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

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 onedimensional 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.

Möbius bands, unstretchable material sheets and developable surfaces

“…Subsequent to the papers of Sadowsky [1][2][3][4][5][6] and Wunderlich [7,8], the practice of restricting attention to surfaces that lie on rectifiable developables has dominated the literature concerning Möbius bands made from unstretchable flat material sheets, as evidenced by earlier studies [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. [24], who used homotopy methods to prove that a flat rectangular strip of half-width b and length l admits an isometric immersion as a Möbius band in three-dimensional Euclidean point space if and only if π b < l and, moreover, conjectured that such a strip can be isometrically embedded as a Möbius band in three-dimensional Euclidean point space only if the more restrictive inequality 2 √ 3b < l holds.…”

“…Quantifying the magnitude of the error incurred by using (2.6) to approximate an isometric mapping of a narrow ribbon remains an interesting question. A satisfactory answer to this question would provide useful guidelines for numerical strategies based on minimizing the bending energy functional (2.10), such as those used by Starostin & van der Heijden [22] and Shen et al [23]. Even so, there is no reason to believe, a priori, that the class of mappings (2.6) is rich enough to include all possible equilibrium shapes of such a sheet.…”

More views of a one-sided surface: mechanical models and stereo vision techniques for Möbius strips

“…A prominent feature the shape of a Möbius strip is the manifestation of nearly flat triangular regions bounded by creases [38]. Such phenomena are of interest in studying energy localization and are understood to result from coupling between bending and twisting deformation modes [11].…”

A convenient formulation of Sadowsky's model for elastic ribbons

“…In general, DAE systems are more difficult to solve than ordinary differential equations (ODEs) and numerical integrators are less commonly found, and typically less optimized. Singularities in the solutions are encountered in the Möbius problem as well as in other geometries: they typically require the integration interval to be arranged (and sometimes broken down) manually in such a way that the singularities lie at their endpoints, as done in past analyses of the Möbius problem [4,5].…”