DSpace at VNU: Non-linear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects

International Journal of Non-Linear Mechanics 46 (2011) 1195–1204

Contents lists available at ScienceDirect

International Journal of Non-Linear Mechanics
journal homepage: www.elsevier.com/locate/nlm

Non-linear axisymmetric response of functionally graded shallow spherical
shells under uniform external pressure including temperature effects
Dao Huy Bich a, Hoang Van Tung b,Ã
a
b

Vietnam National University, Ha Noi, Viet Nam
Faculty of Civil Engineering, Hanoi Architectural University, Ha Noi, Viet Nam

a r t i c l e i n f o

abstract

Article history:
Received 5 April 2011
Received in revised form
23 May 2011
Accepted 26 May 2011
Available online 6 June 2011

This paper presents an analytical approach to investigate the non-linear axisymmetric response of
functionally graded shallow spherical shells subjected to uniform external pressure incorporating the
effects of temperature. 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 shallow spherical shells are derived by using
the classical shell theory and specialized for axisymmetric deformation with both geometrical nonlinearity and initial geometrical imperfection are taken into consideration. One-term deflection mode is
assumed and explicit expressions of buckling loads and load–deflection curves are determined due to
Galerkin method. Stability analysis for a clamped spherical shell shows the effects of material and
geometric parameters, edge restraint and temperature conditions, and imperfection on the behavior of
the shells.
& 2011 Elsevier Ltd. All rights reserved.

Keywords:
Nonlinear response
Functionally graded materials
Imperfection
Temperature effects
Shallow spherical shells

1. Introduction
Shallow spherical shells constitute an important portion in
many engineering structures. They can find applications in the
aircraft, missile and aerospace components. These shell elements
also be widely used in other industries such as shipbuilding,
underground structures and building constructions. As a result,
the problems relating to buckling and postbuckling behaviors
bring major importance in the design of this type of shell
structure and have attracted attention of many researchers.
Huang [2] reported an investigation on unsymmetrical buckling
of thin isotropic shallow spherical shells under external pressure.
He pointed out that unsymmetrical deformation may be the
source of discrepancy in critical pressures between axisymmetrical buckling theory and experiment. Tillman [3] investigated
the buckling behavior of clamped shallow spherical caps under a
uniform pressure load. He also considered the effect of shell

geometric parameters on the axisymmetric and asymmetric postbifurcation behavior of a clamped perfect and imperfect isotropic
shallow spherical cap under uniform external pressure both
theoretically and experimentally. Uemura [4] employed a twoterm approximation of deflection to treat axisymmetrical snap
buckling of a clamped imperfect isotropic shallow spherical shell

à Corresponding author.

E-mail address: (H. Van Tung).
0020-7462/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijnonlinmec.2011.05.015

subjected to uniform external pressure. This work also assessed
the effect of shell shallowness on the non-uniqueness of solution
by using the second variation of total potential energy. Non-linear
static and dynamic responses of spherical shells with simply
supported and clamped immovable edge have been analyzed by
Nath and Alwar [5] by making use of Chebyshev series expansion.
Non-linear free vibration response, static response under uniformly distributed load, and the maximum transient response
under uniformly distributed step load of orthotropic thin spherical caps on elastic foundations have been obtained by Dumir [6].
Static and dynamic axisymmetric snap-through buckling of
orthotropic shallow spherical shells subjected to uniform pressure have also been investigated by Chao and Lin [7] using the
classical thin shell theory and finite difference method. Buckling
and postbuckling behaviors of laminated spherical caps subjected
to uniform external pressure have been analyzed by Xu [8] and
Muc [9]. The former employed non-linear shear deformation
theory and a Fourier-Bessel series solution to determine load–
deflection curves of spherical shell under axisymmetric deformation, whereas the latter applied the classical shell theory and
Rayleigh–Ritz procedure to obtain upper and lower pressures and
postbuckling equilibrium paths without considering axisymmetry. Alwar and Narasimhan [10] used method of global interior
collocation to study axisymmetric non-linear behavior of laminated orthotropic annular spherical shells. Subsequently, a static

and dynamic non-linear axisymmetric analysis of thick shallow
spherical and conical orthotropic caps has been reported by Dube


1196

D. Huy Bich, H. Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204

et al. [11] employing Galerkin method and the first order shear
deformation theory. Also, Nie [12] proposed the asymptotic
iteration method to treat non-linear buckling of externally pressurized isotropic shallow spherical shells with various boundary
conditions incorporating the effects of imperfection, edge elastic
restraint and elastic foundation. Recently, Eslami et al. [13] and
Shahsiah and Eslami [14] used an analytical approach to study
linear buckling of isotropic shallow spherical shells with and
without imperfection under thermal and mechanical loads. In
these works the equilibrium and linear stability equations are
derived using the variational method and analytically solved to
obtain closed-form solutions of critical buckling loads for simply
supported shallow spherical shells. Li et al. [15] adopted the
modified iteration method to solve non-linear stability problem of
shear deformable isotropic shallow spherical shells under uniform external pressure.
Due to advanced characteristics in comparison with traditional
metals and conventional composites, Functionally Graded Materials (FGMs) consisting of metal and ceramic constituents have
received increasingly attention in structural applications recent
years. Smooth and continuous change in material properties
enable FGMs to avoid interface problems and unexpected thermal
stress concentrations. By high performance heat resistance capacity, FGMs are now chosen to use as structural components
exposed to severe temperature conditions such as aircraft, aerospace structures, nuclear plants and other engineering applications. Despite the evident importance in practical applications, it
is fact from the open literature that investigations on the buckling

and postbuckling behaviors of FGM spherical shells are comparatively scarce. Shahsiah et al. [16] extended their previous works
for isotropic material to analyze linear stability of FGM shallow
spherical shells subjected to three types of thermal loading. This
study is performed on the point of view of small deflection and
the existence of type-bifurcation buckling of thermally loaded
spherical shells. Recently, the non-linear axisymmetric dynamic
stability of clamped FGM shallow spherical shells has been
analyzed by Prakash et al. [17] and Ganapathi [18] using the first
order shear deformation theory and finite element method. To
best of authors’ knowledge, there is no analytical investigation on
the non-linear stability of FGM shallow spherical shells.
In this paper, the non-linear axisymmetric response of FGM
shallow spherical shells under uniform external pressure with
and without the effects of temperature is investigated by an
analytical approach. The properties of constituent materials are
assumed to be temperature-independent and the effective properties of FGMs are graded in thickness direction according to a
power law function of thickness coordinate. Equilibrium and
compatibility equations of a spherical shell are established by
using the classical shell theory. Then these equations are specialized for axisymmetrically deformed shallow spherical shells
taking into account geometric non-linearity and initial geometrical imperfection. One-term approximation of deflection is
assumed and explicit expressions of extremum buckling loads
and load–deflection curves for a clamped spherical shell are
determined by Galerkin method. An analysis is carried out to
assess the effects of material and geometric parameters, edge
restraint and temperature conditions, and imperfection on the
non-linear response of the shells.

2. Functionally graded shallow spherical shells
Consider a functionally graded shallow spherical shell with
radius of curvature R, base radius a and thickness h as shown in

Fig. 1. The shell is made from a mixture of ceramics and metals,
and is defined in coordinate system ðj, y,zÞ whose origin is located

Fig. 1. Configuration and the coordinate system of a shallow spherical shell.

on the middle surface of the shell, j and y are in the meridional
and circumferential directions, respectively, of the shell and z is
perpendicular to the middle surface and points outwards
ðÀh=2 r z rh=2Þ. Suppose that the material composition of the
shell varies smoothly along the thickness in such a way that the
inner surface is metal-rich and the outer surface is ceramic-rich
by following a simple power law in terms of the volume fractions
of the constituents as

k
2z þ h
Vc ðzÞ ¼
, Vm ðzÞ ¼ 1ÀVc ðzÞ,
ð1Þ
2h
where Vc and Vm are the volume fractions of ceramic and metal
constituents, respectively, and volume fraction index k is a nonnegative number that defines the material distribution.
It is assumed that the effective properties Preff of FGM
spherical shell change in the thickness direction z and can be
determined by the linear rule of mixture as
Preff ðzÞ ¼ Prc Vc ðzÞ þ Prm Vm ðzÞ,

ð2Þ

where Pr denotes a temperature-independent material property,

and subscripts m and c represent the metal and ceramic constituents, respectively.
From Eqs. (1) and (2) the effective properties of FGM shallow
spherical shell such as modulus of elasticity E, the coefficient of
thermal expansion a, and the coefficient of thermal conduction K
can be defined as

k
2z þh
,
ð3Þ
½EðzÞ, aðzÞ,Kðzފ ¼ ½Em , am ,Km Š þ ½Ecm , acm ,Kcm Š
2h
whereas Poisson ratio n is assumed to be constant and
Ecm ¼ Ec ÀEm ,

acm ¼ ac Àam , Kcm ¼ Kc ÀKm :

ð4Þ

It is evident that E ¼ Ec , a ¼ ac , K ¼ Kc at z ¼h/2 and E ¼ Em , a ¼ am ,
K ¼ Km at z¼ Àh/2.

3. Governing equations
In the present study, the classical shell theory is used to obtain
the equilibrium and compatibility equations as well as expressions of buckling loads and non-linear load–deflection curves of
FGM shallow spherical shells.
For a shallow spherical shell it is convenient to introduce a
variable r referred to as the radius of parallel circle and defined by
r ¼ Rsinj. Moreover, due to shallowness of the shell it is approximately assumed that cosj ¼ 1 and Rdj ¼ dr.
The strains across the shell thickness at a distance z from the

mid-plane are

er ¼ erm Àzwr , ey ¼ eym Àzwy , gry ¼ grym Àzwry ,

ð5Þ

where erm and eym are the normal strains, grym is the shear strain
at the middle surface of the spherical shell, whereas wr , wy , wry are
the change of curvatures and twist.


D. Huy Bich, H. Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204

According to the classical shell theory, the strains at the
middle surface and the change of curvatures and twist are related
to the displacement components u,v,w in the j, y,z coordinate
directions, respectively, as [1]

erm ¼ u,r À
grym ¼ r

w 1 2
þ w ,
R 2 ,r

v
r

,r

þ

eym ¼

v, y þu w
1
À þ 2 w2, y ,
r
R 2r

u, y 1
þ w,r w, y ,
r
r

wr ¼ w,rr , wy ¼

w, yy w,r
,
þ
r
r2

ð6Þ
1
r

wry ¼ w,ry À

w, y

,
r2

ð7Þ

where geometrical non-linearity in case of small strain and
moderately small rotation is accounted for, also, subscript (,)
indicates the partial derivative.
Hooke law for a spherical shell including temperature effect is
defined as
ðsr , sy Þ ¼

E
½ðer , ey Þ þ nðey , er ÞÀð1 þ nÞaDT ð1,1ފ,
1Àn2

E
sry ¼
g ,
2ð1 þ nÞ ry



1
1
r
ðrM r Þ,rr þ2 Mry,ry þ Mry, y þ My, yy ÀMy,r þ ðNr þ Ny Þ
r
r
R



1
þ ðrN r w,r þ Nry w, y Þ,r þ Nry w,r þ Ny w, y
þ qr ¼ 0,
r
,y

where q is uniform external pressure positive inwards.
The first two of Eqs. (13) are identically satisfied by introducing a stress function f as
 
f
1
1
ð14Þ
Nr ¼ f,r þ 2 f, yy , Ny ¼ f,rr , Nry ¼ À , y :
r
r ,r
r
Introduction of Eqs. (10), (11) and (14) into the third of Eqs. (13)
gives the following equation:




1
1
1
2 1
1

f,r þ 2 f, yy w,rr þ
f,ry À 2 f, y w,ry
DD2 wÀ Df À
R
r
r r
r
r


1
1
2 1
1
À 2 f,rr w, yy À f,rr w,r þ 2 2 f, y À f,ry w, y Àq ¼ 0,
ð15Þ
r
r
r
r r

ð8Þ

D¼

E1 E3 ÀE22
,
E1 ð1Àn2 Þ

1

r

DðÞ ¼ ðÞ,rr þ ðÞ,r þ

1
ðÞ :
r 2 , yy

1
1
1
1
Dw
þ w2ry Àwr wy :
e
À erm,r þ 2 ðr 2 eym,r Þ,r À 2 ðr grym Þ,ry ¼ À
R
r 2 rm, yy r
r
r
ð17Þ

Introduction of Eqs. (3), (5) and (8) into Eqs. (9) gives the
constitutive relations as

From the constitutive relations (10) one can write

Nr ¼

E1

E2
Fm
ðerm þ neym ÞÀ
ðw þ nwy ÞÀ
,
1Àn
1Àn2
1Àn2 r

ðerm , eym Þ ¼

Ny ¼

E1
E2
Fm
ðe þ nerm ÞÀ
ðw þ nwr ÞÀ
,
1Àn
1Àn2 ym
1Àn2 y

grym ¼

E1
E
g À 2 w :
Nr y ¼
2ð1 þ nÞ rym 1 þ n ry

Mr ¼

E2
E3
F
ðerm þ neym ÞÀ
ðw þ nwy ÞÀ b ,
1Àn
1Àn2
1Àn2 r

My ¼

E2
E3
F
ðe þ nerm ÞÀ
ðw þ nwr ÞÀ b ,
1Àn
1Àn2 ym
1Àn2 y
E2
E
g À 3 w ,
2ð1þ nÞ rym 1 þ n ry

ð10Þ

ð11Þ

where
E1 ¼ Em h þ Ecm h=ðk þ 1Þ,

E2 ¼ Ecm h2 ½1=ðk þ 2ÞÀ1=ð2kþ 2ފ,

E3 ¼ Em h3 =12 þEcm h3 ½1=ðk þ 3ÞÀ1=ðk þ 2Þ þ 1=ð4k þ4ފ,

k #
2z þh
Em þ Ecm
ðFm , Fb Þ ¼
2h
Àh=2
"

k #
2z þ h
 am þ acm
DTð1,zÞ dz:
2h
h=2

ð16Þ

Eq. (15) is the equilibrium equation of FGM shallow spherical
shells in terms of two dependent unknowns, that is deflection of
shell w and stress function f. To obtain a second equation relating
these two unknowns, the compatibility equation may be used.
The geometrical compatibility equation of a shallow spherical
shell is written as

Àh=2

Z

ð13Þ

where

where DT denotes the change of environment temperature from
stress free initial state or temperature difference between the
surfaces of FGM spherical shell.
The force and moment resultants of an FGM spherical shell are
expressed in terms of the stress components through the thickness as
Z h=2
ðNij ,Mij Þ ¼
sij ð1,zÞ dz, ij ¼ r, y,ry:
ð9Þ

Mry ¼

1197

"

ð12Þ

The non-linear equilibrium equations of a perfect shallow spherical shell based on the classical shell theory are given by [1,16]

1

½ðNr ,Ny ÞÀnðNy ,Nr Þ þ E2 ðwr , wy Þ þ Fm ð1,1ފ,
E1

2
½ð1 þ nÞNry þE2 wry Š:
E1

ð18Þ

Substituting the above equations into Eq. (17), with the aid of
Eqs. (7) and (14), leads to the compatibility equation of a perfect
FGM shallow spherical shell as

2


1 2
Dw
1
1
1
1
þ
w,ry À 2 w, y Àw,rr 2 w, yy þ w,r :
D f ¼À
ð19Þ
E1
R
r
r

r
r
Eqs. (15) and (19) are equilibrium and compatibility equations,
respectively, of an FGM shallow spherical shell in the case of
asymmetric deformation. As shown by Huang [2], unsymmetrical
buckling may occur for shallow spherical shells and the presence
of unsymmetric deformation reduces the critical pressures and
has influences on the postbuckling behavior of the shell. However,
the analysis of such a problem is very complicated. Therefore,
with the purpose of obtaining explicit expressions of buckling
pressures and load–deflection curves, the present study restricts
to considering axisymmetrically deformed FGM shallow spherical
shells. Specialization of Eqs. (15) and (19) for an FGM shallow
spherical shell under axisymmetric deformation gives equilibrium equation
1
1
1
DD2s wÀ Ds f À f 0 w00 À f 00 w0 Àq ¼ 0
R
r
r

ð20Þ

and compatibility equation

ðrN r Þ,r þ Nry, y ÀNy ¼ 0,

1 2
Dw 1

D f ¼ À s À w0 w00 ,
E1 s
R
r

ðrN ry Þ,r þ Ny, y þNry ¼ 0,

where Ds ðÞ ¼ ðÞ00 þ ðÞ0 =r and prime indicates differentiation with
respect to r, i.e. ðÞ0 ¼ dðÞ=dr.

ð21Þ


1198

D. Huy Bich, H. Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204

For an imperfect spherical shell, let wn(r) denotes a known
small axisymmetric imperfection. This parameter represents a
small initial deviation of the shell surface from a spherical shape.
When imperfection is accounted for, Eqs. (20) and (21) are,
respectively, modified into forms as
1
1
1
DD2s wÀ Ds f À f 0 ðw00 þ w00n ÞÀ f 00 ðw0 þ w0n ÞÀq ¼ 0,
R
r
r

ð22Þ

1 2
Dw 1
1
1
D f ¼ À s À w0 w00 À w00 w0n À w0 w00n :
E1 s
R
r
r
r

ð23Þ

Eqs. (22) and (23) are the basic equations used to investigate the
non-linear axisymmetric stability of functionally graded shallow
imperfect spherical shells. These are non-linear equations in
terms of two dependent unknowns w and f.

4. Stability analysis
In this section, an analytical approach is used to investigate the
non-linear axisymmetric response of FGM shallow spherical
shells under uniform external pressure with and without the
effects of temperature. The FGM spherical shells are assumed to
be clamped along the periphery and subjected to external
pressure uniformly distributed on the outer surface of the shells
and, in some cases, exposed to temperature conditions. Depending on the in-plane behavior at the edge, two cases of boundary
conditions, labeled Cases (1) and (2), will be considered [4,15].
Case (1). The edge is clamped and freely movable (FM) in the

meridional direction. The associated boundary conditions are
w0 ¼ 0,

r ¼ 0,

w ¼ W,

r ¼ a,

w ¼ w0 ¼ 0,

Nr ¼ 0:

w ¼ W,

r ¼ a,

w ¼ w0 ¼ 0,

0

ð25Þ

where W is the amplitude of deflection (i.e. radial maximum
displacement) and Nr0 is the fictitious compressive edge load
rendering the edge immovable.
With the consideration of the boundary conditions (24) and
(25), the deflection w is approximately assumed as follows
w¼W

ða2 Àr 2 Þ2
:
a4

ð26Þ

Also considering the boundary conditions (24) and (25), the
imperfections of the shallow spherical shells are assumed to be
the same form of the deflection as
wn ¼ mh

ða2 Àr 2 Þ2
a4

E1 W
E1 WðW þ 2mhÞ
rþ
r þ Nr0 r,
3R
2a2

ð27Þ

where the small coefficient m (i.e. À1 r m r1) represents imperfection size. Solution form (26) is similar to first part of two-term
solution presented in [4].
Introduction of Eqs. (26), (27) into Eq. (23) and integration of
the resulting equation give stress function f with





E1 W r 5 a2 r 3
E1 WðW þ 2mhÞ r 7 2a2 r 5
f0 ¼ À 4
À
À
À
þ a4 r 3
8
2
6
3
a R 6
a
C1 r
C2 r C3
ln r þ
þ
,
ð28Þ
þ
2
2
r
where C1,C2,C3 are constants of integration. Due to the finiteness
of the strains and resultants at the apex of shallow spherical shell,
i.e. at r ¼0, the constants C1 and C3 must be zero. After determining the constant C2 from in-plane restraint condition on the

ð29Þ

where Nr0 ¼0 for the spherical shells with movable clamped edge.
Substituting Eqs. (26), (27) and (29) into Eq. (22) and applying
Galerkin method for the resulting equation yield


64D 3E1
976E1
409E1
q¼
WðW þ mhÞÀ
WðW þ 2mhÞ
þ
WÀ
a4
7R2
693a2 R
693a2 R
þ

848E1
40Nr0
2
WðW þ mhÞðW þ 2mhÞ þ
ðW þ mhÞÀ Nr0 :
R
429a4
7a2

ð30Þ

Eq. (30) is used to determine the buckling loads and non-linear
equilibrium paths of FGM shallow spherical shells under uniform
external pressure with and without the effects of temperature
conditions.
4.1. Mechanical stability analysis
The clamped FGM shallow spherical shell with freely movable
edge (that is, Case (1)) is assumed to be subjected to external
pressure q (in Pascals) uniformly distributed on the outer surface
of the shell in the absence of temperature conditions. In this case
Nr0 ¼0 and Eq. (30) leads to
!
64D
3E 1
E1
þ
WÀ
W ð1385W þ1794mÞ
q¼
R4h R4a 7R2h
693R3h R2a
þ

848E 1
W ðW þ mÞðW þ2mÞ,
429R4h R4a

ð31Þ

where
Rh ¼ R=h,

w ¼ 0,
Nr ¼ Nr0 ,

À

ð24Þ

Case (2). The edge is clamped and immovable (IM). For this case,
the boundary conditions are
r ¼ 0,

boundary, i.e. Nr ðr ¼ aÞ ¼ Nr0 , the stress function f is obtained
such that




E1 W r 5 a2 r 3
E1 WðW þ2mhÞ r 7 2a2 r 5
4 3
À
À
À
þa
f0 ¼ À 4
r
2
6
3

a R 6
a8

Ra ¼ a=R,

D ¼ D=h3 ,

E 1 ¼ E1 =h,

W ¼ W=h:

ð32Þ

For a perfect spherical shell, i.e. m ¼ 0, it is deduced from Eq. (31)
that
!
64D
3E 1
1385E 1
848E 1
2
3
þ
WÀ
W þ
W :
ð33Þ
q¼
R4h R4a 7R2h
693R3h R2a

429R4h R4a
It is evident from Eqs. (31) and (33) that qðW Þ curves originate
from coordinate origin and there is no bifurcation-type buckling
for both perfect and imperfect FGM shallow spherical shells under
uniform external pressure. This indicates that externally pressure-loaded shallow spherical shells are axisymmetrically
deflected at the onset of loading and buckling, if occurs, only
may be extremum-type buckling. For perfect spherical shells,
extremum points of qðW Þ curves are obtained from condition
dq
2
¼ AÀ2BW þ CW ¼ 0,
dW

ð34Þ

which yields
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
B2 ÀAC
W 1,2 ¼ B 8
C

ð35Þ

provided
B2 ÀAC 40,

ð36Þ

where
A¼

64D
3E 1
þ
,
R4h R4a 7R2h

B¼

1385E 1
,
693R3h R2a

C¼

848E 1
:
143R4h R4a

ð37Þ

It is easy to examine that if condition (36) is satisfied qðW Þ curve
of perfect shell reaches maximum at W 1 and minimum at W 2


D. Huy Bich, H. Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204

with respective load values are
1
qu ¼ qðW 1 Þ ¼

½Bð3ACÀ2B2 Þ þ2ðB2 ÀACÞ3=2 Š,
3C 2

þ
ð38Þ

1199

ð2637nÀ4331ÞE 1
W ðW þ 2mÞ
2772ð1ÀnÞR3h R2a

ð42 833À27 103nÞE 1
W ðW þ mÞðW þ2mÞ
9009ð1ÀnÞR4h R4a
!
40P DT
2
40m P DT
W
þ
À
,
À
Rh 7R2h R2a 1Àn
7ð1ÀnÞR2h R2a
þ

ql ¼ qðW 2 Þ ¼

1
½Bð3ACÀ2B2 ÞÀ2ðB2 ÀACÞ3=2 Š:
3C 2

ð39Þ

Eqs. (38) and (39) represent upper and lower limit buckling loads of
perfect FGM spherical shell, respectively, under uniform external
pressure. This analysis immediately indicates that if material and
geometric parameters of the shell satisfy the condition (36), the
shell will exhibit a snap-through behavior whose intensity is
measured by difference between upper and lower buckling loads,
that is by 4ðB2 ÀACÞ3=2 =3C 2 . If B2 ¼AC, load–deflection curve has
only an inflexion and for the shells such that B2 oAC, the equilibrium paths are stable, that is, the deflection is monotonically
increased when the pressure increases. A similar behavior trend of
imperfect FGM spherical shells also may be predicted. This nonlinear axisymmetric response of FGM shallow spherical shells is
relatively similar to non-linear behavior of FGM cylindrical panels
under uniform external pressure presented in [19].

ð43Þ

where
P ¼ Em am þ

Em acm þ Ecm am Ecm acm
þ
,
kþ 1
2kþ 1

E 2 ¼ E2 =h2 :

ð44Þ

4.2. Thermomechanical stability analysis

4.2.2. Through the thickness temperature gradient
In this case, the temperature through the thickness is governed
by the one-dimensional Fourier equation of steady-state heat
conduction established in spherical coordinate system whose
origin is the center of complete sphere as


d
dT
2 KðzÞ dT
KðzÞ
þ
¼ 0,
dz
dz
z dz
Tðz ¼ RÀh=2Þ ¼ Tm , Tðz ¼ R þ h=2Þ ¼ Tc ,
ð45Þ

A clamped FGM shallow spherical shell with immovable edge
(that is, Case (2)) under simultaneous action of uniform external
pressure q (in Pascals) and thermal load is considered. The
condition expressing the immovability on the boundary edge,
i.e. u ¼0 on r¼a, is fulfilled on the average sense as

Z pZ a
@u
r dr dy ¼ 0:
ð40Þ
0
0 @r

where Tc and Tm are temperatures at ceramic-rich and metal-rich
surfaces, respectively. In Eq. (45), z is radial coordinate of a point
which is distant z from the shell middle surface with respect to
the center of sphere, i.e. z ¼ R þz and RÀh=2 r z rR þ h=2.
The solution of Eq. (45) can be obtained as follows:
Z z
DT
dz
TðzÞ ¼ Tm þ R R þ h=2
,
ð46Þ
2
dz
RÀh=2
z KðzÞ
RÀh=2 2

From Eqs. (6) and (10) one can obtain the following relation in
which Eq. (14), imperfection and axisymmetry have been
included


@u

1 f0
E2
1
w Fm
¼
Ànf 00 þ w00 À ðw0 Þ2 Àw0 w0n þ þ
:
ð41Þ
@r
E1 r
2
R
E1
E1
Substituting Eqs. (26), (27) and (29) into Eq. (41) and then putting
the result into Eq. (40) give


Fm
ð5nÀ7ÞE1
2E2
À
Nr0 ¼ À
þ
W
1Àn
36ð1ÀnÞR ð1ÀnÞa2
þ

ð35À13nÞE1

WðW þ 2mhÞ
72ð1ÀnÞa2

ð42Þ

which represents the compressive stress making the edge
immovable.
In what follows, specific expressions of thermomechanical
load–deflection curves of FGM shallow spherical shells under
uniform external pressure and two types of thermal loads will be
determined.
4.2.1. Uniform temperature rise
Environment temperature is assumed to be uniformly raised
from initial value Ti, at which the shell is thermal stress free, to
final one Tf and temperature change DT ¼ Tf ÀTi is independent to
thickness variable.
The thermal parameter Fm can be expressed in terms of the DT
due to Eqs. (12). Subsequently, employing this expression Fm in
Eq. (42) and then substitution of the result into Eq. (30) lead to
"
#
64D
ð103À89nÞE 1
4E 2
q¼ 4 4 þ
þ
W
Rh Ra
126ð1ÀnÞR2h
ð1ÀnÞR3h R2a

"
#
ð1526nÀ1746ÞE 1
80E 2
À
W ðW þ mÞ
þ
693ð1ÀnÞR3h R2a
7ð1ÀnÞR4h R4a

z KðzÞ

where, in this case of thermal loading, DT ¼ Tc ÀTm is defined as
the temperature difference between ceramic-rich and metal-rich
surfaces of the FGM spherical shell. Due to mathematical difficulty, this section only considers linear distribution of metal and
ceramic constituents, i.e. k¼1, and


2ðzÀRÞ þ h
KðzÞ ¼ Km þKcm
:
ð47Þ
2h
Introduction of Eq. (47) into Eq. (46) gives temperature distribution across the shell thickness as
(

DTh
4Kcm
ðKc þ Km Þh þ 2Kcm z
TðzÞ ¼ Tm þ

ln
I
2hK m
ðKc þ Km À2Kcm Rh Þ2 h


2ðR þzÞ
2ð2z þ hÞ
Àln
þ
,
ð48Þ
2RÀh
ðKc þ Km À2Kcm Rh ÞðR þ zÞð2RÀhÞ
where z has been replaced by z þR after integration, and
I¼

4Kcm
Kc ð2Rh À1Þ
ln
ðKc þ Km À2Kcm Rh Þ2 Km ð2Rh þ 1Þ
8
:
þ
ðKc þ Km À2Kcm Rh Þð4R2h À1Þ

ð49Þ

Assuming the metal surface temperature as reference temperature and substituting Eq. (48) into Eq. (12) give Fm ¼ DThL=I,
where



Kcm b
b
2
L ¼ À 2 ½xðRh þ 1=2ÞÀ1ŠÀ
xÀ
2Rh À1
2J
J


b
Kc
Kcm d
þ 2 Km ÀKc þ Kc ln
À 2 ½Rh ÀðR2h À1=4ÞxŠ
Km
J
J


d
d
Kc
2
Km ÀKc2 þ2Km Kc ln
À ð1ÀRh xÞÀ 2
J
Km

2J Kcm
"
!#

2
3
Kcm Ecm acm 1 4Rh
1 4Rh
2Ecm acm
1
þ
þ
À
x
þ
þ
9
6
6ð2Rh À1Þ
3
3
J
J2


1200

D. Huy Bich, H. Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204





Ecm acm
Kc
3
2
þRh ÀR2h x þ 2 2 4ðKm
ÀKc3 Þ þ3Kc ðKc2 þ 3Km
Þln
Km
9J Kcm

ð50Þ

and
J ¼ Kc þ Km À2Kcm Rh ,

x ¼ ln

2Rh þ1
,
2Rh À1

b ¼ ðac þ am ÞðEc þ Em Þ, d ¼ Ecm ðac þ am Þ þ acm ðEc þ Em Þ:

To illustrate the proposed approach, we consider a ceramicmetal functionally graded spherical shell that consists of aluminum and alumina with the following properties [20]:
Em ¼ 70 GPa,

am ¼ 23 Â 10À6 J CÀ1 , Km ¼ 204 W=mK,

Ec ¼ 380 GPa,

ac ¼ 7:4 Â 10À6 J CÀ1 , Kc ¼ 10:4 W=mK,

ð52Þ

ð51Þ

By following the same procedure as the preceding loading case we
obtain thermomechanical qðW Þ curves for the case of through the
thickness temperature gradient as Eq. (43), provided P is replaced by
L/I. Such a expression is omitted here for sake of brevity.
Eq. (43) and its temperature gradient counterpart are explicit
expressions of external pressure–deflection curves incorporating
the effects of temperature. These expressions can be used to
consider the non-linear axisymmetric response of immovably
clamped FGM spherical shells subjected to external pressure
and exposed to temperature conditions. Inversely, the temperature difference DT may be obtained in terms of q,W as well as
material and geometric properties due to these expressions.

whereas Poisson’s ratio is chosen to be 0.3. The effects of material
and geometric parameters on the non-linear response of the
perfect and imperfect FGM shallow spherical shells with movable
clamped edge under uniform external pressure are considered in
Figs. 3–5. It is noted that in all figures W/h denotes the dimensionless maximum deflection of the shell.
Fig. 3 shows the effects of volume fraction index k (¼0, 1 and 5)
on the non-linear response of FGM spherical shells subjected to
external pressure. As can be seen, the load–deflection curves
become lower when k increases. This is expected because the
volume percentage of ceramic constituent, which has higher

0.04
perfect

0.035

imperfect (μ = 0.1)

5. Results and discussion

0.03
R/h = 80, a/R = 0.3
0.025
q (GPa)

In this section, the non-linear response of the axisymmetrically deformed FGM shallow spherical shells is analyzed. The
shells are assumed to be clamped along boundary edge and,
unless otherwise specified, edge is freely movable. In characterizing the behavior of the spherical shells, deformations in which the
central region of a shell moves toward the plane that contains the
periphery of the shell are referred to as inward deflections
(positive deflections). Deformations in the opposite direction are
referred to as outward deflection (negative deflections).
As part of the validation of the present method, the buckling
behavior of an isotropic thin shallow spherical shell under uniform external pressure is analyzed, which was also considered by
Li et al. [15] using an updated iteration method and first order
shear deformation shell theory. The dimensionless upper buckling
loads of a clamped immovable shell versus a2/Rh ratio are
compared in Fig. 2 with result of Ref. [15]. As can be seen, a good
agreement is achieved in this comparison study.

0.02

k=0

0.015
k=1

0.01
0.005

k=5

0
0

1

2

3

4

5

W/h
Fig. 3. Effects of volume fraction index k on the non-linear response of FGM
shallow spherical shells.

100

0.06

Present
Li et al. [15]

perfect

60

imperfect (μ = 0.1)

0.05

isotropic thin shallow spherical shell
clamped immovable edge, ν = 0.3

a/R = 0.3, k = 1.0
1: R/h = 60
2: R/h = 70
3: R/h = 80

0.04
q (GPa)

qua4/Eh4

80

40

0.03

0.02
1
20
0.01

2
3

0

0
2

4

6

8

10

12

14

a2/Rh
Fig. 2. Comparison of dimensionless upper buckling loads for clamped immovable
isotropic thin shallow spherical shells.

0

1

2
W/h

3

4

Fig. 4. Effects of curvature radius-to-thickness ratio on the non-linear response of
FGM shallow spherical shells.


D. Huy Bich, H. Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204

upper and lower loads are almost identical and non-linear
behavior of shells is predicted to be very mild. However, when
a/R ratio become higher, i.e. for deeper shells, the difference
between upper and lower loads to be bigger. This shows that
deeper spherical shells experience an unstable postbuckling
behavior and a snap-through phenomenon with increasing intensity. Although the buckling loads of immovable edge shells are
higher, variation tendency of buckling loads is very similar for
both movable and immovable edge shells.
The effects of environment temperature on the thermomechanical behavior of FGM shallow spherical shells with immovable
clamped edge are analyzed in Fig. 8. In this case, the shells are
exposed to temperature field prior to applying external pressure. It
is evident that the spherical shells exhibit a bifurcation-type buckling behavior due to the presence of temperature field. Specifically,

no deflection occurring until external pressure reaches a bifurcation

0.05
perfect
imperfect (μ = 0.1)
0.04
R/h = 80, a/R = 0.3, k = 1.0
1: FM
q (GPa)

elasticity modulus, is dropped with increasing values of k. However, the increase in the extremum-type buckling load and load
carrying capacity of the ceramic-rich spherical shells is paid by a
more severe snap-through behavior, i.e. a bigger difference
between upper and lower buckling loads.
Fig. 4 depicts the effects of curvature radius-to-thickness ratio
R/h ( ¼60, 70 and 80) on the non-linear behavior of the externally
pressurized FGM spherical shells. As can be observed, the load
bearing capability of the spherical shell is considerably enhanced
as R/h ratio decreases. Furthermore, the increase in R/h ratio is
accompanied by a drop of non-linear equilibrium paths and a
more severe snap-through response.
Fig. 5 analyzes the effects of radius of base-to-curvature radius
ratio a/R ( ¼0.3, 0.4 and 0.5) on the non-linear response of FGM
spherical shells subjected to uniform external pressure. It is
shown that the non-linear response of spherical shells is very
sensitive with change of a/R ratio characterizing the shallowness
of spherical shell. Specifically, the enhancement of the upper
buckling loads and the load carrying capacity in small range of
deflection as a/R increases is followed by a very severe snapthrough behavior. In other words, in spite of possessing higher
limit buckling loads, deeper spherical shells exhibit a very

unstable response from the postbuckling point of view.
Fig. 6 considers the effects of in-plane restraint conditions on
the behavior of FGM spherical shells under external pressure. In
this figure the load–deflection curves of clamped shells with
freely movable (FM) edge are plotted in comparison with those
of shells with immovable (IM) edge. As can be seen, although the
spherical shells with immovable clamped edge have a comparatively higher capability of carrying external pressure, their
response is unstable. That is, these shells experience a snapthrough with much higher intensity than their movable clamped
edge counterparts. This trend of response is very similar for both
perfect and imperfect shells. Furthermore, above figures also
show that the effect of initial imperfection on the non-linear
response of FGM spherical shells under external pressure is not
very pronounced. This indicates the imperfection insensitivity of
externally pressure-loaded spherical shells.
Fig. 7 illustrates the variation of the upper and lower buckling
loads versus radius of base-to-curvature radius ratio a/R for both
movable and immovable edge spherical shells. As can be
observed, in small range of a/R, i.e. for very shallow shells, the

1201

0.03

2: IM

0.02
2
0.01

1

0
0

1

2

3

4

W/h
Fig. 6. Effects of in-plane restraint on the non-linear response of FGM shallow
spherical shells under uniform external pressure.

0.25
Movable

0.06
0.2

perfect
0.05

imperfect (μ = 0.1)
R/h = 80, k = 1.0

0.04

0.15
1: a/R = 0.3
qu & ql (GPa)

q (GPa)

2: a/R = 0.4
3: a/R = 0.5

0.03
3

0.02

Immovable
R/h = 80, k = 1.0

0.1
qu

0.05
0

2
0.01

ql

−0.05

1
0

−0.1

−0.01

−0.15
0

1

2

3
W/h

4

5

6

Fig. 5. Effects of radius of base-to-curvature radius ratio a/R on the non-linear
response of FGM shallow spherical shells.

0.2

0.4

0.6

0.8

1

a/R
Fig. 7. Effects of the a/R ratio on the upper and lower buckling loads of FGM
shallow spherical shells.


1202

D. Huy Bich, H. Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204

value represented by intersection of curves with q axis. The
bifurcation point external pressure depending on the temperature
difference may be predicted by the last term in Eq. (43). This
behavior of the spherical shells can be explained as follows. The
temperature field makes the shell to deflect outwards (negative
deflection) prior to application of mechanical load. With the action
of uniform external pressure, the outward deflection is reduced and
when external pressure exceeds bifurcation point load an inward
deflection occurs. The enhancement of temperature difference is
accompanied by the increase in bifurcation points, load bearing
capability in the small region of deflection and the intensity of snapthrough behavior of the spherical shells. In addition, it is interesting
to note that there exists an intersection point where all qðW Þ curves
with various values of temperature difference pass. The coordinates
n
n

of this node can be determined from Eq. (43) are ðW ,qðW ÞÞ, where
n
2
W ¼ 7Rh Ra =20Àm which is independent of thermal loading
conditions.

Fig. 9 analyzes the effects of through the thickness temperature gradient on the non-linear response of clamped immovable
FGM shallow spherical shells. In this figure, the curves are plotted
for three various values of temperature at ceramic surface Tc
(¼ 27, 400 and 800 1C), whereas temperature at metal surface is
retained at Tm ¼27 1C (room temperature). It seems that bifurcation points are lower and the intensity of snap-through is weaker
under temperature gradient in comparison with their uniform
temperature counterparts. Subsequently, Fig. 10 assesses the
effects of the side of gradient on the behavior of immovable edge
FGM perfect spherical shells under uniform external pressure.
Beside the present model of FGM spherical shell, another type of
FGM shell with interchanged properties whose inner surface (i.e.
concave side) is ceramic-rich and outer surface (i.e. convex side)
is metal-rich is considered. The temperature is conducted through
the thickness from ceramic pure surface while metal surface
temperature is maintained at reference value. It is shown that

0.08

0.1

ceramic−rich outer surface
ceramic−rich inner surface

perfect

imperfect (μ = 0.1)

0.08

0.06
R/h = 80, a/R = 0.3, k = 1.0
2: ΔT = 200°C

3
0.04

q (GPa)

q (GPa)

R/h = 80, a/R = 0.3, k = 1.0, μ = 0

1: ΔT = 0

0.06

3: ΔT = 400°C

2

1: Tc = 400°C

2

0.04

2: Tc = 800°C

1
0.02

1
0.02
1

2
0

0
3

−0.02

−0.02
0

1

2

3

4

5

0

1

2

W/h
Fig. 8. Effects of temperature field on the non-linear response of FGM shallow
spherical shells under uniform external pressure.

4

5

Fig. 10. Effects of the side of temperature gradient on the non-linear response of
FGM shallow spherical shells (immovable edge).

0.03

0.1

R/h = 80, a/R = 0.3, k = 1.0

perfect
imperfect (μ = 0.1)

0.08

1: μ = − 0.3

0.025

2: μ = − 0.2

R/h = 80, a/R = 0.3, k = 1.0
0.06

0.02

1: Tc = 27°C
q (GPa)

q (GPa)

3
W/h

2: Tc = 400°C
0.04

3

3: Tc = 800°C

1

2

3: μ = − 0.1

3
6
7

0.015

4: μ = 0

54
21

7: μ = 0.3

0.01

0.02
1

5: μ = 0.1
6: μ = 0.2

2

0

0.005

3

0

−0.02
0

1

2

3

4

5

W/h
Fig. 9. Effects of temperature gradient on the non-linear response of FGM shallow
spherical shells under external pressure.

0

1

2

3

4

5

W/h
Fig. 11. Effects of imperfection on the non-linear response of FGM shallow
spherical shells (movable edge).


D. Huy Bich, H. Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204

0.08

0.07
R/h = 80, a/R = 0.3, k = 1.0

1: μ = 0, ΔT = 0

1: μ = − 0.3

0.06
0.05

2: μ = 0.1, ΔT = 200

0.06

2: μ = − 0.2

3: μ = 0.2, ΔT = 400

5

3: μ = − 0.1

0.04

1

4
7

0.03

6

5

4: μ = − 0.1, ΔT = 200

4: μ = 0
q (GPa)

q (GPa)

1203

5: μ = 0.1

2
3

6: μ = 0.2

3

0.04

5: μ = − 0.2, ΔT = 400

4
2
0.02

7: μ = 0.3

1

0.02
0
0.01
R/h = 80, a/R = 0.3, k = 1.0
−0.02

0
0

1

2

3

4

0

1

2

W/h

both bifurcation point loads and the intensity of snap-through
behavior are increased when temperature is conducted from
concave side. In contrast, bifurcation points become lower and
the postbuckling behavior of the shells is milder when conduction
is performed from the convex side. These thermomechanical nonlinear behaviors of FGM shallow spherical shells are similar in
trend to the response of FGM cylindrical panels subjected to
simultaneous action of external pressure and temperature conditions reported in [19].
The effects of initial imperfection on the non-linear axisymmetric response of externally pressurized FGM spherical shells
with movable and immovable clamped edge are illustrated in
Figs. 11 and 12, respectively. In these figures, the non-linear
equilibrium paths are plotted for various values of imperfection
size m in which negative or positive imperfections produce
perturbations in the spherical shell geometry that move the
central region of a shell outward or inward, respectively [21]. As
can be seen, the load carrying capacity of shell in the small range
of deflection is reduced when positive imperfection increases. In
contrast, the capability of load bearing of spherical shells is
remarkably improved due to the presence of negative initial
imperfection. However, accompanied with this enhancement of
upper buckling loads and load–deflection curves is an unstable

postbuckling behavior.
Fig. 13 depicts the interactive effects of imperfection and
temperature field on the thermomechanical response of FGM
spherical shells. It is shown that the perfect spherical shells
without the effect of environment temperature exhibit a more
benign snap-through response and a more stable postbuckling
behavior. In contrast, the presence of temperature field is followed by both higher bifurcation buckling loads and an increase
in the intensity of snap-through, even though the shells possessing inward imperfection. Specially, the bifurcation buckling
external pressures of the negative imperfect spherical shells are
considerably higher than their positive imperfect counterparts in
the same temperature condition and the amplitude of imperfection. Nevertheless, this increase in the buckling load is paid by a
very severe snap-through response and, as a result, very unstable
equilibrium paths from the postbuckling point of view. This fact
may be explained that the initial imperfection in conjunction with
temperature field produces an effective imperfection [21] and the
amplitude of this effective imperfection is enhanced due to high

4

5

Fig. 13. Interactive effects of imperfection and temperature field on the non-linear
response of FGM shallow spherical shells (immovable edge).

2500
perfect
imperfect (μ = 0.1)
2000
R/h = 80, a/R = 0.3, k = 1.0

ΔT (°C)

Fig. 12. Effects of imperfection on the non-linear response of FGM shallow
spherical shells (immovable edge).

3
W/h

1500
3
2

1000
1: q = 0

1
2: q = 0.05 (GPa)

500

3: q = 0.1 (GPa)
0
−3

−2.5

−2

−1.5
W/h

−1

−0.5

0

Fig. 14. Effects of pre-existent external pressure on the thermal non-linear
response of FGM shallow spherical shells (immovable edge).

temperature difference and negative initial imperfection. It seems
that the real curvature of spherical shell is increased because
of combination between temperature field and negative
imperfection.
Finally, Fig. 14 assesses the effects of pre-existent external
pressure on the thermal non-linear response of immovable edge
FGM spherical shells exposed to temperature field. Shown in this
figure is a monotonically increasing non-linear response with
outward deflection of the shells. The shells exhibit a bifurcation
buckling behavior when they are subjected to external pressure
prior to the application of thermal load. Both the bifurcation
buckling loads and the capability of temperature resistance are
enhanced with the increase in pre-existent uniform external
pressure. However, with all values of mechanical load, the
thermal postbuckling behavior is very stable, i.e. without a
snap-through. In fact, the shell deflects inwards under external
pressure and when temperature reaches a specific value, i.e.
bifurcation point temperature, the shell surface returns to initial

1204

D. Huy Bich, H. Van Tung / International Journal of Non-Linear Mechanics 46 (2011) 1195–1204

state. When the temperature exceeds bifurcation point, the
spherical shell is monotonically deflected outwards. In addition,
Fig. 14 also shows that the effect of initial imperfection on the
non-linear behavior of the shells under this condition of loading is
almost immaterial.

6. Concluding remarks
This paper presents an analytical approach to investigate the
non-linear axisymmetric response of FGM shallow spherical
shells subjected to uniform external pressure with and without
including the effects of temperature conditions. The effective
properties of functionally graded material are expressed as power
functions of thickness variable whereas the properties of constituents are assumed to be temperature-independent. Formulation
for axisymmetrically deformed spherical shells is based on the
classical shell theory with both geometrical non-linearity and
initial geometrical imperfection are incorporated. One-term
deflection mode is approximately assumed and explicit expressions of buckling loads and load–deflection curves for a clamped
spherical shell under mentioned loads are determined by applying Galerkin procedure. From these explicit expressions, the nonlinear axisymmetric response of the shells is analyzed and the
results are illustrated in graphic form. The results shows that the
non-linear response of the FGM spherical shells is complex and
greatly influenced by the material and geometric parameters and
in-plane restraint. The study also reveals important role of preexistent temperature conditions and weak effect of initial imperfection on the non-linear response of FGM shallow spherical
shells under uniform external pressure.

Acknowledgements
This work was supported by the National Foundation for

Science and Technology Development of Vietnam – NAFOSTED,
project code 107.02.07.09. The authors are grateful for this
financial support.

References
[1] D.O. Brush, B.O. Almroth, Buckling of Bars, Plates and Shells, McGraw-Hill,
New York, 1975.
[2] N.C. Huang, Unsymmetrical buckling of thin shallow spherical shells, J. Appl.
Mech. Trans. ASME 31 (1964) 447–457.
[3] S.C. Tillman, On the buckling behavior of shallow spherical caps under a
uniform pressure load, Int. J. Solids Struct. 6 (1970) 37–52.
[4] M. Uemura, Axisymmetrical buckling of an initially deformed shallow
spherical shell under external pressure, Int. J. Nonlinear Mech. 6 (1971)
177–192.
[5] N. Nath, R.S. Alwar, Non-linear static and dynamic response of spherical
shells, Int. J. Nonlinear Mech. 13 (1978) 157–170.
[6] P.C. Dumir, Nonlinear axisymmetric response of orthotropic thin spherical
caps on elastic foundations, Int. J. Mech. Sci. 27 (1985) 751–760.
[7] C.C. Chao, I.S. Lin, Static and dynamic snap-through of orthotropic spherical
caps, Composite Struct. 14 (1990) 281–301.
[8] C.S. Xu, Buckling and post-buckling of symmetrically laminated moderately
thick spherical caps, Int. J. Solids Struct. 28 (1991) 1171–1184.
[9] A. Muc, Buckling and postbuckling behavior of laminated shallow spherical
shells subjected to external pressure, Int. J. Nonlinear Mech. 27 (3) (1992)
465–476.
[10] R.S. Alwar, M.C. Narasimhan, Axisymmetric non-linear analysis of laminated
orthotropic annular spherical shells, Int. J. Nonlinear Mech. 27 (4) (1992)
611–622.
[11] G.P. Dube, S. Joshi, P.C. Dumir, Nonlinear analysis of thick shallow spherical
and conical orthotropic caps using Galerkin’s method, Appl. Math. Modelling

25 (2001) 755–773.
[12] G.H. Nie, Asymptotic buckling analysis of imperfect shallow spherical shells
on non-linear elastic foundation, Int. J. Mech. Sci. 43 (2001) 543–555.
[13] M.R. Eslami, H.R. Ghorbani, M. Shakeri, Thermoelastic buckling of thin
spherical shells, J. Therm. Stresses 24 (2001) 1177–1198.
[14] R. Shahsiah, M.R. Eslami, Thermal and mechanical instability of an imperfect
shallow spherical cap, J. Therm. Stresses 26 (7) (2003) 723–737.
[15] Q.S. Li, J. Liu, J. Tang, Buckling of shallow spherical shells including the effects
of transverse shear deformation, Int. J. Mech. Sci. 45 (2003) 1519–1529.
[16] R. Shahsiah, M.R. Eslami, R. Naj, Thermal instability of functionally graded
shallow spherical shell, J. Therm. Stresses 29 (8) (2006) 771–790.
[17] T. Prakash, N. Sundararajan, M. Ganapathi, On the nonlinear axisymmetric
dynamic buckling behavior of clamped functionally graded spherical caps,
J. Sound Vibrat. 299 (2007) 36–43.
[18] M. Ganapathi, Dynamic stability characteristics of functionally graded
materials shallow spherical shells, Composite Struct. 79 (2007) 338–343.
[19] N.D. Duc, H. Van Tung, Nonlinear response of pressure-loaded functionally
graded cylindrical panels with temperature effects, Composite Struct. 92
(2010) 1664–1672.
[20] H. Van Tung, N.D. Duc, Nonlinear analysis of stability for functionally graded
plates under mechanical and thermal loads, Composite Struct. 92 (2010)
1184–1191.
[21] L. Librescu, M.P. Nemeth, J.H. Starnes, W. Lin, Nonlinear response of flat and
curved panels subjected to thermomechanical loads, J. Therm. Stresses 23
(2000) 549–582.