A discrete differential geometry-based numerical framework for extensible ribbons

Abstract

As a typical mechanical structure, ribbons are characterized with three distinctly different dimensions, i.e., length ≫ width ≫ thickness, which leads to their exclusive behaviors different from the one-dimensional (1D) case of slender rods and the two-dimensional (2D) case of thin plates. In this paper, we report a discrete differential geometry (DDG)-based numerical method to simulate the geometrically nonlinear deformations of extensible ribbons. With a cross section-dependent regularized parameter introduced, the proposed 1D framework allows both in-plane stretching and out-of-plane bending in the ribbon mid-surface, aimed at bridging the gap between the linear Kirchhoff rod theory and the developable Sadowsky ribbon model. Instead of solving the ordinary differential equations (ODEs) directly with the associated boundary conditions, the mechanical object is discretized into a mass–spring system, and its equilibrium configuration is obtained through a dynamic relaxation method. The numerical framework is applied to seven typical occasions based on either experimental or published datasets in order to verify its performance. Quantitative agreements demonstrate the effectiveness and accuracy of the proposed discrete approach for extensible ribbons, which, as a computationally efficient numerical simulator, could provide a pivotal understanding of a batch of slender structures, and further inspire the simulation-guided design of man-made systems.