Counting wrinkles on cones

Wrinkles on curved surfaces have a rich phenomenology and are relevant to various biological and engineering structures. Consider the problem of a soft elastic cone covered by a stiff thin sheet with excess circumferential length. When moving down along the surface of the cone away from the tip, the changing circumference leads to an incompatibility between wrinkles of fixed wavelength (favored by a local balance of the elastic energy of the sheet with the substrate) and a fixed number of wrinkles (which would avoid the energy of inserting new wrinkles into the pattern). This can create a library of very rich wrinkling patterns, which might be connected to hierarchical microchannels found on the Sarracenia trichome. Here, we carry out large-scale finite element simulations to systematically examine the effect of the cone angle on the wrinkle splitting along the axial and circumferential directions. For small cone angles, we observe a hierarchical pattern with defect-free zones (constant wrinkle wavelength) separated by defect-rich zones where new wrinkles emerge. Our findings closely resemble the mesoscale structure of wrinkle patterns in an annulus film floating on water and highlight the generality of the multiscale wrinkling structures governed by geometry incompatibility.