Second-order analysis for thin structures.
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.
scientific article; zbMATH DE number 2056770