A tale of two nested elastic rings

Elastic rods in contact provide a rich paradigm for understanding shape and deformation in interacting elastic bodies. Here, we consider the problem of determining the static solutions of two nested elastic rings in the plane. If the inner ring is longer than the outer ring, it will buckle creating a space between the two rings. This deformation can be further influenced by either adhesion between the rings or if pressure is applied externally or internally. We obtain an exact solution of this problem when both rings are assumed inextensible and unshearable. Through a variational formulation of the problem, we identify the boundary conditions at the contact point and use the Kirchhoff analogy to give exact solutions of the problems in terms of elliptic functions. The role of both adhesion and pressure is explored.