Rotationally symmetric solutions for shallow membrane caps
Author:
R. W. Dickey
Journal:
Quart. Appl. Math. 47 (1989), 571-581
MSC:
Primary 73K15
DOI:
https://doi.org/10.1090/qam/1012280
MathSciNet review:
MR1012280
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Abstract: We discuss the displacements and stresses in a shallow membrane cap with either displacement or stress prescribed on the boundary and subjected to a small applied pressure. It is shown that if the prescribed boundary stress is sufficiently large there is a rotationally symmetric solution and if the applied pressure is large there is a solution to the problem with zero displacement prescribed on the boundary. However, there are situations in which there are no rotationally symmetric solutions. One case is described in which there are no rotationally symmetric solutions unless the prescribed boundary stress or prescribed boundary displacement is large.
- E. Bromberg and J. J. Stoker, Non-linear theory of curved elastic sheets, Quart. Appl. Math. 3 (1945), 246–265. MR 13355, DOI https://doi.org/10.1090/S0033-569X-1945-13355-7
- Eric Reissner, Rotationally symmetric problems in the theory of thin elastic shells., Proceedings of the Third U.S. National Congress of Applied Mechanics, Brown University, Providence, R.I., June 11-14, American Society of Mechanical Engineers, New York, 1958, pp. 51–69. MR 0101672
- R. W. Dickey, Membrane caps, Quart. Appl. Math. 45 (1987), no. 4, 697–712. MR 917020, DOI https://doi.org/10.1090/S0033-569X-1987-0917020-X
- R. W. Dickey, Membrane caps under hydrostatic pressure, Quart. Appl. Math. 46 (1988), no. 1, 95–104. MR 934684, DOI https://doi.org/10.1090/S0033-569X-1988-0934684-8
H. Hencky, Über de 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
- H. J. Weinitschke, On finite displacements of circular elastic membranes, Math. Methods Appl. Sci. 9 (1987), no. 1, 76–98. MR 881554, DOI https://doi.org/10.1002/mma.1670090108
M. A. Goldberg, An iterative solution for rotationally symmetric non-linear membrane problems, Internat J. Non-linear Mech. 1, 169–178 (1966)
- John V. Baxley, A singular nonlinear boundary value problem: membrane response of a spherical cap, SIAM J. Appl. Math. 48 (1988), no. 3, 497–505. MR 941097, DOI https://doi.org/10.1137/0148028
J. T. Schwartz, Non-linear Functional Analysis, Gordon and Breach, New York, 1969
- Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR 0069338
E. Bromberg and J. J. Stoker, Non-linear theory of curved elastic sheets, Quart. Appl. Math. 3, 246–265 (1945/46)
E. Reissner, Rotationally symmetric problems in the theory of thin elastic shells, 3rd U.S. Math. Congress of Applied Mechanics, 51–69, (1958)
R. W. Dickey, Membrane caps, Quart. Appl. Math. 45, 697–712 (1987); Erratum, Quart. Appl. Math. 46, 192 (1988)
R. W. Dickey, Membrane caps under hydrostatic pressure, Quart. Appl. Math. 46, 95–104 (1988)
H. Hencky, Über de 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, 219–236 (1967)
A. J. Callegari and E. L. Reiss, Non-linear boundary value problems for the circular membrane, Arch. Rational Mech. Anal. 31, 390–400 (1968)
A. J. Callegari, H. B. Keller, and E. L. Reiss, Membrane buckling: a study of solution multiplicity, Comm. Pure Appl. Math. 24, 499–527 (1971)
H. J. Weinitschke, On finite displacements of circular elastic membranes, Math. Methods Appl. Sci. 9, 76–98 (1987)
M. A. Goldberg, An iterative solution for rotationally symmetric non-linear membrane problems, Internat J. Non-linear Mech. 1, 169–178 (1966)
J. V. Baxley, A singular nonlinear boundary value problem: Membrane response of a spherical cap, SIAM J. Appl. Math. 48, 497–585 (1988)
J. T. Schwartz, Non-linear Functional Analysis, Gordon and Breach, New York, 1969
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955
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© Copyright 1989
American Mathematical Society