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Journal of Convex Analysis 24 (2017), No. 3, 819--855
Copyright Heldermann Verlag 2017

Thin Elastic Plates Supported over Small Areas. II: Variational-Asymptotic Models

Giuseppe Buttazzo
Dip. di Matematica, UniversitÓ di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy

Giuseppe Cardone
Dept. of Engineering, UniversitÓ del Sannio, Corso Garibaldi 107, 82100 Benevento, Italy

Sergei A. Nazarov
Mathematics and Mechanics Faculty, St. Petersburg State University, Universitetskaya nab. 7-9, St. Petersburg 199034, Russia


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 $h\ll 1$. 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 $ch^{1/2}\left\vert \ln h \right\vert$. Ignoring this boundary layer effect reduces the precision order down to $\left\vert \ln h\right\vert ^{-1/2}$. 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 $h^{1/2}\left\vert \ln h\right\vert$ in certain norms. Several formulations and applications of the model are discussed.

Keywords: Kirchhoff plate, small support zone, asymptotic analysis, self-adjoint extensions, variational model.

MSC: 74K20, 74B05

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