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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Singular perturbations and the transition from thin plate to membrane


Author: Zeev Schuss
Journal: Proc. Amer. Math. Soc. 58 (1976), 139-147
MSC: Primary 35B25
MathSciNet review: 0412571
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Abstract | References | Similar Articles | Additional Information

Abstract: The equation

$\displaystyle \frac{{E{h^2}}}{{12(1 - {\sigma ^2})}}{\Delta ^2}w - \sum\limits_... ..._i}}}\left( {{\sigma _{ij}}\frac{{\partial w}}{{\partial {x_j}}}} \right) = f} $

describing the normal displacement $ w$ of a thin elastic plate of thickness $ h$ under uniform tension in equilibrium is considered. It is shown that if the displacement and its normal derivative on the edge of the plate are bounded uniformly with respect to $ h$ then the solution $ u$ of the membrane equation

$\displaystyle - \sum\limits_{ij = 1}^2 {\frac{\partial }{{\partial {x_i}}}\left( {{\sigma _{ij}}\frac{{\partial u}}{{\partial {x_j}}}} \right)} = f$

with the same boundary values as $ w$ approximates the displacement throughout the plate in the $ {L^2}$ sense. Herein, the rate

$\displaystyle ( \ast )\quad \iint\limits_\Omega {\vert w(x) - u(x){\vert^2}dx \in C{h^2}}\iint\limits_\Omega {\vert f(x){\vert^2}}dx$

is given, where $ C$ is a constant independent of $ h$ and $ f$, and $ \Omega $ in the face of the plate. This extends the results of A. Friedman [6] and F. John [10] up to the boundary and improves the rate of convergence in ($ ( \ast )$) given by J. L. Lions [12] and W. M. Greenlee [7] from $ h$ to $ {h^2}$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0412571-6
PII: S 0002-9939(1976)0412571-6
Keywords: Singular perturbations, higher order elliptic boundary value problems, thin plates and membranes
Article copyright: © Copyright 1976 American Mathematical Society