Buckling and zipping of a magnetic ring under gravity

Assemblies of magnetic beads, also called magnetostructures, exhibits interesting mechanical properties, adapting orientations of the dipoles to minimize the dipolar energy. A ring made of $N$ spherical magnetic beads behaves like an elastic annulus. This elastic-like property is due to the dipolar nature of the particles. When submitted to gravity and as a function of its size, the initial circular shape of a magnetic ring is seen to experience flattening. This capsule-like shape appears when the number $N$ of magnets reaches a critical point scaling with the Bond number $N_f \propto \rm Bo^{-1/3}$. When the number of magnets increases more and more, the ring starts to buckle and a flat object appears at a second critical point $N_z \propto {\rm Bo}^{-1}+b$. There, a zipping state corresponding to the attraction of two opposite sides is formed. We propose a theoretical approach to capture these scaling laws in agreement with experimental data. All shapes are also numerically obtained in a Discrete Element Model confirming our findings.