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The dynamics of elastic closed curves under uniform high pressure

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Abstract

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.

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  1. Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan

    Shinya Okabe

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  1. Shinya Okabe
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Okabe, S. The dynamics of elastic closed curves under uniform high pressure. Calc. Var. 33, 493–521 (2008). https://doi.org/10.1007/s00526-008-0179-0

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  • DOI: https://doi.org/10.1007/s00526-008-0179-0

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