31 research outputs found
Isometric immersions, energy minimization and self-similar buckling in non-Euclidean elastic sheets
The edges of torn plastic sheets and growing leaves often display
hierarchical buckling patterns. We show that this complex morphology (i)
emerges even in zero strain configurations, and (ii) is driven by a competition
between the two principal curvatures, rather than between bending and
stretching. We identify the key role of branch-point (or "monkey-saddle")
singularities in generating complex wrinkling patterns in isometric immersions,
and show how they arise naturally from minimizing the elastic energy.Comment: 6 pages, 6 figures. This article supersedes arXiv:1504.0073
Pattern Universes
In this essay we explore analogies between macroscopic patterns, which result
from a sequence of phase transitions/instabilities starting from a homogeneous
state, and similar phenomena in cosmology, where a sequence of phase
transitions in the early universe is believed to have separated the fundamental
forces from each other, and also shaped the structure and distribution of
matter in the universe. We discuss three distinct aspects of this analogy: (i)
Defects and topological charges in macroscopic patterns are analogous to spins
and charges of quarks and leptons; (ii) Generic (3+1) stripe patterns carry an
(energy) density that accounts for phenomena that are currently attributed to
dark matter; (iii) Space-time patterns of interacting nonlinear waves display
behaviors reminiscent of quantum phenomena including inflation, entanglement
and dark energy.Comment: 15 Pages, Essay with 3 technical appendice
Mass Exchange Dynamics of Surface and Subsurface Oil in Shallow-Water Transport
We formulate a model for the mass exchange between oil at and below the sea
surface. This is a particularly important aspect of modeling oil spills.
Surface and subsurface oil have different chemical and transport
characteristics and lumping them together would compromise the accuracy of the
resulting model. Without observational or computational constraints, it is thus
not possible to quantitatively predict oil spills based upon partial field
observations of surface and/or sub-surface oil. The primary challenge in
capturing the mass exchange is that the principal mechanisms are on the
microscale. This is a serious barrier to developing practical models for oil
spills that are capable of addressing questions regarding the fate of oil at
the large spatio-temporal scales, as demanded by environmental questions. We
use upscaling to propose an environmental-scale model which incorporates the
mass exchange between surface and subsurface oil due to oil droplet dynamics,
buoyancy effects, and sea surface and subsurface mechanics. While the mass
exchange mechanism detailed here is generally applicable to oil transport
models, it addresses the modeling needs of a particular to an oil spill model
[1]. This transport model is designed to capture oil spills at very large
spatio-temporal scales. It accomplishes this goal by specializing to
shallow-water environments, in which depth averaging is a perfectly good
approximation for the flow, while at the same time retaining mass conservation
of oil over the whole oceanic domain.Comment: 18 pages, 6 figure
Distributed branch points and the shape of elastic surfaces with constant negative curvature
We develop a theory for distributed branch points and investigate their role
in determining the shape and influencing the mechanics of thin hyperbolic
objects. We show that branch points are the natural topological defects in
hyperbolic sheets, they carry a topological index which gives them a degree of
robustness, and they can influence the overall morphology of a hyperbolic
surface without concentrating energy. We develop a discrete differential
geometric (DDG) approach to study the deformations of hyperbolic objects with
distributed branch points. We present evidence that the maximum curvature of
surfaces with geodesic radius containing branch points grow
sub-exponentially, in contrast to the exponential growth
for surfaces without branch points. We argue that, to optimize
norms of the curvature, i.e. the bending energy, distributed branch points are
energetically preferred in sufficiently large pseudospherical surfaces.
Further, they are distributed so that they lead to fractal-like recursive
buckling patterns.Comment: 59 pages, 20 figures. Major revisions including new proofs with
weakened hypotheses, expanded discussion and additional references. Some
images are not at their original resolution to keep them at a reasonable
size. Comments are very welcome and much appreciate