Poisson's ratio in composite elastic media with rigid rods

TL;DR

This paper investigates how adding rigid rods to elastic media affects their overall elastic properties, revealing fixed points for Poisson's ratio and providing formulas that match experimental observations of biological composites.

Contribution

It introduces a micro-mechanical model for composite elastic media with rods, identifying fixed points for Poisson's ratio and deriving an approximate formula for elastic constants at various rod densities.

Findings

Poisson's ratio has fixed points at 1/2 and 1/4 with rod addition

Derived an approximate elastic constant formula valid at all densities

Results align with experiments on microtubule-F-actin composites

Abstract

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.