INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2002; 54:1683–1716 (DOI: 10.1002/nme.486) An objective nite element approximation of the kinematics of geometrically exact rods and its use in the formulation of an energy–momentum conserving scheme in dynamics I. Romero and F. Armero ∗; † Department of Civil and Environmental Engineering; Structural Engineering; Mechanics and Materials; University of California; Berkeley; CA 94720; U.S.A. SUMMARY We present in this paper a new nite element formulation of geometrically exact rod models in the three- dimensional 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 nite rotations of the director frame, in contrast with some existing formulations. This goal is accomplished through a direct nite element interpolation of the director elds dening the motion of the rod’s cross- section. 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. Copyright ? 2002 John Wiley & Sons, Ltd. KEY WORDS: geometrically exact rods; dynamics; nite elements; objectivity; energy–momentum conserving algorithms 1. INTRODUCTION A rod can be dened as a slender solid with two characteristic dimensions much smaller than the third one, the latter dening the axial direction. This characterization leads to the consideration of a general curve in the three-dimensional space with a plane section attached to each of its material points. The plane of the section is then dened through two vectors, the so-called directors. Non-linear models of rods as directed continua have been considered by a large number of authors in the past; see the classical works of Cosserat and Cosserat [1] Green and Laws [2], Ericksen and Truesdell [3] Reissner [4] and the review of Antman [5], among others. This geometric characterization allows the denition of the dierent components ∗ Correspondence to: F. Armero, Structural Engineering, Mechanics and Materials, Department of Civil and Envi- ronmental Engineering, 713 Davis Hall, University of California, Berkeley, CA 94720-1710, U.S.A. † E-mail: armero@ce.berkeley.edu Contract=grant sponsor: AFOSR; contract=grant number: F49620-00-1-0360 Received 5 February 2001 Copyright ? 2002 John Wiley & Sons, Ltd. Revised 21 September 2001