Rotationally invariant elasticity in a planar fractal network

We develop a tensorial description of rotationally invariant elasticity in two-dimensional tenuously connected structures, in analogy with the theory of resistor networks. This approach is used to investigate the scaling behavior of elasticity on fractals, the structure of which can be described in terms of nodes, links, and blobs. We retrieve the nonscalar behavior of the stretching compliancy, and we point out the influence on the elastic response of a new geometrical parameter, the ``eccentricity.'' This parameter defines the strength of a coupling effect between displacements and rotations, which tends to rigidify the loops. We show on a deterministic fractal that the scaling exponent of the stretching compliancy becomes smaller than the conjecture ζ₀+2 (ζ₀ is the electric resistance exponent) because of this eccentricity effect. This suggests that the critical exponent of the elastic moduli in percolation is equal to T=t+2{ν}-{Δ} (t is the conductivity exponent), {Δ} being an eccentricity correction. This result is also discussed in the context of the micropolar theory of elasticity.