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.

Automated summary

We investigate the equilibrium configurations of an elastic fiber growing in a confined, flexible ring. This system is widely applicable to biological, medical, and engineering problems. Using a simplified geometry where the initial container is a circular ring of radius R, we study the quasi-static growth by solving equilibrium equations as the fiber length l increases from l = 2R. By considering the fiber and ring as inextensible and unshearable, we find that the fiber buckles beyond a critical length that depends on the relative bending stiffness. Additionally, as the fiber continues to grow, it folds and distorts the ring, breaking the mirror symmetry at l > 2 pi R. We determine that the equilibrium shapes depend on two dimensionless parameters: the length ratio mu = l/R and the bending stiffnesses ratio kappa. These findings are corroborated by finite element simulations. Experimental validation shows excellent quantitative agreement with the observed buckling and folding regimes at varying geometrical parameters.