Available online at www.sciencedirect.com ScienceDirect Comput. Methods Appl. Mech. Engrg. 278 (2014) 388–403 www.elsevier.com/locate/cma Discrete double directors shell element for the functionally graded material shell structures analysis M. Wali ∗ , A. Hajlaoui, F. Dammak Mechanical Modelization and Manufacturing Laboratory (LA2MP), National Engineering School of Sfax, B.P W3038, Sfax, University of Sfax, Tunisia Received 24 December 2013; received in revised form 28 March 2014; accepted 7 May 2014 Available online 11 June 2014 Abstract In this paper, the accuracy and the efficiency of the 3d-shell model based on a double directors shell element for the functionally graded material (FGM) shell structures analysis is studied. The vanishing of transverse shear strains on top and bottom faces is considered in a discrete form. Thus, the third-order shear deformation plate theory (TSDT) is a particular case of the discrete double directors shell model (DDDSM) used in the present work. The DDDSM is introduced to remove the shear correction factors, when using the first-order shear deformation theory (FSDT), and improve an excellent performance when compared with other works. This model can be used for static, free vibration and buckling analyses of FGM. The convergence of the proposed model is compared to other well-known formulations found in the literature. c ⃝ 2014 Elsevier B.V. All rights reserved. Keywords: FGM; Shell element; Third-order deformation theory 1. Introduction In recent years, shell structures made of FGMs are widely used in many engineering fields such as aerospace, gas turbines, nuclear fusions, electronics, etc. because they present many advantages. Indeed, they are used in systems which need high heat-resistance, high rigidity and eventually absence of the interface problem unlike laminate structures. The material properties of FGMs are inhomogeneous and vary continuously in one or more directions. Typical FGMs are made from a mixture of ceramic and metal, or a combination of different metals or different ceramics that are appropriate to achieve the desired objective. The importance of this kind of materials motivates many contemporary researchers to study their properties and behaviors. Among these studies, we mention works of Vel and Batra [1] who presented the three dimensional exact solution for free and forced vibrations of functionally graded rectangular plates, Ferreira et al. [2] who studied the static deformations of functionally graded plates using the radial basis function collocation method and a higher-order shear deformation theory, they selected the shape parameter in the radial basis functions by an optimization procedure based on the cross validation technique. Matsunaga [3] calculated the natural frequencies and buckling stresses of plates made of FGMs using a 2-D higher- order deformation theory. Carrera et al. [4] evaluated the effect of thickness stretching in plate/shell structures made by FGM in the thickness directions. Neves et al. [5] presented a quasi-3d hyperbolic shear deformation theory for the ∗ Corresponding author. Tel.: +216 20385960; fax: +216 74666535. E-mail address: mondherwali@yahoo.fr (M. Wali). http://dx.doi.org/10.1016/j.cma.2014.05.011 0045-7825/ c ⃝ 2014 Elsevier B.V. All rights reserved.