DSpace at VNU: Nonlinear analysis of stability for functionally graded plates under mechanical and thermal loads

Composite Structures 92 (2010) 1184–1191

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Composite Structures
journal homepage: www.elsevier.com/locate/compstruct

Nonlinear analysis of stability for functionally graded plates under mechanical
and thermal loads
Hoang Van Tung a,*, Nguyen Dinh Duc b
a
b

Faculty of Civil Engineering, Hanoi Architectural University, Ha Noi, VietNam
College of Technology, Vietnam National University, Ha Noi, VietNam

a r t i c l e

i n f o

Article history:
Available online 20 October 2009
Keywords:
Nonlinear analysis
Functionally graded materials
Postbuckling
Imperfection

a b s t r a c t
This paper presents a simple analytical approach to investigate the stability of functionally graded plates
under in-plane compressive, thermal and combined loads. Material properties are assumed to be temperature-independent, and graded in the thickness direction according to a simple power law distribution in

terms of the volume fractions of constituents. Equilibrium and compatibility equations for functionally
graded plates are derived by using the classical plate theory taking into account both geometrical nonlinearity in von Karman sense and initial geometrical imperfection. The resulting equations are solved
by Galerkin procedure to obtain explicit expressions of postbuckling load–deflection curves. Stability
analysis of a simply supported rectangular functionally graded plate shows the effects of the volume fraction index, plate geometry, in-plane boundary conditions, and imperfection on postbuckling behavior of
the plate.
Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction
Functionally Graded Materials (FGMs) are microscopically inhomogeneous composites usually made from a mixture of metals and
ceramics. By gradually varying the volume fraction of constituent
materials, their material properties exhibit a smooth and continuous
change from one surface to another, thus eliminating interface problems and mitigating thermal stress concentrations. By high performance heat resistance capacity, FGMs are now developed for
general use as structure components in ultrahigh temperature environments and extremely large thermal gradients such as aircraft,
space vehicles, nuclear plants, and other engineering applications.
Buckling and postbuckling behaviors are one of main interest in
design of structural components such as plates, shells and panels
for optimal and safe usage. Therefore, it is important to study the
buckling and postbuckling behaviors of FGM plates under mechanical, thermal and combined thermomechanical loads for accurate
and reliable design. Some works about the stability of FGM structures relating to present study are introduced in the following.
Javaheri and Eslami [2–4] and Shariat and Eslami [5] reported
mechanical and thermal buckling of rectangular functionally
graded plates by using the classical plate theory [2,3] and higher
order shear deformation plate theory [4,5]. They used energy
method to derive governing equations that analytically solved to
obtain the closed-form solutions of critical loading. The same
* Corresponding author.
E-mail address: (H.V. Tung).
0263-8223/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compstruct.2009.10.015

authors and Shariat [6–8] extended their these studies when influences of initial geometrical imperfection on the critical buckling
loading are taken into consideration. Lanhe [9] used the first order
shear deformation theory to derive closed-form relations for buckling temperature difference of simply supported moderately thick
rectangular FGM plates. Three dimensional thermal buckling analysis of functionally graded composite plates, using finite element
method, is reported by Na and Kim [10]. The research on thermoelastic stability of FGM cylindrical shells is introduced by Eslami
and his co-workers [12–14] and Lanhe et al. [15]. Except [10],
above mentioned works used analytical approach to study buckling of FGM plates and shells. Furthermore, by linear buckling analysis effects of prebuckling deformation and postbuckling behavior
have not been considered in these works. Recently, Darabi et al.
[16] presented nonlinear analysis of dynamic stability for functionally graded cylindrical shells under periodic axial loading by
analytical approach. Some investigations about postbuckling
behavior of functionally graded plates are also reported by Liew
et al. [17,18] using differential quadrature method, Shen [19,20]
using perturbation asymptotic method, and Zhao and Liew [21]
using the element-free kp-Ritz method. The influences of shear
deformation, initial imperfection, piezoelectric actuators, and temperature-dependent properties on postbuckling behavior of FGM
plates are also taken into consideration in these works.
This paper presents a simple analytical approach to investigate
buckling and postbuckling behaviors of functionally graded plates
subjected to in-plane compressive, thermal, and combined loads.
The motivation of this study results from practical significance of


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H.V. Tung, N.D. Duc / Composite Structures 92 (2010) 1184–1191

relatively simple closed-form expressions of buckling load and
postbuckling load–deflection curves in the design. Formulation is
based on the classical plate theory with both von Karman type of
kinematic nonlinearity and initial geometrical imperfection are accounted for. By Galerkin procedure, the resulting equations are

solved to obtain closed-form expressions of postbuckling equilibrium paths. Stability analysis is carried out for a rectangular FGM
plate simply supported on all edges and effects of material and
geometric parameters, in-plane boundary conditions, and imperfection on the postbuckling behavior are discussed.

where geometrical nonlinearity in von Karman sense is accounted
for and subscript (,) indicates the partial derivative. Hooke law for
a plate is defined as

ðrx ; ry Þ ¼ ½E=ð1 À m2 ފ½ðex ; ey Þ þ mðey ; ex Þ À ð1 þ mÞaDTð1; 1ފ;

rxy ¼ ½E=2ð1 þ mފcxy

The force and moment resultants of a plate are expressed in
terms of the stress components through the thickness as

ðNij ; M ij Þ ¼

Consider a rectangular functionally graded plate of length a,
width b, and thickness h, referred to the rectangular Cartesian
coordinates ðx; y; zÞ, where ðx; yÞ plane coincides with middle surface of the plate and z is the thickness coordinate ðÀh=2 6 z 6 h=2Þ.
By applying a simple power law distribution, the volume fractions
of metal and ceramic, V m and V c , are obtained as follows [3,4,9,11]:


k
2z þ h
;
2h

V m ðzÞ ¼ 1 À V c ðzÞ

ð1Þ

where volume fraction index k is a nonnegative number that defines
the material distribution and can be chosen to optimize the structural response.
It is assumed that the effective properties P eff of functionally
graded plate, such as the modulus of elasticity E, the coefficient
of thermal expansion a, and the coefficient of thermal conduction
K, change in the thickness direction z and can be determined by the
linear rule of mixture as [3,4,9,11]

Peff ¼ P c V c ðzÞ þ Pm V m ðzÞ

ð2Þ

where P denotes a temperature-independent material property, and
subscripts m and c stand for the metal and ceramic constituents,
respectively.
From Eqs. (1) and (2), the effective properties of FGM plate can
be written as follows in which Poisson’s ratio m is assumed to be
constant.


k
2z þ h
½EðzÞ; aðzÞ; Kðzފ ¼ ½Em ; am ; K m Š þ ½Ecm ; acm ; K cm Š
2h

h=2

rij ð1; zÞdz; ij ¼ x; y; xy

Nx ¼

E1
E2
Um
ðexm þ meym Þ þ
ðkx þ mky Þ À
1 À m2
1 À m2
1Àm

Ny ¼

E1
E2
Um
ðeym þ mexm Þ þ
ðky þ mkx Þ À
1 À m2
1 À m2
1Àm

Nxy ¼

E1
E
c þ 2 kxy
2ð1 þ mÞ xym 1 þ m

Mx ¼

E2
E3
Ub
ðexm þ meym Þ þ
ðkx þ mky Þ À
1 À m2
1 À m2
1Àm

My ¼

E2
E3
Ub
ðeym þ mexm Þ þ
ðky þ mkx Þ À
1 À m2
1 À m2
1Àm

M xy ¼

acm ¼ ac À am ; K cm ¼ K c À K m

ð10Þ

E2

E
c þ 3 kxy
2ð1 þ mÞ xym 1 þ m

where

E1 ¼ Em h þ Ecm h=ðk þ 1Þ;
3

2

E2 ¼ Ecm h ½1=ðk þ 2Þ À 1=ð2k þ 2ފ;

3

E3 ¼ Em h =12 þ Ecm h ½1=ðk þ 3Þ À 1=ðk þ 2Þ þ 1=ð4k þ 4ފ;

k #
Z h=2 "
2z þ h
Em þ Ecm
ðUm ; Ub Þ ¼
2h
Àh=2
"

k #
2z þ h
DTð1; zÞdz
 am þ acm

2h
ð11Þ

ð4Þ

Nx;x þ Nxy;y ¼ 0
Nxy;x þ Ny;y ¼ 0

ð12Þ

M x;xx þ 2M xy;xy þ M y;yy þ Nx w;xx þ 2Nxy w;xy þ Ny w;yy ¼ 0

3. Governing equations
In the present study, the classical plate theory is used to obtain
the equilibrium and compatibility equations as well as expressions
of buckling loads and postbuckling equilibrium paths of FGM
plates.
The strains across the plate thickness at a distance z from the
mid-plane are [1]

ex ¼ exm þ zkx ; ey ¼ eym þ zky ; cxy ¼ cxym þ 2zkxy

ð5Þ

where exm and eym are the normal strains, cxym is the shear strain at
the middle surface of the plate, and kij are the curvatures.
In the framework of classical plate theory, the strains at the
middle surface and the curvatures are related to the displacement
components u; v ; w in the coordinates as [1]

exm ¼ u;x þ w2;x =2; eym ¼ v ;y þ w2;y =2; cxym ¼ u;y þ v ;x þ w;x w;y ;
ky ¼ Àw;yy ;

ð9Þ

The nonlinear equilibrium equations of a perfect plate based on
the classical plate theory are given by

where

kx ¼ Àw;xx ;

ð8Þ

Substituting Eqs. (3), (5) and (7) into Eq. (8) gives the constitutive relations

ð3Þ

mðzÞ ¼ m
Ecm ¼ Ec À Em ;

Z

Àh=2

2. Functionally graded plates

V c ðzÞ ¼

ð7Þ

kxy ¼ Àw;xy
ð6Þ

If the temperature distributes uniformly in x and y directions
and when Eqs. (9) and (10) are substituted into Eq. (12), the equilibrium equations can be written in terms of deflection variable w
and force resultants as

Nx;x þ Nxy;y ¼ 0
Nxy;x þ Ny;y ¼ 0

ð13Þ

4

Dr w À ðNx w;xx þ 2N xy w;xy þ N y w;yy Þ ¼ 0
where r2 ¼ @ 2 =@x2 þ @ 2 =@y2 , and

D¼

E1 E3 À E22
E1 ð1 À m2 Þ

ð14Þ

For an imperfect plate, let wà ðx; yÞ denotes a known small
imperfection. This parameter represents a small initial deviation
of the plate plane from a flat shape. When imperfection is considered, the equilibrium Eq. (13) is modified into form as [6–8]

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H.V. Tung, N.D. Duc / Composite Structures 92 (2010) 1184–1191

N x;x þ N xy;y ¼ 0

w ¼ M xx ¼ Nxy ¼ 0;

N xy;x þ Ny;y ¼ 0
h
i
Dr4 w À Nx ðw;xx þ wÃ;xx Þ þ 2Nxy ðw;xy þ wÃ;xy Þ þ Ny ðw;yy þ wÃ;yy Þ ¼ 0

w ¼ v ¼ Myy ¼ 0;

ð15Þ
Considering the first two of Eqs. (15), a stress function f may be
defined as

Nx ¼ f;yy ;

Ny ¼ f;xx ;

Nxy ¼ Àf;xy

ð16Þ

Substituting Eq. (16) in the third of Eqs. (15) leads to

h

i
Dr4 w À f;yy ðw;xx þ wÃ;xx Þ À 2f ;xy ðw;xy þ wÃ;xy Þ þ f;xx ðw;yy þ wÃ;yy Þ ¼ 0
ð17Þ
The Eq. (17) includes two dependent unknowns, w and f. To obtain a second equation relating these two unknowns, the compatibility equation may be used.
The geometrical compatibility equation is written as [1]

exm;yy þ eym;xx À cxym;xy ¼ w2;xy À w;xx w;yy

ð18Þ

For a imperfect plate, the above equation may be modified into
form as

exm;yy þ eym;xx À cxym;xy ¼ w2;xy À w;xx w;yy þ 2w;xy wÃ;xy À w;xx wÃ;yy
À w;yy wÃ;xx

ð19Þ

From the constitutive relations (9), one can write

ðexm ; eym Þ ¼

cxym

1
½ðNx ; Ny Þ À mðNy ; Nx Þ À E2 ðkx ; ky Þ þ Um ð1; 1ފ;
E1

2
¼ ½ð1 þ mÞNxy À E2 kxy Š

E1

N y ¼ Ny0

on x ¼ 0; a

ð24Þ

on y ¼ 0; b

where Nx0 ; Ny0 are prebuckling force resultants in directions x and y,
respectively, for Case (1) and the first of Case (3), and are fictitious
compressive edge loads rendering the edges immovable for Case (2)
and the second of Case (3). To solve two Eqs. (17) and (21) for unknowns w and f, and with the consideration of the boundary conditions (22)–(24), we assume the following approximate solutions
[22–24]

w ¼ W sin km x sin ln y
f ¼ A1 cos 2km x þ A2 cos 2ln y þ A3 cos 2km x cos 2ln y
1
1
þ A4 sin km x sin ln y þ N x0 y2 þ Ny0 x2
2
2

ð25Þ

where km ¼ mp=a; ln ¼ np=b; m; n ¼ 1; 2; . . . are number of half
waves in x and y directions, respectively, and W is amplitude of
deflection. Also, Ai ði ¼ 1 Ä 4Þ are coefficients to be determined.
Considering the boundary conditions (22)–(24), the imperfections of the plate are assumed as [1,6–8]

wà ¼ lh sin km x sin ln y;

m; n ¼ 1; 2; . . .

ð26Þ

where the coefficient l varying between 0 and 1 represents imperfection size. By substituting Eqs. (25) and (26) into Eq. (21), the
coefficients Ai are determined as

E1 l2n

A1 ¼
ð20Þ

Nx ¼ Nx0

32k2m

WðW þ 2lhÞ;

A2 ¼

E1 k2m
WðW þ 2lhÞ;
32l2n

A3 ¼ A4 ¼ 0

ð27Þ

Introduction of Eqs. (25) and (26) into Eq. (17) and applying
Galerkin method for the resulting equation yield

Substituting the above equations in Eq. (19), with the aid of Eqs.
(6) and (16), leads to the compatibility equation of an imperfect
FGM plate as

Â
Ã
Dðk2m þ l2n Þ2 W þ 2k2m l2n ðA1 þ A2 Þ þ k2m Nx0 þ l2n Ny0 ðW þ lhÞ ¼ 0



r4 f À E1 w2;xy À w;xx w;yy þ 2w;xy wÃ;xy À w;xx wÃ;yy À w;yy wÃ;xx ¼ 0

Eq. (28), derived for odd values of m; n, is used to determine
buckling loads and postbuckling curves of rectangular FGM plates
under mechanical, thermal, and combined loads.

ð21Þ
Eqs. (17) and (21) are the basic equations used to investigate
the stability of functionally graded plates. They are nonlinear equations in terms of two dependent unknowns w and f.
4. Stability analysis
In this section, an analytical approach is used to investigate the
stability of FGM plates subjected to mechanical, thermal, and combined loads. Depending on the in-plane behavior at the edges,
three cases of boundary conditions, labelled Cases (1), (2) and (3)
will be considered [22].
Case (1). The edges are simply supported and freely movable
(FM). The associated boundary conditions are

w ¼ M xx ¼ N xy ¼ 0;
w ¼ M yy ¼ Nxy ¼ 0;

Nx ¼ Nx0 ;
Ny ¼ Ny0 ;

on x ¼ 0; a
on y ¼ 0; b

Nx ¼ Nx0

on x ¼ 0; a

w ¼ v ¼ M yy ¼ 0;

Ny ¼ N y0

on y ¼ 0; b

4.1. Mechanical stability analysis
The simply supported FGM plate with freely movable edges
(that is, Case (1)) is assumed to be under in-plane compressive
loads Px and Py (in Pascals), uniformly distributed along the edges
x ¼ 0; a and y ¼ 0; b, respectively.
The prebuckling force resultants are [1,2]

Nx0 ¼ ÀPx h;

N y0 ¼ ÀPy h

ð29Þ

Introduction of Eqs. (27) and (29) into Eq. (28) gives



Px ¼

p2 D m2 B2a þ n2

2



p2 E1 m4 B4a þ n4

W






 WðW þ 2lÞ
þ
B2h m2 B2a þ bn2 W þ l 16B2h m2 B2a þ bn2

ð22Þ

Case (2). The edges are simply supported and immovable (IM).
The associated boundary conditions are

w ¼ u ¼ Mxx ¼ 0;

ð28Þ

ð23Þ

Case (3). The edges are simply supported. Uniaxial edge loads
are applied in the direction of the x-coordinate. The edges
x ¼ 0; a are considered freely movable, the remaining two edges
being unloaded and immovable. For this case, the boundary conditions are

ð30Þ
where

Ba ¼ b=a;

3

Bh ¼ b=h;

D ¼ D=h ;

b ¼ Py =Px

E1 ¼ E1 =h;

W ¼ W=h;

ð31Þ

For a perfect plate, l ¼ 0, Eq. (30) leads to equation from which
buckling compressive load P xb may be obtained as



Pxb ¼

p2 D m2 B2a þ n2

2



B2h m2 B2a þ bn2

ð32Þ


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H.V. Tung, N.D. Duc / Composite Structures 92 (2010) 1184–1191

The above equation has been reported by Javaheri and Eslami
[2] when they analyze linear buckling of perfect FGM plates under
in-plane compressive loadings. The critical buckling loads P xcr is
obtained for values of m and n that make the preceding expression
a minimum. In contrast, when l – 0, imperfection sensitivity of
the plates may be predicted. Specifically, no bifurcation-type buckling occurs, and the plates start to deflect at the onset of

compression.
Eq. (30) may be used to trace postbuckling load–deflection
curves of FGM plates subjected to in-plane compressive loads.

4.2.1. Uniform temperature rise
Under mentioned boundary conditions, temperature can be
uniformly raised from initial value T i to final one T f and temperature difference DT ¼ T f À T i is a constant.
The thermal parameter Um can be expressed in terms of the DT
from Eq. (11) and then introduction of the result into Eq. (37) one
obtains

DT ¼

4.2. Thermal stability analysis
A simply supported FGM plate with immovable edges (that is,
Case (2)) under thermal loads is considered. The condition expressing the immovability on the edges, u ¼ 0 (on x ¼ 0; a) and v ¼ 0 (on
y ¼ 0; b), is fulfilled on the average sense as [20,22]

Z
0

b

Z

a

0

@u

dxdy ¼ 0;
@x

Z

a

Z

0

0

b

ð33Þ

@u
1
E2
1
Um
¼ ðf;yy À mf;xx Þ þ w;xx À w2;x À w;x wÃ;x þ
@x E1
2
E1
E1
@v
1
E2

1
Um
¼ ðf;xx À mf;yy Þ þ w;yy À w2;y À w;y wÃ;y þ
2
@y E1
E1
E1

Um

þ
Ny0 ¼ À

1Àm

þ

Um

ð36Þ

1Àm

which is derived by Javaheri and Eslami [3] by solving the membrane form of equilibrium equations and using the method proposed by Meyers and Hyer [25].
By substituting Eqs. (27) and (35) into Eq. (28) we obtain the
following expression for thermal parameter





W
W
þ lh
b


4E2 m4 B4a þ 2mm2 n2 B2a þ n4

 W
þ
2
mnð1 þ mÞb m2 B2a þ n2

Um ¼

2

h

þ

p2 E1 ð3 À m2 Þðm4 B4a þ n4 Þ þ 4mm2 n2 B2a
2

16ð1 þ mÞb ðm2 B2a þ n2 Þ

E2 ¼ E2 =h

2

ð39Þ



p2 Dð1 À mÞ m2 B2a þ n2


ð40Þ

B2h P

This equation has been derived by Javaheri and Eslami [3] when
they analyze linear buckling of perfect FGM plates under uniform
temperature rise. When minimization methods are carried, the
critical buckling temperature difference of perfect plates is obtained for m ¼ n ¼ 1. In addition, with this buckling mode Eq.
(38) may be used to trace postbuckling curves of FGM plates subjected to thermal load under consideration.
4.2.2. Nonlinear temperature change across the thickness
In this case, the temperature through thickness is governed by the
one-dimensional Fourier equation of steady-state heat conduction

Eq. (35) represents the compressive stresses making the edges
immovable and depending on thermal parameter and prebuckling
deflection. It should be noted that when prebuckling deflection is
ignored Eq. (35) leads to

p2 Dð1 À mÞ m2 B2a þ n2

Em acm þ Ecm am Ecm acm
þ

;
kþ1
2k þ 1

ð35Þ

À 2
Á
4E2
l þ mk2m W
mnp2 ð1 À m2 Þ n

À 2
Á
E1
l þ mk2m WðW þ 2lhÞ
8ð1 À m2 Þ n

Nx0 ¼ N y0 ¼ À

ð38Þ

When imperfection is not taken into consideration, Eq. (38)
leads to expression from which bifurcation-type buckling temperature difference DT b may be obtained as

DT b ¼

À 2
Á
4E2

k þ ml2n W
mnp2 ð1 À m2 Þ m

E1
ðk2 þ ml2n ÞWðW þ 2lhÞ
8ð1 À m2 Þ m

Um

þ

þ

P ¼ Em am þ

ð34Þ

Substituting Eqs. (25) and (26) into Eq. (34) and then into Eq.
(33) yield

1Àm

W
W þl


4E2 m4 B4a þ 2mm2 n2 B2a þ n4

 W
þ

mnð1 þ mÞB2h P m2 B2a þ n2
h


i
p2 E1 ð3 À m2 Þ m4 B4a þ n4 þ 4mm2 n2 B2a
WðW þ 2lÞ
þ
16ð1 þ mÞB2h Pðm2 B2a þ n2 Þ
B2h P

where

@v
dydx ¼ 0
@y

From Eqs. (6) and (9) one can obtain the following relations in
which Eq. (16) and imperfection have been accounted for.

Nx0 ¼ À

p2 Dð1 À mÞðm2 B2a þ n2 Þ

!
d
dT
KðzÞ
¼ 0;
dz

dz

Tðz ¼ h=2Þ ¼ T c ;

n
P
k
cm =K m Þ
r 5 ðÀr Knkþ1
TðzÞ ¼ T m þ DT Pn¼0
5
ðÀK cm =K m Þn

n¼0

ð37Þ

By Eq. (37) the postbuckling behavior of rectangular FGM plates
under two types of thermal loads will be analyzed.

ð42Þ

nkþ1

where r ¼ ð2z þ hÞ=2h and DT ¼ T c À T m is defined as the temperature
difference between ceramic-rich and metal-rich surfaces of the plate.
By following the same procedure as the preceding loading case,
and assuming the metal surface temperature as reference temperature, yields

DT ¼

WðW þ 2lhÞ

ð41Þ

where T c and T m are temperatures at ceramic-rich and metal-rich
surfaces, respectively. The solution of Eq. (41) can be obtained by
means of polynomial series. Taking the first seven terms of the series, the solution for temperature distribution across the plate thickness becomes [3,7,9]



i

Tðz ¼ Àh=2Þ ¼ T m

p2 Dð1 À mÞ m2 B2a þ n2



W
W þl


4 4
2 2 2
4E2 m Ba þ 2mm n Ba þ n4

 W
þ
mnð1 þ mÞB2h H m2 B2a þ n2

h


i
p2 E1 ð3 À m2 Þ m4 B4a þ n4 þ 4mm2 n2 B2a


WðW þ 2lÞ
þ
16ð1 þ mÞB2h H m2 B2a þ n2
B2h H

ð43Þ


1188

H.V. Tung, N.D. Duc / Composite Structures 92 (2010) 1184–1191

where

P5

ðÀK cm =K m Þn
n¼0
nkþ1

H¼

h

Em am
nkþ2

P5

acm þEcm am
Ecm acm
þ Emðnþ1Þkþ2
þ ðnþ2Þkþ2

5. Results and discussion

i
ð44Þ

ðÀK cm =K m Þn
n¼0
nkþ1

It is similar to preceding loading case, when initial imperfection
is ignored Eq. (43) is reduced to expression from which buckling
temperature change may be obtained as Eq. (40), provided P is replaced by H. Such a result has been reported by Javaheri and Eslami
[3] by linear buckling analysis of the perfect FGM plates under nonlinear temperature gradient. The imperfection sensitivity of the
plates under thermal loads may be predicted from Eqs. (38) and
(43), that is, postbuckling curves originate from coordinate origin
because no bifurcation buckling point exists when l – 0.
4.3. Thermomechanical stability analysis
A simply supported plate with movable edges x ¼ 0; a and
immovable edges y ¼ 0; b (that is, Case (3)) and subjected to the

simultaneous action of a thermal field and an uniaxial compressive
loading Px, uniformly distributed along the edges x = 0, a is
considered.
Employing N x0 ¼ ÀP x h, Eq. (27) and the second of Eqs. (33), (34)
in Eq. (28) yields



Px ¼

p2 D m2 B2a þ n2

2

W
4E n3


 2
W
þ
2
2
m2 Ba þ mn2 W þ l Bh m m2 B2a þ mn2


p2 E1 m4 B4a þ 3n4
n2 P DT

 WðW þ 2lÞ À

þ
2
2
2
m B2a þ mn2
16Bh m2 Ba þ mn2

To validate the present formulation in buckling and postbuckling of plates under mechanical, thermal and combined loads, the
postbuckling of a homogeneous isotropic plate under uniaxial
compression is considered, which was also analyzed by Shen [19]
using the perturbation asymptotic method and Reddy’s higher-order shear deformation theory. The plate is simply supported on all
edges (Case (1)). The postbuckling load–deflection curves of an isotropic plate ðm ¼ 0:326Þ with and without initial imperfection are
compared in Fig. 1 with Shen’s results. It is evident that good
agreement is achieved in this comparison study. As a second comparison study, postbuckling of a simply supported isotropic plate
with all immovable edges (Case (2)) under uniform temperature
rise is considered, which was also analyzed by Bhimaraddi and
Chandrashekhara [26] using the single mode approach and the parabolic shear deformation theory. The postbuckling paths of perfect

ΔT/ΔT
cr
3

B2h

2
ð45Þ

1.5

Eq. (45) is employed to trace postbuckling curves of the FGM

plates under combined mechanical and thermal loads. Specifically,
it is used to determine the dependence of the in-plane compressive
edge loads vs. total deflection (for given uniform temperature rise)
and conversely, the variation of the temperature rise vs. total deflection (for given compressive edge load). Obviously, temperature
changes can shift Px ðWÞ curves along the Px - axis by an amount
DPx ¼ Àn2 P DT=ðm2 B2a þ mn2 Þ and conversely, DTðWÞ curves can be
displaced along the DT- axis an amount Àðm2 B2a þ mn2 ÞP x =ðn2 PÞ due
to the presence of axial compressive load.

1

P /P
x

2.5

b/a = 1.0, b/h = 10

0.5
0

0

0.2

0.4

W/h

0.6

0.8

1

Fig. 2. Comparisons of postbuckling curves for isotropic plates under uniform
temperature rise.

x

1.6

Shen [19], μ = 0
Shen [19], μ = 0.1
Present, μ = 0
Present, μ = 0.1

1.4
1.2

1.5

perfect
imperfect (μ=0.1)
k=0

b/a = 1.0, b/h = 40, β = 0

1
0.8

1

k=1

0.6

isotropic thin plate (ν = 0.326)
b/a = 1.0, (m,n) = (1,1)

0.5
0

isotropic thick plate

P (GPa)

xcr

2

Ref. [26], μ = 0
Ref. [26], μ = 0.1
Present, μ = 0
Present, μ = 0.1

2.5

0.4

k=5

0.2
0

0.5

1
W/h

1.5

2

Fig. 1. Comparisons of postbuckling curves for isotropic thin plates under uniaxial
compression.

0
0

0.5

W/h

1

1.5

Fig. 3. Postbuckling curves of FGM plates under uniaxial compressive load vs. k.

1189

H.V. Tung, N.D. Duc / Composite Structures 92 (2010) 1184–1191

and imperfect isotropic plates are compared in Fig. 2 with results
in Ref. [26]. As can be observed, a good agreement is obtained in
this comparison study.
To illustrate the present approach, we consider a ceramic–metal
functionally graded plate that consist of aluminum and alumina
with the following properties [3,5,9]

Em ¼ 70 GPa; am ¼ 23:10À6  CÀ1 ;
Ec ¼ 380 GPa; ac ¼ 7; 4:10

À6 

K m ¼ 204 W=mK

À1

C ;

K c ¼ 10; 4 W=mK

ð46Þ

In the case of mechanical stability, a simply supported square
FGM plate under uniaxial compression is considered as a example.
In this case, the critical buckling load of perfect plates corresponds

to m ¼ n ¼ 1, which is the first buckling mode.
Fig. 3 shows variation of postbuckling equilibrium paths of a
FGM plate with side-to-thickness ratio b=h ¼ 40 under uniaxial
compressive load vs. three different values of volume fraction in-

P (GPa)
x
1.4

ΔT (oC)
1000

1: FM, μ=0.0
2: FM, μ=0.1
3: IM, μ=0.0
4: IM, μ=0.1

1.2
1

b/a = 1.0, b/h = 40

k=1
400

1

0.4
0.2

2
0

3

4
0.5

W/h

1

1.5

0

W/h

1

1.5

Fig. 6. Postbuckling curves of FGM plates under nonlinear temperature change
vs. k.

perfect
imperfect (μ = 0.1)

800

b/a = 1.0, b/h = 40

k=0

b/h = 40, k = 1.0

600

2: b/a = 1.5
400

k=5

100

0.5

W/h

1

3

1: b/a = 1.0

k=1
200

0

0.5

ΔT ( C)
1000
perfect
imperfect (μ = 0.1)

300

0

o

ΔT (oC)
500
400

k=5

200

Fig. 4. Effect of in-plane boundary conditions on postbuckling behavior of FGM
plates under uniaxial compression.

0

k=0

600

0.6

0

perfect
imperfect (μ = 0.1)

800

b/a = 1.0, b/h = 40, k = 1.0

0.8

dex k (=0, 1, 5). As can be seen, the postbuckling curves become
lower as the k increases as expected, and postbuckling curves of
imperfect plates are lower than those of perfect plates when
deflection is small. Effects of in-plane boundary conditions on postbuckling behavior of FGM plates under uniaxial compression are
illustrated in Fig. 4. Two types of in-plane conditions on edges
y ¼ 0; b, referred to as freely movable (FM) and immovable (IM)
edges, are considered. In Fig. 4, postbuckling curves of the FM
and IM cases are traced by Eq. (30) with b ¼ 0 and Eq. (45) with
DT ¼ 0, respectively. It is shown that postbuckling strength of
the plate is increased when the edges y ¼ 0; b are immovable and
the deflection is sufficiently large.
In the case of thermal stability, the perfect FGM plates buckle
when m ¼ n ¼ 1 for arbitrary aspect ratio b=a. Figs. 5 and 6 give
postbuckling temperature–deflection curves of a square FGM plate
with three various values of k and under two types of thermal loadings. As can be seen, postbuckling curves to be lower with increas-

3: b/a = 2.0

2
1

200

1.5

Fig. 5. Postbuckling curves of FGM plates under uniform temperature rise vs. k.

0

0

0.5

W/h

1

1.5

Fig. 7. Postbuckling curves of FGM plates under uniform temperature rise vs. b=a.


1190

H.V. Tung, N.D. Duc / Composite Structures 92 (2010) 1184–1191

Px (GPa)

2.5

Px (GPa)
1

perfect
imperfect (μ = 0.1)

2

0.8

b/a = 1.0, b/h = 30, k = 0.5

0.6

1.5
1

0.4

2

1

3
0.5

1: ΔT = 0
2: ΔT = 100 (oC)

0.2

o
3: ΔT = 200 ( C)

0
0

0.5

1: μ = 0
2: μ = 0.1
3: μ = 0.2
4: μ = 0.3

1

W/h

1.5

1
2
3

b/a = 1.0, b/h = 40, k = 1.0

4

0
0

0.5

1

W/h

1.5

Fig. 10. Postbuckling curves of FGM plates under uniaxial compression vs.

l.

Fig. 8. Effects of temperature rise on postbuckling behavior of FGM plates under
uniaxial compression.

o

ΔT ( C)
350

ΔT (oC)
1000
800
600

perfect
imperfect (μ = 0.1)

250

1

150

2
3

100

200

1: P = 0
x

0
0

3: Px = 0.4 GPa

W/h

1

2

1.5

3

4
1

b/a = 1.0, b/h = 40, k = 1.0

50

2: Px = 0.2 GPa

0.5

1: μ = 0
2: μ = 0.1
3: μ = 0.2
4: μ = 0.3

200

b/a = 1.0, b/h = 30, k = 0.5

400

0
0

300

0.5

W/h

1

1.5

Fig. 11. Postbuckling curves of FGM plates under uniform temperature rise vs.

l.

Fig. 9. Effects of compressive load on postbuckling behavior of FGM plates under
uniform temperature rise.

ing values of k as above, and postbuckling loading carrying capability of the plate under nonlinear temperature gradient is higher
than that of plate under uniform temperature rise. Furthermore,
postbuckling strength of imperfect plates is higher than that of
perfect plates when the deflection is sufficiently large. Effects of aspect ratio b=a on thermal postbuckling behavior of FGM plates are
depicted in Fig. 7. It is seen that the postbuckling strength of the
plates under uniform temperature rise is considerably increased
when b=a ratio increases. Fig. 8 shows effects of temperature field
on postbuckling behavior of FGM plates under uniaxial compression. Conversely, the effects of in-plane compressive load on postbuckling behavior of FGM plates under uniform temperature rise
are depicted in Fig. 9. It is shown in these Figs. that the (prestressed) preheated FGM plates exhibit a decreasing tendency in
postbuckling loading carrying capacity when they are subjected
to action of (thermal) compressive loads as mentioned. Finally,
the effects of initial imperfection on postbuckling behavior of

FGM plates with all FM edges subjected to uniaxial compressive
loads are depicted in Fig. 10. It is shown that postbuckling loading
capacity of the plates is reduced with increasing values of imperfection size l when the deflection is small. However, a inverse

trend occurs when the deflection is sufficiently large. Similarly,
variation of postbuckling curves of FGM plates under uniform temperature rise vs. different values of l is plotted in Fig. 11. As can be
observed, when the deflection exceeds a specific value, the curves
become higher when l is increased. In other words, initial imperfection makes FGM plates more stable under temperature field.
6. Concluding remarks
The paper presents a simple analytical approach to investigate
buckling and postbuckling behaviors of functionally graded plates
under in-plane edge compressive, thermal, and combined loads.
The formulation is based the classical plate theory with both von
Karman nonlinear terms and initial imperfection are incorporated.
By using Galerkin method, closed-form expressions of postbuck-


H.V. Tung, N.D. Duc / Composite Structures 92 (2010) 1184–1191

ling load–deflection curves of a simply supported FGM plate are
determined for all mentioned types of load with and without
imperfection. From these explicit expressions, closed-form relations of buckling loads of perfect plates, obtained in foregoing
works by linear buckling analysis, may be derived as particular
cases. The results show that postbuckling behavior of FGM plates
are greatly influenced by material and geometric parameters, and
in-plane boundary conditions. Furthermore, it is also shown that
initial imperfection has significant effects on postbuckling behavior of FGM plates.
Acknowledgement
The authors would like to express their science thank to Professor Dao Huy Bich for offering many valuable suggestions. The
financial support by the research project of Vietnam National University – Ha Noi, coded QGTD.09.01 is gratefully acknowledged.
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