Circularly symmetric deformation of shallow elastic membrane caps

Author:
Kurt N. Johnson

Journal:
Quart. Appl. Math. **55** (1997), 537-550

MSC:
Primary 73K10; Secondary 34B15

DOI:
https://doi.org/10.1090/qam/1466147

MathSciNet review:
MR1466147

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider shallow elastic membrane caps that are rotationally symmetric in their undeformed state, and investigate their deformation under small uniform vertical pressure and a given boundary stress or boundary displacement. To do this we use the small-strain theory developed by Bromberg and Stoker, Reissner, and Dickey. We deal with the two-parameter family of membranes whose undeformed configuration is given in cylindrical coordinates as

**[1]**J. V. Baxley,*A singular nonlinear boundary value problem: Membrane response of a spherical cap*, SIAM J. Appl. Math.**48**, 497-585 (1988) MR**941097****[2]**E. Bromberg and J. J. Stoker,*Non-linear theory of curved elastic sheets*, Quart. Appl. Math.**3**, 246-265 (1945/46) MR**0013355****[3]**A. J. Callegari and E. L. Reiss,*Non-linear boundary value problems for the circular membrane*, Arch. Rat. Mech. Anal.**31**, 390-400 (1968) MR**0233538****[4]**A. J. Callegari, H. B. Keller, and E. L. Reiss,*Membrane buckling: a study of solution multiplicity*, Comm. Pure and Appl. Math.**24**, 499-527 (1971) MR**0290638****[5]**R. W. Dickey,*Membrane caps*, Quart. Appl. Math.**45**, 697-712 (1987);*Erratum*, Quart. Appl. Math.**46**, 192 (1988) MR**917020****[6]**R. W. Dickey,*Membrane caps under hydrostatic pressure*, Quart. Appl. Math.**46**, 95-104 (1988) MR**934684****[7]**R. W. Dickey,*Rotationally symmetric solutions for shallow membrane caps*, Quart. Appl. Math.**47**, 571-581 (1989) MR**1012280****[8]**R. W. Dickey,*The plane circular elastic surface under normal pressure*, Arch. Rat. Mech. Anal.**26**, 219-236 (1967) MR**1553496****[9]**A. Föppl,*Vorlesungen Über Technische Mechanik*, Teubner, Leipzig, 1907**[10]**M. A. Goldberg,*An iterative solution for rotationally symmetric non-linear membrane problems*, Internat. J. Non-linear Mech.**1**, 169-178 (1966)**[11]**H. Hencky,*Über den Spannungszustand in kreisrunden Platten*, Z. Math. Phys.**63**, 311-317 (1915)**[12]**K. N. Johnson,*Circularly Symmetric Deformations of Shallow Elastic Membrane Caps*, Ph.D. Thesis, University of Wisconsin-Madison, 1994 MR**2691744****[13]**H. B. Keller,*Numerical Solution of Two Point Boundary Value Problems*, SIAM, Philadelphia, 1976 MR**0433897****[14]**M. H. Protter and H. F. Weinberger,*Maximum Principles in Differential Equations*, Prentice-Hall, Englewood Cliffs, New Jersey, 1967 MR**0219861****[15]**E. Reissner,*Rotationally symmetric problems in the theory of thin elastic shells*, 3rd U.S. Natl. Congress of Applied Mechanics, 1958 MR**0101672****[16]**H. J. Weinitschke,*On finite displacements of circular elastic membranes*, Math. Meth. in the Appl. Sci.**9**, 76-98 (1987) MR**881554**

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Additional Information

DOI:
https://doi.org/10.1090/qam/1466147

Article copyright:
© Copyright 1997
American Mathematical Society