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.
image of force chains inside granular 
material

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 cell.

[Diagram of
the apparatus used.] 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.
[calibration data]
The forces against a particular surface are normalized to the average 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.
Force Distribution
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 not reveal any force correlations in the plane normal to the force component measured.