Second-order analysis for thin structures.
type: Item
instance of: scholarly article
title: Second-order analysis for thin structures.
zbMATH Open document ID: 1044.49014
author: Pedro Miguel Santos
author: Elvira Zappale
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-03-14T00:00:00Z
review text: The purpose of the paper is to study thin films by means of dimensional reduction through asymptotic analysis from \(3D\) models to \(2D\) ones. The models taken into account have nonconvex bulk energies involving second-order derivatives of the deformations, and generalize those presented in [\textit{K. Bhattacharya} and \textit{R. D. James}, J. Mech. Phys. Solids 47, 531--576 (1999; Zbl 0960.74046)] by considering more general interfacial energies. By using \(\Gamma\)-convergence techniques, a general result for \(2D\) models is first established for energy densities allowing material heterogeneities and varying profiles. Then, the study is specialized by considering transversally inhomogeneous thin domains, and deriving the homogeneous model from those. Finally, a model is also considered in which microstructure and profile oscillate on a scale which is comparable to the thickness of the domain. An integral representation result for the limit energy is provided under quite general assumptions on the initial density. The limit energy is determined by two vector fields \(u\) and \(b\) defined on a plane sheet, where \(u\) is associated to the deformation of the middle surface and \(b\) is the Cosserat vector associated with transverse shear and normal compression, and which keeps memory of the rotation of the original normal vector to the section \(\omega\) in the \(3D\) approximating thick film. Since the limit model is not convex, and takes into account both \(u\) and \(b\), it requires a more general notion of convexity, the \(\mathcal A\)-quasiconvexity, for a suitable operator \(\mathcal A\). The paper concludes with an Appendix containing an integral representation result relying on the so-called global method techniques.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 74K35
Mathematics Subject Classification ID: 74B20
zbMATH DE Number: 2056770
zbMATH Keywords: dimension reduction
zbMATH Keywords: thin films
zbMATH Keywords: \(\mathcal A\)-quasiconvexity
zbMATH Keywords: \(\Gamma\)-convergence techniques
zbMATH Keywords: relaxation
zbMATH Keywords: Lipschitz continuation
MaRDI profile type: Publication
full work available at URL: j.na.2003.10.007
OpenAlex ID: W2017789591
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DOI: 10.1016/J.NA.2003.10.007
description: scientific article; zbMATH DE number 2056770