A discrete mechanics approach to the Cosserat rod theory-Part 1: static equilibria
A theory of discrete Cosserat rods is formulated in the language of discrete Lagrangian mechanics. By exploiting Kirchhoff's kinetic analogy, the potential energy density of a rod is a function on the tangent bundle of the configuration manifold and thus formally corresponds to the Lagrangian function of a dynamical system. The equilibrium equations are derived from a variational principle using a formulation that involves null-space matrices. In this formulation, no Lagrange multipliers are necessary to enforce orthonormality of the directors. Noether's theorem relates first integrals of the equilibrium equations to Lie group actions on the configuration bundle, so-called symmetries. The symmetries relevant for rod mechanics are frame-indifference, isotropy, and uniformity. We show that a completely analogous and self-contained theory of discrete rods can be formulated in which the arc-length is a discrete variable ab initio. In this formulation, the potential energy density is defined directly on pairs of points along the arc-length of the rod, in analogy to Veselov's discrete reformulation of Lagrangian mechanics. A discrete version of Noether's theorem then identifies exact first integrals of the discrete equilibrium equations. These exact conservation properties confer the discrete solutions accuracy and robustness, as demonstrated by selected examples of application.
© 2010 John Wiley & Sons, Ltd. Received 13 July 2009; Revised 13 March 2010; Accepted 21 April 2010. Article first published online: 21 Jun. 2010. Michael Ortiz is grateful to the Office of Naval Research (ONR) for financial support made available through Caltech's Multidisciplinary University Research Initiative (MURI) on 'Mechanics and Mechanisms of Impulse Loading', grant N00014-06-1-0730. We thank P. Betsch for providing the data used for comparison in Section 7.2. Furthermore, we also thank the referee and the editor for their very concise and constructive remarks.