Shooting method for free vibration of FGM ReissnerMindlin circular plates resting on elastic foundation in thermal environments
QingLu Li^{1} , WeiDi Luan^{2} , ZuoQuan Zhu^{3}
^{1, 2, 3}Department of Engineering Mechanics, Lanzhou University of Technology, Lanzhou, Gansu 730050, P. R. China
^{1}Corresponding author
Journal of Vibroengineering, Vol. 19, Issue 6, 2017, p. 44234439.
https://doi.org/10.21595/jve.2017.17815
Received 7 October 2016; received in revised form 11 April 2017; accepted 13 April 2017; published 30 September 2017
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.
Keywords: functionally graded materials, moderately thick plate, free vibration, frequency, thermal environment.
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. 1. Geometry 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. 2. Variations 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 1. Temperaturedependent 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 2. The first fourth dimensionless natural frequencies ${\omega}_{n}$ 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 3. Dimensionless natural frequencies ${\omega}_{n}$ of FGM Mindlin plates with different graded index $\delta =$ 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 4. Dimensionless natural frequencies ${\omega}_{n}$ of FGM thick circular plates with clamped edge ($\nu =$ 0.3, ${T}_{c}={T}_{m}=$ 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. 3. Fundamental dimensionless frequency parameter ${\omega}_{1}$ of clamped plate versus $\delta $ in room temperature field
Fig. 4. Fundamental dimensionless frequency parameter ${\omega}_{1}$ of clamped plate versus $\delta $ subjected to heat conduction
Table 5. Dimensionless natural frequencies ${\omega}_{n}$ of FGM thick circular plates with clamped edge ($\nu =$ 0.3, $\delta =$ 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 6. Fundamental dimensionless frequencies ${\omega}_{1}$ of FGM thick circular plates with clamped edge subject to temperature rise and heat conduction ($\nu =$ 0.3, $\delta =$ 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 7. Dimensionless natural frequencies ${\omega}_{n}$ of FGM thick circular plates with clamped edge ($\nu =$ 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. 5. Anterior 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 8. Dimensionless natural frequencies ${\omega}_{n}$ 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 9. Dimensionless natural frequencies ${\omega}_{n}$ 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 10. Dimensionless natural frequencies ${\omega}_{n}$ of FGM thick circular plates with free edge ($\nu =0.3$, $Kw=Kg=$ 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 11. Dimensionless natural frequencies ${\omega}_{n}$ of FGM thick circular plates with free edge ($\nu =$ 0.3, ${T}_{c}=$ 300 K, ${T}_{m}=$ 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. 7. Anterior thirdorder modes of FGM moderately thick circular plate with simply supported edge
Fig. 8. Anterior 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.
Acknowledgements
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.
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