Surface tension and the Mori-Tanaka theory of non-dilute soft composite solids

TL;DR

This paper extends Eshelby's theory within the Mori-Tanaka framework to account for interfacial stress in non-dilute soft composites, revealing how droplet size and interfacial tension influence composite stiffness.

Contribution

It introduces a novel extension of Eshelby's theory to non-dilute composites with interfacial stress using the Mori-Tanaka scheme, focusing on liquid droplet inclusions.

Findings

Interfacial tension significantly affects elastic moduli when R/L ≲ 100.

Droplet size relative to L determines stiffening or cloaking effects.

Theoretical predictions match observed influence of droplet size on composite mechanics.

Abstract

Eshelby's theory is the foundation of composite mechanics, allowing calculation of the effective elastic moduli of composites from a knowledge of their microstructure. However it ignores interfacial stress and only applies to very dilute composites -- i.e. where any inclusions are widely spaced apart. Here, within the framework of the Mori-Tanaka multiphase approximation scheme, we extend Eshelby's theory to treat a composite with interfacial stress in the non-dilute limit. In particular we calculate the elastic moduli of composites comprised of a compliant, elastic solid hosting a non-dilute distribution of identical liquid droplets. The composite stiffness depends strongly on the ratio of the droplet size, $R$, to an elastocapillary length scale, $L$. Interfacial tension substantially impacts the effective elastic moduli of the composite when $R/L\lesssim 100$. When $R < 3L/2$ ($R=3L/2$) liquid inclusions stiffen (cloak the far-field signature of) the solid.