The nonlinear circular membrane under a vertical force

Dickey, R. W.

doi:10.1090/qam/721423

Author: R. W. Dickey
Journal: Quart. Appl. Math. 41 (1983), 331-338
MSC: Primary 73K15
DOI: https://doi.org/10.1090/qam/721423
MathSciNet review: 721423
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Abstract: The exact theory for the deformation of a plane circular membrane under a vertical force is derived. It is shown that the system of equations can be reduced to a single, nonlinear, ordinary differential equation. In addition it is shown that the Foppl approximation is the first term in an asymptotic expansion of the exact theory.

A. Foppl, Vorlesungen über technische Mechanik, Bd. 5, P. Teubner, Leipzig, 1907

H. Hencky, Über der 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
Andrew J. Callegari and Edward L. Reiss, Non-linear boundary value problems for the circular membrane, Arch. Rational Mech. Anal. 31 (1968/69), 390–400. MR 233538, DOI https://doi.org/10.1007/BF00251421
Andrew J. Callegari, Edward L. Reiss, and Herbert B. Keller, Membrane buckling: A study of solution multiplicity, Comm. Pure Appl. Math. 24 (1971), 499–527. MR 290638, DOI https://doi.org/10.1002/cpa.3160240405