Soft beams: when capillarity induces axial compression

TL;DR

This paper investigates how capillarity affects the axial compression of elastic beams, deriving modified equilibrium equations that incorporate surface energy effects and analyzing the behavior at the triple phase point.

Contribution

It introduces a variational framework that accounts for surface energy in elastic beam mechanics, extending classical Hooke's law to include capillarity effects.

Findings

Surface energy modifies the beam's constitutive relation.

At the triple point, the external force aligns with the liquid-vapor interface.

The extension remains continuous at the triple point under certain conditions.

Abstract

We study the interaction of an elastic beam with a liquid drop in the case where bending and extensional effects are both present. We use a variational approach to derive equilibrium equations and constitutive relation for the beam. This relation is shown to include a term due to surface energy in addition of the classical Young's modulus term, leading to a modification of Hooke's law. At the triple point where solid, liquid, and vapor phases meet we find that the external force applied on the beam is parallel to the liquid-vapor interface. Moreover, in the case where solid-vapor and solid-liquid interface energies do not depend on the extension state of the beam, we show that the extension in the beam is continuous at the triple point and that the wetting angle satisfy the classical Young-Dupr\'e relation.