Document Type : Research Paper
Authors
^{1} Young Researchers and Elite Club, Islamic Azad University, Arak Branch, Arak, Iran
^{2} Department of Mechanical Engineering, Islamic Azad University, Arak Branch, Arak, Iran
Abstract
Keywords

Mechanics of Advanced Composite Structures 2 (2015) 95111


Semnan University 
Mechanics of Advanced Composite Structures journal homepage: http://macs.journals.semnan.ac.ir 
Free Vibration Analysis of 2D Functionally Graded Annular Plate considering the Effect of Material Composition via 2D Differential Quadrature Method
M. Nejati^{a}, H. Mohsenimonfared^{b*}, A. Asanjarani^{a}
^{a}Young Researchers and Elite Club, Islamic Azad University, Arak Branch, Arak, Iran
^{b}Department of Mechanical Engineering, Islamic Azad University, Arak Branch, Arak, Iran
Paper INFO 

ABSTRACT 
Paper history: Received 18 January 2015 Received in revised form 13 December 2015 Accepted 2 January 2016 
This study investigates the free vibration of the TwoDimensional Functionally Graded Annular Plates (2DFGAP). The theoretical formulations are based on the threedimensional elasticity theory with small strain assumption. The TwoDimensional Generalized Differential Quadrature Method (2DGDQM) as an efficient and accurate semianalytical approach is used to discretize the equations of motion and to implement the various boundary conditions. The fast rate of convergence for this method is shown and the results are compared with the existing results in the literature. The material properties are assumed to be continuously changing along thickness and radial directions simultaneously, which can be varied according to the power law and exponential distributions, respectively. The effects of the geometrical parameters, the material graded indices in thickness and radial directions, and the mechanical boundary conditions on the frequency parameters of the twodimensional functionally graded annular plates are evaluated in detail. The results are verified to be against those given in the literature.




Keywords: Functionally graded material Generalized differential quadrature Free vibration Annular plate



© 2015 Published by Semnan University Press. All rights reserved. 
In the last decade, the Functionally Graded Materials (FGMs) were employed in various engineering applications and were first reported in 1984 by Japanese material scientists [1]. The FGMs are heterogeneous materials in which, the thermomechanical properties vary continuously and generally as a function of position throughout the material. A typical FGM is created by smoothly changing the volume fraction of its materials constituents.
This new type of materials can be applied to avoid interfacial stress concentration appeared in laminated structures, and therefore, a great promise is proposed by FGMs in applications where the working conditions are severe[2], including spacecraft heat shields, heat exchanger tubes, plasma facings for fusion reactors, engine components and highpower electrical contacts or even magnets.
Compared with the analysis of functionally graded rectangular plates [311] and functionally graded spheres [12] as well as functionally graded cylindrical shells [1317], the investigations of functionally graded annular plates are limited in number. Moreover, the proposed semianalytical 2DGDQ has led to obtain more accurate results with appropriate rate of convergence. Prakash and Ganapathi [18] investigated the asymmetric free vibration behaviours and the thermoelastic stability of the functionally graded circular plates based on the firstorder shear deformation theory using the finite element method. The material properties were supposed graded in the thickness direction according to the simple power law distribution. Eraslan and Akis [19] obtained the closed form solution of the functionally graded rotating solid shaft and rotating solid disks under the generalized plane strain and plane stress assumptions, respectively. Efraim and Eisenberger [20] investigated the FGM plate with the material properties varying smoothly through the thickness of the plate. The equations of motion including the effect of shear deformations using the Firstorder Shear Deformation Theory (FSDT) were derived and solved exactly for various arrangements of the boundary conditions. The solution was obtained.
Nie and Zhong [21] studied the three – dimensional vibration of the functionally graded circular plates using the semianalytical approach. The dynamic analysis of the multidirectional functionally graded annular plates was obtained using the state spacebased differential quadrature method based on the threedimensional elastic theory and on the assumption that the material properties have an exponentlaw variation along the thickness, radial direction or both directions. Dong [22] studied the threedimensional free vibration of functionally graded annular plates with various boundary conditions using the Chebyshev–Ritz method. Two types of variations were considered for the material properties in the thickness direction of the plates. Yas and Tahouneh [23] studied the free vibration of the functionally graded annular plates based on the elastic foundations, using the differential quadrature method for various boundary conditions. The foundation was defined by Pasternak or a twoparameter model. A semianalytical method composed of Differential Quadrature Method (DQM) and series solution are employed to solve the equations of motions. The material properties were continuously varied through the thickness of the plate. HosseiniHashemi et al. [24] presented a solution to investigate the free vibration analysis of the radially functionally graded circular and annular sector thin plates of variable thickness based on the Classical Plate Theory (CPT) using the Differential Quadrature Method (DQM).
Jodaei et al. [25] presented a threedimensional analysis of functionally graded annular plates using StateSpace based Differential Quadrature Method (SSDQM) and a comparative behaviour modelling using Artificial Neural Network (ANN) for various boundary conditions. The material properties were supposed to have an exponentlaw variation along the thickness. Malekzadeh et al. [26] studied the free vibration of Functionally Graded (FG) thick annular plates subjected to thermal environment based on the 3D elasticity theory. The material properties were assumed to be temperature dependent and graded in the thickness direction. The Differential Quadrature Method (DQM) as an efficient and accurate numerical tool was used to solve both the thermoelastic equilibrium and the free vibration equations.
Behravan Rad and Shariyat [27] investigated the bending and stress analyses of the twodirectional Functionally Graded (FG) annular plates resting on the nonuniform twoparameter WinklerPasternak foundations, subjected to the normal and inplaneshear tractions using the exact threedimensional theory of elasticity. The solution was obtained by employing the state space and differential quadrature methods. The material properties were assumed to vary in both transverse and radial directions. Three different types of variations of the stiffness of the foundation were considered in the radial direction: linear, parabolic, and sinusoidal. Malekzadeh and Safaeian [28] employed a threedimensional (3D) discrete layer approach coupled with the Differential Quadrature Method (DQM) to investigate the free vibration analysis of the laminated Functionally Graded (FG) annular plates subjected to a thermal environment. The formulations were derived from the elasticity theory, which included the effects of the initial thermal stresses. Liang et al. [29] studied the transient response of the FGM annular sector plate with arbitrary circular boundary conditions based on the semianalytical methodology. The FGM annular sector plate was simply supported at the radial edges. At the circular edges, four kinds of boundary conditions were considered: Clamped–Clamped, Clamped–Simply supported, Clamped–Free and Simply supported–Simply supported. The results obtained in this study can be served as the benchmark data for further researches.
To the best of the author's knowledge, the 2D generalized quadrature method has not been yet employed to study the free vibration of the 2D functionally graded annular plates. Therefore, the purpose of this study is to investigate the free vibration analysis of the considered 2DFG annular plate. It is supposed that the plate rests on different boundary conditions namely simplysimply, clampedclamped, simplyclamped as well as freeclamped on the inner and outer edges, respectively. The equations of the motion are obtained according to a 3Delasticity theory which ultimately. They are reduced to the coupled differential equations. Consequently, a semianalytical formulation based on a twodimensional GDQ approach for the considered system of differential equations is derived, which results in determination of the frequency parameters. The material properties are assumed to be continuously changing along the thickness and radial directions simultaneously and affect the material composition, which can be varied according to the power law and exponential distributions, which in turn are evaluated in details and compared with each other. Also, the rate of convergence for the 2DGDQ is compared with other numerical methods.
2. Basic Equations
2.1 Governing equations
The equations of motion in the cylindrical coordinate system have to be formulated, in order to employ the socalled numerical procedure 2DGDQ. For this purpose, it is supposed that the 2DFG annular plate has the inner radial r_{i}, outer radial r_{o} and thickness h, with respect to the cylindrical coordinates (r, θ, z) attached on the bottom of the plate (Fig. 1). In the form of the cylindrical coordinate system, the linear straindisplacement relations are employed as follows:
(1) 
where and are displacement components toward radial, circumferential and thickness directions and are the corresponding axial strain components and are inplane and outofplane shear strain components, respectively. In the absence of the body forces, the equations of motion are obtained according to Eq. (2):
(2) 
where and are axial stress components, and and are shear stress components. Apparently, denotes material density and t refers to time, here.
Figure 1. The geometry and coordinate system of the annular plate
The stressstrain relations for the mentioned isotropic FGM are obtained according to the generalized Hook’s law as follows:
(3) 
where denotes compliance for the isotropic FGM. The displacement fields and for the free vibration of 2DFG annular plates are represented as unknown functions along the radial and thickness directions and known trigonometric functions along the circumferential direction as follows:
(4) 
where m denotes the circumferential wavenumber which could specify any positive integer values (m=0,1,…, ). is the natural frequency parameter and i is the imaginary number. It is obvious that m = 0 means axisymmetric vibration. Combining the straindisplacement relations Eq. (1) into the Hook’s law Eq. (3) and substituting into the equations of motion Eq. (2) and then using the trigonometric functions Eq. (4) leads to what follows:
(5)
(6)
(7) 
Eqs. (5)(7) denote 3D elasticity solution for the equations of motion in terms of the displacement components and must be solved for known compliances and assumptive displacement fields which have been supposed in Eq. (4). Then, it must be solved by employing the existing numerical procedure. Here, 2DGDQ procedure has been considered to discretize the linear system of the equations Eqs. (5)(7).
The Privileges of the 2DGDQ approach compared with onedimensional procedure are the types of boundary conditions and the grids distribution toward two directions which are illustrated in the following sections. In order to investigate the free vibration of 2DFG annular plate resting on different foundations, four different kinds of boundary conditions are considered namely Clamped–Clamped (CC), Simply supported –Clamped (SC), FreeClamped (FC) and Simply supported–Simply supported(SS). The boundary conditions at edges for the above mentioned foundations are as follows:
ClampedClamped (CC):
(8) 
Simply supported – Clamped (SC):
(9) 
FreeClamped (FC):
(10) 
Simply supported – Simply supported (SS):
(11) 
2.2 Micromechanics of the 2DFGM annular plate
A typical FGM is made of two randomly distributed isotropic constituents, i.e. a metal and a ceramic. The macroscopic response of the FGM is isotropic and the compositions of the considered FGM vary throughout the radial and thickness directions.
Such a material allows continuous variation of the material composition. One important step in determining the micromechanical model of FG circular plate is homogenization. The approach should be chosen based on the gradient of composition relative to the size of a typical Representative Volume Element (RVE). Here, two kinds of materials distributions are employed. The first type is based on the exponential law distribution, and the second one is according to the power law distribution. The material elastic coefficients (C_{ij}) and the mass density ( ) are assumed to have the following exponential law distribution along the thickness and radial directions of the plate as the following [30]:
(12) 
where and are the reference values at the center point of the bottom plane, and and denote the material property graded indexes in the thickness and radial directions, respectively.
In the first step, the material properties of the annular plate are assumed according to the exponentlaw variation in the thickness and radial directions. Young’s modulus at the center of the bottom plane is E = 380 (GPa) and Poisson’s ratio is chosen as constant ( ). The material density at the center of the bottom plane equals .
Defining the nondimensional parameter in the thickness direction and in the radial directions, the density variation for different values of material property graded indexes is demonstrated in Fig. 2.
Where R is the average radius of annular plate and H=r_{o}r_{i}. In case of power law distribution, we have annular plate which its inner and outer surfaces are made of two distinct ceramics and two distinct metals, respectively. C_{1}, C_{2}, M_{1} and M_{2} denote ﬁrst ceramic, second ceramic, ﬁrst metal and second metal, respectively. The volume fraction of the ﬁrst ceramic material is varied from 100% at the lower surface to zero at the upper surface by a power law function. Also, this volume fraction is continuously varied from inner surface to the outer surface.
(a) 
(b) 
(c) 
Figure 2. The density variation for (a) (b) and (c)
The volume fractions of the other materials are changed in a similar way to the mentioned one in two directions. The volume fraction function of each material can be described as follows [31]:
(13) 
where are parameters representing the basic constituent distribution in r and z directions.
The effective material properties including elasticity modulus and density at each point can be obtained using the simple linear rule of mixtures, in which the mentioned material properties, E and , at any arbitrary point (r, z) in the 2DFGM annular plate are determined by a linear mixture of the volume fractions. The material properties of the basic materials can be described as Eq. (14). It must be noted that, the Poisson's ratio is assumed to have a constant value.
(14) 
where m and c subscripts stand for the metal and ceramic, respectively. The five different models section is configured in Fig. 3.
In this step, the effects of the FGM conﬁguration are studied by investigating the natural frequencies of the five different FGM models as follows:
 Model (1): The 2DFGM annular plate has ceramics on its inner surface and metals on its outer surface
 Model (2): The 2DFGM annular plate has metal on its inner surface and ceramics on its outer surface
 Model (3): The 2DFGM annular plate has ceramics on its bottom surface and metals on its upper surface
 Model (4): The 2DFGM annular plate has metal on its bottom surface and ceramics on its upper surface
 Model (5): The 2DFGM annular plate has ceramics on the corner at the left on the bottom surface, in the corner at the right on the upper surface and metals on the corner at the right on the bottom surface, on the corner at the left on the upper surface.
The basic constituents of the 2DFG annular plate for two kinds of distributions (exponential and power law) are presented in Table. 1 and it should be noted that the Poisson’s ratio is assumed to be constant ( ).
Fig. 4 illustrates the distribution of the effective density of the 2DFG plate versus two nondimensional parameters and for different values of the power exponents and .
Figure 3. The FGM annular plate with the twodimensional material distribution (Models 15)
Table 1. The material properties of the FGM plate for different distributions
Distribution 
Constituents 
E(GPa) 
(kg/m^{3}) 
Powerlaw distribution [31] 
M_{1} 
115 
2715 
M_{2} 
69 
4515 

C_{1} 
440 
3210 

C_{2} 
150 
3470 

Exponentlaw distribution [30] 
center point of the bottom plane 
380 
3800 
Figure 4. The distribution of the effective density for
It is difficult to solve the equations of motion analytically. Hence, one should use a numerical procedure in order to discretise the partial differential equations in terms of displacement components. Then, the set of discretized algebraic equation must be solved as an Eigenvalue problem to find the natural frequencies. Apparently, Eqs. (5)(7) are a linear set of the partial differential equations of motion which ought to be solved with one of the numerical solver. Here, the 2DGDQ procedure is used to discretize the motion equations and followed by related boundary conditions. Choosing the solution procedure i.e. the 2DGDQ method is first based on the type of boundary conditions and secon meshing the plate in two directions (Here the grids are distributed toward the material variations namely, the radial and thickness directions). The differential quadrature method (DQM) is a semianalytical solution technique for initial and/or boundary value problems [32]. It was ﬁrst developed by Richard Bellman and his associates in the early 1970s [33]. The DQ method is originated from the idea of conventional integral quadrature and approximates the derivative of a function at any discrete points by a weighted linear summation of the functional values in the whole domain [34]. A simple and accepted choice of the grid distribution is the uniform grid spacing rule; however, it is found that the nonuniform grid spacing yields results with a better accuracy. Hence, in this study, the Chebyshev–Gauss–Lobatto quadrature points are employed, the following equations are derived [32]:
(15) 
where and are the distribution of nodes along the r and z directions, respectively. As one can observe, with this nonhomogeneous distribution of nodes, by approaching to the internal and external boundaries of the FGM annular plate with various boundary conditions and also the upper and lower boundaries with free boundary condition, the density of nodes increases while it causes the boundary conditions to be better satisfied. Getting far from boundaries, less density in the node of grid is generated. In the 2DGDQ method, in order to approximate the firstorder derivatives of the function f (r,z) with respect to r in a node of grid, according to Eq. (16), the summation of multiplication of function values in all nodes in the r direction is parallel to the target node with the weighting coefficients in the direction r.
(16) 
In order to obtain the weighting coefficients, a set of test functions should be employed. For the polynomial basis functions DQ, a set of Lagrange polynomials are employed as the test functions. The weighting coefficients for the firstorder derivatives in rdirection can be described as follows [32]:
(17) 
where is the length of domain along the rdirection and
(18) 
According to the DQ method, for higherorder derivatives, higherorder weighting coefficient should be employed, i.e. for secondorder derivatives, the weighting coefficient can be expressed as below:
(19) 
In the similar way, the weighting coefficients can be obtained for the zdirection.
Using the 2DGDQ method for the spatial derivatives, the discretized form of the equations of motion at each domain grid point can be obtained as shown in Appendix A. Similarly, the boundary conditions can be discretized. For this purpose, using the 2DGDQ discretization rules for spatial derivatives, one can obtain:
For clamped edge:
(20) 
For free edge at r=r_{i}:
(21) 
For simply support edge at r= r_{i}:
(22) 
For z=0, h:
(23) 
where , and are the ﬁrst and secondorder DQ weighting coefﬁcients in the r and z directions, respectively. The expressions of the , and formula are available in the literature [32, 34].
In order to carry out the eigenvalue analysis, the domain and boundary degrees of freedom are separated and described in vector forms. They are denoted by {d} and {b}, respectively. Based on these definitions, the discretized form of the motion equations and the related boundary conditions take the following forms:
Equations of motion (Appendix A):
(24) 
Boundary conditions (Eqs. (20)(23)):
(25) 
Eliminating the boundary degrees of freedom in Eq. (24), using Eq. (25), this equation turns into:
(26) 
where . The above eigenvalue system of equations can be solved to find the natural frequencies and mode shapes of the annular plates.
4.1 Verification of results
Free vibration analysis of TwoDimensional Functionally Graded annular plates (2DFG annular plate) is performed via general Differential Quadrature Method (GDQ). The equations of motion are obtained based on the 3D elasticity theory.
For this purpose, a 2DFG annular plate with arbitrary dimension must be considered. All values including dimensions of the plate as well as frequency parameters are presented as nondimensional form for ease of use. In order to ensure the accuracy of the results, comparisons have been performed by the literature.
Verification of the results is performed in two stages. Firstly, a homogenous and isotropic annular plate with clampedclamped boundary condition is considered and then a freeclamped FG annular plate with conventional material distribution (Eq. (27)) is investigated.
In Tables 2 and 3, the presented results are compared with the similar ones in the literature, namely the homogenous isotropic annular plate and onedimensional FG annular plate ( ).
The results obtained by analysis of the 2DGDQ for both cases are shown in Tables 2 and 3. A faster rate of convergence for the proposed 2DGDQ procedure is obvious in comparison with Chebyshev–Ritz method ([22, 35]). Hence, a good agreement is achieved within the results. For this verification, the number of grids in radial and thickness directions for the 2DGDQ is assumed . The constituents of the FG annular plate in the second case are distributed according to Eq. 32 as follows:
(27) 
4.2 Parameter studies
In this section the effects of different parameters such as BCs as well as power exponent coefficients are studied on the natural frequency parameters. There are several efforts in which the mechanical dynamic response of FGMs has been studied, but in the case of vibration analysis of the 2DFG plate using the 2DGDQ method, a lack of knowledge is perceived.
Table 2. The first eight nondimensional frequencies for the isotropic annular plates with the clampedclamped condition
( )
m 
method 

0 
present 
4.6582 
8.9505 
11.3915 
14.3382 
17.2782 
19.2184 
19.4718 
21.8688 

Ref. [22] 
4.661 
8.952 
11.40 
14.34 
17.29 
19.22 
19.48 
21.87 

Ref. [35] 
4.660 
8.950 
11.40 
14.34 
17.28 
19.22 
19.48 
21.87 
1 
present 
4.7482 
8.5359 
9.1248 
11.2232 
12.7365 
14.4148 
16.7438 
17.6333 

Ref. [22] 
4.752 
8.536 
9.127 
11.23 
12.74 
14.42 
16.75 
17.64 

Ref. [35] 
4.750 
8.536 
9.126 
11.23 
12.74 
14.42 
16.75 
17.64 
2 
Present 
5.0865 
9.0963 
9.7069 
11.1450 
13.6461 
14.6436 
16.5603 
18.1813 

Ref. [22] 
5.089 
9.097 
9.709 
11.15 
13.65 
14.65 
16.56 
18.18 

Ref. [35] 
5.088 
9.096 
9.708 
11.15 
13.65 
14.65 
16.56 
18.18 
3 
Present 
5.7206 
9.7439 
10.5762 
11.4287 
14.7313 
15.0190 
16.5436 
18.7558 

Ref. [22] 
5.723 
9.745 
10.58 
11.43 
14.73 
15.02 
16.55 
18.76 

Ref. [35] 
5.722 
9.744 
10.58 
11.43 
14.73 
15.02 
16.55 
18.76 
Table 3. The first eight nondimensional frequencies for the FGM annular plates having the freeclamped condition
( )
P 
m 
method 

1 
0 
present 
3.6499 
8.6838 
10.6489 
15.0467 
16.7138 
17.7982 
18.0660 
21.0876 


Ref. [22] 
3.652 
8.684 
10.650 
15.045 
16.711 
17.795 
18.063 
21.087 

1 
present 
4.4508 
7.2983 
9.2262 
10.6320 
13.6344 
14.1947 
15.2558 
16.6110 


Ref. [22] 
4.453 
7.298 
9.226 
10.632 
13.633 
14.193 
15.255 
16.608 

2 
present 
6.2567 
7.7562 
10.5431 
11.5508 
14.5422 
15.7763 
16.1538 
17.0964 


Ref. [22] 
6.259 
7.756 
10.543 
11.550 
14.542 
15.775 
16.152 
17.094 

3 
present 
8.4036 
8.9101 
12.3127 
13.2659 
15.3383 
16.6047 
17.5216 
18.2774 


Ref. [22] 
8.406 
8.910 
12.313 
13.265 
15.338 
16.602 
17.519 
18.275 
5 
0 
present 
4.4804 
11.2575 
12.5651 
19.5573 
21.0189 
23.0737 
23.5991 
28.3433 


Ref. [22] 
4.482 
11.254 
12.565 
19.548 
21.007 
23.058 
23.589 
28.359 

1 
present 
5.4668 
9.1670 
11.7041 
12.9115 
17.2524 
18.8533 
19.6915 
20.9746 


Ref. [22] 
5.468 
9.165 
11.701 
12.910 
17.244 
18.843 
19.684 
20.960 

2 
present 
7.6872 
9.8504 
12.9084 
14.7923 
18.6025 
20.0166 
21.4650 
21.7318 


Ref. [22] 
7.689 
9.848 
12.904 
14.788 
18.596 
20.006 
21.456 
21.715 

3 
present 
10.3284 
11.4778 
14.9668 
17.4532 
19.5565 
21.5712 
22.8608 
23.2461 


Ref. [22] 
10.329 
11.475 
14.962 
17.447 
19.550 
21.558 
22.851 
23.231 
Table 4. The convergence study of the first three nondimensional frequencies for the FGM annular plates having the freeclamped condition
( )



The number of Chebyshev polynomial terms as shown in Ref. [22] 
I*J=8*4 
4.474 
7.317 
9.278 
I*J=20*12 
4.453 
7.298 
9.227 

I*J=30*15 
4.453 
7.298 
9.226 

The number of nodes in GDQ method 
Nr*Nz=7*7 
4.446 
7.300 
9.226 
Nr*Nz=9*9 
4.449 
7.298 
9.225 

Nr*Nz=11*11 
4.450 
7.298 
9.226 

Nr*Nz=13*13 
4.450 
7.298 
9.226 
Two studies dealing with the free vibration analysis of homogenous and FG isotropic annular plates are presented in previous section in order to validate the results. Hence, we are satisfied with the results which are presented in previous section. Two functions of constituent distribution namely exponential and power law introduced and presented in previous section, reveal the functionality of material distribution with respect to the radial and thickness directions.
It is assumed that the plate stands in ambiance temperature with different mechanical boundary conditions are introduced in section 2. The significant difference within the numerical results is revealed in the rate of convergence. The generalized differential quadrature method due to the few number of grids, for convergence and also the accurate solutions in comparison with other numerical methods is more popular.
Fig. 5 shows the rate of convergence for the first natural frequency of the annular plate obtained in terms of different values of when has specified constant values.
Figure 5. The effect of the GDQ grids on the rate of convergence for the first natural frequency, h=0.2, , r_{i}/r_{o}=0.2
According to the figure, reasonable converged results and accurate natural frequency are obtained in at least 15 nodes along the thickness and 9 nodes along the radius directions and that would be acceptable in differential quadrature method. Moreover, Table 4 compares accuracy and convergence rate of the presented method and a popular numerical method for frequency analysis of the FGM annular plates. In this table, 30*15 number of Chebyshev polynomial terms in ChebyshevRitz is tantamount to 11*11 nodes in the GDQ method for analysis of the first three natural frequencies in the FGM annular plates which means the GDQ method solves fewer equations and consequently it leads to accurate answers faster than the other method.
Meshing the plate in two directions and using the 2DGDQ procedure lead to obtain more accurate result with a fast rate of convergence. Also, the 3D graphs are employed in order to investigate the rate of convergence for this algorithm. The number of points required for convergence in radius and thickness directions is studied and the rate of convergence for the first three nondimensional natural frequencies in terms of different numbers of nodes in radial ( ) and thickness ( ) directions is shown in Fig. 6. The flat part of each figure in Fig. 6 (ac) shows the first three frequencies achieve convergence for a certain number of nodes along the r and z directions.
In other words, significant changes in response to increasing the number of nodes in the twodimensional grid have not been observed. For example, in Fig. 6 (a) per nine nodes in the r direction and 15 nodes along the z direction the first frequency converges, or in the case of Fig. 6 (c) per 9 nodes along the r direction and 11 nodes along the z direction the third frequency converges. As it is observed, the convergence is fast and a few numbers of grid points along the radial and thickness directions are required ( ). Variations of the first nondimensional natural frequencies versus the material graded index (or power coefficients) in case of conventional FG annular plate along the thickness or radial and the 2DFG annular plate in both directions, for different boundary conditions are demonstrated in Figs. (79) (r_{i}=0.2, r_{o}=1, h=0.2). Figs. 78 depict the first nondimensional natural frequency variations of conventional FG annular plate versus the variation of the power coefficients along the thickness and radial directions, in terms of different boundary conditions, respectively.
Fig. 7 shows variation of the frequency versus the power coefficient for conventional FG annular plate, in which the materials are graded throughout the thickness direction. It is observed that the frequency tends to decrease when the power coefficient increases. Also, for a plate with fully clamped edges, the highest natural frequency is obtained. Additionally, the slop of frequency graph decreases for a plate with fewer constraints i.e. here the FC plate is less than other types of boundary conditions.
In Fig. 9, it is assumed . It is observed that the boundary conditions have significant effects on natural frequency parameter of the FG annular plate. For the case of functionally graded plate along the radial direction, the frequency variation is quiet different. As it is observed, the first nondimensional natural frequency of the annular plate under all boundary conditions except (SS) increases with an increase in the functionally graded index in radial direction, but under (SS) condition, it reduces.
(a) 
(b) 
(c) 
Figure 6. The rate of convergence for the first three natural frequencies in case of simply supported–clamped condition, h=0.2, , r_{i}/r_{o}=0.2
Fig. 9 for the 2DFGM shows a same behavior for all boundary conditions, the first non dimensional natural frequency gradually increases with an increase in the ratio of the radial to thickness graded indices. It should be noticed, for an annular plate with a material property graded index in radial direction lower than 2 (Fig. 8), and for a 2DFGM with a material graded index ratio almost less than 0.5 (Fig. 9), the first nondimensional natural frequency under (SS) boundary condition is higher than the natural frequency under (FC) condition. For the 2DFGM in case of (SS) condition, the first nondimensional natural frequency is almost constant for a material graded index ratio greater than 1. The effects of plate thickness on the first nondimensional natural frequency for conventional FG annular plate graded along thickness, or radial direction and 2DFG annular plate under different boundary conditions including (CC),(FC),(SC) and (SS) are shown in Figs. 1012, respectively (r_{i} =0.2, r_{o}=1). Figs. 10 and 11 show the nondimensional natural frequency variations of unidirectional FGM in thickness or radial directions, respectively, versus the material property graded index under different boundary conditions. The aforementioned graded index in the thickness and radial directions has an insignificant effect on the first nondimensional natural frequency of thin plates (h/R_{o}=0.05). In case of the thin 2Dfunctionally graded annular plate in Fig. 12, the first nondimensional natural frequency gradually increases and then remains almost constant for the ratio of the radial to thickness graded indices around 1, it remains almost constant. Fig. 13 (ae) shows the first nondimensional natural frequency variations versus n_{r }and n_{z }parameters for 5 considered models illustrated in Fig. 3. It should be noted, the dynamic behavior of two models (1) and (3) are similar and the first natural frequency increases with the increase in n_{r }and n_{z} parameters. It could be as a result of the increase in the plate stiffness. In model (5), a similar behavior is shown. However, more variation of the first natural frequency is observed with the increase in n_{z }at constant n_{r}. The dynamic behavior of model (2) with respect to n_{r} and n_{z} parameters variation is similar with model (4) with respect to n_{z} and n_{r}, respectively. According to Fig. 13 (ae), one can obtain an appropriate design for the 2DFGM arrangement among different possible cases for a variety range of applications. The behaviors of the first and third kinds of these arrangements are similar, but the other kinds present different characteristics which can be used for different goals.

Figure 7. The variations of the first nondimensional natural frequency parameter for the 1DFGM versus the graded index in the thickness direction ( )
Figure 8. The variations of the first nondimensional natural frequency parameter for the 1DFGM versus the graded index in the radial direction ( )
Figure 9. The variations of the first nondimensional natural frequency parameter versus the ratio of the radial graded index to the thickness graded index for the 2DFGM
(a) 
(b) 
(c) 
(d) 
Figure 10. The variations of the first nondimensional natural frequency parameter versus the graded index in the thickness direction ( ) under a(CC), b(FC), c(SC), d(SS) boundary conditions
(a) 
(b) 
(c) 
(d) 
Figure 11. The variations of the first nondimensional natural frequency parameter versus the graded index in the radial direction ( ) under a(CC), b(FC), c(SC), d(SS) boundary conditions
(a) 
(b) 
(c) 
(d) 
Figure 12. The variations of the first nondimensional natural frequency parameter versus the ratio of the radial graded index to the thickness graded index for the 2D FGM under a(CC), b(FC), c(SC), and d(SS) boundary conditions
(a) 
(b) 
(c) 
(d) 
(e) 
Figure 13. The behavior of the first nondimensional natural frequency versus the material property graded indexes for different models, amodel (1), bmodel (2), cmodel (3), dmodel (4), and emodel (5)
5. Conclusions
The mechanical free vibration of the 2DFG annular plates was investigated using the Generalized Differential Quadrature method (GDQ) based on the twodimensional meshing. The threedimensional elasticity theory was employed in order to derive equations of motion for the 2DFG annular plates. The mechanical properties of the 2DFG annular plate were assumed to be temperature independent since only the mechanical vibration analysis was performed. The material constituents were distributed as the function of r and z coordinates simultaneously by the power law and exponential formulation.
A semianalytical approach composed of twodimensional generalized differential quadrature methods (2DGDQ) and series solution were adopted to solve the equations of the motion. The effect of the functionality of the material properties on the position, boundary conditions including simply supportedclamped, clampedclamped, freeclamped and simply supportedsimply supported in inner and outer radii, respectively as well as the aspect ratio were taken into account, too.
The results for the natural frequency parameter of the 2DFG annular plate with different parameters have illustrated the fast convergence and high accuracy of the proposed approach in terms of using 2D grids. From this study, the following conclusion can be remarked:
 The convergence of the 2DGDQ method is fast and the results have a good agreement with the previous investigations.
 The different distributions of four kinds of materials, according to the rule of mixture to form an annular plate with variable properties in two directions are employed. The results show different or same dynamic behaviors in terms of different arrangement of materials and it could be seen.
For the exponential distribution, it is concluded that:
 In terms of the functionally graded indexes in thickness or radial directions increase, the first nondimensional natural frequencies increase for all kinds of boundary conditions, but (ss) boundary condition for the unidirectional FGM annular plate in radial direction, shows subtractive trend.
 Increasing the geometrical thickness parameter leads to an increase in the first dimensionless frequency for all kinds of boundary conditions.
 The nondimensional natural frequencies of thin unidirectional functionally graded along the radial or thickness directions as well as the thin 2DFGM annular plate according to Figs.(1012) depend on the functionally graded indices in thickness and radial directions, however, their variation against these parameters is small.
 For the 2DFGM under (SS) condition, the first nondimensional natural frequency is almost constant for the material graded index ratio which is greater than 1.
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Appendix A
(A1)
(A2) 
(A3)