Keywords: unified formulation, principle of virtual displacements, functionally graded materials, closed form solutions, finite element method, equivalent single layer models, layer wise models.

This paper illustrates the extension of Unified Formulation (UF) by Carrera [1,2] to Functionally Graded Materials (FGM) plates. The Principle of Virtual Displacements is applied to derive the governing equations. Analytical and FE solutions are compared with three dimensional ones [3] to validate the proposed models, different theories are reported in form of table. The results concern the static analysis of a simply supported FGM plate subjected to a transverse mechanical load. UF permits to implement both Layer Wise and Equivalent Single Layer models, and the order of expansion for displacements ranges from 1 up to 4. The results, are improved respect to classical theories such as Classical Lamination Theory, First order Shear Deformation Theory and High order Shear Deformation Theory, where the transverse displacement is assumed constant, and so normal stress-strain are zero [4]. In particular high order of expansion are requested for thick plates.

In literature different laws through the thickness are possible for the elastic properties in Functionally Graded Materials: in general Young's modulus, Poisson's ratio and mass density, depending on volume fraction of the constituent materials, have a variation through the thickness that is exponential or in polynomail functions. So via UF and Legendre polynomials (a linear combination of them) is possible a general methodology to describe all sort of evaluations through the z direction, without need of the discrete layer methodology. The use of Legendre polynomials permits to approximate the thickness law with order of expansions that goes from linear up to 9th order. The good agreement with 3D solutions and the improvements respect to classical theories suggest future implementations, such as the extension to Reissner's Mixed Variational Theorem (that will permit to obtain a priori the transverse and normal stresses from the model), and to shells and multifields problems.

1

E. Carrera, "A Class of Two Dimensional Theories for Multilayered Plates Analysis", Atti Accademia delle Scienze di Torino, Mem Sci. Fis., 19-20, 49-87, 1995.

2

E. Carrera, "Theories and finite elements for multilayered anisotropic, composite plates and shells", Archives of Computational Method in Engineering, State of the Art Reviews, 9, 87-140, 2002. doi:10.1007/BF02736649

3

M. Kashtalyan, "Three-dimensional elasticity solution for bending of functionally graded rectangular plates", European Journal of Mechanics A/Solids, 23, 853-864, 2004. doi:10.1016/j.euromechsol.2004.04.002

4

A.M. Zenkour, "Generalized shear deformation theory for bending analysis of functionally graded plates", Applied Mathematical Modelling, 30, 67-84, 2006. doi:10.1016/j.apm.2005.03.009

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