Active filaments I: Curvature and torsion generation

Abstract

In many filamentary structures, such as hydrostatic arms, roots, and stems, the active or growing part of the material depends on contractile or elongating fibers. Through their activation by muscular contraction or growth, these fibers will generate internal stresses that are partially relieved by the filament acquiring intrinsic torsion and curvature. This process is fundamental in morphogenesis but also in plant tropism, nematic solid activation, and muscular motion of filamentary organs such as elephant trunks and octopus arms. Here, we provide a general theory that links the activation of arbitrary fibers at the microscale to the generation of curvature and torsion at the macroscale. This theory is obtained by dimensional reduction from the full anelastic description of three-dimensional bodies to morphoelastic Kirchhoff rods. Hence, it links the geometry and material properties of embedded fibers to the shape and stiffness of the rod. The theory is applied to fibers that are wound helically around a central core in tapered and untapered filaments.