Hysteretic wavelength selection in isometric, unsupported radial wrinkling

In [A. S. Pal, L. Pocivavsek and T. A. Witten, arXiv, DOI: 10.48550/arXiv.2206.03552], the authors discuss how an unsupported flat annulus contracted at its inner boundary by fraction Δ, buckles into a radial wrinkling pattern that is asymptotically isometric and tension-free. What selects the wavelength in such a pure-bending configuration, in the absence of any competing sources of work? In this paper, with the support of numerical simulations, we argue that competition between stretching and bending energies at local, mesoscopic scales leads to the selection of a wavelength scale λ* sensitive to both the width w and thickness t of the sheet: λ* ∼ w^2/3t^1/3Δ^-1/6. This scale λ* corresponds to a kinetic arrest criterion for wrinkle coarsening starting from any finer wavelength λ ≲ λ*. However, the sheet can support coarser wavelengths: λ ≳ λ*, since there is no penalty to their existence. Since this wavelength selection mechanism depends on the initial value of λ, it is path-dependent or hysteretic.