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Publication 99-CNA-15 3D-2D Asymptotic Analysis for Inhomogeneous Thin Films

Andrea Braides
S.I.S.S.A, 34014 Trieste, Italy
email: A. Braides

Irene Fonseca
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213, USA
email: fonseca@andrew.cmu.edu


Gilles Francfort
L.P.M.T.M.
Universite Paris-Nord
93430 Villetaneuse, France
email: G. Francfort

Abstract: A dimension reduction analysis is undertaken using $\Gamma$-convergence techniques within a relaxation theory for 3D nonlinear elastic thin domains of the form

\begin{displaymath}\Omega_{\varepsilon} := \{(x_1,x_2,x_3): (x_1,x_2) \in \omega, \vert x_3\vert < \varepsilon f_\varepsilon(x_1,x_2)\}, \end{displaymath}

where $\omega$ is a bounded domain of ${{\mathbb R}}^2$ and $f_\varepsilon$ is an $\varepsilon$-dependent profile. An abstract representation of the effective 2D energy is obtained, and specific characterizations are found for nonhomogeneous plate models, periodic profiles, and within the context of optimal design for thin films.

Keywords : dimension reduction, $\Gamma$-convergence, plate models,
periodicity, relaxation.

1998 Mathematics Subject Classification: 35E99, 35M10, 49J45, 74B20, 74K15, 74K20, 74K35.

 

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