Nonlinear Mechanics, Morphology and Instability of Ribbons, Plates and Rods

Abstract

Mechanical forces play a key role in the shaping of versatile morphologies of thin structures in natural and synthetic systems. The study of large deformation and instability of thin objects will facilitate understanding of morphology generation in these systems,

and benefit the ongoing efforts in developing programmable microfabrication techniques and novel functional devices including artificial muscles, stretchable electronics and bio-inspired robots. For this purpose, in this thesis, the morphology and deformation of thin ribbons, plates and rods and their instabilities are systematically investigated, through both theoretical modeling and table-top experiments.

First, a theoretical model based on linear elasticity theory, differential geometry and stationarity principles is developed for the spontaneous bending and twisting of ribbons with tunable geometries in presence of mechanical anisotropy. It is shown that helicity arises from mechanical anisotropy and the mis-orientation between the principal axes of effective surface stresses and geometric axes of the ribbon. Closed form analytic predictions are obtained from this theory with no adjustable parameters, and validated with simple, table-top experiments that are in excellent agreement with the theoretical predictions. Since the ribbon bends in two directions with non-zero Gauss curvature, this approach for thin, narrow ribbons goes beyond the scope of the classical Stoney formulation of planar bending of ribbons under surface stress.

For large deformation of ribbons and plates, a more general theory is developed to account for mechanical instability induced by geometric nonlinearity, due to the competition between inhomogeneous bending and mid-plane stretching energy. This

comprehensive, reduced parameter model leads to unique predictions that are validated with a series of table-top experiments. Moreover, it is shown that edge effects can alter the energy profile of the two locally stable states when the in-plane dimensions are asymmetric (i.e., when the length does not equal the width).

Yet another related topic, large deformation and instability of rods, is also addressed in this thesis. A model is proposed to investigate buckling behavior of rods embedded in an elastic medium and in the presence of external torques applied at the two ends. Such a study is relevant for instability-related phenomena in a broad spectrum of natural and synthetic systems, including helical growth of plant roots, buckling of cytoskeletal tissues and mechanotransduction in cells, and helical buckling of oil pipes in wellbores.

This thesis complements the reviving efforts in studying mechanics and geometry of thin objects, and will promote understanding of morphology and pattern formations in a variety of natural and engineered systems, and meet the emergent needs for developing programmable micro-/nano-fabrication techniques and designing bioinspired devices with smart, functional responses to external stimuli.