Stability and folds in an elastocapillary system

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.