{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/101500"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/101500","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Manipulation and mechanics of thin elastic objects","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Borum, Andy"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Aerospace Engineering","degree_department":null,"school":null,"contributors":["Bretl, Timothy","Balajewicz, Maciej","Belabbas, Mohamed","Namachchivaya, Sri"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2018,"date_issued":"2018-09-27T16:17:30Z","date_published":"2018-09-27T16:17:30Z","updated_at":"2026-07-22T22:24:40Z","subjects":["Manipulation","Stability","Elasticity","Optimal Control"],"languages":["en"],"rights":["Copyright 2018 Andy Borum"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/101500","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bretl, Timothy","Balajewicz, Maciej","Belabbas, Mohamed","Namachchivaya, Sri"]},{"key":"dc:creator","label":"Author","values":["Borum, Andy"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2018-09-27T16:17:30Z","2018-06-21","2018-08"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Aerospace Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Manipulation","Stability","Elasticity","Optimal Control"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2018 Andy Borum"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/101500"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2018-09-27 without embargo terms","The student, Andy Borum, accepted the attached license on 2018-06-20 at 18:00.","The student, Andy Borum, submitted this Dissertation for approval on 2018-06-20 at 18:17.","This Dissertation was approved for publication on 2018-06-21 at 14:28.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12655 on 2018-09-27 at 10:45:26","Made available in DSpace on 2018-09-27T16:17:30Z (GMT). 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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.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2018-09-27 without embargo terms","The student, Andy Borum, accepted the attached license on 2018-06-20 at 18:00.","The student, Andy Borum, submitted this Dissertation for approval on 2018-06-20 at 18:17.","This Dissertation was approved for publication on 2018-06-21 at 14:28.","DSpace SAF Submission Ingestion Package generated from Vireo submission #12655 on 2018-09-27 at 10:45:26","Made available in DSpace on 2018-09-27T16:17:30Z (GMT). No. of bitstreams: 2 BORUM-DISSERTATION-2018.pdf: 18813461 bytes, checksum: 7cf1a6494d85b917ac1fc8765686f5f2 (MD5) LICENSE.txt: 4207 bytes, checksum: 49405066bbedd1cc082e73f896c7a901 (MD5) Previous issue date: 2018-06-21"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/101500"],"dc:language":["en"],"dc:rights":["Copyright 2018 Andy Borum"],"dc:subject":["Manipulation","Stability","Elasticity","Optimal Control"],"dc:title":["Manipulation and mechanics of thin elastic objects"],"dc:type":["text"],"thesis:degree_discipline":["Aerospace Engineering"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:40Z"}