Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



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

$\displaystyle z\left( x \right) = C\left( {1 - {x^\gamma }} \right), \qquad \left( 1 \right)$

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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DOI: https://doi.org/10.1090/qam/1466147
Article copyright: © Copyright 1997 American Mathematical Society

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