11 research outputs found
Isostatic phase transition and instability in stiff granular materials
In this letter, structural rigidity concepts are used to understand the
origin of instabilities in granular aggregates. It is shown that: a) The
contact network of a noncohesive granular aggregate becomes exactly isostatic
in the limit of large stiffness-to-load ratio. b) Isostaticity is responsible
for the anomalously large susceptibility to perturbation of these systems, and
c) The load-stress response function of granular materials is critical
(power-law distributed) in the isostatic limit. Thus there is a phase
transition in the limit of intinitely large stiffness, and the resulting
isostatic phase is characterized by huge instability to perturbation.Comment: RevTeX, 4 pages w/eps figures [psfig]. To appear in Phys. Rev. Let
Strain versus stress in a model granular material: a Devil's staircase
The series of equilibrium states reached by disordered packings of rigid,
frictionless discs in two dimensions, under gradually varying stress, are
studied by numerical simulations. Statistical properties of trajectories in
configuration space are found to be independent of specific assumptions ruling
granular dynamics, and determined by geometry only. A monotonic increase in
some macroscopic loading parameter causes a discrete sequence of
rearrangements. For a biaxial compression, we show that, due to the statistical
importance of such events of large magnitudes, the dependence of the resulting
strain on stress direction is a Levy flight in the thermodynamic limit.Comment: REVTeX, 4 pages, 5 included PostScript figures. New version altered
throughout text, very close to published pape
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
Cracking Piles of Brittle Grains
A model which accounts for cracking avalanches in piles of grains subject to
external load is introduced and numerically simulated. The stress is
stochastically transferred from higher layers to lower ones. Cracked areas
exhibit various morphologies, depending on the degree of randomness in the
packing and on the ductility of the grains. The external force necessary to
continue the cracking process is constant in wide range of values of the
fraction of already cracked grains. If the grains are very brittle, the force
fluctuations become periodic in early stages of cracking. Distribution of
cracking avalanches obeys a power law with exponent .Comment: RevTeX, 6 pages, 7 postscript figures, submitted to Phys. Rev.
Anisotropy in granular media: classical elasticity and directed force chain network
A general approach is presented for understanding the stress response
function in anisotropic granular layers in two dimensions. The formalism
accommodates both classical anisotropic elasticity theory and linear theories
of anisotropic directed force chain networks. Perhaps surprisingly, two-peak
response functions can occur even for classical, anisotropic elastic materials,
such as triangular networks of springs with different stiffnesses. In such
cases, the peak widths grow linearly with the height of the layer, contrary to
the diffusive spreading found in `stress-only' hyperbolic models. In principle,
directed force chain networks can exhibit the two-peak, diffusively spreading
response function of hyperbolic models, but all models in a particular class
studied here are found to be in the elliptic regime.Comment: 34 pages, 17 figures (eps), submitted to PRE, figures amended,
partially to compare better to recent exp. wor
Footprints in Sand: The Response of a Granular Material to Local Perturbations
We experimentally determine ensemble-averaged responses of granular packings
to point forces, and we compare these results to recent models for force
propagation in a granular material. We used 2D granular arrays consisting of
photoelastic particles: either disks or pentagons, thus spanning the range from
ordered to disordered packings. A key finding is that spatial ordering of the
particles is a key factor in the force response. Ordered packings have a
propagative component that does not occur in disordered packings.Comment: 5 pages, 4 eps figures, Phys. Rev. Lett. 87, 035506 (2001
Shearing of loose granular materials: A statistical mesoscopic model
A two-dimensional lattice model for the formation and evolution of shear
bands in granular media is proposed. Each lattice site is assigned a random
variable which reflects the local density. At every time step, the strain is
localized along a single shear-band which is a spanning path on the lattice
chosen through an extremum condition. The dynamics consists of randomly
changing the `density' of the sites only along the shear band, and then
repeating the procedure of locating the extremal path and changing it. Starting
from an initially uncorrelated density field, it is found that this dynamics
leads to a slow compaction along with a non-trivial patterning of the system,
with high density regions forming which shelter long-lived low-density valleys.
Further, as a result of these large density fluctuations, the shear band which
was initially equally likely to be found anywhere on the lattice, gets
progressively trapped for longer and longer periods of time. This state is
however meta-stable, and the system continues to evolve slowly in a manner
reminiscent of glassy dynamics. Several quantities have been studied
numerically which support this picture and elucidate the unusual system-size
effects at play.Comment: 11 pages, 15 figures revtex, submitted to PRE, See also:
cond-mat/020921
Internal states of model isotropic granular packings. I. Assembling process, geometry and contact networks
This is the first paper of a series of three, reporting on numerical
simulation studies of geometric and mechanical properties of static assemblies
of spherical beads under an isotropic pressure. Frictionless systems assemble
in the unique random close packing (RCP) state in the low pressure limit if the
compression process is fast enough, slower processes inducing traces of
crystallization, and exhibit specific properties directly related to
isostaticity of the force-carrying structure. The different structures of
frictional packings assembled by various methods cannot be classified by the
sole density. While lubricated systems approach RCP densities and coordination
number z^*~=6 on the backbone in the rigid limit, an idealized "vibration"
procedure results in equally dense configurations with z^*~=4.5. Near neighbor
correlations on various scales are computed and compared to available
laboratory data, although z^* values remain experimentally inaccessible. Low
coordination packings have many rattlers (more than 10% of the grains carry no
force), which should be accounted for on studying position correlations, and a
small proportion of harmless "floppy modes" associated with divalent grains.
Frictional packings, however slowly assembled under low pressure, retain a
finite level of force indeterminacy, except in the limit of infinite friction.Comment: 29 pages. Published in Physical Review
Force distribution in frictionless granular packings at rigidity threshold
With a new simulation method, involving viscous lubricating forces, we identify
the exact list of active contacts in a model two-dimensional
frictionless granular packing at equilibrium under an isotropic load. Within a
well-justified approximation of infinitesimal displacements, efforts are
localized on a minimal central-force structure that satisfies a geometric
optimization criterion. Size and boundary effects are analysed. The maximum
packing density of discs is evaluated to first order in their small
polydispersity parameter. We find no diverging correlation length in the
thermodynamic limit, and accurately compute the distribution of contact forces
in large systems