Modelling, Analysis and Computational Methods for a One-Dimensional Formulation of Developable Elastic Ribbons.

This thesis proposes a kinematic reduction for Kirchhoff-Love shells in the special case of developable base surfaces that deform isometrically. These specifications apply (approximately) whenever large deformations with small material strains occur, as for example during paper bending or the installation of flexible flat cables in consumer electronics. The model resulting from this reduction considers a curve and a director field along it, reducing the shell to items with one-dimensional parameter domain and thereby decreasing the number of degrees of freedom involved. A thorough study of the differential geometry of this model culminates in a set of geometric constraints on the base curve and the director vector field which are proven to be equivalent to isometric surface deformation. Thus, the equilibrium state of a developable reference ribbon under given boundary conditions is the minimiser of the elastic shell energy with respect to the constraints. Different optimisation methods are compared to each other, and the numerical behaviour of the approach is studied. Eventually, several examples illustrate the applicability of the model – among them, the famous Möbius strip.