Abstract
We establish the stability of axial motions (steady motions along the lengthwise direction) of nonlinearly elastic loops of string. A key observation here is that a linear combination of the total energy and the total circulation of the string, both of which are conserved quantities, yields an appropriate Liapunov function. From our previous work [5], we know that there are uncountably many shapes corresponding to a given axial speed. Accordingly, we establish “orbitai” stability (modulo this collection of relative equilibria). For a well-defined class of “soft” materials, there is an upper bound on the axial speed sufficient for stability; “stiff” materials are shown to be orbitally stable at any axial speed.
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Healey, T.J. Stability of axial motions of nonlinearly elastic loops. Z. angew. Math. Phys. 47, 809–816 (1996). https://doi.org/10.1007/BF00915277
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DOI: https://doi.org/10.1007/BF00915277