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

The purpose of this paper is to determine Gphom, 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. Modélisation numérique des panneaux structuraux légers. 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. Méc. 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 ∂Gphom is given thanks to the ππ-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 ∂Gphom. For a laminated plate described with a yield function of the form F(x3,σ)=σu(x3)F^(σ), where σuσu is a positive even function of the out-of-plane coordinate x3x3 and F^ is a convex function of the local stress σσ, two effective constants and a normalization procedure are introduced. A symmetric sandwich plate consisting of two Von-Mises materials (σu=σ1u in the skins and σu=σ2u in the core) is studied. It is found that, for small enough contrast ratios (r=σ1u/σ2u≤5), the normalized strength domain G^phom is close to the one corresponding to a homogeneous Von-Mises plate [Ilyushin, A.-A., 1956. Plasticité. Eyrolles, Paris].