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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
buttazzo@dm.unipi.it

Giuseppe Cardone
Dept. of Engineering, UniversitÓ del Sannio, Corso Garibaldi 107, 82100 Benevento, Italy
giuseppe.cardone@unisannio.it

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



[Abstract-pdf]

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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