A phenomenological one-dimensional model for elastic ribbons

Abstract

Ribbons exhibit fascinating buckling-dominated behavior under mechanical loading because of a unique combination of geometric dimensions. The recent interest in examining engineering applications of ribbon-like structures underscores the need for dedicated structural mechanics models to predict their complex behavior. In this thesis, we deal with ribbons that have at unstressed con figurations. Due to their physical appearance, such ribbons are typically modeled either as rods with highly anisotropic cross-sections (width of the cross-section is much larger than the thickness) or narrow plates. We speci fically examine the predictive capabilities of the Geometrically exact two-director Cosserat rod and Geometrically exact one-director Cosserat plate models. We measure ribbon shapes in various bending-dominated experiments and compare them with predictions computed using detailed finite element simulations of these models. We nd the plate theory to be particularly useful under a broad range of loading conditions, mainly because it captures nontrivial (and nonlinear) curvature distributions realized in the material bers oriented along the ribbon's width. This feature, which is noticeably absent in rod models, contributes to their poor predictive capabilities. We then propose a phenomenological one-dimension ribbon model by dimensional reduction from the Cosserat plate theory. Speci fically, we impose kinematic assumptions on the displacement field's dependence along the width direction of a ribbon to permit non-trivial lateral surface curvatures observed in the Cosserat plate solutions corresponding to various experiments. We speci fically examine polynomial dependences for the displacement field on the coordinate along the width. In principle, we expect a quadratic dependence to suffice since it helps to reproduce non-zero curvatures along the width. However, we nd that the resulting restricted kinematics is prone to membrane locking. Presuming a cubic dependence helps circumvent the issue. Alternately, resorting to selective reduced integration techniques during numerical approximation using finite element methods helps alleviate the issue.