The filament-bundle elastica

Abstract

Filament-bundles are ubiquitous in nature. They are composed by an assembly of flexible rods held together by elastic springs, such as those found in ciliary systems and flagella. We study the static, post-transient, post-buckled configurations of a generalized filament-bundle elastica or flagella. We recur to linear and weakly nonlinear analysis, as well as geometrically exact numerical solutions. The bundle cross-linking mechanics is characterized by non-local moments affecting distant parts of the structure. This induces a bimodal post-buckling response sensitive to the interfilament sliding at the base. We report the occurrence of a novel reversed cusp catastrophe, reminiscent of the counterbend phenomenon, that folds and suppresses the saddle-node bifurcation back a pitchfork bistability landscape, found in classical elastica systems. The filament-bundle elastica can thus prevent violent jumps, non-uniqueness and hysteresis. This non-trivial folding of the imperfection-sensitivity diagram may impact bundle systems with naturally occurring buckling phenomena.