Fibered by fibers: the geometry and elasticity of frustrated filaments

Filamentous and columnar assemblies are a ubiquitous motif in materials, from microscopic and biological materials, like discotic liquid crystals and biopolymer bundles, to familiar macroscopic materials like yarns, cables, and ropes. Ordered ground states in filament bundles, however, are highly geometrically constrained. We show that only two families of filament textures permit equidistance between the constituent filaments: the developable domains, which can bend, but not twist, and the helical domains, which can twist uniformly, but not bend. The elastic response of non-equidistant filament bundles is then frustrated, and cannot adequately be described by a linearized energy. To describe nonequidistant configurations, we derive a geometrically nonlinear, coordinate invariant, gauge-like theory for the elasticity of filamentous materials. Within this framework, we discuss the impact of filament texture on bundle elasticity, and calculate the stable states for non-equidistant bundles with small curvature.