Impact on floating thin elastic sheets: A mathematical model

Abstract

We investigate impact of a sphere onto a floating elastic sheet and the resulting formation and evolution of wrinkles in the sheet. Following impact, we observe a radially propagating wave, beyond which the sheet remains approximately planar but is decorated by a series of radial wrinkles whose wavelength grows in time. We develop a mathematical model to describe these phenomena by exploiting the asymptotic limit in which the bending stiffness is small compared to stresses in the sheet. The results of this analysis show that, at a time t after impact, the transverse wave is positioned a radial distance r ∼ t 1/2 from the impactor, in contrast to the classic r ∼ t 2/3 scaling observed for capillary-inertia ripples produced by dropping a stone into a pond. We describe the shape of this wave, starting from the simplest case of a point impactor and subsequently addressing a finite-radius spherical impactor. We show also that the coarsening of wrinkles in the flat portion of the sheet is controlled by the inertia of the underlying liquid: short-wavelength, small-amplitude wrinkles form at early times since they accommodate the imposed solid deformation without significantly displacing the underlying liquid. As time progresses, the liquid accelerates and the wrinkles grow larger and coarsen. We explain this coarsening quantitatively using numerical simulations and scaling arguments. arXiv:1811.11288v2 [cond-mat.soft] 21 Dec 2018 2

Dynamics of wrinkling in ultrathin elastic sheets

“…6 and 7 and illustrated in Fig. 3 appear to be robust, there is some spread in the prefactor; we show elsewhere (33) that this spread in prefactor is largely accounted for by the size of the impactor.…”

“…The results presented correspond to approximately constant impact velocities, with data only shown up to the time at which the predicted contact radius becomes comparable with the sphere radius (33). The only exception to this is in Fig.…”

“…10 ; numerical solutions presented in ref. 33 demonstrate that the scaling of Eq. 10 may have a logarithmic-type correction.…”

“…The measured values of the instantaneous wavelength were distributed around some mean value rather than being a single well-defined wavelength ( SI Appendix ); the average wavelength reported herein was determined by counting visible wrinkles in images and averaging this number over the length of a particular material circle. We investigate the distribution of wavelengths in more detail in another paper (33).…”

Dynamic Reversible Evolution of Wrinkles on Floating Polymer Films under Magnetic Control

“…Therefore, it is reasonable to say that this work is at least experimentally novel. Additionally, it is worthy to note that the dynamic and reversible evolution of surface wrinkles in our system causes no damage to the polymer film, and no hysteretic effect on the evolution process, which is fundamentally different from a steel ball impacting the floating film, and dripping a water droplet on the floating film [21,31].…”