A finite element for nonlinear three-dimensional Kirchhoff rods

This paper presents the formulation of a finite element method for nonlinear Kirchhoff rods (i.e. without transverse shear strain) in the general three-dimensional setting defined by a Cosserat director treatment of the cross sections attached to the rod's axis. The new element is based on a $G^1$ interpolation of the rod's geometry in terms of Hermite shape functions of the rod's axis (including its tangent defining the tangential director), while the transversal directors defining the different bending and torsional responses of the rod consider a Lagrangian interpolation of the section directors. This direct interpolation of the directors, as opposed of underlying rotation vectors, assures the objectivity of the proposed formulation. In fact, the invariance properties of the resulting finite element are analyzed in detail, assuring the correct resolution of the local fundamental equilibrium relations between forces and moments, hence avoiding the so-called “self-straining” associated to separate treatments of the rod's geometry and its kinematics. Several representative numerical simulations are presented illustrating these properties as well as the appropriateness of the proposed formulation for the analysis of thin rods undergoing large finite deformations in the three-dimensional range.