Poisson's Ratio in Composite Elastic Media with Rigid Rods

We calculate both the micromechanical response and bulk elastic constants of composites of rods embedded in elastic media. We find two fixed points for Poisson's ratio with respect to rod density: there is an unstable fixed point for Poisson's ratio = 1/2 (an incompressible system) and a stable fixed point for Poisson's ratio = 1/4 (a compressible system). We also derive approximate expressions for the elastic constants for arbitrary rod density, which agree with exact results for both low and high density. These results may help to explain recent experiments [Phys. Rev. Lett. 102, 188303 (2009)] that reported compressibility for composites of microtubules in filamentous actin networks.