Force Propagation in Granular Materials

We use photo-elastic discs to study the force networks within vertically confined granular materials (for example, a grain silo). In granular materials, force is rarely transmitted uniformly, but rather preferentially along a network forming force chains -- often right next to regions where there is little or no force.

When photo-elastic discs are viewed through circular polarizers, high forces show up as bright regions. When the force is very large, the polarization is rotated through multiple phases of p, showing fringes.

There are several models for describing the force chains observed experimentally in two-dimensional granular materials. Most models start with lattice-like descriptions of grains connected by specific contacts to argue for continuum equations of force propagation.

q-Model	Socolar's a-Model	Experiment

Left two images modified from J.E.S. Socolar, PRE 57, 3204-3215 (1997)

References:

The q-model: This is the first lattice model, where (in two-dimensions) each grain, under a total force w from above, rests on two lower neighbors, one receiving a force q(w+1) from the grain and the other a force (1-q)(w+1). The continuum version of this model describes heat-like or diffusive propagation of force, and there are no force chains.
S. N. Coppersmith, C.-H. Liu, S. Majumadar, O. Narayan, and T. A. Witten, Phys. Rev. E 53, 4673 (1996)

The tripod model: The next model that was developed described each grain as having three lower neighbors upon which it could be resting. One neighbor is on-axis, directly below the grain, and the other two are to either side of that grain at the same angle y. This model allows for balance of force vectors and predicts that force propagates outward like a wave, with depth into the material corresponding to time. Still no force chains, but the angle allows a preferred force propagation direction to be determined.
P. Claudin, J.-P. Bouchaud, M.E. Cates, and J.P. Wittmer Phys. Rev. E 57, 4441 (1998)