Persistent Link:
http://hdl.handle.net/10150/223311
Title:
Shape Selection in the Non-Euclidean Model of Elasticity
Author:
Gemmer, John Alan
Issue Date:
2012
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
In this dissertation we investigate the behavior of radially symmetric non-Euclidean plates of thickness t with constant negative Gaussian curvature. We present a complete study of these plates using the Föppl-von Kármán and Kirchhoff reduced theories of elasticity. Motivated by experimental results, we focus on deformations with a periodic profile. For the Föppl-von Kármán model, we prove rigorously that minimizers of the elastic energy converge to saddle shaped isometric immersions. In studying this convergence, we prove rigorous upper and lower bounds for the energy that scale like the thickness t squared. Furthermore, for deformation with n-waves we prove that the lower bound scales like nt² while the upper bound scales like n²t². We also investigate the scaling with thickness of boundary layers where the stretching energy is concentrated with decreasing thickness. For the Kichhoff model, we investigate isometric immersions of disks with constant negative curvature into R³, and the minimizers for the bending energy, i.e. the L² norm of the principal curvatures over the class of W^2,2 isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H² into R³. In elucidating the connection between these immersions and the nonexistence/ singularity results of Hilbert and Amsler, we obtain a lower bound for the L^∞ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W^2,2, and numerically have lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grow approximately exponentially with the radius of the disc.
Type:
text; Electronic Dissertation
Keywords:
non-Euclidean elasticity; nonlinear elasticity; thin elastic sheets; Applied Mathematics; calculus of variations; morphogenesis of soft tissue
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Graduate College; Applied Mathematics
Degree Grantor:
University of Arizona
Advisor:
Venkataramani, Shankar C.

Full metadata record

DC FieldValue Language
dc.language.isoenen_US
dc.titleShape Selection in the Non-Euclidean Model of Elasticityen_US
dc.creatorGemmer, John Alanen_US
dc.contributor.authorGemmer, John Alanen_US
dc.date.issued2012-
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractIn this dissertation we investigate the behavior of radially symmetric non-Euclidean plates of thickness t with constant negative Gaussian curvature. We present a complete study of these plates using the Föppl-von Kármán and Kirchhoff reduced theories of elasticity. Motivated by experimental results, we focus on deformations with a periodic profile. For the Föppl-von Kármán model, we prove rigorously that minimizers of the elastic energy converge to saddle shaped isometric immersions. In studying this convergence, we prove rigorous upper and lower bounds for the energy that scale like the thickness t squared. Furthermore, for deformation with n-waves we prove that the lower bound scales like nt² while the upper bound scales like n²t². We also investigate the scaling with thickness of boundary layers where the stretching energy is concentrated with decreasing thickness. For the Kichhoff model, we investigate isometric immersions of disks with constant negative curvature into R³, and the minimizers for the bending energy, i.e. the L² norm of the principal curvatures over the class of W^2,2 isometric immersions. We show the existence of smooth immersions of arbitrarily large geodesic balls in H² into R³. In elucidating the connection between these immersions and the nonexistence/ singularity results of Hilbert and Amsler, we obtain a lower bound for the L^∞ norm of the principal curvatures for such smooth isometric immersions. We also construct piecewise smooth isometric immersions that have a periodic profile, are globally W^2,2, and numerically have lower bending energy than their smooth counterparts. The number of periods in these configurations is set by the condition that the principal curvatures of the surface remain finite and grow approximately exponentially with the radius of the disc.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectnon-Euclidean elasticityen_US
dc.subjectnonlinear elasticityen_US
dc.subjectthin elastic sheetsen_US
dc.subjectApplied Mathematicsen_US
dc.subjectcalculus of variationsen_US
dc.subjectmorphogenesis of soft tissueen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorVenkataramani, Shankar C.en_US
dc.contributor.committeememberVenkataramani, Shankar C.en_US
dc.contributor.committeememberNewell, Alanen_US
dc.contributor.committeememberTabor, Michaelen_US
dc.contributor.committeememberGlickenstein, Daviden_US
All Items in UA Campus Repository are protected by copyright, with all rights reserved, unless otherwise indicated.