Invariant isogeometric formulations for three-dimensional Kirchhoff rods

Abstract

In the isogeometric formulation for three-dimensional Kirchhoff rods, the invariance is required such that rigid-body modes associated with translations and infinitesimal rotations can be exactly represented. By enforcing the invariance in the isogeometric formulation of the rod, the equilibrium of forces and moments at fixed ends is preserved, and the performance is improved considerably for elements of finite sizes. We recognize that not all existing isogeometric formulations are invariant for general geometry of rods. Though an alternative formulation based on the Frenet–Serret frame and rotation angle representation was shown to be invariant, it does not work for rods with vanishing curvature, while its accuracy is affected by the low continuity of the angle of twist used. Motivated by this, two new invariant isogeometric formulations are developed, through different interpolations for the angle of twist. One uses the derivatives of the NURBS functions as the basis for the angle-of-twist interpolation, and the other uses a mixed interpolation incorporating both the Lagrange and NURBS functions. Several representative numerical examples were prepared to verify the accuracy and robustness of the two new formulations. Particularly, for rods of variable curvatures simulated by elements of finite or large sizes, the proposed formulations are demonstrated to outperform the two existing ones mentioned in predicting the displacements, end reactions and stress resultants.