Keywords: multilayered plates, FEM, mixed formulation, stress treatment, layer-wise theories, Reissner's Mixed Variational Theorem.

The subject of the present work consists of multilayered finite elements that are able to furnish an accurate description of strain/stress fields in multilayer flat structure analysis (for a complete historical review of multilayered structures see [1]). The formulation of the finite elements is based upon Reissner's Mixed Variational Theorem (see [2] and [3]), which allows one to assume two independent fields for displacements and transverse stress variables. The resulting advanced finite element can describe, a priori, the interlaminar continuous transverse shear and normal stress fields, and the so called -requirements can be satisfied. An indicial notation that leads to the writing of all matrices in terms of a few arrays is used. As a fundamental property of such indicial notation, all finite element matrices are written in terms of the fundamental nuclei, which have the dimension of . For more details see [4] and [5].
This paper is mainly concerned about the treatment of stress variables in the mixed formulation. In particular, two layer-wise finite elements are compared. The first finite element examined, called LMN (see [4] and [5]), uses a Layer-wise Mixed formulation and the displacements and transverse stresses are expanded along the thickness of a generic layer using a Legendre polynomial of N degree. In the assembling process, the transverse stress variables are eliminated at element level (static-condensation technique) after the generation of the element multilayered matrices. In the second finite element, called LMNF, the approach taken in the first finite element (LMN) is used, except that the static-condensation technique is not applied, and both displacement and stress variables appear as problem unknowns. This last approach guarantees that the transverse stresses between two adjacent elements are continuous functions (this does not happen if the static-condensation method is used). This paper demonstrates that the first finite element (LMN) furnishes very good results (even for thick plates) and that the transverse stresses (when calculated a priori) are not continuous functions over the plate surface, as expected. The full mixed case does not show this problem and fulfills the -requirements a priori (for example see figure 1 for the square plate analyzed in [6]). Therefore, it can be concluded that LMNF is a powerful 2-dimensional tool to analyze very thick multilayered plates.

1

Carrera E., "Historical Review of Zig-Zag Theories for Multilayered Plates and Shells", Applied Mechanics Reviews Vol. 56, No 3, May 2003. doi:10.1115/1.1557614

2

Reissner E., "On a Certain Mixed Variational Theory and a Proposed Applications", International Journal for Numerical Methods in Engineering Vol. 20, 1366-1368, 1984. doi:10.1002/nme.1620200714

3

Reissner E., "On a Mixed Variational Theorem and on a Shear Deformable Plate Theory", International Journal for Numerical Methods in Engineering Vol. 20, 193-198, 1986. doi:10.1002/nme.1620230203

4

Carrera E., Demasi L., "Classical and Advanced Multilayered Plate Elements Based Upon PVD and RMVT. Part 1: Derivation of Finite Element Matrices", International Journal for Numerical Methods in Engineering Vol. 55, 191-231, 2002. doi:10.1002/nme.492

5

Carrera E., Demasi L., "Classical and Advanced Multilayered Plate Elements Based Upon PVD and RMVT. Part 2: Numerical Implementation", International Journal for Numerical Methods in Engineering Vol. 55, 253-291, 2002. doi:10.1002/nme.493

6

Pagano N. J., Hatfield S. J., "Elastic Behaviour of Multilayered Bidirectional Composites", American Institute of Aeronautics and Astronautics Journal Vol. 10, 931-933, 1972. doi:10.2514/3.50249

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