Limit analysis of multi-layered plates. Part I: The homogenized Love-Kirchhoff model

摘要

The purpose of this paper is to determine G_p~(hom), the overall homogenized Love-Kirchhoff strength domain of a rigid perfectly plastic multi-layered plate, and to study the relationship between the 3D and the homogenized Love-Kirchhoff plate limit analysis problems. In the Love-Kirchhoff model, the generalized stresses are the in-plane (membrane) and the out-of-plane (flexural) stress field resultants. The homogenization method proposed by Bourgeois [1997. Modelisation numerique des panneaux structuraux legers. Ph.D. Thesis, University Aix-Marseille] and Sab [2003. Yield design of thin periodic plates by a homogenization technique and an application to masonry wall. C. R. Mec. 331, 641-646] for in-plane periodic rigid perfectly plastic plates is justified using the asymptotic expansion method. For laminated plates, an explicit parametric representation of the yield surface partial derivG_p~(hom) is given thanks to the π-function (the plastic dissipation power density function) that describes the local strength domain at each point of the plate. This representation also provides a localization method for the determination of the 3D stress components corresponding to every generalized stress belonging to partial derivG_p~(hom). For a laminated plate described with a yield function of the form F(x_3, σ) = σ~u(x_3)F(σ), where σ~u is a positive even function of the out-of-plane coordinate x_3 and F is a convex function of the local stress σ, two effective constants and a normalization procedure are introduced. A symmetric sandwich plate consisting of two Von-Mises materials (σ~u = σ_1~u in the skins and σ~u = σ_2~u in the core) is studied. It is found that, for small enough contrast ratios (r = σ_1~u/σ_2~u ≤ 5), the normalized strength domain G_p~(hom) is close to the one corresponding to a homogeneous Von-Mises plate [Ilyushin, A.-A., 1956. Plasticite. Eyrolles, Paris].