One-Dimensional Modelling of Developable Elastic Strips by Geometric Constraints and their Link to Surface Isometry

Abstract

The goal of this paper is to introduce a kinematical reduction for the structural model of Kirchhoff-Love shells with developable base surfaces. The dimensional reduction to a curve and a vector field along it decreases the involved number of degrees of freedom. Local coordinates in form of a relatively parallel frame allow us to simplify the geometric constraints occurring in the model and prevent instabilities caused by points or segments of zero curvature. The core of this work is to prove equivalence of these requirements and the isometry of the transformation. Subsequently, we derive the one-dimensional bending energy functional for rectangular strips. In order to compute the equilibrium state of a static shell, we minimise a penalised version of this functional over the finitely many degrees of freedom stemming from an isogeometric discretisation. Several example strips clamped at both ends illustrate the feasibility of this approach.