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Nonlinear dynamical theory of the elastica

Published online by Cambridge University Press:  14 November 2011

Russel E. Caflisch  and
John H. Maddocks
Russel E. Caflisch
Affiliation:
Courant Institute, New York University, 251 Mercer Street, New York, NY 10012, U.S.A.
John H. Maddocks
Affiliation:
Mathematics Department, Stanford University, Stanford, CA 94305, U.S.A.
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Extract

The dynamical behaviour of a slender rod is analyzed here in terms of a generalization of Euler's elastica theory. The model includes a linear stress-strain relation but nonlinear geometric terms. Properties of the rod may vary along its length and various boundary conditions are considered. A rotational inertia term that is neglected in many theories is retained, and is essential to the analysis. By use of the equivalence of an energy and a Sobolev norm, and by reformulation of the equations as a semilinear system, global existence of solutions is proved for any smooth initial data. Equilibrium solutions that are stable in the static sense of minimizing the potential energy are then proved to be stable in the dynamic sense due to Liapounov.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 99 , Issue 1-2 , 1984 , pp. 1 - 23
Copyright
Copyright © Royal Society of Edinburgh 1984

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