Vol.2, No.9, 968-978 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.29118
Copyright © 2010 SciRes. OPEN ACCESS
Thermal buckling analysis of ceramic-metal functionally
graded plates
Ashraf M. Zenkour1,2*, Daoud S. Mashat1
1Department of Mathematics, Faculty of Science, King AbdulAziz University, Jeddah, Saudi Arabia;
*Corresponding Author: zenkour@kau.edu.sa
2Department of Mathematics, Faculty of Science, Kafrelsheikh University, Kafr El-Sheikh, Egypt
Received 7 March 2010; revised 20 April 2010; accepted 26 April 2010.
ABSTRACT
Thermal buckling response of functionally
graded plates is presented in this paper using
sinusoidal shear deformation plate theory (SPT).
The material properties of the plate are assumed
to vary according to a power law form in the
thickness direction. Equilibrium and stability
equations are derived based on the SPT. The
non-linear governing equations are solved for
plates subjected to simply supported boundary
conditions. The buckling analysis of a function-
ally graded plate under various types of thermal
loads is carried out. The influences of many
plate parameters on buckling temperature dif-
ference will be investigated. Numerical results
are presented for the SPT, demonstrating its
importance and accuracy in comparison to
other theories.
Keywords: Thermal Buckling; Non-Linear Strains;
Functionally Graded Material; Sinusoidal Plate
Theory; Thermal Load
1. INTRODUCTION
The rapid development of composite materials and
structures in recent years has drawn increased attention
from many engineers and researchers. These materials
are broadly used in aerospace, mechanical, nuclear, ma-
rine, and structural engineering. In conventional lami-
nated composite structures, homogeneous elastic laminas
are bonded together to obtain enhanced mechanical and
thermal properties. However, the abrupt change in mate-
rial properties across the interface between different ma-
terials can result in large inter-laminar stresses leading to
delimitation, cracking, and other damage mechanisms
which result from the abrupt change of the mechanical
properties at the interface between the layers. To remedy
such defects, functionally graded materials (FGMs),
within which material properties vary continuously, have
been proposed.
The concept of FGM was proposed in 1984 by a
group of materials scientists, in Sendai, Japan, for ther-
mal barriers or heat shielding properties. Initially FGM
was designed as a thermal barrier material for aerospace
application and fusion reactors. Later on FGM was de-
veloped for the military, automotive, biomedical and
semiconductor industries, and as a general structural
element in high thermal environments. FGM is one of
the advanced high temperature materials capable of
withstanding extreme temperature environments. FGMs
are composite and microscopically heterogeneous in
which the mechanical properties vary smoothly and con-
tinuously from one surface to the other. This is achieved
by gradually varying the volume fraction of the con-
stituent materials. Typically, these materials are made
from a mixture of ceramics and metal or a combination
of different materials. The ceramic constituent of the
material provides the high-temperature resistance due to
its low thermal conductivity and protects the metal from
oxidation. The ductile metal constituent, on the other
hand, prevents fracture caused by stresses due to
high-temperature gradient in a very short period of time.
Further, a mixture of a ceramic and a metal with a con-
tinuously varying volume fraction can be easily manu-
factured [1-4].
A comprehensive work on the FGMs was presented in
the literature. The response of FG ceramic-metal plates
has been investigated by Praveen and Reddy [5] using a
plate finite element. They investigated the static and
dynamic thermoelastic responses of the FGMs by vary-
ing the volume fraction using a simple power law distri-
bution. Reddy [6] developed the Navier’s solutions for
FG plates using the third-order shear deformation plate
theory (TSDT) and an associated finite element model.
Amini et al. [7] described a method for three-dimensio-
nal free vibration analysis of rectangular FGM plates
A. M. Zenkour et al. / Natural Science 2 (2010) 968-978
Copyright © 2010 SciRes. OPEN ACCESS
969
969
resting on an elastic foundation using Chebyshev poly-
nomials and Ritz’s method. This analysis has been based
on a linear, small-strain, three-dimensional elasticity
theory. Analysis of FG plates under static and dynamic
loads has been presented by Sladek et al. [8] using the
meshless local Petrov-Galerkin method and Reissner-
Mindlin theory to describe the plate bending problem.
Kim et al. [9] investigated finite element computation of
fracture parameters in FGM assemblages of arbitrary
geometry with stationary cracks. In Altenbach and Ere-
meyev [10], a viscoelastic FG polymer foam has been
studied using a new plate theory based on the direct ap-
proach. The large deflection response of simply sup-
ported rectangular FG plates under normal pressure
loading has been analyzed by Ovesy and Ghannadpour
[11] using a finite strip method. In Han [12], a numerical
method was proposed for analyzing transient waves in
plates of FGM excited by impact loads. The bending
problem of transverse load acting on FGM rectangular
plate using both two-dimensional trigonometric and
three-dimensional elasticity solutions was investigated
by Zenkour [13]. Zenkour [14,15] studied the bending
response, buckling and free vibration of simply sup-
ported FG sandwich plate using the SPT. Zenkour [16]
presented the derivation of equations for free vibration
of FG plates expressing the displacement components by
trigonometric series representation through the plate
thickness. Other researches into FGMs have included the
nonlinear analysis of FG plates [17], large deformation
analysis of FG shells [18], static and vibration analysis
of FG beams [19,20].
In view of the advantages of FGMs, a number of in-
vestigations dealing with thermal buckling had been
published in the scientific literature. In recent years, the
mechanical and thermal buckling analysis of FG ce-
ramic-metal plates has been presented by Zhao et al. [21]
using the first-order shear deformation plate theory, in
conjunction with the Ritz method. A two-dimensional
global higher-order deformation theory has been em-
ployed by Matsunaga [22] for thermal buckling of plates
made of FGMs. Morimoto et al. [23] presented the ther-
mal buckling analysis of FG rectangular plates subjected
to partial heating in a plane and uniform temperature rise
through its thickness. In Ref. [24], Shariat and Eslami
presented the thermal buckling analysis of rectangular
FG plates with geometrical imperfections using the clas-
sical plate theory to derive the equilibrium, stability, and
compatibility equations of an imperfect FGM. Thermal
buckling of rectangular and circular plates compose of
FGM was also studied based on the first- and higher-order
shear deformation plate theory [25-27].
Various plate theories, depending upon the through-
thickness displacement pattern considered, have been
used to determine thermal buckling loads of composite
plates. The classical plate theory [24], which is based on
Kirchhoff’s hypothesis, overestimates the thermal buck-
ling load when applied to even moderately thick plates.
This is particularly true for composite plates in which
transverse shear moduli are small in comparison to the
in-plane Young’s moduli [28]. In such cases, it becomes
necessary to take into account shear deformation effects.
Thus, various improved plate theories such as first-order
shear deformation [25,26], higher order shear deforma-
tion [5,6] and sinusoidal shear deformation [13-16,29-
31] plate theories have been developed to predict the
behavior of plates with thickness shear deformation. In
this article, thermal buckling analysis of rectangular FG
ceramic-metal plates is investigated. The material prop-
erties of the FG plates are assumed to vary continuously
through the thickness, according to a simple power law
distribution of the volume fraction of the constituents.
The SPT is used to obtain the buckling of the plate under
different types of thermal loads. The thermal loads are
assumed to be uniform, linear and non-linear distribution
through the thickness. Additional numerical results are
presented for FGM plates that show the effects of vari-
ous parameters on thermal buckling response.
2. MATHEMATICAL MODEL
Consider a rectangular plate of length a, width b and
thickness h made of FGM. The plate is subjected to a
thermal load (, ,)Txyz. The material properties of the
FGM plate, such as Young’s modulus E and thermal ex-
pansion coefficients
are assumed to be functions of
the volume fraction of the constituent materials. The
FGM plate is supported at four edges defined in the
(, ,)
x
yz coordinate system with x- and y-axes located in
the middle plane (0)z
and its origin placed at the
corner of the plate.
The modulus of elasticity E, the coefficient of thermal
expansion
and Poisson’s ratio
are assumed as
[5]
mcm
mcm 0
() ,
(), (),
k
k
EzEE V
zVvzv


  (1)
where
cmc m
cmc m
,
,
1,
2
EEE
z
Vh






(2)
and m
E and m
denote the elastic moduli and the
coefficient of thermal expansion of metal; c
E and c
denote the elastic moduli and the coefficient of thermal
A. M. Zenkour et al. / Natural Science 2 (2010) 968-978
Copyright © 2010 SciRes. OPEN ACCESS
970
expansion of ceramic, and k is the volume fraction ex-
ponent. The value of k equal to zero represents a fully
ceramic plate. The above power law assumption reflects
a simple rule of mixtures used to obtain the effective
properties of the ceramic-metal plate. The rule of mix-
tures applies only to the thickness direction. The density
of the plate varies according to the power law, and the
power law exponent may be varied to obtain different
distributions of the component materials through the
thickness of plate. Note that the volume fraction of the
metal is high near the bottom surface of the plate, and
that of ceramic high near the top surface. In addition,
Eq.1 indicates that the bottom surface of the plate
(/2)zh is metal-rich whereas the top surface
(/2)zh of the plate is ceramic-rich. For simplicity,
is assumed constant across the plate thickness.
The displacements of a material point located at (x, y,
z) in the FGM plate might better be illustrated as [29,
30]:
11
22
3
(, ,)(),
(, ,)(),
(, ,),
w
uxyzu zz
x
w
uxyz vzz
y
uxyzw
 
 
(3)
where u, v and w are the displacements of the middle
surface along the axes x, y and z, respectively, and 1
and 2
are the rotations about the y and x-axes and
account for the effect of transverse shear. The coefficient
of 1
or 2
which is given by ()z should be odd
function of z. All of the generalized displacements
12
(,, ,,)uvw
are functions of the (x, y). The dis-
placement of the classical thin plate theory (CPT) can
easily be obtained by setting () 0z. The displace-
ments of the first-order shear deformation plate theory
(FPT) are obtained by setting ()zz
. In addition, the
higher-order shear deformation plate theory (HPT) [6] is
obtained by setting
2
4
()1 3
z
zz h


 





. (4)
Also, the SPT is obtained by setting [14,15]:
() sin
hz
zh

 

. (5)
Note that the present SPT, as well as HPT, is simpli-
fied by enforcing traction-free boundary conditions at
the plate faces. The SPT accounts according to a co-
sine-law distribution of the transverse shear deformation
through the thickness of the FGM plate. The SPT, HPT
and FPT contain the same number of dependent un-
knowns. No transversal shear correction factors are
needed for both SPT and HPT because a correct repre-
sentation of the transversal shearing strain is given.
The non-linear strain components ij
compatible
with the displacement field in Eq.3 are
0
11 111111
0
22 222222
0
12 121212
() ,zz
 
 
 






  
 
  
  
  
  
 

(6)
0
23 23
33 ,3
0
13 13
0,( ),z






 
(7)
where
020 200
1
11,1,1 22,2,2232 131
2
1
, , , ,
2
uwv w


0
12,1,2,1,2
11,11 22,22 12,12
,
, , 2,
vu ww
ww w
 


111,122 2,2122,11,2
, , .

 (8)
The stress-strain relations for the FGM plate are given
by
11 11
2
22 22
23131223 13 12
1
() ,
11
()
{,,} {,,},
2(1 )
T
Ez
T
Ez


 
 

 

 

 


 
(9)
where (,,)Txyz is the temperature rise through the
thickness.
The stress and moment resultants of the FGM plate
can be obtained by integrating Eq.9 over the thickness,
and are written as
0
2
11
,
1
1
ikkkiT
ikkkiT
ikkkiT
NABC A
MBDF B
SCFGC

 

 

 
 




 

 

 
 

(10)
0
12 12
12 12
12 12
1,
2(1 )
kkk
kkk
kk k
NABC
MBDF
SCFG


 
 










 

(11)
and
A. M. Zenkour et al. / Natural Science 2 (2010) 968-978
Copyright © 2010 SciRes. OPEN ACCESS
971
971
0
13 13
0
23 23
,
2(1 )
k
QH
Q


 
 
 
 
(12)
where 1, 2i and
00 0000
1 11222 2211
1 112222211
1 112222211
,,
,,
,.
 
 
 
 
 
 
(13)
In Eqs.10-1 2, 12
,,NN and 12
N and 12
,,
M
M and
12
M
are the basic components of stress resultants and
stress couples; 12
,,SS and 12
S are additional stress
couples associated with the transversal shear effects; and
13
Q and 23
Q are transversal shear stress resultants.
The coefficients ,,,...
kkk
ABC etc. are defined by
/2 2
/2
/2
/2
/2
/2
/2 2
,3
/2
{, ,}(){1,,}d,
{,,}() (){1,,()}d,
{,,}()() (,,){1,,()}d,
()( ())d.
h
kk kh
h
kk kh
h
TTT h
h
kh
ABDEzzzz
CFGzEzzzz
A
BCzEzTxyzzzz
HEzzz
 


(14)
3. EQUILIBRIUM AND STABILITY
EQUATIONS
The total potential energy of a plate subjected to thermal
loads is defined as [27]
,
mbcT
VUU U U (15)
where ,,
mbc
UUU and T
U are membrane strain en-
ergy, bending strain energy, coupled strain energy and
thermal strain energy. The strain energy for FGM plate
based on the SPT is defined as given below in Eq.16.
The equilibrium and stability equations of FGM plates
may be derived by the variational approach. The expan-
sion of V about the equilibrium state by the Taylor series
is
23
11
2!3! ... .VVVV
 
  (17)
The governing equations of equilibrium can be de-
rived by using the first variation .V
The non-linear
equilibrium equations associated with the present SPT
are
1,112,2 0,NN
12,12,2 0,NN
1,1112,122,221 ,112,2212,12
220,MMMNwNwNw
 
1,112,213 0,SS Q

12,12,223 0.SSQ

(18)
To establish the stability equations, the condition
20V
used to derive the stability equations of many
practical plate buckling problems is also used here. The
external load acting on the original configuration is con-
sidered to be the critical buckling temperature if the
above equation 2
(0)V
is satisfied. Assuming that
the state of stable equilibrium of a general plate under
thermal load may be designated by 000 0 0
12
,, ,,uvw
.
The displacements of the neighboring state are
0101
111
010 1
222
01
,,
,,
,
uu u
vv v
ww w


 
 

(19)
where 11 1 1
1
,, ,uvw
and 1
2
are arbitrarily small
increment of displacements. The stability equations are
represented by using the above total displacement given
in Eq.19 in the equation 20V
and collecting the
second-order terms. They read
11
1,112,20,NN
11
12,12,2 0,NN
111010101
1,1112,122,221,112,2212 ,12
220,MMMNwNwNw
 
11 1
1,112,213 0,SS Q

111
12,12,223 0,SSQ

(20)
where the superscript 1 refers to the state of stability and
the superscript 0 refers to the state of equilibrium condi-
tions. The terms 00
12
,NN
and 0
12
N are the pre-buckling
force resultants obtained as
000
1112
,,0
11
TT
AA
NNN


. (21)
4. EXACT SOLUTIONS FOR THERMAL
BUCKLING OF FGM PLATES
Rectangular plates are generally classified in accordance

1111222212 12232313 13
v
1dv.
2
VTT
  



 (16)
A. M. Zenkour et al. / Natural Science 2 (2010) 968-978
Copyright © 2010 SciRes. OPEN ACCESS
972
with the type support used in the absent of the body
forces and lateral loads except the external temperature
load. The following boundary conditions are imposed at
the side edges
111 111
21 11
111 111
12 22
0at0, ,
0at0, ,
vwNM Sxa
uwNM Syb

  (22)
Following Navier solution procedure, we assume the
following solution form for 1111 1
12
(, , ,,)uvw
that
satisfies the simply-supported boundary conditions,
1
1
1
1
1
1
,1
1
1
1
1
1
2
cos( )sin()
sin() cos()
,
sin()sin()
cos( )sin()
sin() cos()
mn
mn
mn
mn
mn
mn
Uxy
u
Vxy
v
Wxy
w
X
xy
Yxy






 
 
 
 
 

 
 
 
 
 
 
 
 
 
(23)
where ma
, nb
; m and n are mode num-
bers; 1111
,, ,,
mn mnmnmn
UVWX and 1
mn
Y are arbitrary pa-
rameters to be determined subjected to the condition that
the solution in Eq.23 satisfies the conditions in Eq.22.
Substituting Eq.23 into Eq.20 , one obtains
[]{} 0,L (24)
where {} denotes the column
11 111
{} {,,,,},
t
mnmnmnmn mn
UVWXY (25)
and elements rs sr
LL of the coefficient matrix []L
are given by:
22
11 [2(1)],
k
LA

 
12 (1 ),
k
LA


22
13 2( ),
k
LB


22
14 [2(1) ],
k
LC

 
15 (1 ),
k
LC


22
22 [(1)2 ],
k
LA

 
22
23 2( ),
k
LB


2415 ,LL
22
25 [(1)2],
k
LC

 
22 22
33 2()[ ()(1)],
kT
LDA
 
 
22
34 2( ),
k
LF


22
35 2( ),
k
LF


22
44 [2(1) ](1),
kk
LG H
 
 
45 (1 ),
k
LG


22
55 [(1)2](1).
kk
LG H
 
 (26)
For non-trivial solutions of Eq.24, the determinant
L should be zero. This equation (0)L is stated for
the determination of the lowest critical load. In the fol-
lowing, the solutions of the equation 0L for differ-
ent types of thermal loading conditions are presented.
The plate is assumed simply supported in bending and
rigidly fixed in extension. The temperature change is
varied only in the thickness direction.
4.1. Thermal Buckling for FGM Plates under
Uniform Temperature Rise
The initial uniform temperature of the plate is assumed
to be i
T. The temperature is uniformly raised to a final
value
f
T in which the plate buckles. The temperature
change is
f
i
TTT
. Substituting Eq.26 into the
equation 0L
, the buckling temperature change using
the shear deformation theories is obtained as
222 22222 2
12
222222
11 2
()[(1)()]
(1 )[(1)()]
T
nsm PaPnsm
TaAPaPns m



  
,
(27)
where
2
12
, 2(),
kkkk k
PAHPAGC
2
11
2
22
,
22(2),
kkk
kkk kkkkk
PPDBH
PPD AFBBG FC

  (28)
/2
1/2 ()()d, /.
h
Th
A
zEzz sa b

The critical buckling temperature change cr
T, is the
smallest value of T
which is obtained when m =1
and n = 1. Therefore,
22 222
12
2222
11 2
(1)[(1)(1)]
.
(1 )[(1)(1)]
cr
T
sPaPs
TaAPaPs




(29)
For the classical plate theory, the critical buckling
temperature difference cr
T is given as
22
2
1
(1)()
(1 )
kkk
cr
kT
s
ADB
TaAA

. (30)
4.2. Thermal Buckling for FGM Plates
Subjected to a Graded Temperature
Change across the Thickness
For an FG plate, the temperature change is not uniform.
The temperature varies according to the power law
variation. Usually, the temperature rises much higher at
A. M. Zenkour et al. / Natural Science 2 (2010) 968-978
Copyright © 2010 SciRes. OPEN ACCESS
973
973
the ceramic side than that in the metal side of the plate.
In this case, the temperature through the thickness is
given by
m
()TzTV T
, (31)
where m
T is the temperature of the bottom surface
which is metal-rich and
is the power law exponent
(0
).
Similar to the previous loading case, solving the equa-
tion 0L, the buckling temperature difference
(/2)( /2)TTh Th  using the shear deformation
plate theories can be determined, and then we can obtain
the critical buckling cr
T as
22222
1
12
2222
2
21 2
(1)[(1) (1)],
(1 )[(1)(1)]
mT
cr
T
T
TA
sPaPs
TA
aAPaPs





(32)
where
/2
2/2 ()()d.
h
Th
A
zVE zz
(33)
Also, the critical buckling temperature difference cr
T
for the classical plate theory, is deduced as
22
1
2
2
2
(1)() .
(1 )
kk kmT
cr
T
kT
s
ADBT A
TA
aAA


(34)
Note that the value of
equal to unity represents a
linear temperature change across the thickness. While
the value of
excluding unity represents a non-linear
temperature change through the thickness.
5. RESULTS AND DISCUSSION
The general approach outlined in the previous sections
for the thermal buckling analysis of the homogeneous
and FGM plates under uniform, linear and non-linear
temperature rises through the thickness is illustrated in
this section using the SPT. The correlation between the
present theory and different higher- and first-order shear
deformation theories and classical plate theory is estab-
lished. To illustrate the proposed method, a ceramic-metal
FG plate is considered. The combination of materials
consists of aluminum and alumina. The Young’s modu-
lus and the coefficient of thermal expansion for alumina
are c380 GPa,E 6
c7.410/C,
 and for alu-
minum are m70 GPa,E 6
m23 10/C,
 respec-
tively. Note that, Poisson's ratio is selected constant for
both aluminum and alumina and it equal to 0.3. The
shear correction factor for FPT is set equal to 5/6. For
the linear and non-linear temperature rises through the
thickness, the temperature rises 5C
in the metal-rich
surface of the plate (i.e. m5CT). We will assume in
all analyzed cases (unless otherwise stated) that /2,ab
/10,ah
and 3.
Numerical results of the present investigation are
given in Tables 1-4 and Figures 1-4. In Tables 1 and 2,
the side to thickness ratio of the plate is set as /100ah
.
In these tables the critical buckling temperature differ-
ence cr
T of the plate under uniform and linear tem-
perature rises is shown for different values of the power
law index k using various plate theories. The results ob-
tained as per the present HPT and CPT are compared
with the corresponding ones presented by Javaheri and
Eslami [32]. Excellent agreement is achieved between
the two solutions. It is seen that, for all theories, the
critical temperature difference increases monotonically
as the aspect ratio /ab increases. Moreover, the criti-
cal buckling cr
T decreases until it reaches minimum
values and then increases as the values of the volume
fraction exponent k increases. Tables 3 and 4 exhibit the
critical temperature difference 3
10
cr cr
tT
for differ-
ent values of the aspect ratio /ab, the temperature ex-
ponent
and the power law index k under non-linear
temperature loads at /10ah
and 5, respectively. The
nonlinearity temperature exponent
is taken here as 2,
5 and 10. The effect of /ab on the critical buckling
cr
t is similar to that in the case of uniform and linear
temperature difference across the thickness. As the
power law index k increases, the critical buckling cr
t
decreases to reach lowest values and then increases ex-
cluding cr
t of the rectangular plates for 10
. Also,
it is noticed that cr
t increases as the nonlinearity index
increases. In general, the values of the critical tem-
perature difference calculated by using the shear defor-
mation theories are lower than those calculated by using
the classical plate theory, indicating the shear deforma-
tion effect. The SPT without using any shear correction
factor gives results very close to HPT and closer than
those obtained using FPT.
The critical buckling temperature difference cr
t of
the ceramic-metal FG rectangular plate (5)k versus
the side-to-thickness ratio /ah calculated by all theo-
ries under a uniform, linear and non-linear temperature
load are shown in Figure 1. For plates with small /ah
ratio, very large differences between the results of both
SPT and HPT and those of both FPT and CPT are ob-
served. Moreover, the differences between the higher-
order shear deformation theories (SPT and HPT) and
FPT are lower than those between any of them and CPT.
However, for a large value of the side-to-thickness ratio
the difference between the values predicted by the shear
deformation theories and CPT is low significant because
the plate is essentially thin. Because of permitting shear
deformation in SPT, HPT and FPT, the plate becomes
more flexible and thus the critical buckling temperatures
calculated by these theories are smaller than those cal-
A. M. Zenkour et al. / Natural Science 2 (2010) 968-978
Copyright © 2010 SciRes. OPEN ACCESS
974
Table 1. Critical buckling temperature change cr
T of FGM plate under uniform temperature rise for different values of power law
index k and aspect ratio /ab.
k Theory /1ab /2ab /3ab
/4ab
/5ab
0
SPT
HPT
FPT
CPT
17.0894
17.0894 (17.088)
17.0894
17.0991 (17.099)
42.6876
42.6875 (42.688)
42.6875
42.7477 (42.747)
85.2554
85.2551 (85.252)
85.2551
85.4955 (85.495)
144.6500
144.6490 (144.648)
144.6489
145.3424 (145.342)
220.6729
220.6706 (220.667)
220.6704
222.2883 (222.288)
1
SPT
HPT
FPT
CPT
7.9400
7.9400 (7.939)
7.9400
7.9437 (7.943)
19.8359
19.8358 (19.835)
19.8358
19.8594 (19.859)
39.6249
39.6248 (39.624)
39.6248
39.7188 (39.718)
67.2510
67.2506 (67.250)
67.2506
67.5220 (67.522)
102.6365
102.6356 (102.634)
102.6355
103.2690 (103.269)
2
SPT
HPT
FPT
CPT
7.0390
7.0390
7.0392
7.0426
17.5840
17.5840
17.5853
17.6065
35.1233
35.1234
35.1285
35.2130
59.6034
59.6037
59.6184
59.8621
90.9501
90.9508
91.9850
91.5538
5
SPT
HPT
FPT
CPT
7.2606
7.2606 (7.260)
7.2615
7.2657 (7.265)
18.1324
18.1327 (18.132)
18.1380
18.1642 (18.164)
36.2014
36.2025 (36.203)
36.2236
36.3285 (36.328)
61.3921
61.3951 (61.395)
61.4559
61.7585 (61.758)
93.5999
93.6069 (93.605)
93.7481
94.4542 (94.454)
10
SPT
HPT
FPT
CPT
7.4634
7.4634 (7.462)
7.4644
7.4692 (7.469)
18.6365
18.6366 (18.636)
18.6427
18.6731 (18.673)
37.2001
37.2006 (37.200)
37.2246
37.3463 (37.346)
63.0673
63.0687 (63.068)
63.1378
63.4888 (63.488)
96.1183
96.1213 (96.120)
96.2820
97.1005 (97.100)
The results in parenthesis are obtained in [32].
Table 2. Critical buckling temperature change cr
T of FGM plate under linear temperature rise for different values of power law
index k and aspect ratio /ab.
k Theory /1ab /2ab /3ab
/4ab
/5ab
0
SPT
HPT
FPT
CPT
24.1789
24.1789 (24.177)
24.1789
24.1982 (24.198)
75.3753
75.3751 (75.376)
75.3751
75.4955 (75.495)
160.5109
160.5102 (160.505)
160.5102
160.9910 (160.991)
279.3000
279.2980 (279.297)
279.2979
280.6848 (280.684)
431.3459
431.3412 (431.334)
431.3409
434.5767 (434.576)
1
SPT
HPT
FPT
CPT
5.5138
5.5138 (5.513)
5.5138
5.5209 (5.520)
27.8242
27.8242 (27.823)
27.8242
27.8683 (27.868)
64.9379
64.9376 (64.936)
64.9376
65.1140 (65.114)
116.7498
116.7490 (116.748)
116.7490
117.2580 (117.258)
183.1140
183.1123 (183.110)
183.1122
184.3002 (184.300)
2
SPT
HPT
FPT
CPT
3.5893
3.5893
3.5897
3.5956
22.1521
22.1522
22.1544
22.1916
53.0271
53.0273
53.0363
53.1850
96.1203
96.1209
96.1467
96.5757
151.3011
151.3023
151.3624
152.3637
5
SPT
HPT
FPT
CPT
3.8911
3.8912 (3.891)
3.8927
3.8999 (3.899)
22.6047
22.6052 (22.604)
22.6143
22.6595 (22.659)
53.7068
53.7086 (53.710)
53.7450
53.9256 (53.925)
97.0673
97.0725 (97.073)
97.1771
97.6980 (97.698)
152.5063
152.5184 (152.516)
152.7615
153.9769 (153.977)
10
SPT
HPT
FPT
CPT
4.3653
4.3653 (4.364)
4.3670
4.3757 (4.375)
24.1648
24.1650 (24.165)
24.1757
24.2297 (24.229)
57.0607
57.0615 (57.061)
57.1041
57.3198 (57.319)
102.8991
102.9015 (102.901)
103.0240
103.6459 (103.646)
161.4674
161.4729 (161.471)
161.7575
163.2080 (163.208)
The results in parenthesis are obtained in [32].
culated by CPT.
The critical buckling temperature difference cr
t as a
function of /ab for various values of the power law
index k under a uniform, linear and non-linear tem-
perature loads is depicted in Figure 2. It is observed that,
with increasing the plate aspect ratio /ab, the critical
buckling temperature difference also increases gradually,
whatever the material gradient index k is. Since the ce-
ramic plate is weaker than the metallic one, thus the
critical buckling temperature of the first plate is higher
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Table 3. Critical buckling temperature change cr
t of FGM plate under non-linear temperature rise for different values of index k,
aspect ratio /ab, and temperature exponent

/10ah.
/1ab /2ab
/3ab
k Theory
2
5
10
2
5
10
2
5
10
0
SPT
HPT
FPT
CPT
4.8414
4.8410
4.8408
5.1147
9.6829
9.6821
9.6817
10.2294
17.7520
17.7505
17.7498
18.7540
11.2294
11.2269
11.2246
12.8093
22.4589
22.4538
22.4492
25.6186
41.1747
41.1654
41.1568
46.9675
20.0164
20.0066
19.9919
25.6336
40.0328
40.0133
39.9838
51.2673
73.3935
73.3577
73.3037
93.9900
1
SPT
HPT
FPT
CPT
2.1068
2.1066
2.1065
2.2072
4.3182
4.3179
4.3178
4.5241
8.1906
8.1900
8.1898
8.5812
4.9517
4.9508
4.9499
5.5391
10.1496
10.1476
10.1458
11.3534
19.2512
19.2475
19.2440
21.5346
8.9711
8.9673
8.9615
11.0921
18.3880
18.3802
18.3684
22.7355
34.8774
34.8626
34.8402
43.1235
2
SPT
HPT
FPT
CPT
1.6765
1.6766
1.6812
1.7627
3.2736
3.2738
3.2828
3.4419
6.1232
6.1235
6.1404
6.4379
3.9243
3.9246
3.9493
4.4256
7.6627
7.6633
7.7116
8.6417
14.3327
14.3339
14.4242
16.1638
7.0655
7.0659
7.1433
8.8640
13.7962
13.7970
13.9483
17.3080
25.8051
25.8066
26.0895
32.3737
5
SPT
HPT
FPT
CPT
1.5955
1.5964
1.6141
1.7083
2.8485
2.8500
2.8816
3.0498
4.9990
5.0017
5.0571
5.3522
3.6479
3.6521
3.7444
4.2885
6.5126
6.5202
6.6849
7.6562
11.4292
11.4425
11.7317
13.4363
6.3635
6.3755
6.6569
8.5888
11.3609
11.3822
11.8847
15.3337
19.9377
19.9751
20.8569
26.9097
10
SPT
HPT
FPT
CPT
1.6766
1.6770
1.6974
1.8092
2.8844
2.8851
2.9202
3.1126
4.7717
4.7728
4.8310
5.1492
3.7953
3.7970
3.9016
4.5414
6.5293
6.5322
6.7122
7.8130
10.8015
10.8062
11.1040
12.9250
6.5362
6.5402
6.8510
9.0951
11.2448
11.2515
11.7862
15.6470
18.6022
18.6134
19.4980
25.8848
Table 4. Critical buckling temperature change cr
t of FGM plate under non-linear temperature rise for different values of index k,
aspect ratio /ab, and temperature exponent

/5ah
.
/1ab /2ab
/3ab
k Theory
2
5
10
2
5
10
2
5
10
0
SPT
HPT
FPT
CPT
16.7416
16.7353
16.7270
20.5039
33.4833
33.4706
33.4541
41.0078
61.3861
61.3628
61.3325
75.1810
32.8985
32.8633
32.7842
51.2823
65.7971
65.7266
65.5685
102.5646
120.6281
120.4989
120.2090
188.0351
48.6540
48.5388
48.1978
102.5796
97.3080
97.0776
96.3955
205.1592
178.3980
177.9756
176.7252
376.1253
1
SPT
HPT
FPT
CPT
7.4586
7.4561
7.4529
8.8709
15.2878
15.2827
15.2762
18.1827
28.9971
28.9875
28.9751
34.4879
15.0945
15.0800
15.0476
22.1983
30.9390
30.9094
30.8430
45.4997
58.6835
58.6274
58.5014
86.3014
22.9714
22.9214
22.7734
44.4106
47.0843
46.9819
46.6785
91.0281
89.3070
89.1127
88.5373
172.6573
2
SPT
HPT
FPT
CPT
5.8880
5.8885
5.9430
7.0886
11.4979
11.4981
11.6045
13.8415
21.5048
21.5065
21.7057
25.8898
11.7774
11.7751
11.9755
17.7406
22.9970
22.9923
23.3838
34.6407
43.0146
43.0059
43.7381
64.7936
17.7227
17.7018
18.0864
35.4938
34.6058
34.5650
35.3160
69.3061
64.7282
64.6519
66.0566
129.6333
5
SPT
HPT
FPT
CPT
5.3654
5.3742
5.5741
6.8687
9.5789
9.5945
9.9515
12.2627
16.8104
16.8378
17.4644
21.5203
10.1426
10.1682
10.8794
17.1895
18.1076
18.1534
19.4230
30.6885
31.7779
31.8582
34.0863
53.8566
14.4932
14.5269
15.9245
34.3909
25.8748
25.9349
28.4301
61.3982
45.4087
45.5142
49.8932
107.7502
10
SPT
HPT
FPT
CPT
5.5369
5.5400
5.7630
7.2736
9.5255
9.5308
9.9144
12.5134
15.7580
15.7669
16.4014
20.7009
10.2387
10.2435
11.0005
18.2025
17.6144
17.6226
18.9250
31.3150
29.1395
29.1530
31.3076
51.8043
14.3554
14.3463
15.7723
36.4172
24.6965
24.6810
27.1342
62.6510
40.8554
40.8297
44.8880
103.6433
than that of the second. For the FGM plate, cr
t de-
creases as the metallic constituent in the plate increases.
Figure 3 investigates the critical buckling temperature
difference cr
t of homogeneous and FG plates versus
the side-to-thickness ratio /ah under various types of
temperature loads. Figure 4 gives similar for FG plates
versus the aspect ratio /ab. The buckling temperature
of the homogeneous plate is considerably higher than
A. M. Zenkour et al. / Natural Science 2 (2010) 968-978
Copyright © 2010 SciRes. OPEN ACCESS
976
Figure 1. Critical buckling temperature difference
cr
t due to uniform, linear and non-linear tempera-
ture rise across the thickness versus the side-to-
thickness ratio /ah.
Figure 2. Critical buckling temperature difference
cr
t due to uniform, linear and non-linear tempera-
ture rise across the thickness versus the aspect ratio
/ab.
A. M. Zenkour et al. / Natural Science 2 (2010) 968-978
Copyright © 2010 SciRes. OPEN ACCESS
977
977
Figure 3. Critical buckling temperature difference
cr
t due to uniform, linear and non-linear tempera-
ture rise across the thickness versus the side-to-
thickness ratio /ah
.
Figure 4. Critical buckling temperature difference
cr
t due to uniform, linear and non-linear tempera-
ture rise across the thickness versus the aspect ratio
/ab.
that for the FGM one, especially for the comparatively
thicker plates. Again, because of the thicker plates are
stronger than the thinner ones, thus the critical buckling
temperature of the first type is higher than that of the
second one. Note that cr
t of the plate under uniform
temperature rise is smaller than that of the plate under
linear temperature rise and the latter is smaller than that
of the plate under non-linear temperature rise.
6. CONCLUSIONS
The thermal buckling analysis for ceramic-metal FG plates
under uniform, linear and nonlinear thermal loading
through the thickness is investigated in this paper. The
constituent materials are graded from the ceramic sur-
face to the metallic surface according to the power law
variation. The SPT is used to deduce the equilibrium and
stability equations for a simply supported functionally
graded rectangular plate under thermal loading. The re-
sults obtained using SPT are compared with those ob-
tained using HPT, FPT and CPT; and compared with
published ones. The numerical results of critical buck-
ling temperature difference using SPT are very close to
those of HPT and the two theories have similar trends
for all cases of loading. The critical buckling tempera-
ture difference is proportional to the plate aspect ratio.
The thicker plates need a temperature to buckle higher
than that the thinner plates need it. The critical buckling
temperature differences of functionally graded plates are
generally lower than the corresponding ones for homo-
geneous ceramic plates.
7. ACKNOWLEDGEMENTS
The investigators would like to express their appreciation to the Dean-
ship of Scientific Research at King AbdulAziz University for its finan-
cial support of this study, Grant No. 3-038/429.
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