Non-linear elasticity of packings

Before a packing is compressed many contacts between grains are still open. When it is then slowly loaded the contacts close one by one making that the global elastic response of the system increases faster than the response of one single contact. In 1986 experiments in Rennes with stapled cylinders showed that for very small displacements the stresses increased with a power-law in the strains with a high exponent. With Dietrich Stauffer and Stephane Roux we did some simulations finding indeed exponents of the order of four as published in J. Physique Vol.48, p.347-351 (1987). A theoretical argument is given in a twin paper and a letter in Europhys. Lett. Vol.3, p.265-267 (1987) summarized the results.

The same effect of non-linear response can also be obtained in an electrical analog model in which a network of resistors is considered which only conduct above a threshold which is randomly chosen. In this case one can analytically find that the current increases like the square of the voltage as shown in a paper published with Stephane Roux in Europhys. Lett. Vol.4, p.1227-1232 (1987).

With Gadi Oron we also did later some exact calculations of the forces inside a small heap of discs and of spheres where under their own weight they do deform the packing in such a way that some of the contacts are open and some are closed. These results were published in two dimensions in Phys. Rev. E, Vol.58, p.2079-2089 (1998) and in three dimensions in Physica A, Vol.260 p.1-5 (1999). We studied the dependence of the texture of the packing and discovered an astonishing fact, namely that with finite probability there are contacts that are closed but transmit no forces since they just touch.