Developable zig-zag wrinkling in a thin annulus contracted inwards*

The Lamé system – an annulus subjected to radial tensile loads at both boundaries so that the material circles wrinkle – serves as a prototype for studying two-dimensional (2d) wrinkling. While this system has been extensively studied [1], an open question is whether it can attain a fully strain-free (or developable) state post-buckling. In this work, we analyse a novel geometric Lamé system that displays such a strain-free state of azimuthal wrinkling. Here, we pull in the inner boundary of a thin, unsupported elastic annulus by a radial distance Δ, leaving the outer boundary free. Using finite-element simulations, we show that, as Δ is increased, the annulus transitions from a smooth radially wrinkled configuration to a faceted developable morphology of regularly alternating flat triangles and radial cones, with the cone tips located on the inner boundary, and with wavelength much shorter than the radius. We account theoretically for the observed scaling behaviour of the energy and stress, and offer a novel kinetic mechanism for the wavelength selection based on the structure of wrinklons [2].

References –
[1] Davidovitch et al, PNAS 2011
[2] Vandeparre et al, PRL 2011