Evolving wrinkles: time-dependent buckling of an elastic sheet on a liquid substrate

TL;DR

This work develops a dynamic, spectrally-formed model for wrinkles on a floating elastic sheet under uniaxial compression, capturing the transition from inertia-dominated growth to gravity-dominated equilibrium. By applying nondimensionalisation, modal decomposition, and a coupled beam–fluid framework, the authors derive a differential-algebraic system for the amplitude spectrum $a(k,t)$ that evolves under bending, inertia, gravity, and a nonlinear length constraint. The study reveals that a spectrum of wrinkling modes, rather than a single evolving mode, governs the dynamics; energy transfer between kinetic and potential forms and between modes drives coarsening, and dissipation narrows the spectrum while enabling convergence to the equilibrium wavenumber $k=1$ when present. These findings illuminate the dynamic pattern selection and energy pathways in hydroelastic wrinkle formation, with implications for controlling wrinkle patterns in flexible devices and materials that interact with liquid substrates.

Abstract

We model the formation and evolution of wrinkles in a floating elastic sheet under uniaxial compression. This is a canonical setup in the study of wrinkling, and whilst its static equilibrium configuration is well characterised, its dynamics are not. In this work, we focus on modelling the transition from early, inertia-dominated wrinkle growth to late-time gravity-moderated equilibrium. For an initial configuration in which the sheet is flat, an initial disturbance will first grow at the shortest available wavelengths, because this requires the least kinetic energy, but will subsequently transition to a longer preferred wavelength that minimises potential energy. We observe that the evolving wave pattern must be a spectrum, as opposed to a fundamental wrinkle mode whose wavelength evolves in time. Our results demonstrate that changes in the dominant wrinkle wavelength are coupled to a decay in the compressive force, which is to be expected from equilibrium theory.

Figures (11)

• Figure 1: (a) The base-state configuration of an elastic beam, which at time $t^* = 0$ is subject to a prescribed end-to-end displacement of size $\Delta^*$, which buckles the beam out-of-plane, displacing the underlying viscous fluid leading to (b) a highly-wrinkled beam configuration, induced by viscous and inertial restoring forces. As time evolves, the wrinkles coarsen, eventually leading to (c) the late-time steady state for which hydrostatic and spring restoring forces overcome visco-inertial forces and combine to balance bending effects.
• Figure 2: Numerical solutions to mode amplitude equations \ref{['eq:DAE_system_1']}--\ref{['eq:DAE_system_3']}, for Gaussian initial data in form of \ref{['eq:init_cond_Gaussian']} with $\alpha=2$ and $\beta=1/10$. Left: contour plot of $ka(k,t)$, the length accommodated across wavenumbers $k$ as a function of time $t$. Right: $ka$ as a function of $k$ at times $t = 0$, $1$, $10$ and $100$. The dissipation coefficient $\mu$ is set to {0, 0.005, 0.05} top to bottom.
• Figure 3: Numerical solutions to mode amplitude equations \ref{['eq:DAE_system_1']}--\ref{['eq:DAE_system_3']}, for rectangular initial data in form of \ref{['eq:init_cond_rectangular']} with $\alpha=1.5$ and $\beta=1$. Left: contour plot of $ka(k,t)$, the length accommodated across wavenumbers $k$ as a function of time $t$. Right: $ka$ as a function of $k$ at times $t = 0$, $1$, $10$ and $100$. The dissipation coefficient $\mu$ is set to {0, 0.005, 0.05} top to bottom.
• Figure 4: The magnitude of the compressive force, $-F(t)$, the dominant wavenumber $k_{\mathrm{dom}}$, and the moments $\mathscr{M}_i$ for $i=1,2,3$ for the Gaussian (left) and rectangular (right) initial conditions. Dashed lines are $\sim t^{-2/5}$.
• Figure 5: Left: Potential energy density contours. Right: Energy density at chosen times. The dissipation values $\mu$ are {0, 0.005, 0.05} top to bottom. Gaussian initial data in form of \ref{['eq:init_cond_Gaussian']} with $\alpha=2$ and $\beta=1/10$.