An isogeometric one-dimensional Kirchhoff–Love type model for developable elastic ribbons

Abstract

This paper proposes a kinematical reduction for Kirchhoff–Love shells in the special case of developable base surfaces. The resulting model considers a curve and a director field along it, reducing the shell to one-dimensional items and thereby decreasing the number of degrees of freedom involved. The presented overview of the geometry of developable surfaces covers dissolution of two types of singularities: a relatively parallel frame allows for points or segments of zero curvature along the curve and an explicit condition accounts for local self-intersections of the ruled surface description. We generalise earlier results after which the geometric constraints proposed are equivalent to isometric shell deformation. We also give a more general update of the elastic bending energy of the ribbon, which yields equilibrium states as minimisers. Subsequently, we discuss the numerical details of the approach and illustrate the feasibility at hand of several examples, among them the famous Möbius ribbon.