Invariant Hermitian finite elements for thin Kirchhoff rods. II: The linear three-dimensional case

This paper considers the formulation of finite elements for thin Kirchhoff rods in the three-dimensional case, extending our work on plane rods presented in [5]. The new issue to be addressed, besides the basic consideration of the three-dimensional geometry, is the approximation of the rod's torsion, still in the geometrically linear setting of infinitesimal strains. In particular, it is the goal of this paper to identify approximations that preserve the basic invariance properties of the underlying structural theory, in the sense that the fundamental rigid-body modes of translations and infinitesimal rotations are represented exactly without straining, accounting fully for the rod's curvature and associated Kirchhoff's kinematics in contrast to common straight (framework) element approximations based on the simplified Euler-Bernoulli beam kinematics. To this purpose, different basic and new interpolations of the angle of twist rotation as well as interpolations of directors increments in the setting provided by a Cosserat treatment of the rod's kinematics are explored and studied in detail, including rigorous analytical characterizations of the aforementioned rigid-body modes. These different interpolations are combined with the new C1-continuous Hermitian interpolation developed in that reference for the axis displacements, extended to general three-dimensionally curved rods, as well as with an assumed strain mixed treatment to improve the overall performance of the finite elements. Several representative numerical simulations are presented to illustrate the properties and the numerical performance of the different new finite elements.