Asymmetric equilibria of two nested elastic rings

Abstract

The packing of soft elastic structures is an important and challenging problem due to the possibility of multiple discrete and continuous zones of contact between different parts of the material. To address this problem, we consider the simplest possible packing problem of a thin elastic ring confined within another shorter flexible ring. The elastic properties as well as the dimensionality of both structures, combined with the contact condition yield a wide a variety of possible equilibrium shapes. When the rings are assumed to be inextensible and unshearable, the equilibrium shapes depend only on their relative bending stiffness κ, and on their relative length μ. Whereas the symmetric equilibria for such a problem have been completely determined, the possibility of asymmetric equilibria with lower energy has not yet been considered. For a fixed value of the relative bending stiffness, we explore these symmetry-breaking equilibria as the length of the inner ring increases. We show that, for μ ≃ 1.9 there is a symmetry-breaking bifurcation and asymmetric equilibria are preferred in order to relax the elastic energy.