Effects of Viscosity and Loading Rate on Wrinkling Dynamics and Coarsening of Floating Sheets

Elastic sheets floating on a fluid substrate buckle into equilibrium wrinkles under quasi-static compression, selecting a wavelength that reflects the energy balance between the sheet and the fluid substrate. However, when the fluid is viscous and the compression has a finite strain rate, the wrinkling instability and development are drastically different from those under quasi-static compression. Here, we investigate the effects of viscosity and loading rate on the wrinkling dynamics and coarsening of floating elastic sheets by theoretical analyses, numerical simulations, and experiments. We find that such wrinkling instability and development are viscosity and loading rate dependent. A theoretical solution to a critical time is obtained at which the compressive force in the sheet reaches a maximum indicating the onset of the dynamic wrinkling instability. The critical time shows a scaling law with an exponent 1/3 of viscosity and an exponent -2/3 of loading rate. The solution to the critical wavelength of the wrinkle patterns follows a scaling law with an exponent -1/6 of both viscosity and loading rate, which is verified by our experiments. Moreover, numerical and theoretical results show that coarsening of such wrinkles of floating sheets is significantly altered by viscosity, loading rate, and substrate stiffness, different from those under quasi-static compression. Our results provide insights into the rate-dependent instability and coarsening of dynamic wrinkling, which may be applicable for the usage of shape morphing structures via dynamic wrinkling.