# 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.