141 research outputs found
Isostaticity at Frictional Jamming
Amorphous packings of frictionless, spherical particles are isostatic at
jamming onset, with the number of constraints (contacts) equal to the number of
degrees of freedom. Their structural and mechanical properties are controlled
by the interparticle contact network. In contrast, amorphous packings of
frictional particles are typically hyperstatic at jamming onset. We perform
extensive numerical simulations in two dimensions of the geometrical asperity
(GA) model for static friction, to further investigate the role of
isostaticity. In the GA model, interparticle forces are obtained by summing up
purely repulsive central forces between periodically spaced circular asperities
on contacting grains. We compare the packing fraction, contact number,
mobilization distribution, and vibrational density of states using the GA model
to those generated using the Cundall-Strack (CS) approach. We find that static
packings of frictional disks obtained from the GA model are mechanically stable
and isostatic when we consider interactions between asperities on contacting
particles. The crossover in the structural and mechanical properties of static
packings from frictionless to frictional behavior as a function of the static
friction coefficient coincides with a change in the type of interparticle
contacts and the disappearance of a peak in the density of vibrational modes
for the GA model. These results emphasize that mesoscale features of the model
for static friction play an important role in determining the properties of
granular packings.Comment: 4.5 pages, 5 figures, http://prl.aps.org/covers/110/1
Decompositions, Packings, and Coverings of Complete Directed Gaphs with a 3-Circuit and a Pendent Arc.
In the study of Graph theory, there are eight orientations of the complete graph on three vertices with a pendant edge, K3 ∪ {e}. Two of these are the 3-circuit with a pendant arc and the other six are transitive triples with a pendant arc. Necessary and sufficient conditions are given for decompositions, packings, and coverings of the complete digraph with the two 3-circuit with a pendant arc orientations
Restricted and Unrestricted Coverings of Complete Bipartite Graphs with Hexagons
A minimal covering of a graph G with isomorphic copies of graph H is a set {H1, H2, H3, ... , Hn} where Hi is isomorphic to H, the vertex set of Hi is a subset of G, the edge set of G is a subset of the union of Hi\u27s, and the cardinality of the union of Hi\u27s minus G is minimum. Some studies have been made of covering the complete graph in which case an added condition of the edge set of Hi is the subset of the edge set of G for all i which implies no additional restrictions. However, if G is not the complete graph, then this condition may have implications. We will give necessary and sufficient conditions for minimal coverings of complete bipartite graph with 6-cycles, which we call minimal unrestricted coverings. We also give necessary and sufficient conditions for minimal coverings of the complete bipartite graph with 6-cycles with the added condition the edge set of Hi is a subset of G for all i, and call these minimal restricted coverings
Polygons vs. clumps of discs: a numerical study of the influence of grain shape on the mechanical behaviour of granular materials
We performed a series of numerical vertical compression tests on assemblies
of 2D granular material using a Discrete Element code and studied the results
with regard to the grain shape. The samples consist of 5,000 grains made from
either 3 overlapping discs (clumps - grains with concavities) or six-edged
polygons (convex grains). These two grain type have similar external envelopes,
which is a function of a geometrical parameter .
In this paper, the numerical procedure applied is briefly presented followed
by the description of the granular model used. Observations and mechanical
analysis of dense and loose granular assemblies under isotropic loading are
made. The mechanical response of our numerical granular samples is studied in
the framework of the classical vertical compression test with constant lateral
stress (biaxial test). The comparison of macroscopic responses of dense and
loose samples with various grain shapes shows that when is considered
a concavity parameter, it is therefore a relevant variable for increasing
mechanical performances of dense samples. When is considered an
envelope deviation from perfect sphericity, it can control mechanical
performances for large strains. Finally, we present some remarks concerning the
kinematics of the deformed samples: while some polygon samples subjected to a
vertical compression present large damage zones (any polygon shape), dense
samples made of clumps always exhibit thin reflecting shear bands. This paper
was written as part of a CEGEO research project www.granuloscience.comComment: This version of the paper doesn't include figures. Visit the journal
web site to download the final version of the paper with the figure
Dynamical partitions of space in any dimension
Topologically stable cellular partitions of D dimensional spaces are studied.
A complete statistical description of the average structural properties of such
partition is given in term of a sequence of D/2-1 (or (D-1)/2) variables for D
even (or odd). These variables are the average coordination numbers of the
2k-dimensional polytopes (2k < D) which make the cellular structure. A
procedure to built D dimensional space partitions trough cell-division and
cell-coalescence transformations is presented. Classes of structures which are
invariant under these transformations are found and the average properties of
such structures are illustrated. Homogeneous partitions are constructed and
compared with the known structures obtained by Voronoi partitions and sphere
packings in high dimensions.Comment: LaTeX 5 eps figures, submetted to J. Phys.
Packings and Coverings of Complete Graphs with a Hole with the 4-Cycle with a Pendant Edge
In this thesis, we consider packings and coverings of various complete graphs with the 4-cycle with a pendant edge. We consider both restricted and unrestricted coverings. Necessary and sufficient conditions are given for such structures for (1) complete graphs Kv, (2) complete bipartite graphs Km,n, and (3) complete graphs with a hole K(v,w)
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