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

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 $R\mathrm{/}L\lesssim 100$. When $R<3L\mathrm{/}2$ ($R=3L\mathrm{/}2$) liquid inclusions stiffen (cloak the far-field signature of) the solid.