Elastic and superelastic percolation networks: Imperfect duality, critical Poisson ratios, and relations between microscopic models

The connections between the elasticity and the superelasticity percolation problems are investigated in the case of the two-dimensional granular model. An imperfect mapping between two dual square lattices leads to the inequality f+2{ν}, f and c being the elasticity and superelasticity exponents. Exact calculations on deterministic fractals suggest that this inequality is strict and yield new estimates of f and c. The elasticity critical Poisson ratio ηE and the superelasticity ratio ηS are also estimated and are related to simple geometrical properties of the link and blob distributions: ηE is governed by the positional anisotropy of the links, while ηS is reduced to a weak blob contribution. In addition, the connections between the bond-bending and granular models are described: Both are special cases of a more general (micropolar) model and lead to the same scaling behavior. However, strong corrections to scaling are to be expected in the superelasticity problem as well as in the elasticity one. These corrections are similar to those observed in the anisotropic conductivity problem.