228 research outputs found
Discrete models of force chain networks
A fundamental property of any material is its response to a localized stress
applied at a boundary. For granular materials consisting of hard, cohesionless
particles, not even the general form of the stress response is known. Directed
force chain networks (DFCNs) provide a theoretical framework for addressing
this issue, and analysis of simplified DFCN models reveal both rich
mathematical structure and surprising properties. We review some basic elements
of DFCN models and present a class of homogeneous solutions for cases in which
force chains are restricted to lie on a discrete set of directions.Comment: 17 pages, 6 figures, dcds-B.cls; Minor corrections to version 2, but
including an important factor of 2; Submitted to Discrete and Continuous
Dynamical Systems B for special issue honoring David Schaeffe
Controlling spatiotemporal dynamics with time-delay feedback
We suggest a spatially local feedback mechanism for stabilizing periodic
orbits in spatially extended systems. Our method, which is based on a
comparison between present and past states of the system, does not require the
external generation of an ideal reference state and can suppress both absolute
and convective instabilities. As an example, we analyze the complex
Ginzburg-Landau equation in one dimension, showing how the time-delay feedback
enlarges the stability domain for travelling waves.Comment: 4 pages REVTeX + postscript file with 3 figure
Forcing nonperiodicity with a single tile
An aperiodic prototile is a shape for which infinitely many copies can be
arranged to fill Euclidean space completely with no overlaps, but not in a
periodic pattern. Tiling theorists refer to such a prototile as an "einstein"
(a German pun on "one stone"). The possible existence of an einstein has been
pondered ever since Berger's discovery of large set of prototiles that in
combination can tile the plane only in a nonperiodic way. In this article we
review and clarify some features of a prototile we recently introduced that is
an einstein according to a reasonable definition. [This abstract does not
appear in the published article.]Comment: 18 pages, 10 figures. This article has been substantially revised and
accepted for publication in the Mathematical Intelligencer and is scheduled
to appear in Vol 33. Citations to and quotations from this work should
reference that publication. If you cite this work, please check that the
published form contains precisely the material to which you intend to refe
Directed force chain networks and stress response in static granular materials
A theory of stress fields in two-dimensional granular materials based on
directed force chain networks is presented. A general equation for the
densities of force chains in different directions is proposed and a complete
solution is obtained for a special case in which chains lie along a discrete
set of directions. The analysis and results demonstrate the necessity of
including nonlinear terms in the equation. A line of nontrivial fixed point
solutions is shown to govern the properties of large systems. In the vicinity
of a generic fixed point, the response to a localized load shows a crossover
from a single, centered peak at intermediate depths to two propagating peaks at
large depths that broaden diffusively.Comment: 18 pages, 12 figures. Minor corrections to one figur
Local growth of icosahedral quasicrystalline tilings
Icosahedral quasicrystals (IQCs) with extremely high degrees of translational
order have been produced in the laboratory and found in naturally occurring
minerals, yet questions remain about how IQCs form. In particular, the
fundamental question of how locally determined additions to a growing cluster
can lead to the intricate long-range correlations in IQCs remains open. In
answer to this question, we have developed an algorithm that is capable of
producing a perfectly ordered IQC, yet relies exclusively on local rules for
sequential, face-to-face addition of tiles to a cluster. When the algorithm is
seeded with a special type of cluster containing a defect, we find that growth
is forced to infinity with high probability and that the resultant IQC has a
vanishing density of defects. The geometric features underlying this algorithm
can inform analyses of experimental systems and numerical models that generate
highly ordered quasicrystals.Comment: 13 pages, 15 figures, 1 tabl
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