Percolation and Cosserat elasticity: Exact results on a deterministic fractal

The elastic problem in two-dimensional percolation is investigated on a hierarchical lattice, in the context of Cosserat elasticity (or granular elasticity). For this regular fractal, the exponent of elastic moduli ($\frac{T}{v}$) is slightly smaller than the expected value ($\frac{t}{v}+2\mathrm{}$) deduced from the conductivity exponent ($\frac{t}{v}$). The correction is related to the distribution of a new geometric parameter (the "eccentricity"). We also discuss the inequalities $1+dv<T<t+2v$, the anisotropy of the elastic response, and the case of scalar elasticity.