Frictionless Beads

Can a Pile of Beads Hold Itself Up?

What sort of load will a frictionless pack of smooth beads support? Ordinary solid grains, like rice or mustard seeds, will support loads in much the same fashion as a solid does. If you pour these out on a table they make a pile that holds itself up, despite the weight trying to flatten the pack and spread it like a liquid. You can even stand on a pile of grains without sinking down to the bottom. The pack can support the local load from your feet because forces between the beads hold them in place and keep them from moving. Some of the forces are frictional. But what would happen without friction? We should be able to answer such a basic question. Besides, many real bead-like materials are essentially frictionless. Dense colloidal suspensions are an example.

Removing friction surely makes the pack less stable. But according to one argument, it should still be able to hold itself up. Suppose we make a pile on a rough surface and then let it settle down as it wishes. In order to settle, the beads need to slide over each other. However, they are packed against one another and cannot slide without dilating--their volume must expand. That is, in order to slide sideways the pile must move upward to some extent. Of course, gravity resists this upward motion and opposes the sliding motion as well. Thus the pile contains forces that oppose flattening. But are these forces sufficient? Surprisingly, scientists and engineers have not reached agreement on a quantitative description of how piles with or without friction hold themselves up.

In a recent work, members of the Materials Center provided a partial resolution to this puzzle. They showed that if a bead pack is piled up one bead at a time it can't hold itself up without rearrangements of the beads, despite the dilation argument above. They also showed that the forces in frictionless packs can be described in a much simpler way than those in a pack with friction. If a small force is added to one of the beads (even a buried one), this force gets carried to the bottom through only some of the contacts; the others are unaffected.

To develop an intuitive feel of how frictionless packs work, we decided to do an experiment. However, in the physical world it's difficult to make macroscopic beads that are frictionless. Thus we used the computer to simulate the properties of frictionless beads. The computer experiment below calculates precisely the forces at each contact owing to the beads that have been added. The theoretical work mentioned above says that it is impossible to make a pack of arbitrary size. In order to stay in place, some beads would have to pull on each other instead of just pressing on each other--some contacts would have to be tensile. Since tensile contacts are impossible for ordinary beads, these beads would have to start moving. The pack will quickly become unstable. Though large piles must become unstable, surely very small piles will remain stable. This raises our question: How large can a pile become without going unstable? You can try to answer this question by playing with the experiment below. In the process, you will see how the forces propagate downward from each newly added bead, and how tensile forces arise.

A Piling Experiment

Piling Experiment

OBJECT: Add as many beads as you can without causing an avalanche.

Click on a bead to add another bead on top of it:

Thin lines going down from the added bead show how its weight is being held up. A short line means a weak force; a long line means a strong force. A thick line shows the smallest total contact force from all beads. If the line is red, the force is a pull (tensile force): beads would fall--avalanche. Place the mouse over a bead and type f to see the total force on its two supporting contacts from all beads. (how it works)

Click here to start this java applet. This beta version has been tested with Netscape 4.0 and Microsoft Internet Explorer 4.0. Here is a list of known bugs

Reference:

  1. "Stress Propagation through Frictionless Granular Material" A. V. Tkachenko, T. A. Witten. cond-mat/. 9811171,, 13 Nov. 1998. Accepted 4/2/99 Physical Review E. Available at: http://xxx.lanl.gov/abs/cond-mat/9811171

For our more recent work, see http://jfi.uchicago.edu/~tten/Glasperlenspiel