 1...
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| Veröffentlicht in: | Nature materials 2007-08, Vol.6 (8), p.563-567 |
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| container_title | Nature materials |
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| creator | Starostin, E. L. van der Heijden, G. H. M. |
| description | The Möbius strip, obtained by taking a rectangular strip of plastic or paper, twisting one end through 180
∘
, and then joining the ends, is the canonical example of a one-sided surface. Finding its characteristic developable shape has been an open problem ever since its first formulation in refs
1,2
. Here we use the invariant variational bicomplex formalism to derive the first equilibrium equations for a wide developable strip undergoing large deformations, thereby giving the first non-trivial demonstration of the potential of this approach. We then formulate the boundary-value problem for the Möbius strip and solve it numerically. Solutions for increasing width show the formation of creases bounding nearly flat triangular regions, a feature also familiar from fabric draping
3
and paper crumpling
4
,
5
. This could give new insight into energy localization phenomena in unstretchable sheets
6
, which might help to predict points of onset of tearing. It could also aid our understanding of the relationship between geometry and physical properties of nano- and microscopic Möbius strip structures
7
,
8
,
9
. |
| doi_str_mv | 10.1038/nmat1929 |
| format | Article |
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∘
, and then joining the ends, is the canonical example of a one-sided surface. Finding its characteristic developable shape has been an open problem ever since its first formulation in refs
1,2
. Here we use the invariant variational bicomplex formalism to derive the first equilibrium equations for a wide developable strip undergoing large deformations, thereby giving the first non-trivial demonstration of the potential of this approach. We then formulate the boundary-value problem for the Möbius strip and solve it numerically. Solutions for increasing width show the formation of creases bounding nearly flat triangular regions, a feature also familiar from fabric draping
3
and paper crumpling
4
,
5
. This could give new insight into energy localization phenomena in unstretchable sheets
6
, which might help to predict points of onset of tearing. It could also aid our understanding of the relationship between geometry and physical properties of nano- and microscopic Möbius strip structures
7
,
8
,
9
.</description><identifier>ISSN: 1476-1122</identifier><identifier>EISSN: 1476-4660</identifier><identifier>DOI: 10.1038/nmat1929</identifier><identifier>PMID: 17632519</identifier><language>eng</language><publisher>London: Nature Publishing Group UK</publisher><subject>Biomaterials ; Chemistry and Materials Science ; Civil engineering ; Condensed Matter Physics ; Equilibrium ; Fabrics ; letter ; Materials Science ; Microstructure ; Nanotechnology ; Numerical analysis ; Optical and Electronic Materials ; Physical properties ; Physics</subject><ispartof>Nature materials, 2007-08, Vol.6 (8), p.563-567</ispartof><rights>Springer Nature Limited 2007</rights><rights>Copyright Nature Publishing Group Aug 2007</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c342t-8023fc8dfb38334a8b6bbff617a7583307aa6e19d97f5aaa8c7fa4e9c11db0a03</citedby><cites>FETCH-LOGICAL-c342t-8023fc8dfb38334a8b6bbff617a7583307aa6e19d97f5aaa8c7fa4e9c11db0a03</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1038/nmat1929$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1038/nmat1929$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids><backlink>$$Uhttps://www.ncbi.nlm.nih.gov/pubmed/17632519$$D View this record in MEDLINE/PubMed$$Hfree_for_read</backlink></links><search><creatorcontrib>Starostin, E. L.</creatorcontrib><creatorcontrib>van der Heijden, G. H. M.</creatorcontrib><title>The shape of a Möbius strip</title><title>Nature materials</title><addtitle>Nature Mater</addtitle><addtitle>Nat Mater</addtitle><description>The Möbius strip, obtained by taking a rectangular strip of plastic or paper, twisting one end through 180
∘
, and then joining the ends, is the canonical example of a one-sided surface. Finding its characteristic developable shape has been an open problem ever since its first formulation in refs
1,2
. Here we use the invariant variational bicomplex formalism to derive the first equilibrium equations for a wide developable strip undergoing large deformations, thereby giving the first non-trivial demonstration of the potential of this approach. We then formulate the boundary-value problem for the Möbius strip and solve it numerically. Solutions for increasing width show the formation of creases bounding nearly flat triangular regions, a feature also familiar from fabric draping
3
and paper crumpling
4
,
5
. This could give new insight into energy localization phenomena in unstretchable sheets
6
, which might help to predict points of onset of tearing. It could also aid our understanding of the relationship between geometry and physical properties of nano- and microscopic Möbius strip structures
7
,
8
,
9
.</description><subject>Biomaterials</subject><subject>Chemistry and Materials Science</subject><subject>Civil engineering</subject><subject>Condensed Matter Physics</subject><subject>Equilibrium</subject><subject>Fabrics</subject><subject>letter</subject><subject>Materials Science</subject><subject>Microstructure</subject><subject>Nanotechnology</subject><subject>Numerical analysis</subject><subject>Optical and Electronic Materials</subject><subject>Physical properties</subject><subject>Physics</subject><issn>1476-1122</issn><issn>1476-4660</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2007</creationdate><recordtype>article</recordtype><sourceid>BENPR</sourceid><recordid>eNpl0MtKxDAUBuAgiqOj4AOIFBeii2pO0ibNUgZvMOJmXIfTNnE6TC8m7cIX8wV8MSNTGdBVQvLxn8NPyAnQa6A8u2lq7EExtUMOIJEiToSgu-MdgLEJOfR-RSmDNBX7ZAJScJaCOiCni6WJ_BI7E7U2wuj56zOvBh_53lXdEdmzuPbmeDyn5PX-bjF7jOcvD0-z23lc8IT1cUYZt0VW2pxnnCeY5SLPrRUgUabhhUpEYUCVStoUEbNCWkyMKgDKnCLlU3Kxye1c-z4Y3-u68oVZr7Ex7eC1yIAqKpMAz__AVTu4JuymGWNhmEogoMsNKlzrvTNWd66q0X1ooPqnLv1bV6BnY96Q16bcwrGfAK42wIev5s247cB_Yd_gBXHB</recordid><startdate>20070801</startdate><enddate>20070801</enddate><creator>Starostin, E. 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L.</au><au>van der Heijden, G. H. M.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The shape of a Möbius strip</atitle><jtitle>Nature materials</jtitle><stitle>Nature Mater</stitle><addtitle>Nat Mater</addtitle><date>2007-08-01</date><risdate>2007</risdate><volume>6</volume><issue>8</issue><spage>563</spage><epage>567</epage><pages>563-567</pages><issn>1476-1122</issn><eissn>1476-4660</eissn><abstract>The Möbius strip, obtained by taking a rectangular strip of plastic or paper, twisting one end through 180
∘
, and then joining the ends, is the canonical example of a one-sided surface. Finding its characteristic developable shape has been an open problem ever since its first formulation in refs
1,2
. Here we use the invariant variational bicomplex formalism to derive the first equilibrium equations for a wide developable strip undergoing large deformations, thereby giving the first non-trivial demonstration of the potential of this approach. We then formulate the boundary-value problem for the Möbius strip and solve it numerically. Solutions for increasing width show the formation of creases bounding nearly flat triangular regions, a feature also familiar from fabric draping
3
and paper crumpling
4
,
5
. This could give new insight into energy localization phenomena in unstretchable sheets
6
, which might help to predict points of onset of tearing. It could also aid our understanding of the relationship between geometry and physical properties of nano- and microscopic Möbius strip structures
7
,
8
,
9
.</abstract><cop>London</cop><pub>Nature Publishing Group UK</pub><pmid>17632519</pmid><doi>10.1038/nmat1929</doi><tpages>5</tpages></addata></record> |
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| ispartof | Nature materials, 2007-08, Vol.6 (8), p.563-567 |
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| subjects | Biomaterials Chemistry and Materials Science Civil engineering Condensed Matter Physics Equilibrium Fabrics letter Materials Science Microstructure Nanotechnology Numerical analysis Optical and Electronic Materials Physical properties Physics |
| title | The shape of a Möbius strip |
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