Structure of Fluctuating Thin Sheets Under Random Forcing

Research from both wave turbulence [1] and the physics of crumpled paper [2] has made it clear that understanding the spectra of deformations of thin sheets is crucial to understanding their structural and dynamic properties. The Föppl-von Kármán (FvK) equations which describe such deformations however have long been recognised as extremely challenging to solve. Here, we argue for a noisy over-damped FvK equation, in which a conserved random force keeps a fluctuating thin sheet out of equilibrium. By applying a novel method known as the Self-Consistent Expansion, a method which both borrows from and extends the renormalisation group, we have been able to obtain a concise integral equation for the two-point function of the deformations of such thin sheets. Surprisingly, this integral equation is amenable to analytic methods and provides precise analytic predictions. Of particular note is that unlike previous claims [3], we find that no exotic roughness exponents are needed to characterise the spectrum, rather, simple rational functions suffice. Simulations further confirm the validity of our solution.

[1] G. Düring, C. Josserand and S. Rica, Physica D 347, 42 (2017)
[2] D.L Blair and A. Kudrolli, PRL 94, 166107 (2005)
[3] F. Plouraboué and S. Roux, Physica A, 227, 173 (1996)