Analysis of functionally graded plates using an edge-based smoothed finite element method H. Nguyen-Xuan a,b,⇑ , Loc V. Tran b , T. Nguyen-Thoi a,b , H.C. Vu-Do a a Department of Mechanics, Faculty of Mathematics & Computer Science, University of Science Ho Chi Minh City, Viet Nam b Division of Computational Mechanics, Ton Duc Thang University Ho Chi Minh City, Viet Nam article info Article history: Available online 30 April 2011 Keywords: FGM Reissner/Mindlin plates Mesh-free methods Finite element method (FEM) Edge-based smoothed finite element method (ES-FEM) Discrete shear gap (DSG) method Stabilization technique abstract An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap (DSG) tech- nique using triangular meshes (ES-DSG) was recently proposed to enhance the accuracy of the existing FEM with the DSG for analysis of isotropic Reissner/Mindlin plates. In this paper, the ES-DSG is further formulated for static, free vibration and buckling analyses of functionally graded material (FGM) plates. The thermal and mechanical properties of FGM plates are assumed to vary across the thickness of the plate by a simple power rule of the volume fractions of the constituents. In the ES-DSG, the stiffness matrices are obtained by using the strain smoothing technique over the smoothing domains associated with the edges of the elements. The present formulation uses only linear approximations and its imple- mentation into finite element programs is quite simple. Several numerical examples are given to demon- strate the performance of the present formulation for FGM plates. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Functionally graded materials (FGM) were first discovered by a group of scientist in Sendai, Japan in 1984 and then developed rap- idly over the world [1–8]. Functionally graded materials are often formed by two or more materials whose volume fractions are changing continuously along certain dimensions of the structure. The volume fractions of FGM plates are derived from a function of position through their thickness. It is well known that function- ally graded materials are capable of resisting high temperature environments or extremely large temperature gradients and there- fore they are more suitable to use in aerospace structure applica- tions and nuclear plants. With the advantageous features of FGMs in many practical applications and the limitations of analytical methods [9–11], practical and effective numerical methods have been devised to analyze and simulate the behavior of FGMs in the structural com- ponents. Among these numerical approaches, the earliest ones were the layered element methods [7] working well for the FGM plates. The finite element method (FEM) was also used later by many authors [12–24,76]. Nowadays the FEM has become the most powerful and reliable tool for analysis of the FGM plates. Recently, mesh-free methods have already been developed for analyzing FGM plates [25–30] and specially, a survey on the developments of meshless methods for laminated and functionally graded plates and shells has been very recently reviewed in [31]. One of major advantages of meshless approaches compared to FEM is capable of handling large deformations, nonlinear problems with moving discontinuities [32–35]. However, the computational cost of mesh- less methods is still quite expensive. In the effort to the development of advanced numerical meth- ods, Liu et al. have combined the strain smoothing technique, which was originally proposed by Chen et al. [36] to stabilize a di- rect nodal integration in mesh-free methods (or SCNI), into the FEM to formulate a family of smoothed FEM (S-FEM) models with different applications such as a cell/element-based smoothed FEM (SFEM or CS-FEM) [37], a node-based smoothed FEM (NS-FEM) [47], an edge-based smoothed FEM (ES-FEM) [51] and a face-based smoothed FEM (FS-FEM) [57]. Each of these smoothed FEM has dif- ferent characters and properties, and has been used successfully in solid mechanics. Similar to the standard FEM, these S-FEM models also use a mesh of elements. In these S-FEM models, the discrete weak form is evaluated using smoothed strains over smoothing domains instead of using compatible strains over the elements as in the traditional FEM. The smoothed strains are computed by inte- grating the weighted (smoothed) compatible strains. The smooth- ing domains are created based on the features of the element mesh such as nodes [47], or edges [51] or faces [57]. These smoothing do- mains are linear independent and hence stability and convergence of the S-FEM models are ensured. They cover parts of adjacent 0263-8223/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2011.04.028 ⇑ Corresponding author at: Department of Mechanics, Faculty of Mathematics & Computer Science, University of Science Ho Chi Minh City, Viet Nam. Tel.: +84 8 22137144. E-mail address: nxhung@hcmuns.edu.vn (H. Nguyen-Xuan). URL: http://www.math.hcmuns.edu.vn/~nxhung/ (H. Nguyen-Xuan). Composite Structures 93 (2011) 3019–3039 Contents lists available at ScienceDirect Composite Structures journal homepage: www.elsevier.com/locate/compstruct