Accepted Manuscript

Three-dimensional exact solution for the free vibration of arbitrarily thick func-tionally graded rectangular plates with general boundary conditions

Guoyong Jin, Zhu Su, Shuangxia Shi, Tiangui Ye, Siyang Gao

PII: S0263-8223(13)00498-4DOI: http://dx.doi.org/10.1016/j.compstruct.2013.09.051Reference: COST 5383

To appear in: Composite Structures

Please cite this article as: Jin, G., Su, Z., Shi, S., Ye, T., Gao, S., Three-dimensional exact solution for the freevibration of arbitrarily thick functionally graded rectangular plates with general boundary conditions, CompositeStructures (2013), doi: http://dx.doi.org/10.1016/j.compstruct.2013.09.051

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1

Three-dimensional exact solution for the free vibration of arbitrarily

thick functionally graded rectangular plates with general boundary

conditions

Guoyong Jin, Zhu Su*, Shuangxia Shi, Tiangui Ye, Siyang Gao

College of Power and Energy Engineering, Harbin Engineering University,

Harbin, 150001, P. R. China

Corresponding author: Zhu Su

College of Power and Energy Engineering

Harbin Engineering University

Harbin, 150001, P. R. China

86-451-82569458 (phone),

xiuzhu0403@163.com

2

Abstract

A new three-dimensional exact solution for the free vibrations of arbitrarily thick

functionally graded rectangular plates with general boundary conditions is presented. The

three-dimensional elasticity theory is employed to formulate the theoretical model. According

to a power law distribution of the volume of the constituents, the material properties change

continuously through the thickness of the functionally graded plates. Each of displacements

of the plates, regardless of boundary conditions, is expanded as a three-dimensional (3-D)

Fourier cosine series supplemented with closed-form auxiliary functions introduced to

eliminate all the relevant discontinuities with the displacements and its derivatives at the

edges. Since the displacement fields are constructed adequately smooth throughout the entire

solution domain, an exact solution is obtained based on Rayleigh-Ritz procedure by the

energy functions of the plate. The excellent accuracy and reliability of the current solutions

are demonstrated by numerical examples and comparison of the present results with those

available in the literature, and numerous new results for thick FG plates with elastic boundary

conditions are presented. The effects of gradient indexes are also illustrated.

Keywords: functionally graded plates; three-dimensional elasticity theory; elastically

restrained edge; free vibration

3

1. Introduction

Functionally graded materials (FGMs) are a new class of composites structures that are

designed so that material properties vary smoothly and continuously along desired directions.

This is achieved by gradually varying the volume fraction of the constituent materials,

usually in the thickness direction only. The continuity of the material properties can

overcome problems arising from conventional laminated composite structures such as lager

inter-laminar stresses. Therefore, FGMs have attracted much interest since they were first

introduced by Japanese scientists in 1984. In recent years, the increasing utilization of FG

materials has motivated more intensive research activities. The FG plates are widely used in

various engineering applications, such as aircraft, space vehicles and nuclear, and in some

case they are frequently subjected to dynamic loads, which may cause fatigue damage and

severe reductions in the strength and stability of the structures. Therefore a thorough

understanding of their vibration characteristics is essential for designers and engineers.

As far as the plate deformation theories in previous studies are concerned, there are a

significant number of two-dimensional (2-D) theories. A partial classification of 2-D models

is briefly introduced here, including classical plate theory (CPT), first-order shear

deformation plate theory (FSDT), higher-order deformation plate theory (HSDT). More

detailed descriptions regarding the development of researches on this subject may be found in

several monographs respectively by Quta [1], Reddy [2], Carrera [3] and some survey articles

[4-7]. The CPT which is the simplest theory neglects the effects of shear and normal

deformations in the thickness direction. There are some studies regarding FG plates which are

4

based on the CPTs [8-13]. Although this theory gives sufficiently accurate results for thin

plates, it is not valid for the vibration analysis of the moderately thick and thick plates. In

order to eliminate the deficiency of the CPT, the effects of transverse shear deformations is

considered, and the FSDT were developed. There exist a large number of studies regarding

FG plates based on FSDTs [14-21]. Since the transverse shear strains in the FSDT are

assumed to be constant in the plate thickness, the shear correction factors are introduced to

adjust the transverse shear stiffness. However, the value of the shear correction factor is not a

constant but changes with material properties, loading case, boundary conditions. To avoid

the use of the shear correction factors and have a better prediction the FG thick plates, a

number of HSDTs based on the assumption of high-order variations of in-plane

displacements through the plate thickness are used [22-28]. Actually, those two-dimensional

theories reduce the dimensions of the plate problems from three to two by making certain

hypotheses on the stress and strain in the thickness direction. These assumptions greatly

simplify the formulation and solution in the analytical and computational methods, but they

also introduce errors at the same time. Since no hypotheses are assumed for the distribution

field of the deformations and stresses in the three-dimensional (3-D) elasticity theory and

contributions of all stresses and strains are considered by accounting for all the elastic

constants, the most accurate representation for the vibration analysis of plates can be obtained

by using the 3-D elasticity theory. Some investigations were carried out based on 3-D

elasticity theory [29-40]. In present work, in order to obtain highly accurate vibration result

for thick FG plates, the three-dimensional elasticity theory is just employed to formulate the

5

theoretical model.

Apart from the aforementioned plate theories, it has also been of great interest for

researchers to develop an accurate and efficient method which can be used to determine the

vibration behaviors of FG plates. Great research efforts have been devoted to this subject in

the past few years. So far, many computational methods are available for the vibration

analysis of FG plates, such as Ritz method [14,34, 36, 38], meshless method [18, 26],

extended Kantorovich method (EKM) [20], finite element method (FEM) [21, 22, 40],

meshless local Petrov-Galerkin (MLPG) method [27, 28], power series method [33],

differential quadrature (DQ) method [37], etc. Most of these methods were subsequently

extended to analyze the dynamic behaviors of FG plates from isotropic plates.

From the review of the literature, most of the previous studies on the FG plates are

confined to the classical boundary conditions, such as free, simply-supported and clamped

and their combinations. However, a variety of possible boundary restraining encountered in

practice engineering applications may not always be classical in nature, and there will always

be some elasticity along the supports, and there is a considerable lack of corresponding

researches regarding the free vibrations of the FG rectangular plates subjected to elastic

boundary conditions.

The objective of the present paper is to provide an accurate and reliable method for

three-dimensional vibration analysis of arbitrarily thick FG rectangular plates with arbitrary

boundary conditions including classical boundary conditions and elastic boundary conditions.

The present work can be considered as an extension of the authors’ previous works [41-43].

6

In this present work, the formulations are based on the three-dimensional elasticity theory.

According to a power law distribution of the volume of the constituents, the material

properties change continuously through the thickness of the functionally graded plates. Each

of displacements of the plates, regardless of boundary conditions, is expanded as a standard

three-dimensional Fourier cosine series supplemented with closed-form auxiliary functions

introduced to eliminate all the relevant discontinuities with the displacement and its

derivatives at the edges. Since the displacement field is constructed adequately smooth

throughout the entire solution domain, an exact solution can be obtained based on

Rayleigh-Ritz method by the energy functions of the plate. Compared with most of the

existing methods, the current method can be applicable to arbitrary boundary conditions

without requiring any special procedures or schemes. The excellent accuracy and reliability

of the current solutions are demonstrated by numerical examples and comparison of the

present results with those available in the literature, and numerous new results for FG plates

with elastic boundary conditions are presented, which can serve as the benchmark solutions

for other numerical methods in the future. The effects of gradient indexes are also illustrated.

2. Theoretical formulations

2.1 Description of the model

The physical dimensions and co-ordinate system of a 3-D FG plate under consideration

are shown in Fig.1(a). The length, width, thickness of functionally graded plate are

represented by a, b and h respectively. The plate geometry and dimensions are defined with

respect to a Cartesian coordinate system (x, y, z), introduced such that the bottom and top

7

surfaces of the plate lie in the plane z=0 and z=h. The displacements of the plate in the x, y

and z directions are denoted by u, v and w, respectively. The general boundary conditions are

represented as three of independent linear springs (ku, kv and kw) placed at the ends, and

different boundary conditions can be obtained by setting proper spring stiffness. For example

a clamped boundary can be obtained by assigning the springs’ stiffness at infinity, and a free

boundary can be obtained by assigning the springs’ stiffness at zero.

Consider FG plates made from a mixture of two material phases, for example, a metal

and a ceramic as shown in Fig. 1(b). Herein, the top surface of the plate is ceramics-rich

whereas the bottom surface is metal-rich. Young’s modulus and density per unit volume are

assumed to vary continuously through the plate thickness according to a power-law

distribution as

( ) ( )c m c mE z E E V E= − + (1)

( ) ( )c m c mz Vρ ρ ρ ρ= − + (2)

in which the subscripts c and m represent the ceramic and metallic constituents, respectively,

and the volume fraction Vc may be given by

(0 )( ) pc

zV z h

h= ≤ ≤ (3)

where z is the thickness coordinate , and p is the gradient index and takes only positive values.

The value of p equal to zero represents a fully ceramic, whereas infinite p indicates a fully

metallic plate. Since the small variations of the Poisson’s ratio, µ, it is assumed to be constant.

Typical values for metal and ceramics used in the FG plates are listed in Table 1.

2.2 Kinematic relations and Constitutive relation

8

According to the linear, small-strain elasticity theory, the strain components, εij (i,j=x, y,

z), are defined in terms of displacements as:

, , ,

, ,

xx yy zz

xy xz yz

u v w

x y z

u v u w v w

y x z x z y

ε ε ε

γ γ γ

∂ ∂ ∂= = =

∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂= + = + = +

∂ ∂ ∂ ∂ ∂ ∂

(4)

Based on Hooke’s law, the stress-displacement relations are defined as:

( ) 2 ( ) ( ) ( ) 0 0 0

( ) ( ) 2 ( ) ( ) 0 0 0

( ) ( ) ( ) 2 ( ) 0 0 0

0 0 0 ( ) 0 0

0 0 0 0 ( ) 0

0 0 0 0 0 ( )

xx xx

yy yy

zz zz

xy xy

xz xz

yz yz

z G z z z

z z G z z

z z z G z

G z

G z

G z

λ λ λ

λ λ λ

λ λ λ

σ ε

σ ε

σ ε

σ γ

σ γ

σ γ

+

+ +

= (5)

for FG plate, since the properties vary through the thickness of the plate with a desired

variation of volume fractions, λ(z) and G(z) are expressed as:

( ) ( )( ) , ( )

(1 )(1 2 ) 2(1 )

E z E zz G z

µλ

µ µ µ= =

+ − + (6)

In this study, the general boundary condition along each edge will be described in term

of three sets of restraining springs. Accordingly, the boundary conditions become

0 0 0, , ( 0)xx xy xzux vx wxk u k v k w xσ σ σ= = = = (7)

, , ( )uxa xx vxa xy wxa xzk u k v k w x aσ σ σ= = = = (8)

0 0, , ( 0)yy xy wyo yzuy vyk u k v k w yσ σ σ= = = = (9)

, , ( )yy xy yzuyb vyb wybk u k v k w y bσ σ σ= = = = (10)

where kux0, kvx0, kwx0, kuxa, kvxa and kwxa denote the six sets springs stiffness along the

9

edges x=0 and x=a while kuy0, kvy0, kwy0, kuyb, kvyb and kwyb denote the another six sets

springs stiffness along the edges y=0 and y=b.

2.3 Admissible displacement functions

Constructing of appropriate admissible displacement functions is of crucial importance

in the present method. In practice, the displacements of the plate are often expanded in the

terms of the beam functions under the same boundary conditions. Thus, there requires a

specific customized set of beam functions for each type of boundary conditions. This results

in that the use of this approach is inconvenient and will result in very tedious calculations.

Instead of the beam functions, one can also use other forms of admissible functions such as

simple or orthogonal polynomial and trigonometric functions. The lower order polynomials

cannot form a complete set, and the higher order polynomials trend to become numerically

unstable due to the computer round-off errors. Since the Fourier functions constitute a

complete set and exhibit an excellent numerical stability, a Fourier series representation may

be able to avoid these difficulties. However, the conventional Fourier series expression will

have a convergence problem along the boundary conditions except for a few simple boundary

edges. When the displacements of the plate are expressed in standard Fourier series, there

may be discontinuities in the displacements’ derivatives at the edges. Thus, the solution may

not converge or converge slowly. In order to overcome the difficulty, a modified Fourier

series method was previously proposed for the vibration analysis of elastically supported

isotropic beams [44] and thin plates [45, 46]. Most recently, it was extended by the authors to

analyze the composite laminated shell structures [41, 42] and double-panel cavity structure

10

[43] with general boundary conditions. In this present work, this method is further extended

to the vibration analysis of arbitrarily thick functionally graded rectangular plates. Therefore,

to satisfy the general boundary conditions of the thick 3-D rectangular plates, three

displacement functions of the plate are expanded as a modified 3-D Fourier series with

closed-form auxiliary functions:

0 0 0

1 1 2 20 0

3 1 4 20 0

5 1 6 20 0

cos cos cos

[ ( ) ( )]cos cos

( , , , )

[ ( ) ( )]cos cos

[ ( ) ( )]cos cos

mnt mx ny tzm n t

mx nymn z mn zm n

mx tzmt y mt ym t

ny tznt x nt xn t

A x y z

a z a z x y

u x y z t

a y a y x z

a x a x y z

λ λ λ

ξ ξ λ λ

ξ ξ λ λ

ξ ξ λ λ

∞ ∞ ∞

= = =

∞ ∞

= =

∞ ∞

= =

∞ ∞

= =

+ +

=

+ +

+ +

∑∑∑

∑∑

∑∑

∑∑

jwte

(11.a)

0 0 0

1 1 2 20 0

3 1 4 20 0

5 1 6 20 0

cos cos cos

[ ( ) ( )]cos cos

( , , , )

[ ( ) ( )]cos cos

[ ( ) ( )]cos cos

mnt mx ny tzm n t

mx nymn z mn zm n

mx tzmt y mt ym t

ny tznt x nt xn t

B x y z

b z b z x y

v x y z t

b y b y x z

b x b x y z

λ λ λ

ξ ξ λ λ

ξ ξ λ λ

ξ ξ λ λ

∞ ∞ ∞

= = =

∞ ∞

= =

∞ ∞

= =

∞ ∞

= =

+ +

=

+ +

+ +

∑∑∑

∑∑

∑∑

∑∑

jwte

(11.b)

0 0 0

1 1 2 20 0

3 1 4 20 0

5 1 6 20 0

cos cos cos

[ ( ) ( )]cos cos

( , , , )

[ ( ) ( )]cos cos

[ ( ) ( )]cos cos

mnt mx ny tzm n t

mx nymn z mn zm n

mx tzmt y mt ym t

ny tznt x nt xn t

C x y z

c z c z x y

w x y z t

c y c y x z

c x c x y z

λ λ λ

ξ ξ λ λ

ξ ξ λ λ

ξ ξ λ λ

∞ ∞ ∞

= = =

∞ ∞

= =

∞ ∞

= =

∞ ∞

= =

+ +

=

+ +

+ +

∑∑∑

∑∑

∑∑

∑∑

jwte

(11.c)

where ω denotes the natural frequency of the plate and 1j = − , /mx m aλ π= , /ny n bλ π= ,

/tz t hλ π= , and Amnt, Bmnt, Cmnt are the Fourier coefficients of three-dimensional Fourier

series expansions for the three displacement, respectively. a1mn, a2mn, a3mt, a4mt, a5nt, a6nt, b1mn,

11

b2mn, b3mt, b4mt, b5nt, b6nt, c1mn, c2mn, c3mt, c4mt, c5nt and c6nt are the supplemented coefficients of

the auxiliary functions. All of them need to be determined in future. The closed-form

auxiliary functions 1xξ , 2xξ , 1yξ , 2 yξ , 1zξ and 2zξ are respectively used to remove any

discontinuities potentially exhibited by the original displacement functions and their

derivatives at the edges x=0, x=a, y=0, y=b, z=0 and z=h. According to three-dimensional

elasticity theory, it is required that at least two-order derivatives of the three displacement

functions exist and continuous at any point on the plate. Therefore two auxiliary functions in

every direction are supplemented as demonstrated in Eqs.11(a)-(c). The closed-form auxiliary

functions are given as follow

2

2

1 2( ) ( 1) , ( ) ( 1)x xa

x x xx x x

a aζ ζ= − = − (12.a)

2

2

1 2( ) ( 1) , ( ) ( 1)y yb

y y yy y y

b bζ ζ= − = − (12.b)

2

2

1 2( ) ( 1) ( ) ( 1),z zh

z z zz z z

h hζ ζ= − = − (12.c)

It is easy to verify that

1 1

' '1 1(0) ( ) ( ) 0, (0) 1

x xx x a aζ ζ ζ ζ= = = = (13.a)

2 2

' '2 2(0) ( ) (0) 0, ( ) 1

x xx x a aζ ζ ζ ζ= = = = (13.b)

1 1

' '1 1(0) ( ) ( ) 0, (0) 1

y yy y b bζ ζ ζ ζ= = = = (13.c)

2 2

' '2 2(0) ( ) (0) 0, ( ) 1

y yy y b bζ ζ ζ ζ= = = = (13.d)

1 1

' '1 1(0) ( ) ( ) 0, (0) 1

z zz z h hζ ζ ζ ζ= = = = (13.e)

2 2

' '2 2(0) ( ) (0) 0, ( ) 1

z zz z h hζ ζ ζ ζ= = = = (13.f)

It can be proven mathematically that the series expansion given in Eq. (11) can be simply

12

differentiated, through term-by-term, to obtain uniformly convergent series expansions for up

to the second-order derivatives. It should be noted that in numerical calculations the three

series solutions have to be truncated numerically to M, N and T.

2.4 Energy functions and the solution procedure

The strain energy Π for FG rectangular plates is given in integral form by:

0 0 0

1( )

2

h b a

xx xx yy yy zz zz xy xy xz xz yz yz dxdydzσ ε σ ε σ ε σ γ σ γ σ γ+ + + + +Π = ∫ ∫ ∫ (14)

Substituting to Eq. (4), Eq. (5) and Eq. (6) into Eq. (14), the strain energy Π can be

written as follows:

2 2 2

0 0 0

2 2 2 2 2 2

1( 2 )[( ) ( ) ( ) ]

2

2 ( )

[( ) ( ) ( ) ( ) ( ) ( ) ]

2 ( )

h b au v w

Gx y z

u v u w v w

x y x z y z

u v u w v wG

y x z x z y

u v u w v wG dxdydz

y x z x z y

λ

λ

∂ ∂ ∂+ + +

∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂+ + +

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂+ + + + + +

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂+ + +

∂ ∂ ∂ ∂ ∂ ∂

Π = ∫ ∫ ∫

(15)

The potential energy P stored in the boundary springs can be expressed as:

2 2 20 0 0

0 0

2 2 20

0 0

1[( ) ( ) ( ) ]

2

1[ ) ( ) ( ) ]

2

h b

uxa vxa wxaux vx wx

h a

uyo vyo wyuyb vyb wyb

P k k u k k v k k w dydz

k k u k k v k k w dxdz

= + + + + +

+ + + + + +

∫ ∫

∫ ∫ (16)

The kinetic energy T of the FG plates is depicted as:

2 2 2

0 0 0

1[( ) ( ) ( ) ]

2

h b au v w

T dxdydzt t t

ρ∂ ∂ ∂

= + +∂ ∂ ∂∫ ∫ ∫ (17)

In order to determine the coefficients in the functions, the Rayleigh-Ritz method is

applied in present work, which is a powerful tool in the vibration analysis of structural

13

elements because of its simplicity and high accuracy. The Lagrangian energy function of the

plate can be expressed in terms of strain energy, kinetic energy and potential energy stored in

boundary edges as:

L T P= −Π− (18)

Substituting the above Eq. (11) into Eq. (18), and minimizing the Lagrangian energy

functional L with respect to the unknown coefficients

1 2 1 2 1 2 6, , , , , , , , , , )0 ( u mnt mn mn mnt mn mn mnt mn mn nt

u

A a a B b b C c c cL

αα

=∂

=∂

(19)

The problem will be transformed into a eigenvalue and eigenvector problem, and the

following governing eigenvalue equation in matrix form can be achieved

2[[ ] ] 0K M Xω− = (20)

where [ ] [ ] [ ]s pK K K= + , [ ]sK is the symmetric stiffness matrix obtained from the strain

energy, [ ]pK is the symmetric stiffness matrix obtained from the potential energy stored in

boundary edge, and [ ]M is the symmetric mass matrix obtained from the kinetic energy.

They can be written as:

[

[ ] 0 0

] 0 [ ] 0

0 0 [ ]

uu

vv

ww

M

M M

M

= (21.a)

[ ] [ ] [ ]

[ ] [ ] [ ]

[ ]

s

suu suv suw

svv svw

sww

K K K

K K K

sym K

= (21.b)

[ ] 0 0

[ ] 0 [ ] 0

0 0 [ ]

p

puu

pvv

pww

K

K K

K

=

(21.c)

The element of the stiffness matrix [ ]K and mass matrix [ ]M are given in Appendix

14

A. X is the column matrix composed of the unknown coefficients expressed in the

following form:

[ , , ]u v wX X X X Τ= (22)

where

000 100 1 200 2 300 3

5400 4 500 600 6

[ , , , , , , , , , , , , , ,

, , , , , , , , ]

umnt MNT MN MN MT

NTMT NT

X A A A a a a a a a

a a a a a a

=

000 100 1 200 2 300 3

5400 4 500 600 6

[ , , , , , , , , , , , , , ,

, , , , , , , , ]

vmnt MNT MN MN MT

NTMT NT

X B B B b b b b b b

b b b b b b

=

000 100 1 200 2 300 3

5400 4 500 600 6

[ , , , , , , , , , , , , , ,

, , , , , , , , ]

wmnt MNT MN MN MT

NTMT NT

X C C C c c c c c c

c c c c c c

=

Eq. (20) represents a standard characteristic equation. All the natural frequencies and

mode shapes can be obtained by solving Eq. (20). Each eigenvector is the set of Fourier series

coefficients for the corresponding mode. Then the displacements and force of the plate can be

obtained easily.

3. Numerical examples

In this section, numerical examples for the free vibration analysis of FG plates with

various gradient index and arbitrary boundary conditions are presented to validate the

accuracy and reliability of the present method. First, the determination of the spring stiffness

is investigated, and the convergence of the present method is checked by analyzing the FG

plates with four simply supported edges. Then, FG plates with various classical boundary

conditions and elastic boundary conditions are studied. Finally, the effects of gradient indexes

are discussed as well.

3.1 Determination of the spring stiffness

15

In this study, three sets of linear springs are used to simulate the given or typical

boundary conditions, and a clamped edge can be obtained by assigning the spring stiffness at

infinity. However, numerical computations cannot deal with infinite values, and it is

significant for the accuracy of the result to setting proper spring stiffness to clamped edge.

Therefore the determination of the spring stiffness is investigated. The effects of the spring

stiffness (ku, kv and kw) on the convergence of the first six non-dimensional frequency

parameters 2/ /c ca h Eω ρΩ = for SSSS square FG plates are shown in Table 2. The

thickness-side ratios (h/a) are take as 0.1, 0.2 and 0.5. The gradient indexes are taken to be 1.

The spring parameters vary from 5×1015

to 5×1019

. From the comparison, it shows that good

convergence of the solution is obtained by increasing the spring stiffness. The spring stiffness

185 10u v wk k k= = = × is the proper stiffness.

3.2 Convergence study

Theoretically, there are infinite terms in the modified Fourier series solution. However,

the series is numerically truncated and only finite terms are counted in actual calculations.

The convergence of the present method will be checked. The non-dimensional fundamental

frequency parameters 2/ /c ca h Eω ρΩ = of the FG plates with four simply supported edges

are presented in Table 3 for different truncation schemes. The geometry of the plates is given

as: a=b=1m, thickness-side ratios h/a= 0.1, 0.2 and 0.5, and the values of the gradient indexes

are taken to be 0, 1, 2, 4, 8 and 10. It is found that the results of this study show a monotonic

convergence trend. When the truncated numbers change from 14×14×7 to 16×16×8, the

difference of the frequency parameters does not exceed 0.3136% for the worst case, which is

16

acceptable. More accurate results may be obtained by further truncated numbers, but the

computational cost will be increased. Therefore, for the sake of both accuracy and

computational cost, the truncated number of the displacement expressions will be uniformly

selected as 14 14 7M N T× × = × × in the following numerical examples.

3.3 FG plates with classical boundary conditions

There are many possible boundary conditions at the ends of a rectangular FG plate and it

is impossible to undertake an all-encompassing survey of the vibrations for every case of

boundary condition. In this section, three types of classical boundary conditions often

encountered in practice are studied. Take edge x=0 for example, they are defined as follows:

Free boundary condition (F):

0, 0, 0xy xzxxσ σ σ= = =

Simply-supported boundary condition (S):

0, 0, 0xx v wσ = = =

Clamped boundary conditions (C):

0, 0, 0u v w= = =

The corresponding spring stiffness for the three types of classical boundaries is given in

Table 4. First, the accuracy of the present solution is demonstrated in Tables 5-7. Table 5

shows the first seven frequency parameters 2 /b h Dϖ ω ρ= for square isotropic plates with

different boundary conditions. The results are compared with other published solutions by

using the 3-D Ritz method with simple algebraic polynomials [29], the 3-D Ritz method with

general orthogonal polynomials using the Gram-Schmidt process [30], and the 3-D Ritz

17

method with general orthogonal polynomials [32]. The first five non-dimensional frequency

parameters 2/ /c ca h Eω ρΩ = of Al/Al2O3 square plates under simply-supported boundary

conditions with various thickness-side ratios and gradient indexes are presented in Table 6.

The present results are compared with those obtained by Huang et al [38] employing 3-D

elastic theory and a variational Ritz methodology. And the difference is very small, and does

not exceed 0.053% for the worst case. The non-dimensional natural frequency parameters

/m mh Eψ ω ρ= of SSSS square Al/ZrO2 plates are shown in Table 7. It is seen that the

present results are in good agreement with those obtained by higher-order theory [25] and

three-dimensional theory by employing the powers series method [33]. In general, the present

solutions agree very well with those available in the literature.

In Tables 8 and 9, the frequency parameters /m mh Eψ ω ρ= of FG plate with different

boundary conditions are illustrated. The symmetric boundary conditions, including SSSS,

SCSC and SFSF boundary conditions, and asymmetric boundary conditions, including SCSF,

SSSC and SSSF boundary conditions, are considered. It can be observed from Tables that the

values of the non-dimensional frequency parameters corresponding to different boundary

conditions have obvious difference. The frequencies of the FG plates with SSSF, SFSF and

SCSF boundary conditions is smaller, while the frequencies of the FG plate with SCSC and

SSSC boundary conditions is obviously larger. This is due to the fact that higher constraints

at the edges increase the flexural rigidity of the plate, leading to higher frequency response.

For the FG plates with h/a=0.5 and b/a=2, the frequency parameters of the plates under SSSS,

SCSC and SSSC are same. This is maybe due to that the effects of the free edges and

18

clamped edges at y=0 and y=b have little different with the increase of the thickness-side

ratios. 3-D mode shapes for SSSS, SFSF and SCSC square thick FG plate are depicted in

Figs. 2-4.

3.4 FG plates with elastic boundary conditions

The above numerical examples are presented as FG plates with classical boundary

conditions. The frequency parameters for arbitrary classical boundaries can be obtained easily

by setting the spring stiffness a proper value. What’s more the present method also can offer a

unified solution for elastic boundary conditions. In this section, five types of elastic boundary

conditions often encountered in practice are studied. There are five types of elastic boundary

conditions which are considered for the FG plates. Fist type of elastic boundary E1

is

considered to be that only x-axial displacement along the plate edge are elastically restrained

(i.e. u ≠ 0, v = w =0). Second type of elastic boundary E2 allows elastically restrained

displacement in the y axes direction (i.e. v ≠ 0, u = w =0). Only z-axial displacement are

elastically restrained, which is defined as third type of elastic boundary E3 (i.e. w ≠ 0, v = u =

0). When both x-axial displacement and y-axial displacement along the plate edge are

elastically restrained (i.e. u ≠ 0, v ≠ 0, w=0), the edge support is donated by E4. Fifth type of

elastic boundary E5 allows elastically restrained rotation in the all axes directions (i.e. u ≠ 0, v

≠ 0, v ≠ 0). The corresponding spring stiffness for five types of elastic boundaries is given as

follows:

E1: 10 181 10 , 5 10u v wk k k= × = = ×

E2: 10 181 10 , 5 10v u wk k k= × = = ×

19

E3: 10 181 10 , 5 10w u vk k k= × = = ×

E4: 10 181 10 , 5 10u v wk k k= = × = ×

E5:

101 10u v wk k k= = = ×

There are 125 different combinations of the five types of elastic boundary conditions due to

the rectangular plates have four ends. It is strenuous to analyze all conditions, and we only

consider that the four ends conditions of the FG plates are same.

The non-dimensional frequency parameters /m mh Eψ ω ρ= of the FG plates with

different elastic boundary conditions are shown in Tables 8 and 9. Five types of elastic

boundary conditions (i.e. E1 E

1 E

1 E

1, E

2 E

2 E

2 E

2, E

3 E

3 E

3 E

3, E

4 E

4 E

4 E

4 and E

5 E

5 E

5 E

5)

are studied. The thickness-side ratios are 0.1, 0.2, 0.5, and the gradient indexes are 0, 1, 2, 5.

Table 8 shows the frequency parameters of Al/Al2O3 plates and Table 9 shows the frequency

parameters of Al/ZrO2 plates. From the tables, we can see that the frequency parameters of

FG square plates with E1 E

1 E

1 E

1 and E

2 E

2 E

2 E

2 boundary conditions are similar. The two

sets of linear springs have same effects on the frequency of the FG square plates. For the FG

plates with b/a=2, the frequency parameters of FG square plates with E1 E

1 E

1 E

1 and E

2 E

2

E2 E

2 boundary conditions are also similar besides the frequency parameters of the FG plates

with h/a=0.1. This is maybe due to the effect of the thickness-side ratios on the frequency

parameters of the FG plates.

In contrast to most existing techniques, the current method can be universally

applicable to variety of boundary conditions including all the classical cases, elastic restraints.

The change of the boundary conditions is as easy as modifying material properties and

20

geometry dimensions without the need of making any change to the solution procedure. In

addition, the current solution can be readily applied to plates with more complex boundary

conditions like point supports, non-uniform elastic restraints, mix boundaries, partial supports

and their combinations.

3.5 Parametric studies

The last section, the present method is applied to investigate the free vibration of

rectangular plates with different gradient indexes. According to the Eqs. (3) and (4), the

variation of the volume fraction through the thickness is presented in Fig. 5. It can be seen

that the volume fraction Vc varies quickly near the lowest surface for p<1 and increases

quickly near the top surface for p>1. First, the variation of the fundamental natural frequency

parameter with the power law index for square Al/Al2O3 plates with h/a=0.5 is presented in

Fig. 6(a). SSSS, SCSC, SFSF, SSSF, SSSC and SSSF boundary conditions are studied. It can

be seen that the six curves show a similar behavior. The frequencies decrease as the gradient

index increases. When the gradient index is minor, the frequencies decrease quickly, while

when the gradient index gradually raises, the change of the frequencies decrease slowly. Fig.

6(b) shows the variation of the fundamental frequency parameters with the power law index

for square Al/Al2O3 plates with h/a=0.2. Figs. 6(c) and 6(d) show the variation of the

fundamental natural frequency parameter with the power law index for square Al/ZrO2 plates

when h/a=0.5 and 0.2. The frequency characteristics shown in these three Figs are similar to

those in Fig. 6(a).

21

4. Conclusions

In this paper, a unified and accurate solution method has been developed to deal with the

free vibration analysis of arbitrarily thick functionally graded plate with general boundary

based on the linear, small-strain 3D elasticity theory. Each of displacements of the plates,

regardless of boundary conditions, is expanded as a standard three-dimensional Fourier

cosine series supplemented with closed-form auxiliary functions introduced to eliminate all

the relevant discontinuities with the displacement and its derivatives at the edges.

Mathematically, such series expansions are capable of representing any functions, including

the exact displacement solutions. Rayleigh-Ritz method is applied to obtain solution by the

energy functions of the plates. One of the advantages of the current method is that it can be

applicable to a variety of boundary conditions achieved by setting the stiffness of restraining

springs without requiring any special procedures or schemes. The excellent accuracy of the

current solutions is demonstrated by numerical examples and comparison of the present

results with those available in the literature, and the effects of gradient indexes are illustrated.

New results for thick FG plates with elastic boundary conditions are presented, which may be

serve as benchmark solutions.

Appendix A. Detailed expressions for the stiffness matrix and mass matrix

The detailed expressions of the mass matrix and the first row of the stiffness matrix in

Eq. (20) are given as follows. To make the expressions simple and clear, some indexes are

pre-defined.

1 1 1( 1)( 1)( 1) ( 1)( 1) ( 1)( 1)( 1) ( 1)( 1)s t M N n M m q t M N n M m= − + + + − + + = − + + + − + +

22

1 1 1 1 1( 1)( 1)( 1) ( 1)( 1) ( 1)( 1)( 1) ( 1)( 1)s l M N n M m q l M N n M m= − + + + − + + = − + + + − + +

2 2 1 1 1( 1)( 1)( 1) ( 1)( 1) ( 1)( 1)( 1) ( 1)( 1)s l M T t M m q l M T t M m= − + + + − + + = − + + + − + +

3 3 1 1 1( 1)( 1)( 1) ( 1)( 1) ( 1)( 1)( 1) ( 1)( 1)s l N T t N n q l N T t N n= − + + + − + + = − + + + − + +

1 1 1

11,

0 0 0

( )cos cos cos cos cos cosa b h

uu s q mx ny tz m x n y t zM z x y z x y zdxdydzρ λ λ λ λ λ λ= ∫ ∫ ∫

11 1

12

, 1

0 0 0

( )( )cos cos cos cos cosuu s q l z

a b h

mx ny tz m x n yM zz x y z x y dxdydzζρ λ λ λ λ λ= ∫ ∫ ∫

2 11 1

13

,

0 0 0

( )( )cos cos cos cos cosuu s q l y z

a b h

mx ny tz m x tM yz x y z x zdxdydzζρ λ λ λ λ λ= ∫ ∫ ∫

3 1 1 1

14

,

0 0 0

( )( )cos cos cos cos cosuu s q l x y z

a b h

mx ny tz n tM xz x y z y zdxdydzζρ λ λ λ λ λ= ∫ ∫ ∫

1 1 1 1

21,

0 0 0

( ) ( )cos cos cos cos coslz

a b h

uu s q mx ny m x n y t zzM z x y x y zdxdydzζρ λ λ λ λ λ= ∫ ∫ ∫

1 1 11 1

22

,

0 0 0

( ) ( )( )cos cos cos cosuu s q lz l z

a b h

mx ny m x n yM z zz x y x y dxdydzζ ζρ λ λ λ λ= ∫ ∫ ∫

1 2 1 11

23

,

0 0 0

( ) ( ) cos( )cos cos cosuu s q lz l y t z

a b h

mx ny m xM z y zz x y x dxdydzζ ζ λρ λ λ λ= ∫ ∫ ∫

1 3 1 1 1

24

,

0 0 0

( ) ( )( )cos cos cos cosuu s q lz l x y z

a b h

mx ny n tM z xz x y y zdxdydzζ ζρ λ λ λ λ= ∫ ∫ ∫

2 1 1 1

31,

0 0 0

( ) ( )cos cos cos cos cosly

a b h

uu s q mx tz m x n y t zyM z x z x y zdxdydzζρ λ λ λ λ λ= ∫ ∫ ∫

2 1 11 1

32

,

0 0 0

( ) ( )( )cos cos cos cosuu s q ly l z

a b h

mx tz m x n yM y zz x z x y dxdydzζ ζρ λ λ λ λ= ∫ ∫ ∫

2 2 1 11

33

,

0 0 0

( ) ( )cos( )cos cos cosuu s q ly l y t z

a b h

mx tz m xM y y zz x z x dxdydzζ ζ λρ λ λ λ= ∫ ∫ ∫

23

2 3 1 1 1

34

,

0 0 0

( ) ( )( )cos cos cos cosuu s q ly l x y z

a b h

mx tz n tM y xz x z y zdxdydzζ ζρ λ λ λ λ= ∫ ∫ ∫

3 1 1 1

41,

0 0 0

( ) ( )cos cos cos cos cosa b h

uu s q ny tz m x n y t zlxM z x y z x y zdxdydzρ ζ λ λ λ λ λ= ∫ ∫ ∫

3 1 11 1

42

,

0 0 0

( )( ) ( )cos cos cos cosuu s q l z

a b h

ny tz m x n ylxM zz x y z x y dxdydzζρ ζ λ λ λ λ= ∫ ∫ ∫

3 2 1 11

43

,

0 0 0

( )cos( ) ( )cos cos cosuu s q l y t z

a b h

ny tz m xlxM y zz x y z x dxdydzζ λρ ζ λ λ λ= ∫ ∫ ∫

3 3 1 1 1

44

,

0 0 0

( )( ) ( )cos cos cos cosuu s q l x y z

a b h

ny tz n tlxM xz x y z y zdxdydzζρ ζ λ λ λ λ= ∫ ∫ ∫

uu vv wwM MM = =

1 1 1

1 1 1 1

1

1 1

1

0

0

11,

0 0 0

( 2 ) cos

( 1) )

(

[( 2 ) sin cos cos sin cos cos

sin cos cos sin cos

( cos cos cos cos

(

mx

m m

ux uxa

uy uyb

a b h

uu s q mx m x mx ny tz m x n y t z

ny n y ny tz m x n y t z

ny tz n y t z

G x

k

K G x y z x y z

y z x y z

k k y z y z

k

λ λ

λ λ λ λ λ λ λ λ λ

λ λ λ λ λ λ λ

λ λ λ λ+

+

−

+

= +

+

+ +

+

∫ ∫ ∫

1

1 11) )cos cos cos cos ]n n

mx tz m x t zx z x z dxdydzλ λ λ λ+−

1

1

1

1 1

1 1

1 1 1

12

, 1

0

0

1

0 0 0

( )

( 2 ) cos ( )

( 1) ) ( )

( 1

( cos cos cos

(

[( 2 ) sin cos cos sin cos

sin cos cos sin

uu s q l z

mx l z

m m

ux uxa l z

uy uyb

ny tz n y

a b h

mx m x mx ny tz m x n y

ny n y ny tz m x n y

K z

G x z

z

k

k k y z y

k

G x y z x y

y z x y

ζ

λ λ ζ

ζλ λ λ

λ λ λ λ λ λ λ λ

λ λ λ λ λ λ+

+ +

+ −

+ −

+

+

= +∫ ∫ ∫

1

1 1) ) cos ( )]cos cosn n

mx l ztz m xx zz x dxdydzλ ζλ λ+

2 1

1

1

1 1

1

1 1

1 1

13

,

'

0

0

1

0 0 0

( )

( 2 )( ) cos ( )

( 1) ) ( )

(0)

( cos cos cos

(

[( 2 ) sin cos cos sin cos

sin cos cos cosl y

uu s q l y z

mx z

m m

ux uxa l y

uy l y

ny tz t z

a b h

mx m x mx ny tz m x t

ny ny tz m x t

K y

G x y

yk k y z z

k

G x y z x z

y z x z

ζ

λ λ ζ

ζ

ζ

λ λ λ

λ λ λ λ λ λ λ λ

λ λ λ λ λ+

+ −

−

+

+ +

+

= +

+

∫ ∫ ∫

1 1 1cos ( ))cos cos cos cos ]

uyb l y mx tz m x t znyk b b x z x z dxdydzζ λ λ λ λλ

24

3 1

1

1 1 1 1

1 1

1 1 1

14 '

,

0

0 0 0

) ( )

( 2 ) cos ( )

(0) cos ( ))( cos cos cos cos

[( 2 )( sin cos cos cos cos

sin cos sin cos

uu s q l x y z

y mx l x y z

ux l x uxa mx l x ny tz n y t z

a b h

mx mx ny tz n t

ny n ny tz n t

K x

G x x

a ak k y z y z

G x y z y z

y z y z

ζ

λ λ ζ

ζ λ ζ λ λ λ λ

λ λ λ λ λ λ λ

λ λ λ λ λ λ+

+ +

= + −

+

∫ ∫ ∫

1

1 10 ( 1) )cos( cos cos cos ]n n

uy uyb mx tz m x t zk xk z x z dxdydzλ λ λ λ++ −+

1 1 1

1 1 1 1

1

11,

0 0 0

2 cos

[ sin cos cos cos sin cos

sin cos sin cos cos ]mx

a b h

uv s q mx n y mx ny tz m x n y t z

ny m x ny tz m x n y t zG x

K x y z x y z

y z x y z dxdydzλ

λλ λ λ λ λ λ λ λ

λ λ λ λ λ λ λ

=

+

∫ ∫ ∫

1

1

1 1

1 1 1

12

, 1 1

0 0 0

( )

2 cos ( )]

[ sin cos cos cos sin

sin cos sin cos

uv s q l z

mx l z

a b h

mx n y mx ny tz m x n y

ny m x ny tz m x n y

K z

G x z z

x y z x y

y z x y dxdydz

ζ

λ ζ

λλ λ λ λ λ λ λ

λ λ λ λ λ λ+

= ∫ ∫ ∫

2 1

1

1 1

1 1 1

13 '

,

0 0 0

) ( )

2 cos ( ) ]

[ ( sin cos cos cos cos

sin cos sin cos

l yuv s q z

mx l y z

a b h

mx mx ny tz m x t

ny m x ny tz m x t

K y

G x y

x y z x z

y z x z dxdydz

ζ

λ ζ

λ λ λ λ λ λ λ

λ λ λ λ λ λ

= −

+

∫ ∫ ∫

3 1

1

1 1

1 1

14

,

'

1

0 0 0

( )

2 ( )cos ( ) ]

[ sin cos cos sin cos

sin cos cos cos

uv s q l x y z

mx l x y z

a b h

mx n y mx ny tz n t

ny ny tz n t

K x

G x x

x y z y z

y z y z dxdydz

ζ

λ ζ

λλ λ λ λ λ λ λ

λ λ λ λ λ−

=

+

∫ ∫ ∫

1 1 1

1 1 1 1

1

11,

0 0 0

2 cos

[ sin cos cos cos cos sin

cos sin sin cos cos ]

t z

mx

a b h

uw s q mx t z mx ny tz m x n y

tz m x ny tz m x n y t zG x

K x y z x y z

y z x y z dxdydzλ

λλ λ λ λ λ λ λ λ

λ λ λ λ λ λ λ

=

+

∫ ∫ ∫

1

1

1 1

1 1 1

12 '

, 1

0 0 0

) ( )

2 cos ( )]

[ ( sin cos cos cos sin

cos sin sin cos

uw s q l z

mx l z

a b h

mx mx ny tz m x n y

tz m x ny tz m x n y

K z

G x z z

x y z x y

y z x y dxdydz

ζ

λ ζ

λ λ λ λ λ λ λ

λ λ λ λ λ λ+

= −∫ ∫ ∫

2 1

1

1 1

1 1 1

13

, 1

0 0 0

( )

2 cos ( ) ]

[ sin cos cos cos sin

cos sin sin cos

uw s q l y z

mx l y z

a b h

mx t z mx ny tz m x t

tz m x ny tz m x t

K y

G x y

x y z x z

y z x z dxdydz

ζ

λ ζ

λλ λ λ λ λ λ λ

λ λ λ λ λ λ

=

+

∫ ∫ ∫

3 1

1

1 1

1 1

14

,

'

1

0 0 0

( )

2 ( ) cos ( ) ]

[ sin cos cos cos sin

cos sin cos cos

uw s q l x y z

mx l x y z

a b h

mx t z mx ny tz n t

tz ny tz n t

K x

G x x

x y z y z

y z y z dxdydz

ζ

λ ζ

λλ λ λ λ λ λ λ

λ λ λ λ λ−

=

+

∫ ∫ ∫

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31

List of Collected Table and Figure Captions

Table 1 Material properties of the used FG plate.

Table 2 First six frequency parameters 2/ /c ca h Eω ρΩ = for SSSS square Al/Al2O3 plates

with different thickness-side ratio when p=1.

Table 3 The fundamental frequency parameters 2/ /c ca h Eω ρΩ = for simply-supported

square Al/Al2O3 plates.

Table 4 Value of spring stiffness for three types of classical boundary conditions

Table 5 The first seven frequency parameters 2 /b h Dϖ ω ρ= of square isotropic plates with

different boundary conditions.

Table 6 The non-dimensional frequency parameters 2/ /c ca h Eω ρΩ = of Al/Al2O3 square

plate with simply-supported boundary conditions.

Table 7 The non-dimensional frequency parameters /m mh Eψ ω ρ= of Al/ZrO2 square

plates with simple-supported boundary conditions (τ=h/a).

Table 8 The non-dimensional frequency parameters /m mh Eψ ω ρ= of Al/Al2O3 plates with

different classical boundary conditions.

Table 9 The non-dimensional frequency parameters /m mh Eψ ω ρ= of Al/ZrO2 plates with

different classical boundary conditions.

Table 10 The non-dimensional frequency parameters /m mh Eψ ω ρ= of Al/Al2O3 plates

with different elastic boundary conditions.

Table 11 The non-dimensional frequency parameters /m mh Eψ ω ρ= of Al/ZrO2 plates with

different elastic boundary conditions.

32

Fig. 1. Schematic diagram of the 3-D rectangular functionally graded plates: (a) the geometry

and coordinates; (b) FG plate of two material phases; (c) the boundary restraining springs.

Fig.2. Modes shapes of Al/Al2O3 plates with SSSS boundary condition when h/a=0.5 and

p=1: (a) m = 1; (b) m = 2; (c) m = 3; (d) m = 4; (e) m=5; (f) m=6.

Fig.3. Modes shapes of Al/Al2O3 plates with SFSF boundary condition when h/a=0.5 and

p=1: (a) m = 1; (b) m = 2; (c) m = 3; (d) m = 4; (e) m=5; (f) m=6.

Fig.4. Modes shapes of Al/Al2O3 plates with SCSC boundary condition when h/a=0.5 and

p=1: (a) m = 1; (b) m = 2; (c) m = 3; (d) m = 4; (e) m=5; (f) m=6.

Fig. 5. Variation of volume fraction Vc through the non-dimensions thickness of FG plates

Fig.6 Variation of the frequency parameters with the gradient index for FG square plates:(a)

Al/Al2O3 plates with h/a=0.5; (b) Al/Al2O3 plates with h/a=0.2; (c) Al/ZrO2 plates with

h/a=0.5; (d) Al/ZrO2 plates with h/a=0.2.

Table 1

Material properties of the used FG plate

Material Properties

E(GPa) μ ρ (kg/m3)

Aluminum(Al) 70 0.3 2702

Alumina(Al2O3) 380 0.3 3800

Zirconia(ZrO2) 200 0.3 5700

Table 2

First six frequency parameters 2 / /c ca h E for SSSS square Al/Al2O3 plates with

different thickness-side ratio when p=1.

h/a ku=kv

=kw

Ω1 Ω2 Ω3 Ω4 Ω5 Ω6

= kw

0.1 5×1015

4.4278 10.6305 10.6305 16.1998 16.1998 16.4065

5×1016

4.4281 10.6309 10.6309 16.2007 16.2007 16.4075

5×1017

4.4281 10.6309 10.6309 16.2008 16.2008 16.4076

5×1018

4.4281 10.6309 10.6309 16.2008 16.2008 16.4076

5×1019

4.4280 10.6309 10.6309 16.2008 16.2008 16.4076

0.2 5×1015

4.0994 8.0881 8.0881 9.1083 9.1083 11.4166

5×1016

4.0996 8.0886 8.0886 9.1085 9.1085 11.4166

5×1017

4.0996 8.0886 8.0886 9.1086 9.1086 11.4166

5×1018

4.0996 8.0886 8.0886 9.1086 9.1086 11.4166

5×1019

4.0996 8.0886 8.0886 9.1086 9.1086 11.4166

0.5 5×1015

2.9901 3.2011 3.2011 4.4672 4.4672 5.6301

5×1016

2.9902 3.2013 3.2013 4.4673 5.6303 5.6303

5×1017

2.9902 3.2013 3.2013 4.4673 5.6303 5.6303

5×1018

2.9902 3.2013 3.2013 4.4673 5.6303 5.6303

5×1019

2.9902 3.2013 3.2013 4.4673 5.6303 5.6303

Table 3

The fundamental frequency parameters 2 / /c ca h E for simply-supported square

Al/Al2O3 plates.

h/a Truncated

p=0 p=1 p=2 p=4 p=8 p=10

number

0.1 8×8×4 5.7989 4.4450 4.0325 3.8277 3.7058 3.6431

9×9×4 5.7978 4.4439 4.0314 3.8268 3.7050 3.6425

10×10×5 5.7855 4.4335 4.0212 3.8171 3.6955 3.6328

12×12×6 5.7818 4.4330 4.0179 3.8140 3.6927 3.6300

14×14×7 5.7794 4.4281 4.0158 3.8121 3.6909 3.6281

16×16×8 5.7786 4.4273 4.0151 3.8114 3.6902 3.6275

18×18×9 5.7779 4.4267 4.0145 3.8108 3.6897 3.6269

20×20×10 5.7776 4.4264 4.0142 3.8106 3.6895 3.6266

0.2 8×8×4 5.3097 4.1042 3.7013 3.4608 3.3142 3.2576

9×9×4 5.3096 4.1011 3.7011 3.4601 3.3141 3.2575

10×10×5 5.3060 4.1011 3.6982 3.4579 3.3110 3.2542

12×12×6 5.3050 4.1001 3.6972 3.4570 3.3101 3.2532

14×14×7 5.3043 4.0996 3.6967 3.4565 3.3096 3.2627

16×16×8 5.3041 4.0993 3.6964 3.4563 3.3094 3.2525

18×18×9 5.3039 4.0992 3.6963 3.4561 3.3093 3.2523

20×20×10 5.3038 4.0991 3.6962 3.4560 3.3092 3.2522

0.5 8×8×4 3.7616 2.9913 2.6672 2.3992 2.2236 2.1819

9×9×4 3.7616 2.9913 2.6672 2.3992 2.2236 2.1819

10×10×5 3.7608 2.9906 2.6665 2.3985 2.2226 2.1808

12×12×6 3.7605 2.9903 2.6662 2.3983 2.2224 2.1804

14×14×7 3.7603 2.9902 2.6661 2.3981 2.2222 2.1802

16×16×8 3.7602 2.9902 2.6660 2.3981 2.2221 2.1801

18×18×9 3.7602 2.9901 2.6660 2.3980 2.2221 2.1801

20×20×10 3.7602 2.9901 2.6660 2.3980 2.2221 2.1801

Table 4

Values of spring stiffness for three types of classical boundary conditions

Boundary Spring stiffness

condition ku kv kw

Free edge (F) 0 0 0

Simply-supported edge (S) 0

5×1018

5×1018

Clamped edge (C) 5×1018

5×1018

5×1018

Table 5

The first seven frequency parameters 2 /b h D of square isotropic plates with

different boundary conditions.

BC h/a method Modes

1 2 3 4 5 6 7

SSSS 0.1 present 19.098 45.636 45.636 64.384 64.384 70.149 85.500

Ref.[29] 19.090 45.622 45.622 64.383 64.383 70.112 85.502

Ref.[30] 19.090 45.619 45.619 64.383 64.383 70.104 85.488

0.2 present 17.528 32.192 32.192 38.488 38.488 45.526 55.802

Ref.[29] 17.528 32.192 32.192 38.502 38.502 45.526 55.843

Ref.[30] 17.526 32.192 32.192 38.483 38.483 45.526 55.787

0.5 present 12.426 12.877 12.877 18.210 23.009 23.009 25.753

Ref.[30] 12.426 12.877 12.877 18.210 23.007 23.007 25.753

CCCC 0.1 present 33.009 63.043 63.043 88.411 104.28 105.29 123.73

Ref.[29] 32.797 62.672 62.672 87.941 103.71 104.70 123.60

Ref.[30] 32.782 62.630 62.630 87.869 103.61 104.60 123.59

0.2 present 27.065 47.346 47.346 62.000 62.000 63.635 72.604

Ref.[29] 26.974 47.253 47.253 61.944 61.944 63.570 72.568

Ref.[30] 26.906 47.103 47.103 61.917 61.917 63.348 72.286

0.5 present 15.358 24.136 24.136 24.866 24.866 29.379 31.578

Ref.[30] 15.294 24.078 24.078 24.823 24.823 29.377 31.210

FFFF 0.1 present 12.728 18.956 23.346 31.965 31.965 55.493 55.493

Ref.[31] 12.726 18.955 23.347 31.965 31.965 55.493 55.493

0.2 present 11.710 17.433 21.252 27.648 27.648 40.192 42.775

Ref.[31] 11.710 17.433 21.252 27.647 27.647 40.191 42.776

0.5 present 8.7801 12.515 14.962 16.072 17.030 17.030 17.632

Ref.[31] 8.7802 12.515 14.962 16.073 17.030 17.030 17.631

Table 6

The non-dimensional frequency parameters 2 / /c ca h E of Al/Al2O3 square plate

with simply-supported boundary conditions.

h/a p method Mode

0.1 1 2 3 4 5

0 present 5.779 13.81 13.81 19.48 19.48

Ref.[38] 5.777 13.81 13.81 19.48 19.48

Diff.% 0.0346 0.0000 0.0000 0.0000 0.0000

1 present 4.428 10.63 10.63 16.20 16.20

Ref.[38] 4.426 10.63 10.63 16.20 16.20

Diff.% 0.0452 0.0000 0.0000 0.0000 0.0000

5 present 3.774 8.931 8.931 12.64 12.64

Ref.[38] 3.772 8.927 8.927 12.64 12.64

Diff.% 0.0530 0.0448 0.0448 0.000 0.000

0.2 0 present 5.304 9.742 9.742 11.65 11.65

Ref.[38] 5.304 9.742 9.742 11.65 11.65

Diff.% 0.0000 0.0000 0.0000 0.0000 0.0000

1 present 4.100 8.089 8.089 9.108 9.108

Ref.[38] 4.099 8.089 8.089 9.107 9.107

Diff.% 0.0244 0.0000 0.0000 0.0110 0.0110

5 present 3.405 6.296 6.296 7.344 7.344

Ref.[38] 3.405 6.296 6.296 7.343 7.343

Diff.% 0.0000 0.0000 0.0000 0.0136 0.0136

Table 7

The non-dimensional frequency parameters /m mh E of Al/ZrO2 square plates with

simple-supported boundary conditions (τ=h/a).

Method p=0 n=1 τ=0.2

1/ 10 τ=0.1 τ=0.05 τ=0.1 τ=0.2 n=2 n=3 n=5

Present 0.4658 0.0578 0.0159 0.0620 0.2285 0.2264 0.2271 0.2281

Ref.[25] 0.4658 0.0577 0.0158 0.0619 0.2285 0.2264 0.2270 0.2281

Ref.[33] 0.4658 0.0577 0.0153 0.0596 0.2192 0.2197 0.2211 0.2225

Table 8

The non-dimensional frequency parameters /m mh E of Al/Al2O3 plates with

different classical boundary conditions.

b/a h/a p Boundary conditions

SSSS SCSC SFSF SCSF SSSC SSSF

1 0.1 0 0.1135 0.1604 0.0562 0.0731 0.1339 0.0677

1 0.0870 0.1236 0.0430 0.0559 0.1029 0.0518

2 0.0789 0.1118 0.0390 0.0507 0.0932 0.0470

5 0.0741 0.1038 0.0368 0.0477 0.0871 0.0443

0.2 0 0.4169 0.5402 0.2141 0.2713 0.4731 0.2550

1 0.3222 0.4236 0.1645 0.2092 0.3681 0.1962

2 0.2905 0.3799 0.1488 0.1889 0.3310 0.1773

5 0.2676 0.3412 0.1388 0.1749 0.3014 0.1649

0.5 0 1.8470 1.9139 1.0652 0.9570 1.9139 0.9570

1 1.4687 1.5724 0.8342 0.7937 1.5499 0.7937

2 1.3095 1.4026 0.7464 0.7149 1.3796 0.7149

5 1.1450 1.2072 0.6687 0.6168 1.1961 0.6168

2 0.1 0 0.0719 0.0793 0.0568 0.0607 0.0751 0.0600

1 0.0550 0.0608 0.0435 0.0465 0.0575 0.0459

2 0.0499 0.0552 0.0395 0.0422 0.0522 0.0417

5 0.0471 0.0519 0.0372 0.0398 0.0492 0.0393

0.2 0 0.2713 0.2941 0.2166 0.1914 0.2814 0.1914

1 0.2088 0.2271 0.1665 0.1592 0.2169 0.1592

2 0.1888 0.2050 0.1507 0.1438 0.1960 0.1438

5 0.1754 0.1895 0.1405 0.1243 0.1817 0.1243

0.5 0 0.9570 0.9570 0.9570 0.4785 0.9570 0.4785

1 0.7937 0.7937 0.7937 0.3978 0.7937 0.3978

2 0.7149 0.7149 0.7149 0.3591 0.7149 0.3591

5 0.6168 0.6168 0.6168 0.3101 0.6168 0.3101

Table 9

The non-dimensional frequency parameters /m mh E of Al/ZrO2 plates with different

classical boundary conditions.

b/a h/a p Boundary conditions

SSSS SCSC SFSF SCSF SSSC SSSF

1 0.1 0 0.0673 0.0950 0.0333 0.0432 0.0793 0.0401

1 0.0620 0.0878 0.0306 0.0398 0.0732 0.0369

2 0.0617 0.0872 0.0306 0.0397 0.0728 0.0368

5 0.0629 0.0882 0.0312 0.0405 0.0740 0.0376

0.2 0 0.2469 0.3200 0.1268 0.1607 0.2803 0.1510

1 0.2285 0.2981 0.1170 0.1486 0.2602 0.1395

2 0.2264 0.2934 0.1164 0.1475 0.2570 0.1386

5 0.2281 0.2916 0.1180 0.1489 0.2572 0.1403

0.5 0 1.0941 1.1337 0.6310 0.5669 1.1337 0.5669

1 1.0258 1.0842 0.5882 0.5424 1.0789 0.5424

2 1.0013 1.0577 0.5779 0.5294 1.0507 0.5294

5 0.9804 1.0227 0.5730 0.5119 1.0227 0.5119

2 0.1 0 0.0426 0.0470 0.0337 0.0360 0.0445 0.0356

1 0.0392 0.0433 0.0310 0.0331 0.0410 0.0327

2 0.0391 0.0432 0.0309 0.0330 0.0408 0.0327

5 0.0399 0.0440 0.0316 0.0337 0.0417 0.0333

0.2 0 0.1607 0.1742 0.1283 0.1134 0.1667 0.1134

1 0.1484 0.1611 0.1184 0.1085 0.1540 0.1085

2 0.1474 0.1598 0.1178 0.1059 0.1529 0.1059

5 0.1492 0.1613 0.1194 0.1024 0.1546 0.1024

0.5 0 0.5669 0.5669 0.5669 0.2834 0.5669 0.2834

1 0.5424 0.5424 0.5424 0.2712 0.5424 0.2712

2 0.5294 0.5294 0.5294 0.2647 0.5294 0.2647

5 0.5119 0.5119 0.5119 0.2560 0.5119 0.2560

Table 10

The non-dimensional frequency parameters /m mh E of Al/Al2O3 plates with

different elastic boundary conditions.

b/a h/a p Boundary conditions

E1E

1E

1E

1 E

2E

2E

2E

2 E

3E

3E

3E

3 E

4E

4E

4E

4 E

5E

5E

5E

5

1 0.1 0 0.0635 0.0635 0.0615 0.0633 0.0536

1 0.0684 0.0684 0.0644 0.0681 0.0511

2 0.0703 0.0703 0.0649 0.0699 0.0496

5 0.0722 0.0722 0.0655 0.0717 0.0489

0.2 0 0.1270 0.1270 0.1257 0.1267 0.1209

1 0.1369 0.1369 0.1343 0.1363 0.1247

2 0.1405 0.1405 0.1371 0.1398 0.1250

5 0.1444 0.1444 0.1397 0.1433 0.1257

0.5 0 0.3175 0.3175 0.3163 0.3167 0.3141

1 0.3421 0.3421 0.3399 0.3405 0.3353

2 0.3511 0.3511 0.3482 0.3491 0.3421

5 0.3606 0.3606 0.3560 0.3578 0.3485

2 0.1 0 0.0549 0.0549 0.0515 0.0544 0.0410

1 0.0592 0.0592 0.0524 0.0544 0.0382

2 0.0556 0.0607 0.0522 0.0495 0.0368

5 0.0532 0.0623 0.0519 0.0497 0.0361

0.2 0 0.1099 0.1098 0.1075 0.1089 0.0974

1 0.1183 0.1183 0.1136 0.1167 0.0978

2 0.1214 0.1214 0.1153 0.1193 0.0973

5 0.1246 0.1245 0.1166 0.1219 0.0972

0.5 0 0.2746 0.2746 0.2728 0.2723 0.2668

1 0.2957 0.2956 0.2922 0.2916 0.2806

2 0.3033 0.3031 0.2985 0.2982 0.2844

5 0.3112 0.3109 0.3041 0.3044 0.2879

Table 11

The non-dimensional frequency parameters /m mh E of Al/ZrO2 plates with different

elastic boundary conditions.

b/a h/a p Boundary conditions

E1E

1E

1E

1 E

2E

2E

2E

2 E

3E

3E

3E

3 E

4E

4E

4E

4 E

5E

5E

5E

5

1 0.1 0 0.0517 0.0517 0.0487 0.0514 0.0390

1 0.0600 0.0600 0.0546 0.0595 0.0406

2 0.0637 0.0637 0.0572 0.0599 0.0418

5 0.0683 0.0683 0.0605 0.0609 0.0436

0.2 0 0.1033 0.1033 0.1014 0.1028 0.0944

1 0.1199 0.1199 0.1164 0.1191 0.1046

2 0.1275 0.1275 0.1231 0.1264 0.1094

5 0.1366 0.1366 0.1312 0.1353 0.1156

0.5 0 0.2584 0.2584 0.2566 0.2571 0.2532

1 0.2996 0.2996 0.2966 0.2975 0.2906

2 0.3184 0.3184 0.3145 0.3158 0.3073

5 0.3414 0.3414 0.3360 0.3380 0.3275

2 0.1 0 0.0446 0.0446 0.0397 0.0420 0.0292

1 0.0436 0.0517 0.0433 0.0389 0.0300

2 0.0435 0.0550 0.0450 0.0388 0.0307

5 0.0444 0.0589 0.0473 0.0396 0.0319

0.2 0 0.0893 0.0893 0.0858 0.0880 0.0742

1 0.1035 0.1035 0.0973 0.1014 0.0809

2 0.1099 0.1099 0.1024 0.1074 0.0843

5 0.1178 0.1177 0.1086 0.1147 0.0887

0.5 0 0.2232 0.2232 0.2204 0.2199 0.2121

1 0.2587 0.2586 0.2538 0.2534 0.2402

2 0.2747 0.2746 0.2686 0.2685 0.2529

5 0.2943 0.2941 0.2862 0.2866 0.2686

(a)

(b)

(c)

Fig. 1. Schematic diagram of the rectangular 3-D functionally graded plates: (a) the geometry

and coordinates; (b) FG plate of two material phases; (c) the boundary restraining springs.

(a) (b)

(c) (d)

(e) (f)

Fig.2. Modes shapes of Al/Al2O3 plates with SSSS boundary condition when h/a=0.5 and

p=1: (a) m = 1; (b) m = 2; (c) m = 3; (d) m = 4; (e) m=5; (f) m=6.

(a) (b)

(c) (d)

(e) (f)

Fig.3. Modes shapes of Al/Al2O3 plates with SFSF boundary condition when h/a=0.5 and

p=1: (a) m = 1; (b) m = 2; (c) m = 3; (d) m = 4; (e) m=5; (f) m=6.

(a) (b)

(c) (d)

(d)

Fig.4. Modes shapes of Al/Al2O3 plates with SCSC boundary condition when h/a=0.5 and

p=1: (a) m = 1; (b) m = 2; (c) m = 3; (d) m = 4; (e) m=5; (f) m=6.

Fig. 5. Variation of volume fraction Vc through the non-dimensions thickness of FG plates

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

p=10

p=5

p=3

p=1

p=0.5

p=0.3

p=0.1

Vc

z/h

(a) (b)

(c) (d)

Fig. 6. Variation of the frequency parameters with the gradient index for FG square plates:(a)

Al/Al2O3 plates with h/a=0.5; (b) Al/Al2O3 plates with h/a=0.2; (c) Al/ZrO2 plates with

h/a=0.5; (d) Al/ZrO2 plates with h/a=0.2.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

The gradient index

Th

e fre

qu

en

cy p

ara

me

ter

SCSC

SSSC

SSSS

SCSF

SSSF

SFSF

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

The gradient index

Th

e fre

qu

en

cy p

ara

me

ter

SCSC

SSSC

SSSS

SCSF

SSSF

SFSF

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

The gradient index

Th

e fre

qu

en

cy p

ara

me

ter

SCSC

SSSC

SSSS

SCSF

SSSF

SFSF

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

The gradient index

Th

e fre

qu

en

cy p

ara

me

ter

SCSC

SSSC

SSSS

SCSF

SSSF

SFSF