# 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.

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**ABSTRACT:**Free vibration analysis of moderately thick rectangular FG plates on elastic foundation with various combinations of simply supported and clamped boundary conditions are studied. Winkler model is considered to describe the reaction of elastic foundation on the plate. Governing equations of motion are obtained based on the Mindlin plate theory. A semi-analytical solution is presented for the governing equations using the extended Kantorovich method together with infinite power series solution. Results are compared and validated with available results in the literature. Effects of elastic foundation, boundary conditions, material, and geometrical parameters on natural frequencies of the FG plates are investigated.Archive of Applied Mechanics 01/2013; 83(2). · 1.44 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The nonlinear forced vibration of infinitely long functionally graded cylindrical shells is studied using the Lagrangian theory and multiple scale method. The equivalent properties of functionally graded materials are described as a power-law distribution in the thickness direction. The energy approach is applied to derive the reduced low-dimensional nonlinear ordinary differential equations of motion. Using the multiple scale method, a special case is investigated when there is a 1:2 internal resonance between two modes and the excitation frequency is close to the higher natural frequency. The amplitude–frequency curves and the bifurcation behavior of the system are analyzed using numerical continuation method, and the path leading the system to chaos is revealed. The evolution of symmetry is depicted by both the perturbation method and the numerical Poincaré maps. The effect of power-law exponent on the amplitude response of the system is also discussed.Thin-Walled Structures 05/2014; 78:26–36. · 1.23 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**This paper presents an efficient shear deformation theory for vibration of functionally graded plates. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The mechanical properties of functionally graded plate are assumed to vary according to a power law distribution of the volume fraction of the constituents. Equations of motion are derived from the Hamilton’s principle. Analytical solutions of natural frequency are obtained for simply supported plates. The accuracy of the present solutions is verified by comparing the obtained results with those predicted by classical theory, first-order shear deformation theory, and higher-order shear deformation theory. It can be concluded that the present theory is not only accurate but also simple in predicting the natural frequencies of functionally graded plates.Archive of Applied Mechanics 01/2013; 83(1). · 1.44 Impact Factor

Page 1

Free vibration of functionally graded rectangular plates using

first-order shear deformation plate theory

Sh. Hosseini-Hashemia, H. Rokni Damavandi Taherb, H. Akhavana,*, M. Omidia

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

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

a r t i c l e i n f o

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

a b s t r a c t

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 ? 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 ? 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 Modelling

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

Page 2

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

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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

EðzÞ ¼ ðEc? EmÞVfðzÞ þ Em;

qðzÞ ¼ ðqc?qmÞVfðzÞ þqm;

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

be given by

?

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/ZrO2rectangular plate with various values of gradient index a, is shown in Fig. 2. For a = 0 and a = 1, the plate is

ð1Þ

VfðzÞ ¼

z

hþ1

2

?a

;

ð2Þ

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

Page 4

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 Ux, Uyand Uz, may be expressed as

Ux¼ ?zwxðx;y;tÞ;

Uy¼ ?zwyðx;y;tÞ;

Uz¼ wzðx;y;tÞ;

where wxand wyare the rotational displacements about the y and x axes at the middle surface of the plate, respectively, wzis

the transverse displacement, and t is the time variable. By neglecting ezzin the stress–strain relations, the general strain–dis-

placement relations for small deformation are defined as

exx¼ ?zwx;x;

cxy¼ ?z

22

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

ative. For example, wx,yis equivalent to @wx/@y while wx,yymeans @2wx/@y2. Hook’s law for a plate may be expressed as

EðzÞ

1 ?m2ðexxþmeyyÞ;

sxy¼ GðzÞcxy;

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

Zh=2

Mxy¼

Zh=2

ð3Þ

eyy¼ ?zwy;y;

?

ezz¼ 0;

wx? wz;x

wx;yþ wy;x

?

;

cxz¼ ?

??

;

cyz¼ ?

wy? wz;y

2

??

;

ð4Þ

rxx¼

ryy¼

EðzÞ

1 ?m2ðeyyþmexxÞ;

syz¼ GðzÞcyz;

rzz¼ 0;

sxz¼ GðzÞcxz;

ð5Þ

Mii¼

?h=2riizdz;

Zh=2

i ¼ x;y

?h=2sxyzdz;

Qj¼ j2

?h=2rjzdz;

j ¼ x;y;

ð6Þ

Fig. 2. Variation of Young’s modulus through the dimensionless thickness of Al/ZrO2plate.

Table 1

Material properties used in the FG plate.

Properties MetalCeramic

Aluminum (Al)Zirconia (ZrO2) Alumina (Al2O3)

E (GPa)

q (kg/m3)

70 200

5700

380

3800 2702

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

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in which j2is the transverse shear correction coefficient, applied to the transverse shear forces due to the fact that the trans-

verse shear strains (exzand eyz) have a nearly parabolic dependency to the thickness coordinate. Substituting Eqs. (4) and (5)

into Eq. (6) gives the resultant bending moments (Mxxand Myy), twisting moment (Mxy), and the transverse shear forces (Qx

and Qy) per unit length as follows:

Mxx¼ ?Aðwx;xþmwy;yÞ;

Mxy¼ ?ð1 ?mÞ

Qx¼ ?j2hBðwx? wz;xÞ;

in which

Myy¼ ?Aðwy;yþmwx;xÞ;

2

Aðwx;yþ wy;xÞ;

Qy¼ ?j2hBðwy? wz;yÞ;

ð7Þ

A ¼

h3

ð1 ?m2Þ

1

2ð1 þmÞ

að8 þ 3a þa2ÞEmþ 3ð2 þa þa2ÞEc

12ð1 þaÞð2 þaÞð3 þaÞ

EcþaEm

1 þa

??

;

B ¼

:

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

Mxx;xþ Mxy;y? Qx¼ ?1

Mxy;xþ Myy;y? Qy¼ ?1

Qx;xþ Qy;y? P ¼ Dh€wz;

in which

12Ch3€wx;

12Ch3€wy;

ð9Þ

C ¼að8 þ 3a þa2Þqmþ 3ð2 þa þa2Þqc

ð1 þaÞð2 þaÞð3 þaÞ

D ¼qcþaqm

1 þa

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

;

;

ð10Þ

P ¼ KSðwz;xxþ wz;yyÞ ? KWwz;

in which KSand KWare the shear and Winkler foundation coefficients, respectively.

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

ð11Þ

X ¼x

a;

Y ¼y

a;

d ¼h

a;

g ¼a

b;

KS¼KSa2

A

;

KW¼KWa4

A

;

b ¼ xa2

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

wxðx;y;tÞ ¼?wxðX;YÞejxt;

wyðx;y;tÞ ¼?wyðX;YÞejxt;

wzðx;y;tÞ ¼1

where x denotes the natural frequency of vibration in radians and j ¼

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

a

?wzðX;YÞejxt;

ð13Þ

ffiffiffiffiffiffiffi

?1

p

. It should be noted that each parameter having

?wx;xxþg2?wx;yyþm2

?wy;xxþg2?wy;yyþm2

?wz;xxþg2?wz;yy? ð?wx;xþg?wy;yÞ ?Fd2

where F = A/h3, m1= (1 ? m)/2, and m2= (1 + m)/2.

m1ð?wx;xxþg?wy;xyÞ ?j2B

m1gð?wx;xyþg?wy;yyÞ ?j2B

Fd2m1

ð?wx??wz;xÞ þb2d2

ð?wy?g?wz;yÞ þb2d2

12m1

?wx¼ 0;

Fd2m1

12m1

?wy¼ 0;

??DFb2d2

j2B

?KW?wzþ KSð?wz;xxþg2?wz;yyÞ

?

BCj2

?wz¼ 0;

ð14Þ

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

Page 6

2.5. Boundary conditions

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

Mxx¼?wy¼?wz¼ 0:

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

ð15Þ

– for a free edge

Mxx¼ Mxy¼ 0;

Qy¼ ?KS

A

a2g

??

?wz;y;

ð16Þ

– for a simply-supported edge

Myy¼?wx¼?wz¼ 0;

– for a clamped edge

ð17Þ

?wx¼?wy¼?wz¼ 0:

ð18Þ

2.6. Solution of governing equations

The general solutions to Eq. (14) in terms of the three dimensionless potentials Wx, Wyand Wzmay be expressed as

?wx¼ C1Wx;xþ C2Wy;x?gWz;y;

?wy¼ C1gWx;yþ C2gWy;y? Wz;x;

?wz¼ Wxþ Wy;

ð19Þ

where

C1¼

B2

a2

1? B1;

C2¼

B3

a2

2? B1;

ð20Þ

in which B1, B2, and B3along witha2

given after mathematical manipulation by

1anda2

2are the coefficients that may be determined using equations of motion and can be

B1¼H

m1?b2d2

m2

m1

?

12m1;

?

m2

m1

B2¼

?

1 ?Fd2

k2B

KS

!

!

a2

1þFd2

k2B

KWþDFd2

k2BC

b2

"

"

#

#

þ

k2B

Fd2m1

k2B

Fd2m1

(

(

)

)

;

B3¼

?

1 ?Fd2

k2B

KS

a2

2þFd2

k2B

KWþDFd2

k2BC

b2

þ

;

ð21Þ

and

a2

1;a2

v

u

2¼?12KW? 12HKSþ b212 D=C þ d2KSþ H

24 KSþ H

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

?12KW? 12HKSþ b212 D=C þ d2KSþ H

KSþ H

where H = Bj2/Fd2. The Eq. (14) can be restated in terms of the three dimensionless potentials as

????

??

?

1

24

?48 b2d2? 12H

?

??

KSþ H

??

?KWþ Db2=C

?

?2

?þ

?????2

?

u

u

u

t

u

;

ð22Þ

Wx;xxþg2Wx;yy¼ ?a2

Wy;xxþg2Wy;yy¼ ?a2

Wz;xxþg2Wz;yy¼ ?a2

where a2

1Wx;

2Wy;

3Wz;

ð23Þ

3¼ ?B1. One set of solutions to Eq. (23) are taken as

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

1281

Page 7

Wx¼ A1Sinðk1YÞ þ A2Cosðk1YÞ

Wy¼ A5Sinhðk2YÞ þ A6Coshðk2YÞ

Wz¼ A9Sinhðk3YÞ þ A10Coshðk3YÞ

in which Aiare the arbitrary coefficients, kjand ljare related to the ajby

½

½

½

?Sinðl1XÞ þ A3Sinðk1YÞ þ A4Cosðk1YÞ

?Sinðl2XÞ þ A7Sinhðk2YÞ þ A8Coshðk2YÞ

?Sinðl3XÞ þ A11Sinhðk3YÞ þ A12Coshðk3YÞ

½?Cosðl1XÞ;

?Cosðl2XÞ;

?Cosðl3XÞ;

½

½

ð24Þ

a2

1¼ l2

1þg2k2

1;

a2

2¼ l2

2?g2k2

2;

a2

3¼ l2

3?g2k2

3

ð25Þ

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

Wx¼ A1Sinðk1YÞ þ A2Cosðk1YÞ

Wy¼ A5Sinhðk2YÞ þ A6Coshðk2YÞ

Wz¼ A9Sinhðk3YÞ þ A10Coshðk3YÞ

in which l = l1= l2= l3= mp (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).

½

½

½

?SinðlXÞ;

?SinðlXÞ;

?SinðlXÞ:

ð26Þ

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 (j2) 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

j2¼5 þ 5m

6 þ 5m:

ð27Þ

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

j2¼

5

6 ? ðmmVmþmcVcÞ;

ð28Þ

in which Vmand Vcare 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/ZrO2and Al/Al2O3, 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 j2for a wide range of the gradient indices a and thickness to

length ratios h/a are plotted for Al/Al2O3(Fig. 3a) and Al/ZrO2(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 j2for 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 j2initially 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 ZrO2show higher

resistance to this pattern in comparison with the Al/Al2O3plates. In other words, it is evident from Figs. 3b and 4b that

for the Al/ZrO2plates, 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 j2than thinner plates. In fact, the error in calculating the frequency param-

eter is more tangible for thinner FG plates if the shear correction factor j2is assumed to be constant (e.g., j2= 5/6).

1282

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

Page 8

In order to obtain a formula for the shear correction factors j2, 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

j2shown in Fig. 3. Finally, the following function is proposed as

j2ða;dÞ ¼5

where Ci(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 j2(i.e., Eq.

(29)).

6þ C1 e?C2a? e?C3a

??ð10d ? 2Þ ? C4 e?C5a? e?C6a

??

10d ? 1

ðÞ;

ð29Þ

Fig. 3. Variation of the shear correction factors j2against the gradient index a and thickness to length ratio h/a for (a) Al/Al2O3; (b) Al/ZrO2SSSS square

plates, a/b = 1.

Fig. 4. Variation of the shear correction factors j2versus the gradient index a for (a) Al/Al2O3; (b) Al/ZrO2SSSS 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 j2formula for two FG materials.

FGMsConstant coefficients

C1

C2

C3

C4

C5

C6

Al/Al2O3

Al/ZrO2

0.750

0.560

0.025

0.001

2.000

5.450

0.640

0.420

0.060

0.095

1.000

1.175

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

1283

Page 9

3.2. Comparison studies

To demonstrate the efficiency and accuracy of the present solution along with new shear correction factor j2, 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 (Al2O3). 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

j2in their study work. For convenience in comparison, a new frequency parameter is defined as b ¼ xh

ffiffiffiffiffiffiffiffiffiffiffiffi

qc=Ec

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 (j2= 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 (j2= 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/Al2O3and Al/

ZrO2). Fundamental frequency parameters are given in Table 4 in the form of^b ¼ xa2

the results of FSDT [13] for the Al/Al2O3square plate are in good agreement with the present analytical solution but for the

Al/ZrO2square plate they reveal a great deviation from the present analytical solution and 2D HAPT [10] due to the reasons

mentioned in Example 1.

ffiffiffiffiffiffiffiffiffiffiffiffi

qc=Ec

p

=h. Table 4 shows this fact that

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=

10 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 (ZrO2). All fundamental frequency parameters presented in Table 5 are defined as~b ¼ xh2

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

ffiffiffiffiffiffip

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qm=Em

p

. Com-

Table 3

Comparison of fundamental frequency parameter?b ¼ xh

d = h/a

Method

ffiffiffiffiffiffiffiffiffiffiffiffi

Gradient index (a)

qc=Ec

p

for SSSS Al/Al2O3square plates (a/b = 1).

j2

00.51410

1

0.05Present solution

FSDT [13]

Eq. (29)

5/6

0.01480

0.01464

0.01281

0.01241

0.01150

0.01118

0.01013

0.00970

0.00963

0.00931–

0.1Present solution

2D HAPT [10]

FSDT [10]

FSDT [13]

Eq. (29)

–

1

5/6

Eq. (27)

Eq. (28)

0.05769

0.05777

0.06382

0.05673

0.05713

0.05711

0.04920

0.04917

0.05429

0.04818

0.04849

0.04847

0.04454

0.04427

0.04889

0.04346

0.04371

0.04370

0.03825

0.03811

0.04230

0.03757

0.03781

0.03779

0.03627

0.03642

0.04047

0.03591

0.03619

0.03618

0.02936

0.02933

–

–

–

–

0.2Present solution

2D HAPT [10]

FSDT [10]

FSDT [13]

Eq. (29)

–

1

5/6

Eq. (27)

Eq. (28)

0.2112

0.2121

0.2334

0.2055

0.2098

0.2096

0.1806

0.1819

0.1997

0.1757

0.1790

0.1788

0.1650

0.1640

0.1802

0.1587

0.1616

0.1614

0.1371

0.1383

0.1543

0.1356

0.1383

0.1382

0.1304

0.1306

0.1462

0.1284

0.1313

0.1312

0.1075

0.1077

–

–

–

–

Table 4

Comparison of fundamental frequency parameter^b ¼ xa2

FGMs Method

ffiffiffiffiffiffiffiffiffiffiffiffi

qc=Ec

p

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

Gradient index (a)

0 0.5125810

Al/Al2O3

Present solution

FSDT [13]

5.7693

5.6763

4.9207

4.8209

4.4545

4.3474

4.0063

3.9474

3.7837

3.7218

3.6830

3.6410

3.6277

3.5923

Al/ZrO2

Present solution

2D HAPT [10]

FSDT [13]

5.7693

5.7769

5.6763

5.3176

–

5.1105

5.2532

5.3216

4.8713

5.3084

–

4.6977

5.2940

–

4.5549

5.2312

–

4.4741

5.1893

–

4.4323

1284

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

Page 10

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 xb2

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.

ffiffiffiffiffiffiffiffiffiffiffi

Ch=A

p

=p2. The present results are compared with those

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 ðKS¼ 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 xa

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

ffiffiffiffiffiffiffiffiffiffiffi

Ch=A

p

. It can be seen from Table 7 that there

Table 5

Comparison of fundamental frequency parameter~b ¼ xh2

Method

a = 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

qm=Em

p

for SSSS Al/ZrO2square plates (a/b = 1).

a = 1

d = 0.2

d ¼ 1=

0.4618

0.4658

0.4658

0.4619

ffiffiffiffiffiffi

10

p

d = 0.1

d = 0.05

d = 0.1

d = 0.2

a = 2

a = 3

a = 5

Present solution

2D HAPT [10]

HSDT [9]

FSDT [9]

0.0576

0.0578

0.0578

0.0577

0.0158

0.0158

0.0157

0.0162

0.0611

0.0618

0.0613

0.0633

0.2270

0.2285

0.2257

0.2323

0.2249

0.2264

0.2237

0.2325

0.2254

0.2270

0.2243

0.2334

0.2265

0.2281

0.2253

0.2334

Table 6

Comparison of fundamental frequency parameter b ¼ xb2

d = h/a

Method

ffiffiffiffiffiffiffiffiffiffiffi

Fundamental frequency parameter b ¼ xb2

ðKW;KSÞ

(100, 0)

2.2413

2.2413

2.2413

Ch=A

p

=p2for homogeneous SSSS square plates (a/b = 1).

ffiffiffiffiffiffiffiffiffiffiffi

Ch=A

p

=p2

ðKW;KSÞ

(500, 0)

3.0215

3.0215

3.0214

ðKW;KSÞ

(100, 10)

2.6551

2.6551

2.6551

ðKW;KSÞ

(500, 10)

3.3400

3.3400

3.3398

0.01Present solution

Mindlin theory [17]

3D method [20]

(200, 0)

2.3989

2.3989

2.3951

(1000, 0)

3.7212

3.7212

3.7008

(200, 10)

2.7842

2.7842

2.7756

(1000, 10)

3.9805

3.9805

3.9566

0.1Present solution

Mindlin theory [17]

3D method [20]

(0, 10)

2.2505

2.2505

2.2334

(10, 10)

2.2722

2.2722

2.2539

(100, 10)

2.4590

2.4591

2.4300

(1000, 10)

3.8567

3.8567

3.7111

0.2Present solution

Mindlin theory [17]

3D method [20]

Table 7

Comparison of fundamental frequency parameter b ¼ xa

h/a = 0.05.

ffiffiffiffiffiffiffiffiffiffiffi

Ch=A

p

for homogeneous square plates (a/b = 1) with different boundary conditions when KS¼ 0 and

KW

MethodBoundary conditions

SSSS SSSCSCSCSSSF SFSFSCSF

0 Present solution

Mindlin theory [18]

Exact CPT [19]

Mindlin theory [22]

19.737

19.737

19.740

19.739

23.643

23.643

23.650

23.646

28.944

28.944

28.950

28.951

11.680

11.680

11.680

11.684

9.630

9.630

9.630

9.631

12.681

12.681

12.690

12.686

100 Present solution

Mindlin theory [18]

Exact CPT [19]

22.126

22.126

22.130

25.671

25.671

25.670

30.623

30.623

30.630

15.376

15.367

15.380

13.883

13.878

13.880

16.149

16.138

16.150

1000 Present solution

Mindlin theory [18]

Exact CPT [19]

Mindlin theory [22]

37.276

37.276

37.280

37.278

39.483

39.483

39.490

39.486

42.869

42.869

42.870

42.873

33.710

33.667

33.710

33.712

33.056

33.037

31.620

31.623

34.070

34.018

34.070

34.073

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

1285

Page 11

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 xh

parameter) in Tables 8 and 9, and as b ¼ xa2

that the eigenfrequency parameter b ¼ xa2

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

ffiffiffiffiffiffiffiffiffiffiffiffi

qc=Ec

p

(called the frequency

ffiffiffiffiffiffiffiffiffiffiffi

Ch=A

ffiffiffiffiffiffiffiffiffiffiffi

p

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

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

Ch=A

p

4.1. List of the frequency parameters?b for the FG plates

Fundamental frequency parameters?b of the Al/Al2O3square 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 ðKW;KSÞ. Furthermore,

fundamental frequency parameters?b are given in Table 9 for the Al/ZrO2rectangular 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/ZrO2

instead of Al/Al2O3.

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 KWfor

SCSS rectangular Al/Al2O3plates 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 KW, 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

KS¼ 0;125;250 and 500. It can be seen from Fig. 5a that the eigenfrequency parameter b increases as the KStakes the higher

Table 8

Fundamental frequency parameter?b ¼ xh

ðKW;KSÞ

ffiffiffiffiffiffiffiffiffiffiffiffi

qc=Ec

p

for the Al/Al2O3square Mindlin plate when h/a = 0.15.

g = a/b

a

Boundary conditions

SSSS SSSCSCSCSSSFSFSFSCSF

(0, 0) 0.50

0.25

1

5

1

0

0.25

1

5

1

0

0.25

1

5

1

0

0.25

1

5

1

0

0.25

1

5

1

0

0.25

1

5

1

0.08006

0.07320

0.06335

0.05379

0.04100

0.12480

0.11354

0.09644

0.08027

0.06352

0.28513

0.25555

0.20592

0.16315

0.14591

0.08325

0.07600

0.06541

0.05524

0.04263

0.14378

0.12974

0.10725

0.08720

0.07318

0.35045

0.30709

0.23262

0.17691

0.17921

0.08729

0.07950

0.06790

0.05695

0.04443

0.16713

0.14927

0.11955

0.09479

0.08507

0.41996

0.36112

0.26091

0.19258

0.21375

0.06713

0.06145

0.05346

0.04568

0.03417

0.07537

0.06890

0.05968

0.05078

0.03836

0.10065

0.09170

0.07851

0.06610

0.05123

0.06364

0.05829

0.05080

0.04349

0.03239

0.06290

0.05761

0.05021

0.04301

0.03202

0.06217

0.05695

0.04970

0.04262

0.03164

0.06781

0.06205

0.05391

0.04600

0.03451

0.08062

0.07351

0.06308

0.05322

0.04104

0.13484

0.12160

0.10066

0.08226

0.06863

1

2

(100, 10)0.50.12870

0.11842

0.10519

0.09223

0.06591

0.17020

0.15599

0.13652

0.11786

0.08663

0.32768

0.29612

0.24674

0.20359

0.16773

0.13097

0.12040

0.10659

0.09318

0.06708

0.18550

0.16892

0.14483

0.12296

0.09442

0.38719

0.34264

0.26994

0.21501

0.19805

0.13376

0.12280

0.10824

0.09426

0.06808

0.20450

0.18463

0.15431

0.12854

0.10409

0.45150

0.39216

0.29496

0.22830

0.22981

0.11917

0.10978

0.09781

0.08594

0.06065

0.13016

0.11986

0.10635

0.09282

0.06625

0.16797

0.15448

0.13539

0.11603

0.08550

0.11513

0.10603

0.09465

0.08338

0.05860

0.11517

0.10611

0.09465

0.08343

0.05862

0.11593

0.10651

0.09476

0.08338

0.05901

0.11984

0.11037

0.09824

0.08622

0.06100

0.13529

0.12432

0.10945

0.09485

0.06886

0.19829

0.18035

0.15251

0.12705

0.10093

1

2

1286

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

Page 12

Table 9

Fundamental frequency parameter?b ¼ xh

ðKW;KSÞ

ffiffiffiffiffiffiffiffiffiffiffiffi

qc=Ec

p

for the Al/ZrO2rectangular Mindlin plate (a/b = 1.5).

d = h/a

a

Boundary conditions

SSSSSSSCSCSCSSSFSFSF SCSF

(0, 0) 0.050

0.25

1

5

1

0

0.25

1

5

1

0

0.25

1

5

1

0

0.25

1

5

1

0

0.25

1

5

1

0

0.25

1

5

1

0.02392

0.02269

0.02156

0.02180

0.02046

0.09188

0.08603

0.08155

0.08171

0.07895

0.32284

0.31003

0.29399

0.29099

0.27788

0.03129

0.02899

0.02667

0.02677

0.02689

0.11639

0.10561

0.09734

0.09646

0.10001

0.37876

0.36117

0.33549

0.32783

0.32545

0.04076

0.03664

0.03250

0.03239

0.03502

0.14580

0.12781

0.11453

0.11234

0.12528

0.43939

0.41624

0.37962

0.36695

0.37755

0.01024

0.00981

0.00948

0.00963

0.00880

0.04001

0.03810

0.03679

0.03718

0.03438

0.14871

0.14354

0.13851

0.13888

0.12779

0.00719

0.00692

0.00674

0.00685

0.00618

0.02835

0.02717

0.02641

0.02677

0.02426

0.10795

0.10436

0.10127

0.10200

0.09276

0.01249

0.01185

0.01132

0.01146

0.01073

0.04817

0.04532

0.04327

0.04352

0.04139

0.17323

0.16671

0.15937

0.15878

0.14885

0.1

0.2

(250, 25)0.05 0.03421

0.03285

0.03184

0.03235

0.02937

0.13365

0.12771

0.12381

0.12533

0.11484

0.49945

0.48327

0.46997

0.47400

0.43001

0.04021

0.03786

0.03577

0.03615

0.03455

0.15131

0.14271

0.13550

0.13611

0.13150

0.54079

0.52073

0.49952

0.49979

0.46469

0.04815

0.04412

0.04035

0.04053

0.04138

0.17690

0.16008

0.14848

0.14792

0.15200

0.58657

0.56189

0.53159

0.52779

0.50402

0.02291

0.02218

0.02172

0.02203

0.01968

0.09079

0.08773

0.08586

0.08699

0.07801

0.35225

0.34182

0.33473

0.33853

0.30268

0.01877

0.01821

0.01789

0.01821

0.01613

0.07472

0.07240

0.07114

0.07234

0.06420

0.29172

0.28124

0.27785

0.28469

0.25067

0.02508

0.02411

0.02334

0.02360

0.02155

0.09838

0.09409

0.09118

0.09196

0.08453

0.37193

0.36007

0.35005

0.35237

0.31959

0.1

0.2

Fig. 5. Variation of the eigenfrequency parameter b versus KWfor SCSS rectangular Al/Al2O3plates with different values of (a) KS, (b)g, (c) d, and (d) m when

a = 5.

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

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

eigenfrequency parameter b becomes constant for any values of the KW.

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

foundation ðKS¼ 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 KW. 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 KS¼ 10, m = 1 and h/a = 0.001, 0.1, 0.15, and 0.2. As it is expected, for a certain value of KW, 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 KW 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/Al2O3plates, 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 bn.

Fig. 6a and b depict the relation between the normalized eigenfrequency parameter bnand 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 KS¼ 0 and KW¼ 0;50;250, and 500 and in

Fig. 6b KW¼ 10 and KS¼ 0;5;10 and 25. The results in Fig. 6a and b indicate that the normalized eigenfrequency parameter

bnincreases 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

bnis minimized for a specific value of the gradient indexa, herein called the critical gradient index and denoted byacr. Due to

the importance of the value of acr, Section 4.5 is devoted to determining the critical values of the gradient index a.

Fig. 6. Variation of the normalized eigenfrequency parameter bnversus a for SSSS rectangular Al/Al2O3plates with different values of (a) KW, (b) KS, (c) g,

and (d) d when m = 1.

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The effect of the aspect ratio a/b on the normalized eigenfrequency parameter bnis 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;ðKW;KSÞ ¼ ð0;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 bn. For each value of the aspect ratio a/b, there is also a minimum value for the normalized eigenfrequency parameter bn,

as one can see in Fig. 6c.

Fig. 7. Variation of the normalized eigenfrequency parameter bnversus 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 KW¼ KS¼ 5 and m = 1.

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

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The graph of the normalized eigenfrequency parameter bnagainst 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 ðKW;KSÞ ¼ ð0; 0Þ

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

increases for a > acr, while an inverse behavior is experienced for a < acr.

4.4. Effect of different FGMs on the eigenfrequency parameters b

The influence of the two different FGMs (i.e., Al/Al2O3and Al/ZrO2) on the normalized eigenfrequency parameter bnis

shown in Fig. 7 for a square FG Mindlin plate (h/a = 0.18 and a/b = 1), resting on Pasternak elastic foundation ðKW¼

KS¼ 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 bnis 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?3to 103, the FG Al/Al2O3material has a higher effect on

the normalized eigenfrequency parameter bnwhen compared with the FG Al/ZrO2material. It can also be figured out that for

the most ranges of the gradient index a, the normalized eigenfrequency parameters bnof the Al/ZrO2square plate are higher

than those of the Al/Al2O3one. 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/ZrO2plate, a peak at the point around a = 10

is more evident for the eigenfrequency parameter bn.

4.5. Determination of critical gradient index acr

Fig. 8 represents the behavior of the critical gradient index acrversus 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 acris 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 acrfor the Al/Al2O3SSSS square plate (h/

a = 0.2 and a/b = 1) are greater than those for the Al/ZrO2one 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/ZrO2square plate will

occur at the smaller values of the gradient index a. It is worthwhile to mention that in contrast with the FG Al/ZrO2material,

the minimum value of the eigenfrequency parameter b for the FG Al/Al2O3material considerably shifts to the lower values of

Fig. 8. Variation of the critical gradient indexacrversus (a) KW, (b) KS, (c) g, and (d) KWfor rectangular FG Mindlin plates with different boundary conditions

when m = 1.

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