“…at is the reason why numerous researchers focus on investigating the dynamic as well as static behaviors of the FGM structures, for example, beams [5][6][7][8][9][10], plates [11][12][13][14][15][16][17][18], shells [19][20][21], and other structures [22,23]. Many plate theories have been established and/or applied to investigate these structures, for example, classical plate theory (CPT), first-order shear deformation (FSDT), higherorder shear deformation theory (HSDT), and quasi-3D shear deformation theory.…”
A new mixed four-node quadrilateral element (MiQ4) is established in this paper to investigate functionally graded material (FGM) plates with variable thickness. The proposed element is developed based on the first-order shear deformation and mixed finite element technique, so the new element does not need any selective or reduced numerical integration. Numerous basic tests have been carried out to demonstrate the accuracy and convergence of the proposed element. Besides, the numerical examples show that the present element is free of shear locking and is insensitive to the mesh distortion, especially for the case of very thin plates. The present element can be applied to analyze plates with arbitrary geometries; it leads to reducing the computation cost. Several parameter studies are performed to show the roles of some parameters such as the power-law index, side-to-thickness ratio, boundary conditions (BCs), and variation of the plate thickness on the static bending behavior of the FGM plates.
“…at is the reason why numerous researchers focus on investigating the dynamic as well as static behaviors of the FGM structures, for example, beams [5][6][7][8][9][10], plates [11][12][13][14][15][16][17][18], shells [19][20][21], and other structures [22,23]. Many plate theories have been established and/or applied to investigate these structures, for example, classical plate theory (CPT), first-order shear deformation (FSDT), higherorder shear deformation theory (HSDT), and quasi-3D shear deformation theory.…”
A new mixed four-node quadrilateral element (MiQ4) is established in this paper to investigate functionally graded material (FGM) plates with variable thickness. The proposed element is developed based on the first-order shear deformation and mixed finite element technique, so the new element does not need any selective or reduced numerical integration. Numerous basic tests have been carried out to demonstrate the accuracy and convergence of the proposed element. Besides, the numerical examples show that the present element is free of shear locking and is insensitive to the mesh distortion, especially for the case of very thin plates. The present element can be applied to analyze plates with arbitrary geometries; it leads to reducing the computation cost. Several parameter studies are performed to show the roles of some parameters such as the power-law index, side-to-thickness ratio, boundary conditions (BCs), and variation of the plate thickness on the static bending behavior of the FGM plates.
“…In addition, the beam is composed by two materials, which are gradually varied along the thickness direction from metal at the bottom surface to ceramic at the top surface. According to a HSDBT, the axial u and the transverse w displacements of the beam can be written as 53–55 Figure 1.Geometry of the beam made of FG materials.…”
This paper investigates the vibrational behavior of beams made of functionally graded materials using a mixed formulation. Unlike the other high order shear deformation theories (HSDTs), the proposed formulation is elaborated within a double field of displacements and stresses which offers the possibility of the development of low order linear elements with enhanced accuracy. As well as, the effect of the transverse shear strains and the zero condition of the transverse shear stresses on the top and bottom surfaces are verified. The material characteristics of the beams are described via a power law distribution in order to take into account the continuous variation of the volume fraction of its constituents along the thickness direction. Numerical simulations are conducted to show the influence of power law index, slenderness ratios, and boundary conditions on natural frequencies of functionally graded beams. Results demonstrate the efficiency and the applicability of the model based on a refined mixed formulation and its ability to predict the vibrational behavior of functionally graded beams with good accuracy.
“…This leads to the proposition of the third-order beam theory (TBT), higher-order beam deformation theories (HOBT) and quasi-3D theories. The effectiveness of these theories dealing with macro structures can be found in some recent contributions [20][21][22][23][24][25][26][27][28][29]. As regarded the micro scales, Nateghi et al [30] applied the generalized differential quadrature method (GDQM) to examine the size-dependent buckling behaviour of FG beams using the CBT, FOBT and TBT.…”
In this paper, the free vibration behaviour of bi-dimensional functionally graded (BDFG) microbeams under arbitrary boundary conditions (BCs) is studied. Based on the frame work of the modified couple stress theory and Hamilton's principle, governing equations of motion are developed for the BDFG microbeams using a quasi-3D theory. The formula then can be reduced to a higher-order beam theory (HOBT) of conventional functionally graded (FG) microbeams with the material properties varying along the thickness direction only. Two types of BDFG microbeams with different patterns of material volume distribution are considered. The material properties used in this study are assumed to vary exponentially along both longitudinal and thickness directions of microbeams. Based on the state-space concept, the governing equations are solved for natural frequencies and vibration mode shapes of microbeams under various BCs. The effects of material distribution, geometric parameters and BCs are also investigated to examine the size-dependent behaviour of BDFG microbeams.
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