Hysteretic wavelength selection in isometric, unsupported radial wrinkling

In [Pal et al., arXiv:2206.03552], the authors discuss how an unsupported flat annulus contracted at its inner boundary by a factor $\mathrm{\Delta }$, buckles into a radial wrinkling pattern that is fully 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 ${\lambda }^{\ast }$ sensitive to both the width $w$ and thickness $t$ of the sheet: ${\lambda }^{\ast }\sim w^{2\mathrm{/}3}{t}^{1\mathrm{/}3}{\mathrm{\Delta }}^{-1\mathrm{/}6}$. This scale ${\lambda }^{\ast }$ corresponds to an arrest criterion for wrinkle coarsening starting from any wavelength $\lambda \lesssim {\lambda }^{\ast }$, which can be interpreted in terms of both size and energetic barriers to further coarsening. However, the sheet can support coarser wavelengths: $\lambda \gtrsim {\lambda }^{\ast }$, since there is no penalty to their existence. Since this wavelength selection mechanism depends on the value of $\lambda$ itself, it is hysteretic.