  We study the elastic response of composites of rods embedded in elastic
media. We calculate the micro-mechanical response functions, and bulk elastic
constants as functions of rod density. We find two fixed points for Poisson's
ratio with respect to the addition of rods in 3D composites: 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 an approximate expression for the elastic constants for arbitrary rod
density that yields exact results for both low and high density. These results
may help to explain recent experiments [Physical Review Letters 102, 188303
(2009)] that reported compressibility for composites of microtubules in F-actin
networks.
