On the dynamics of rods in the theory of Kirchhoff and Clebsch

Coleman, Bernard D.; Dill, Ellis H.; Lembo, Marzio; Li, Zheng; Tobias, Irwin

doi:10.1007/bf00375625

Cited by 138 publications

(109 citation statements)

References 6 publications

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“…(1.5), which is 6 th order in the spatial derivatives of u. For a physical closed loop of wire with a small circular cross section of radius ρ > 0, a more realistic expression for the elastic potential energy is given by 16) where λ and µ are the Lamé constants for a homogeneous isotropic hyperelastic material (see [14]). (For a wire made of material of a fixed density, the kinetic energy is of order ρ 2 .…”

Section: Related Geometrical and Physical Problemsmentioning

confidence: 99%

“…(1.14). Maddocks and Dichmann [15], Coleman et al [16] and others consider director theories, originated by Kirchhoff and Clebsch, in which there are further stress-strain relations between the tangent and two independent normal vectors (see [17]). The above choices for the potential and kinetic energy will not be pursued here.…”

Section: Related Geometrical and Physical Problemsmentioning

confidence: 99%

“…In the proof of the lemma, we will construct a transformation (u, ∂ t u) → (P, Q, β) between H r+2 × H r and H r × H r+1 × R which is continuous and has a continuous inverse for all r ≥ −1. 16) and P 0 and Q 0 are given by…”

Section: Remark 24mentioning

confidence: 99%

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On the Cauchy Problem for a Dynamical Euler's Elastica

Burchard

Thomas

2003

Communications in Partial Differential Equations

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The dynamics for a thin, closed loop inextensible Euler's elastica moving in three dimensions are considered. The equations of motion for the elastica include a wave equation involving fourth order spatial derivatives and a second order elliptic equation for its tension. A Hasimoto transformation is used to rewrite the equations in convenient coordinates for the space and time derivatives of the tangent vector. A feature of this elastica is that it exhibits time-dependent monodromy. A frame frame parallel-transported along the elastica is in general only quasi-periodic, resulting in time-dependent boundary conditions for the coordinates. This complication is addressed by a gauge transformation, after which a contraction mapping argument can be applied. Local existence and uniqueness of elastica solutions are established for initial data in suitable Sobolev spaces.

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Section: Related Geometrical and Physical Problemsmentioning

confidence: 99%

Section: Related Geometrical and Physical Problemsmentioning

confidence: 99%

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On the Cauchy Problem for a Dynamical Euler's Elastica

Burchard

Thomas

2003

Communications in Partial Differential Equations

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“…В работе [7] в замкну-той форме получены решения типа бегущих волн в линейно растяжимом и бесс-двиговом (тонком) плоском стержне. В работе [8] авторы представили описание нерастяжимых стержней с кручением и получили выражения для пространствен-ных (трехмерных) бегущих волн при наличии крутящего момента. В работе [9] обсуждается динамика изгиба плоского нерастяжимого стержня при условии пре-небрежения вращательной частью кинетической энергии, которая мала по сравне-нию с полной энергией стержня.…”

Section: Introductionunclassified

“…Сохранение импульса и углового момента и уравнения состояния (2.1) приводят к уравнениям для силы и момента [3], [8]:…”

Section: Introductionunclassified