Non-Euclidean Ribbons: Generalized Sadowsky Functional for Residually-Stressed Thin and Narrow Bodies

This paper presents a generalization of the Sadowsky functional, which measures the bending energy of narrow ribbons, for the case of incompatible ribbons (having no stress-free configuration). Specific solutions to special cases where the reference normal curvatures vanish, and for a naturally curved developable ribbon are presented and the resulting twist-stretch relations are discussed.

The classical theory of ribbons as developed by Sadowsky and Wunderlich has received much attention in recent years. It concerns the equilibrium conformations of thin and narrow ribbons whose intrinsic structure favors a rectangular and flat state. However, the intrinsic structure of naturally formed ribbons will often be more complicated; Spatial variations in the in-plane distance metric can give rise to both geodesic curvature and Gaussian curvature, curving the ribbon in and out of its plane. Moreover, metric variation across the thickness of the ribbon may result in nontrivial reference normal curvatures. The resulting geometric structure is likely to have no zero-energy (stress-free) realizations in Euclidean space.