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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.
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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.
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Sh. Hosseini-Hashemi et al./Applied Mathematical Modelling 34 (2010) 1276–1291
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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)
70200
5700
380
38002702
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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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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
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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).
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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.
FGMs Constant 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
0 0.514 10
1
0.05 Present 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–
0.1 Present 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
FGMsMethod
ffiffiffiffiffiffiffiffiffiffiffiffi
qc=Ec
p
=h for SSSS square plates (a/b = 1) when h/a = 0.1.
Gradient index (a)
00.51258 10
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
SSSSSSSC SCSC SSSF SFSFSCSF
0Present 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
100Present 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
1000Present 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 SSSCSCSC SSSF SFSF SCSF
(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
SSSSSSSC SCSC SSSFSFSF 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
1287
Page 13
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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the gradient index a as the foundation stiffness parameters increase. It should be noted that KSand KWare taken as 10 in
Fig. 8a and b, respectively.
Fig. 8c shows the behavior of the critical gradient index acragainst the aspect ratios a/b for the SSSS FG rectangular plate
(h/a = 0.2) resting on the Pasternak elastic foundations ðKW¼ KS¼ 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 acrto the FG Al/ZrO2
material is much more than that to the FG Al/Al2O3material. In fact, the minimum value of the eigenfrequency parameter
b for the Al/Al2O3square 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 acris illustrated for an FG
square plate (h/a = 0.15 and a/b = 1) when KS¼ 20 and KWvaries 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 acrtakes the higher values. It can
also be concluded from Fig. 8d that the variation of the critical gradient index acragainst the KWis almost negligible for the
Al/ZrO2plates especially with SFSF and SSSS boundary conditions. A rapid decrease in the values of the critical gradient index
acris observed for the SSSS Al/Al2O3square plate when KW< 900 and for the SFSF one when KW< 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 j2for FG rectangular plates was derived in which the effect of material properties
and the geometry of the plate on the shear correction factor j2were considered. It was demonstrated that the shear correc-
tion factor j2deviates 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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