Geometry and Mechanics of Growing, Nonlinearly Elastic Plates and Membranes

Persistent Link:
http://hdl.handle.net/10150/194028
Title:
Geometry and Mechanics of Growing, Nonlinearly Elastic Plates and Membranes
Author:
McMahon, Joseph Brian
Issue Date:
2009
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:
Until the twentieth century, theories of elastic rods and shells arose from collections of geometric and mechanical assumptions and approximations. These theories often lacked internal consistency and were appropriate for highly proscribed and sometimes unknown geometries and deformation sizes. The pioneering work of Truesdell, Antman, and others converted mechanical intuition into rigorous mathematical statements about the physics and mechanics of rods and shells. The result is the modern, geometrically exact theory of finite deformations of rods and shells.In the latter half of the twentieth century, biomechanics became a major focus of both experimental and theoretical mechanics. The genesis of residual stress by non-elastic growth has significant impact on the shape and mechanical properties of soft tissues. Inspired by the geometry of blood vessels and adopting a formalism found in elasto-plasticity, mechanicians have produced rigorous and applied results on the effect of growth on finite elastic deformations of columns and hollow tubes. Less attention has been paid to shells.A theory of growing elastic plates has been constructed in the context of linear elasticity. It harnessed many results in the theory of Riemann surfaces and has produced solutions that are surprisingly similar to experimental observations. Our intention is to provide a finite-deformation alternative by combining growth with the geometrically exact theory of shells. Such a theory has a clearer and more rigorous foundation, and it is applicable to thicker structures than is the case in the current theory of growing plates.This work presents the basic mathematical tools required to construct this alternative theory of finite elasticity of a shell in the presence of growth. We make clear that classical elasticity can be viewed in terms of three-dimensional Riemannian geometry, and that finite elasticity in the presence of growth must be considered in this way. We present several examples that demonstrate the viability and tractability of this approach.
Type:
text; Electronic Dissertation
Keywords:
Finite elasticity; Incompatible growth; Riemannian geometry; Solid mechanics
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Applied Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Tabor, Michael; Goriely, Alain
Committee Chair:
Tabor, Michael; Goriely, Alain

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titleGeometry and Mechanics of Growing, Nonlinearly Elastic Plates and Membranesen_US
dc.creatorMcMahon, Joseph Brianen_US
dc.contributor.authorMcMahon, Joseph Brianen_US
dc.date.issued2009en_US
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.abstractUntil the twentieth century, theories of elastic rods and shells arose from collections of geometric and mechanical assumptions and approximations. These theories often lacked internal consistency and were appropriate for highly proscribed and sometimes unknown geometries and deformation sizes. The pioneering work of Truesdell, Antman, and others converted mechanical intuition into rigorous mathematical statements about the physics and mechanics of rods and shells. The result is the modern, geometrically exact theory of finite deformations of rods and shells.In the latter half of the twentieth century, biomechanics became a major focus of both experimental and theoretical mechanics. The genesis of residual stress by non-elastic growth has significant impact on the shape and mechanical properties of soft tissues. Inspired by the geometry of blood vessels and adopting a formalism found in elasto-plasticity, mechanicians have produced rigorous and applied results on the effect of growth on finite elastic deformations of columns and hollow tubes. Less attention has been paid to shells.A theory of growing elastic plates has been constructed in the context of linear elasticity. It harnessed many results in the theory of Riemann surfaces and has produced solutions that are surprisingly similar to experimental observations. Our intention is to provide a finite-deformation alternative by combining growth with the geometrically exact theory of shells. Such a theory has a clearer and more rigorous foundation, and it is applicable to thicker structures than is the case in the current theory of growing plates.This work presents the basic mathematical tools required to construct this alternative theory of finite elasticity of a shell in the presence of growth. We make clear that classical elasticity can be viewed in terms of three-dimensional Riemannian geometry, and that finite elasticity in the presence of growth must be considered in this way. We present several examples that demonstrate the viability and tractability of this approach.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_US
dc.subjectFinite elasticityen_US
dc.subjectIncompatible growthen_US
dc.subjectRiemannian geometryen_US
dc.subjectSolid mechanicsen_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US
thesis.degree.disciplineApplied Mathematicsen_US
thesis.degree.disciplineGraduate Collegeen_US
thesis.degree.grantorUniversity of Arizonaen_US
dc.contributor.advisorTabor, Michaelen_US
dc.contributor.advisorGoriely, Alainen_US
dc.contributor.chairTabor, Michaelen_US
dc.contributor.chairGoriely, Alainen_US
dc.contributor.committeememberTabor, Michaelen_US
dc.contributor.committeememberGoriely, Alainen_US
dc.contributor.committeememberVenkataramani, Shankaren_US
dc.identifier.proquest10756en_US
dc.identifier.oclc659753591en_US
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