Friction, force chains, and falling fruit supplemental material

Videos of fruit-pile-collapse experiments

 

 

 

 

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Isostatic packing

Particles that are packed together—apples in a grocery display, for example—are subject to forces whenever they are in contact with a neighbor. The independent force components may be viewed as degrees of freedom for the system. In addition, the system has constraints; those may be taken as force and torque balance for each particle in the system. The number of constraints, however, can never exceed the number of degrees of freedom. For the special case in which the number of degrees of freedom and the number of constraints are equal, the packing is said to be in an isostatic state. As discussed in the accompanying Quick Study, one associates the onset of jamming with the creation of the isostatic state.

The average number of contacts per particle, Z, is an important property for any packing. After all, the behavior of the particles near jamming is tied to the number density of the packing. To illustrate how Z can be calculated, consider the isostatic state for N hard disks in two dimensions. If there is friction between the disks, the number of degrees of freedom is 2ZN/2. The 2 in the numerator arises because each force has a normal and tangential component. But Newton's third law reduces the number of unknown forces by 2, hence the 2 in the denominator. The constraints of force and torque balance total 3N, two force constraints and one torque constraint for each particle. Equating the number of degrees of freedom with the number of constraints gives Z=3.

Figure 1

For frictionless particles, the interparticle force must be normal and the particles are not subject to torques. (Figure 1 contrasts the cases with and without friction.) Thus the number of degrees of freedom is ZN/2, the number of constraints is 2N, and the isostatic condition implies Z=4. In the real world, of course, friction is never entirely absent. But if frictional forces are tiny, then one might expect that the resulting minuscule torques would not affect the properties of the packing. That idea has been explored by Chaoming Song and colleagues.¹ In their study, as friction turns on, Z decreases rapidly from the friction-free value of 4 toward the value of 3, which may be thought of as an infinite-friction limit.

In addition to force-balance constraints, the disks are subject to geometric constraints. In two dimensions, one has 2N degrees of freedom corresponding to the locations of the disk’s centers. That the disks are just touching introduces ZN/2 constraints. Since the constraints must number less than the location degrees of freedom, Z≤4. The friction-free isostatic result is just barely allowed by geometric considerations.

Figure 2

Figure 2 shows why Z falls from the isostatic value to 0 (at least for homogeneous systems of frictionless particles) when the number density drops to less than the isostatic value. Start at the isostatic density, at which grains are just touching as illustrated in the left image. If the system is mechanically stable, then Z will be at the isostatic value too. If you decrease the density by shrinking the particles by a tiny bit, as in the right image, then Z will vanish everywhere.

Figure 3

The shape of the packed particles can also drastically change Z. Consider, for example, two-dimensional frictionless ellipses, as illustrated in figure 3. The particles feel only normal contact forces, but those forces generally exert torques. In this case, the system’s ZN/2 force degrees of freedom are subject to 3N constraints of force and torque balance. In the isostatic state, then, Z=6. Again, in the real world, very nearly circular particles will experience tiny torques, but those will not be relevant for analyzing the isotropic state. As particles become more elliptical, Z continuously increases from 4 to 6.

References

1. C. Song, P. Wang, H. A. Makse, Nature 453, 629 (2008)

Supplemental resources

1. T. Wakabayashi, “Photo-Elastic Method for Determination of Stress in Powdered Mass,” J. Phys. Soc. Jpn. 5, 383 (1950)
2. P. Dantu, “Contribution à l’étude mécanique et géométrique des milieux pulvérulents,” in Proceedings of the Fourth International Conference on Soil Mechanics and Foundation Engineering, vol. 1, Butterworths, London (1957), p. 144
3. Liu Research Group, “Jamming and the Glass Transition,” http://www.physics.upenn.edu/liugroup/jamming.html