The motion of elastic planar closed curves under the area-preserving condition

Okabe, Shinya

doi:10.1512/iumj.2007.56.3015

Cited by 28 publications

(33 citation statements)

References 12 publications

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“…∞ in the C ∞ -topology. The result can be extended to the following case: (i) L 2 gradient flow for E under the area-preserving condition and (C) [27] and (ii) L 2 gradient flow for Tadjbakhsh-Odeh energy functional under the constraint (C) [28]. Moreover, the result [26] was also extended to the case of space curves [29].…”

Section: Equation Of Motionmentioning

confidence: 99%

Shape Memory Wires in R3

Okabe¹,

Suzuki²,

Yoshikawa³

2017

Shape Memory Alloys - Fundamentals and Applications

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We propose a new model describing the dynamics of wire made of shape memory alloys, by combining an elastic curve theory and the Ginzburg-Landau theory. The wire is assumed to be a closed curve and is not to be stretched with deformation. The derived system of nonlinear partial differential equations consists of a thermoelastic system and a geometric evolution equation under the inextensible condition. We also show that the system has dual variation structure as well as a straight material case. The structure implies stability of infinitesimally stable stationary state in the Lyapunov sense.

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Section: Equation Of Motionmentioning

confidence: 99%

Shape Memory Wires in R3

Okabe¹,

Suzuki²,

Yoshikawa³

2017

Shape Memory Alloys - Fundamentals and Applications

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“…On the other hand, we studied the motion of an elastic planar closed curve with constant enclosed area [13]. In [13] we considered the following system of equations:…”

Section: Lemma 31 Let γ (X T) Be a Solution Of (Gf) Then The Centementioning

confidence: 99%

“…In [13], the energy is also the total squared curvature, but constraints are (3.2) and the area-preserving condition. We proved that the system (GT) has a classical solution for all time t > 0, and the solution converges to a stationary solution as t → ∞ in the C ∞ topology.…”

Section: Lemma 31 Let γ (X T) Be a Solution Of (Gf) Then The Centementioning

confidence: 99%

“…Various problems concerning an elastica have been studied, for instance, on a closed elastica [7,8], a L 2 gradient flow for the total squared curvature [17], a curve straightening flow [9,11,21], the motion of an elastic curve [6], the motion of an elastic curve with constant enclosed area [13], as well as an area-preserving elastica in spectral geometry [12,22,23]. Furthermore, the study on the buckling behavior of an elastic wire also belongs to this S. Okabe (B) Mathematical Institute, Tohoku University, Sendai 980-8578, Japan e-mail: okabe@math.tohoku.ac.jp field.…”

Section: Introductionmentioning

confidence: 99%

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The dynamics of elastic closed curves under uniform high pressure

Okabe

2008

Calc. Var.

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We consider the dynamics of an inextensible elastic closed wire in the plane under uniform high pressure. In 1967, Tadjbakhsh and Odeh (J. Math. Anal. Appl. 18:59-74, 1967) posed a variational problem to determine the shape of a buckled elastic ring under uniform pressure. In order to comprehend a dynamics of the wire, we consider the following two mathematical questions: (i) can we construct a gradient flow for the Tadjbakhsh-Odeh functional under the inextensibility condition?; (ii) what is a behavior of the wire governed by the gradient flow near every critical point of the Tadjbakhsh-Odeh variational problem? For (i), first we derive a system of equations which governs the gradient flow, and then, give an affirmative answer to (i) by solving the system involving fourth order parabolic equations. For (ii), we first prove a stability and instability of each critical point by considering the second variation formula of the Tadjbakhsh-Odeh functional. Moreover, we give a lower bound of its Morse index. Finally we prove a dynamical aspects of the wire near each equilibrium state. Mathematics Subject Classification (2000)Primary: 74H40 · Secondary: 74H55, 74G60, 74G65, 34B15, 35K55, 74B20

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“…In [Koi96] the evolution of closed, inextensible curves in R 3 under the curve straightening flow with the same curvature functional as in the present study was considered, but without making use of the indicatrix representation. In the same way in [Oka07] the motion of elastic planar closed curves under the additional constraint of area-preservation was considered. Finally in [LS05] and in [DKS02] efficient ways to compute the evolution towards the stationary points called elasticae were investigated.…”

Section: Introductionmentioning

confidence: 99%

On the curve straightening flow of inextensible, open, planar curves

Oelz

2011

SeMA

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Abstract. We consider the curve straightening flow of inextensible, open, planar curves generated by the Kirchhoff bending energy. It can be considered as a model for the motion of elastic, inextensible rods in a high friction regime. We derive governing equations, namely a semilinear fourth order parabolic equation for the indicatrix and a second order elliptic equation for the Lagrange multiplier. We prove existence and regularity of solutions, compute the energy dissipation, prove its coercivity and conclude convergence to equilibrium, namely to a straight curve, at an exponential rate.

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The motion of elastic planar closed curves under the area-preserving condition

Cited by 28 publications

References 12 publications

Shape Memory Wires in R3

Shape Memory Wires in R3

The dynamics of elastic closed curves under uniform high pressure

On the curve straightening flow of inextensible, open, planar curves

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