Nonlinear mechanics of non-Euclidean solids
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Sozio, Fabio
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Abstract
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.
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Date
2022-02-21
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Dissertation