Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-30T21:27:08.021Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Antman, S. S. and Kenney, C. S.. Large, buckled states of nonlinearly elastic rods under torsion, thrust and gravity. Arch. Rational Mech. Anal. 76 (1981), 289–338.Google Scholar
2Antman, S. S. and Liu, T.-P.. Travelling waves in hyperelastic rods. Quart. Appl. Math. 36 (1979), 377–399.Google Scholar
3Antman, S. S. and Rosenfeld, G.. Global behaviour of buckled states of nonlinearly elastic rods. SIAM Rev. 20 (1978), 513; Corrigenda 22 (1980), 186.CrossRefGoogle Scholar
4Ball, J. M., Knops, R. J. and Marsden, J. E.. Two examples in nonlinear elasticity. In Proc. of “Journees d'Analyse Non Lineaire”, Besancon, 1977. Lecture Notes in Mathematics 665 (Berlin: Springer, 1978).Google Scholar
5Ball, J. M. and Marsden, J. E.. Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Rational Mech. Anal., submitted for publication.Google Scholar
6Browne, R. C.. Dynamic stability of one-dimensional viscoelastic bodies. Arch. Rational Mech. Anal. 68 (1979), 257–282.CrossRefGoogle Scholar
7Coffman, C. V.. The nonhomogeneous classical elastica. Tech. Report, Dept. of Math., Carnegie-Mellon Univ., 1976.Google Scholar
8Gelfand, I. M. and Fomin, S. V.. Calculus of Variations (New Jersey: Prentice Hall, 1963).Google Scholar
9Knops, R.J. and Wilkes, E.W.. Elastic stability. In Handbuch der Physik, VI a/3, p. 125 (Berlin: Springer, 1973).Google Scholar
10Koiter, W. T.. A basic open problem in the theory of elastic stability. In Proc. of IUTAM-IMU Symposium on Applications of Functionals to Problems of Mechanics. Lecture Notes in Mathematics 503 (Berlin: Springer, 1976).Google Scholar
11Maddocks, J. H.. Restricted quadratic forms and their application to bifurcation and stability in constrained variational principles. SIAM J. Math. Anal. (1984), 16, in press.CrossRefGoogle Scholar
12Maddocks, J. H.. Stability of nonlinearly elastic rods. Arch. Rational Mech. Anal. (1984), 311–354.CrossRefGoogle Scholar
13Potier-Ferry, M.. On the mathematical foundations of elastic stability theory, I. Arch. Rational Mech. Anal. 78 (1982), 55–72.Google Scholar
14Rauch, J. and Reed, M. C.. Propagation of singularities for semilinear hyperbolic equations in one space variable. Ann. Math. 111 (1980), 531–532.CrossRefGoogle Scholar
15Wather, W. W. and Levinson, M.. Destabilization of a nonconservative loaded elastic system due to rotary inertia. Canad. Aero. Space Inst. Trans. 1 (1968), 91–93.Google Scholar
16Weinberger, H. F.. Variational methods for eigenvalue approximation (C.B.M.S. Conference Series 15) (Philadelphia: SIAM, 1974).Google Scholar