A variational model for anisotropic and naturally twisted ribbons

TL;DR

This paper develops a new variational model for naturally twisted anisotropic ribbons by extending classical theories to include geometrical frustration and residual stresses, using Gamma-convergence.

Contribution

It introduces a novel one-dimensional variational model for twisted ribbons that accounts for anisotropy and natural curvature, generalizing the classical Sadowsky energy.

Findings

Derivation of a new model via Gamma-convergence

Generalization of the classical Sadowsky energy

Inclusion of residual stresses and anisotropy

Abstract

We consider thin plates whose energy density is a quadratic function of the difference between the second fundamental form of the deformed configuration and a "natural" curvature tensor. This tensor either denotes the second fundamental form of the stress-free configuration, if it exists, or a target curvature tensor. In the latter case, residual stress arises from the geometrical frustration involved in the attempt to achieve the target curvature: as a result, the plate is naturally twisted, even in the absence of external forces or prescribed boundary conditions. Here, starting from this kind of plate energies, we derive a new variational one-dimensional model for naturally twisted ribbons by means of Gamma-convergence. Our result generalizes, and corrects, the classical Sadowsky energy to geometrically frustrated anisotropic ribbons with a narrow, possibly curved, reference configuration.