An investigation of models for elastic ribbons: Simulations & experiments

Understanding the feature-rich buckling-dominated behavior of thin elastic ribbons is ripe with opportunities for fundamental studies exploring the nexus between geometry and mechanics, and for conceiving of engineering applications that exploit geometric nonlinearity as a functioning principle. Predictive mechanical models play an instrumental role to this end. As a direct consequence of their physical appearance, ribbons are usually modeled either as one-dimensional rods having wide cross sections, or as narrow two-dimensional plates/shells. These models employ drastically different kinematic assumptions, which in turn play a decisive role in their predictive capabilities. Here, we critically examine three modeling approaches for elastic ribbons using detailed measurements of their complex three-dimensional deformations realized in quasistatic experiments with annulus-shaped ribbons. We find that simple and practically realizable ribbon deformations contradict assumptions underlying strain-displacement relationships in nonlinear rod and von Kármán plate models. These observations do not point at shortcomings of the theories themselves, but highlight fallacies in their application to modeling ribbon-like structures that are capable of undergoing large displacements and rotations. We identify and validate, seemingly for the first time, the 1-director Cosserat plate theory as a model for elastic ribbons over a useful range of loading conditions. In the process, we demonstrate annular ribbons to be prototypical systems for studying the mechanics of elastic ribbons. Annular ribbons exhibit a tunable degree of geometric nonlinearity in response to simple displacement and rotation boundary conditions— a feature that we exploit here for highlighting the consequences of kinematic assumptions underlying different ribbon models. We additionally provide experimental evidence for the existence of multiple stable equilibria, bifurcation phenomena correlated with the number of zero crossings in the mean curvature, and localization of energy, thus making annular ribbons interesting mechanical systems to study in their own right.