Stability and folds in an elastocapillary system

TL;DR

This paper analyzes the stability of an elastocapillary system involving a circular membrane and menisci, revealing how capillary and elastic forces interact to cause or prevent structural instabilities.

Contribution

It introduces a variational and spectral method framework to determine stability criteria for elastocapillary systems, highlighting the role of membrane and meniscus perturbations.

Findings

Membrane out-of-plane and in-plane spectra are positive at large elastocapillary numbers.

Meniscus perturbations can induce instability, especially at small elastocapillary numbers.

Configurations exist where individual components are stable but the combined system is unstable.

Abstract

We examine the equilibrium and stability of an elastocapillary system to model drying-induced structural failures. The model comprises a circular elastic membrane with a hole at the center that is deformed by the capillary pressure of simply connected and doubly connected menisci. Using variational and spectral methods, stability is related to the slope of equilibrium branches in the liquid content versus pressure diagram for the constrained and unconstrained problems. The second-variation spectra are separately determined for the membrane and meniscus, showing that the membrane out-of-plane spectrum and the in-plane spectrum at large elatocapillary numbers are both positive, so that only meniscus perturbations can cause instability. At small elastocapillary numbers, the in-plane spectrum has a negative eigenvalue, inducing wrinkling instabilities in thin membranes. In contrast, the smallest eigenvalue of the meniscus spectrum always changes sign at a pressure turning point where stability exchange occurs in the unconstrained problem. We also examine configurations in which the meniscus and membrane are individually stable, while the elastocapillary system as a whole is not; this emphasizes the connection between stability and the coupling of elastic and capillary forces.