Unsymmetric deformation of the circular membrane
Author:
R. W. Dickey
Journal:
Quart. Appl. Math. 44 (1986), 81-90
MSC:
Primary 73K15
DOI:
https://doi.org/10.1090/qam/840445
MathSciNet review:
840445
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Abstract: If the Föppl equations for the deformation of a plane circular membrane under normal pressure are linearized about the radially symmetric solution, it is shown that the resulting linear theory has infinitely many nontrivial angle-dependent solutions. If the prescribed normal pressure and the prescribed (angle-independent) boundary stress are allowed to approach zero in the appropriate way, these nontrivial solutions are retained. Thus the linear theory indicates that, in addition to the solution with radial symmetry, there are infinitely many angle-dependent solutions for arbitrarily small values of the prescribed pressure and prescribed boundary stress.
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- J. J. Stoker, Nonlinear elasticity, Gordon and Breach Science Publishers, New York-London-Paris, 1968. MR 0413654
H. Hencky, Über den spannungszustand in kreisrunden Platten, Z. Math. Phys. 63, 311–317 (1915)
- R. W. Dickey, The plane circular elastic surface under normal pressure, Arch. Rational Mech. Anal. 26 (1967), no. 3, 219–236. MR 1553496, DOI https://doi.org/10.1007/BF00281971
A. J. Callegari and E. L. Reiss, Nonlinear boundary value problems for the circular membrane, Arch. Rat. Mech. Anal. 31, 390–400 (1970)
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E. B. Van Vleck, A determination of the number of real and imaginary roots of the hypergeometric series, Trans. Amer. Math. Soc. 3, 110–131 (1902)
A. Föppl, Vorlesungen über technische Mechanik, Bd. 5, G. Teubner, Leipzig, 1905
J. J. Stoker, Nonlinear elasticity, Gordon and Breach, New York, 1968
H. Hencky, Über den spannungszustand in kreisrunden Platten, Z. Math. Phys. 63, 311–317 (1915)
R. W. Dickey, The plane circular elastic surface under normal pressure, Arch. Rat. Mech. Anal. 26, 219–236 (1967)
A. J. Callegari and E. L. Reiss, Nonlinear boundary value problems for the circular membrane, Arch. Rat. Mech. Anal. 31, 390–400 (1970)
A. J. Callegari, H. B. Keller and E. L. Reiss, Membrane buckling: A study in solution multiplicity, Comm. Pure Appl. Math. 24, 499–521 (1971)
N. N. Lebedev, Special functions and their applications, Prentice-Hall, Englewood Cliffs, N. J., 1965
E. A. Coddington and N. Levinson, Ordinary differential equations, McGraw-Hill, New York, 1955
F. Klein, Ueber die Nullstellen der hypergeometrischen reihe, Math. Ann. 37, 573–590 (1890)
E. B. Van Vleck, A determination of the number of real and imaginary roots of the hypergeometric series, Trans. Amer. Math. Soc. 3, 110–131 (1902)
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Article copyright:
© Copyright 1986
American Mathematical Society
