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Equilibrium Shapes with Stress Localisation for Inextensible Elastic Mobius and Other Strips

Starostin, EL; van der Heijden, GHM; (2015) Equilibrium Shapes with Stress Localisation for Inextensible Elastic Mobius and Other Strips. Journal of Elasticity , 119 (1-2) 67 - 112. 10.1007/s10659-014-9495-0. Green open access

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Abstract

We formulate the problem of finding equilibrium shapes of a thin inextensible elastic strip, developing further our previous work on the Möbius strip. By using the isometric nature of the deformation we reduce the variational problem to a second-order one-dimensional problem posed on the centreline of the strip. We derive Euler–Lagrange equations for this problem in Euler–Poincaré form and formulate boundary-value problems for closed symmetric one- and two-sided strips. Numerical solutions for the Möbius strip show a singular point of stress localisation on the edge of the strip, a generic response of inextensible elastic sheets under torsional strain. By cutting and pasting operations on the Möbius strip solution, followed by parameter continuation, we construct equilibrium solutions for strips with different linking numbers and with multiple points of stress localisation. Solutions reveal how strips fold into planar or self-contacting shapes as the length-to-width ratio of the strip is decreased. Our results may be relevant for curvature effects on physical properties of extremely thin two-dimensional structures as for instance produced in nanostructured origami.

Type: Article
Title: Equilibrium Shapes with Stress Localisation for Inextensible Elastic Mobius and Other Strips
Open access status: An open access version is available from UCL Discovery
DOI: 10.1007/s10659-014-9495-0
Publisher version: http://dx.doi.org/10.1007/s10659-014-9495-0
Language: English
Additional information: This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Keywords: Mobius strip, Inextensible ribbon, Developable surface, Switching point, Equilibrium, Invariant variational formulation, Stress localisation, Conical surface
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Civil, Environ and Geomatic Eng
URI: https://discovery.ucl.ac.uk/id/eprint/1468757
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