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Gamma-Limit of a Model for the Elastic Energy of an Inextensible Ribbon

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Abstract

A Γ-convergence result involving the elastic bending energy of a narrow inextensible ribbon is established. As a consequence of the result, the energy is reduced to a one-dimensional integral, over the centerline of the ribbon, in which the aspect ratio of the ribbon appears as a small parameter. That integral is observed to increase monotonically with the aspect ratio. The Γ-limit of the family of energies is taken in a Sobolev space of centerlines with nonvanishing curvature. In that space, it is shown that the Γ-limit is a functional first proposed by Sadowsky in the context of narrow ribbons that form Möbius bands. The results obtained here do not apply to such ribbons, since the centerline of a Möbius band must have at least one inflection point. As a first step toward dealing with such inflection points, a result concerning the lower semicontinuity of the Sadowsky functional with inflection points comprising a set of measure zero within the domain of an arclength parameterization is presented.

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Washington, Seattle, WA, 98195, USA

    Nicholas O. Kirby

  2. Mathematical Soft Matter Unit, Okinawa Institute of Science and Technology, 1919-1 Tancha, Onna-son, Kunigami-gun, Okinawa, 904-0495, Japan

    Eliot Fried

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  1. Nicholas O. Kirby
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  2. Eliot Fried
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Correspondence to Eliot Fried.

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Kirby, N.O., Fried, E. Gamma-Limit of a Model for the Elastic Energy of an Inextensible Ribbon. J Elast 119, 35–47 (2015). https://doi.org/10.1007/s10659-014-9475-4

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  • DOI: https://doi.org/10.1007/s10659-014-9475-4

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