Shear induced supercritical pitchfork bifurcation of pre-buckled bands, from narrow strips to wide plates

Abstract We combine discrete differential geometry (DDG)-based models and desktop experiments to study the supercritical pitchfork and bifurcation phenomena of a pre-compressed elastic plate under lateral end translation, with a focus on its width effect. Based on the ratio among length, width, and thickness, the elastic structures in our study fall into three different structural categories: rods, ribbons, and plates. In order to numerically simulate the mechanical response of these structures, we employ two DDG-based numerical frameworks–Discrete Anisotropic Rods method and Discrete Elastic Plates method. Even though the multi-stability and bifurcation of a narrow strip can be precisely captured by a naive one dimensional rod model, it fails to match with experiments as the ribbon increases in width. A two dimensional approach using a plate model, on the other hand, accurately predicts the geometrically nonlinear deformations and the supercritical pitchfork points for plates even when the width is as large as half of the length. Exploiting the efficiency and robustness of the simulator, we perform a systematic parameter sweep on plate size and lateral displacement to build a phase diagram of different configurations of the elastic plates. We find that the deformed configuration of the nearly developable strips can be described, up to a very good approximation, using the bending and twisting of the centerline. This indicates that a one dimensional energy model for the simulation of nearly developable strips can potentially be developed in the future. The results can serve as a benchmark for future numerical investigations into modeling of ribbons. Our study can also provide guidelines on the choice of the appropriate structural model – rod vs. ribbon vs. plate – in simulation of thin elastic structures.