A discrete mechanics approach to Cosserat rod theory - Part I: static equilibria

Jung P, Leyendecker S, Linn J, Ortiz M (2010)

Publication Language: English

Publication Type: Journal article

Publication year: 2010

Journal

International Journal for Numerical Methods in Engineering Wiley

Book Volume: Vol. 85

Pages Range: 31-60

DOI: 10.1002/nme.2950

Abstract

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. Copyright © 2010 John Wiley & Sons, Ltd.

 

Authors with CRIS profile

Sigrid Leyendecker

Additional Organisation(s)

Institute of Applied Dynamics

Involved external institutions

Fraunhofer-Institut für Techno- und Wirtschaftsmathematik (ITWM) / Fraunhofer Institute for Industrial Mathematics DE Germany (DE) Technische Universität Kaiserslautern DE Germany (DE) California Institute of Technology (Caltech) US United States (USA) (US)

How to cite

APA:

Jung, P., Leyendecker, S., Linn, J., & Ortiz, M. (2010). A discrete mechanics approach to Cosserat rod theory - Part I: static equilibria. International Journal for Numerical Methods in Engineering, Vol. 85, 31-60. https://dx.doi.org/10.1002/nme.2950

MLA:

Jung, Pascal, et al. "A discrete mechanics approach to Cosserat rod theory - Part I: static equilibria." International Journal for Numerical Methods in Engineering Vol. 85 (2010): 31-60.

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