Bifurcations, Multi-stability, and Localization in Thin Structures

Thin structures exist as one dimensional slender objects (hairs, tendrils, telephone cords, etc.) and two dimensional thin sheets (tree leaves, Mobius bands, eggshells, etc.). Geometric and material nonlinearities can conspire together to create complex phenomena in thin structures. This dissertation studies snap-through, multi-stability, and localization in thin rods and sheets through a combination of experiments and numerics.

The first work experimentally explores the multi-stability and bifurcations of buckled elastic strips subject to clamping and lateral end translations, and compares these results with numerical continuation of a perfectly anisotropic Kirchhoff rod model. It is shown that this naive Kirchhoff rod model works surprisingly well as an organizing framework for thin bands with various widths.

Thin sheets prefer to bend rather than to stretch because of the high cost of stretching energy. Knowing the bending response of thin sheets can aid in simulating deformations such as creasing. The second work introduces an exact pure bending linkage mechanism for potential use in a bend tester that measures the moment-curvature relationship of soft sheets and filaments.

Mechanical rotary pleating is a bending-deformation-dominant process that deforms nonwoven materials into zigzag filter structures. The third work studies what combinations of processing and material parameters lead to successful rotary pleating. The rotary pleating process is formulated as a multi-point variable-arc-length boundary value problem for an inextensible rod, with a moment-curvature constitutive law, such as might be measured by a bend tester, as input. Through parametric studies, this work generates pleatability surfaces that may help avoid pleating failure in the real pleating process.

Creased thin sheets are generally bistable. The final work of this dissertation studies bistability of creased thin disks under the removal of singularities. A hole is cut in the disk and, through numerical continuation of an inextensible strip model, this work studies how the crease stiffness, crease angle, and hole geometry affect the bistability.