The heterogeneous three-dimensional (3D) structure is then replaced by an orthotropic two dimensional (2D) layered plate structure, on which it will be possible to perform calculations with reduced computing time. This transformation is enabled by using the homogenisation method of periodic media based on multi-scale asymptotic expansions [1,2]. A previous study [3] allowed us to determine the overall effective behaviour of the equivalent plate by solving a set of six problems posed on a Representative Elementary Volume R.E.V. These problems correspond to the six elementary loadings constituting a basis for the generalized stresses and . Hence, we obtain for each component of and the 3D stress distribution field on the R.E.V. The results for an industrial sandwich panel obtained with the orthotropic layered plate model, meshed by DKT finite elements, are very close to those given by the experimental tests. This proves the validity of the homogenisation method used and the equivalent plate model proposed.

The localisation of the 3D stress solution for the sandwich structure is determined by combining the results of the equivalent plate model and those of the homogenisation procedure [4,5]. The 3D displacement solution is obtained from the displacement of the equivalent plate by using the Kirchhoff-Love assumptions. The comparison between the 3D finite element solution considered as a reference and the solution obtained by homogenisation is performed by defining two error indicators in terms of stress and displacement. As examples, we have considered a sandwich structure in the following cases: pure traction and pure bending. The quality of this reconstituted solution is confirmed by the low values of the error for each loading.

Due to their excellent mechanical performances, the sandwich shells are excellent candidates in many domains of industry. To model the equivalent behaviour of these structures, we have applied the homogenisation method of periodic media which has already been implemented for the sandwich plates. The equivalent model is an orthotropic layered shell made up of the superposition of three shells; it exactly reproduces the homogenised behaviour. The quality of the results given by this model is estimated by simulating several mechanical tests. The error indicators in terms of stress and displacement are then calculated.

1

D. Caillerie, Thin elastic and periodic plates, Math. Meth. Appl. Sci. 6, 159-191, 1984. doi:10.1002/mma.1670060112

2

E. Sanchez Palencia, In: Homogenization Techniques for Composites Media, Lecture Note in Physics, 272, 122-189, 1985, Springer, Berlin. doi:10.1007/3-540-17616-0

3

A. Saidi, M. Guessasma, P. Coorevits, Two-dimensional equivalent model for the analysis of sandwich structures: (application to reinforced double thin sheets). Second International Conference on Advanced Computational Methods in Engineering, Liège (Belgium), 2002.

4

A. Saidi, P. Coorevits, H. Jaffal, Solution 3D par homogénéisation d'une structure sandwich et retour à l'échelle macroscopique, 6ème Colloque National en Calcul de Structures, Giens, 2003.

5

A. Saidi, P. Coorevits, Détermination de la solution 3D d'une structure sandwich par homogénéisation, 16ème Congrès Français de Mécanique, Nice, 2003.

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