On the Sadowsky functional for anisotropic ribbons

TL;DR

This paper proves that the Sadowsky functional accurately models the bending energy of anisotropic ribbons, including complex shapes like Möbius strips, as their width approaches zero.

Contribution

It extends the Gamma-convergence result of the Sadowsky functional to anisotropic ribbons with curved reference configurations and boundary conditions.

Findings

Gamma-convergence holds for anisotropic ribbons with curved references.

The result applies to ribbons with affine boundary conditions, including Möbius strips.

The bending energy is accurately captured by the Sadowsky functional in the thin limit.

Abstract

The equilibrium shape of a thin, elastic, inextensible ribbon minimizes its bending energy. It has been shown that, as the width of the ribbon tends to zero, the bending energy Gamma-converges to the so called Sadowsky functional. In this paper we consider geometrically frustrated anisotropic ribbons with a possibly curved reference configuration. We prove that the Gamma-convergence remains valid under prescribed affine boundary conditions, including, in particular, those satisfied by a M\"obius strip.