Singular perturbations and the transition from thin plate to membrane

Abstract:

The equation \[ \frac {{E{h^2}}}{{12(1 - {\sigma ^2})}}{\Delta ^2}w - \sum \limits _{ij = 1}^2 {\frac {\partial }{{\partial {x_i}}}\left ( {{\sigma _{ij}}\frac {{\partial w}}{{\partial {x_j}}}} \right ) = f} \] 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 \[ - \sum \limits _{ij = 1}^2 {\frac {\partial }{{\partial {x_i}}}\left ( {{\sigma _{ij}}\frac {{\partial u}}{{\partial {x_j}}}} \right )} = f\] with the same boundary values as $w$ approximates the displacement throughout the plate in the ${L^2}$ sense. Herein, the rate \[ ( \ast )\quad \iint \limits _\Omega {|w(x) - u(x){|^2}dx \in C{h^2}}\iint \limits _\Omega {|f(x){|^2}}dx\] 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 ${h^2}$.