Thin elastic plates supported over small areas. I. Korn's inequalities and boundary layers

Abstract

A thin anisotropic elastic plate clamped along its lateral side and also supported at a small area ${\theta }_h$ of one base is considered; the diameter of ${\theta }_h$ is of the same order as the plate relative thickness $h\ll 1$. In addition to the standard Kirchhoff model with the Sobolev point condition, a three-dimensional boundary layer is investigated in the vicinity of the support ${\theta }_h$, which with the help of the derived weighted inequality of Korn's type, will provide an error estimate with the bound $c{h}^{1/2}|\ln h|$. Ignoring this boundary layer effect reduces the precision order down to $|\ln h{|}^{-1/2}$.

35 pages, 1 figure