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.

Liquid-bridge stability and breakup on surfaces with contact-angle hysteresis

“…(1) with the total grand canonical potential, the Lagrange multiplier is determined λ = (p l − p g )/2γ gl = −q/2 (see Akbari et al 14 and Neumann et al 42 ). In general, ψ 1 (t 1 ) ̸ = 0, so equilibrium requires γ gl Φ r ′ | t 1 + R 1 (γ sl − γ sg ) = 0, furnishing the contact-line constraint…”

“…Studying the stability and dynamics of liquid bridges is motivated by a broad range of applications, including crystal growth in microgravity 1 , surface patterning, nano-printing, and nano-lithography [2][3][4] , aggregation and coalescence of flexible fibres [5][6][7][8] , and capillary induced collapse of elastic structures [9][10][11][12][13][14] . Quantifying liquid-bridge and jet breakup upon stability loss dates to the works of Plateau 15 and Rayleigh 16 , 17 .…”

An elastocapillary model of wood-fibre collapse

“…We derived a variational principle for the stability and equilibrium of the elastocapillary model shown in figure 1 [34]. Neglecting the bending contribution to the elastic strain energy, the membrane in-plane and out-of-plane displacement profiles are (see appendix A)…”

“…Equations (2.1) and (2.2) are derived using von Kármán's plate theory, which requires dw/dr p ∼ H/R 1 [34]. Therefore, we restrict our analysis to cases for which H/R 1, so that the membrane displacements can be accurately approximated by equations (2.1) and (2.2).…”

“…Here, the meniscus geometry is exactly treated for the interfacial energies to account for the significant role of the Laplace pressure when collapsed conformations are approached. Using the stability criteria we recently derived based on a spectral analysis of the potential energy [34], the dry-state conformation is determined from the stability of equilibrium branches. The model is also relevant to structural failures and stiction in micro-electromechanical systems during wet etching, where capillary surfaces undergo catastrophic transitions.…”