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3D-2D asymptotic analysis for inhomogeneous thin films. (English) Zbl 0987.35020
The authors present in the context of fully nonlinear elasticity, a general approach that allows for material heterogeneity as well as rapidly varying profiles. A dimension reduction analysis is undertaken using \(\Gamma\)-convergence techniques within a relaxation theory for 3D nonlinear elastic thin domains of the form \[ \Omega_\varepsilon =\bigl\{ (x_1, x_2, x_3)\mid (x_1,x_2) \in\omega,\;|x_3|<\varepsilon f_\varepsilon (x_1,x_2) \bigr\}, \] where \(\omega\) is a bounded domain of \(\mathbb{R}^2\) and \(f_\varepsilon\) is an \(\varepsilon\)-dependent profile. Moreover 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.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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