Manipulation and mechanics of thin elastic objects

Borum, Andy

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https://hdl.handle.net/2142/101500 Copy

Description

Title

Manipulation and mechanics of thin elastic objects

Author(s)

Borum, Andy

Issue Date

2018-06-21

Director of Research (if dissertation) or Advisor (if thesis)

Bretl, Timothy

Doctoral Committee Chair(s)

Bretl, Timothy

Committee Member(s)

• Balajewicz, Maciej
• Belabbas, Mohamed
• Namachchivaya, Sri

Department of Study

Aerospace Engineering

Discipline

Aerospace Engineering

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Manipulation
• Stability
• Elasticity
• Optimal Control

Abstract

In this thesis, multiple problems concerning the equilibrium and stability properties of thin deformable objects are considered, with particular focus given to the analysis of thin elastic rods. The problems considered can be divided into two related categories: manipulation and mechanics. First, a few results concerning symmetries in geometric optimal control theory are derived, which are later used in the analysis of thin elastic objects. Then the problem of quasi-statically manipulating an elastic rod from an initial configuration into a goal configuration is considered. Based upon an analysis of symmetries, geometric and topological characterizations of the set of all stable equilibrium configurations of an elastic rod are derived. Specifically, under a few regularity assumptions, it is shown that the set of all stable equilibrium configurations without conjugate points of an extensible, shearable, anisotropic, and uniform Cosserat elastic rod subject to conservative body forces is a smooth six-dimensional manifold parameterized by a single global coordinate chart. Furthermore, in the case of an inextensible, unshearable, anisotropic, uniform, and intrinsically straight Kirchhoff elastic rod without body forces, this six-dimensional manifold is shown to be path-connected. In addition to their applications to manipulation, the geometric and topological results described above can be used to answer questions concerning the mechanics of elastic rods and other deformable objects. For an inextensible, unshearable, isotropic, and uniform Kirchhoff elastic rod, it is shown that the closure of the set of all stable equilibria with helical centerlines is star-convex, and this property is used to compute and visualize the boundary between stable and unstable helical rods. Finally, two applications of geometric optimal control theory to the analysis of constitutive equations for thin elastic objects are considered. In the first application, the Pontryagin maximum principle is used to analyze curvature discontinuities observed in inextensible surfaces. In the second application, the Pontryagin maximum principle is used to derive constitutive equations for an elastic rod subject to a local injectivity constraint, and the use of this model for analyzing highly flexible helical springs with contact between neighboring coils is considered.

Graduation Semester

2018-08

Type of Resource

text

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http://hdl.handle.net/2142/101500

Copyright and License Information

Copyright 2018 Andy Borum

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