In a recent series of papers [1--3], a statistical model that accounts for correlations between topological and geometrical properties of a two-dimensional shuffled foam has been proposed and compared with experimental and numerical data. Here, the v
arious assumptions on which the model is based are exposed and justified: the equiprobability hypothesis of the foam configurations is argued. The range of correlations between bubbles is discussed, and the mean field approximation that is used in the model is detailed. The two self-consistency equations associated with this mean field description can be interpreted as the conservation laws of number of sides and bubble curvature, respectively. Finally, the use of a Grand-Canonical description, in which the foam constitutes a reservoir of sides and curvature, is justified.
The rigidity of a network of elastic beams crucially depends on the specific details of its structure. We show both numerically and theoretically that there is a class of isotropic networks which are stiffer than any other isotropic network with same
density. The elastic moduli of these textit{stiffest elastic networks} are explicitly given. They constitute upper-bounds which compete or improve the well-known Hashin-Shtrikman bounds. We provide a convenient set of criteria (necessary and sufficient conditions) to identify these networks, and show that their displacement field under uniform loading conditions is affine down to the microscopic scale. Finally, examples of such networks with periodic arrangement are presented, in both two and three dimensions.
The methods of statistical mechanics are applied to two-dimensional foams under macroscopic agitation. A new variable -- the total cell curvature -- is introduced, which plays the role of energy in conventional statistical thermodynamics. The probabi
lity distribution of the number of sides for a cell of given area is derived. This expression allows to correlate the distribution of sides (topological disorder) to the distribution of sizes (geometrical disorder) in a foam. The model predictions agree well with available experimental data.
