Persistent Link:
http://hdl.handle.net/10150/195031
Title:
Morphoelasticity: The Mechanics and Mathematics of Elastic Growth
Author:
Vandiver, Rebecca Marie
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:
Growth plays a key role in many fundamental biological processes. In many cylindrical structures in biology, residual stress fields are created through differential growth. We present a general formulation of growth for a three-dimensional nonlinear elastic body and apply it to specific geometries relevant in many physiological and biological systems. The goal of this work is to study the development of residual stress induced by differential growth of biological cylindrical structures and elucidate its possible mechanical role in modifying material properties.As a tissue grows, it is not only subject to stresses but it also develops stresses by itself. These stresses play an important role in the evolution and regulation of growth, both in physiological and pathological conditions. We explore the interplay between growth and stress and the time evolution it generates. In particular, we show that in the case of spatially homogeneous growth, a general form of time evolution can be obtained leading to a dynamical system coupling the growth and stresses.The effect of tissue tension on the stability is studied through an analysis of the buckling properties of residually stressed cylindrical tubes. The general method to study stability is through a perturbation expansion in which an incremental deformation is superimposed on some finite deformation. If a solution is found to the incremental equation with the appropriate boundary conditions, then the possibility of instability exists. This method allows us to understand how residual stresses affect the overall stability of the system in the context of plant stem rigidity and arterial buckling.Lastly, we study the problem of elastic cavitation, the opening of a void in elastic materials. For a particular class of materials, the existence of a bifurcated solution has been shown in which a sphere supports the trivial spherical solution and a cavitated solution with spherical symmetry whose cavity radius vanishes at some critical external pressure. It naturally leads to interesting questions regarding the opening of cavities in residually stressed systems. We show that residual stresses induced by differential growth can induce cavitation and it can also play an important role in microvoid opening.
Type:
text; Electronic Dissertation
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Applied Mathematics; Graduate College
Degree Grantor:
University of Arizona
Advisor:
Goriely, Alain
Committee Chair:
Goriely, Alain

Full metadata record

DC FieldValue Language
dc.language.isoENen_US
dc.titleMorphoelasticity: The Mechanics and Mathematics of Elastic Growthen_US
dc.creatorVandiver, Rebecca Marieen_US
dc.contributor.authorVandiver, Rebecca Marieen_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.abstractGrowth plays a key role in many fundamental biological processes. In many cylindrical structures in biology, residual stress fields are created through differential growth. We present a general formulation of growth for a three-dimensional nonlinear elastic body and apply it to specific geometries relevant in many physiological and biological systems. The goal of this work is to study the development of residual stress induced by differential growth of biological cylindrical structures and elucidate its possible mechanical role in modifying material properties.As a tissue grows, it is not only subject to stresses but it also develops stresses by itself. These stresses play an important role in the evolution and regulation of growth, both in physiological and pathological conditions. We explore the interplay between growth and stress and the time evolution it generates. In particular, we show that in the case of spatially homogeneous growth, a general form of time evolution can be obtained leading to a dynamical system coupling the growth and stresses.The effect of tissue tension on the stability is studied through an analysis of the buckling properties of residually stressed cylindrical tubes. The general method to study stability is through a perturbation expansion in which an incremental deformation is superimposed on some finite deformation. If a solution is found to the incremental equation with the appropriate boundary conditions, then the possibility of instability exists. This method allows us to understand how residual stresses affect the overall stability of the system in the context of plant stem rigidity and arterial buckling.Lastly, we study the problem of elastic cavitation, the opening of a void in elastic materials. For a particular class of materials, the existence of a bifurcated solution has been shown in which a sphere supports the trivial spherical solution and a cavitated solution with spherical symmetry whose cavity radius vanishes at some critical external pressure. It naturally leads to interesting questions regarding the opening of cavities in residually stressed systems. We show that residual stresses induced by differential growth can induce cavitation and it can also play an important role in microvoid opening.en_US
dc.typetexten_US
dc.typeElectronic Dissertationen_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.advisorGoriely, Alainen_US
dc.contributor.chairGoriely, Alainen_US
dc.contributor.committeememberTabor, Michaelen_US
dc.contributor.committeememberVande Geest, Jonathanen_US
dc.identifier.proquest10293en_US
dc.identifier.oclc659750914en_US
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