109 research outputs found
Isostaticity in two dimensional pile of rigid disks
We study the static structure of piles made of polydisperse disks in the
rigid limit with and without friction using molecular dynamic simulations for
various elasticities of the disks and pile preparation procedures. The
coordination numbers are calculated to examine the isostaticity of the pile
structure. For the frictionless pile, it is demonstrated that the coordination
number converges to 4 in the rigid limit, which implies that the structure of
rigid disk pile is isostatic. On the other hand, for the frictional case with
the infinite friction constant, the coordination number depends on the
preparation procedure of the pile, but we find that the structure becomes very
close to isostatic with the coordination number close to 3 in the rigid limit
when the pile is formed through the process that tends to make a pile of random
configuration.Comment: 3 pages, 3 figures, Submitted to J. Phys. Soc. Jp
A mean field description of jamming in non-cohesive frictionless particulate systems
A theory for kinetic arrest in isotropic systems of repulsive,
radially-interacting particles is presented that predicts exponents for the
scaling of various macroscopic quantities near the rigidity transition that are
in agreement with simulations, including the non-trivial shear exponent. Both
statics and dynamics are treated in a simplified, one-particle level
description, and coupled via the assumption that kinetic arrest occurs on the
boundary between mechanically stable and unstable regions of the static
parameter diagram. This suggests the arrested states observed in simulations
are at (or near) an elastic buckling transition. Some additional numerical
evidence to confirm the scaling of microscopic quantities is also provided.Comment: 9 pages, 3 figs; additional clarification of different elastic moduli
exponents, plus typo fix. To appear in PR
Infinite-cluster geometry in central-force networks
We show that the infinite percolating cluster (with density P_inf) of
central-force networks is composed of: a fractal stress-bearing backbone (Pb)
and; rigid but unstressed ``dangling ends'' which occupy a finite
volume-fraction of the lattice (Pd). Near the rigidity threshold pc, there is
then a first-order transition in P_inf = Pd + Pb, while Pb is second-order with
exponent Beta'. A new mean field theory shows Beta'(mf)=1/2, while simulations
of triangular lattices give Beta'_tr = 0.255 +/- 0.03.Comment: 6 pages, 4 figures, uses epsfig. Accepted for publication in Physical
Review Letter
Jamming transition in emulsions and granular materials
We investigate the jamming transition in packings of emulsions and granular
materials via molecular dynamics simulations. The emulsion model is composed of
frictionless droplets interacting via nonlinear normal forces obtained using
experimental data acquired by confocal microscopy of compressed emulsions
systems. Granular materials are modeled by Hertz-Mindlin deformable spherical
grains with Coulomb friction. In both cases, we find power-law scaling for the
vanishing of pressure and excess number of contacts as the system approaches
the jamming transition from high volume fractions. We find that the
construction history parametrized by the compression rate during the
preparation protocol has a strong effect on the micromechanical properties of
granular materials but not on emulsions. This leads the granular system to jam
at different volume fractions depending on the histories. Isostaticity is found
in the packings close to the jamming transition in emulsions and in granular
materials at slow compression rates and infinite friction. Heterogeneity of
interparticle forces increases as the packings approach the jamming transition
which is demonstrated by the exponential tail in force distributions and the
small values of the participation number measuring spatial localization of the
forces. However, no signatures of the jamming transition are observed in
structural properties, like the radial distribution functions and the
distributions of contacts.Comment: Submitted to PR
Exact results and scaling properties of small-world networks
We study the distribution function for minimal paths in small-world networks.
Using properties of this distribution function, we derive analytic results
which greatly simplify the numerical calculation of the average minimal
distance, , and its variance, . We also discuss the
scaling properties of the distribution function. Finally, we study the limit of
large system sizes and obtain some analytic results.Comment: RevTeX, 4 pages, 5 figures included. Minor corrections and addition
Geometric origin of mechanical properties of granular materials
Some remarkable generic properties, related to isostaticity and potential
energy minimization, of equilibrium configurations of assemblies of rigid,
frictionless grains are studied. Isostaticity -the uniqueness of the forces,
once the list of contacts is known- is established in a quite general context,
and the important distinction between isostatic problems under given external
loads and isostatic (rigid) structures is presented. Complete rigidity is only
guaranteed, on stability grounds, in the case of spherical cohesionless grains.
Otherwise, the network of contacts might deform elastically in response to load
increments, even though grains are rigid. This sets an uuper bound on the
contact coordination number. The approximation of small displacements (ASD)
allows to draw analogies with other model systems studied in statistical
mechanics, such as minimum paths on a lattice. It also entails the uniqueness
of the equilibrium state (the list of contacts itself is geometrically
determined) for cohesionless grains, and thus the absence of plastic
dissipation. Plasticity and hysteresis are due to the lack of such uniqueness
and may stem, apart from intergranular friction, from small, but finite,
rearrangements, in which the system jumps between two distinct potential energy
minima, or from bounded tensile contact forces. The response to load increments
is discussed. On the basis of past numerical studies, we argue that, if the ASD
is valid, the macroscopic displacement field is the solution to an elliptic
boundary value problem (akin to the Stokes problem).Comment: RevTex, 40 pages, 26 figures. Close to published paper. Misprints and
minor errors correcte
Random spread on the family of small-world networks
We present the analytical and numerical results of a random walk on the
family of small-world graphs. The average access time shows a crossover from
the regular to random behavior with increasing distance from the starting point
of the random walk. We introduce an {\em independent step approximation}, which
enables us to obtain analytic results for the average access time. We observe a
scaling relation for the average access time in the degree of the nodes. The
behavior of average access time as a function of , shows striking similarity
with that of the {\em characteristic length} of the graph. This observation may
have important applications in routing and switching in networks with large
number of nodes.Comment: RevTeX4 file with 6 figure
Stressed backbone and elasticity of random central-force systems
We use a new algorithm to find the stress-carrying backbone of ``generic''
site-diluted triangular lattices of up to 10^6 sites. Generic lattices can be
made by randomly displacing the sites of a regular lattice. The percolation
threshold is Pc=0.6975 +/- 0.0003, the correlation length exponent \nu =1.16
+/- 0.03 and the fractal dimension of the backbone Db=1.78 +/- 0.02. The number
of ``critical bonds'' (if you remove them rigidity is lost) on the backbone
scales as L^{x}, with x=0.85 +/- 0.05. The Young's modulus is also calculated.Comment: 5 pages, 5 figures, uses epsfi
Degeneracy Algorithm for Random Magnets
It has been known for a long time that the ground state problem of random
magnets, e.g. random field Ising model (RFIM), can be mapped onto the
max-flow/min-cut problem of transportation networks. I build on this approach,
relying on the concept of residual graph, and design an algorithm that I prove
to be exact for finding all the minimum cuts, i.e. the ground state degeneracy
of these systems. I demonstrate that this algorithm is also relevant for the
study of the ground state properties of the dilute Ising antiferromagnet in a
constant field (DAFF) and interfaces in random bond magnets.Comment: 17 pages(Revtex), 8 Postscript figures(5color) to appear in Phys.
Rev. E 58, December 1st (1998
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