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

Article (PDF Available)inApplied Mathematical Modelling 34:1276-1291 · May 2010with192 Reads
DOI: 10.1016/j.apm.2009.08.008
Abstract
The main objective of this research work is to present analytical solutions for free vibration analysis of moderately thick rectangular plates, which are composed of functionally graded materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The proposed rectangular plates have two opposite edges simply-supported, while all possible combinations of free, simply-supported and clamped boundary conditions are applied to the other two edges. In order to capture fundamental frequencies of the functionally graded (FG) rectangular plates resting on elastic foundation, the analysis procedure is based on the first-order shear deformation plate theory (FSDT) to derive and solve exactly the equations of motion. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. First, a new formula for the shear correction factors, used in the Mindlin plate theory, is obtained for FG plates. Then the excellent accuracy of the present analytical solutions is confirmed by making some comparisons of the results with those available in literature. The effect of foundation stiffness parameters on the free vibration of the FG plates, constrained by different combinations of classical boundary conditions, is also presented for various values of aspect ratios, gradient indices, and thickness to length ratios.
Free vibration of functionally graded rectangular plates using
first-order shear deformation plate theory
Sh. Hosseini-Hashemi
a
, H. Rokni Damavandi Taher
b
, H. Akhavan
a,
*
, M. Omidi
a
a
School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran
b
School of Engineering, University of British Columbia Okanagan, Kelowna, BC, Canada V1V 1V7
article info
Article history:
Received 1 June 2009
Received in revised form 5 August 2009
Accepted 18 August 2009
Available online 22 August 2009
Keywords:
Free vibration
FGM
Mindlin theory
Elastic foundation
abstract
The main objective of this research work is to present analytical solutions for free vibration
analysis of moderately thick rectangular plates, which are composed of functionally graded
materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The
proposed rectangular plates have two opposite edges simply-supported, while all possible
combinations of free, simply-supported and clamped boundary conditions are applied to
the other two edges. In order to capture fundamental frequencies of the functionally
graded (FG) rectangular plates resting on elastic foundation, the analysis procedure is
based on the first-order shear deformation plate theory (FSDT) to derive and solve exactly
the equations of motion. The mechanical properties of the FG plates are assumed to vary
continuously through the thickness of the plate and obey a power law distribution of
the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. First,
a new formula for the shear correction factors, used in the Mindlin plate theory, is obtained
for FG plates. Then the excellent accuracy of the present analytical solutions is confirmed
by making some comparisons of the results with those available in literature. The effect of
foundation stiffness parameters on the free vibration of the FG plates, constrained by dif-
ferent combinations of classical boundary conditions, is also presented for various values of
aspect ratios, gradient indices, and thickness to length ratios.
Crown Copyright Ó 2009 Published by Elsevier Inc. All rights reserved.
1. Introduction
In recent years, astonishing advances in science and technology have motivated researchers to work on new structural
materials. Functionally graded materials (FGMs) are classified as novel composite materials which are widely used in aero-
space, nuclear, civil, automotive, optical, biomechanical, electronic, chemical, mechanical, and shipbuilding industries. Due
to smoothly and continuously varying material properties from one surface to the other, FGMs are usually superior to the
conventional composite materials in mechanical behavior. FGMs may possess a number of advantages such as high resis-
tance to temperature gradients, significant reduction in residual and thermal stresses, and high wear resistance.
A few researchers employed classical plate theory (CPT) to analyze vibration and static behavior of thin FG plates. Natural
frequencies of FG simply-supported and clamped rectangular thin plates were obtained by Abrate [1] using the CPT. Free
vibration, buckling and deflection analysis of the FG thin plates were presented by Zhang and Zhou [2] on the basis of the
physical neutral surface. Woo et al. [3] provided an analytical solution for the nonlinear free vibration behavior of FG square
thin plates using the von-Karman theory.
0307-904X/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier Inc. All rights reserved.
doi:10.1016/j.apm.2009.08.008
* Corresponding author. Tel.: +98 912 547 0654; fax: +98 217 724 0488.
E-mail address: hamedakhavan@mecheng.iust.ac.ir (H. Akhavan).
Applied Mathematical Modelling 34 (2010) 1276–1291
Contents lists available at ScienceDirect
Applied Mathematical Mo delling
journal homepage: www.elsevier.com/locate/apm
Higher-order shear deformation plate theory (HSDT) and 3D methods were used by some investigators for analyzing thick
FG plates. Early research efforts for harmonic vibration analysis of an FG simply-supported rectangular plate, using a 3D
asymptotic theory, date back to the work of Reddy and Cheng [4]. Then, Qian et al. [5] conducted an investigation on free
and forced vibrations and static deformations of an FG thick simply-supported square plate by using a higher-order shear
and normal deformable plate theory and a meshless local Petrov–Galerkin method. Vel and Batra [6] described an excellent
investigation on the analytical solution for free and forced vibrations of FG simply-supported square plates based on the 3D
elasticity solution. Zhong and Yu [7] used a state-space approach to analyze free and forced vibrations of an FG piezoelectric
rectangular thick plate simply-supported at its edges. Roque et al. [8] investigated the free vibration of FG plates with
different combinations of boundary conditions by the multiquadric radial basis function method and the HSDT. Free vibra-
tion analysis of FG simply-supported square plates was carried out by Pradyumna and Bandyopadhyay [9] using a higher-
order finite element formulation, as a small part of their study work. Recently, Matsunaga [10] studied natural frequencies
and buckling stresses of FG simply-supported rectangular plates based on 2D higher-order approximate plate theory (2D
HAPT).
Due to its high efficiency and simplicity, first-order shear deformation theory (FSDT) was used for analyzing moderately
thick FG plates. An excellent work on the free vibration, buckling, and static deflections of FG square, circular, and skew
plates with different combinations of boundary conditions was carried out by Abrate [11] on the basis of the CPT, FSDT,
and TSDT. Ferreira et al. [12] employed the collocation method with multiquadric radial basis functions along with the FSDT
and third-order shear deformation plate theory (TSDT) to find natural frequencies of FG square plates with different bound-
ary conditions at the edges. Very recently, Zhao et al. [13] presented a free vibration analysis for FG square and skew plates
with different boundary conditions using the element-free kp-Ritz method on the basis of the FSDT.
One necessary term, used in the FSDT, is a shear correction factor that amends the effect of uniform transverse stress in
shear forces. In isotropic homogeneous plates, shear correction factor is mainly equal to 5/6. However, in FG plates, material
properties of which vary in thickness direction, an error in frequency results will arise in practice owing to the use of a con-
stant shear correction factor. In the past, some investigators tried to improve shear correction factor to yield more exact re-
sults for vibration of the plate. Early efforts by Timoshenko [14] showed that shear correction factor is dependent on
Poisson’s ratio. Recently, a well-known work by Efraim and Eisenberger [15] proposed a formula for shear correction factor
in terms of Poisson’s ratio and volume fractions of both gradients in an FG plate. Furthermore, Nguyen et al. [16] presented
the shear correction factor in terms of ceramic-to-metal Young’s modulus ratio and gradient indices in order to examine sta-
tic analysis of an FG plate. As one can see, there is no more investigation on the vibration of FG plates based on an improved
shear correction factor in the FSDT.
Plates resting on elastic foundations have found considerable applications in structural engineering problems. Rein-
forced-concrete pavements of highways, airport runways, foundation of storage tanks, swimming pools, and deep walls to-
gether with foundation slabs of buildings are well-known direct applications of these kinds of plates. The underlying layers
are modeled by a Winkler-type elastic foundation. The most serious deficiency of the Winkler foundation model is to have no
interaction between the springs. In other words, the springs in this model are assumed to be independent and unconnected.
The Winkler foundation model is fairly improved by adopting the Pasternak foundation model, a two-parameter model, in
which the shear stiffness of the foundation is considered.
Although a few studies on the vibration and buckling analysis of isotropic homogeneous rectangular plates resting on
elastic foundation have been carried out (see for example, Xiang et al. [17], Xiang [18], Lam et al. [19], Zhou et al. [20], Akh-
avan et al. [21,22] and their cited references), research studies on the dynamic behavior of their corresponding FG plates
have received very little attention. Cheng and Kitipornchai [23] proposed a membrane analogy to derive an exact explicit
eigenvalue for compression buckling, hydrothermal buckling, and vibration of FG plates on a Winkler–Pasternak foundation
based on the FSDT. Yang and Shen [24] studied both free vibration and transient response of initially stressed FG rectangular
thin plates subjected to impulsive lateral loads, resting on Pasternak elastic foundation, based on the CPT. The second-order
statistics of the buckling of clamped FG rectangular plates that are resting on Pasternak elastic foundations and subjected to
uniform edge compression was studied by Yang et al. [25] in the framework of the FSDT. Ying et al. [26] treated 2D elasticity
solutions for bending and free vibration of FG beams resting on Winkler–Pasternak elastic foundations. Huang et al. [27]
used a benchmark 3D elasticity solution to study the bending behavior of FG thick simply-supported square plates on a Win-
kler–Pasternak foundation.
Three points can apparently be raised from the literature survey. First, all aforementioned research works have been car-
ried out within the last decade. Second, analytical solutions have been employed by a very few of these research studies due
to the mathematical complexity. It is also well known that an exact solution may be achieved for rectangular plates having at
least one pair of opposite edges simply-supported. Third, there is no work on the analytical solutions for free vibration anal-
ysis of FG moderately thick rectangular plates resting on Winkler–Pasternak elastic foundation. In this paper, the analysis
procedure is based on the first-order shear deformation plate theory, including plate-foundation interaction. For the first
time, a brand-new formula for shear correction factors, used in the Mindlin plate theory, is proposed to guarantee good accu-
racy. In addition, this formula is suitable for different FGMs and easy to implement. In order to validate the obtained results,
the
authors
compare their results with existing data available from other analytical and numerical techniques. The effect of
the plate parameters such as foundation stiffness coefficients, aspect ratios, thickness to length ratios, and gradient indices
on the natural frequencies of FG rectangular plates is presented for six combinations of classical boundary conditions,
namely SSSS, SCSS, SCSC, SSSF, SFSC and SFSF.
Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
1277
2. Mathematical formulation
2.1. Geometrical configuration
A flat, isotropic and moderately thick FG rectangular plate of length a, width b, and uniform thickness h, resting on two-
parameter elastic foundation, is depicted in Fig. 1. The plate has two opposite edges simply-supported along y axis (i.e. along
the edges x = 0 and x = a), while the other two edges may be free, simply-supported, or clamped. The Cartesian coordinate
system (x,y,z) is considered to extract mathematical formulations when x and y axes are located in the undeformed mid-
plane of the plate.
2.2. Material properties
FGMs are composite materials, the mechanical properties of which vary continuously due to gradually changing the vol-
ume fraction of the constituent materials, usually in the thickness direction. In this study, the FG plate is made from a mix-
ture of ceramics and metal and the composition varies from the top to the bottom surface. In fact, the top surface (z = h/2) of
the plate is ceramic-rich whereas the bottom surface (z = h/2) is metal-rich. Young’s modulus and density per unit volume
are assumed to vary continuously through the plate thickness according to a power-law distribution as
EðzÞ¼ðE
c
E
m
ÞV
f
ðzÞþE
m
;
q
ðzÞ¼ð
q
c
q
m
ÞV
f
ðzÞþ
q
m
;
ð1Þ
in which the subscripts m and c represent the metallic and ceramic constituents, respectively, and the volume fraction V
f
may
be given by
V
f
ðzÞ¼
z
h
þ
1
2

a
; ð2Þ
where
a
is the gradient index and takes only positive values. Poisson’s ratio is taken as 0.3 throughout the analyses. Typical
values for metal and ceramics used in the FG plate are listed in Table 1.
In order to gain a better understanding of Eqs. (1) and (2), the variation of Young’s modulus E in the thickness direction z,
for the Al/ZrO
2
rectangular plate with various values of gradient index
a
, is shown in Fig. 2. For
a
= 0 and
a
= 1, the plate is
Fig. 1. Geometry of an SCSF rectangular FG plate with coordinate convention.
1278 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
fully ceramic and metallic, respectively; whereas the composition of metal and ceramic is linear for
a
= 1. It is also observed
from Fig. 2 that the Young’s modulus of the FG plate quickly approaches ceramic’s one for
a
< 1 especially within 0.5 6 z/
h 6 0. For
a
> 1, the FG plate is made from a mixture in which the metal is used more than the ceramic.
2.3. Constitutive relations
According to the Mindlin plate theory, the displacement components of the middle surface along the x, y, and z axes, des-
ignated by U
x
, U
y
and U
z
, may be expressed as
U
x
¼zw
x
ðx; y; tÞ;
U
y
¼zw
y
ðx; y; tÞ;
U
z
¼ w
z
ðx; y; tÞ;
ð3Þ
where
w
x
and
w
y
are the rotational displacements about the y and x axes at the middle surface of the plate, respectively,
w
z
is
the transverse displacement, and t is the time variable. By neglecting
e
zz
in the stress–strain relations, the general strain–dis-
placement relations for small deformation are defined as
e
xx
¼zw
x;x
;
e
yy
¼zw
y;y
;
e
zz
¼ 0;
c
xy
¼z
w
x;y
þ w
y;x
2

;
c
xz
¼
w
x
w
z;x
2

;
c
yz
¼
w
y
w
z;y
2

;
ð4Þ
where
e
and
c
denote the normal and shear strains, respectively. Here, the symbol ‘‘,” is used to indicate the partial deriv-
ative. For example,
w
x,y
is equivalent to @
w
x
/@y while
w
x,yy
means @
2
w
x
/@y
2
. Hook’s law for a plate may be expressed as
r
xx
¼
EðzÞ
1
m
2
ð
e
xx
þ
me
yy
Þ;
r
yy
¼
EðzÞ
1
m
2
ð
e
yy
þ
me
xx
Þ;
r
zz
¼ 0;
s
xy
¼ GðzÞ
c
xy
;
s
xz
¼ GðzÞ
c
xz
;
s
yz
¼ GðzÞ
c
yz
;
ð5Þ
where G(z)=E(z)/[2(1 +
m
)] is the shear modulus and
m
is the Poisson’s ratio. The stress resultant–displacement relations are
given by
M
ii
¼
Z
h=2
h=2
r
ii
zdz; i ¼ x; y
M
xy
¼
Z
h=2
h=2
s
xy
zdz;
Q
j
¼
j
2
Z
h=2
h=2
r
jz
dz; j ¼ x; y;
ð6Þ
Fig. 2. Variation of Young’s modulus through the dimensionless thickness of Al/ZrO
2
plate.
Table 1
Material properties used in the FG plate.
Properties Metal Ceramic
Aluminum (Al) Zirconia (ZrO
2
) Alumina (Al
2
O
3
)
E (GPa) 70 200 380
q
(kg/m
3
) 2702 5700 3800
Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
1279
in which
j
2
is the transverse shear correction coefficient, applied to the transverse shear forces due to the fact that the trans-
verse shear strains (
e
xz
and
e
yz
) have a nearly parabolic dependency to the thickness coordinate. Substituting Eqs. (4) and (5)
into Eq. (6) gives the resultant bending moments (M
xx
and M
yy
), twisting moment (M
xy
), and the transverse shear forces (Q
x
and Q
y
) per unit length as follows:
M
xx
¼Aðw
x;x
þ
m
w
y;y
Þ; M
yy
¼Aðw
y;y
þ
m
w
x;x
Þ;
M
xy
¼
ð1
m
Þ
2
Aðw
x;y
þ w
y;x
Þ;
Q
x
¼
j
2
hBðw
x
w
z;x
Þ; Q
y
¼
j
2
hBðw
y
w
z;y
Þ;
ð7Þ
in which
A ¼
h
3
ð1
m
2
Þ
a
ð8 þ 3
a
þ
a
2
ÞE
m
þ 3ð2 þ
a
þ
a
2
ÞE
c
12ð1 þ
a
Þð2 þ
a
Þð3 þ
a
Þ

;
B ¼
1
2ð1 þ
m
Þ
E
c
þ
a
E
m
1 þ
a
:
ð8Þ
2.4. Equations of motion
On the basis of the Mindlin plate theory, the governing differential equations of motion for the plate can be given in terms
of the stress resultants by
M
xx;x
þ M
xy;y
Q
x
¼
1
12
Ch
3
w
x
;
M
xy;x
þ M
yy;y
Q
y
¼
1
12
Ch
3
w
y
;
Q
x;x
þ Q
y;y
P ¼ Dh
w
z
;
ð9Þ
in which
C ¼
a
ð8 þ 3
a
þ
a
2
Þ
q
m
þ 3ð2 þ
a
þ
a
2
Þ
q
c
ð1 þ
a
Þð2 þ
a
Þð3 þ
a
Þ
;
D ¼
q
c
þ
aq
m
1 þ
a
;
ð10Þ
where dot-overscript convention represents the differentiation with respect to the time variable t. Since the Pasternak elastic
foundation provides force components in z direction for a deflected plate, normal transverse load per unit area can be written
as
P ¼ K
S
ðw
z;xx
þ w
z;yy
ÞK
W
w
z
; ð11Þ
in which K
S
and K
W
are the shear and Winkler foundation coefficients, respectively.
For coding and derivational convenience, the following non-dimensional parameters are introduced:
X ¼
x
a
; Y ¼
y
a
; d ¼
h
a
;
g
¼
a
b
;
K
S
¼
K
S
a
2
A
;
K
W
¼
K
W
a
4
A
; b ¼
x
a
2
ffiffiffiffiffi
Ch
A
r
; ð12Þ
in which d and
g
are named as the thickness to length ratio and aspect ratio, respectively, and b is also called the eigenfre-
quency parameter. For a harmonic solution, the rotational and transverse displacements are assumed to be
w
x
ðx; y; tÞ¼
w
x
ðX; YÞe
j
x
t
;
w
y
ðx; y; tÞ¼
w
y
ðX; YÞe
j
x
t
;
w
z
ðx; y; tÞ¼
1
a
w
z
ðX; YÞe
j
x
t
;
ð13Þ
where
x
denotes the natural frequency of vibration in radians and j ¼
ffiffiffiffiffiffi
1
p
. It should be noted that each parameter having
the over-bar is non-dimensional. Substitution of Eqs. (11)–(13) into Eq. (9) leads to
w
x;xx
þ
g
2
w
x;yy
þ
m
2
m
1
ð
w
x;xx
þ
g
w
y;xy
Þ
j
2
B
Fd
2
m
1
ð
w
x
w
z;x
Þþ
b
2
d
2
12
m
1
w
x
¼ 0;
w
y;xx
þ
g
2
w
y;yy
þ
m
2
m
1
g
ð
w
x;xy
þ
g
w
y;yy
Þ
j
2
B
Fd
2
m
1
ð
w
y
g
w
z;y
Þþ
b
2
d
2
12
m
1
w
y
¼ 0;
w
z;xx
þ
g
2
w
z;yy
ð
w
x;x
þ
g
w
y;y
Þ
Fd
2
j
2
B
K
W
w
z
þ K
S
ð
w
z;xx
þ
g
2
w
z;yy
Þ

DFb
2
d
2
BC
j
2
w
z
¼ 0;
ð14Þ
where F = A/h
3
,
m
1
=(1
m
)/2, and
m
2
=(1+
m
)/2.
1280 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
2.5. Boundary conditions
The boundary conditions along the edges X = 0 and X = 1, considered to remain simply-supported, are as follow:
M
xx
¼
w
y
¼
w
z
¼ 0: ð15Þ
The boundary conditions along the edges Y = 0 and Y = 1 are as follows:
for a free edge
M
xx
¼ M
xy
¼ 0; Q
y
¼K
S
A
a
2
g

w
z;y
; ð16Þ
for a simply-supported edge
M
yy
¼
w
x
¼
w
z
¼ 0; ð17Þ
for a clamped edge
w
x
¼
w
y
¼
w
z
¼ 0: ð18Þ
2.6. Solution of governing equations
The general solutions to Eq. (14) in terms of the three dimensionless potentials W
x
, W
y
and W
z
may be expressed as
w
x
¼ C
1
W
x;x
þ C
2
W
y;x
g
W
z;y
;
w
y
¼ C
1
g
W
x;y
þ C
2
g
W
y;y
W
z;x
;
w
z
¼ W
x
þ W
y
;
ð19Þ
where
C
1
¼
B
2
a
2
1
B
1
; C
2
¼
B
3
a
2
2
B
1
; ð20Þ
in which B
1
, B
2
, and B
3
along with
a
2
1
and
a
2
2
are the coefficients that may be determined using equations of motion and can be
given after mathematical manipulation by
B
1
¼
H
m
1
b
2
d
2
12
m
1
;
B
2
¼
m
2
m
1

1
Fd
2
k
2
B
K
S
!
a
2
1
þ
Fd
2
k
2
B
K
W
þ
DFd
2
k
2
BC
b
2
"#
þ
k
2
B
Fd
2
m
1
()
;
B
3
¼
m
2
m
1

1
Fd
2
k
2
B
K
S
!
a
2
2
þ
Fd
2
k
2
B
K
W
þ
DFd
2
k
2
BC
b
2
"#
þ
k
2
B
Fd
2
m
1
()
;
ð21Þ
and
a
2
1
;
a
2
2
¼
12K
W
12HK
S
þ b
2
12 D=C þ d
2
K
S
þ H

24
K
S
þ H

1
24
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
48 b
2
d
2
12H

K
S
þ H

K
W
þ Db
2
=C

þ
12
K
W
12HK
S
þ b
2
12 D=C þ d
2
K
S
þ H

2
K
S
þ H

2
v
u
u
u
u
u
t
;
ð22Þ
where
H
= B
j
2
/Fd
2
. The Eq. (14) can be restated in terms of the three dimensionless potentials as
W
x;xx
þ
g
2
W
x;yy
¼
a
2
1
W
x
;
W
y;xx
þ
g
2
W
y;yy
¼
a
2
2
W
y
;
W
z;xx
þ
g
2
W
z;yy
¼
a
2
3
W
z
;
ð23Þ
where
a
2
3
¼B
1
. One set of solutions to Eq. (23) are taken as
Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
1281
W
x
¼ A
1
Sinðk
1
YÞþA
2
Cosðk
1
YÞ½Sinð
l
1
XÞþ A
3
Sinðk
1
YÞþA
4
Cosðk
1
YÞ½Cosð
l
1
XÞ;
W
y
¼ A
5
Sinhðk
2
YÞþA
6
Coshðk
2
YÞ½Sinð
l
2
XÞþ A
7
Sinhðk
2
YÞþA
8
Coshðk
2
YÞ½Cosð
l
2
XÞ;
W
z
¼ A
9
Sinhðk
3
YÞþA
10
Coshðk
3
YÞ½Sinð
l
3
XÞþ A
11
Sinhðk
3
YÞþA
12
Coshðk
3
YÞ½Cosð
l
3
XÞ;
ð24Þ
in which A
i
are the arbitrary coefficients, k
j
and
l
j
are related to the
a
j
by
a
2
1
¼
l
2
1
þ
g
2
k
2
1
;
a
2
2
¼
l
2
2
g
2
k
2
2
;
a
2
3
¼
l
2
3
g
2
k
2
3
ð25Þ
On the assumption of simply-supported conditions at edges X = 0 and X = 1, Eq. (24) is given by
W
x
¼ A
1
Sinðk
1
YÞþA
2
Cosðk
1
YÞ
½
Sinð
l
XÞ;
W
y
¼ A
5
Sinhðk
2
YÞþA
6
Coshðk
2
YÞ½Sinð
l
XÞ;
W
z
¼ A
9
Sinhðk
3
YÞþA
10
Coshðk
3
YÞ½Sinð
l
XÞ:
ð26Þ
in which
l
=
l
1
=
l
2
=
l
3
= m
p
(m = 1,2,...).
Introducing Eq. (26) in Eq. (19) and substituting the results into the appropriate boundary conditions along the edges
Y = 0 and 1, leads to six homogeneous equations. To obtain non-trivial solution of these equations, the determinant of coef-
ficients matrix must be zero, which yields characteristic equations for rectangular Mindlin plates, with six combinations of
boundary conditions, namely, SSSS, SSSC, SCSC, SSSF, SFSF, and SCSF, resting on two-parameter foundation. It should be
noted that notation SCSC, for example, indicates that edges X = 0 and X = 1 are simply-supported (S), and edges Y = 0 and
Y = 1 are clamped (C).
3. Numerical results
3.1. Shear correction factor
Due to the simplicity of the analysis and programming, the model of Mindlin plates is strongly recommended by many
researchers to analyze the dynamic behavior of plates. However, a shear correction factor (
j
2
) is needed to correctly compute
transverse shear forces (see Eq. (6)). The shear correction factor is typically taken to be 5/6 for homogeneous plates. On the
other hand, this constant shear correction factor is not appropriate for FG plates (Nguyen et al. [16]), since it may be a func-
tion of material properties and the geometric dimension of an FG plate. Timoshenko [14] assumed that the shear correction
factor is dependent upon the Poisson’s ratio
m
as
j
2
¼
5 þ 5
m
6 þ 5
m
: ð27Þ
Efraim and Eisenberger [15] presented a shear correction factor for FG plates as
j
2
¼
5
6 ð
m
m
V
m
þ
m
c
V
c
Þ
; ð28Þ
in which V
m
and V
c
are the volume fraction of metal and ceramic, respectively, in the entire cross-section. However, as it is
seen from Eqs. (27) and (28), the effect of mechanical properties of the FG plate, including Young’s modulus E and density per
unit volume
q
, and the geometric dimension of the FG plate such as thickness to length ratio h/a has not been considered.
In order to achieve a comprehensive form of the shear correction factor, frequency parameters of FG SSSS square plates (a/
b = 1), made of Al/ZrO
2
and Al/Al
2
O
3
, are obtained for a significant number of gradient indices and thickness to length ratios,
using the present analytical solution and the finite element method (FEM). A well-known commercially available FEM pack-
age is used for the extraction of the frequency parameters. After ensuring the high accuracy of the FEM results, by solving
some problems of the literature, the results of the present analytical solution are compared with those obtained by the
FEM results. The authors used the shear correction factors in which the results of the present analytical solution become
identical to those acquired by the finite element solution.
In Fig. 3, the various values of the shear correction factors
j
2
for a wide range of the gradient indices
a
and thickness to
length ratios h/a are plotted for Al/Al
2
O
3
(Fig. 3a) and Al/ZrO
2
(Fig. 3b) SSSS square plates (a/b = 1). Their corresponding 2D
plots are also shown in Fig. 4 for h/a = 0.05, 0.1, 0.15, and 0.2 while gradient index
a
varies from 0 to 20. It is well known that
the shear correction factor
j
2
for fully ceramic (
a
= 0) and metallic (
a
= 1) plates is the same and equal to 5/6. From Figs. 3
and 4, it can be observed that the shear correction factors
j
2
initially decrease for smaller values of gradient index (i.e.
a
<2)
and then increase in order to approach the constant value 5/6. However, the plates composed of Al and ZrO
2
show higher
resistance to this pattern in comparison with the Al/Al
2
O
3
plates. In other words, it is evident from Figs. 3b and 4b that
for the Al/ZrO
2
plates, highly large values of the gradient index
a
are needed for shear correction factor to get close to the
constant value of 5/6 especially for thinner plates. Another interesting point about Figs. 3 and 4 is that the thicker plates have
lower sensitivity to the shear correction factors
j
2
than thinner plates. In fact, the error in calculating the frequency param-
eter is more tangible for thinner FG plates if the shear correction factor
j
2
is assumed to be constant (e.g.,
j
2
= 5/6).
1282 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
In order to obtain a formula for the shear correction factors
j
2
, in which material properties and the geometric dimension
of an FG plate are considered, too much running time has been taken to fit a function to the obtained shear correction factors
j
2
shown in Fig. 3. Finally, the following function is proposed as
j
2
ð
a
; dÞ¼
5
6
þ C
1
e
C
2
a
e
C
3
a

ð10d 2ÞC
4
e
C
5
a
e
C
6
a

10d 1ðÞ; ð29Þ
where C
i
(i = 1,2,...,6) are the constant coefficients, values of which are listed in Table 2. It should be noted that the best curve
fitting was carried out to minimize the sum of squares of errors between the data and the above function. Hereafter, all re-
sults presented in the next sections are obtained by considering the new function of the shear correction factors
j
2
(i.e., Eq.
(29)).
Fig. 3. Variation of the shear correction factors
j
2
against the gradient index
a
and thickness to length ratio h/a for (a) Al/Al
2
O
3
; (b) Al/ZrO
2
SSSS square
plates, a/b =1.
Fig. 4. Variation of the shear correction factors
j
2
versus the gradient index
a
for (a) Al/Al
2
O
3
; (b) Al/ZrO
2
SSSS square plates, a/b = 1, when h/a = 0.05, 0.1,
0.15, and 0.2.
Table 2
The values of the constant coefficients used in
j
2
formula for two FG materials.
FGMs Constant coefficients
C
1
C
2
C
3
C
4
C
5
C
6
Al/Al
2
O
3
0.750 0.025 2.000 0.640 0.060 1.000
Al/ZrO
2
0.560 0.001 5.450 0.420 0.095 1.175
Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
1283
3.2. Comparison studies
To demonstrate the efficiency and accuracy of the present solution along with new shear correction factor
j
2
, some illus-
trative examples are solved and the results are compared with the existing data available in the literature.
Example 1. Fundamental frequency parameters of the SSSS square FG plates (a/b = 1) for different values of the thickness to
length ratios (h/a = 0.05, 0.1, and 0.2) are presented in Table 3 when
a
= 0, 0.5, 1, 4, 10, and 1. The plates are made of a
mixture of aluminum (Al) and alumina (Al
2
O
3
). It should be noted that the results reported by Matsunaga [10] were based on
the both FSDT and 2D HAPT; whereas Zhao et al. [13] employed the FSDT and used different values of shear correction factors
j
2
in their study work. For convenience in comparison, a new frequency parameter is defined as b ¼
x
h
ffiffiffiffiffiffiffiffiffiffiffiffi
q
c
=E
c
p
.
From Table 3, it can be observed that the present results are in excellent agreement with those acquired by the 2D HAPT
[10]. It is worth noting that all results obtained on the basis of the FSDT [10] are inappropriate since the value of shear cor-
rection factor is taken to be constant (
j
2
= 1) for any values of thickness to length ratios and gradient indices. In addition, the
effect of truncated power series to approximate displacement, strain components, and in-plane stress [10] on these apparent
discrepancies can not be neglected. The results obtained by the FSDT [13] are also different from those acquired by the pres-
ent analytical solution and the 2D HAPT [10], particularly for the cases in which the value of shear correction factor is as-
sumed to be constant (
j
2
= 5/6). Another reason of this difference is due to the fact that Zhao et al. [13] employed a
numerical solution (element-free kp-Ritz method) to obtain the natural frequencies of the FG plates.
Example 2. Fundamental frequency parameters of the SSSS square FG Mindlin plate (a/b = 1 and h/a = 0.1) for different
values of the gradient indices
a
= 0, 0.5, 1, 2, 5, 8, and 10 are presented in Table 4 for two FG materials (i.e., Al/Al
2
O
3
and Al/
ZrO
2
). Fundamental frequency parameters are given in Table 4 in the form of
^
b ¼
x
a
2
ffiffiffiffiffiffiffiffiffiffiffiffi
q
c
=E
c
p
=h. Table 4 shows this fact that
the results of FSDT [13] for the Al/Al
2
O
3
square plate are in good agreement with the present analytical solution but for the
Al/ZrO
2
square plate they reveal a great deviation from the present analytical solution and 2D HAPT [10] due to the reasons
mentioned in Example 1.
Example 3. Fundamental frequencies of the FG square plate (a/b = 1) with simply-supported boundary conditions at four
edges for h/a = 0.1, 0.2, and 1=
ffiffiffiffiffi
10
p
are listed in Table 5 when
a
= 0, 1, 2, 3, and 5. The plate is made of a mixture of aluminum
(Al) and zirconia (ZrO
2
). All fundamental frequency parameters presented in Table 5 are defined as
~
b ¼
x
h
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q
m
=E
m
p
. Com-
paring the present results with those obtained by the 2D HAPT [10] and HSDT [9] shows that all results are in excellent
Table 3
Comparison of fundamental frequency parameter
b ¼
xh
ffiffiffiffiffiffiffiffiffiffiffiffi
q
c
=E
c
p
for SSSS Al/Al
2
O
3
square plates (a/b = 1).
d = h/a Method
j
2
Gradient index (
a
)
0 0.5 1 4 10 1
0.05 Present solution Eq. (29) 0.01480 0.01281 0.01150 0.01013 0.00963
FSDT [13] 5/6 0.01464 0.01241 0.01118 0.00970 0.00931
0.1 Present solution Eq. (29) 0.05769 0.04920 0.04454 0.03825 0.03627 0.02936
2D HAPT [10] 0.05777 0.04917 0.04427 0.03811 0.03642 0.02933
FSDT [10] 1 0.06382 0.05429 0.04889 0.04230 0.04047
FSDT [13] 5/6 0.05673 0.04818 0.04346 0.03757 0.03591
Eq. (27) 0.05713 0.04849 0.04371 0.03781 0.03619
Eq. (28) 0.05711 0.04847 0.04370 0.03779 0.03618
0.2 Present solution Eq. (29) 0.2112 0.1806 0.1650 0.1371 0.1304 0.1075
2D HAPT [10] 0.2121 0.1819 0.1640 0.1383 0.1306 0.1077
FSDT [10] 1 0.2334 0.1997 0.1802 0.1543 0.1462
FSDT [13] 5/6 0.2055 0.1757 0.1587 0.1356 0.1284
Eq. (27) 0.2098 0.1790 0.1616 0.1383 0.1313
Eq. (28) 0.2096 0.1788 0.1614 0.1382 0.1312
Table 4
Comparison of fundamental frequency parameter
^
b ¼
xa
2
ffiffiffiffiffiffiffiffiffiffiffiffi
q
c
=E
c
p
=h for SSSS square plates (a/b = 1) when h/a = 0.1.
FGMs Method Gradient index (
a
)
00.5125810
Al/Al
2
O
3
Present solution 5.7693 4.9207 4.4545 4.0063 3.7837 3.6830 3.6277
FSDT [13] 5.6763 4.8209 4.3474 3.9474 3.7218 3.6410 3.5923
Al/ZrO
2
Present solution 5.7693 5.3176 5.2532 5.3084 5.2940 5.2312 5.1893
2D HAPT [10] 5.7769 5.3216
FSDT [13] 5.6763 5.1105 4.8713 4.6977 4.5549 4.4741 4.4323
1284 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
agreement with each other. It is also seen that the present analytical solution for the FG square plates under consideration
provides the results lower than those obtained by the 2D HAPT [10] and greater than those acquired on the basis of the HSDT
[9]. In addition, the discrepancy between the FSDT [9] and three other methods (i.e., the present analytical solution, 2D HAPT
[10], and HSDT [9] ) is also considerable.
Example 4. Fundamental frequencies of the homogenous SSSS square plates (a/b = 1 and
a
= 0), resting on the elastic foun-
dation with different values of the thickness to length ratios and foundation stiffness parameters, are tabulated in Table 6.
Fundamental frequencies listed in Table 6 are defined as
b ¼
x
b
2
ffiffiffiffiffiffiffiffiffiffi
Ch=A
p
=
p
2
. The present results are compared with those
obtained by Xiang et al. [17] using exact Mindlin solution and Zhou et al. [20] using 3D elasticity theory. Table 6 proves
the fact that all results are in excellent agreement with each other.
Example 5. Table 7 shows a comparison of fundamental frequencies for the homogenous square plates (a/b =1,h/a = 0.05
and
a
= 0), resting on the Winkler elastic foundation ðK
S
¼ 0Þ with different combinations of boundary conditions, with those
obtained by Lam et al. [19] using exact thin plate theory, Xiang [18] and Akhavan et al. [22] using exact Mindlin solution.
Note that fundamental frequencies listed in this table are defined as
b ¼
x
a
ffiffiffiffiffiffiffiffiffiffi
Ch=A
p
. It can be seen from Table 7 that there
is an excellent agreement among these results confirming the high accuracy of the present analytical solution.
Table 5
Comparison of fundamental frequency parameter
~
b ¼
xh
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
q
m
=E
m
p
for SSSS Al/ZrO
2
square plates (a/b = 1).
Method
a
=0
a
=1 d = 0.2
d ¼ 1=
ffiffiffiffiffi
10
p
d = 0.1 d = 0.05 d = 0.1 d = 0.2
a
=2
a
=3
a
=5
Present solution 0.4618 0.0576 0.0158 0.0611 0.2270 0.2249 0.2254 0.2265
2D HAPT [10] 0.4658 0.0578 0.0158 0.0618 0.2285 0.2264 0.2270 0.2281
HSDT [9] 0.4658 0.0578 0.0157 0.0613 0.2257 0.2237 0.2243 0.2253
FSDT [9] 0.4619 0.0577 0.0162 0.0633 0.2323 0.2325 0.2334 0.2334
Table 6
Comparison of fundamental frequency parameter
b ¼ xb
2
ffiffiffiffiffiffiffiffiffiffi
Ch=A
p
=
p
2
for homogeneous SSSS square plates (a/b = 1).
d = h/a Method
Fundamental frequency parameter
b ¼ xb
2
ffiffiffiffiffiffiffiffiffiffi
Ch=A
p
=
p
2
ðK
W
; K
S
ÞðK
W
; K
S
ÞðK
W
; K
S
ÞðK
W
; K
S
Þ
(100, 0) (500, 0) (100, 10) (500, 10)
0.01 Present solution 2.2413 3.0215 2.6551 3.3400
Mindlin theory [17] 2.2413 3.0215 2.6551 3.3400
3D method [20] 2.2413 3.0214 2.6551 3.3398
(200, 0) (1000, 0) (200, 10) (1000, 10)
0.1 Present solution 2.3989 3.7212 2.7842 3.9805
Mindlin theory [17] 2.3989 3.7212 2.7842 3.9805
3D method [20] 2.3951 3.7008 2.7756 3.9566
(0, 10) (10, 10) (100, 10) (1000, 10)
0.2 Present solution 2.2505 2.2722 2.4590 3.8567
Mindlin theory [17] 2.2505 2.2722 2.4591 3.8567
3D method [20] 2.2334 2.2539 2.4300 3.7111
Table 7
Comparison of fundamental frequency parameter
b ¼ xa
ffiffiffiffiffiffiffiffiffiffi
Ch=A
p
for homogeneous square plates (a/b = 1) with different boundary conditions when K
S
¼ 0 and
h/a = 0.05.
K
W
Method Boundary conditions
SSSS SSSC SCSC SSSF SFSF SCSF
0 Present solution 19.737 23.643 28.944 11.680 9.630 12.681
Mindlin theory [18] 19.737 23.643 28.944 11.680 9.630 12.681
Exact CPT [19] 19.740 23.650 28.950 11.680 9.630 12.690
Mindlin theory [22] 19.739 23.646 28.951 11.684 9.631 12.686
100 Present solution 22.126 25.671 30.623 15.376 13.883 16.149
Mindlin theory [18] 22.126 25.671 30.623 15.367 13.878 16.138
Exact CPT [19] 22.130 25.670 30.630 15.380 13.880 16.150
1000 Present solution 37.276 39.483 42.869 33.710 33.056 34.070
Mindlin theory [18] 37.276 39.483 42.869 33.667 33.037 34.018
Exact CPT [19] 37.280 39.490 42.870 33.710 31.620 34.070
Mindlin theory [22] 37.278 39.486 42.873 33.712 31.623 34.073
Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
1285
4. Parametric studies
After verifying the merit and accuracy of the present analytical solution, the following new results for the vibration anal-
ysis of rectangular Mindlin FG plates, resting on elastic foundation, can be used as a benchmark for future research studies.
Natural frequencies of the plate are obtained and considered to be dimensionless as
b ¼
x
h
ffiffiffiffiffiffiffiffiffiffiffi
q
c
=E
c
p
(called the frequency
parameter) in Tables 8 and 9, and as b ¼
x
a
2
ffiffiffiffiffiffiffiffiffiffi
Ch=A
p
(called the eigenfrequency parameter) in Figs. 5–8. It should be noted
that the eigenfrequency parameter b ¼
x
a
2
ffiffiffiffiffiffiffiffiffiffi
Ch=A
p
is defined for the first time in this form (see Eq. (12)) and has its special
own characteristics, as it will be shown in Sections 4.2–4.5.
4.1. List of the frequency parameters
b for the FG plates
Fundamental frequency parameters
b of the Al/Al
2
O
3
square Mindlin plate are listed in Table 8 for various values of aspect
ratio (a/b = 0.5, 1, and 2), gradient index (
a
= 0, 0.25, 1, 5, and 1), and foundation stiffness parameters ðK
W
; K
S
Þ. Furthermore,
fundamental frequency parameters
b are given in Table 9 for the Al/ZrO
2
rectangular Mindlin plate (a/b = 1.5) with different
values of foundation stiffness parameters and gradient index as well as thickness to length ratio (h/a = 0.05, 0.1, and 0.2). The
primary conclusion, drawn from Tables 8 and 9, is to enhance the frequency parameters
b with the increase in foundation
stiffness parameters and thickness to length ratio, while all other parameters are considered to be fixed. It can also be ob-
served in both tables that the frequency parameters
b enhance as higher degree of edge constraints (in the order from free to
simply-supported to clamped) is applied to the other two edges of the plate. In Table 8, the frequency parameters
b are found
to diminish with an increase in gradient indices. However, this trend is violated in Table 9 when the plate is made of Al/ZrO
2
instead of Al/Al
2
O
3
.
4.2. Effect of foundation on the eigenfrequency parameters b
Fig. 5 shows the variation of the eigenfrequency parameter b versus the Winkler foundation stiffness parameter
K
W
for
SCSS rectangular Al/Al
2
O
3
plates with different modes m. Note that in Fig. 5 the gradient index
a
is considered to be 5. From
Fig. 5a–d, it can obviously be seen that with the increase of the
K
W
, the eigenfrequency parameter b increases.
All curves in Fig. 5a are plotted for a rectangular FG Mindlin plate (h/a = 0.15 and a/b = 0.4) when m = 1 and
K
S
¼ 0; 125; 250 and 500. It can be seen from Fig. 5a that the eigenfrequency parameter b increases as the K
S
takes the higher
Table 8
Fundamental frequency parameter
b ¼
xh
ffiffiffiffiffiffiffiffiffiffiffiffi
q
c
=E
c
p
for the Al/Al
2
O
3
square Mindlin plate when h/a = 0.15.
ðK
W
; K
S
Þ
g
= a/b
a
Boundary conditions
SSSS SSSC SCSC SSSF SFSF SCSF
(0, 0) 0.5 0 0.08006 0.08325 0.08729 0.06713 0.06364 0.06781
0.25 0.07320 0.07600 0.07950 0.06145 0.05829 0.06205
1 0.06335 0.06541 0.06790 0.05346 0.05080 0.05391
5 0.05379 0.05524 0.05695 0.04568 0.04349 0.04600
1 0.04100 0.04263 0.04443 0.03417 0.03239 0.03451
1 0 0.12480 0.14378 0.16713 0.07537 0.06290 0.08062
0.25 0.11354 0.12974 0.14927 0.06890 0.05761 0.07351
1 0.09644 0.10725 0.11955 0.05968 0.05021 0.06308
5 0.08027 0.08720 0.09479 0.05078 0.04301 0.05322
1 0.06352 0.07318 0.08507 0.03836 0.03202 0.04104
2 0 0.28513 0.35045 0.41996 0.10065 0.06217 0.13484
0.25 0.25555 0.30709 0.36112 0.09170 0.05695 0.12160
1 0.20592 0.23262 0.26091 0.07851 0.04970 0.10066
5 0.16315 0.17691 0.19258 0.06610 0.04262 0.08226
1 0.14591 0.17921 0.21375 0.05123 0.03164 0.06863
(100, 10) 0.5 0 0.12870 0.13097 0.13376 0.11917 0.11513 0.11984
0.25 0.11842 0.12040 0.12280 0.10978 0.10603 0.11037
1 0.10519 0.10659 0.10824 0.09781 0.09465 0.09824
5 0.09223 0.09318 0.09426 0.08594 0.08338 0.08622
1 0.06591 0.06708 0.06808 0.06065 0.05860 0.06100
1 0 0.17020 0.18550 0.20450 0.13016 0.11517 0.13529
0.25 0.15599 0.16892 0.18463 0.11986 0.10611 0.12432
1 0.13652 0.14483 0.15431 0.10635 0.09465 0.10945
5 0.11786 0.12296 0.12854 0.09282 0.08343 0.09485
1 0.08663 0.09442 0.10409 0.06625 0.05862 0.06886
2 0 0.32768 0.38719 0.45150 0.16797 0.11593 0.19829
0.25 0.29612 0.34264 0.39216 0.15448 0.10651 0.18035
1 0.24674 0.26994 0.29496 0.13539 0.09476 0.15251
5 0.20359 0.21501 0.22830 0.11603 0.08338 0.12705
1 0.16773 0.19805 0.22981 0.08550 0.05901 0.10093
1286 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
Table 9
Fundamental frequency parameter
b ¼
xh
ffiffiffiffiffiffiffiffiffiffiffiffi
q
c
=E
c
p
for the Al/ZrO
2
rectangular Mindlin plate (a/b = 1.5).
ðK
W
; K
S
Þ
d = h/a
a
Boundary conditions
SSSS SSSC SCSC SSSF SFSF SCSF
(0, 0) 0.05 0 0.02392 0.03129 0.04076 0.01024 0.00719 0.01249
0.25 0.02269 0.02899 0.03664 0.00981 0.00692 0.01185
1 0.02156 0.02667 0.03250 0.00948 0.00674 0.01132
5 0.02180 0.02677 0.03239 0.00963 0.00685 0.01146
1 0.02046 0.02689 0.03502 0.00880 0.00618 0.01073
0.1 0 0.09188 0.11639 0.14580 0.04001 0.02835 0.04817
0.25 0.08603 0.10561 0.12781 0.03810 0.02717 0.04532
1 0.08155 0.09734 0.11453 0.03679 0.02641 0.04327
5 0.08171 0.09646 0.11234 0.03718 0.02677 0.04352
1 0.07895 0.10001 0.12528 0.03438 0.02426 0.04139
0.2 0 0.32284 0.37876 0.43939 0.14871 0.10795 0.17323
0.25 0.31003 0.36117 0.41624 0.14354 0.10436 0.16671
1 0.29399 0.33549 0.37962 0.13851 0.10127 0.15937
5 0.29099 0.32783 0.36695 0.13888 0.10200 0.15878
1 0.27788 0.32545 0.37755 0.12779 0.09276 0.14885
(250, 25) 0.05 0 0.03421 0.04021 0.04815 0.02291 0.01877 0.02508
0.25 0.03285 0.03786 0.04412 0.02218 0.01821 0.02411
1 0.03184 0.03577 0.04035 0.02172 0.01789 0.02334
5 0.03235 0.03615 0.04053 0.02203 0.01821 0.02360
1 0.02937 0.03455 0.04138 0.01968 0.01613 0.02155
0.1 0 0.13365 0.15131 0.17690 0.09079 0.07472 0.09838
0.25 0.12771 0.14271 0.16008 0.08773 0.07240 0.09409
1 0.12381 0.13550 0.14848 0.08586 0.07114 0.09118
5 0.12533 0.13611 0.14792 0.08699 0.07234 0.09196
1 0.11484 0.13150 0.15200 0.07801 0.06420 0.08453
0.2 0 0.49945 0.54079 0.58657 0.35225 0.29172 0.37193
0.25 0.48327 0.52073 0.56189 0.34182 0.28124 0.36007
1 0.46997 0.49952 0.53159 0.33473 0.27785 0.35005
5 0.47400 0.49979 0.52779 0.33853 0.28469 0.35237
1 0.43001 0.46469 0.50402 0.30268 0.25067 0.31959
Fig. 5. Variation of the eigenfrequency parameter b versus K
W
for SCSS rectangular Al/Al
2
O
3
plates with different values of (a) K
S
, (b)
g
, (c) d, and (d) m when
a
=5.
Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
1287
values. Another interesting conclusion can be inferred from Fig. 5a is that when extremely high values of the K
S
are taken, the
eigenfrequency parameter b becomes constant for any values of the
K
W
.
The eigenfrequency parameters b of FG rectangular Mindlin plates (h/a = 0.15 and m = 1) resting on the Winkler elastic
foundation ð
K
S
¼ 0Þ for different values of the aspect ratios a/b = 0.4, 1, 2, and 3 are plotted in Fig. 5b for a wide range of
the
K
W
. It is observed, in Fig. 5b, that the eigenfrequency parameter b enhances with increasing the aspect ratio a/b.
In Fig. 5c, the eigenfrequency parameter b is given for rectangular FG plates (a/b = 2.5) resting on the Pasternak elastic
foundation when
K
S
¼ 10, m = 1 and h/a = 0.001, 0.1, 0.15, and 0.2. As it is expected, for a certain value of K
W
, the eigenfre-
quency parameter b rises as the plate thickness diminishes.
The same plate parameters as those used in Fig. 5b are considered for Fig. 5d except that a/b = 1 and the modes m are
taken to be 1, 2, 3, and 4. It is worth noting that the effect of the
K
W
on the growth rate of the eigenfrequency parameter
b is more tangible for lower modes m.
4.3. Effect of gradient index
a
on the eigenfrequency parameters b
The variation of the eigenfrequency parameter b versus the gradient index
a
, for SSSS rectangular Al/Al
2
O
3
plates, is
shown in Fig. 6. Note that the eigenfrequency parameter b is normalized in Fig. 6 by dividing its maximum value, and de-
noted by b
n
.
Fig. 6a and b depict the relation between the normalized eigenfrequency parameter b
n
and the gradient index
a
for a
square FG Mindlin plate (h/a = 0.2 and a/b = 1) with m = 1 when in Fig. 6a
K
S
¼ 0 and K
W
¼ 0; 50; 250, and 500 and in
Fig. 6b
K
W
¼ 10 and K
S
¼ 0; 5; 10 and 25. The results in Fig. 6a and b indicate that the normalized eigenfrequency parameter
b
n
increases with the increasing values of the foundation stiffness parameters. Another interesting point attracting one’s
attention is that, regardless of the values of the foundation stiffness parameters, the normalized eigenfrequency parameter
b
n
is minimized for a specific value of the gradient index
a
, herein called the critical gradient index and denoted by
a
cr
. Due to
the importance of the value of
a
cr
, Section 4.5 is devoted to determining the critical values of the gradient index
a
.
Fig. 6. Variation of the normalized eigenfrequency parameter b
n
versus
a
for SSSS rectangular Al/Al
2
O
3
plates with different values of (a) K
W
, (b) K
S
, (c)
g
,
and (d) d when m =1.
1288 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
The effect of the aspect ratio a/b on the normalized eigenfrequency parameter b
n
is investigated for rectangular FG Mind-
lin plates (h/a = 0.2) with various values of the gradient index
a
when a/b = 0.5, 1, 1.5, and 3; ðK
W
; K
S
Þ¼ð0; 0Þ, and m =1,as
shown in Fig. 6c. It is found that the higher values of the aspect ratio a/b will reduce the normalized eigenfrequency param-
eter b
n
. For each value of the aspect ratio a/b, there is also a minimum value for the normalized eigenfrequency parameter b
n
,
as one can see in Fig. 6c.
Fig. 7. Variation of the normalized eigenfrequency parameter b
n
versus
a
for a square FG Mindlin plate (a/b = 1 and h/a = 0.18) with (a) SFSF, (b) SSSS, and
(c) SCSC boundary conditions when
K
W
¼ K
S
¼ 5 and m =1.
Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
1289
The graph of the normalized eigenfrequency parameter b
n
against the gradient index
a
is plotted in Fig. 6d for square FG
Mindlin plates (a/b = 1) for different values of the thickness to length ratio (h/a = 0.08, 0.1, and 0.2) when ð
K
W
; K
S
Þ¼ð0; 0Þ
and m = 1. It is seen that as the thickness to length ratio h/a decreases, the normalized eigenfrequency parameter b
n
mainly
increases for
a
>
a
cr
, while an inverse behavior is experienced for
a
<
a
cr
.
4.4. Effect of different FGMs on the eigenfrequency parameters b
The influence of the two different FGMs (i.e., Al/Al
2
O
3
and Al/ZrO
2
) on the normalized eigenfrequency parameter b
n
is
shown in Fig. 7 for a square FG Mindlin plate (h/a = 0.18 and a/b = 1), resting on Pasternak elastic foundation ð
K
W
¼
K
S
¼ 5Þ when m = 1. The boundary conditions at the edges of the plate are considered to be SFSF, SSSS, and SCSC in
Fig. 7a–c, respectively. Herein, it should be noted that the normalized eigenfrequency parameter b
n
is obtained by dividing
its corresponding b into the eigenfrequency parameter in which
a
=10
3
.
Fig. 7 proves the fact that as the gradient index
a
varies from 10
3
to 10
3
, the FG Al/Al
2
O
3
material has a higher effect on
the normalized eigenfrequency parameter b
n
when compared with the FG Al/ZrO
2
material. It can also be figured out that for
the most ranges of the gradient index
a
, the normalized eigenfrequency parameters b
n
of the Al/ZrO
2
square plate are higher
than those of the Al/Al
2
O
3
one. It is worthy of mention that as lower degrees of edge constraints (in the order from clamped
to simply-supported to free) are applied to the other two edges of the square Al/ZrO
2
plate, a peak at the point around
a
=10
is more evident for the eigenfrequency parameter b
n
.
4.5. Determination of critical gradient index
a
cr
Fig. 8 represents the behavior of the critical gradient index
a
cr
versus the plate parameters, including foundation stiffness
parameters, aspect ratio, and different boundary conditions, for FG rectangular Mindlin plates when m = 1. It is emphasized
that the critical gradient index
a
cr
is defined by finding the gradient index
a
in which the eigenfrequency parameter b is
minimized.
From Fig. 8a and b, it can be seen that the values of the critical gradient index
a
cr
for the Al/Al
2
O
3
SSSS square plate (h/
a = 0.2 and a/b = 1) are greater than those for the Al/ZrO
2
one when any values of the foundation stiffness parameters are
taken into account. In other words, the minimum value of the eigenfrequency parameter b for the Al/ZrO
2
square plate will
occur at the smaller values of the gradient index
a
. It is worthwhile to mention that in contrast with the FG Al/ZrO
2
material,
the minimum value of the eigenfrequency parameter b for the FG Al/Al
2
O
3
material considerably shifts to the lower values of
Fig. 8. Variation of the critical gradient index
a
cr
versus (a) K
W
, (b) K
S
, (c)
g
, and (d) K
W
for rectangular FG Mindlin plates with different boundary conditions
when m =1.
1290 Sh. Hosseini-Hashemi et al. / Applied Mathematical Modelling 34 (2010) 1276–1291
the gradient index
a
as the foundation stiffness parameters increase. It should be noted that K
S
and K
W
are taken as 10 in
Fig. 8a and b, respectively.
Fig. 8c shows the behavior of the critical gradient index
a
cr
against the aspect ratios a/b for the SSSS FG rectangular plate
(h/a = 0.2) resting on the Pasternak elastic foundations ð
K
W
¼ K
S
¼ 10Þ, when the aspect ratios a/b varies from 0.4 to 3.0. It is
evident from Fig. 8c that as the aspect ratio a/b increases, the reaction of the critical gradient index
a
cr
to the FG Al/ZrO
2
material is much more than that to the FG Al/Al
2
O
3
material. In fact, the minimum value of the eigenfrequency parameter
b for the Al/Al
2
O
3
square plates can hardly move on the b
a
diagram with increasing the aspect ratio a/b.
In Fig. 8d, the influence of the different boundary conditions on the critical gradient index
a
cr
is illustrated for an FG
square plate (h/a = 0.15 and a/b = 1) when
K
S
¼ 20 and K
W
varies from 0 to 1500. It is found that as higher degrees of edge
constraints are applied to the other two edges of the FG plate, the critical gradient index
a
cr
takes the higher values. It can
also be concluded from Fig. 8d that the variation of the critical gradient index
a
cr
against the K
W
is almost negligible for the
Al/ZrO
2
plates especially with SFSF and SSSS boundary conditions. A rapid decrease in the values of the critical gradient index
a
cr
is observed for the SSSS Al/Al
2
O
3
square plate when K
W
< 900 and for the SFSF one when K
W
< 150.
5. Conclusions
In the present study, an analytical solution for free vibration analysis of moderately thick FG rectangular plates, resting on
either Winkler or Pasternak elastic foundations, was presented for all six possible combinations of boundary conditions. A
new form of the shear correction factor
j
2
for FG rectangular plates was derived in which the effect of material properties
and the geometry of the plate on the shear correction factor
j
2
were considered. It was demonstrated that the shear correc-
tion factor
j
2
deviates from 5/6 as the thickness to length ratio h/a decreases. All comparison studies demonstrated that the
present solution is highly efficient for exact analysis of the vibration of FG rectangular plates on the basis of the Mindlin plate
theory. The influence of the foundation stiffness parameters on the natural frequencies of the FG plates with different com-
binations of boundary conditions was investigated for various values of aspect ratios, gradient indices, and thickness to
length ratios. Due to the inherent features of the current analytical solution, the present findings will be a useful benchmark
for evaluating other analytical and numerical methods which will be developed by researchers in the future.
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    • "Their procedure was based on the Reissner–Mindlin plate theory. Then, Hosseini-Hashemi et al. [13] worked on free vibration analysis of moderately thick rectangular plates, which were composed of functionally graded materials and supported by either Winkler or Pasternik elastic foundations. The finite element based simulation of the dynamic response is regularly employed by the FGM manufacturers in order to improve their products' comfort and reliability. "
    Full-text · Article · May 2016
    • "This new type of materials can be applied to avoid in-terfacial stress concentration appeared in laminated struc-tures, and therefore, a great promise is proposed by FGMs in applications where the working conditions are severe[2], including spacecraft heat shields, heat ex-changer tubes, plasma facings for fusion reactors, engine components and highpower electrical contacts or even magnets. Compared with the analysis of functionally graded rectangular plates34567891011 and functionally graded spheres [12] as well as functionally graded cylindrical shells1314151617, the investigations of functionally graded annular plates are limited in number. Moreover , the proposed semi-analytical 2D-GDQ has led to obtain more accurate results with appropriate rate of convergence. "
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    Full-text · Article · Mar 2016 · Composite Structures
    • "[58,59] . There are excellent reviews on this subject , including different approaches, such as60616263646566. For slender beams, it is well-known that rather than shear deformations or axial deformations, the foremost reason of displacements and rotation is bending. "
    [Show abstract] [Hide abstract] ABSTRACT: In this paper, arbitrarily large in-plane deflections of planar curved beams made of Functionally Graded Materials (FGM) are examined. Geometrically exact beam theory is revisited, but the material properties are considered as an arbitrary function of the position on the cross-section of the beam, to derive the governing differential equation system. Axial, and shear deformations are taken into account. Equations are solved by the method called Variational Iterational Method (VIM). Solution steps are given explicitly. Presented formulation is validated by solving some examples existing in the literature. It is seen that the solution method is easy, and efficient. Deflection values, and deflected shapes of half, and quarter circular cantilever beams made of FGM are given for different variations of the material. Snap-through, and bifurcation buckling of pinned–pinned circular arches made of FGM are examined. Effects of material variation on the deflections, and bifurcation buckling load are examined. New results are also given for arbitrarily large in-plane deflections of planar curved beams made of FGM.
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