Calculus of Variations and Geometric Measure Theory

G. Buttazzo - G. Cardone - S. Nazarov

Thin elastic plates supported over small areas II. Variational-asymptotic models.

created by buttazzo on 24 Jan 2017


Submitted Paper

Inserted: 24 jan 2017
Last Updated: 24 jan 2017

Year: 2017


An asymptotic analysis is performed for thin anisotropic elastic plate clamped along its lateral side and also supported at a small area $\theta_h$ of one base with diameter of the same order as the plate thickness $h\ll1.$ A three-dimensional boundary layer in the vicinity of the support $\theta_h$ is involved into the asymptotic form which is justified by means of the previously derived weighted inequality of Korn's type provides an error estimate with the bound $ch^{1/2}
\ln h
$. Ignoring this boundary layer effect reduces the precision order down to $
\ln h
^{-1/2}$. A two-dimensional variational-asymptotic model of the plate is proposed within the theory of self-adjoint extensions of differential operators. The only characteristics of the boundary layer, namely the elastic logarithmic potential matrix of size $4\times4$, is involved into the model which however keeps the precision order $h^{1/2}
\ln h
$ in certain norms. Several formulations and applications of the model are discussed.

Keywords: variational model, Asymptotic Analysis, Kirchhoff plate, small support zone, self-adjoint extensions