Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Equilibria of the circular elastica under a uniform central force field

Authors: R. W. Dickey and J. J. Roseman
Journal: Quart. Appl. Math. 51 (1993), 201-216
MSC: Primary 73H05
DOI: https://doi.org/10.1090/qam/1218364
MathSciNet review: MR1218364
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A mathematical model for the problem of an inextensible circular elastica under a uniform centrally directed force field is derived and studied. It is shown analytically that stable large amplitude solutions exist at forces $ P < {P_1}$, the first eigenpressure for the linearized model, and it is shown numerically that these solutions have only one axis of symmetry. These results agree with experiment. In addition, numerical solutions are calculated for states with more than one axis of symmetry which resemble those found in the literature on elastic rings under hydrostatic pressure.

References [Enhancements On Off] (What's this?)

  • [1] I. Tadjbakhsh and F. Odeh, Equilibrium states of elastic rings, J. Math. and Appl. 18, 59-74 (1967) MR 0205528
  • [2] S. S. Antman, A note on a paper of Tadjbakhsh and Odeh, J. Math. Anal. Appl. 21, 132-135 (1968) MR 0221811
  • [3] S. S. Antman, The shape of buckled nonlinear elastic rings, Z. angew. Math. Phys. 21, 422-438 (1970) MR 0277144
  • [4] J. J. Stoker, Nonlinear Elasticity, Gordon and Breach, New York, 1968 MR 0413654
  • [5] R. W. Dickey, Nonlinear bending of circular plates, SIAM J. Appl. Math. 30, 1-9 (1976) MR 0403377
  • [6] R. W. Dickey Minimum energy solution for the spherical shell, Quart. Appl. Math. 48, 321-339 (1990) MR 1052139
  • [7] E. Isaacson and H. B. Keller, Analysis of numerical methods, Wiley, New York, 1966 MR 0201039
  • [8] B. J. Struik, Lectures on classical differential geometry, Addison-Wesley, Reading, MA, 1961
  • [9] R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, Interscience, New York, 1953 MR 0065391
  • [10] E. L. Reiss, Column buckling--an elementary example of bifurcation, Bifurcation Theory and Nonlinear Eigenvalue Problems (J. B. Keller and S. Antman eds.), W. A. Benjamin, New York, 1969

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73H05

Retrieve articles in all journals with MSC: 73H05

Additional Information

DOI: https://doi.org/10.1090/qam/1218364
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society