Surface tension and a self-consistent theory of soft composite solids with elastic inclusions

TL;DR

This paper develops a self-consistent theory for soft composite solids with elastic inclusions, highlighting the significant influence of surface tension on their elastic properties and introducing a method to determine surface tension and inclusion stiffness.

Contribution

It extends inclusion theory to account for surface tension effects in soft composites, providing analytical expressions for stiffness and elastic cloaking.

Findings

Surface tension significantly affects composite stiffness even with stiff inclusions.

The theory predicts elastic cloaking of inclusions within the composite.

A method to determine surface tension and inclusion stiffness from measurements.

Abstract

The importance of surface tension effects is being recognized in the context of soft composite solids, where it is found to significantly affect the mechanical properties, such as the elastic response to an external stress. It has recently been discovered that Eshelby's inclusion theory breaks down when the inclusion size approaches the elastocapillary length $L\equiv\gamma/E$, where $\gamma$ is the inclusion/host surface tension and $E$ is the host Young's modulus. Extending our recent results for liquid inclusions, here we model the elastic behavior of a non-dilute distribution of isotropic elastic spherical inclusions in a soft isotropic elastic matrix, subject to a prescribed infinitesimal far-field loading. Within our framework, the composite stiffness is uniquely determined by the elastocapillary length $L$, the spherical inclusion radius $R$, and the stiffness contrast parameter ${C}$, which is the ratio of the inclusion to the matrix stiffness. We compare the results with those from the case of liquid inclusions, and we derive an analytical expression for elastic cloaking of the composite by the inclusions. Remarkably, we find that the composite stiffness is influenced significantly by surface tension even for inclusions two orders of magnitude more stiff than the host matrix. Finally, we show how to simultaneously determine the surface tension and the inclusion stiffness using two independent constraints provided by global and local measurements.