Abstract
This paper presents a free vibration analysis of functionally graded ReissnerMindlin circular plates with various supported boundaries in thermal environments. A FGM consisting of metal and ceramic was considered in the study. Based on the geometric equation, physical equation and equilibrium equation of thick plate, taking into account the transverse shearing deformation, the free vibration equation of the axisymmetric FGM moderately thick circular plates was derived in terms of the middle surface angles of rotation and lateral displacement. The material properties of the plate were assumed to vary continuously in the thickness direction according a power law. By using shooting method to solve the coupled ordinary differential equations with different boundary conditions, the natural frequencies of FGM thick circular plates were obtained numerically. The effects of material gradient property, thickness ratio and boundary conditions on the natural frequencies were discussed in detail.
1. Introduction
Functionally graded materials(FGMs), a novel generation of microscopically composites in which the mechanical properties vary smoothly and continuously from one surface to another, proposed in the early 1980s [1, 2]. Structures made from these materials can effectively reduce thermal stress concentration so that they can be used in the hightemperature environments such as aerospace, optics, nuclear and civil engineering. So, the studies of the mechanical behaviors of FGM structures under the mechanical and thermal loadings have being attracted more and more attentions of scientists and also have become a new research field of solid mechanics.
Comprehensive studies on the vibration responses of FGM plates are available in the literatures, but most of which have been limited to analysis of beams and thin circular plates. For example, Xiang and Yang [3] carried out the free and forced vibration analysis of a laminated FGM Timoshenko beam with variable thickness under heat conduction. Li and Fan [4] proposed natural frequencies solutions for the free vibration of a FGM beam with a throughwidth delamination based on the Timoshenko beam theory. Leissa and Narita [5] investigated natural frequencies of a simply supported circular plate using Classical plate theory with ordinary and modified Bessel functions of the first kind. Kermani at al. [6] studied the free vibration analysis of multidirectional FGM circular and annular plates. Dong studied [7] threedimensional free vibration of FGM annular plates based on ChebyshevRitz method. Wang at al. [8] investigated the axisymmetric free vibration of FGM thin circular plates based on the threedimensional theory. Malekzadeh at al. [9] derived governing equations and boundary conditions by Hamilton principle of FGM circular arches with temperaturedependent properties in thermal environment and then gave the inplane free vibration solutions. Iman at al. [10] provided an analytic solution for the free vibration of multidirectional FGM circular and annular plates. They used statespace differential quadrature method to find the semianalytical or numerical solutions based on the threedimensional theory of elasticity. Lee at al. [11] provided a semianalytical solution for the free vibration analysis of circular plate with multiple holes by using the indirect boundary integral method. Ramesh and Mohan Rao [12] presented the natural frequencies of vibration of a rotating pretwisted functionally graded cantilever beam. Ying et al. [13] presented solutions for bending and free vibration of FGM beams resting on a WinklerPasternak elastic foundation. By using classical beam theory, Aydogdu and Taskin [14] investigated the free vibration behavior of a simply supported FGM beam. Pradhan and Chakraverty [15] carried out the free vibration analysis of FGM thin elliptic plates with various edge supports.
Compared with the free vibration analysis of FGM beams and thin circular plates, the free vibration analysis of FGM thick plates is rare. Chen at al. [16] investigated the thermal behavior of a thick transversely isotropic FGM rectangular plate based on the threedimensional elasticity theory. Kim and Lee [17] developed the geometrically nonlinear isogeometric analysis of FGM plates based on physical neutral surface and firstorder shear deformation theory. Foroughi Hamid and Azhari Mojtaba [18] analyzed the mechanical buckling and free vibration of thick FGM plates resting on elastic foundation using the higher order Bspline finite strip method. Jha at al. [19] presented deformation and stress analyses of functionally graded thick plates based on the twodimensional theory. Liu and Lee [20] provided a vibrations analysis of thick circular and annular plates based on threedimensional theory with finite element method.
To the author’s best knowledge, in the previous works only the mechanical vibrations of FGM circular plates are investigated and there have been few researches dealing with the free vibration behavior of moderately thick circular plates, especially the effects of thermal environment are not considered. The few researches are Malekzadeh [21], who studied the free vibration characteristic of tapered Mindlin plates using DQM. Malekzadeh et al. [22] presented the vibration analysis of FGM thick annular plates subjected to thermal environment based on 3D elasticity theory. Considering the thermal environment effects and using Hamilton's principle, differential quadrature method is adopted to solve the equations of motion and the frequency parameters are obtained. Furthermore, Malekzadeh et al. [23] extended their work to the free vibration of FGM elastically supported FGM annular plates in thermal environment. Shen and Wang [24] reported a free vibration analysis of FGM rectangular plates resting on elastic foundations in thermal environments based on Voigt model and MoriTanaka model respectively.
Shooting method is found to be a simple and efficient numerical technique for solving nonlinear governing equations. Some works have successfully used shooting method to analyze postbuckling, bending and vibration analysis. For example, Li and Zhou [25] carried out the nonlinear vibration and thermal buckling of heated orthotropic circular plates by shooting method. Ma and Lee [26] investigated the nonlinear mechanical behaviors inplane thermal loading of a beam made of FGM. Sun and Li [27] presented the thermal postbuckling of FGM circular plates subjected to transverse poingspace constraints, but they cannot consider the effect of thermal loading.
Therefore, in the present study, free vibration of a FGM moderately thick circular plate in thermal environment is presented by employing the numerical shooting technique. The material properties are thus assumed temperaturedependent and graded in the thickness direction. The equations of motion are derived using the Hamilton’s principle based on the first order shear deformation theory and three different boundary conditions are considered. After that, we employ the shooting method to solve the equations of motion with the related boundary conditions. The effects of uniform and nonuniform temperature rise parameters, gradients of materials properties, thickness ratio and boundary conditions on the natural frequencies are analyzed in detail.
2. Problem formulation
Consider a FGM moderately thick circular plate on Winkler foundation with radius $R$ and uniform thickness $h$, which is made from a mixture of ceramic and metals. A cylindrical coordinate system $(r,\theta ,z)$ with its origin at the center of the midplane of the plate is defined, where $r$, $\theta $ and $z$ represent coordinates in the radial, circumferential and thickness directions, respectively. The coordinate system is illustrated in Fig. 1.
Fig. 1Geometry and coordinates of an FGM circular plate
2.1. Temperature field
Since functionally graded materials are most commonly used in high temperature environment, it is essential to take into consideration this temperaturedependency. Thus, the effective material properties $P$ of the metal and ceramic, such as elasticity modulus, $E\text{,}$ thermal expansion coefficient, $\alpha $, thermal conductivity, $K$, Poisson’s ratio, $\nu $, may be expressed as a nonlinear function of temperature:
In which $T={T}_{0}+\mathrm{\Delta}T$ and ${T}_{0}=$ 300 K (room temperature), the temperaturedependence coefficients ${P}_{1}$, ${P}_{0}$, ${P}_{1}$, ${P}_{2}$ and ${P}_{3}$ are unique to the constituents.
In this section, we assume that the temperature variation is uniform or occurs in the thickness only. Therefore, the temperature distribution along the thickness direction can be obtained by solving a steadystate onedimensional heat transfer equation:
This equation is solved by imposing boundary condition of $T\left(\frac{h}{2}\right)={T}_{m}$, $T\left(\frac{h}{2}\right)={T}_{c}$:
In which $\mathrm{\Delta}T={T}_{c}{T}_{m}$, $C=\sum _{i=0}^{\infty}({K}_{c}{K}_{m}{)}^{i}/(i\times p+1){{K}_{m}}^{i}$, where ${K}_{c}$ and ${K}_{m}$ denotes the ceramic and metal thermal conductivity.
2.2. Effective material properties of FGMs
According to Voigt rule, the effective material properties $P$ of FGMs may be written as:
where the subscripts $c$ and $m$ denote the ceramic and metallic constituents, respectively, ${V}_{c}$ and ${V}_{m}$ denotes the ceramic and metal volume fractions of and they can be expressed by ${V}_{c}+{V}_{m}=1$, the volume fraction ${V}_{c}$ defined a simple power low by:
where $p$ is the gradient index of FGM. Fig. 2 shows the variations of volume fraction of ceramic phase through the thickness of plate for various values of $p$ calculated from Eq. (5).
Fig. 2Variations of the volume fraction of ceramic phase versus the dimensionless thickness of the FGM circular plate for different values of p
For simplicity, the Poisson’s ratio $v$ is assumed to be a constant for functionally graded materials. The effective material properties follow the distribution law of Eqs. (4) and (5), namely:
2.3. Governing equations
According to oneorder shear plate theory, it can be assumed that the axisymmetric displacement field are:
In which ${U}_{r}$, ${U}_{\theta}$ and ${U}_{z}$ are total displacement components along the coordinates $r$, $\theta $ and $z$, respectively, $u$ and $w$ are the displacements in the midplane of the plate along the coordinates $r$ and $z$, respectively, and $\psi $ denotes the slope at $z=$ 0 of the deformed line that was straight in the undeformed plate, $t$ is the time variable.
The strains can be expressed as follow:
Considering nonlinear straindisplacement relationships, one then obtains:
where ${\epsilon}_{r}$ and ${\epsilon}_{\theta}$ are the normal strains and ${\gamma}_{rz}$ is the shear strain.
Based on Hooke’s law, the stressstrain relations are expressed by following formulations:
The stress can then be determined in terms of the midplane displacement through Eqs. (9a)(9c), they can be calculated as:
The membrane forces, the bending moments and the shear resultant force can be deduced from thickness integration of Eqs. (13), (14) and Eq. (15) respectively:
Here ${\kappa}_{s}$ is the shear correction factor, for Mindlin plate ${\kappa}_{s}=12/{\pi}^{2}$, Reissner plate ${\kappa}_{s}=$ 6/5. ${A}_{11}$, ${B}_{11}$, ${D}_{11}$ and $S$ are the stretching, bendingstretching coupling, bending and shear rigidity coefficient respectively; and the thermal membrane force ${N}_{T}$ and thermal bending moment ${M}_{T}$ can be calculated by following expressions:
Substitution of Eq. (6a) and Eq. (3) into Eq. (17) and Eq. (18) gives following statements:
Note that${E}_{r}={E}_{m}/{E}_{c}\text{,}$${\alpha}_{r}={\alpha}_{m}/{\alpha}_{c}\text{,}$${K}_{r}={K}_{m}/{K}_{c}$, ${f}_{i}(i=1,2,3,4,5)$ are coefficients determined as follows:
2.4. Motion equations
The kinetic energy $T$ and the elastic potential ${V}_{e}$ of the FGM plate may be written as:
where $\rho =\rho (z,T)$ is the mass density of the FGM plate, and $\mathrm{\Omega}$denotes domain of the FGM plate.
The loaddisplacement relationship of the foundation is assumed to be $q={k}_{w}w{k}_{g}{\nabla}^{2}w$, where $q$ is the force per unit area, and $\nabla $ is Laplace differential operator; ${k}_{w}$ is the Winkler foundation stiffness and ${k}_{g}$ is a constant showing the effect of the shear interactions of the vertical elements.
The elastic foundation energy having shear deformable layers ${V}_{k}$ of the plate is expressed as follows:
2.5. Hamilton’s principle
The free vibration equations of motion will be derived according Hamilton’s principle, which has the following form:
where ${t}_{1}$ and ${t}_{2}$ are the beginning and end of motion time, respectively, in which the virtual kinetic energy is:
The virtual elastic potential and elastic foundation energy may be expressed as follows:
In which:
where:
Applying integrating by parts and collecting the coefficients of $\delta \psi $, $\delta w$, the governing equations of motion can be expressed as:
Inserting Eqs. (16a)(16e) into Eqs. (24a, b) and regardless of nonlinear terms, the governing equations of motion can be expressed as:
Assume that the vibration of circular plate is harmonic. Then, the dynamic response can be expressed as:
where $\mathrm{\Omega}$ is the natural frequency, and $\stackrel{~}{w}\left(r\right)$ and $\stackrel{~}{\psi}\left(r\right)$ are the shape functions. Substituting Eqs. (28a) and (28b) into Eqs. (27a) and (27b) gives the governing equations of motion of the problem as follow:
Nondimensional variables are introduced as follows:
${\omega}_{n}=\mathrm{\Omega}{R}^{2}\sqrt{\frac{12(1{\nu}^{2}){\rho}_{c}h}{{E}_{c}{h}^{3}}}.$
Substitution of the above nondimensional transformation into Eqs. (29a) and (29b) gives the governing equations of the FGM ReissnerMindlin circular plates in dimensionless forms as:
In which:
where:
In the present study, three traditional boundary conditions are considered. Boundary conditions in dimensions form can be written as:
Clamped edge (C):
Simply supported edge (S):
Free edge (F):
3. Shooting method
Due to complexity and the coupling of these partial differential equations, it is very different to obtain any analytical solution of this problem. In what follows, a shooting method [2527] is employed to numerically solve the problems. Here, the governing Eqs. (30) and (31) and boundary conditions Eqs. (32)(33), (32)(34) and (32)(35) can be rewritten in the standard form:
where:
Expressions of ${\mathrm{\Delta}}_{1}$, ${\mathrm{\Delta}}_{2}$ and ${b}_{1}$, ${b}_{2}$ are as follows:
where $\eta $ is a dimensionless deflection parameter.
${\mathbf{B}}_{0}$ and ${\mathbf{B}}_{1}$ are matrixes of order 3×5 and 2×5 respectively, they are:
For the clamped boundary:
For the Simply supported boundary:
For the free boundary:
Consider the initial value problem:
where $G={\left\{{g}_{1},{g}_{2},{g}_{3},{g}_{4},{g}_{5}\right\}}^{T}$, $I={\left\{0,{d}_{1},\eta ,0,{d}_{2}\right\}}^{T}$, and $D={\left\{{d}_{1},{d}_{2}\right\}}^{T}$ is the initial parameter vector.
The unique solution for the initial value problem must exist, namely:
For given value of $\eta $, we seek components of ${D}^{*}={\left\{{d}_{1}^{*},{d}_{2}^{*}\right\}}^{T}$ such that a solution of Eq. (38) satisfies boundary condition Eq. (37b), that is:
Obviously, if $\mathbf{D}={\mathbf{D}}^{*}$ is a root of Eq. (39), the solution of the boundaryvalue problem Eq. (36) is then obtained as:
We employ the RungeKutta method to integrate the system Eq. (38) of ordinary differential equations, and the NewtonRaphson iteration method to search for a root ${\mathbf{D}}^{*}$ of Eq. (39) to find a numerical solution of the boundaryvalue problem Eq. (36). This approach is called a shooting method.
4. Shooting method for approximate solutions of free vibration of FGM ReissnerMindlin plates
In what follows, the material properties of ceramic, $Zr{O}_{2}$, and metallic, Ti6Al4V, as given in Table 1, are used in the numerical computations.
4.1. Comparison studies
To ensure the accuracy and effectiveness of shooting method, two examples are solved for free vibration of homogeneous and FGM Mindlin circular plate.
Table 1Temperaturedependent coefficients for ceramic and metals from Reddy and Chin [28]
Material  Proprieties  ${P}_{1}$  ${P}_{0}$  ${P}_{1}$  ${P}_{2}$  ${P}_{3}$ 
Ti6Al4V  $E$ (GPa)  0  122.7e+9  –4.605e4  –4.6054  0 
$Zr{O}_{2}$  $\alpha $ (1/K)  0  7.5788e6  6.638e4  –3.147e6  0 
$\kappa $ (W/mK)  0  1.0  1.704e2  0  0  
$\rho $ (kg·m^{3})  0  4229  0  0  0  
$E$ (GPa)  0  244.27e+9  –1.371e3  1.214e6  –3.681e10  
$\alpha $ (1/K)  0  12.766e6  –1.491e3  1.006e5  0  
$\kappa $ (W/m K)  0  1.7  1.276e4  6.648e8  0  
$\rho $ (kg·m^{3})  0  3000  0  0  0 
Example 1. When the power law index $p$ equals zero, the functionally graded material is reduced to the homogeneous ceramics Mindlin circular plate. In order to show the accuracy of the present numerical method, by giving $p=$ 0, $\nu =$ 0.3, ${K}_{w}={K}_{g}=$ 0, $\delta =$ 0.2，the comprehensive comparison between the shooting method results and those in Ref. [29] is given in Table 2, which includes the cases of abovementioned three sets of boundary conditions. Obviously, the first four lowerorder dimensionless natural frequencies show good agreement with the existing results.
Example 2. The first three order dimensionless fundamental frequency parameters $\omega $ of the FGM thick circular plates for the three traditional boundary conditions are calculated and compared in Table 3 with a set of results [15] based on RayleighRitz method. The present results are in good agreement with existing results.
Tables 23 show that the present results agree well existing results, and thus the accuracy of the shooting method technique are confirmed.
Table 2The first fourth dimensionless natural frequencies ωn for thick circular plates under three sets of boundary conditions
Boundary condition  ${\omega}_{1}$  ${\omega}_{2}$  ${\omega}_{3}$  ${\omega}_{4}$  
$C$  Present  9.2401  30.211  56.683  85.572 
[29]  9.2400  30.211  56.682  85.571  
$S$  Present  4.7773  24.995  52.514  82.767 
[29]  4.7777  24.994  52.514  82.766  
$F$  Present  –  8.5051  31.111  59.646 
[29]  –  8.5050  31.111  59.645 
Table 3Dimensionless natural frequencies ωn of FGM Mindlin plates with different graded index δ= 0.01
Boundary condition  ${\omega}_{n}$  
$p=$ 0.0  $p=$ 0.1  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  
$C$  1  Present  10.213  9.9836  9.4503  9.1331  8.815  8.591 
[15]  10.216  9.8510  8.973  8.500  8.125  7.576  
2  Present  39.734  38.843  36.767  35.532  33.423  33.423  
[15]  39.773  38.352  34.933  33.093  29.496  29.496  
$S$  1  Present  4.935  4.823  4.5673  4.4133  4.2684  4.1426 
[15]  4.935  4.759  4.335  4.106  3.925  3.659  
2  Present  29.704  29.036  27.486  26.564  25.416  24.667  
[15]  29.736  28.674  26.118  24.742  23.651  22.052  
$F$  2  Present  8.9686  8.7673  8.2986  8.0204  7.7737  5.9883 
[15]  9.003  8.682  7.908  7.491  7.161  6.677  
3  Present  37.787  36.939  34.964  33.793  32.754  25.2045  
[15]  37.564  36.221  32.993  31.225  29.877  27.857 
4.2. Numerical results and discussions
In this section, the free vibration of a FGM thick circular plate is numerically analyzed in thermal environment. Table 4 shows the first three order dimensionless fundamental frequency parameters $\omega $ of the FGM thick circular plates for clamped boundary condition changing with the power law index $p$ in room temperature fields without elastic foundation. It can be seen that all the frequencies decrease with the increasing of power law index $p$, which is due to the fact that the decrease of rigidity of the plate, or the increase of the component of metallic in the plate. Also, along with the increase in the value of $\delta $, the frequencies decrease.
Table 4Dimensionless natural frequencies ωn of FGM thick circular plates with clamped edge (ν= 0.3, Tc=Tm= 300 K, Kw=Kg= 0)
$\delta $  $\omega $  
Mindlin solution  Reissner solution  
$p=$ 0.0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  $p=$ 0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  
0.2  1  8.6948  7.9758  7.3639  6.8684  6.4509  6.0217  8.7051  7.9850  7.3724  6.8763  6.4587  6.0291 
2  29.464  27.028  24.908  23.111  21.529  19.931  29.538  27.095  24.969  23.168  21.585  19.983  
3  55.752  51.185  47.158  43.659  40.514  37.355  55.944  51.358  47.317  43.808  40.655  37.488  
0.3  1  8.1034  7.3952  6.8503  6.3628  5.9388  5.5085  8.1201  7.4468  6.8642  6.3759  5.9514  5.5206 
2  24.245  22.211  20.516  18.984  17.599  16.213  24.333  22.347  20.589  19.053  17.665  16.274  
3  43.037  39.484  36.446  33.674  31.129  28.594  43.226  39.731  36.604  33.822  31.267  28.723 
Subjected to room temperature or heat conduction, the fundamental frequencies decrease significantly with increasing the value of $\delta $ in Fig. 3 and 4. It can be found that for a giving $\delta $, Ceramic plate provide the largest fundamental frequency while the metal plate hold the smallest one. Table 5 shows the effect of uniform temperature rise on the first threeorder dimensionless frequency parameters for FGM Mindlin plate with $\delta =$ 0.2 and clamped ends that frequencies monotonously decrease as uniform temperature rise increase.
Fig. 3Fundamental dimensionless frequency parameter ω1 of clamped plate versus δ in room temperature field
Fig. 4Fundamental dimensionless frequency parameter ω1 of clamped plate versus δ subjected to heat conduction
Table 5Dimensionless natural frequencies ωn of FGM thick circular plates with clamped edge (ν= 0.3, δ= 0.2, Kw=Kg= 0)
$\delta $  $\omega $  
${T}_{c}={T}_{m}=$ 400 K  ${T}_{c}={T}_{m}=$ 500 K  
$p=$ 0.0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  $p=$ 0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  
1  8.5046  7.8108  7.2219  6.7486  6.3529  5.9435  8.3095  7.6418  7.0768  6.6245  6.2531  5.8641 
2  29.211  26.808  24.719  22.950  21.397  19.823  28.956  26.588  24.528  22.788  21.263  19.715 
3  55.439  50.914  46.924  43.460  40.347  37.219  55.124  50.642  46.689  43.259  40.180  37.084 
In order to compare uniform and nonuniform temperature rise effects, average temperature of nonuniform parameters equal to the temperature of uniform case are considered. Table 6 shows the effect of uniform temperature rise and heat conduction on the free vibration of FGM Mindlin circular plates without elastic foundation. It can be observed that for the same value of both uniform and nonuniform rise, the firstthree frequency parameters of FGM Mindlin plate decrease with increasing the power law index $p$. One can see that the frequency in case of uniform heating is smaller than that of the nonuniform temperature rise.
Table 6Fundamental dimensionless frequencies ω1 of FGM thick circular plates with clamped edge subject to temperature rise and heat conduction (ν= 0.3, δ= 0.2, Kw=Kg= 0)
$\omega $  
$p=$ 0  $p=$ 0.1  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  $p=$ 100  
${T}_{c}={T}_{m}=$ 325 K  8.6484  8.4774  8.0844  7.8528  7.6488  7.4601  7.2487 
${T}_{c}={T}_{m}=$ 350 K  8.6008  8.4334  8.0468  7.8198  7.6210  7.4355  7.2278 
${T}_{c}={T}_{m}=$ 375 K  8.5528  8.3881  8.0091  7.7875  7.5922  7.4117  7.2069 
${T}_{c}=$ 350 K, ${T}_{m}=$ 300 K  8.6873  8.5146  8.1137  7.8788  7.6715  7.4789  7.2664 
${T}_{c}=$ 400 K, ${T}_{m}=$ 300 K  8.6797  8.5063  8.1067  7.8719  7.6648  7.4740  7.2632 
${T}_{c}=$ 450 K, ${T}_{m}=$ 300 K  8.6437  8.4988  8.0996  7.8650  7.6589  7.4691  7.2592 
The effects of twoparameter elastic foundation on the firstthree frequency parameters of the FGM thick circular plates in thermal environment are exhibited in Table 7. As we expect, the elastic foundation has significant effects on the firstthree frequencies of FGM Mindlin plates subjected to uniform temperature and heat conduction, the results are bigger than the frequencies without elastic foundation. Obviously, additional elastic foundation leads frequency rise due to the decrease of the flexibility and the deformation capacity.
Table 7Dimensionless natural frequencies ωn of FGM thick circular plates with clamped edge (ν= 0.3, Kw= 50, Kg= 5)
$\delta $  $\omega $  
${T}_{c}={T}_{m}=$ 325 K  ${T}_{c}=$ 350 K, ${T}_{m}=$ 300 K  
$p=$ 0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  $p=$ 0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  
0.2  1  12.509  11.772  11.141  10.630  10.155  9.7098  12.524  11.784  11.149  10.618  10.161  9.7499 
2  32.575  30.159  28.065  26.322  24.729  23.177  32.603  30.180  28.081  26.299  24.739  23.256  
3  59.129  54.584  50.595  47.184  44.047  40.973  59.165  54.612  50.615  47.154  44.061  41.079  
0.3  1  10.214  9.5429  8.9591  8.4667  8.0138  7.5773  10.221  9.5485  8.9631  8.4609  8.0163  7.5963 
2  27.813  25.843  24.112  22.624  21.259  19.937  27.827  25.854  24.119  22.613  21.265  19.977  
3  47.875  44.399  41.319  38.630  36.150  33.742  47.894  44.414  41.330  38.614  36.157  33.798 
Figs. 5 shows the anterior threeorder modes of vibration for a FGM thick circular plate with clamped edge. It can be seen that the first three order mode diagrams with $\delta =$ 0.2 are very close to the one with $\delta =$ 0.3. Also, the mode diagrams are in good agreement with the boundary conditions.
Fig. 5Anterior thirdorder modes of FGM moderately thick circular plate with clamped edge
Consider a FGM Mindlin circular plate with and without elastic foundation which is simply supported or free edge. Table 8 shows the effect of uniform temperature rise and heat conduction on the free vibration of FGM Mindlin circular plates with simply supported edge without elastic foundation. It can be found that for the same value of both uniform and nonuniform rises, the firstthree frequency parameters of FGM Mindlin plate are decreased by increasing the power law index $p$. Subjected to the same temperature loads, the firstthree frequency parameters decrease with increasing the ratio of the thickness to the radius in Table 4 and 7, but it is not always true for the FGM moderately thick circular plates with simply supported edge due to effect of heat conduction in Table 8. Also, the influence of twoparameter elastic foundation on the firstthree frequencies of FGM circular plates in room and uniform temperature rise is exhibited in Table 9. As we expect, the elastic foundation has significant effects on the firstthree frequencies of FGM Mindlin plates subjected to uniform temperature and heat conduction, the results are bigger than the frequencies without elastic foundation. Subjected to room temperature, uniform temperature rises and heat conduction, the second and thirdorder frequencies for FGM moderately thick circular plates with free edge are calculated (see Tables 1011). It also can be found that the frequency in case of uniform heating is smaller than that of the nonuniform temperature rise. Also, one can see that increasing the thickness of the circular plate, the firstthree frequency parameters decrease.
Table 8Dimensionless natural frequencies ωn of FGM thick circular plates with simply supported edge (Kw=Kg= 0)
$\delta $  $\omega $  
${T}_{c}={T}_{m}=$ 325 K  ${T}_{c}=$ 350 K, ${T}_{m}=$ 300 K  
$p=$ 0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  $p=$ 0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  
0.2  1  3.6883  3.4191  3.2044  3.0511  2.9396  2.8112  3.7344  3.4556  3.2311  3.0718  3.9562  2.8257 
2  24.046  22.038  20.323  18.909  17.703  16.467  24.082  22.066  20.344  18.926  17.717  16.479  
3  51.437  47.174  43.468  40.312  37.526  34.705  51.478  47.206  43.492  40.331  37.542  34.719  
0.25  1  4.0256  3.7072  3.4449  3.2441  3.0857  2.9158  4.0525  3.7286  3.4607  3.2564  3.0957  2.9246 
2  22.613  20.728  19.105  17.746  16.568  15.368  22.638  20.747  19.119  17.758  16.578  15.377  
3  46.006  42.222  38.901  36.027  33.451  30.856  56.035  42.245  38.918  36.040  33.462  30.866  
0.3  4.1439  3.8063  3.5247  3.3033  3.1234  2.9345  4.1619  3.8207  3.5354  3.3116  3.1302  2.9406  
21.119  19.365  17.844  16.556  15.423  14.275  21.137  19.379  17.855  16.564  15.430  14.281  
41.296  37.925  34.941  32.325  29.951  27.572  41.318  37.943  34.955  32.335  29.960  27.580 
Table 9Dimensionless natural frequencies ωn of FGM thick circular plates with simply supported edge (Kw= 50, Kg= 5)
$\delta $  $\omega $  
${T}_{c}={T}_{m}=$ 325 K  ${T}_{c}=$ 350 K, ${T}_{m}=$ 350 K  
$p=$ 0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  $p=$ 0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  
0.2  1  9.5876  9.1729  8.7978  8.5058  8.2371  7.9820  9.4763  9.0761  8.7311  8.4373  8.1825  7.9404 
2  27.798  25.792  24.051  22.663  21.450  20.238  27.601  25.618  23.929  22.534  21.344  20.155  
3  55.194  50.936  47.210  44.113  41.359  38.603  54.960  50.725  47.060  43.953  41.226  38.498 
Table 10Dimensionless natural frequencies ωn of FGM thick circular plates with free edge (ν=0.3, Kw=Kg= 0)
$\delta $  $\omega $  
${T}_{c}=$ 300 K, ${T}_{m}=$ 300 K  ${T}_{c}=$ 400 K, ${T}_{m}=$ 400 K  
$p=$ 0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  $p=$ 0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  
0.2  2  8.0032  7.3311  6.7677  6.3221  5.9564  5.5768  7.8281  7.1795  6.6373  6.2119  5.8657  5.5042 
3  30.343  27.779  25.596  23.800  22.264  20.689  30.082  27.555  25.402  23.635  22.126  20.578  
0.25  2  7.9352  7.2618  6.6958  6.2437  5.8689  5.4817  7.8210  7.1629  6.6107  6.1715  5.8092  5.4335 
3  28.075  25.707  23.681  21.995  20.536  19.044  27.895  25.552  23.547  21.881  20.440  18.966  
0.3  2  7.7676  7.1050  6.5474  6.0994  5.7257  5.3399  7.6855  7.0340  6.4862  6.0473  5.6824  5.3048 
3  25.879  23.702  21.832  20.262  18.889  17.488  25.745  23.586  21.732  20.176  18.817  17.429 
Table 11Dimensionless natural frequencies ωn of FGM thick circular plates with free edge (ν= 0.3, Tc= 300 K, Tm= 400 K, Kw=Kg= 0)
$\delta $  $\omega $  
Mindlin solution  Reissner solution  
$p=$ 0.0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  $p=$ 0  $p=$ 0.2  $p=$ 0.5  $p=$ 1  $p=$ 2  $p=$ 5  
0.2  2  7.8281  7.1795  6.6373  6.2119  5.8657  5.5042  7.8321  7.1831  6.6405  6.2149  5.8685  5.5069 
3  30.082  27.555  25.402  23.635  22.126  20.578  30.131  27.598  25.442  23.672  22.163  20.613  
0.25  2  7.8210  7.1629  6.6107  6.1715  5.8092  5.4335  7.8262  7.1675  6.6148  6.1753  5.8129  5.4370 
3  27.895  25.552  23.547  21.881  20.440  18.966  27.952  25.604  23.594  21.925  20.483  19.007  
0.3  2  7.6855  7.0340  6.4862  6.0473  5.6824  5.3048  7.6917  7.0395  6.4912  6.0519  5.6869  5.3092 
3  25.745  23.586  21.732  20.176  18.817  17.429  25.679  23.643  21.832  20.225  18.864  17.475 
Figs. 7 and 8 show the anterior threeorder modes of vibration for FGM circular plates with simply supported and free edge. Also, the mode diagrams are in good agreement with the boundary conditions.
Fig. 7Anterior thirdorder modes of FGM moderately thick circular plate with simply supported edge
Fig. 8Anterior thirdorder modes of the FGM moderately thick circular plate free edge
5. Conclusions
In this paper, the free vibration of functionally graded moderately thick circular plates resting on twoparameter elastic foundations in thermal environment has been presented by employing the numerical shooting technique. Investigations on vibration of FGM thick circular plates in thermal environment with three different boundaries are also introduced. Material primary parameters are assumed temperaturedependent and vary along the thickness of the plate. By employing a shooting method, the nonlinear governing equations are solved numerically and the natural frequencies of the circular plates are obtained. The effects of uniform temperature rise and heat conduction, material constant, boundary conditions and elastic foundations on the natural frequency parameters are discussed in detail. Looking into the present results, one may conclude as follows.
1) The natural frequency parameters of vibration are decreases with increasing the ratio of the thickness to the radius for both the homogeneous ceramics plate and FGM plate without thermal environment, but it is not always true for a FGM plate with simply supported edge due to effect of heat conduction. The uniform temperature rise has more effect than the nonuniform temperature rises on the frequency parameters.
2) All the firstthree order frequencies of the FGM moderately thick circular plates in thermal environment decrease monotonously with the increase of the value of $p$, the reason is that the increase of the volume fraction of the metal reduces the bending stiffness of the whole plate.
3) The effect of elastic foundation on the free vibration is significant. The natural frequency of vibration for the plate is lower than the plates with elastic foundation. Also, the boundary conditions have large influence on vibration frequency of the thick plates. The natural frequency parameters are maximum with fixed boundary condition but it is minimum under free boundary condition.
4) The solutions can be used as benchmark for other numerical methods.
References

Yamanouchi M., Hirai T., Shiota I., et al. Proceeding of the First International Symposium on Functionally Gradient Materials, Sendai Japan, 1990.

Koizumi M. The concept of FGM. Ceramic Transactions Functionally Gradient Materials, Vol. 34, 1993, p. 310.

Xiang H. J., Yang J. Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction. Composites, Part B: Engineering, Vol. 39, 2008, p. 292303.

Li S. R., Fan L. L. Free vibration of FGM Timoshenko beams with throughwidth delamination. Science China Physics, Mechanics and Astronomy, Vol. 57, 2014, p. 927934.

Leissa A. W., Narita. Y. Natural frequencies of simply supported circular plates. Journal of Sound and Vibration, Vol. 70, 1990, p. 221229.

Kermani I. D., Ghayour M., Mirdamadi H. R. Free vibration analysis of multidirectional functionally graded circular and annular plates. Journal of Mechanical and Technology, Vol. 26, 2012, p. 33993410.

Dong C.Y. Threedimensional free vibration analysis of functionally graded annular plates using the ChebyshevRitz method. Materials and Design, Vol. 29, 2008, p. 15181525.

Wang Y., Xu R. Q., Ding H. J. Free axisymmetric vibration of FGM circular plates. Applied Mathematics and Mechanics (English Edition), Vol. 30, 2009, p. 10771082.

Malekzadeh P., Atashi M. M., Karami G. Inplane free vibration of functionally graded circular arches with temperaturedependent properties under thermal environment. Sound and Vibration, Vol. 326, 2009, p. 837851.

Iman D.K., Mostafa G., Hamid R. M. Free vibration analysis of multidirectional functionally graded circular and annular plates. Journal of Mechanical Science and Technology, Vol. 26, 2012, p. 33993410.

Lee W. M., Chen J. T., Lee Y. T. Free vibration analysis of circular plates with multiple circular holes using indirect BIEMs. Journal of Sound and Vibration, Vol. 304, 2007, p. 811830.

Ramesh M. N. V., Mohan Rao N. Free vibration analysis of pretwisted rotating FGM beams. International Journal of Mechanics and Materials in Design, Vol. 9, 2013, p. 367383.

Ying J., Lu C. F., Chen W. Q. Twodimensional elasticity solutions for functionally graded beams resting on elastic foundations. Composite Structure, Vol. 84, 2008, p. 209219.

Aydogdu M., Taskin V. Free vibration analysis of functionally graded beams with simply supported edges. Materials Design, Vol. 28, 2007, p. 16511656.

Pradhan K. K., Chakraverty S. Free vibration of functionally graded thin elliptic plates with various edge supports. Structural Engineering and Mechanics, Vol. 53, Issue 2, 2015, p. 337354.

Chen W. Q., Bian Z. G., Ding H. J. Threedimensional analysis of a thick FGM rectangular plate in thermal environment. Journal of Zhejiang University, Science A, Vol. 4, 2003, p. 14.

Kim NIl., Lee J. H. Geometrically nonlinear isogeometric analysis of functionally graded plates based on firstorder shear deformation theory considering physical neutral surface. Composite Structures, Vol. 153, 2016, p. 804814.

Foroughi H., Azhari M. Mechanical buckling and free vibration of thick functionally graded plates resting on elastic foundation using the higher order Bspline finite strip method. Meccanica, Vol. 49, 2014, p. 981993.

Jha D. K., Kant T., Singh R. K. An accurate twodimensional theory for deformation and stress analyses of functionally graded thick plates. International of Advance Structure Engineering, Vol. 62, 2014, p. 111.

Liu Z., Lee Y. T. Finite element analysis of threedimensional vibrations of thick circular and annular plates. Journal of Sound and Vibration, Vol. 233, Issue 1, 2000, p. 6380.

Malekzadeh P. Nonlinear free vibration of tapered Mindlin plates with edges elastically restrained against rotation using DQM. ThinWalled Structure, Vol. 46, 2008, p. 1126.

Malekzadeh P., Shahpari S. A. Ziaee H. R. Threedimensional free vibration off thick functionally graded annular plates in thermal environment. Journal of Sound and Vibration, Vol. 329, 2010, p. 425442.

Malekzadeh P., Golbahar Haghighi M. R., Atashi M. M. Free vibration analysis of elastically supported functionally graded annular plates subjected to thermal environment. Meccanica, Vol. 46, 2011, p. 893913.

Shen H. S., Wang Z. X. Assessment of Voigt and MoriTanaka models for vibration analysis of functionally graded plates. Composite Structures, Vol. 94, 2012, p. 21972208.

Li S. R., Zhou Y. H. Shooting method for nonlinear vibration and thermal buckling of heated orthotropic circular plates. Journal of Sound and Vibration, Vol. 248, 2001, p. 379386.

Ma L. S., Lee D. W. A further discussion of nonlinear mechanical behavior for FGM beams under inplane thermal loading. Composite Structures, Vol. 93, 2011, p. 831842.

Yun S., Li S. R. Thermal postbuckling of functionally graded material circular plates subjected to transverse pointspace constraints. Journal of Thermal Stresses, Vol. 37, 2014, p. 11531172.

Reddy J. N., Chin C. D. Theromechanical analysis functionally graded cylinders and plates. Journal of Thermal Stresses, Vol. 21, 1998, p. 593626.

Irie T., Yamada G., Takagi K. Natural frequencies of thick annular plates. Journal of Applied of Mechanics, Vol. 49, Issue 3, 1982, p. 633638.
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This research was supported by the National Natural Science Foundation of China (11262010, 11272278) and the Fundamental Research Funds for the Universities of Gansu (201405056001). The authors gratefully acknowledge both of the supports.