On the incremental equations in surface elasticity

TL;DR

This paper derives incremental equations for hyperelastic solids with surface tension effects, providing a unified framework applicable to any geometry and demonstrating its use in analyzing instabilities in solid cylinders.

Contribution

It introduces a general form of incremental equations incorporating surface energy effects for hyperelastic materials, extending previous models to more complex geometries and surface behaviors.

Findings

Derived simple incremental equations including surface tension effects.

Expressed surface elastic moduli in terms of surface energy and principal stretches.

Applied theory to analyze Plateau--Rayleigh and Wilkes instabilities.

Abstract

We derive the incremental equations for a hyperelastic solid that incorporate surface tension effect by assuming that the surface energy is a general function of the surface deformation gradient. The incremental equations take the same simple form as their purely mechanical counterparts and are valid for any geometry. In particular, for isotropic materials, the extra surface elastic moduli are expressed in terms of the surface energy function and the two surface principal stretches. The effectiveness of the resulting incremental theory is illustrated by applying it to study the Plateau--Rayleigh and Wilkes instabilities in a solid cylinder.