The mode branching route to localization of the finite-length floating elastica

The beam on elastic foundation is a general model used in physical, biological, and technological problems to study delamination, wrinkling, or pattern formation. Recent focus has been given to the buckling of beams deposited on liquid baths, and in the regime where the beam is soft compared to hydrostatic forces the wrinkling pattern observed at buckling has been shown to lead to localization of the deformation when the confinement is increased. Here we perform a global study of the general case where the intensity of the liquid foundation and the confinement are both varied. We compute equilibrium and stability of the solutions and unravel secondary bifurcations that play a major role in the route to localization. Moreover we classify the post-buckling solutions and shed light on the mechanism leading to localization. Finally, using an asymptotic technique imported from fluid mechanics, we derive an approximated analytical solution to the problem.