Born in the Wrong Geometry: Longitudinal and transverse frustration in non-parallel filament bundles

Assemblies of one-dimensional, filamentous materials are commonplace in physical systems, from microscopic materials, such as columnar liquid crystals and biopolymer bundles, to familiar, macroscopic materials like wires, cables, and ropes. The geometry of continuus filament bundles with cross sectional ordering in three dimensions is highly constrained, and we show that only two families of such filament bundles permit equidistant configurations of their constituent filaments: those with bend, but no twist, and those with twist, but no bend. The elastic response of bent and twisted bundles with no such ordered ground state, such as those formed by DNA plasmids under confinement, is then doubly geometrically frustrated: the presence of twist frustrates crystalline order in the cross-section, and the presence of bend couples this compromise structure to the filament tangents. In filaments, as in many frustrated systems, the global response to deformations can not be adequately described by a linear energy. We present a fully nonlinear theory for the elasticity of bundles with one-dimensional components which can slide freely along their tangents, and discuss the response of Euclidean filament packings to the imposition of non-equidistant geometries.