Free vibration analysis of solar functionally graded plates with temperature-dependent material properties using second order shear deformation theory

Abstract

Second-order shear deformation theory (SSDT) is employed to analyze vibration of temperature-dependent solar functionally
graded plates (SFGP’s). Power law material properties and linear steady-state thermal loads are assumed to be graded along
the thickness. Two different types of SFGP’s such as ZrO2/Ti-6Al-4V and Si3N4/SUS304 are considered. Uniform, linear, nonlinear, heat-flux and sinusoidal thermal conditions are imposed at the upper and
lower surface for simply supported SFGPs. The energy method is applied to derive equilibrium equations, and solution is based
on Fourier series that satisfy the boundary conditions (Navier’s method). Non-dimensional results are compared for temperature-dependent
and temperature-independent SFGP’s and validated with known results in the literature. Numerical results indicate the effect
of material composition, plate geometry, and temperature fields on the vibration characteristics and mode shapes. The results
obtained using the SSDT are very close to results from other shear deformation theories.

KeywordsFGM–Solar functionally graded plate–Second-order shear deformation–Temperature-dependent properties–Vibration analysis

Full text

Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209
www.springerlink.com/content/1738-494x
DOI 10.1007/s12206-011-0610-x
Free vibration analysis of solar functionally graded plates with temperature-
dependent material properties using second order shear deformation theory†
A. Shahrjerdi1,2,*, F. Mustapha2, M. Bayat3 and D. L. A. Majid2
1Department of Mechanical Engineering, MalayerUniversity, Malayer, 65719, Iran
2Department of Aerospace Engineering, Universiti Putra Malaysia, 43400, Selangor, Malaysia
3Department of Civil Engineering, Aalborg University, 9000 Aalborg, DK
(Manuscript Received February 26, 2011; Revised May 22, 2011; Accepted June 6, 2011)
----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Abstract
Second-order shear deformation theory (SSDT) is employed to analyze vibration of temperature-dependent solar functionally graded
plates (SFGP’s). Power law material properties and linear steady-state thermal loads are assumed to be graded along the thickness. Two
different types of SFGP’s such as ZrO2/Ti-6Al-4V and Si3N4/SUS304 are considered. Uniform, linear, nonlinear, heat-flux and sinusoi-
dal thermal conditions are imposed at the upper and lower surface for simply supported SFGPs. The energy method is applied to derive
equilibrium equations, and solution is based on Fourier series that satisfy the boundary conditions (Navier's method). Non-dimensional
results are compared for temperature-dependent and temperature-independent SFGP’s and validated with known results in the literature.
Numerical results indicate the effect of material composition, plate geometry, and temperature fields on the vibration characteristics and
mode shapes. The results obtained using the SSDT are very close to results from other shear deformation theories.
Keywords: FGM; Solar functionally graded plate; Second-order shear deformation; Temperature-dependent properties; Vibration analysis
-
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
1. Introduction
In recent years, solar energy has become an increasingly
important area in applied engineering. One of the most signifi-
cant parts of this field is the solar plate. A solar plate is used to
focus solar radiation onto an absorber located at the focal
point in a parabolic dish concentrator to provide solar energy.
A concentrating solar collector consists of a reflector over the
solar plate, an absorber and housing. A parabolic disc is made
from pieces of solar plates. The performance of a solar plate in
terms efficiency, service life and optical alignment depends on
the material and operating conditions. Normally, a solar plate
can be fabricated from polished pure material or a coated plate
with a special coating [1]. However, some specific applica-
tions (such as solar satellites, solar power towers and solar
power heat engines) demand low weight and high temperature
resistance to avoid undesirable deformation [2]. Functionally
graded materials (FGMs) are designed to be thermal barrier
materials for aerospace structural applications and fusion reac-
tors [3]. Flexible properties and high thermal resistance pro-
vides suitable stiffness to avoid unsought deformation to better
optical alignment. It has been reported that FG plates have
much lower weight and better heat resistance than pure mate-
rial plate products of similar size. FGMs usually are composed
of a ceramic and a metal, where the volume fraction of the two
materials is variable [4]. Based on the literature, a consider-
able amount of work has been reported on the vibration char-
acteristics of isotropic plates and composite laminates [5, 6].
To determine the small and large deflections, static, linear and
non-linear dynamic behavior of FG plates, shear deformation
theories of the first, second and third order can be effective
tools [7].
Finite element methods (FEM) and first-order shear defor-
mation theory (FSDT) were employed by Praveen and Reddy
[8] for nonlinear transient thermo-elastic responses of FG
plates. Mokhtar et al. [9] investigated thermal buckling analy-
ses of S-FG plates by using FSDT. The thermal buckling was
carried out under uniform, linear and sinusoidal temperature
rise across the thickness. Shukla and Huang [10] presented
nonlinear static and dynamic responses of the temperature-
dependent FG rectangular plate using FSDT. Ibrahim et al.
[11] used nonlinear FEM model to assess the nonlinear ran-
dom response of FG panel subject to combined thermal and
acoustic loads by applying FSDT and Von-Karman geometric
nonlinearity. Park and Kim [12] presented thermal post buck-
ling and vibration behavior of the FG plates based on FSDT.
† This paper was recommended for publication in re
vised form by Associate Editor
Ohseop Song
*Corresponding author. Tel.: +6089466400, Fax.: +60386567125
E-mail address: alishahrjerdi2000@yahoo.com
© KSME & Springer 2011

2196 A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209
Initial displacement and initial stress for nonlinear tempera-
ture-dependent material properties were adopted in their re-
search. Zhao et al. [13] employed a element-free kp-Ritz
method to analyze the free vibration of temperature-dependent
FG plates based on FSDT. Ferreira and Batra [14] provided a
global collocation method for natural frequencies of FG plates
by a meshless method and applying FSDT. Bayat et al. [15]
examined small and large deflection of FG rotating disc by
applying FSDT and Von Karman theory. Later, Bayat et al.
[16] obtained small deflection of an FG rotating disc with
constant and variable thickness profile under thermo-
mechanical load using FSDT. Batra and Jin [17] used the
FSDT with the FEM to study free vibrations of an FG anisot-
ropic rectangular plate with various boundary conditions.
Several studies were conducted on three-dimensional exact
solutions, and other shear deformation theories (such as a
third-order-shear deformation theory (TSDT)) for FG plates
have been considered [18]. To determine the effects of non-
linearity, Huang et al. [19] compared vibration and dynamic
response of temperature dependent FG plates in thermal envi-
ronment based on the higher-order shear deformation theory.
Matsunaga [20] conducted the thermal buckling of tempera-
ture-independent FG plates according to a 2D higher-order
shear deformation theory. Kim [21] demonstrated vibration
characteristics of pre-stressed temperature-dependent FG rec-
tangular plates in thermal environment. In his major study,
Kim [21], identified two thermal conditions, linear and nonli-
near, with the Rayleigh-Ritz method based on TSDT. Sunda-
rarajan et al. [22] employed nonlinear formulation based on
von Karman’s assumptions to study the free vibration charac-
teristics of temperature dependent FG plates subjected to uni-
form thermal environments. Chen et al. [23] derived nonlinear
partial differential equations for the vibration motion of an
initially stressed temperature-independent FGP. Yang and
Shen [24] found the dynamic response of initially stressed
temperature-dependent FG rectangular thin plates based on
Reddy’s higher-order shear deformation plate theory and in-
cluded the thermal effects due to uniform temperature varia-
tion. They applied the same method [25] to study the dynamic
stability and free vibration of FG cylindrical panels subjected
to combined loads and uniform thermal environment.
Some studies apply second-order shear deformation theory
(SSDT). Khdeir and Reddy [26] studied the free vibration of
laminated composite plates using SSDT. A general formula-
tion for FG circular and annular plates presented by Saidi and
Sahraee [27] using SSDT and developed the bending solution
that accounts for deflections and various boundary conditions.
Shahrjerdi et al. [28] presented free vibration analysis of a
quadrangle FG plate by using Navier’s method based on
SSDT, and then used SSDT for stress analysis of FG solar
plates subjected to in-plane and out-plane mechanical loads
[29]. Bahtuei and Eslami [30] also conducted the coupled
thermo-elastic response of an FG circular cylindrical based on
SSDT. Thermal loads were not considered in these studies [26,
28-30].
Many of the above-mentioned papers deal with tempera-
ture-independent materials with shear deformation theories.
Temperature-dependent materials in a constant temperature
field and temperature variations with surface-to-surface heat
flow through the thickness direction were considered in other
research by applying first, third and higher order shear defor-
mation theories. To the authors’ knowledge, few works have
been done in the area of dynamic stability of FGM plate by
using SSDT. In this paper, the analytical solution is provided
for the vibration characteristics of SFGPs under temperature
field and applying SSDT. The temperature is assumed to be
constant in the plane of the plate. The variation of temperature
is assumed to occur in the thickness direction only. The
SFGPs are assumed to be simply supported with temperature-
dependent and independent material properties with a power-
law distribution in terms of the volume fractions of the con-
stituents and subjected to uniform, linear, nonlinear, heat flux
and sinusoidal temperature rise. The frequency equation is
obtained using Navier's method based on SSDT. This work
aims to show the effect of material compositions, plate ge-
ometry and temperature fields on the vibration characteristics.
Furthermore, the effects of temperature dependency of SFGPs
for some types of thermal condition are investigated.
2. Gradation relations
There are some models in the literature that express the var-
iation of material properties in FGMs [9, 16]. The most com-
monly used is the power law distribution of the volume frac-
tion. Here, an SFGP rectangular in Cartesian coordinate sys-
tem (
1 2 3
, ,
x x x
) with constant thickness
,
h
width
,
a
and
length
b
is considered as shown in Fig. 1. The materials at
the top surface ( 3
/ 2
x h
=
) and bottom surface ( 3
/ 2
x h
= −
)
are, respectively, pure ceramic and metal. Between these two
pure materials, a power-law distribution of material is applied.
According to this model, the material properties of SFGPs are
assumed to be position and temperature- dependent and can be
expressed as the following [21]:
( ) ( )
3 3
, ( ( ) ( )) / 1/ 2 ( )
p
c m m
x T T T x h T
Γ = Γ = Γ − Γ + + Γ (1)
Fig. 1. Solar functionally graded plate.

A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209 2197
where
Γ
denotes a generic material property such as elastic
modulus
,
E
the Poisson’s ratio
,
ν
mass density
ρ
and
thermal expansion coefficient
α
of SFGPs; furthermore
subscripts
m
and
c
refer to the pure metal and ceramic
plates, respectively. The expression
( )
3
/ 1/ 2
p
x h + denotes
the ceramic volume fraction, where
0
p
≥
is a namely grad-
ing index that is the volume fraction exponent. The non-linear
SFGP’s material can be expressed as the following [24]:
(
)
1 2 3
0 1 1 2 3
( 1 )
T T T T T
ρ ρ ρ ρ ρ ρ
−
−
= + + + + (2)
where
ρ
denotes material property and
0 3
( )
T T T x
= + ∆
indicates the environmental temperature; 0
300( )
T K
=
is
room temperature;
1 0 1 2
, , ,
ρ ρ ρ ρ
−and
3
ρ
are the coeffi-
cients of temperature-dependent material properties unique to
the constituent materials, and
3
( )
T x
∆
is the temperature rise
only through the thickness direction, whereas thermal conduc-
tivity
k
is temperature-independent. Temperature-dependent
typical values for some SFGP material components such as
silicon nitride and stainless steel are in Table 1 [21].
3. Elastic equations
3.1 Displacement field and strains
The displacement field based on the second-order shear de-
formation theory (SSDT) can be represented as [26, 31]:
2
1 3 1 3 2
u u x x
φ φ
= + + , (3a)
2
2 3 1 3 2
u v x x
ψ ψ
= + + , (3b)
3
u w
=
(3c)
where
1 2
,
u u
and
3
u
denote the displacement components in
the
1 2
,
x x
and
3
x
directions, respectively;
,
u v
and
w
de-
fine the displacements of a point on the mid plane ( 1 2
, ,0
x x
);
1 2 1
, ,
φ φ ψ
and
2
ψ
are the rotations of a transverse normal
about
2 1
, .
x x
The displacement components
1
, , , ,
u v w
φ
2 1 2
, ,
φ ψ ψ
and
2
ψ
are functions of in-plane coordinate vari-
ables
1 2
( , )
x x
and time
t
. The strain-displacement relations
are given by [26, 32]:
0 '
11 11
11 11
0 2 '
22 22 3 22 3 22
0 1 2
12
12 12 12
x x
ε κ
ε κ
ε ε κ κ
γγ γ γ
   
 
 
   
 
     
= + +
       
       
   
   
   
,
(4a)
0 1
23 23 23
3
0 1
13
13 13
x
γ γ γ
γ
γ γ
   
 
   
= +
     
 
   
   
(4b)
where
0 '
1 2
11 11 11
1 1 1
0 '
1 2
22 22 22
2 2 2
0 1 2
1 1 2 2
12 12 12
2 1 2 1 2 1
0 0 1 1
23 1 13 1 23 2 13 2
2 1
, ,
, ,
( ), ( ), ( )
( ), ( ), 2 , 2 .
u
x x x
v
x x x
u v
x x x x x x
w w
x x
φ φ
ε κ κ
ψ ψ
ε κ κ
φ ψ φ ψ
γ γ γ
γ ψ γ φ γ ψ γ φ
∂ ∂ ∂
= = =
∂ ∂ ∂
∂ ∂ ∂
= = =
∂ ∂ ∂
∂ ∂ ∂ ∂ ∂ ∂
= + = + = +
∂ ∂ ∂ ∂ ∂ ∂
∂ ∂
= + = + = =
∂ ∂
(5)
3.2 Stress-strain relations
The stress-strain relation of plates is [26, 29]:
11 12
11 11
12 22
22 22
44
23 23
55
13 13
66
12 12
0 0 0
0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
q q
q q
q
q
q
σ ε
σ ε
σ γ
σ γ
σ γ
 
   
 
   
 
   
    
=
    
    
    
    
   
 
(6)
where
12 12
2
γ ε
=,
13 13
2
γ ε
=,
23 23
2
γ ε
= and each
ij
q
is a
function of position and temperature as follows:
3
11 22 2
3
12 3 11
3
44 55 66
3
( , )
1 ( , )
( , )
( , )
.
2(1 ( , ))
E x T
q q x T
q x T q
E x T
q q q
x T
ν
ν
ν
= = −
=
= = = +
(7)
3.3 Equations of motion
The total strain energy of SFGP is given by
,
P T
U U U
= +
where UP and UT are the strain energies due to mechanical and
thermal effects, respectively. Considering the heating process,
it can be physically observed that the plate is heated to the
steady state condition after which vibration occurs in the
thermally-static plate. If the temperature change
T
∆
is ap-
plied, the plate will undergo an initial deflection, and the cor-
responding initial thermal stresses occur because the plate is
Table 1. Temperature-dependent coefficients for ZrO2/Ti-6Al-
4V and
Si3N4/SUS304.

2198 A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209
constrained of the boundary supports. The strain energies UP
and UT are given by [32-34]:
11 11 22 22 12 12 23 23
13 13
1[ 2 2
2
2] 0
PV
U
dV
σ ε σ ε σ ε σ ε
σ ε
= + + +
+ =
∫ (8)
11 11 12 12 22 22
1
[ 2 ]
2
T T T
T
V
U d d d dV
σ σ σ
= + +
∫ (9a)
where
,( . 1,2)
ij
d i j
=
is the nonlinear strain-displacement rela-
tionship [34]. By substituting
ij
d
into Eq. (9a) the following
equation is obtained:
2 2 2
3
1 2
11
1 1 1
3 3
1 1 2 2
12
1 2 1 2 1 2
2 2 2
31 2
22
2 2 2
1{ [( ) ( ) ( ) ]
2
2 [( )( ) ( )( ) ( )( )]
[( ) ( ) ( ) ]} 0 .
T
TV
T
T
uu u
Uxx x
u uu u u u
x x x x x x
u
u u dV
x x x
σ
σ
σ
∂∂ ∂
= + + +
∂ ∂ ∂
∂ ∂∂ ∂ ∂ ∂
+ + +
∂ ∂ ∂ ∂ ∂ ∂
∂
∂ ∂
+ + =
∂ ∂ ∂
∫
(9b)
The matrix representation of the thermal conductivity tensor
is diagonal because the material is assumed to be isotropic. By
considering this view and the plane stress assumption, the
temperature terms
11
T
σ
and
22
T
σ
are the only remaining
terms [8, 33].
The kinetic energy of plate is given by:
2 2 2
3
1
( , )[ ] 0
2V
K x T u v w dV
ρ
= + + =
∫ɺ ɺ ɺ . (10)
Hamilton's principle for an elastic body can be represented as:
( )
2
1
( ) 0
t
t
K U V dt
δ δ
− + =
∫. (11)
The inertias are also defined as:
( )
( )
2
0 3 3
2
0,1, 2,...,6
h
i
i
h
I x dx i
ρ
−
= =
∫. (12)
By substituting Eq. (4) into Eq. (6) and applying Eqs. (11)
and (3) Navier’s equations for SFGP can be obtained as fol-
lows:
( )
2 2 2
11 66 12 66
2 2
1 2
1 2
2 2 2 2
1 1 2 2
11 66 11 66
2 2 2 2
1 2 1 2
u u v
A A A A x x
x x
B B D D
x x x x
φ φ φ φ
∂ ∂ ∂
+ + + +
∂ ∂
∂ ∂
∂ ∂ ∂ ∂
+ + + +
∂ ∂ ∂ ∂
( ) ( )
2 2
1 2
12 66 12 66
1 2 1 2
2 2 2 2
1 2
11 11 11 22
2 2 2 2
1 1 1 2
2 2
1 2
22 22 0 2 2 1 1
2 2
2 2
T T T T
T T
B B D D
x x x x
u u
A B D A
x x x x
B D I u I I
x x
ψ ψ
φ φ
φ φ φ φ
∂ ∂
+ + + +
∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
+ + + +
∂ ∂ ∂ ∂
∂ ∂
+ = + +
∂ ∂
ɺɺ ɺɺ
ɺɺ
(13a)
2 2 2 2
12 66 22 66
2 2
1 2 1 2 2 1
2 2
1 2
12 66 12 66
1 2 1 2
2 2 2 2
1 1 2 2
66 22 66 22
2 2 2 2
1 2 1 2
2 2 2 2
1 2
11 11 11 22
2 2 2 2
1 1 1 2
2
1
22 2
2
( ) ( )
T T T T
T
u u v v
A A A A
x x x x x x
B B D D
x x x x
B B D D
x x x x
v v
A B D A
x x x x
B D
x
φ φ
ψ ψ ψ ψ
ψ ψ
ψ
∂ ∂ ∂ ∂
+ + + +
∂ ∂ ∂ ∂ ∂ ∂
∂ ∂
+ + + +
∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
+ + + +
∂ ∂ ∂ ∂
∂ ∂ ∂ ∂
+ + + +
∂ ∂ ∂ ∂
∂+
∂
2
2
22 0 2 2 1 1
2
2
TI v I I
x
ψψ ψ
∂= + +
∂ɺɺ ɺɺ
ɺɺ
(13b)
2
1 2 1
55 55 55 44
2
1 1 2
1
2 2 2
2
44 44 11 22 0
2 2 2
2
2 1 2
2
2T T
w
A A B A
x x x
x
w w w
A B A A I w
x
x x x
φ φ ψ
ψ
∂ ∂ ∂ ∂
+ + + +
∂ ∂ ∂
∂
∂ ∂ ∂ ∂
+ − − =
∂
∂ ∂ ∂
ɺɺ
(13c)
( )
( ) ( )
2 2 2
11 66 12 66
2 2
1 2
1 2
2 2 2 2
1 1 2 2
11 66 11 66
2 2 2 2
1 2 1 2
2 2
1 2
12 66 12 66
1 2 1 2
2
55 55 1 55 2 11 2
11
2 2 2
1 2
11 11 22
2 2 2
1 1 2
2T
T T T
u u v
B B B B x x
x x
D D E E
x x x x
D D E E
x x x x
w u
A A B B
xx
u
D E B
x x x
φ φ φ φ
ψ ψ
φ φ
φ φ
∂ ∂ ∂
+ + + +
∂ ∂
∂ ∂
∂ ∂ ∂ ∂
+ + + +
∂ ∂ ∂ ∂
∂ ∂
+ + + −
∂ ∂ ∂ ∂
 
∂ ∂
− − − −
 
∂∂
 
∂ ∂ ∂
− −
∂ ∂ ∂
2
1
22 2
2
2
2
22 2 1 1 3 2
2
2
T
T
Dx
E I I u I
x
φ
φφ φ
∂
− −
∂
∂= + +
∂
ɺɺ ɺɺ
ɺɺ
(13d)
( ) ( )
2 2 2
11 66 12 66
2 2
1 2
1 2
2 2 2 2
1 1 2 2
11 66 11 66
2 2 2 2
1 2 1 2
2 2
1 2
12 66 12 66
1 2 1 2
2
55 55 2 55 1 11 2
11
2 2
1 2
11 11
2 2
1 1
( )
2 2 T
T T
u u v
D D D D x x
x x
E E F F
x x x x
E E F F
x x x x
w u
B D B D
x
x
E F
x x
φ φ φ φ
ψ ψ
φ φ
φ φ
∂ ∂ ∂
+ + + +
∂ ∂
∂ ∂
∂ ∂ ∂ ∂
+ + + +
∂ ∂ ∂ ∂
∂ ∂
+ + + −
∂ ∂ ∂ ∂
 
 
∂ ∂
+ + − −
 
 
 
∂∂
 
 
∂ ∂
− −
∂ ∂
2 2
1
22 22
2 2
2 2
2
2
22 2 4 2 3 1
2
2
T T
T
u
D E
x x
F I u I I
x
φ
φφ φ
∂ ∂
− −
∂ ∂
∂= + +
∂
ɺɺ ɺɺ
ɺɺ
(13e)

A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209 2199
( )
2 2 2
12 66 22 66
2 2
1 2 2 1
2
1
44 12 66
2 1 2
2 2 2
2 1 1
12 66 66 22
2 2
1 2 1 2
2 2
2 2
66 22 44 1 44 2
2 2
1 2
2 2 2 2
1 2
11 11 11 22
2 2 2 2
1 1 1 2
( )
( )
2
T T T T
u v v
B B B B
x x x x
w
A D D
x x x
E E D D
x x x x
E E A B
x x
v v
B D E B
x x x x
D
φ
φ ψ ψ
ψ ψ ψ ψ
ψ ψ
∂ ∂ ∂
+ + + −
∂ ∂ ∂ ∂
∂ ∂
+ + +
∂ ∂ ∂
∂ ∂ ∂
+ + + +
∂ ∂ ∂ ∂
∂ ∂
+ − − −
∂ ∂
∂ ∂ ∂ ∂
− − − −
∂ ∂ ∂ ∂
2 2
1 2
22 22 2 1 1 3 2
2 2
2 2
T T
E I I v I
x x
ψ ψ ψ ψ
∂ ∂
− = + +
∂ ∂ ɺɺ ɺɺ
ɺɺ
(13f)
( )
( )
2 2 2
12 66 66 22
2 2
1 2 1 2
2
1
44 12 66
2 1 2
2 2 2
2 1 1
12 66 66 22
2 2
1 2 1 2
2 2
2 2
66 22 44 1 44 2
2 2
1 2
2 2 2 2
1 2
11 11 11 22
2 2 2 2
1 1 1 2
2 ( )
2 4
T T T T
uv v
D D D D
x x x x
w
B E E
x x x
F F E E
x x x x
F F B D
x x
v v
D E F D
x x x x
E
φ
φ ψ ψ
ψ ψ ψ ψ
ψ ψ
∂ ∂ ∂
+ + + −
∂ ∂ ∂ ∂
∂ ∂
+ + +
∂ ∂ ∂
∂ ∂ ∂
+ + + +
∂ ∂ ∂ ∂
∂ ∂
+ − − −
∂ ∂
∂ ∂ ∂ ∂
− − − −
∂ ∂ ∂ ∂
2 2
1 2
22 22 2 4 2 3 1
2 2
2 2
T T
F I v I I
x x
ψ ψ ψ ψ
∂ ∂
− = + +
∂ ∂
ɺɺ ɺɺ
ɺɺ
(13g)
where , , ,
ij ij ij ij ij
A B D E and F
are the plate stiffness and
, , ,
T T T T T
ii ii ii ii ii
A B D E and F
are the plate temperature stiffness.
( )
2
234
3 3 3 3 3
2
, , , , 1, , , ,
h
ij ij ij ij ij ij
h
A B D E F q x x x x dx
−
=∫ (14a)
for
(
)
( )
, , , 1,2,4,5, 6
, , 1,2, 6
ij ij ij
ij ij
A D F i j
E B i j
 =

=


(14b)
( )
2
234
3 3 3 3 3
2
, , , , 1, , , ,
( 1,2)
h
T T T T T T
ii ii ii ii ii ii
h
A B D E F x x x x dx
i
σ
−


=



=


∫
(14c)
11 11 12
3
22 12 22 3
3
66
12
0 1 0 ( , )
0 0 1 ( ).
( , )
0 0 0 0
T
T
T
q q x T
q q T x
x T
q
σα
σα
σ
     
   
     
= − ∆
   
     
     
  
 
 
(14d)
It should be noted that first-order shear deformation equa-
tions for SFGP can be obtained by considering zero values
for
2 2
&
φ ψ
in Eq. (13).
4. Boundary conditions
4.1 Mechanical boundary conditions
For an SFGP with simply support boundary conditions at
the edges as shown in Fig. 2, the following relation can be
written:
10,
x a
= ⇒
(
)
( )
( )
( )
( )
( )
2
2
1 2
1 2
2 2
2 2
0, , 0
, , 0
0, , 0
, , 0
0, , 0
, , 0
v x t
v a x t
x t
a x t
x t
a x t
ψ
ψ
ψ
ψ
 =
=


=


=


=


=

,
( )
( )
( )
( )
11 2
11 2
11 2
11 2
0, , 0
0, , 0
, , 0
, , 0
M x t
N x t
M a x t
N a x t

=

=


=


=

,
(
)
( )
2
2
0, , 0
, , 0
w x t
w a x t

=


=


(15a)
20,
x b
= ⇒
(
)
( )
( )
( )
( )
( )
1
1
1 1
1 1
2 1
2 1
,0, 0
, , 0
,0, 0
, , 0
,0, 0
, , 0
u x t
u x b t
x t
x b t
x t
x b t
φ
φ
φ
φ

=

=


=


=


=


=

,
( )
( )
( )
( )
22 1
22 1
22 1
22 1
,0, 0
,0, 0
, , 0
, , 0
M x t
N x t
M x b t
N x b t

=

=


=


=

,
(
)
( )
1
1
,0, 0
, , 0
w x t
w x b t

=


=


(15b)
where
N
and
M
are the stress resultants.
4.2 Thermal conditions
Five cases of one-dimensional temperature distribution
through the thickness (vertical,
3
x
) are considered, with
3
( ).
T T x
=
4.2.1 Uniform temperature
In this case, a uniform temperature field of
3 0 3
( ) ( )
T x T T x
= + ∆
(16)
Fig. 2. Simply supported boundary condition in SFGP.

2200 A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209
is used where
3
( )
T x
∆
denotes the temperature change and
0
300
T K
=
is room temperature.
4.2.2 Linear temperature
Assuming temperatures
b
T
and
t
T
are imposed at the
bottom and top of the plate, the temperature field under linear
temperature rise along the thickness can be obtained as
3
3 0
1
( ) ( )
2
b
x
T x T T T h
= + + ∆ +
(17)
where 3
( )
t b
T x T T
∆ = −
is the temperature gradient.
4.2.3 Nonlinear temperature
Consider a temperature load at the upper and lower surfaces
such as
0
t
T T T
= +
at 3
2
h
x
=
and 0
b
T T T
= +
at 3
2
h
x
= −
.
These thermal loads can be used to solve the steady-state
heat transfer Eq. (18):
3
3
[ ( ) ] 0
d dT
k x
dx dx
−
=
. (18)
The temperature distribution along the thickness can be ob-
tained:
3
3
3
/ 2
3 0 / 2
3
3
/ 2
1
( )
( ) ( )( )
1
( )
x
h
b t bh
h
dx
k x
T x T T T T
dx
k x
−
−
= + − − ∫
∫
. (19)
In the case of power-law SFGP, the solution of Eq. (18) also
can be expressed by means of a polynomial series [35]:
3 0 3
( ) ( ) ( )
b t b
T x T T T T T x
= + + − ∆
(20)
1
3 3
2 1
2
3
2
3 1
3
3
33
4 1
4
3
4
5 1
5
3
5
2 2
2 ( 1) 2
2
2
(2 1)
2
1
( ) 2
(3 1)
2
2
(4 1)
2
2
(5 1)
N
tb
b
N
tb
b
N
tb
b
N
tb
b
N
tb
b
x h k x h
h N k h
k x h
h
N k
k x h
T x C h
N k
k x h
h
N k
k x h
h
N k
+
+
+
+
+
 
+ +
   
− +
 
   
+
   
 
 
+
 
 
−
 
 
+ 
 
+
 
∆ = − +
 
+ 

+
 
−
 
+ 

+
 
 
+
 
 










(21)
2 3
2 3
4 5
4 5
1( 1) (2 1) (3 1)
(4 1) (5 1)
tb tb tb
b
b b
tb tb
b b
k k k
N k
N k N k
C
k k
N k N k
 
− + −
 
++ +
 
=
 
 
+ −
 
+ +
 
(22)
where
.
tb t b
k k k
= −
(23)
For an isotropic material plate, such as pure metal or ce-
ramic plates, the temperature rise through the thickness is
3 0 3
( ) 2
t b t b
T T T T
T x T x
h
+ −
= + + . (24)
4.2.4 Heat transfer rate by heat-flux
In this thermal condition, the heat flow from the upper sur-
face to the lower one is assumed to be
2
( / )
q W m
and the
lower surface is held at 3 0
( )
b
T x T T
= +
. The heat transfer rate
per unit area (heat flux)
q
is
3
3
3
( )
( )
dT x
q k x
dx
= − . (25)
By solving Eq. (25) for an SFGP with the thermal conduc-
tivity varying through the thickness, the temperature rise
through the thickness can be found:
3
3 0 3
3
/ 2
1
( )
( )
x
b
h
T x T T q dx
k x
−
= + + ∫. (26)
For an isotropic material plate (pure metal or ceramic) with
constant thermal conductivity, the temperature rise through the
thickness is:
3 0 3
( ) ( )
2
b
q h
T x T T x
k
= + + + . (27)
4.2.5 Sinusoidal temperature rise
The temperature field under sinusoidal temperature rise
across the thickness is assumed as [9]:
3
3 0
( ) ( ) 1 cos 2 4
t b b
x
T x T T T T
h
π π
 
 
= + − − + +
 
 
 
  . (28)
5. Method of solution
Based on Navier’s method with simply supported boundary
conditions, the displacement fields are expressed as [32]:
( ) ( )
1 2
1 1
cos sin ,
i t
mn mn
n m
u u t x x u t Ue
ω
α β
∞ ∞
−
= =
= =
∑∑ , (29a)
( ) ( )
1 2
1 1
sin cos ,
i t
mn mn
n m
v v t x x v t Ve
ω
α β
∞ ∞
−
= =
= =
∑∑ , (29b)

A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209 2201
( ) ( )
1 2
1 1
sin sin ,
i t
mn mn
n m
w w t x x w t We
ω
α β
∞ ∞
−
= =
= =
∑∑ , (29c)
( ) ( )
1 1 1 2 1 1
1 1
cos sin ,
i t
mn mn
n m
t x x t e
ω
φ φ α β φ
∞ ∞
−
= =
= = Φ
∑∑ , (29d)
( ) ( )
2 2 1 2 2 2
1 1
cos sin ,
i t
mn mn
n m
t x x t e
ω
φ φ α β φ
∞ ∞
−
= =
= = Φ
∑∑ , (29e)
( ) ( )
1 1 1 2 1 1
1 1
sin cos ,
i t
mn mn
n m
t x x t e
ω
ψ ψ α β ψ
∞ ∞
−
= =
= = Ψ
∑∑ , (29f)
( ) ( )
2 2 1 2 2 2
1 1
sin cos ,
i t
mn mn
n m
t x x t e
ω
ψ ψ α β ψ
∞ ∞
−
= =
= = Ψ
∑∑ , (29g)
where
ω
is the natural frequency and
,
m n
a b
π π
α β
= = . (30)
Substituting the displacement fields (29) into equations of
motion (13), the following frequency equation is obtained:
[ ] [ ]
2
1 1
2 2
1 1
2 2
0
0
0
0
0
0
0
U U
V V
W W
C M
ω
ψ ψ
ψ ψ
   
 
   
 
   
 
   
 
   
 
− =
Φ Φ
     
     
Φ Φ
     
     
     
 
   
(31)
where stiffness matrix [C] and mass matrix [M] can be written
as:
2 2
11 11 11 66 22
12 12 66
13
2 2
14 11 11 66 22
2 2
15 11 11 66 22
16 12 66
17 12 66
( ) ( )
( )
0
() ( )
( ) ( )
( )
( )
T T
T T
T T
C A A A A
C A A
C
C B B B B
C D D D D
C B B
C D D
α β
αβ
α β
α β
αβ
αβ
= − + −
= +
=
= − + −
= − + −
= +
= +
21 12
2 2
22 66 11 22 22
23
24 12 66
25 12 66
2 2
26 66 11 22 22
2 2
27 66 11 22 22
( ) ( )
0
( )
()
( ) ( )
( ) ( )
T T
T T
T T
C C
C A A A A
C
C B B
C D D
C B B B B
C D D D D
α β
αβ
αβ
α β
α β
=
= − + −
=
= +
= +
= − + −
= − + −
31
32
2 2
33 55 11 44 22
0
0
( ) ( )
T T
C
C
C A A A A
α β
=
=
= − + −
34 55
35 55
36 44
37 44
2
2
C A
C B
C A
C B
α
α
β
β
=
=
=
=
41 14
42 24
43 34
2 2
44 11 11 66 22 55
2 2
45 11 11 66 22 55
46 12 66
47 12 66
( ) ( )
( ) ( ) 2
( )
( )
T T
T T
C C
C C
CC
C D D D D A
C E E E E B
C D D
C E E
α β
α β
αβ
αβ
=
=
=
= − + − +
= − + − +
= +
= +
51 15
52 25
53 35
54 45
2 2
55 11 11 66 22 55
56 12 66
57 12 66
( ) ( ) 4
( )
( )
T T
C C
C C
CC
C C
C F F F F D
C E E
C F F
α β
αβ
αβ
=
=
=
=
= − + − +
= +
= +
,
61 16
62 26
63 36
64 46
65 56
2 2
66 66 11 22 22 44
2 2
67 66 11 22 22 44
( ) ( )
( ) ( ) 2
T T
T T
C C
C C
CC
C C
C C
C D D D D A
C E E E E B
α β
α β
=
=
=
=
=
= − + − +
= − + − +
71 17
72 27
73 37
74 47
75 57
76 67
2 2
77 66 11 22 22 44
( ) ( ) 4 .
T T
C C
C C
CC
C C
C C
C C
C F F F F D
α β
=
=
=
=
=
=
= − + − +
(32)
Natural frequencies and the corresponding mode shapes of
SFGPs are obtained by solving the eigenvalue problem (Eq.
31).
6. Material properties in thermal conditions
The variation of Young modulus in SFGPs through the
thickness in room temperature, uniform, linear, nonlinear and
sinusoidal thermal conditions is presented in Figs. 3-8, respec-
tively. Room temperature is defined at 0
300( )
T K
= for all
thermal conditions. The temperature rise in uniform tempera-
ture is
600( ),
b t
T T K
= = the nonlinear thermal conditions are
0( )
b
T K
= and
600( ),
t
T K
= the heat-flux thermal condi-
tions are
0( )
b
T K
= and
4 2
5 10 ( / )
q W m
= × and the sinu-
soidal thermal conditions are
300( )
b
T K
= and
500( )
t
T K
=.
It is seen from Figs. 3 and 4 that Young’s modulus is simi-

2202 A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209
lar for conditions with room temperature and uniform tem-
perature, but the graphs move to smaller values with the uni-
form temperature rise. It is clear that Young’s modulus de-
creases with the increase of the grading index. Figs. 3-8 show
that the behavior of Young’s modulus in nonlinear, heat flux
and sinusoidal thermal loads is completely different from that
in room and uniform temperature cases.
In nonlinear and sinusoidal thermal conditions (Figs. 5 and
7), the value of Young’s modulus increases close to the lower
surface, then decreases when
1,
p
<
and the modulus de-
creases when
1 10.
p≤ < However, Young’s modulus de-
creases then increases close to upper surface for the large
value of grading index (
10
p
>
) in nonlinear and sinusoidal
thermal conditions.
The nonlinear thermal conditions are
0( )
b
T K
= and
600( ),
t
T K
= which means that the temperature at the upper
surface (ceramic-rich) is higher than the temperature of the
lower surface (metal-rich). This may result from the fact that
the elastic modulus decreases in the ceramic-rich surface. It is
clearly extracted by considering Eq. 1 so that if 3
/ 0.5
x h
=
then the elastic modulus can be expressed as
(
)
3
,
E E x T
=
( ).
c
E T
=
It is shown that when 3
/ 0.5,
x h
=
the elastic
modulus is equal to the elastic modulus of ceramic-rich mate-
rial at 600(K), which is less than the elastic modulus of metal-
rich at 0(K).
In heat-flux thermal condition (Fig. 6) the elastic modulus at
the upper surface increases with the increase of the grading
index. These phenomena can be explained in that as the value
of the grading index increases (see Eq. (1)), the volume frac-
tion of ceramic is increased and the heat flow is more re-
stricted. Comparison of Young’s modulus has been made for
uniform, linear, nonlinear, heat-flux and sinusoidal thermal
conditions in Fig. 8. It can be seen that Young’s modulus in-
creases close to the lower surface for all thermal loads and
then decreases through the thickness except in the uniform
thermal condition.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
104
106
108
110
112
114
116
118
Non-Dimensional thickness (z/h)
Ro om T emp era ture E la stic m odu lus e (GP a)
p=0.1 p=0.5 p= 1 p=2 p=10
Fig. 3. Variation of Elastic modulus versus non-
dimensional thickness
of SFGP in room temperature field and different values of grading
index
( ).
p
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
71
72
73
74
75
76
77
78
79
80
81
Non-Dimensional thickness (z/h)
Uniform temperature Elastic moduluse (GP a)
p=0.1 p=0.5 p=1 p=2 p=10
Fig. 4. Variation of Elastic modulus versus non-
dimensional thickness
of SFGP in linear temperature field and di
fferent values of grading
index
( ).
p
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0. 5
70
75
80
85
90
95
100
105
110
115
Non-Dimensional thickness (z/h)
Temperature dependent Elastic moudulus (GP a)
p=0.1
p=0.2
p=0.5
p=1
p=2
p=5
p=10
p=100
Fig. 5. Variation of Elastic modulus versus non-
dimensional thickness
of SFGP in nonlinear temperature field and di
fferent values of grading
index
( ).
p
-0.5 -0.4 -0.3 -0.2 -0.1 0 0. 1 0.2 0.3 0.4 0.5
85
90
95
100
105
110
115
Non-dimensional thi ckness (z/h)
Elastic modulus (GPa)
p=0.1
p=0.2
p=0.5
p=1
p=2
p=5
p=10
Fig. 6. Variation of Elastic modulus versus non-
dimensional thickness
of SFGP in heat flux temperature field and
different values of grading
index
( ).
p

A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209 2203
7. Validation and numerical results
7.1 Validation
The results for temperature-dependent SFGP obtained by
applying SSDT in this study are compared by adapting the
higher-order shear deformation theory results [19]. The fol-
lowing non-dimensional fundamental frequencies in Table 2
are obtained by considering a combination of ZrO2/Ti–6Al–
4V where the upper surface is ceramic-rich and the lower
surface is metal-rich [19]. In accordance with [19], the dimen-
sion-less natural frequency parameter is 2
( / )
a h
ω ω
=
2 1 / 2
[ (1 ) / ]
b b
E
ρ ν
− where
b
E
and
b
ρ
are at T0 = 300(K).
Validation has been done by considering the values of thick-
ness, side, Poisson's ratio, density and thermal conductivity in
the ceramic and metal as:
0.025
h m
=,
0.2
a m
=,
0.3
ν
=
,
3
3000 /
C
kg m
ρ
=
1.80 /
C
W mK
κ
=,
3
4429 /
m
kg m
ρ
=,7.82 /
m
W mK
κ
=.
The same value of Poisson's ratio
υ
is considered for the
ceramic and metal. Young's modulus and thermal expansion
coefficient of these materials are assumed to be temperature-
dependent and listed in Ref. [19]. Table 2 shows the natural
frequencies obtained from the present study using SSDT and
Huang [19].
There is a considerable agreement between the presented
results and those from Ref. [19], especially for thick plates.
The main reason which can explain the difference between
presented results in Table 2 and Ref. [19] is the difference
between SSDT and HSDT. The results from the present study
by SSDT are greater than those from the higher-order shear
deformation theory (HSDT, Ref. [19]). This phenomenon can
be described as that the transverse shear and rotary inertia
have more effect on thick plate. Moreover, the transverse
shear strains in HSDT are assumed to be parabolically distrib-
uted across the plate thickness. Due to greater accuracy
HSDTs use higher-order polynomials in the expansion of the
displacement components through the thickness of the plate.
The HSDTs introduce additional unknowns that are often
difficult to interpret in physical terms. In principle, one may
expand the displacement field of a plate in terms of the thick-
ness variable up to any desired degree. However, due to the
algebraic complexity and computational effort involved with
HSDTs in return for a marginal gain in accuracy, theories
higher than second order have not been attempted in this re-
search. For the thick plates considered in this case, there is less
difference between the result predicted by SSDT and HSDT;
the SSDT slightly over-predicts the frequencies.
The natural frequencies can be decreased by increasing
3
( )
T x
∆
because the modulus of elasticity decreases with
rising temperatures. It can be noted that the natural frequency
in temperature-dependent SFGPs is greater than those in tem-
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
80
82
84
86
88
90
92
94
96
98
Non-dimensional thi ckness (z/h)
Elasti c modulu s (GPa)
p=0.1
p=0.5
p=1
p=2
p=5
p=10
Fig. 7. Variation of Elastic modulus versus non-
dimensional thickness
of SFGP in sinusoidal temperature field and differ
ent values of grading
index
( ).
p
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
70
75
80
85
90
95
100
105
110
Non-dimensional thickness (z/h)
Elastic modulus (GPa)
Uniform temperature
Linear temperature
Non-linear temperature
Heat flux temperature
Sinusoidal temperature
T0=300k
Tt=500k
Tb=300k
q=5e4W/m2
Fig. 8. Variation of Elastic modulus versus non-
dimensional thickness
of SFGP in uniform, linear, nonlinear, heat flux and sinusoidal tem-
perature field and di
fferent values of grading index
( ).
p
Table 2. Non-dimensional natural frequency parameter (ZrO2/Ti-6Al-
4V) SFGP for simply supported in thermal environments.
300
b
T k
=
400
t
T k
=
600
t
T k
=
Mode (1,1)
Natural fre
quency of
SFGP (ZrO2 and
Ti-6Al-4V)
300
t
T k
=
Tempera-
ture-
dependent
Tempera-
ture-
independ-
ent
Tempera-
ture-
dependent
Tempera-
ture-
independ-
ent
Present 8.333 7.614 7.892 5.469 6.924
ZrO2
Ref. [19]
8.273 7.868 8.122 6.685 7.686
Present 7.156 6.651 6.844 5.255 6.175
p=0.5
Ref. [19]
7.139 6.876 7.154 6.123 6.776
Present 6.700 6.281 6.446 5.167 5.904
p=1 Ref. [19]
6.657 6.437 6.592 5.819 6.362
Present 6.333 5.992 6.132 5.139 5.711
p=2 Ref. [19]
6.286 6.101 6.238 5.612 6.056
Present 5.439 5.103 5.333 4.836 5.115
Ti-6Al-
4V Ref. [19]
5.400 5.322 5.389 5.118 5.284

2204 A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209
perature-independent SFGPs. As expected, natural frequencies
in pure ceramic are greater than those in pure metal and results
for SFGPs are in between.
7.2 Numerical results
Tables 3 and 4 show the natural frequencies in ZrO2/Ti–
6Al–4V and Si3N4/SUS304 for different thermal loads, re-
spectively. The non-dimensional natural frequency parameter
is defined as
2 2 1/ 2
( / )[ (1 ) / ]
b b
a h E
ω ω ρ ν
= − where
b
E
and
b
ρ
are at T0 = 300 (K) [19]. The effect of volume fraction
index
p
on the frequencies can be seen by considering the
same value of thermal load and shape mode. The result for
SFGPs is in between those for pure material plates, because
Young’s modulus decreases from pure ceramic to pure metal.
The frequencies are decreased by increasing the temperature
difference between top and bottom surfaces for the same
value of grading index and shape mode that represent the
effects of thermal loads. The comparison between tempera-
ture-dependent and independent SFGPs in Tables 3 and 4
reveals the smaller frequencies in temperature-dependent
SFGPs, which demonstrates the accuracy and effectiveness of
temperature-dependent material properties.
The behavior of natural frequencies for Si3N4/SUS304 in
Table 4 is similar to those for ZrO2/Ti–6Al–4V in Table 3.
The value of natural frequency for Si3N4/SUS304 is greater
than that for ZrO2/Ti–6Al–4V due to higher modulus of elas-
ticity of Si3N4 compared with ZrO2. It is worth mentioning that
the difference of natural frequency for the same temperature
and shape mode is decreased with the increase of grading
index. This phenomenon is because the stiffnesses for Ti–
6Al–4V and SUS304 are close to each other.
Table 5 shows the natural frequencies in Si3N4/SUS304 for
large value of volume fraction index (
p
) and different values
of thermal loads. As described before, the non-dimensional
natural frequency parameter is defined as 2
( / )
a h
ω ω
=
Table 3. Non-dimensional frequency parameter (ZrO2/Ti-6Al-
4V)
SFGP for simply supported in thermal environments.
300
b
T k
=
400
t
T k
=
600
t
T k
=
Mode numbers of
SFGP (ZrO2&
Ti-6Al-4V)
300
t
T k
=
Tempera-
ture-
dependent
Tempera-
ture-
independ-
ent
Tempera-
ture-
dependent
Tempera-
ture-
independ-
ent
(1,1) 8.333 7.614 7.892 5.469 6.924
(1,2) 19.613
18.585
19.151
16.092
18.191
(2,2) 29.785
28.471
29.300
25.518
28.305
(1,3) 36.084
34.588
35.584
31.331
34.562
ZrO2
(2,3) 44.942
43.192
44.420
39.506
43.359
(1,1) 7.156 6.651 6.844 5.255 6.175
(1,2) 16.852
16.126
16.527
14.431
15.859
(2,2) 25.604
24.677
25.263
22.647
24.567
(1,3) 31.036
29.981
30.684
27.734
29.970
p=0.5
(2,3) 38.670
37.439
38.304
34.887
37.562
(1,1) 6.700 6.281 6.446 5.167 5.904
(1,2) 15.774
15.161
15.507
13.765
14.960
(2,2) 23.959
23.172
23.679
21.482
23.109
(1,3) 29.040
28.143
28.752
26.264
28.167
p=1
(2,3) 36.176
35.128
35.876
32.985
35.268
(1,1) 6.333 5.992 6.132 5.139 5.711
(1,2) 14.896
14.383
14.684
13.260
14.253
(2,2) 22.608
21.942
22.386
20.557
21.935
(1,3) 27.392
26.630
27.163
25.077
26.700
p=2
(2,3) 34.106
33.211
33.867
31.425
33.384
(1,1) 5.439 5.103 5.333 4.836 5.115
(1,2) 12.801
12.130
12.689
11.655
12.463
(2,2) 19.440
18.467
19.323
17.802
19.085
(1,3) 23.551
22.390
23.430
21.604
23.185
Ti-6Al-4V
(2,3) 29.333
27.907
29.206
26.954
28.951
Table 4. Non-dimensional natural frequency parameter (Si3N4/
SUS304) SFGP for simply supported in thermal environments.
300
b
T k
=
400
t
T k
=
600
t
T k
=
Mode numbers of
SFGP (Si3N4&
SUS304)
300
t
T k
=
Tempera-
ture-
dependent
Tempera-
ture-
independ-
ent
Tempera-
ture-
dependent
Tempera-
ture-
independ-
ent
(1,1) 12.506
12.175
12.248
11.461
11.716
(1,2) 29.464
29.030
29.192
28.138
28.641
(2,2) 44.782
44.253
44.496
43.190
43.919
(1,3) 54.277
53.687
53.982
52.514
53.388
Si3N4
(2,3) 67.641
66.967
67.334
65.640
66.715
(1,1) 8.652 8.361 8.405 7.708 7.887
(1,2) 20.355
20.001
20.095
19.233
19.565
(2,2) 30.902
30.491
30.629
29.605
30.076
(1,3) 37.439
36.991
37.157
36.029
36.588
p=0.5
(2,3) 46.621
46.122
46.328
45.055
45.736
(1,1) 7.584 7.306 7.342 6.674 6.834
(1,2) 17.841
17.512
17.587
16.781
17.068
(2,2) 27.082
26.707
26.816
25.871
26.275
(1,3) 32.813
32.407
32.538
31.504
31.981
p=1
(2,3) 40.858
40.411
40.572
39.413
39.993
(1,1) 6.811 6.545 6.575 5.929 6.077
(1,2) 16.017
15.708
15.769
15.002
15.262
(2,2) 24.307
23.958
24.047
23.154
23.517
(1,3) 29.446
29.071
29.177
28.204
28.632
p=2
(2,3) 36.657
36.247
36.376
35.290
35.809
(1,1) 5.410 5.161 5.178 4.526 4.682
(1,2) 12.745
12.471
12.503
11.729
12.004
(2,2) 19.372
19.073
19.117
18.214
18.599
(1,3) 23.479
23.164
23.217
22.229
22.684
SUS304
(2,3) 29.260
28.923
28.987
27.881
28.433

A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209 2205
2 1 / 2
[ (1 ) / ]
b b
E
ρ ν
− where
b
E
and
b
ρ
are at T0 = 300 (K)
[19]. The frequencies are decreased by increasing the volume
fraction index (
p
) and by decreasing the temperature differ-
ence between the top and bottom surfaces for the same value
of grading index and shape mode that represent the effects of
thermal loads. This is expected, because a larger volume frac-
tion index means that a plate has a smaller ceramic component
and that its stiffness is thus reduced. A uniform frequency
distribution for values of (
p
) is virtually predictable, but the
trend of variation is not linear as (
p
) increases.
The following numerical results are obtained by considering
the combination of titanium alloy (Ti–6Al–4V) as the lower
surface and zirconium oxide (ZrO2) as the upper surface in
SFGPs according to Table 1. The geometry of square SFGPs
is as follows:
/ 10
a h
=
(side-to-thickness ratio),
0.2
a m
=(side of the square).
The variation of temperature distribution through the thick-
ness of SFGPs is shown in Figs. 9 and 10 by applying non-
linear and heat-flux thermal conditions, respectively. The up-
per surface is held at Tt = 1000 (K), and the lower surface is
held at Tb = 300 (K) in Fig. 10.
For pure-material plates, temperature distributions are the
same regardless of the material type. The temperature at any
internal point through the thickness of a plate made of pure
material (
0
p
=
and
p
→ ∞
) is always higher than that corre-
sponding to SFGP. In contrast to the linear temperature distri-
bution in a pure plate, the variation of temperature through the
thickness in SFGPs is non-linear.
Fig. 10 shows temperature distributions through the thick-
ness of SFGPs subjected to heat-flux thermal condition. It is
assumed that the lower surface is held at Tb = 0 (K) (T0 = 300
(K)) and the heat-flow from the upper surface to the lower is q
= 5 x 104 W/m2.
The temperature distributions are at a maximum for pure ce-
ramic and a minimum for pure metal with different SFGPs
falling in between. In contrast to the result for nonlinear ther-
mal loads, the trade of temperature variation is linear in
SFGPs.
The temperature distributions of SFGPs under linear, nonli-
near, sinusoidal and heat flux thermal conditions along the
thickness direction are shown in Fig. 11. It is considered that
the upper and lower surfaces are held at Tt = 500 (K) and Tb =
300 (K) and room temperature is T0 = 300 (K). Four cases,
namely linear, non-linear, sinusoidal and heat flux, are consid-
ered. For heat-flux, the source from upper to lower surface is
applied as q = 5 x 104 (W/m2). It is absorbed that the variation
of temperature through the thickness is the lowest for sinusoi-
dal temperature load. Linear thermal load creates the greater
temperature in comparison with non-linear thermal load. As
expected, the variation of temperature in existence of heat flux
is larger than the other results. It is also plotted that the curve
of linear temperature condition is close to nonlinear tempera-
ture condition.
The following dimensionless frequencies are presented by
Table 5. Non-
dimensional natural frequency of temperature dependent
(Si3N4& SUS304) SFGP for different volume fraction index (
p
)
in
thermal environments.
Thermal loads
0
0
T k
=,
0.2
b a
= =
,
0.025
h=
300
b
T k
=
300
b
T k
=
300
b
T k
=
400
t
T k
=
300
b
T k
=
600
b
T k
=
Full ceramic
12.506 12.175 11.461
5
p
=
6.200 5.936 5.328
10
p
=
5.907 5.645 5.031
20
p
=
5.711 5.450 4.825
40
p
=
5.591 5.329 4.694
60
p
=
5.546 5.284 4.645
80
p
=
5.523 5.260 4.619
100
p
=
5.508 5.246 4.603
Non-dimensional natural frequency of tem-
perature dependent SFGP (Si3N4 & SUS304)
for different volume fraction index (p) and
thermal load, Mode (1,1)
Full metal
5.410 5.161 4.526
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
300
400
500
600
700
800
900
1000
1100
1200
1300
Non-dimentional thickness (z/h)
Temperature (k)
p=0 (Isotropic)
p=0.1
p=0.5
p=1
p=5
p=10
p=100
Fig. 9. Variation of nonlinear temperature field versus non-
dimensional
thickness of SFGP for different values of grading index
( )
p
.
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
300
400
500
600
700
800
900
Non-dimensional thickness (z/h)
Temperature (K)
p=0.1
p=0.5
p=1
p=5
p=10
Full ceramic
Full metal
Fig. 10. Variation of heat flux temperature field versus non-
dimensional thickness of SFGP for different values of grading index
( )
p
.

2206 A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209
using non-dimensional parameters [33]:
2
0
2
0
I
b
D
ω
ωπ
=
where
0
I h
ρ
= and
3 2
0
/12(1 )
D Eh
υ
= − .
It is noted that
,
ρ ν
and
E
are chosen to be the values of
Ti-6Al-4V evaluated at the room temperature.
The room temperature is considered 0
300 ( )
T K
=
for all
thermal conditions. The temperature rise is
0 ( ) &
b
T K
=
500 ( )
t
T K
=
in uniform, linear, nonlinear and sinusoidal ther-
mal conditions. The thermal condition in heat-flux case is
500 ( )
b
T K
=
and
3 2
1 10 ( / )
q W m
= × at the bottom surface
toward the upper surface.
Figs. 12-16 show the first four frequencies versus uniform,
linear, nonlinear, heat flux and sinusoidal temperature fields in
simply supported SFGP. The combination of ZrO2/Ti-6Al-4V
as shown in Table 1 is considered with material and geometric
parameters of
1, 0.2
p a
= =
and
/ 10
a h
=
.
As expected, the decrease of Young’s modulus with rising
temperatures leads to the frequencies decreasing with increas-
ing temperature. The decreasing slope of frequencies in higher
modes is greater than those in smaller modes. The difference
between two consequence higher modes is smaller than that in
two consequence lower modes at the same temperature. It is
evident that the temperature rise effect in uniform temperature
condition is more significant than other thermal conditions. It
can be explained in that the decreasing slope frequency in
linear, nonlinear and sinusoidal thermal loads is almost the
same while decreasing slope frequency is greater than other
thermal conditions in uniform thermal load. It is displayed that
the decreasing slope frequency is very small in heat-flux
thermal condition.
Figs. 17 and 18 investigate the effect of side-to-side ratio
versus nonlinear and sinusoidal thermal loads of simply sup-
ported SFGP. The combination of ZrO2/Ti-6Al-4V as shown
in Table 1 is considered while the material and geometric
parameters are
2, 0.2
p a
= =
and
/ 10
a h
=
. It can be dis-
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
600
650
700
750
800
850
900
950
1000
1050
1100
Non-dimensional thi ckness (z/h)
Temperature (K)
linear Temperature
Non-Linear temperature
Heat flux temperature
Sinusoidal temperature
T0=300K
Tt=500K
Tb=300K
p=0.1
q=5e4 W/m2
Fig. 11. Variation of linear, nonlinear, heat flux and sinusoidal tem-
perature field versus non
-dimensional thickness of SFGP for di
fferent
values of grading index
( )
p
.
300 350 400 450 500 550 600 650 700 750 800
1
2
3
4
5
6
7
8
9
Uniform Temperature rise (K)
Non-dimensional Frequency
Mode 1
Mode 2
Mode 3
mode 4
Fig. 12. First four Non-dimensional frequency parameters versus uni-
form temperature field for simply supported (ZrO2/Ti-6Al-
4V) SFGP
when
/ 10
a h
=
and
0.2, 1.
a p
= =
.
300 350 400 450 500 550 600 650 700 750 800
1
2
3
4
5
6
7
8
9
Linear Temperature rise (K)
Non-dimensional Frequency
Mode 1
Mode 2
Mode 3
Mode 4
Fig. 13. First four Non-dimensional frequency paramete
rs versus linear
temperature field for simply supported (ZrO2/Ti-6Al-
4V) SFGP when
/ 10
a h
=
and
0.2, 1.
a p
= =
300 350 400 450 500 550 600 650 700 750 800
1
2
3
4
5
6
7
8
9
Non-linear Temperature rise (K)
No n-dim ensi on al Fre quen cy
Mode 1
Mode 2
Mode 3
Mode 4
Fig. 14. First four Non-dimensional frequency parameters versus non-
linear temperature field for simply supported (ZrO2/Ti-6Al-
4V) SFGP
when
/ 10
a h
=
and
0.2, 1.
a p
= =

A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209 2207
played that in Figs. 17 and 18, due to gradation of stiffness,
the frequencies increase with the increase of the
/
b a
while
/ 2
b a
≤
. As expected, the frequency decreases as
T
∆
increases for nonlinear and sinusoidal temperature fields be-
cause Young’s modulus decreases with rising temperatures. It
is also seen that the decreasing slope frequency for
/ 2
b a
=
is
greater than other side-to-side ratio while side-to-thickness
ratio is equal to ten
( / 10)
a h
=
in SFGP.
8. Conclusion
Temperature-dependent free vibration of solar functionally
graded plates subjected to uniform, linear, nonlinear, heat-flux
and sinusoidal temperature fields are investigated by using an
analytical approach (Navier’s Method) for simply supported
SFGP. The formulations are based on the second-order shear
deformation theory (SSDT) to account for transverse shear
effects through the thickness. Material properties of SFGPs
are assumed to be temperature-dependent and vary along the
thickness by a power-law distribution in terms of volume frac-
tions of constituents. The results are validated by comparing
them with the results of other researchers, with good agree-
ment. Some general conclusions of this study can be summa-
rized as follows:
Free vibration is at the maximum for pure ceramic, the
minimum for pure metal, and degrades gradually as the vol-
ume fraction index p increases.
The frequency decreases as temperature change (
T
∆
) in-
creases in all types of temperature fields.
The uniform and heat flux temperature fields affect the fre-
quencies more significantly than the linear, nonlinear and
sinusoidal temperature fields.
The value of temperature increases with decreasing grading
index (p).
The frequencies of SFGPs with higher grading index are
more sensitive to the temperature rise than those for other
grading index.
The frequencies increase with the increase of the side-to-
300 350 400 450 500 550 600 650 700 750 800
1
2
3
4
5
6
7
8
9
Sinusoidal Temperature rise (K)
N on -di me nsi ona l fre qu enc y
Mode 1
Mode 2
Mode 3
Mode 4
Fig. 15. First four Non-dimensional frequency parameters versus sinu-
soidal temperature field for simply supported (ZrO2/Ti-6Al-
4V) SFGP
when
/ 10
a h
=
and
0.2, 1.
a p
= =
300 350 400 450 500 550 600 650 700 750 800
1
2
3
4
5
6
7
8
9
Heat flux Temperature rise (K)
N on -di mens iona l Fre que nc y
Mode 1
Mode 2
Mode 3
Mode 4
Fig. 16. First four Non-
dimensional frequency parameters versus heat
flux temperature field for simply supported (ZrO2/Ti-6Al-
4V) SFGP
when
/ 10
a h
=
and
0.2, 1.
a p
= =
0 50 100 150 200 250 300 350 400 450 500
2
4
6
8
10
12
14
16
18
Non-linear Temperature rise (K)
Non-dimension al Frequency
b/a=0.5
b/a=0.75
b/a=1
b/a=1.25
b/a=1.5
b/a=2
Fig. 17. Non-dimensional frequency parameters versus nonlinear tem-
perature field for different values of side
-to-side ratio (
/
b a
) and sim-
ply supported
(ZrO2/Ti-6Al-4V) SFGP when
/ 10
a h
=
and
a
=
0.2, 2.
p
=
0 50 100 150 200 250 300 350 400 450 500
2
4
6
8
10
12
14
16
18
Sinusoidal Temperature rise (K)
Non-dimensional Frequency
b/a=0.5
b/a=0.75
b/a=1
b/a=1.25
b/a=1.5
b/a=2
Fig. 18. Non-dimensional frequency parameters versus si
nusoidal
temperature field for different values of side-to-side ratio (
/
b a
) and
simply supported (ZrO2/Ti-6Al-4V) SFGP when
/ 10
a h
=
and
a
=
0.2, 2.
p
=

2208 A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209
side ratio (
/
b a
) while
/ 2
b a
<
.
From the numerical results presented in this study it appears
that the SSDT results are greater than those from higher order
shear deformation theory (HSDT). It is suggested that the
gradation of the constitutive components, geometry and tem-
perature rise are significant parameters in the frequency of
solar functionally graded plates.
Acknowledgment
This work supported by Universiti Putra Malaysia for pro-
viding the research grant (FRGS 07-10-07-398SFR 5523398).
References
[1] Y. Howell and J. Bereny, Engineer's guide to solar energy,
Solar Energy Information Services (1979).
[2] M. Bayat, M. Saleem, B. Sahari, A. Hamouda and E. Mahdi,
Analysis of functionally graded rotating disks with variable
thickness, Mechanics Research Communications, 35 (5)
(2008) 283-309.
[3] M. Bayat, M. Saleem, B. Sahari, A. Hamouda and E. Mahdi,
Mechanical and thermal stresses in a functionally graded ro-
tating disk with variable thickness due to radially symmetry
loads, International Journal of Pressure Vessels and Piping,
86 (6) (2009) 357-372.
[4] R. Naj, M. Sabzikar Boroujerdy and M. Eslami, Thermal
and mechanical instability of functionally graded truncated
conical shells, Thin-Walled Structures, 46 (1) (2008) 65-78.
[5] J. Reddy and N. Phan, Stability and vibration of isotropic,
orthotropic and laminated plates according to a higher-order
shear deformation theory, Journal of Sound and Vibration,
98 (2) (1985) 157-170.
[6] S. Xiang and K. Wang, Free vibration analysis of symmetric
laminated composite plates by trigonometric shear deforma-
tion theory and inverse multiquadric RBF, Thin-Walled
Structures, 47 (3) (2009) 304-310.
[7] R. Batra, Higher-order shear and normal deformable theory
for functionally graded incompressible linear elastic plates,
Thin-Walled Structures, 45 (12) (2007) 974-982.
[8] G. Praveen and J. Reddy, Nonlinear transient thermoelastic
analysis of functionally graded ceramic-metal plates, Inter-
national Journal of Solids and Structures, 35 (1998) 4457-
4476.
[9] B. Mokhtar, T. Abedlouahed, A. El Abbas and M. Abdelka-
der, Buckling Analysis of Functionally Graded Plates with
Simply Supported Edges, Leonardo Journal of Sciences, 8
(2009) 21-32.
[10] K. Shukla, Nonlinear static and dynamic analysis of func-
tionally graded plates Int. J. of Appl. Mech. and Eng., 11 (3)
(2006) 679-698.
[11] H. Ibrahim, H. Yoo, M. Tawfik and K. Lee, Thermo-
acoustic random response of temperature-dependent func-
tionally graded material panels, Computational Mechanics
1-10.
[12] J. Park and J. Kim, Thermal postbuckling and vibration
analyses of functionally graded plates, Journal of Sound and
Vibration, 289 (1-2) (2006) 77-93.
[13] X. Zhao, Y. Lee and K. Liew, Free vibration analysis of
functionally graded plates using the element-free kp-Ritz
method, Journal of Sound and Vibration, 319 (3-5) (2009)
918-939.
[14] A. Ferreira, R. Batra, C. Roque, L. Qian and R. Jorge,
Natural frequencies of functionally graded plates by a mesh-
less method, Composite Structures, 75 (1-4) (2006) 593-600.
[15] M. Bayat, M. Saleem, B. Sahari, A. Hamouda and E. Ma-
hdi, Thermo elastic analysis of a functionally graded rotating
disk with small and large deflections, Thin-Walled Struc-
tures, 45 (7-8) (2007) 677-691.
[16] M. Bayat, B. Sahari, M. Saleem, A. Ali and S. Wong,
Thermoelastic solution of a functionally graded variable
thickness rotating disk with bending based on the first-order
shear deformation theory, Thin-Walled Structures, 47 (5)
(2009) 568-582.
[17] R. Batra and J. Jin, Natural frequencies of a functionally
graded anisotropic rectangular plate, Journal of Sound and
Vibration, 282 (1-2) (2005) 509-516.
[18] C. Dong, Three-dimensional free vibration analysis of func-
tionally graded annular plates using the Chebyshev-Ritz me-
thod, Materials & Design, 29 (8) (2008) 1518-1525.
[19] X. Huang and H. Shen, Nonlinear vibration and dynamic
response of functionally graded plates in thermal environ-
ments, International Journal of Solids and Structures, 41 (9-
10) (2004) 2403-2427.
[20] H. Matsunaga, Thermal buckling of functionally graded
plates according to a 2D higher-order deformation theory,
Composite Structures, 90 (1) (2009) 76-86.
[21] Y. Kim, Temperature dependent vibration analysis of func-
tionally graded rectangular plates, Journal of Sound and Vi-
bration, 284 (3-5) (2005) 531-549.
[22] N. Sundararajan, T. Prakash and M. Ganapathi, Nonlinear
free flexural vibrations of functionally graded rectangular
and skew plates under thermal environments, Finite Ele-
ments in Analysis and Design, 42 (2) (2005) 152-168.
[23] C. Chen, T. Chen and R. Chien, Nonlinear vibration of
initially stressed functionally graded plates, Thin-Walled
Structures, 44 (8) (2006) 844-851.
[24] J. Yang and H. Shen, Vibration characteristics and transient
response of shear-deformable functionally graded plates in
thermal environments, Journal of Sound and Vibration, 255
(3) (2002) 579-602.
[25] J. Yang and H. Shen, Free vibration and parametric resonance
of shear deformable functionally graded cylindrical panels,
Journal of Sound and Vibration, 261 (5) (2003) 871-893.
[26] A. Khdeir and J. Reddy, Free vibrations of laminated com-
posite plates using second-order shear deformation theory,
Computers & Structures, 71 (6) (1999) 617-626.
[27] A. R. Saidi and S. Sahraee, Axisymmetric solutions of
functionally graded circular and annular plates using second-
order shear deformation plate theory, Proceedings of 8th Bi-

A. Shahrjerdi et al. / Journal of Mechanical Science and Technology 25 (9) (2011) 2195~2209 2209
ennial ASME Conference on Engineering Systems Design
and Analysis, ESDA2006 (2006)
[28] A. Shahrjerdi, M. Bayat, F. Mustapha, S. Sapuan and R.
Zahari, Free Vibration Analysis of Functionally Graded
Quadrangle Plates Using Second Order Shear Deformation
Theory, Australian Journal of Basic and Applied Sciences, 4
(5) (2010) 893-905.
[29] A. Shahrjerdi, M. Bayat, F. Mustapha, S. Sapuan and R.
Zahari, Second-Order Shear Deformation Theory to Analyze
Stress Distribution for Solar Functionally Graded Plates,
Mechanics Based Design of Structures and Machines, 38 (3)
(2010) 348-361.
[30] A. Bahtui and M. Eslami, Coupled thermoelasticity of
functionally graded cylindrical shells, Mechanics research
communications, 34 (1) (2007) 1-18.
[31] A. Shahrjerdi, M. Bayat, F. Mustapha, S. Sapuan and R.
Zahari, Stress Analysis a Functionally Graded Quadrangle
Plate Using Second Order Shear Deformation Theory, Inter-
national Review of Mechanical Engineering, 4 (1) (2010)
92-105.
[32] J. N. Reddy, Theory and analysis of elastic plates and shells,
Taylor Print on DEMA (2006).
[33] Q. Li, V. Iu, and K. Kou, Three-dimensional vibration
analysis of functionally graded material plates in thermal en-
vironment, Journal of Sound and Vibration, 324 (3-5) (2009)
733-750.
[34] J. N. Reddy, Mechanics of laminated composite plates and
shells, CRC press (2004).
[35] R. Javaheri and M. Eslami, Thermal buckling of function-
ally graded plates based on higher order theory, Journal of
Thermal Stresses, 25 (7) (2002) 603-625.
Ali Shahrjerdi is a member of the Me-
chanical Engineering department in Ma-
layer University, Iran. He received a Ph.D
in Mechanical Engineering from the Uni-
versity Putra Malaysia. His research inter-
ests include finite element analysis, nu-
merical methods, aircraft design, compos-
ites and functionally graded materials.