Force Chains and Distributions in Bead Packs
Static assemblies of granular material resemble solids in their ability to
support compressive and shear stresses. The manner by which these media
are able to support stress is of great interest in industrial application
as well as in physics, where it is a model non-equilibrium jammed system.
The simplest case, a static pack of spherical beads in a compression cell,
shows surprisingly complex stress patterns.
This image was obtained by viewing pressure-induced changes of the
polarization of light, transmitted through a 3D packing of glass spheres.
Before pressure was applied from the top, crossed polarizers were arranged such that no light was transmitted through the pack.
After the external pressure was applied, force chains appeared as bright lines in the above picture.
To prevent light scattering and reflections, the beads were immersed
in an index-matched fluid.
The high degree of disorder in this system suggests approaching the
problem from a statistical viewpoint. While the system is distinctly
out of equilibrium, and in general one does not have the benefit of
ensemble averaging over time, it is conceivable that spacial averaging may
replace ensemble averaging allowing a statistical approach to succeed.
Thus, in the experiments described below, we measure the distribution
P(F) of the forces F along each surface of a compression
The forces against a particular surface are normalized to the
force <F> at this part of the
system, yielding a normalized force f=F/<F>. The distribution of
normalized forces, P(f), is shown below for each surface.
The apparatus used in these experiments is a uni-axial compression cell
consisting of a 140mm diameter constraining cylinder with pistons at each
end. The region between the pistons is filled with 3.5mm diameter glass
beads. Forces of each bead against the constraining surface(piston or
wall) is measured by lining the surfaces with carbon paper and white
paper. The forces can be determined by calibrating the size and area of
the mark left on the white paper.
The distribution of forces for the pistons and cylinder wall are seen to
be identical within experimental uncertainty. It was also found that
varying the system preparation and boundary conditions in several ways did
not effect P(f). Calcuations of force-force pair correlations did
reveal any force correlations in the plane normal to the force component
- C.-h. Liu, S. R. Nagel, D. Shecter, and
S. Coppersmith, "Force Fluctuations in Bead Packs," Science,
269, 513, 1995.
- S. N. Coppersmith, C.-h. Liu, S. Majumdar, O. Narayan, and T. A. Witten,
"Model of Force Fluctuations in Bead Packs", Physical Review E,
53, 4673 (1996).
- D. M. Mueth, H. M. Jaeger, and S. R. Nagel, "Force Distribution
in a Granular Medium", Physical Review E 57, 3164(1998).
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