Bifurcations, Multi-stability, and Localization in Thin Structures

• Main Author: Yu, Tian
• Other Authors: Engineering Science and Mechanics
• Format: Others
• Published: Virginia Tech 2020
• Subjects: thin structures; multi-stability; localization; pure bending; creases
• Online Access: http://hdl.handle.net/10919/96558

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. === Doctor of Philosophy === Thin structures are those that have at least one dimension smaller than the other dimensions, such as hairs, telephone cords, and tree leaves, to name just a few. They can generate rich mechanical behaviors (e.g., snapping, crumpling) and complex shapes. A simple example is to rotate the two ends of a thin strip that has been deformed into an arch. Snapping will happen at a certain rotation angle. The first work studies snapping behaviors of thin bands subject to rotations and displacements at the two ends. This work employs a mechanical model based on force and moment balance on a spatial curve to solve the shapes of thin strips and capture the rich snapping behaviors. It is much harder to stretch a thin sheet than to bend it, which can be easily seen by deforming a piece of paper. The physics behind this is that stretching requires more energy than bending in thin objects. Knowing the bending response of thin sheets can aid in simulating deformations of thin structures. The second work introduces a new pure bending mechanism that can subject a sheet to pure bending and measure its bending response through a moment-curvature relationship. Thin sheets find broad applications in engineering. Mechanical pleating is a long-standing technique that deforms thin sheets into zigzag filter structures, but the mechanics behind it is unclear. The third work studies a rotary pleating process and aims to answer a basic question: What combinations of processing and material parameters lead to successful pleating? This work employs a one-dimensional model of an inextensible rod, with a moment-curvature constitutive law as input. The moment-curvature relationship of pleating materials can be measured by the pure bending mechanism developed in the second work. Thin sheets with prescribed crease patterns can create complicated and targeted shapes, such as origami (paper folding) and kirigami (paper cutting). A simple creased thin sheet is bistable: A stable configuration can be obtained by inverting the crease, which leads to a conical vertex/singularity. The fourth work of this dissertation finds that the bistability of creased thin sheets will be destroyed if a large hole is made around the vertex. This work studies the loss of bistability of creases under removal of singularities by quantifying how the hole size, hole geometry, and other factors such as the crease angle and crease stiffness affect the bistability.