An objective finite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy–momentum conserving scheme in dynamics

Abstract

SUMMARYWe present in this paper a new ÿnite element formulation of geometrically exact rod models in the threedimensional dynamic elastic range. The proposed formulation leads to an objective (or frame-indi erent under superposed rigid body motions) approximation of the strain measures of the rod involving ÿnite rotations of the director frame, in contrast with some existing formulations. This goal is accomplished through a direct ÿnite element interpolation of the director ÿelds deÿning the motion of the rod's crosssection. Furthermore, the proposed framework allows the development of time-stepping algorithms that preserve the conservation laws of the underlying continuum Hamiltonian system. The conservation laws of linear and angular momenta are inherited by construction, leading to an improved approximation of the rod's dynamics. Several numerical simulations are presented illustrating these properties.

A computational framework for polyconvex large strain elasticity for geometrically exact beam theory

“…Traditional finite element discretisations based on rotations do not satisfy objectivity [1,2,4,6]. In order to circumvent this shortcoming, a formulation in which the nodal degrees of freedom correspond to displacements of the centre line x 0 and the director triad {d 1 , d 2 , d 3 } is considered following reference [1,6].…”

“…In order to circumvent this shortcoming, a formulation in which the nodal degrees of freedom correspond to displacements of the centre line x 0 and the director triad {d 1 , d 2 , d 3 } is considered following reference [1,6]. In this approach, x 0 and the director vectors are interpolated as…”

“…The present manuscript aims to extend the variational and computational framework recently introduced in Bonet et al [12] in the context of three dimensional elasticity to the geometrically exact Simo [13] beam model, the starting point of so many other finite element beam type formulations [1][2][3][4][5][6][7]. The variational approach proposed herein can therefore be viewed as a continuum degenerate formulation emerging from that presented in [12].…”

“…Most classical beam theories [1][2][3][4][5][6][7][8] are based on the definition of so called beam strain measures, namely the axial-shear and torsional-bending strain vectors. Work conjugates to those variables (typically denoted as resultant contact forces and resultant contact couples [9]) can be derived from a hyperelastic energy functional which is defined in terms of the beam strain measures [1][2][3][4][5][6][7][8].…”

“…Work conjugates to those variables (typically denoted as resultant contact forces and resultant contact couples [9]) can be derived from a hyperelastic energy functional which is defined in terms of the beam strain measures [1][2][3][4][5][6][7][8]. Other authors prefer an alternative continuum degenerate approach [10,11] where, typically, the Lagrangian strain tensor is retained as the main strain measure and particularised for the kinematic description of the beam.…”