Membrane caps
Author:
R. W. Dickey
Journal:
Quart. Appl. Math. 45 (1987), 697-712
MSC:
Primary 73K15
DOI:
https://doi.org/10.1090/qam/917020
MathSciNet review:
917020
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Abstract: The exact theory for the rotationally symmetric deformation of a membrane cap under either a gravitational force or a normal force is derived. It is shown that in either case the exact theory can be reduced to a single, second-order, ordinary differential equation for the determination of a quantity related to the radial stress. This equation is specialized to the small strain case. Approximate theories are obtained from the exact equations. In particular, the correct approximate equations are obtained when the applied pressure is small and when the membrane cap is shallow. The shallow spherical cap is treated as a special example.
- 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
M. A. Goldberg, An iterative solution for rotationally symmetric non-linear membrane problems, Int. J. Non-linear Mechs. 1 169–178 (1966)
- R. W. Dickey, The nonlinear circular membrane under a vertical force, Quart. Appl. Math. 41 (1983/84), no. 3, 331–338. MR 721423, DOI https://doi.org/10.1090/S0033-569X-1983-0721423-8
A. Föppl, Vorlesungen über technische Mechanik, Bd. 5, Leipzig: G. Teubner, 1907
- Eric Reissner, On axisymmetrical deformations of thin shells of revolution, Proc. Symposia Appl. Math. v. 3, McGraw-Hill Book Co., New York, N. Y., 1950, pp. 27–52. MR 0039489
- H. J. Weinitschke, On axisymmetric deformations of nonlinear elastic membranes, Mechanics today, Vol. 5, Pergamon, Oxford-New York, 1980, pp. 523–542. MR 591521
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
- 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
H. J. Weinitschke, On finite displacement of circular elastic membranes, Institute of Appl. Math. and Statistics, Univ. British Columbia, Vancouver, Canada, Tech. Rep. No. 85–7, 1985
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. Natn. Congr. of Applied Mechanics, 59–69, 1958
M. A. Goldberg, An iterative solution for rotationally symmetric non-linear membrane problems, Int. J. Non-linear Mechs. 1 169–178 (1966)
R. W. Dickey, The nonlinear circular membrane under a vertical force, Quart. Appl. Math. 41, 331–338 (1983)
A. Föppl, Vorlesungen über technische Mechanik, Bd. 5, Leipzig: G. Teubner, 1907
E. Reissner, On axisymmetric deformation of thin shells of revolution, Proc. Symp. Appl. Math. 3, 27–52 (1950)
H. J. Weinitschke, On axisymmetric deformation of nonlinear elastic membranes, Mechanics Today 5, 523–542. Ed. S. Nemat-Nasser, Oxford and New York: Pergamon Press (1980)
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, Non-linear boundary value problems for the circular membrane, Arch. Rat. Mech. Anal. 31, 390–400 (1968)
H. J. Weinitschke, On finite displacement of circular elastic membranes, Institute of Appl. Math. and Statistics, Univ. British Columbia, Vancouver, Canada, Tech. Rep. No. 85–7, 1985
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Article copyright:
© Copyright 1987
American Mathematical Society