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
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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.
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I. Tadjbakhsh and F. Odeh, Equilibrium states of elastic rings, J. Math. and Appl. 18, 59–74 (1967)
S. S. Antman, A note on a paper of Tadjbakhsh and Odeh, J. Math. Anal. Appl. 21, 132–135 (1968)
S. S. Antman, The shape of buckled nonlinear elastic rings, Z. angew. Math. Phys. 21, 422–438 (1970)
J. J. Stoker, Nonlinear Elasticity, Gordon and Breach, New York, 1968
R. W. Dickey, Nonlinear bending of circular plates, SIAM J. Appl. Math. 30, 1–9 (1976)
R. W. Dickey Minimum energy solution for the spherical shell, Quart. Appl. Math. 48, 321–339 (1990)
E. Isaacson and H. B. Keller, Analysis of numerical methods, Wiley, New York, 1966
B. J. Struik, Lectures on classical differential geometry, Addison-Wesley, Reading, MA, 1961
R. Courant and D. Hilbert, Methods of mathematical physics, vol. 1, Interscience, New York, 1953
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
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© Copyright 1993
American Mathematical Society
