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: 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 \[ z\left ( x \right ) = C\left ( {1 - {x^\gamma }} \right ), \qquad \left ( 1 \right )\] which includes the spherical cap as a special case ($\gamma = 2$ and $C$ small). We show that if $\gamma > 4/3$ then a circularly symmetric deformation is possible for any positive boundary stress (or any boundary displacement) and any positive pressure, but if $1 < \gamma < 4/3$ then no circularly symmetric deformation is possible if the stress and pressure are positive and small (or for non-positive boundary displacement and small positive pressure).
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E. Bromberg and J. J. Stoker, Non-linear theory of curved elastic sheets, Quart. Appl. Math. 3, 246–265 (1945/46)
A. J. Callegari and E. L. Reiss, Non-linear boundary value problems for the circular membrane, Arch. Rat. 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 and Appl. Math. 24, 499–527 (1971)
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)
R. W. Dickey, Rotationally symmetric solutions for shallow membrane caps, Quart. Appl. Math. 47, 571–581 (1989)
R. W. Dickey, The plane circular elastic surface under normal pressure, Arch. Rat. Mech. Anal. 26, 219–236 (1967)
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H. J. Weinitschke, On finite displacements of circular elastic membranes, Math. Meth. in the Appl. Sci. 9, 76–98 (1987)
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© Copyright 1997
American Mathematical Society