Nonlinear mechanics of non-Euclidean solids
Fabio Sozio
Dr. Arash Yavari (CEE)
Dr. Phanish Suryanarayana (CEE), Dr. David McDowell (ME, MSE), Dr. Julian J. Rimoli (AE), Dr.
John Etnyre (Math)
Tuesday, October 19, 2021 at 12:00 PM
Sustainable Education Building (SEB), Room 122
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. 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 this thesis we propose a geometric theory of nonlinear
accretion. The accretion 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 twopotential
approach to include dissipation. We also take into account the nonholonomic constraints that the
lattice puts on the motion of dislocations