A Variational Model for Anisotropic and Naturally Twisted Ribbons

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 energy, 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.