Studies on modeling the mechanics of slender elastic ribbons

Abstract

Ribbons are slender structures characterized by three disparate geometric dimensions: length >> width >> thickness. Such a dimensional disparity enables ribbons to bend, buckle, twist and crease response to simple loading conditions. Their nonlinear deformation behavior, once considered a hindrance, is now routinely exploited in engineering applications related to stretchable electronics and flexible robotics. Such applications demand a systematic understanding of the mechanics of elastic ribbons using experiments, modeling, and simulations. This thesis is a step in this direction.

Experiments using annulus-shaped ribbons and Moebius strips serve as our point of departure. The critical challenge in these experiments lies in measuring complex three-dimensional deformations observed. Routinely used techniques turn out to be inadequate, either due to the compliance of ribbon structures (e.g., contact probes, strain gauges) or due to the large displacements and rotations involved (DIC). We leverage novel computer vision techniques developed in the lab to faithfully digitize shapes and sample deformation maps of ribbons in the experiments. These measurements lead us to the main contributions of this thesis--- a detailed examination of the predictive capabilities of commonly used modeling approaches and the formulation of a dedicated one-dimensional ribbon model.

The physical appearance of ribbons motivates modeling them either as thin elastic plates or as elastic rods having a slender cross-section. Widespread adoption of the von Karman plate theory and the Kirchhoff rod model exemplifies this dichotomy. Somewhat surprisingly, comparing finite element simulations of these models with experimental measurements reveals both approaches to be deficient, even in simpler scenarios than ones where they are routinely used. These studies show that it is essential to permit large displacements and rotations in ribbon models and that compliance in the direction of the width, though small, plays an important role. Indeed, the experiments with annular ribbons and Moebius strips are designed to highlight these deformation features.

We propose adopting the small-strain Cosserat plate theory instead. The model's generality, along with a robust finite element implementation that addresses issues of numerical locking by adopting high order elements and approximating large rotations using exponential maps, translates to excellent agreement with experimental measurements. The model faithfully reproduces measured shapes, displacement fields and curvature distributions, as well as bifurcations and energy localization phenomena observed in experiments.

We then propose a dedicated reduced-order one-dimensional ribbon model by systematically incorporating kinematic assumptions in the plate theory. The model is Sadowsky-type theory that requires one additional field to describe lateral curvatures along the width of the ribbon. We examine the model's predictions through challenging examples, including one involving twist-induced snap-through. The model promises to be a valuable tool to develop insights into the mechanics of ribbons, besides being a compelling alternative to the Sadowsky and Wunderlich ribbon models routinely used in the literature.