Growth of a flexible fibre in a deformable ring

We study the equilibrium configurations related to the growth of an elastic fibre in a confining flexible ring. This system represents a paradigm for a variety of biological, medical, and engineering problems. We consider a simplified geometry in which initially the container is a circular ring of radius $R$. Quasi-static growth is then studied by solving the equilibrium equations as the fibre length $l$ increases, starting from $l = 2R$. Considering both the fibre and the ring as inextensible and unshearable, we find that beyond a critical length, which depends on the relative bending stiffness, the fibre buckles. Furthermore, as the fibre grows further it folds, distorting the ring until it induces a break in mirror symmetry at $l>2 \pi R$. We get that the equilibrium shapes depend only on two dimensionless parameters: the length ratio $\mu = l/R$ and the bending stiffnesses ratio $\kappa$. These findings are also supported by finite element simulation. Finally we experimentally validate the theoretical results showing a very good quantitative prediction of the observed buckling and folding regimes at variable geometrical parameters.