Doctor of Philosophy with a Major in Engineering Science and Mechanics

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Doctor of Philosophy with a Major in Engineering Science and Mechanics

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https://hdl.handle.net/1853/73296

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Degree Series

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School of Civil and Environmental Engineering

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Now showing 1 - 10 of 73

Nonlinear mechanics of non-Euclidean solids

(Georgia Institute of Technology, 2022-02-21) Sozio, Fabio

In this thesis we formulate a geometric theory of the nonlinear mechanics of non-Euclidean solids. The term “non-Euclidean solids” was coined by Henri Poincaré in 1902, and refers to mathematical objects that represent solids with distributed eigenstrains, and hence residual stresses. In particular, we present a theoretical framework for the nonlinear mechanics of accretion (or surface growth) and for continuous dislocation dynamics. Accretion is the growth of a deformable solid by the gradual addition of material on its boundary, resulting in the formation of a residually-stressed structure. Examples of accretion are the growth of biological tissues and crystals, additive manufacturing, the deposition of thin films, etc. Dislocations are crystallographic line defects whose motion is responsible for plastic slip. Both accretion and dislocation dynamics have a close connection with differential geometry; accretion can be seen as the layer-by-layer assembly of non-Euclidean solids, while plasticity concerns the study of the evolution of their geometric structure in time. However, plastic slip is a process that involves more information than the change in distances considered in anelasticity and captured by Riemannian geometry; one must consider the torsion of an associated Weitzenböck manifold as well. In our geometric theory of nonlinear accretion, the anelastic part of the deformation gradient brings each particle to its natural state right before its time of attachment, and depends on both the mass flux and the history of deformation during accretion. This tensor is used to construct a material metric. From a geometric perspective, the presence of residual stresses in an accreted solid is due to a non-vanishing Riemann curvature tensor associated with the material metric, which in turn is related to the incompatibility of the accretion process. In the geometric framework, an accreted solid is represented by a foliated manifold, which allows one to express its 3D geometry in terms of the geometry of its layers and of the mass flux. The theory extends to thermal accretion. A numerical two-step scheme for nonlinear accretion based on a novel matrix formulation for finite differences is also presented. In the setting of geometric anelasticity, we propose a field theory of nonlinear dislocation mechanics in single crystals. The theory relies on the notion of a dislocated lattice structure, described by a triplet of differential 1-forms. Dislocation distributions are represented by a collection of triplets of differential 2-forms. These differential forms constitute a set of internal variables whose evolution equations are formulated in the framework of exterior calculus. This geometric approach allows one to study the integrability of the slip surfaces and its implications on the glide motion. The governing equations are derived using a variational principle of the Lagrange-d’Alembert type with a two-potential approach to include dissipation. We also take into account the nonholonomic constraints that the lattice puts on the motion of dislocations.
Referential and spatial evolutions in nonlinear elasticity

(Georgia Institute of Technology, 2016-11-17) Sadik, Souhayl

In this PhD thesis, we propose a theoretical framework for studying referential and spatial evolutions in nonlinear elasticity. We use the referential evolution-- considering an evolving reference configuration-- to formulate a geometric theory of anelasticity. Indeed, an anelasticity source (such as temperature, defects, or growth) can manifest itself such that the body would fail to find a relaxed state in the Euclidean physical space. However, a reference configuration should by essence be stress-free so that one can properly quantify the strain with respect to it-- and the stress by means of a constitutive equation. Identifying the reference configuration with an abstract manifold-- material manifold-- allows for a rational construction of such a stress-free state which can further accommodate the evolution of the source of anelasticity by allowing the material manifold to have an evolving geometry. In this work, we formulate a general geometric theory of anelasticity for three-dimensional bodies that we apply to the particular case of thermoelasticity; and we also formulate a general theory of anelastic shells that we apply to the particular case of morphoelastic shells, i.e., those subject to growth and remodeling. In the context of anelasticity, as well as in nonlinear elasticity, most exact solutions are obtained by assuming some restrictive class of symmetry for the solution. We propose a theory of small-on-large anelasticity, that is analogous to the small-on-large theory of Green et al. in classical elasticity. It can be used to find exact solutions for non-symmetric distributions of anelasticity sources that are small perturbations of symmetric ones. Finally, motivated by gaining further insights on the theory of nonlinear elasticity as well as the case of a continuum deforming in an evolving ambient space, we formulate a theory of nonlinear elasticity where the geometry of the ambient space is time-dependent.
Pultruded composite materials under shear loading

(Georgia Institute of Technology, 2001-08) Park, Jin Yong
An Electromyographic kinetic model for passive stretch of hypertonic elbow flexors

(Georgia Institute of Technology, 2000-05) Harben, Alan M.
Laser ultrasonic techniques and numerical models for damage and degradation tracking in FRP composites

(Georgia Institute of Technology, 1999-12) Dokun, Olajide David
Development and characterization of a prosthetic intervertebral disc

(Georgia Institute of Technology, 1998-12) Hudgins, Robert Garryl
Laser generation and detection techniques for developing transfer functions to characterize the effect of geometry on elastic wave propagation

(Georgia Institute of Technology, 1996-12) Hurlebaus, Stefan
Simulation of 3-dimensional transient filling and solidification in casting

(Georgia Institute of Technology, 1996-08) Lim, In-Cheol
Ultrasonic characterization of Fiber Reinforced Polymeric (FRP) composites

(Georgia Institute of Technology, 1996-08) Littles, Jerrol W., Jr.
Integral representations of stress intensity factors, the T-stress and the coefficients of Williams function, and a criterion for dynamic crack branching

(Georgia Institute of Technology, 1996-05) Dong, Xiaoyuan