# Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory

**Abstract**

The main objective of this research work is to present analytical solutions for free vibration analysis of moderately thick rectangular plates, which are composed of functionally graded materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The proposed rectangular plates have two opposite edges simply-supported, while all possible combinations of free, simply-supported and clamped boundary conditions are applied to the other two edges. In order to capture fundamental frequencies of the functionally graded (FG) rectangular plates resting on elastic foundation, the analysis procedure is based on the first-order shear deformation plate theory (FSDT) to derive and solve exactly the equations of motion. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. First, a new formula for the shear correction factors, used in the Mindlin plate theory, is obtained for FG plates. Then the excellent accuracy of the present analytical solutions is confirmed by making some comparisons of the results with those available in literature. The effect of foundation stiffness parameters on the free vibration of the FG plates, constrained by different combinations of classical boundary conditions, is also presented for various values of aspect ratios, gradient indices, and thickness to length ratios.

Free vibration of functionally graded rectangular plates using

ﬁrst-order shear deformation plate theory

Sh. Hosseini-Hashemi

a

, H. Rokni Damavandi Taher

b

, H. Akhavan

a,

*

, M. Omidi

a

a

School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran

b

School of Engineering, University of British Columbia Okanagan, Kelowna, BC, Canada V1V 1V7

article info

Article history:

Received 1 June 2009

Received in revised form 5 August 2009

Accepted 18 August 2009

Available online 22 August 2009

Keywords:

Free vibration

FGM

Mindlin theory

Elastic foundation

abstract

The main objective of this research work is to present analytical solutions for free vibration

analysis of moderately thick rectangular plates, which are composed of functionally graded

materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The

proposed rectangular plates have two opposite edges simply-supported, while all possible

combinations of free, simply-supported and clamped boundary conditions are applied to

the other two edges. In order to capture fundamental frequencies of the functionally

graded (FG) rectangular plates resting on elastic foundation, the analysis procedure is

based on the ﬁrst-order shear deformation plate theory (FSDT) to derive and solve exactly

the equations of motion. The mechanical properties of the FG plates are assumed to vary

continuously through the thickness of the plate and obey a power law distribution of

the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. First,

a new formula for the shear correction factors, used in the Mindlin plate theory, is obtained

for FG plates. Then the excellent accuracy of the present analytical solutions is conﬁrmed

by making some comparisons of the results with those available in literature. The effect of

foundation stiffness parameters on the free vibration of the FG plates, constrained by dif-

ferent combinations of classical boundary conditions, is also presented for various values of

aspect ratios, gradient indices, and thickness to length ratios.

Crown Copyright Ó 2009 Published by Elsevier Inc. All rights reserved.

1. Introduction

In recent years, astonishing advances in science and technology have motivated researchers to work on new structural

materials. Functionally graded materials (FGMs) are classiﬁed as novel composite materials which are widely used in aero-

space, nuclear, civil, automotive, optical, biomechanical, electronic, chemical, mechanical, and shipbuilding industries. Due

to smoothly and continuously varying material properties from one surface to the other, FGMs are usually superior to the

conventional composite materials in mechanical behavior. FGMs may possess a number of advantages such as high resis-

tance to temperature gradients, signiﬁcant reduction in residual and thermal stresses, and high wear resistance.

A few researchers employed classical plate theory (CPT) to analyze vibration and static behavior of thin FG plates. Natural

frequencies of FG simply-supported and clamped rectangular thin plates were obtained by Abrate [1] using the CPT. Free

vibration, buckling and deﬂection analysis of the FG thin plates were presented by Zhang and Zhou [2] on the basis of the

physical neutral surface. Woo et al. [3] provided an analytical solution for the nonlinear free vibration behavior of FG square

thin plates using the von-Karman theory.

0307-904X/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier Inc. All rights reserved.

doi:10.1016/j.apm.2009.08.008

* Corresponding author. Tel.: +98 912 547 0654; fax: +98 217 724 0488.

E-mail address: hamedakhavan@mecheng.iust.ac.ir (H. Akhavan).

Applied Mathematical Modelling 34 (2010) 1276–1291

Contents lists available at ScienceDirect

Applied Mathematical Mo delling

journal homepage: www.elsevier.com/locate/apm

Higher-order shear deformation plate theory (HSDT) and 3D methods were used by some investigators for analyzing thick

FG plates. Early research efforts for harmonic vibration analysis of an FG simply-supported rectangular plate, using a 3D

asymptotic theory, date back to the work of Reddy and Cheng [4]. Then, Qian et al. [5] conducted an investigation on free

and forced vibrations and static deformations of an FG thick simply-supported square plate by using a higher-order shear

and normal deformable plate theory and a meshless local Petrov–Galerkin method. Vel and Batra [6] described an excellent

investigation on the analytical solution for free and forced vibrations of FG simply-supported square plates based on the 3D

elasticity solution. Zhong and Yu [7] used a state-space approach to analyze free and forced vibrations of an FG piezoelectric

rectangular thick plate simply-supported at its edges. Roque et al. [8] investigated the free vibration of FG plates with

different combinations of boundary conditions by the multiquadric radial basis function method and the HSDT. Free vibra-

tion analysis of FG simply-supported square plates was carried out by Pradyumna and Bandyopadhyay [9] using a higher-

order ﬁnite element formulation, as a small part of their study work. Recently, Matsunaga [10] studied natural frequencies

and buckling stresses of FG simply-supported rectangular plates based on 2D higher-order approximate plate theory (2D

HAPT).

Due to its high efﬁciency and simplicity, ﬁrst-order shear deformation theory (FSDT) was used for analyzing moderately

thick FG plates. An excellent work on the free vibration, buckling, and static deﬂections of FG square, circular, and skew

plates with different combinations of boundary conditions was carried out by Abrate [11] on the basis of the CPT, FSDT,

and TSDT. Ferreira et al. [12] employed the collocation method with multiquadric radial basis functions along with the FSDT

and third-order shear deformation plate theory (TSDT) to ﬁnd natural frequencies of FG square plates with different bound-

ary conditions at the edges. Very recently, Zhao et al. [13] presented a free vibration analysis for FG square and skew plates

with different boundary conditions using the element-free kp-Ritz method on the basis of the FSDT.

One necessary term, used in the FSDT, is a shear correction factor that amends the effect of uniform transverse stress in

shear forces. In isotropic homogeneous plates, shear correction factor is mainly equal to 5/6. However, in FG plates, material

properties of which vary in thickness direction, an error in frequency results will arise in practice owing to the use of a con-

stant shear correction factor. In the past, some investigators tried to improve shear correction factor to yield more exact re-

sults for vibration of the plate. Early efforts by Timoshenko [14] showed that shear correction factor is dependent on

Poisson’s ratio. Recently, a well-known work by Efraim and Eisenberger [15] proposed a formula for shear correction factor

in terms of Poisson’s ratio and volume fractions of both gradients in an FG plate. Furthermore, Nguyen et al. [16] presented

the shear correction factor in terms of ceramic-to-metal Young’s modulus ratio and gradient indices in order to examine sta-

tic analysis of an FG plate. As one can see, there is no more investigation on the vibration of FG plates based on an improved

shear correction factor in the FSDT.

Plates resting on elastic foundations have found considerable applications in structural engineering problems. Rein-

forced-concrete pavements of highways, airport runways, foundation of storage tanks, swimming pools, and deep walls to-

gether with foundation slabs of buildings are well-known direct applications of these kinds of plates. The underlying layers

are modeled by a Winkler-type elastic foundation. The most serious deﬁciency of the Winkler foundation model is to have no

interaction between the springs. In other words, the springs in this model are assumed to be independent and unconnected.

The Winkler foundation model is fairly improved by adopting the Pasternak foundation model, a two-parameter model, in

which the shear stiffness of the foundation is considered.

Although a few studies on the vibration and buckling analysis of isotropic homogeneous rectangular plates resting on

elastic foundation have been carried out (see for example, Xiang et al. [17], Xiang [18], Lam et al. [19], Zhou et al. [20], Akh-

avan et al. [21,22] and their cited references), research studies on the dynamic behavior of their corresponding FG plates

have received very little attention. Cheng and Kitipornchai [23] proposed a membrane analogy to derive an exact explicit

eigenvalue for compression buckling, hydrothermal buckling, and vibration of FG plates on a Winkler–Pasternak foundation

based on the FSDT. Yang and Shen [24] studied both free vibration and transient response of initially stressed FG rectangular

thin plates subjected to impulsive lateral loads, resting on Pasternak elastic foundation, based on the CPT. The second-order

statistics of the buckling of clamped FG rectangular plates that are resting on Pasternak elastic foundations and subjected to

uniform edge compression was studied by Yang et al. [25] in the framework of the FSDT. Ying et al. [26] treated 2D elasticity

solutions for bending and free vibration of FG beams resting on Winkler–Pasternak elastic foundations. Huang et al. [27]

used a benchmark 3D elasticity solution to study the bending behavior of FG thick simply-supported square plates on a Win-

kler–Pasternak foundation.

Three points can apparently be raised from the literature survey. First, all aforementioned research works have been car-

ried out within the last decade. Second, analytical solutions have been employed by a very few of these research studies due

to the mathematical complexity. It is also well known that an exact solution may be achieved for rectangular plates having at

least one pair of opposite edges simply-supported. Third, there is no work on the analytical solutions for free vibration anal-

ysis of FG moderately thick rectangular plates resting on Winkler–Pasternak elastic foundation. In this paper, the analysis

procedure is based on the ﬁrst-order shear deformation plate theory, including plate-foundation interaction. For the ﬁrst

time, a brand-new formula for shear correction factors, used in the Mindlin plate theory, is proposed to guarantee good accu-

racy. In addition, this formula is suitable for different FGMs and easy to implement. In order to validate the obtained results,

the

authors

compare their results with existing data available from other analytical and numerical techniques. The effect of

the plate parameters such as foundation stiffness coefﬁcients, aspect ratios, thickness to length ratios, and gradient indices

on the natural frequencies of FG rectangular plates is presented for six combinations of classical boundary conditions,

namely SSSS, SCSS, SCSC, SSSF, SFSC and SFSF.

Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

1277

2. Mathematical formulation

2.1. Geometrical conﬁguration

A ﬂat, isotropic and moderately thick FG rectangular plate of length a, width b, and uniform thickness h, resting on two-

parameter elastic foundation, is depicted in Fig. 1. The plate has two opposite edges simply-supported along y axis (i.e. along

the edges x = 0 and x = a), while the other two edges may be free, simply-supported, or clamped. The Cartesian coordinate

system (x,y,z) is considered to extract mathematical formulations when x and y axes are located in the undeformed mid-

plane of the plate.

2.2. Material properties

FGMs are composite materials, the mechanical properties of which vary continuously due to gradually changing the vol-

ume fraction of the constituent materials, usually in the thickness direction. In this study, the FG plate is made from a mix-

ture of ceramics and metal and the composition varies from the top to the bottom surface. In fact, the top surface (z = h/2) of

the plate is ceramic-rich whereas the bottom surface (z = h/2) 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

EðzÞ¼ðE

c

E

m

ÞV

f

ðzÞþE

m

;

q

ðzÞ¼ð

q

c

q

m

ÞV

f

ðzÞþ

q

m

;

ð1Þ

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

f

may

be given by

V

f

ðzÞ¼

z

h

þ

1

2

a

; ð2Þ

where

a

is the gradient index and takes only positive values. Poisson’s ratio is taken as 0.3 throughout the analyses. Typical

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

In order to gain a better understanding of Eqs. (1) and (2), the variation of Young’s modulus E in the thickness direction z,

for the Al/ZrO

2

rectangular plate with various values of gradient index

a

, is shown in Fig. 2. For

a

= 0 and

a

= 1, the plate is

Fig. 1. Geometry of an SCSF rectangular FG plate with coordinate convention.

1278 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

fully ceramic and metallic, respectively; whereas the composition of metal and ceramic is linear for

a

= 1. It is also observed

from Fig. 2 that the Young’s modulus of the FG plate quickly approaches ceramic’s one for

a

< 1 especially within 0.5 6 z/

h 6 0. For

a

> 1, the FG plate is made from a mixture in which the metal is used more than the ceramic.

2.3. Constitutive relations

According to the Mindlin plate theory, the displacement components of the middle surface along the x, y, and z axes, des-

ignated by U

x

, U

y

and U

z

, may be expressed as

U

x

¼zw

x

ðx; y; tÞ;

U

y

¼zw

y

ðx; y; tÞ;

U

z

¼ w

z

ðx; y; tÞ;

ð3Þ

where

w

x

and

w

y

are the rotational displacements about the y and x axes at the middle surface of the plate, respectively,

w

z

is

the transverse displacement, and t is the time variable. By neglecting

e

zz

in the stress–strain relations, the general strain–dis-

placement relations for small deformation are deﬁned as

e

xx

¼zw

x;x

;

e

yy

¼zw

y;y

;

e

zz

¼ 0;

c

xy

¼z

w

x;y

þ w

y;x

2

;

c

xz

¼

w

x

w

z;x

2

;

c

yz

¼

w

y

w

z;y

2

;

ð4Þ

where

e

and

c

denote the normal and shear strains, respectively. Here, the symbol ‘‘,” is used to indicate the partial deriv-

ative. For example,

w

x,y

is equivalent to @

w

x

/@y while

w

x,yy

means @

2

w

x

/@y

2

. Hook’s law for a plate may be expressed as

r

xx

¼

EðzÞ

1

m

2

ð

e

xx

þ

me

yy

Þ;

r

yy

¼

EðzÞ

1

m

2

ð

e

yy

þ

me

xx

Þ;

r

zz

¼ 0;

s

xy

¼ GðzÞ

c

xy

;

s

xz

¼ GðzÞ

c

xz

;

s

yz

¼ GðzÞ

c

yz

;

ð5Þ

where G(z)=E(z)/[2(1 +

m

)] is the shear modulus and

m

is the Poisson’s ratio. The stress resultant–displacement relations are

given by

M

ii

¼

Z

h=2

h=2

r

ii

zdz; i ¼ x; y

M

xy

¼

Z

h=2

h=2

s

xy

zdz;

Q

j

¼

j

2

Z

h=2

h=2

r

jz

dz; j ¼ x; y;

ð6Þ

Fig. 2. Variation of Young’s modulus through the dimensionless thickness of Al/ZrO

2

plate.

Table 1

Material properties used in the FG plate.

Properties Metal Ceramic

Aluminum (Al) Zirconia (ZrO

2

) Alumina (Al

2

O

3

)

E (GPa) 70 200 380

q

(kg/m

3

) 2702 5700 3800

Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

1279

in which

j

2

is the transverse shear correction coefﬁcient, applied to the transverse shear forces due to the fact that the trans-

verse shear strains (

e

xz

and

e

yz

) have a nearly parabolic dependency to the thickness coordinate. Substituting Eqs. (4) and (5)

into Eq. (6) gives the resultant bending moments (M

xx

and M

yy

), twisting moment (M

xy

), and the transverse shear forces (Q

x

and Q

y

) per unit length as follows:

M

xx

¼Aðw

x;x

þ

m

w

y;y

Þ; M

yy

¼Aðw

y;y

þ

m

w

x;x

Þ;

M

xy

¼

ð1

m

Þ

2

Aðw

x;y

þ w

y;x

Þ;

Q

x

¼

j

2

hBðw

x

w

z;x

Þ; Q

y

¼

j

2

hBðw

y

w

z;y

Þ;

ð7Þ

in which

A ¼

h

3

ð1

m

2

Þ

a

ð8 þ 3

a

þ

a

2

ÞE

m

þ 3ð2 þ

a

þ

a

2

ÞE

c

12ð1 þ

a

Þð2 þ

a

Þð3 þ

a

Þ

;

B ¼

1

2ð1 þ

m

Þ

E

c

þ

a

E

m

1 þ

a

:

ð8Þ

2.4. Equations of motion

On the basis of the Mindlin plate theory, the governing differential equations of motion for the plate can be given in terms

of the stress resultants by

M

xx;x

þ M

xy;y

Q

x

¼

1

12

Ch

3

€

w

x

;

M

xy;x

þ M

yy;y

Q

y

¼

1

12

Ch

3

€

w

y

;

Q

x;x

þ Q

y;y

P ¼ Dh

€

w

z

;

ð9Þ

in which

C ¼

a

ð8 þ 3

a

þ

a

2

Þ

q

m

þ 3ð2 þ

a

þ

a

2

Þ

q

c

ð1 þ

a

Þð2 þ

a

Þð3 þ

a

Þ

;

D ¼

q

c

þ

aq

m

1 þ

a

;

ð10Þ

where dot-overscript convention represents the differentiation with respect to the time variable t. Since the Pasternak elastic

foundation provides force components in z direction for a deﬂected plate, normal transverse load per unit area can be written

as

P ¼ K

S

ðw

z;xx

þ w

z;yy

ÞK

W

w

z

; ð11Þ

in which K

S

and K

W

are the shear and Winkler foundation coefﬁcients, respectively.

For coding and derivational convenience, the following non-dimensional parameters are introduced:

X ¼

x

a

; Y ¼

y

a

; d ¼

h

a

;

g

¼

a

b

;

K

S

¼

K

S

a

2

A

;

K

W

¼

K

W

a

4

A

; b ¼

x

a

2

ﬃﬃﬃﬃﬃﬃ

Ch

A

r

; ð12Þ

in which d and

g

are named as the thickness to length ratio and aspect ratio, respectively, and b is also called the eigenfre-

quency parameter. For a harmonic solution, the rotational and transverse displacements are assumed to be

w

x

ðx; y; tÞ¼

w

x

ðX; YÞe

j

x

t

;

w

y

ðx; y; tÞ¼

w

y

ðX; YÞe

j

x

t

;

w

z

ðx; y; tÞ¼

1

a

w

z

ðX; YÞe

j

x

t

;

ð13Þ

where

x

denotes the natural frequency of vibration in radians and j ¼

ﬃﬃﬃﬃﬃﬃﬃ

1

p

. It should be noted that each parameter having

the over-bar is non-dimensional. Substitution of Eqs. (11)–(13) into Eq. (9) leads to

w

x;xx

þ

g

2

w

x;yy

þ

m

2

m

1

ð

w

x;xx

þ

g

w

y;xy

Þ

j

2

B

Fd

2

m

1

ð

w

x

w

z;x

Þþ

b

2

d

2

12

m

1

w

x

¼ 0;

w

y;xx

þ

g

2

w

y;yy

þ

m

2

m

1

g

ð

w

x;xy

þ

g

w

y;yy

Þ

j

2

B

Fd

2

m

1

ð

w

y

g

w

z;y

Þþ

b

2

d

2

12

m

1

w

y

¼ 0;

w

z;xx

þ

g

2

w

z;yy

ð

w

x;x

þ

g

w

y;y

Þ

Fd

2

j

2

B

K

W

w

z

þ K

S

ð

w

z;xx

þ

g

2

w

z;yy

Þ

DFb

2

d

2

BC

j

2

w

z

¼ 0;

ð14Þ

where F = A/h

3

,

m

1

=(1

m

)/2, and

m

2

=(1+

m

)/2.

1280 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

2.5. Boundary conditions

The boundary conditions along the edges X = 0 and X = 1, considered to remain simply-supported, are as follow:

M

xx

¼

w

y

¼

w

z

¼ 0: ð15Þ

The boundary conditions along the edges Y = 0 and Y = 1 are as follows:

– for a free edge

M

xx

¼ M

xy

¼ 0; Q

y

¼K

S

A

a

2

g

w

z;y

; ð16Þ

– for a simply-supported edge

M

yy

¼

w

x

¼

w

z

¼ 0; ð17Þ

– for a clamped edge

w

x

¼

w

y

¼

w

z

¼ 0: ð18Þ

2.6. Solution of governing equations

The general solutions to Eq. (14) in terms of the three dimensionless potentials W

x

, W

y

and W

z

may be expressed as

w

x

¼ C

1

W

x;x

þ C

2

W

y;x

g

W

z;y

;

w

y

¼ C

1

g

W

x;y

þ C

2

g

W

y;y

W

z;x

;

w

z

¼ W

x

þ W

y

;

ð19Þ

where

C

1

¼

B

2

a

2

1

B

1

; C

2

¼

B

3

a

2

2

B

1

; ð20Þ

in which B

1

, B

2

, and B

3

along with

a

2

1

and

a

2

2

are the coefﬁcients that may be determined using equations of motion and can be

given after mathematical manipulation by

B

1

¼

H

m

1

b

2

d

2

12

m

1

;

B

2

¼

m

2

m

1

1

Fd

2

k

2

B

K

S

!

a

2

1

þ

Fd

2

k

2

B

K

W

þ

DFd

2

k

2

BC

b

2

"#

þ

k

2

B

Fd

2

m

1

()

;

B

3

¼

m

2

m

1

1

Fd

2

k

2

B

K

S

!

a

2

2

þ

Fd

2

k

2

B

K

W

þ

DFd

2

k

2

BC

b

2

"#

þ

k

2

B

Fd

2

m

1

()

;

ð21Þ

and

a

2

1

;

a

2

2

¼

12K

W

12HK

S

þ b

2

12 D=C þ d

2

K

S

þ H

24

K

S

þ H

1

24

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

48 b

2

d

2

12H

K

S

þ H

K

W

þ Db

2

=C

þ

12

K

W

12HK

S

þ b

2

12 D=C þ d

2

K

S

þ H

2

K

S

þ H

2

v

u

u

u

u

u

t

;

ð22Þ

where

H

= B

j

2

/Fd

2

. The Eq. (14) can be restated in terms of the three dimensionless potentials as

W

x;xx

þ

g

2

W

x;yy

¼

a

2

1

W

x

;

W

y;xx

þ

g

2

W

y;yy

¼

a

2

2

W

y

;

W

z;xx

þ

g

2

W

z;yy

¼

a

2

3

W

z

;

ð23Þ

where

a

2

3

¼B

1

. One set of solutions to Eq. (23) are taken as

Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

1281

W

x

¼ A

1

Sinðk

1

YÞþA

2

Cosðk

1

YÞ½Sinð

l

1

XÞþ A

3

Sinðk

1

YÞþA

4

Cosðk

1

YÞ½Cosð

l

1

XÞ;

W

y

¼ A

5

Sinhðk

2

YÞþA

6

Coshðk

2

YÞ½Sinð

l

2

XÞþ A

7

Sinhðk

2

YÞþA

8

Coshðk

2

YÞ½Cosð

l

2

XÞ;

W

z

¼ A

9

Sinhðk

3

YÞþA

10

Coshðk

3

YÞ½Sinð

l

3

XÞþ A

11

Sinhðk

3

YÞþA

12

Coshðk

3

YÞ½Cosð

l

3

XÞ;

ð24Þ

in which A

i

are the arbitrary coefﬁcients, k

j

and

l

j

are related to the

a

j

by

a

2

1

¼

l

2

1

þ

g

2

k

2

1

;

a

2

2

¼

l

2

2

g

2

k

2

2

;

a

2

3

¼

l

2

3

g

2

k

2

3

ð25Þ

On the assumption of simply-supported conditions at edges X = 0 and X = 1, Eq. (24) is given by

W

x

¼ A

1

Sinðk

1

YÞþA

2

Cosðk

1

YÞ

½

Sinð

l

XÞ;

W

y

¼ A

5

Sinhðk

2

YÞþA

6

Coshðk

2

YÞ½Sinð

l

XÞ;

W

z

¼ A

9

Sinhðk

3

YÞþA

10

Coshðk

3

YÞ½Sinð

l

XÞ:

ð26Þ

in which

l

=

l

1

=

l

2

=

l

3

= m

p

(m = 1,2,...).

Introducing Eq. (26) in Eq. (19) and substituting the results into the appropriate boundary conditions along the edges

Y = 0 and 1, leads to six homogeneous equations. To obtain non-trivial solution of these equations, the determinant of coef-

ﬁcients matrix must be zero, which yields characteristic equations for rectangular Mindlin plates, with six combinations of

boundary conditions, namely, SSSS, SSSC, SCSC, SSSF, SFSF, and SCSF, resting on two-parameter foundation. It should be

noted that notation SCSC, for example, indicates that edges X = 0 and X = 1 are simply-supported (S), and edges Y = 0 and

Y = 1 are clamped (C).

3. Numerical results

3.1. Shear correction factor

Due to the simplicity of the analysis and programming, the model of Mindlin plates is strongly recommended by many

researchers to analyze the dynamic behavior of plates. However, a shear correction factor (

j

2

) is needed to correctly compute

transverse shear forces (see Eq. (6)). The shear correction factor is typically taken to be 5/6 for homogeneous plates. On the

other hand, this constant shear correction factor is not appropriate for FG plates (Nguyen et al. [16]), since it may be a func-

tion of material properties and the geometric dimension of an FG plate. Timoshenko [14] assumed that the shear correction

factor is dependent upon the Poisson’s ratio

m

as

j

2

¼

5 þ 5

m

6 þ 5

m

: ð27Þ

Efraim and Eisenberger [15] presented a shear correction factor for FG plates as

j

2

¼

5

6 ð

m

m

V

m

þ

m

c

V

c

Þ

; ð28Þ

in which V

m

and V

c

are the volume fraction of metal and ceramic, respectively, in the entire cross-section. However, as it is

seen from Eqs. (27) and (28), the effect of mechanical properties of the FG plate, including Young’s modulus E and density per

unit volume

q

, and the geometric dimension of the FG plate such as thickness to length ratio h/a has not been considered.

In order to achieve a comprehensive form of the shear correction factor, frequency parameters of FG SSSS square plates (a/

b = 1), made of Al/ZrO

2

and Al/Al

2

O

3

, are obtained for a signiﬁcant number of gradient indices and thickness to length ratios,

using the present analytical solution and the ﬁnite element method (FEM). A well-known commercially available FEM pack-

age is used for the extraction of the frequency parameters. After ensuring the high accuracy of the FEM results, by solving

some problems of the literature, the results of the present analytical solution are compared with those obtained by the

FEM results. The authors used the shear correction factors in which the results of the present analytical solution become

identical to those acquired by the ﬁnite element solution.

In Fig. 3, the various values of the shear correction factors

j

2

for a wide range of the gradient indices

a

and thickness to

length ratios h/a are plotted for Al/Al

2

O

3

(Fig. 3a) and Al/ZrO

2

(Fig. 3b) SSSS square plates (a/b = 1). Their corresponding 2D

plots are also shown in Fig. 4 for h/a = 0.05, 0.1, 0.15, and 0.2 while gradient index

a

varies from 0 to 20. It is well known that

the shear correction factor

j

2

for fully ceramic (

a

= 0) and metallic (

a

= 1) plates is the same and equal to 5/6. From Figs. 3

and 4, it can be observed that the shear correction factors

j

2

initially decrease for smaller values of gradient index (i.e.

a

<2)

and then increase in order to approach the constant value 5/6. However, the plates composed of Al and ZrO

2

show higher

resistance to this pattern in comparison with the Al/Al

2

O

3

plates. In other words, it is evident from Figs. 3b and 4b that

for the Al/ZrO

2

plates, highly large values of the gradient index

a

are needed for shear correction factor to get close to the

constant value of 5/6 especially for thinner plates. Another interesting point about Figs. 3 and 4 is that the thicker plates have

lower sensitivity to the shear correction factors

j

2

than thinner plates. In fact, the error in calculating the frequency param-

eter is more tangible for thinner FG plates if the shear correction factor

j

2

is assumed to be constant (e.g.,

j

2

= 5/6).

1282 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

In order to obtain a formula for the shear correction factors

j

2

, in which material properties and the geometric dimension

of an FG plate are considered, too much running time has been taken to ﬁt a function to the obtained shear correction factors

j

2

shown in Fig. 3. Finally, the following function is proposed as

j

2

ð

a

; dÞ¼

5

6

þ C

1

e

C

2

a

e

C

3

a

ð10d 2ÞC

4

e

C

5

a

e

C

6

a

10d 1ðÞ; ð29Þ

where C

i

(i = 1,2,...,6) are the constant coefﬁcients, values of which are listed in Table 2. It should be noted that the best curve

ﬁtting was carried out to minimize the sum of squares of errors between the data and the above function. Hereafter, all re-

sults presented in the next sections are obtained by considering the new function of the shear correction factors

j

2

(i.e., Eq.

(29)).

Fig. 3. Variation of the shear correction factors

j

2

against the gradient index

a

and thickness to length ratio h/a for (a) Al/Al

2

O

3

; (b) Al/ZrO

2

SSSS square

plates, a/b =1.

Fig. 4. Variation of the shear correction factors

j

2

versus the gradient index

a

for (a) Al/Al

2

O

3

; (b) Al/ZrO

2

SSSS square plates, a/b = 1, when h/a = 0.05, 0.1,

0.15, and 0.2.

Table 2

The values of the constant coefﬁcients used in

j

2

formula for two FG materials.

FGMs Constant coefﬁcients

C

1

C

2

C

3

C

4

C

5

C

6

Al/Al

2

O

3

0.750 0.025 2.000 0.640 0.060 1.000

Al/ZrO

2

0.560 0.001 5.450 0.420 0.095 1.175

Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

1283

3.2. Comparison studies

To demonstrate the efﬁciency and accuracy of the present solution along with new shear correction factor

j

2

, some illus-

trative examples are solved and the results are compared with the existing data available in the literature.

Example 1. Fundamental frequency parameters of the SSSS square FG plates (a/b = 1) for different values of the thickness to

length ratios (h/a = 0.05, 0.1, and 0.2) are presented in Table 3 when

a

= 0, 0.5, 1, 4, 10, and 1. The plates are made of a

mixture of aluminum (Al) and alumina (Al

2

O

3

). It should be noted that the results reported by Matsunaga [10] were based on

the both FSDT and 2D HAPT; whereas Zhao et al. [13] employed the FSDT and used different values of shear correction factors

j

2

in their study work. For convenience in comparison, a new frequency parameter is deﬁned as b ¼

x

h

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

q

c

=E

c

p

.

From Table 3, it can be observed that the present results are in excellent agreement with those acquired by the 2D HAPT

[10]. It is worth noting that all results obtained on the basis of the FSDT [10] are inappropriate since the value of shear cor-

rection factor is taken to be constant (

j

2

= 1) for any values of thickness to length ratios and gradient indices. In addition, the

effect of truncated power series to approximate displacement, strain components, and in-plane stress [10] on these apparent

discrepancies can not be neglected. The results obtained by the FSDT [13] are also different from those acquired by the pres-

ent analytical solution and the 2D HAPT [10], particularly for the cases in which the value of shear correction factor is as-

sumed to be constant (

j

2

= 5/6). Another reason of this difference is due to the fact that Zhao et al. [13] employed a

numerical solution (element-free kp-Ritz method) to obtain the natural frequencies of the FG plates.

Example 2. Fundamental frequency parameters of the SSSS square FG Mindlin plate (a/b = 1 and h/a = 0.1) for different

values of the gradient indices

a

= 0, 0.5, 1, 2, 5, 8, and 10 are presented in Table 4 for two FG materials (i.e., Al/Al

2

O

3

and Al/

ZrO

2

). Fundamental frequency parameters are given in Table 4 in the form of

^

b ¼

x

a

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

q

c

=E

c

p

=h. Table 4 shows this fact that

the results of FSDT [13] for the Al/Al

2

O

3

square plate are in good agreement with the present analytical solution but for the

Al/ZrO

2

square plate they reveal a great deviation from the present analytical solution and 2D HAPT [10] due to the reasons

mentioned in Example 1.

Example 3. Fundamental frequencies of the FG square plate (a/b = 1) with simply-supported boundary conditions at four

edges for h/a = 0.1, 0.2, and 1=

ﬃﬃﬃﬃﬃﬃ

10

p

are listed in Table 5 when

a

= 0, 1, 2, 3, and 5. The plate is made of a mixture of aluminum

(Al) and zirconia (ZrO

2

). All fundamental frequency parameters presented in Table 5 are deﬁned as

~

b ¼

x

h

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

q

m

=E

m

p

. Com-

paring the present results with those obtained by the 2D HAPT [10] and HSDT [9] shows that all results are in excellent

Table 3

Comparison of fundamental frequency parameter

b ¼

xh

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

q

c

=E

c

p

for SSSS Al/Al

2

O

3

square plates (a/b = 1).

d = h/a Method

j

2

Gradient index (

a

)

0 0.5 1 4 10 1

0.05 Present solution Eq. (29) 0.01480 0.01281 0.01150 0.01013 0.00963

FSDT [13] 5/6 0.01464 0.01241 0.01118 0.00970 0.00931 –

0.1 Present solution Eq. (29) 0.05769 0.04920 0.04454 0.03825 0.03627 0.02936

2D HAPT [10] – 0.05777 0.04917 0.04427 0.03811 0.03642 0.02933

FSDT [10] 1 0.06382 0.05429 0.04889 0.04230 0.04047 –

FSDT [13] 5/6 0.05673 0.04818 0.04346 0.03757 0.03591 –

Eq. (27) 0.05713 0.04849 0.04371 0.03781 0.03619 –

Eq. (28) 0.05711 0.04847 0.04370 0.03779 0.03618 –

0.2 Present solution Eq. (29) 0.2112 0.1806 0.1650 0.1371 0.1304 0.1075

2D HAPT [10] – 0.2121 0.1819 0.1640 0.1383 0.1306 0.1077

FSDT [10] 1 0.2334 0.1997 0.1802 0.1543 0.1462 –

FSDT [13] 5/6 0.2055 0.1757 0.1587 0.1356 0.1284 –

Eq. (27) 0.2098 0.1790 0.1616 0.1383 0.1313 –

Eq. (28) 0.2096 0.1788 0.1614 0.1382 0.1312 –

Table 4

Comparison of fundamental frequency parameter

^

b ¼

xa

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

q

c

=E

c

p

=h for SSSS square plates (a/b = 1) when h/a = 0.1.

FGMs Method Gradient index (

a

)

00.5125810

Al/Al

2

O

3

Present solution 5.7693 4.9207 4.4545 4.0063 3.7837 3.6830 3.6277

FSDT [13] 5.6763 4.8209 4.3474 3.9474 3.7218 3.6410 3.5923

Al/ZrO

2

Present solution 5.7693 5.3176 5.2532 5.3084 5.2940 5.2312 5.1893

2D HAPT [10] 5.7769 – 5.3216 – – – –

FSDT [13] 5.6763 5.1105 4.8713 4.6977 4.5549 4.4741 4.4323

1284 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

agreement with each other. It is also seen that the present analytical solution for the FG square plates under consideration

provides the results lower than those obtained by the 2D HAPT [10] and greater than those acquired on the basis of the HSDT

[9]. In addition, the discrepancy between the FSDT [9] and three other methods (i.e., the present analytical solution, 2D HAPT

[10], and HSDT [9] ) is also considerable.

Example 4. Fundamental frequencies of the homogenous SSSS square plates (a/b = 1 and

a

= 0), resting on the elastic foun-

dation with different values of the thickness to length ratios and foundation stiffness parameters, are tabulated in Table 6.

Fundamental frequencies listed in Table 6 are deﬁned as

b ¼

x

b

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Ch=A

p

=

p

2

. The present results are compared with those

obtained by Xiang et al. [17] using exact Mindlin solution and Zhou et al. [20] using 3D elasticity theory. Table 6 proves

the fact that all results are in excellent agreement with each other.

Example 5. Table 7 shows a comparison of fundamental frequencies for the homogenous square plates (a/b =1,h/a = 0.05

and

a

= 0), resting on the Winkler elastic foundation ðK

S

¼ 0Þ with different combinations of boundary conditions, with those

obtained by Lam et al. [19] using exact thin plate theory, Xiang [18] and Akhavan et al. [22] using exact Mindlin solution.

Note that fundamental frequencies listed in this table are deﬁned as

b ¼

x

a

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Ch=A

p

. It can be seen from Table 7 that there

is an excellent agreement among these results conﬁrming the high accuracy of the present analytical solution.

Table 5

Comparison of fundamental frequency parameter

~

b ¼

xh

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

q

m

=E

m

p

for SSSS Al/ZrO

2

square plates (a/b = 1).

Method

a

=0

a

=1 d = 0.2

d ¼ 1=

ﬃﬃﬃﬃﬃﬃ

10

p

d = 0.1 d = 0.05 d = 0.1 d = 0.2

a

=2

a

=3

a

=5

Present solution 0.4618 0.0576 0.0158 0.0611 0.2270 0.2249 0.2254 0.2265

2D HAPT [10] 0.4658 0.0578 0.0158 0.0618 0.2285 0.2264 0.2270 0.2281

HSDT [9] 0.4658 0.0578 0.0157 0.0613 0.2257 0.2237 0.2243 0.2253

FSDT [9] 0.4619 0.0577 0.0162 0.0633 0.2323 0.2325 0.2334 0.2334

Table 6

Comparison of fundamental frequency parameter

b ¼ xb

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Ch=A

p

=

p

2

for homogeneous SSSS square plates (a/b = 1).

d = h/a Method

Fundamental frequency parameter

b ¼ xb

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Ch=A

p

=

p

2

ðK

W

; K

S

ÞðK

W

; K

S

ÞðK

W

; K

S

ÞðK

W

; K

S

Þ

(100, 0) (500, 0) (100, 10) (500, 10)

0.01 Present solution 2.2413 3.0215 2.6551 3.3400

Mindlin theory [17] 2.2413 3.0215 2.6551 3.3400

3D method [20] 2.2413 3.0214 2.6551 3.3398

(200, 0) (1000, 0) (200, 10) (1000, 10)

0.1 Present solution 2.3989 3.7212 2.7842 3.9805

Mindlin theory [17] 2.3989 3.7212 2.7842 3.9805

3D method [20] 2.3951 3.7008 2.7756 3.9566

(0, 10) (10, 10) (100, 10) (1000, 10)

0.2 Present solution 2.2505 2.2722 2.4590 3.8567

Mindlin theory [17] 2.2505 2.2722 2.4591 3.8567

3D method [20] 2.2334 2.2539 2.4300 3.7111

Table 7

Comparison of fundamental frequency parameter

b ¼ xa

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Ch=A

p

for homogeneous square plates (a/b = 1) with different boundary conditions when K

S

¼ 0 and

h/a = 0.05.

K

W

Method Boundary conditions

SSSS SSSC SCSC SSSF SFSF SCSF

0 Present solution 19.737 23.643 28.944 11.680 9.630 12.681

Mindlin theory [18] 19.737 23.643 28.944 11.680 9.630 12.681

Exact CPT [19] 19.740 23.650 28.950 11.680 9.630 12.690

Mindlin theory [22] 19.739 23.646 28.951 11.684 9.631 12.686

100 Present solution 22.126 25.671 30.623 15.376 13.883 16.149

Mindlin theory [18] 22.126 25.671 30.623 15.367 13.878 16.138

Exact CPT [19] 22.130 25.670 30.630 15.380 13.880 16.150

1000 Present solution 37.276 39.483 42.869 33.710 33.056 34.070

Mindlin theory [18] 37.276 39.483 42.869 33.667 33.037 34.018

Exact CPT [19] 37.280 39.490 42.870 33.710 31.620 34.070

Mindlin theory [22] 37.278 39.486 42.873 33.712 31.623 34.073

Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

1285

4. Parametric studies

After verifying the merit and accuracy of the present analytical solution, the following new results for the vibration anal-

ysis of rectangular Mindlin FG plates, resting on elastic foundation, can be used as a benchmark for future research studies.

Natural frequencies of the plate are obtained and considered to be dimensionless as

b ¼

x

h

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

q

c

=E

c

p

(called the frequency

parameter) in Tables 8 and 9, and as b ¼

x

a

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Ch=A

p

(called the eigenfrequency parameter) in Figs. 5–8. It should be noted

that the eigenfrequency parameter b ¼

x

a

2

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Ch=A

p

is deﬁned for the ﬁrst time in this form (see Eq. (12)) and has its special

own characteristics, as it will be shown in Sections 4.2–4.5.

4.1. List of the frequency parameters

b for the FG plates

Fundamental frequency parameters

b of the Al/Al

2

O

3

square Mindlin plate are listed in Table 8 for various values of aspect

ratio (a/b = 0.5, 1, and 2), gradient index (

a

= 0, 0.25, 1, 5, and 1), and foundation stiffness parameters ðK

W

; K

S

Þ. Furthermore,

fundamental frequency parameters

b are given in Table 9 for the Al/ZrO

2

rectangular Mindlin plate (a/b = 1.5) with different

values of foundation stiffness parameters and gradient index as well as thickness to length ratio (h/a = 0.05, 0.1, and 0.2). The

primary conclusion, drawn from Tables 8 and 9, is to enhance the frequency parameters

b with the increase in foundation

stiffness parameters and thickness to length ratio, while all other parameters are considered to be ﬁxed. It can also be ob-

served in both tables that the frequency parameters

b enhance as higher degree of edge constraints (in the order from free to

simply-supported to clamped) is applied to the other two edges of the plate. In Table 8, the frequency parameters

b are found

to diminish with an increase in gradient indices. However, this trend is violated in Table 9 when the plate is made of Al/ZrO

2

instead of Al/Al

2

O

3

.

4.2. Effect of foundation on the eigenfrequency parameters b

Fig. 5 shows the variation of the eigenfrequency parameter b versus the Winkler foundation stiffness parameter

K

W

for

SCSS rectangular Al/Al

2

O

3

plates with different modes m. Note that in Fig. 5 the gradient index

a

is considered to be 5. From

Fig. 5a–d, it can obviously be seen that with the increase of the

K

W

, the eigenfrequency parameter b increases.

All curves in Fig. 5a are plotted for a rectangular FG Mindlin plate (h/a = 0.15 and a/b = 0.4) when m = 1 and

K

S

¼ 0; 125; 250 and 500. It can be seen from Fig. 5a that the eigenfrequency parameter b increases as the K

S

takes the higher

Table 8

Fundamental frequency parameter

b ¼

xh

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

q

c

=E

c

p

for the Al/Al

2

O

3

square Mindlin plate when h/a = 0.15.

ðK

W

; K

S

Þ

g

= a/b

a

Boundary conditions

SSSS SSSC SCSC SSSF SFSF SCSF

(0, 0) 0.5 0 0.08006 0.08325 0.08729 0.06713 0.06364 0.06781

0.25 0.07320 0.07600 0.07950 0.06145 0.05829 0.06205

1 0.06335 0.06541 0.06790 0.05346 0.05080 0.05391

5 0.05379 0.05524 0.05695 0.04568 0.04349 0.04600

1 0.04100 0.04263 0.04443 0.03417 0.03239 0.03451

1 0 0.12480 0.14378 0.16713 0.07537 0.06290 0.08062

0.25 0.11354 0.12974 0.14927 0.06890 0.05761 0.07351

1 0.09644 0.10725 0.11955 0.05968 0.05021 0.06308

5 0.08027 0.08720 0.09479 0.05078 0.04301 0.05322

1 0.06352 0.07318 0.08507 0.03836 0.03202 0.04104

2 0 0.28513 0.35045 0.41996 0.10065 0.06217 0.13484

0.25 0.25555 0.30709 0.36112 0.09170 0.05695 0.12160

1 0.20592 0.23262 0.26091 0.07851 0.04970 0.10066

5 0.16315 0.17691 0.19258 0.06610 0.04262 0.08226

1 0.14591 0.17921 0.21375 0.05123 0.03164 0.06863

(100, 10) 0.5 0 0.12870 0.13097 0.13376 0.11917 0.11513 0.11984

0.25 0.11842 0.12040 0.12280 0.10978 0.10603 0.11037

1 0.10519 0.10659 0.10824 0.09781 0.09465 0.09824

5 0.09223 0.09318 0.09426 0.08594 0.08338 0.08622

1 0.06591 0.06708 0.06808 0.06065 0.05860 0.06100

1 0 0.17020 0.18550 0.20450 0.13016 0.11517 0.13529

0.25 0.15599 0.16892 0.18463 0.11986 0.10611 0.12432

1 0.13652 0.14483 0.15431 0.10635 0.09465 0.10945

5 0.11786 0.12296 0.12854 0.09282 0.08343 0.09485

1 0.08663 0.09442 0.10409 0.06625 0.05862 0.06886

2 0 0.32768 0.38719 0.45150 0.16797 0.11593 0.19829

0.25 0.29612 0.34264 0.39216 0.15448 0.10651 0.18035

1 0.24674 0.26994 0.29496 0.13539 0.09476 0.15251

5 0.20359 0.21501 0.22830 0.11603 0.08338 0.12705

1 0.16773 0.19805 0.22981 0.08550 0.05901 0.10093

1286 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

Table 9

Fundamental frequency parameter

b ¼

xh

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

q

c

=E

c

p

for the Al/ZrO

2

rectangular Mindlin plate (a/b = 1.5).

ðK

W

; K

S

Þ

d = h/a

a

Boundary conditions

SSSS SSSC SCSC SSSF SFSF SCSF

(0, 0) 0.05 0 0.02392 0.03129 0.04076 0.01024 0.00719 0.01249

0.25 0.02269 0.02899 0.03664 0.00981 0.00692 0.01185

1 0.02156 0.02667 0.03250 0.00948 0.00674 0.01132

5 0.02180 0.02677 0.03239 0.00963 0.00685 0.01146

1 0.02046 0.02689 0.03502 0.00880 0.00618 0.01073

0.1 0 0.09188 0.11639 0.14580 0.04001 0.02835 0.04817

0.25 0.08603 0.10561 0.12781 0.03810 0.02717 0.04532

1 0.08155 0.09734 0.11453 0.03679 0.02641 0.04327

5 0.08171 0.09646 0.11234 0.03718 0.02677 0.04352

1 0.07895 0.10001 0.12528 0.03438 0.02426 0.04139

0.2 0 0.32284 0.37876 0.43939 0.14871 0.10795 0.17323

0.25 0.31003 0.36117 0.41624 0.14354 0.10436 0.16671

1 0.29399 0.33549 0.37962 0.13851 0.10127 0.15937

5 0.29099 0.32783 0.36695 0.13888 0.10200 0.15878

1 0.27788 0.32545 0.37755 0.12779 0.09276 0.14885

(250, 25) 0.05 0 0.03421 0.04021 0.04815 0.02291 0.01877 0.02508

0.25 0.03285 0.03786 0.04412 0.02218 0.01821 0.02411

1 0.03184 0.03577 0.04035 0.02172 0.01789 0.02334

5 0.03235 0.03615 0.04053 0.02203 0.01821 0.02360

1 0.02937 0.03455 0.04138 0.01968 0.01613 0.02155

0.1 0 0.13365 0.15131 0.17690 0.09079 0.07472 0.09838

0.25 0.12771 0.14271 0.16008 0.08773 0.07240 0.09409

1 0.12381 0.13550 0.14848 0.08586 0.07114 0.09118

5 0.12533 0.13611 0.14792 0.08699 0.07234 0.09196

1 0.11484 0.13150 0.15200 0.07801 0.06420 0.08453

0.2 0 0.49945 0.54079 0.58657 0.35225 0.29172 0.37193

0.25 0.48327 0.52073 0.56189 0.34182 0.28124 0.36007

1 0.46997 0.49952 0.53159 0.33473 0.27785 0.35005

5 0.47400 0.49979 0.52779 0.33853 0.28469 0.35237

1 0.43001 0.46469 0.50402 0.30268 0.25067 0.31959

Fig. 5. Variation of the eigenfrequency parameter b versus K

W

for SCSS rectangular Al/Al

2

O

3

plates with different values of (a) K

S

, (b)

g

, (c) d, and (d) m when

a

=5.

Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

1287

values. Another interesting conclusion can be inferred from Fig. 5a is that when extremely high values of the K

S

are taken, the

eigenfrequency parameter b becomes constant for any values of the

K

W

.

The eigenfrequency parameters b of FG rectangular Mindlin plates (h/a = 0.15 and m = 1) resting on the Winkler elastic

foundation ð

K

S

¼ 0Þ for different values of the aspect ratios a/b = 0.4, 1, 2, and 3 are plotted in Fig. 5b for a wide range of

the

K

W

. It is observed, in Fig. 5b, that the eigenfrequency parameter b enhances with increasing the aspect ratio a/b.

In Fig. 5c, the eigenfrequency parameter b is given for rectangular FG plates (a/b = 2.5) resting on the Pasternak elastic

foundation when

K

S

¼ 10, m = 1 and h/a = 0.001, 0.1, 0.15, and 0.2. As it is expected, for a certain value of K

W

, the eigenfre-

quency parameter b rises as the plate thickness diminishes.

The same plate parameters as those used in Fig. 5b are considered for Fig. 5d except that a/b = 1 and the modes m are

taken to be 1, 2, 3, and 4. It is worth noting that the effect of the

K

W

on the growth rate of the eigenfrequency parameter

b is more tangible for lower modes m.

4.3. Effect of gradient index

a

on the eigenfrequency parameters b

The variation of the eigenfrequency parameter b versus the gradient index

a

, for SSSS rectangular Al/Al

2

O

3

plates, is

shown in Fig. 6. Note that the eigenfrequency parameter b is normalized in Fig. 6 by dividing its maximum value, and de-

noted by b

n

.

Fig. 6a and b depict the relation between the normalized eigenfrequency parameter b

n

and the gradient index

a

for a

square FG Mindlin plate (h/a = 0.2 and a/b = 1) with m = 1 when in Fig. 6a

K

S

¼ 0 and K

W

¼ 0; 50; 250, and 500 and in

Fig. 6b

K

W

¼ 10 and K

S

¼ 0; 5; 10 and 25. The results in Fig. 6a and b indicate that the normalized eigenfrequency parameter

b

n

increases with the increasing values of the foundation stiffness parameters. Another interesting point attracting one’s

attention is that, regardless of the values of the foundation stiffness parameters, the normalized eigenfrequency parameter

b

n

is minimized for a speciﬁc value of the gradient index

a

, herein called the critical gradient index and denoted by

a

cr

. Due to

the importance of the value of

a

cr

, Section 4.5 is devoted to determining the critical values of the gradient index

a

.

Fig. 6. Variation of the normalized eigenfrequency parameter b

n

versus

a

for SSSS rectangular Al/Al

2

O

3

plates with different values of (a) K

W

, (b) K

S

, (c)

g

,

and (d) d when m =1.

1288 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

The effect of the aspect ratio a/b on the normalized eigenfrequency parameter b

n

is investigated for rectangular FG Mind-

lin plates (h/a = 0.2) with various values of the gradient index

a

when a/b = 0.5, 1, 1.5, and 3; ðK

W

; K

S

Þ¼ð0; 0Þ, and m =1,as

shown in Fig. 6c. It is found that the higher values of the aspect ratio a/b will reduce the normalized eigenfrequency param-

eter b

n

. For each value of the aspect ratio a/b, there is also a minimum value for the normalized eigenfrequency parameter b

n

,

as one can see in Fig. 6c.

Fig. 7. Variation of the normalized eigenfrequency parameter b

n

versus

a

for a square FG Mindlin plate (a/b = 1 and h/a = 0.18) with (a) SFSF, (b) SSSS, and

(c) SCSC boundary conditions when

K

W

¼ K

S

¼ 5 and m =1.

Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

1289

The graph of the normalized eigenfrequency parameter b

n

against the gradient index

a

is plotted in Fig. 6d for square FG

Mindlin plates (a/b = 1) for different values of the thickness to length ratio (h/a = 0.08, 0.1, and 0.2) when ð

K

W

; K

S

Þ¼ð0; 0Þ

and m = 1. It is seen that as the thickness to length ratio h/a decreases, the normalized eigenfrequency parameter b

n

mainly

increases for

a

>

a

cr

, while an inverse behavior is experienced for

a

<

a

cr

.

4.4. Effect of different FGMs on the eigenfrequency parameters b

The inﬂuence of the two different FGMs (i.e., Al/Al

2

O

3

and Al/ZrO

2

) on the normalized eigenfrequency parameter b

n

is

shown in Fig. 7 for a square FG Mindlin plate (h/a = 0.18 and a/b = 1), resting on Pasternak elastic foundation ð

K

W

¼

K

S

¼ 5Þ when m = 1. The boundary conditions at the edges of the plate are considered to be SFSF, SSSS, and SCSC in

Fig. 7a–c, respectively. Herein, it should be noted that the normalized eigenfrequency parameter b

n

is obtained by dividing

its corresponding b into the eigenfrequency parameter in which

a

=10

3

.

Fig. 7 proves the fact that as the gradient index

a

varies from 10

3

to 10

3

, the FG Al/Al

2

O

3

material has a higher effect on

the normalized eigenfrequency parameter b

n

when compared with the FG Al/ZrO

2

material. It can also be ﬁgured out that for

the most ranges of the gradient index

a

, the normalized eigenfrequency parameters b

n

of the Al/ZrO

2

square plate are higher

than those of the Al/Al

2

O

3

one. It is worthy of mention that as lower degrees of edge constraints (in the order from clamped

to simply-supported to free) are applied to the other two edges of the square Al/ZrO

2

plate, a peak at the point around

a

=10

is more evident for the eigenfrequency parameter b

n

.

4.5. Determination of critical gradient index

a

cr

Fig. 8 represents the behavior of the critical gradient index

a

cr

versus the plate parameters, including foundation stiffness

parameters, aspect ratio, and different boundary conditions, for FG rectangular Mindlin plates when m = 1. It is emphasized

that the critical gradient index

a

cr

is deﬁned by ﬁnding the gradient index

a

in which the eigenfrequency parameter b is

minimized.

From Fig. 8a and b, it can be seen that the values of the critical gradient index

a

cr

for the Al/Al

2

O

3

SSSS square plate (h/

a = 0.2 and a/b = 1) are greater than those for the Al/ZrO

2

one when any values of the foundation stiffness parameters are

taken into account. In other words, the minimum value of the eigenfrequency parameter b for the Al/ZrO

2

square plate will

occur at the smaller values of the gradient index

a

. It is worthwhile to mention that in contrast with the FG Al/ZrO

2

material,

the minimum value of the eigenfrequency parameter b for the FG Al/Al

2

O

3

material considerably shifts to the lower values of

Fig. 8. Variation of the critical gradient index

a

cr

versus (a) K

W

, (b) K

S

, (c)

g

, and (d) K

W

for rectangular FG Mindlin plates with different boundary conditions

when m =1.

1290 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291

the gradient index

a

as the foundation stiffness parameters increase. It should be noted that K

S

and K

W

are taken as 10 in

Fig. 8a and b, respectively.

Fig. 8c shows the behavior of the critical gradient index

a

cr

against the aspect ratios a/b for the SSSS FG rectangular plate

(h/a = 0.2) resting on the Pasternak elastic foundations ð

K

W

¼ K

S

¼ 10Þ, when the aspect ratios a/b varies from 0.4 to 3.0. It is

evident from Fig. 8c that as the aspect ratio a/b increases, the reaction of the critical gradient index

a

cr

to the FG Al/ZrO

2

material is much more than that to the FG Al/Al

2

O

3

material. In fact, the minimum value of the eigenfrequency parameter

b for the Al/Al

2

O

3

square plates can hardly move on the b

a

diagram with increasing the aspect ratio a/b.

In Fig. 8d, the inﬂuence of the different boundary conditions on the critical gradient index

a

cr

is illustrated for an FG

square plate (h/a = 0.15 and a/b = 1) when

K

S

¼ 20 and K

W

varies from 0 to 1500. It is found that as higher degrees of edge

constraints are applied to the other two edges of the FG plate, the critical gradient index

a

cr

takes the higher values. It can

also be concluded from Fig. 8d that the variation of the critical gradient index

a

cr

against the K

W

is almost negligible for the

Al/ZrO

2

plates especially with SFSF and SSSS boundary conditions. A rapid decrease in the values of the critical gradient index

a

cr

is observed for the SSSS Al/Al

2

O

3

square plate when K

W

< 900 and for the SFSF one when K

W

< 150.

5. Conclusions

In the present study, an analytical solution for free vibration analysis of moderately thick FG rectangular plates, resting on

either Winkler or Pasternak elastic foundations, was presented for all six possible combinations of boundary conditions. A

new form of the shear correction factor

j

2

for FG rectangular plates was derived in which the effect of material properties

and the geometry of the plate on the shear correction factor

j

2

were considered. It was demonstrated that the shear correc-

tion factor

j

2

deviates from 5/6 as the thickness to length ratio h/a decreases. All comparison studies demonstrated that the

present solution is highly efﬁcient for exact analysis of the vibration of FG rectangular plates on the basis of the Mindlin plate

theory. The inﬂuence of the foundation stiffness parameters on the natural frequencies of the FG plates with different com-

binations of boundary conditions was investigated for various values of aspect ratios, gradient indices, and thickness to

length ratios. Due to the inherent features of the current analytical solution, the present ﬁndings will be a useful benchmark

for evaluating other analytical and numerical methods which will be developed by researchers in the future.

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- CitationsCitations75
- ReferencesReferences28

- "Their procedure was based on the Reissner–Mindlin plate theory. Then, Hosseini-Hashemi et al. [13] worked on free vibration analysis of moderately thick rectangular plates, which were composed of functionally graded materials and supported by either Winkler or Pasternik elastic foundations. The finite element based simulation of the dynamic response is regularly employed by the FGM manufacturers in order to improve their products' comfort and reliability. "

- "This new type of materials can be applied to avoid in-terfacial stress concentration appeared in laminated struc-tures, and therefore, a great promise is proposed by FGMs in applications where the working conditions are severe[2], including spacecraft heat shields, heat ex-changer tubes, plasma facings for fusion reactors, engine components and highpower electrical contacts or even magnets. Compared with the analysis of functionally graded rectangular plates34567891011 and functionally graded spheres [12] as well as functionally graded cylindrical shells1314151617, the investigations of functionally graded annular plates are limited in number. Moreover , the proposed semi-analytical 2D-GDQ has led to obtain more accurate results with appropriate rate of convergence. "

[Show abstract] [Hide abstract]**ABSTRACT:**This study investigates the free vibration of the Two-Dimensional Functionally Graded Annular Plates (2D-FGAP). The theoretical formulations are based on the three-dimensional elasticity theory with small strain assumption. The Two-Dimensional Generalized Differential Quadrature Method (2D-GDQM) as an efficient and accurate semi-analytical approach is used to discretize the equations of motion and to implement the various boundary conditions. The fast rate of convergence for this method is shown and the results are compared with the existing results in the literature. The mate-rial properties are assumed to be continuously changing along thickness and radial directions simul-taneously, which can be varied according to the power law and exponential distributions, respec-tively. The effects of the geometrical parameters, the material graded indices in thickness and radial directions, and the mechanical boundary conditions on the frequency parameters of the two-dimensional functionally graded annular plates are evaluated in detail. The results are verified to be against those given in the literature.- "[58,59] . There are excellent reviews on this subject , including different approaches, such as60616263646566. For slender beams, it is well-known that rather than shear deformations or axial deformations, the foremost reason of displacements and rotation is bending. "

[Show abstract] [Hide abstract]**ABSTRACT:**In this paper, arbitrarily large in-plane deflections of planar curved beams made of Functionally Graded Materials (FGM) are examined. Geometrically exact beam theory is revisited, but the material properties are considered as an arbitrary function of the position on the cross-section of the beam, to derive the governing differential equation system. Axial, and shear deformations are taken into account. Equations are solved by the method called Variational Iterational Method (VIM). Solution steps are given explicitly. Presented formulation is validated by solving some examples existing in the literature. It is seen that the solution method is easy, and efficient. Deflection values, and deflected shapes of half, and quarter circular cantilever beams made of FGM are given for different variations of the material. Snap-through, and bifurcation buckling of pinned–pinned circular arches made of FGM are examined. Effects of material variation on the deflections, and bifurcation buckling load are examined. New results are also given for arbitrarily large in-plane deflections of planar curved beams made of FGM.

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