Towards a classification of Euler–Kirchhoff filaments

Nizette, Michel; Goriely, Alain

doi:10.1063/1.532731

Cited by 118 publications

(184 citation statements)

References 35 publications

(52 reference statements)

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“…What could eventually be seen from the fact that (18) and (20) are equivalent can also be derived in a more formal way via Noether's theorem. Assume W = W int that is no external loads act on the rod.…”

Section: Spatial Symmetries and Momentum Mapsmentioning

confidence: 99%

“…The fact that the equations of motion for a Lagrange top are formally equivalent to the equilibrium equations of an isotropic Kirchhoff rod is known in the literature as Kirchhoff's kinetic analogy (see Love [4]; a modern treatment can be found in Nizette and Goriely [18]). In the setting for the Lagrange top, the independent variable denotes time whereas for the Kirchhoff rod it denotes arc-length.…”

Section: Variational Formulation: Uniform Rodsmentioning

confidence: 99%

“…However, the details of the computation are somewhat different (see [2]). The variational principle now yields two equations, (18) being the first and…”

Section: Derivation Of the Equilibrium Equationsmentioning

confidence: 99%

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A discrete mechanics approach to the Cosserat rod theory—Part 1: static equilibria

Jung

Leyendecker

Linn

et al. 2010

Numerical Meth Engineering

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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.

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Section: Spatial Symmetries and Momentum Mapsmentioning

confidence: 99%

Section: Variational Formulation: Uniform Rodsmentioning

confidence: 99%

See 1 more Smart Citation

A discrete mechanics approach to the Cosserat rod theory—Part 1: static equilibria

Jung

Leyendecker

Linn

et al. 2010

Numerical Meth Engineering

View full text Add to dashboard Cite

show abstract

“…Using (30), (46), and the fact that the zeroth order birod moments M jr are zero we obtain expressions for the effective birod moments M ir . With a little algebra it we see there is no ensuing trigonometric dependence on s. We note that M 3r is constant and in the absence of forces N 1r and N 2r , M 1r and M 2r are also constant.…”

Section: D2 Birod Momentsmentioning

confidence: 99%

“…It is known that for rods with the particular linear form of moment vector we use that u 3i = 0 and the twist is linear, e.g., [30]. In this case as the torsion of the rods is constant and this implies that φ 1 = φ 2 = 0, and from (B.3) and (B.6) we have that n 11 (s) = n 12 (s) = 0, which also satisfies (B.4) and (B.7).…”

Section: B Solving the Zeroth Order Equilibrium Equationsmentioning

confidence: 99%