Mechanics and force transmission in soft composites of rods in elastic gels

TL;DR

This paper provides a theoretical analysis of how stiff fibers embedded in elastic gels influence the composite's mechanical properties, revealing a universal fixed point for Poisson's ratio and implications for biological and synthetic materials.

Contribution

It introduces a universal fixed point for Poisson's ratio in fiber-reinforced elastic composites, independent of matrix properties, based on micro-mechanical modeling.

Findings

Poisson's ratio approaches 1/4 in 3D and 1/3 in 2D with increasing fiber density.

Stiff filaments significantly alter the composite's compressibility and shear response.

Results have implications for cell mechanics and engineered composite materials.

Abstract

We report detailed theoretical investigations of the micro-mechanics and bulk elastic properties of composites consisting of randomly distributed stiff fibers embedded in an elastic matrix in two and three dimensions. Recent experiments published in Physical Review Letters [102, 188303 (2009)] have suggested that the inclusion of stiff microtubules in a softer, nearly incompressible biopolymer matrix can lead to emergent compressibility. This can be understood in terms of the enhancement of the compressibility of the composite relative to its shear compliance as a result of the addition of stiff rod-like inclusions. We show that the Poisson's ratio $\nu$ of such a composite evolves with increasing rod density towards a particular value, or {\em fixed point}, independent of the material properties of the matrix, so long as it has a finite initial compressibility. This fixed point is $\nu=1/4$ in three dimensions and $\nu=1/3$ in two dimensions. Our results suggest an important role for stiff filaments such as microtubules and stress fibers in cell mechanics. At the same time, our work has a wider elasticity context, with potential applications to composite elastic media with a wide separation of scales in stiffness of its constituents such as carbon nanotube-polymer composites, which have been shown to have highly tunable mechanics.