Nonlinear analysis of an imperfect radially graded annular plate with a heated edge

Nonlinear analysis of an imperfect radially graded annularplate with a heated edge

P. A. Kadam • Satyajit Panda

Received: 29 August 2013 / Accepted: 3 March 2014

� Springer Science+Business Media Dordrecht 2014

Abstract The present work deals with a geometri-

cally nonlinear finite element analysis of an imperfect

radially graded annular plate with a heated edge. The

geometrical imperfection of the graded annular plate is

assumed in aspect of its little intrinsic transverse

deflection. The analysis is mainly for investigating the

effects of intrinsic geometrical imperfection and

temperature in the graded annular plate on its nonlin-

ear flexural behaviour under a transverse mechanical

load. The temperature is uniformly distributed across

the thickness of the plate and it varies along the radial

direction only. The temperature-dependent material

properties are radially graded according to a simple

power-law that is formed by power-law exponent and

material properties of constituent materials (ceramic

and metal). Based on the Von Karman nonlinear

strain–displacement relations for imperfect annular

plates, the nonlinear finite element equations of

equilibrium are derived employing the principle of

minimum potential energy. A single nodal displace-

ment-control solution strategy is described for numer-

ical solutions of nonlinear finite element equations of

equilibrium. The numerical illustrations show a sig-

nificant role of geometrical imperfection of the

annular plate for its unstable equilibrium and

alteration of structural behaviour under thermo-

mechanical load. The analysis reveals the usefulness

of radially graded annular plate in order to mitigate the

unstable equilibrium of imperfect monolithic annular

plate under thermo-mechanical load. It is found that

the radial location for maximum value of a stress

component insignificantly depends on the magnitudes

of power-law exponent and applied temperature. The

effects of material properties and applied temperature

on the critical mechanical load corresponding to the

unstable equilibrium of the imperfect radially graded

annular plate are also presented.

Keywords Radially graded annular plate �Geometrically nonlinear analysis �Elastic instability

1 Introduction

Composite materials are inevitable in the design of

advanced structures because of certain benefits of high

stiffness to weight ratio, high fatigue strength, etc.

Although those advantages can be achieved by the use

of laminated composite structures, but the mismatch of

the material properties at the interface of two adjacent

layers causes their limited applications under the

severe environment. Under the high thermal environ-

ment, the thermal stresses in the laminated composite

structures may cause their delamination mainly

because of the inconsistency of material properties at

P. A. Kadam � S. Panda (&)

Department of Mechanical Engineering, Indian Institute

of Technology Guwahati, North Guwahati, Guwahati

781039, Assam, India

e-mail: [email protected]

123

Int J Mech Mater Des

DOI 10.1007/s10999-014-9249-y

the interface of two adjacent layers. In order to

alleviate this kind of drawback of laminated structures,

the concept of functionally graded material (FGM)

emerged (Koizumi 1993). FGMs are microscopically

heterogeneous materials. In the macroscopic scale,

those materials are attributed by smooth and contin-

uous variation of material properties along any or all of

the reference coordinate-axes. FGMs are extensively

utilized in the design of rectangular plates under the

high thermal environment. Noda (1999) analysed a FG

rectangular plate for investigating the effects of

various types of cracks in the plate on the thermal

stresses. Reddy and Chin (1998) presented dynamic

thermo-elastic responses of FG cylinders and plates

subjected to thermal load. Praveen and Reddy (1998)

investigated the static and dynamic responses of FG

plates by varying the volume fractions of the ceramic

and metallic constituents. Woo and Meguid (2001)

provided analytical solutions for the large deflection of

FG plates and shallow shells under transverse mechan-

ical loads and a temperature field. Shen (2002)

presented the nonlinear bending analysis of a sim-

ply-supported FG rectangular plate subjected to a

transverse uniform or sinusoidal mechanical load in

thermal environment. Yang and Shen (2003) reported

a nonlinear bending analysis of shear deformable FG

plates subjected to thermo-mechanical loads. Lanhe

(2004) derived the equilibrium and stability equations

of a moderately thick FG rectangular plate under

thermal load. Shariat and Eslami (2007) performed a

buckling analysis of thick rectangular FG plates under

mechanical and thermal loads. Wu et al. (2007)

presented analytical solutions for the post-buckling

response of the FG plate subjected to thermal and

mechanical loads. Zhao et al. (2009) performed the

buckling analysis of FG plates subjected to mechan-

ical and thermal loads. Lee et al. (2010) presented

post-buckling analysis of FG plates under in-plane

compressive load in the presence of a temperature

field. Jagtap et al. (2012) studied the stochastic

nonlinear bending response of heated FG plates with

uncertain system properties.

Similar to rectangular plates, annular and circular

plates are also equally important for their wide

applications in many engineering fields. For their

applications under a varying temperature field across

the thickness of the plate, the concept of FGM is also

substantially utilized. Ma and Wang (2003) studied

axisymmetric large bending deflection of a FG circular

plate under mechanical, thermal and combined thermo-

mechanical loads. In the same report, the axisymmetric

thermal post-buckling behaviour of the FG circular

plate is also studied. Najafizadeh and Heydari (2004)

derived thermo-elastic equilibrium and stability equa-

tions of a FG circular plate using various shear

deformation theories. Prakash and Ganapathi (2006)

developed a finite element model of FG circular plate

for investigating its asymmetric free vibration charac-

teristics and thermo-elastic stability. Allahverdizadeh

et al. (2006) investigated variations of stresses in a thin

circular FG plate due to its large amplitude vibration

under thermal environment. Bayat et al. (2007) studied

the axisymmetric bending of a FG rotating disk under

steady-state thermal load. This study demonstrates the

effects of material properties and thermo-mechanical

load on the stress field in the disk with roller-supported

boundary conditions. Jalali et al. (2010) presented the

thermal stability of laminated FG circular plate of

variable thickness for uniform rise of temperature. The

analysis reveals that the thermal stability of the FG

circular plate is significantly influenced by the thick-

ness variation profile, aspect ratio, volume fraction

index and the core-to-face sheet thickness ratio. Sepahi

et al. (2010) investigated the effects of three-parameter

elastic foundations and thermo-mechanical load on the

axisymmetric large deflection of a simply-supported

annular FG plate. Alibeigloo (2012) analysed a FG

solid and an annular plate subjected to thermo-

mechanical load with various boundary conditions. In

this analysis, the effects of temperature change,

mechanical load, gradient index, boundary conditions

and thickness to radial length ratio on the behavior of

the plate are examined. Fallah and Nosier (2012)

derived the nonlinear equilibrium equations of FG

circular plates under asymmetric transverse loading and

heat conduction through the thickness of the plate. The

analysis shows the snap-through buckling behavior of

the simply-supported FG circular plates which are

immovable along the radial direction under the thermo-

mechanical load. Reddy and Berry (2012) formulated a

microstructure-dependent nonlinear theory for axisym-

metric thermo-elastic bending of FG circular plates

based on a modified stress theory. Golmakani and

Kadkhodayan (2014) investigated the axisymmetric

bending and stretching of circular/annular FG plate

with variable thickness when the plate is subjected to

combined thermo-mechanical loads with various

boundary conditions.

P. A. Kadam, S. Panda

123

The application of FGM for the design of annular/

circular plates is also continued for the case where the

temperature field varies along the radial direction. In

this case, the material properties of the annular/

circular FG plate are graded along the radial direction

while its edge of high temperature is made of ceramic

material. Although a significant number of reports on

the analysis of radially graded annular/circular plates

are available in the literature, but the thermo-elastic

analysis corresponding to the varying temperature

field across the radial span of the plate has been carried

out in few reports. Bakhshi et al. (2006) studied the

thermo-elastic response of radially graded annular

disk subjected to the axisymmetric thermal shock at its

inner surface. Bayat et al. (2009) presented a thermo-

elastic analysis of axisymmetric rotating disks made of

FGM with variable thickness. The temperature-depen-

dent material properties of the disk are graded in the

radial direction. This study addresses the effects of the

thermal field, the material grading index and the

geometry of the disk on its stress field. Chiba (2009)

presented the mean and variance of the temperature in

a FG annular disk with spatially random heat transfer

coefficients on the upper and lower surfaces. The

material properties of the disc are radially graded and

it is subjected to deterministic axisymmetrical heating

at the lateral surfaces. Nie and Batra (2010) studied

axisymmetric deformations of a rotating disk with the

radially varying thickness, mass density, thermal

expansion coefficient and shear modulus. The report

presents the effects of material inhomogeneities on

deformations of a hollow and a solid rotating disk.

Peng and Li (2010) presented an analytical method to

investigate steady thermal stresses in a radially graded

circular hollow disk rotating with constant angular

velocity about its central axis. The hollow disk is

assumed to have varying material properties along the

radial direction. Sepahi et al. (2011) presented

analytical solutions for thermal buckling of FG

annular plates with the varying material properties

along the radial direction.

In the available reports on the thermo-elastic

analysis of radially graded annular plates with a

heated edge, the annular plates are assumed as

geometrically perfect annular plates. Since the tem-

perature at one of the edges of the graded annular plate

usually causes its thermal buckling, the intrinsic

transverse deflection of the plate due to its geometrical

imperfection is an important factor. It (imperfection)

is a usual discrepancy in most of the practical plates.

So, it is a necessary factor for the thermo-elastic

analysis of radially graded annular plates with a heated

edge. Moreover, a plate is subjected to transverse

mechanical loads in most of its engineering applica-

tions. To the best knowledge of authors, the thermo-

elastic analysis of a geometrically imperfect radially

graded annular plate under a transverse mechanical

load in the presence of a high temperature at one of its

edges is not yet addressed in the available literature.

Thus, a nonlinear thermo-elastic analysis of a geo-

metrically imperfect radially graded annular plate is

performed at present. The analysis is mainly for

investigating the flexural deformation characteristics

and elastic instability of a radially graded annular plate

when it operates under a transverse mechanical load in

the presence of a high temperature at one of its edges.

In order to attribute the effect of the geometrical

imperfection of the annular plate on its thermo-elastic

flexural deformations for both low and high temper-

atures, the geometrically nonlinear analysis of the

plate is carried out. One of the edges of the imperfect

graded annular plate is considered to be exposed to a

high temperature while the other edge remains at room

temperature. The geometrical imperfection of the

graded annular plate is assumed in aspect of its little

intrinsic transverse deflection. A nonlinear thermo-

elastic finite element model of an imperfect radially

graded annular plate is developed based on the Von

Karman nonlinear strain–displacement relations. The

temperature is assumed as uniformly distributed

across the thickness of the plate while it varies along

the radial direction only. The temperature-dependent

material properties of the annular plate are radially

graded according to a simple power-law that is formed

by power-law exponent and material properties of

constituent materials (ceramic and metal). For the

numerical solutions of nonlinear finite element equa-

tions of equilibrium, the direct iteration method is

utilized in combination with a new single nodal

displacement-control solution strategy. The numerical

results are evaluated either for inner or for outer

ceramic rich edge of the graded annular plate. The

numerical results illustrate the thermo-elastic insta-

bility and the alteration of structural behaviour of the

graded annular plate under the mechanical load due to

the significant effect of its intrinsic geometrical

imperfection. Since the geometrical impaction of the

annular plates is unavoidable in practice, the analysis

Nonlinear analysis of an imperfect radially graded annular plate

123

shows a possible way to mitigate its detrimental effect

on their (plates) thermo-elastic equilibrium. The

effects of material properties and temperature on the

critical mechanical load corresponding to the thermo-

elastic instability of the imperfect radially graded

annular plate are also presented.

2 Nonlinear finite element formulation

Figure 1 shows a radially graded annular plate made

of ceramic and metal constituent materials. Among the

inner and outer vertical surfaces of the annular plate,

one is fully ceramic surface while the other is fully

metallic surface. The material properties are assumed

to vary along the radial direction only. The middle

plane of the annular plate is considered as the

reference plane and the centre of the middle plane is

the origin of the reference cylindrical coordinate

system (r, h, z). The inner and outer radii of the annular

plate are denoted by, ri and ro, respectively. The

thickness of the plate is represented by the symbol,

h. Since a thin annular plate is considered in the

present study and the equivalent single layer theory is

used for the mathematical formulation of the problem,

the inner and outer vertical surfaces of the annular

plate are denoted as the inner and outer edges of the

same plate. The ceramic rich edge of the radially

graded annular plate is considered to be exposed to a

high temperature (Tc) and the metal rich edge of the

same remains in room temperature Tm = 300 K. A

plane parallel to the inner and outer vertical surfaces of

the annular plate is assumed as an isothermal plane and

the temperature varies along the radial direction only.

The shape of the reference plane due to the intrinsic

geometrical imperfection of the annular plate is

defined by a mathematical function ð �wðr; hÞÞ: The

Von Karman nonlinear strain–displacement relations

for a geometrically imperfect annular plate can be

written as follows (Chia 1980),

er ¼ou

orþ 1

2

ow

or

� �2

þ o �w

or

ow

or;

eh ¼u

rþ 1

r

ov

ohþ 1

2r2

ow

oh

� �2

þ 1

r2

o �w

ohow

oh;

erh ¼1

r

ou

ohþ ov

or� v

rþ 1

r

ow

or

ow

ohþ 1

r

o �w

or

ow

ohþ 1

r

o �w

ohow

or;

erz ¼ou

ozþ ow

or; ehz ¼

ov

ozþ 1

r

ow

oh; ð1Þ

where u, v and w are the displacements at any point

within the domain of the annular plate along r, h and

z directions, respectively. In the present formulation,

the circumferential direction at a radial location (r) is

denoted by, y. According to this notation, the circum-

ferential co-ordinate of a point at an angular location

(h) is, y = (r 9 h). Now, the variation of any function

f(r, h, z) with respect to y co-ordinate can be written as,

of=oy¼fðof=orÞ�ðor=oyÞgþfðof=ohÞ�ðoh=oyÞgþfðof=ozÞ�ðoz=oyÞg: Since y co-ordinate is defined at

a particular radius (r), the various differential terms

are, oh=oy¼1=r;oz=oy¼0 and or=oy¼0: Using those

differential terms, the operator ðo=oyÞ can be written

as, ð1=r�o=ohÞ: Introducing this operator ðo=oyÞ in

Eq. (1), the following nonlinear strain–displacement

relations can be obtained,

er ¼ou

orþ 1

2

ow

or

� �2

þ o �w

or

ow

or;

ey ¼u

rþ ov

oyþ 1

2

ow

oy

� �2

þ o �w

oy

ow

oy;

ery ¼ou

oyþ ov

or� v

rþ ow

or

ow

oyþ o �w

or

ow

oyþ o �w

oy

ow

or;

erz ¼ou

ozþ ow

or; eyz ¼

ov

ozþ ow

oy:

ð2Þ

Since a thin radially graded annular plate is considered

in thepresentanalysis, thekinematicsofdeformationof this

plate is defined according to the first order shear deforma-

tion theory (FSDT). According to this theory, the in-plane

displacements (u and v) at any point are as follows,

uðr; y; zÞ ¼ u0ðr; yÞ þ z/rðr; yÞ and

vðr; y; zÞ ¼ v0ðr; yÞ þ z/yðr; yÞ; ð3Þ

where u0 and v0 are the translational displacements at

any point on the reference plane along the radial and

r

θzo

ro

ri

h

Fig. 1 Schematic diagram of a radially graded annular plate


123

circumferential directions, respectively, /r and /y are

rotations of a normal to the reference plane with

respect to circumferential and radial axes. In case of

the thin annular plates, the rate of change of transverse

deformation with respect to the thickness co-ordinate

(z) is negligibly small. Thus, the transverse deflection

(w) at any point of the annular plate can be assumed as,

wðr; y; zÞ ¼ w0ðr; yÞ; ð4Þ

where w0 is the transverse deflection at any point on

the reference plane. The generalized displacements

(u0, v0, w0, /r, /y) can be expressed in the form of a

generalized displacement vector ({d}) as follows,

fdg ¼ u0 v0 w0 /r /y

� �T: ð5Þ

The state of stress at any point of the annular plate

can be written as,

rbf g ¼ rr ry sry

� �Tand rsf g ¼ srz syz

� �T; ð6Þ

where rr and ry are the normal stresses at any point of

the annular plate along the radial and circumferential

directions, respectively, sry is the in-plane shear stress

in the ry plane, srz and syz are the transverse shear

stresses in the rz and yz planes, respectively. Similarly,

the state of strain at any point of the annular plate can

be written as,

ebf g ¼ er ey ery

� �Tand esf g ¼ erz eyz

� �T; ð7Þ

where er and ey are the normal strains at any point of

the annular plate along the radial and circumferential

directions, respectively, ery is the in-plane shear strain

in the ry plane, erz and eyz are the transverse shear

strains in the rz and yz planes, respectively. Using Eqs.

(2)–(4), the strain vectors (Eq. (7)) can be written as,

ebf g¼ eLb

� �þ eN

b

� �þ z jbf g;

eLb

� �¼ ou0

orþo �w

or

ow0

or

u0

rþov0

oyþo �w

oy

ow0

oy

ou0

oy

�

þov0

or� v0

rþo �w

or

ow0

oyþo �w

oy

ow0

or

T

;

eNb

� �¼ 1

2

ow0

or

� �21

2

ow0

oy

� �2ow0

or

� �ow0

oy

� �" #T

;

jbf g¼o/r

or

/r

rþ

o/y

oy

o/r

oyþ

o/y

or�

/y

r

� T

;

esf g¼ow0

orþ/r

ow0

oyþ/y

� T

:

ð8Þ

The material properties of the radially graded

annular plate can be assumed as (Reddy and Chin

1998),

Pðr; TÞ ¼ PcðTÞ � VcðrÞ þ PmðTÞ � VmðrÞ; ð9Þ

where Pc(T) and Pm(T) are temperature-dependent

material properties of ceramic and metal constituent

materials, respectively, Vc(r) and Vm(r) are volume

fractions of ceramic and metal constituent materials.

The volume fractions of constituent materials vary

along the radial direction according to a simple power-

law as follows,

Vc ¼1

2þ ð�1Þk 2r � ðro þ riÞ

2ðro � riÞ

� �n

;

Vm ¼ 1� Vcð Þ; ð10Þ

where n (0 B n B ?) is the volume fraction index and

k is a positive integer. For odd values of k, the inner

edge of the annular plate is ceramic rich and the outer

edge of the same plate is metal rich. The reverse holds

for even values of k. Thus, the inner edge of the

radially graded annular plate can be modelled either as

ceramic rich edge or as metal rich edge according to

the value of k as 1 or 2, respectively. Substituting

Eq. (10) in Eq. (9), the final expression for the material

properties can be obtained as follows,

Pðr; TÞ ¼ PmðTÞ þ PcðTÞ � PmðTÞh i

� 1

2þ ð�1Þk 2r � ðro þ riÞ

2ðro � riÞ

� �n

: ð11Þ

Equation (11) defines all material properties like

Young’s modulus (E(r, T)), Poisson’s ratio (m(r, T)),

coefficient of thermal expansion (a(r, T)), thermal

conductivity (j(r, T)), etc. However, the constitutive

relations for the graded annular plate can be written as,

rbf g ¼ Cb½ � ebf g � fagDTð Þ; rsf g ¼ Cs½ � esf g;

Cb½ � ¼E

1� m2

1 m 0

m 1 0

0 0 ð1� mÞ=2

264

375;

Cs½ � ¼E

1þ m

1=2 0

0 1=2

� ;

fag ¼ ar ay 0½ �T ; DT ¼ TðrÞ � Tref ; ð12Þ

where ar and ay are the coefficients of thermal

expansion along the radial and circumferential direc-

tions, respectively, Tref is the reference temperature that


123

is considered as the room temperature (Tref = Tm);

T(r) is the temperature distribution along the radial

direction. The temperature distribution (T(r)) along the

radial direction can be obtained by solving one

dimensional heat conduction equation as follows,

1

r

d

drrjðr; TÞ dTðrÞ

dr

� �¼ 0: ð13Þ

Corresponding to a boundary condition ðTðrÞjr¼ri¼

Ti and TðrÞjr¼ro¼ ToÞ; the solution of Eq. (13) can be

written as,

TðrÞ ¼ Ti þ ðTo � TiÞf ðr; TÞ;

f ðr; TÞ ¼

R r

ri

1rjðr; TÞ

�dr

R ro

ri

1rjðr; TÞ

�dr;

Ti ¼ Tc and To ¼ Tm for k ¼ 1;

Ti ¼ Tm and To ¼ Tc for k ¼ 2: ð14Þ

The graded annular plate is subjected to a uniformly

distributed transverse mechanical load in the presence

of a high temperature at one of its edges. Thus, the first

variation of the total potential energy of this plate can

be written as,

dTp ¼Zro

ri

Z2pr

0

" Zh=2

�h=2

debf gT rbf g þ desf gT rsf g�

dz:

�ðdw� pÞ#

dydr; ð15Þ

where d is an operator for first variation and p is the

intensity of applied uniformly distributed transverse

mechanical load. Substituting Eqs. (12), (8) and (14) in

Eq. (15), the first variation of the total potential energy

can be written as,

where the rigidity matrices ([Ab], [As], [Bb], [Db]),

different vectors ({AT}, {BT}) and temperature gradi-

ent (DTg) are as follows,

Ab½ � ¼Zh=2

�h=2

Cb½ �dz; As½ � ¼Zh=2

�h=2

sf Cs½ �dz;

Bb½ � ¼Zh=2

�h=2

Cb½ �zdz; Db½ � ¼Zh=2

�h=2

Cb½ �z2dz;

ATf g ¼Zh=2

�h=2

½1 1 0 �T EðrÞ1� mðrÞaðrÞ 1� f ðr; TÞh idz

fork¼ 1;

ATf g ¼Zh=2

�h=2

½1 1 0 �T EðrÞ1� mðrÞaðrÞ f ðr; TÞh idz

fork¼ 2;

BTf g ¼Zh=2

�h=2

½1 1 0 �T EðrÞ1� mðrÞaðrÞ 1� f ðr; TÞh izdz

fork¼ 1;

BTf g ¼Zh=2

�h=2

½1 1 0 �T EðrÞ1� mðrÞaðrÞ f ðr; TÞh izdz

fork¼ 2;

DTg ¼ Tc� Tmð Þ:ð17Þ

In Eq. (17), sf is the shear correction factor and its value

is 5/6. For the analysis, a finite element mesh over the

domain of the reference plane is generated using the nine-

nodded isoparametric element. In order to generate this

mesh, the radial span (ro – ri) of the annular plane is

divided into m1 number of equal divisions while the

circumference of the same plane is divided into n1 number

of equal divisions. Thus, the finite element mesh over the

domain of the reference plane consists of (m1 9 n1)

number of elements. However, the generalized displace-

ment vector (Eq. (5)) for the ith node of an element is,

dTp ¼Zro

ri

Z2pr

0

deLb

� �Tþ deNb

� �TD E

Ab½ � eLb

� �þ eN

b

� �� þ Bb½ � jbf g � ATf gDTg

� �þ djbf gT

Bb½ � eLb

� �þ eN

b

� �� þ Db½ � jbf g � BTf gDTg

� �þ desf gT

As½ � esf g� �

� dw0 � pð Þ

2664

3775dydr; ð16Þ


123

dif g ¼ u0i v0i w0i /ri /yi

� �T;

ði ¼ 1; 2; 3; . . .; 9Þ:ð18Þ

The generalized displacement vector ({d}) at any

point within an element is,

fdg ¼ ½N� def g; ð19Þ

where the generalized elemental nodal displacement

vector ({de}) and the shape function matrix ([N]) are

given by,

def g ¼ d1f gTd2f gT

d3f gT . . . d9f gT� �T

;

½N� ¼ N1½ � N2½ � N3½ �. . . N9½ �½ �; Ni½ � ¼ ni½I�;ð20Þ

where ni is the shape function of the natural coordi-

nates associated with the ith node and [I] is an identity

matrix. Using Eq. (19), the strain vectors

ðfeLbg; feN

b g; fjbg and fesgÞ and their first variations

ðfdeLbg; fdeN

b g; fdjbg; fdesgÞ can be written as,

eLb

� �¼ BL

b

� �def g; eN

b

� �¼ BN

b

� �def g;

jbf g ¼ Bj½ � def g; esf g ¼ Bs½ � def g;deL

b

� �¼ BL

b

� �ddef g; deN

b

� �¼ BdN

b

� �ddef g;

djbf g ¼ Bj½ � ddef g; desf g ¼ Bs½ � ddef g:

ð21Þ

In Eq. (21), the strain–displacement matrices

ð½BLb �; ½BN

b �; ½Bk�; ½Bs�; ½BdNb �Þ are given by,

BLb

� �¼ LL

b

� �½N�; BN

b

� �¼ LN

b

� �½N�; Bj½ � ¼ Lj½ �½N�;

Bs½ � ¼ Ls½ �½N�; BdNb

� �¼ LdN

b

� �½N�;

LLb

� �¼

oor

0 o �wor

oor

0 0

1r

ooy

o �woy

ooy

0 0

ooy

oor� 1

ro �wor

ooyþ o �w

oyoor

0 0

2664

3775;

LNb

� �¼

0 0 12

ow0

oroor

0 0

0 0 12

ow0

oyooy

0 0

0 0 ow0

orooy

0 0

26664

37775;

Lj½ � ¼

0 0 0 oor

0

0 0 0 1r

ooy

0 0 0 ooy

oor� 1

r

2664

3775;

Ls½ � ¼0 0 o

or1 0

0 0 ooy

0 1

" #;

LdNb

� �¼

0 0 ow0

oroor

0 0

0 0 ow0

oyooy

0 0

0 0 ow0

orooyþ ow0

oyoor

0 0

26664

37775: ð22Þ

Substituting Eq. (21) in Eq. (16), the first variation

of the total potential energy (dTp) for a typical element

can be obtained as,

dTep ¼ ddef gT

KeL

� �þ Ke

N

� �� def g � Fe

T

� �þ Fe

NT

� �� DTg � Fe

M

� �p

� �;

KeL

� �¼Zre

0

rei

Zye2

ye1

BLb

� �TAb½ � BL

b

� �þ BL

b

� �TBb½ � Bj½ � þ Bj½ �T Bb½ � BL

b

� �þ Bj½ �T Db½ � Bj½ � þ Bs½ �T As½ � Bs½ �

�dydr;

KeN

� �¼Zre

0

rei

Zye2

ye1

BLb

� �TAb½ � BN

b

� �þ BdN

b

� �TAb½ � BL

b

� �þ BdN

b

� �TAb½ � BN

b

� �þ BdN

b

� �TBb½ � Bj½ � þ Bj½ �T Bb½ � BN

b

� � �dydr;

FeT

� �¼Zre

0

rei

Zye2

ye1

BLb

� �TATf g þ Bj½ �T BTf g

�dydr; Fe

NT

� �¼Zre

0

rei

Zye2

ye1

BdNb

� �TATf g

�dydr;

FeM

� �¼Zre

0

rei

Zye2

ye1

ð½N�T ½ 0 0 1 0 0 �TÞdydr; ð23Þ


123

where rei and re

o are the inner and outer radial

boundaries of an element, ye1 and ye

2 are the circum-

ferential boundaries of an element. In Eq. (23), the

bending and shear counterparts of the total stiffness

matrix are derived separately for applying the rule of

selective integration in a straight forward manner.

Applying the principle of minimum potential energy

(dTp = 0), the following nonlinear equations of equi-

librium for a typical element can be obtained,

KeL

� �þ Ke

N

� �� def g¼ Fe

M

� �pþ Fe

T

� �þ Fe

NT

� �� DTg:

ð24ÞAssembling the elemental equations of equilibrium

(Eq. (24)) for the whole domain of the plate, the

following global equations of equilibrium can be

obtained,

KL½ � þ KN½ �ð ÞfXg ¼ FMf gpþ FTf g þ FNTf gð ÞDTg;

ð25Þ

where [KL] and [KN] are the linear and nonlinear

counterparts of the global stiffness matrix, {FT} and

{FNT} are the coefficient vectors corresponding to the

linear and nonlinear counterparts of the global thermal

load, {FM} is the coefficient vector for the global

mechanical load, {X} is the global nodal displacement

vector. Equation (25) represents a geometrically

nonlinear finite element model of an imperfect radially

graded annular plate under a uniformly distributed

transverse mechanical load and a temperature gradient

across the radial span of the plate. However, the

nonlinear finite element equations of equilibrium (Eq.

(25)) can also be expressed as,

fXg ¼ f M� �

pþ f T� �

DTg;

f M� �

¼ ½K��1FMf g; f T

� �¼ ½K��1

FTf gþ FNTf gð Þ:ð26Þ

3 Method of solution

The present radially graded annular plate is subjected

to a uniformly distributed transverse mechanical load

(p) in the presence of a temperature gradient (DTg)

across its radial span. First, the solutions ({X}) for the

temperature gradient (DTg) are evaluated without

applying the mechanical load. Next, the variation of

its (plate) transverse deflection with the mechanical

load (p) is evaluated without alteration of the temper-

ature gradient (DTg). For the thermal load only

(DTg = 0, p = 0), the equations of equilibrium (Eq.

(26)) can be written as, {X} = {fT}DTg. These non-

linear equations are solved using the direct iteration

method. In case of a high temperature gradient, the

difficulty in the convergence of solutions may arise or

the number of iterations for the convergence of

solutions may increase. So, the solutions for an

applied temperature gradient (DTg) are evaluated in a

series of steps of temperature gradient. The applied

temperature gradient (DTg) is divided into a number

(q) of small steps ðDTsgÞ as, DTg ¼ q� DTs

g ðq ¼1; 2; 3; . . .Þ: Then, the solutions for every incremental

step (q) are obtained using direct iteration method. The

initial solutions at a particular step are taken as the

converged solutions of previous step. The initial

solutions at the first step (q = 1) may be taken as 0.

This consideration does not cause difficulty in the

convergence of solutions at the first step (q = 1)

because the corresponding temperature gradient

ðDTg ¼ DTsgÞ is very small as compared to the total

temperature gradient (DTg).

The deflection of the graded annular plate due to the

applied temperature gradient (DTg) is taken as the

initial deflection for further application of the mechan-

ical load. The variation of deflection of the graded

annular plate with the mechanical load may be

evaluated by taking the solution-control parameter as

the applied mechanical load. But this method of

solution may fail to evaluate the continuous response

curve because of the snap-through equilibrium of the

graded annular plate. Thus, the thermo-elastic

response of the graded annular plate within a range

of applied mechanical load is evaluated by taking the

solution-control parameter as an element (say, ith

element) of the nodal displacement vector ({X}). This

solution-control parameter may be designated as

single nodal displacement solution-control parameter

(SNDSCP). The solutions at every step of increment of

SNDSCP are obtained using the direct iteration

method. In order to avoid the difficulty in getting

converged solutions, the SNDSCP is considered to be

increased from its initial magnitude corresponding to

the solutions ({X}) for temperature gradient only

(p = 0). However, the computational procedure

for a step of increment of SNDSCP is illustrated as

follows,


123

(1) Select an element (say, ith element) of the nodal

displacement vector ({X}) for SNDSCP.

(2) Take the present solutions ({X}) as the con-

verged solutions of previous step of increment of

SNDSCP or as the converged solutions for the

temperature gradient only (for the first step of

increment of SNDSCP).

(3) Assign a small increment (DXi) of SNDSCP and

determine its new value ðXni ¼ Xi þ DXiÞ corre-

sponding to its magnitude (Xi) as it is obtained

from the presently converged solutions (step 2).

(4) Take this Xni as a constraint for subsequent

iteration process.

(5) Start direct iteration process

(5.1) Take the initial solutions ({X}) for the current

iteration as the solutions given in step 2

(for first iteration) or as the solutions of

the previous iteration.

(5.2) Update the solutions ({X} in step 5.1) by

replacing the assigned value (Xni ) of

SNDSCP.

(5.3) Update the coefficient vectors {fM} and {fT}

(Eq. (26)) and predict the mechanical

load (p) according to Eq. (26) as,

p ¼ ðXni � f T

i � DTgÞ=f Mi :

(5.4) Corresponding to this predicted mechanical

load, determine solutions ({X}) using

Eq. (26).

(5.5) Check convergence of the new solution ({X}

and p).

(5.6) For converged solutions and mechanical

load, go to step 1. Otherwise, go to step

5.1.

In the foregoing solution strategy, the number of

iterations for converged solutions at an incremental

step mainly depends on the corresponding increment

of SNDSCP. For a large increment of SNDSCP, the

number of iterations for converged solutions may

increase. Although any of the elements of nodal

displacement vector can be taken as SNDSCP, the

present solutions are evaluated by choosing it corre-

sponding to a point in the plate where the maximum

transverse deflection occurs. This choice facilitates to

control the required number of iterations for con-

verged solutions at an incremental step. The magni-

tude of increment of SNDSCP may be varied in

different incremental steps based on the difficulty in

the convergence of solutions. However, the present

results are evaluated considering uniform increment of

SNDSCP.

4 Numerical results and discussions

In this section, numerical results are presented for

investigating the geometrically nonlinear flexural

behaviour of an imperfect radially graded annular

plate under a temperature at one of its edges. The

numerical values of inner radius (ri), outer radius (ro)

and thickness (h) of the graded annular plate are

considered as 0.5, 1.5 and 10 mm, respectively. The

inner and outer edges of the annular plate are hinged

(u0 = 0, v0 = 0, w0 = 0, /y = 0). The geometrical

shape of the reference plane due to the intrinsic

geometrical imperfection of the annular plate is

defined according to the following expression,

�w ¼ ð�1Þbðe� hÞ sinpðr � riÞðro � riÞ

� �; ð27Þ

where e is a very small constant and b is a positive

integer. The even or odd value of b signifies the

intrinsic transverse deflection of the annular plate

along the positive or the negative z direction, respec-

tively. Thus, the intrinsic deflection of the annular

plate can be modelled either along the positive or

along the negative z-direction according to the value of

b as 2 or 1, respectively. The following dimensionless

parameters are used for presenting the numerical

results,

Q ¼ p� r4o

Em � h4; W ¼ w=h; a ¼ ro � rið Þ;

�rr ¼rr

jpjh

ro

� �2

; �rh ¼rh

jpjh

ro

� �2

; ð28Þ

where Em is the Young’s modulus of the metal

constituent. The ceramic and metal constituents of

the graded annular plate are considered as zirconia and

aluminium alloy, respectively. For different values of

the volume fraction index (n), the corresponding

variations of ceramic volume fraction [Vc (Eq. (10))]

along the radial direction are demonstrated in Fig. 2a,

b. It may be observed from these figures that the

ceramic constituent in the plate decreases as the

magnitude of volume fraction index (n) increases.

Within a range of temperature (300–600 K), the

temperature dependent material properties of the


123

constituent materials are given by Noda (1999),

Aluminium alloy:

EðTÞ ¼ 74þ 23� 10�3T � 11� 10�5T2�þ51� 10�9T3

GPa;

aðTÞ ¼ 1:6� 10�5þ 3:45� 10�8T � 3:3� 10�11T2�þ2:4� 10�14T3

K�1;

jðTÞ ¼ 218 W=ðmKÞ; m¼ 0:33: ð29Þ

Zirconium:

EðTÞ ¼ 225� 20� 10�2T � 90� 10�6T2�þ4� 10�9T3

GPa;

aðTÞ ¼ 1:48� 10�5 � 2:2� 10�8T�þ1:15� 10�11T2 þ 4� 10�15T3

K�1;

jðTÞ ¼ 1:5 W=ðmKÞ; m ¼ 0:33: ð30Þ

For a specified temperature (Tc = 600 K) of inner

(k = 1) or outer (k = 2) ceramic rich edge of the

graded annular plate, the temperature distribution

along the radial direction is demonstrated in Fig. 3a, b.

It may be observed from Fig. 3a (for k = 1) or 3b (for

k = 2) that the temperature distribution across the

radial span of the graded annular plate is weakly

dependent on the volume fraction index (n). For

constant values of applied temperature (Tc) and

volume fraction index (n), it may also be observed

form these figures (Fig. 3a, b) that the temperature in

the graded annular plate with outer ceramic rich edge

(k = 2) is significantly higher than that in the similar

plate with inner ceramic rich edge (k = 1). Corre-

sponding to any of the temperature distributions, the

material properties at a point within the domain of the

graded annular plate can be computed using Eqs. (11),

(29) and (30).

Since the nonlinear flexural deformation analysis of

similar radially graded annular plate is not available in

the literature, the present nonlinear solutions are

verified considering the graded annular plate as a

perfect (e = 0) isotropic (n = 0) annular plate. For

different values of the dimensionless applied mechan-

ical load (Q), the dimensionless transverse deflections

(W) at the inner edge of the fixed-free isotropic annular

plate are computed. These results are compared with

the published analytical results (Alwar and Reddy

1978) in Fig. 4. It may be observed from Fig. 4 that the

present results are in excellent agreement with the

analytical results (Alwar and Reddy 1978). This

comparison verifies the accuracy of the present

nonlinear finite element solutions.

The ceramic rich edge of the graded annular plate is

exposed to a high temperature. Since the temperature

at any of the edges and the material properties are

uniform along the circumferential direction, it is

reasonable to assume uniform temperature distribution

Fig. 2 Variation of the ceramic volume fraction (Vc) in the radial direction when the annular plate is radially graded a from inner

ceramic rich plate-edge to outer metal rich plate-edge (k = 1) and b vice versa (k = 2)


123

along the same (circumferential) direction at any

radial location. Thus, the graded annular plate is

basically subjected to an axial thermal load along the

radial direction. The effect of this thermal load on the

transverse deflection of the imperfect graded annular

plate is illustrated in Fig. 5a, b when the heated

ceramic rich edge of the annular plate is located either

at inner edge (k = 1) or at outer edge (k = 2). It may

be observed from these figures that the temperature

(Tc) at an edge of the imperfect annular plate causes its

transverse deflection. The deflection versus tempera-

ture slope is negligibly small at a low temperature and

it remains almost constant up to a certain limit of

increasing temperature (Tc). As the temperature (Tc)

exceeds this limit, the deflection versus temperature

slope drastically changes that yields a significant

effect of temperature (Tc) on the transverse deflection

of the plate. However, the imperfect graded annular

plate is basically buckled at this limit of temperature.

So, this limit of temperature may be called as the

critical temperature for buckling of the imperfect

graded annular plate due to its heated edge. It may also

be observed from Fig. 5a, b that the direction of

transverse deflection of the graded annular plate due to

its heated edge is the same as that of the intrinsic

deflection of the same plate. Also, the magnitude of

transverse deflection at an applied temperature (Tc) is

constant for both directions because of uniform

material properties across the thickness of the plate

and constant magnitude of intrinsic deflection along

both directions. For a particular temperature (Tc),

Fig. 5a, b also demonstrate that the thermal bending

deflection of the graded annular plate decreases with

the decrease of volume fraction index (n). In fact, the

increased amount of metal constituent corresponding

to a higher value of n causes more temperature in the

plate even though the applied temperature (Tc)

Fig. 3 Radial distribution of temperature for a the inner (k = 1) and b the outer (k = 2) heated ceramic rich plate-edge of the radially

graded annular plate

Fig. 4 Comparison of dimensionless transverse deflection

(W) at the free-edge of fixed-free isotropic (n = 0) perfect

(e = 0) annular plate with those of an identical annular plate

studied by Alwar and Reddy (1978)


123

remains constant. However, these results show that the

thermal bending deflection at an applied temperature

(Tc) can be reduced by decreasing the magnitude of

volume fraction index. Since the minimum thermal

bending deflection of the graded annular plate with a

heated edge is desired according to the design aspects

of plates, the minimum value of volume fraction index

(n = 0) can be chosen for a pure ceramic annular

plate. The requirement of sufficient stiffness of the

annular plate for sustaining additional mechanical

load eliminates this choice (n = 0) and it suggests to

use the radially graded annular plate (n [ 0). The

volume fraction index of the graded annular plate may

be tuned to achieve desired stiffness of the plate along

with the low magnitude of thermal bending deflection.

This is a significant advantage of radially graded

annular plate over the conventional pure ceramic or

pure metallic annular plate when it (annular plate)

operates under the transverse mechanical load in the

presence of a high temperature at one of its edges.

In the presence of a temperature at the inner

ceramic rich edge of the graded annular plate, Fig. 6a

illustrates thermo-elastic equilibrium curves (Q vs.

W) for different values of the volume fraction index

(n). The direction of the intrinsic deflection of the

annular plate is considered as the negative z direction

(b = 1) and the plate is subjected to a uniformly

distributed transverse mechanical load along the

positive z direction. When the graded annular plate

is subjected to the temperature only (Q = 0, k = 1), it

deflects along the negative z direction (S1 or S2,

Fig. 6a) following the same direction of its intrinsic

deflection. As the additional mechanical load (Q,

along the positive z direction) increases from a value

of 0, the annular plate behaves as a softening structure

due to the significant effect of initial thermal bending

deflection (S1 or S2, Fig. 6a). Because of this struc-

tural behaviour of the annular plate, the magnitude of

load versus deflection slope decreases gradually and it

reaches to 0 (points A1 or A2) at a certain value of

increasing mechanical load. At this point (points A1 or

A2), the plate is in unstable equilibrium. For further

increment of the mechanical load, the plate switches

by a snap-through mode to a new stable equilibrium

configuration (points B1 or B2, Fig. 6a). The struc-

tural behaviour of the plate also switches to hardening

structural behaviour. This type of unstable equilibrium

of a structure is generally called as snap-through

equilibrium and it happens during the increase of

transverse mechanical load on the imperfect radially

Fig. 5 Variation of the dimensionless transverse deflection (W(a/2, h, z)) of the imperfect radially graded annular plate with its a inner

(k = 1) or b outer (k = 2) ceramic rich plate-edge temperature (Tc)


123

graded annular plate because of its initial (Q = 0)

thermal bending deflection. A similar incident also

happens during the decrease of transverse mechanical

load (points C1 or C2, Fig. 6a). This is the reverse snap

of equilibrium of the imperfect radially graded annular

plate. Since the initial mode of equilibrium of the

annular plate is no longer stable when the magnitude

of load versus deflection slope reaches to 0, the

corresponding magnitude of increasing or decreasing

mechanical load may be called as upper or lower

critical mechanical load, respectively.

For a particular temperature (Tc = 400 K) of the

inner ceramic rich edge of the graded annular plate,

Fig. 6a illustrates that the magnitude of critical

mechanical load significantly increases with the

increase of the volume fraction index (n). This may

be due to the fact that the overall nonlinear stiffness of

the graded annular plate corresponding to its initial

thermal bending deflection increases with the increase

of volume fraction index. However, the variations of

upper and lower critical mechanical loads with the

volume fraction index (n) are illustrated in Fig. 6b, c,

Fig. 6 a Variation of the dimensionless transverse deflection (W(a/2, h, z)) of the radially graded annular plate with the mechanical

load (Q); variations of b the upper and c the lower critical mechanical loads with the volume fraction index (n, k = 1, b = 1)


123

respectively for two different applied temperatures

(Tc). It may be observed from these figures that the

magnitude of critical mechanical load nonlinearly

increases with the increase of volume fraction index

(n). Also, the corresponding rate of change of the

critical mechanical load significantly increases when

the inner ceramic rich edge of the graded annular plate

is exposed to a higher temperature. In Fig. 6b, c, a

curve represents the boundary of the applied mechan-

ical load corresponding to the alteration of structural

behaviour of the graded annular plate. When the

magnitude of the applied mechanical load at a

particular value of n exceeds the corresponding critical

value on the curve, the softening structural behaviour

of the graded annular plate switches to the hardening

structural behaviour. So, the upper and lower zones

separated by a curve (Fig. 6b, c) may be designated as

hardening and softening zones, respectively.

For a particular volume fraction index (n = 1) of

the radially graded annular plate, Fig. 7a illustrates

Fig. 7 a Variation of the dimensionless transverse deflection

(W(a/2, h, z)) of the radially graded annular plate with the

mechanical load (Q); variations of b the upper and c the lower

critical mechanical loads with the inner ceramic rich plate-edge

temperature (Tc, k = 1, b = 1)


123

thermo-elastic equilibrium paths (Q vs. W) for two

different temperatures (Tc = 400 and 450 K) of the

inner ceramic rich edge. It may be observed from

Fig. 7a that the load versus deflection slope decreases

slowly with the increasing mechanical load when the

inner ceramic rich edge is exposed to a higher

temperature. So, the magnitude of the critical mechan-

ical load increases with the increase the temperature of

inner ceramic rich edge. However, the variation of this

critical mechanical load (upper or lower) with the

temperature of inner ceramic rich (k = 1) edge is

illustrated in Fig. 7b, c for two different values of

volume fraction index. These figures demonstrate that

the magnitude of critical mechanical load significantly

increases with the increasing temperature for any

value of the volume fraction index. The corresponding

rate of change of the critical mechanical load also

increases for the graded annular plate with a higher

volume fraction index. Similar to Fig. 6b, c, a curve in

Fig. 7b, c signifies the boundary of the applied

mechanical load corresponding to the alteration of

structural behaviour of the graded annular plate.

However, it is clear from the foregoing results

(Figs. 6, 7) that an imperfect radially graded annular

plate behaves either as a softening structure or as a

hardening structure depending on the magnitudes of

mechanical load (p), volume fraction index (n) and

temperature (Tc). It may also be noticed form Figs. 6

and 7 that the deflection of the graded annular plate

during the snap of equilibrium (from A1 to B1 or from

A2 to B2) increases with the increase of initial

(Q = 0) thermal bending deflection. Thus, this type

of unstable equilibrium of the radially graded annular

plate can be mitigated by reducing its initial thermal

bending deflection. The minimum value of the initial

thermal bending deflection corresponding to an

applied temperature (Tc) can be achieved by decreas-

ing the magnitude of volume fraction index to 0

(n = 0, pure ceramic annular plate). Since this option

may not provide sufficient stiffness of the plate for

sustaining the mechanical load, the graded annular

plate (n [ 0) with a low value of volume fraction

index is to be used. This low value of volume fraction

index is to be chosen in such a way that the plate can

sustain the applied mechanical load and its deflection

during the snap of thermo-elastic equilibrium is

sufficiently small (preferably 0). Note that, the con-

ventional imperfect pure ceramic annular plate has not

sufficient stiffness for sustaining mechanical load. The

conventional imperfect pure metallic annular plate is

prone to exhibit snap-through thermo-elastic equilib-

rium because of its low thermal resistance. These

discrepancies of monolithic annular plate can be

mitigated by the use of radially graded annular plate

with an appropriate volume fraction index.

The foregoing results (Figs. 6, 7) are evaluated

considering the intrinsic deflection of the graded

annular plate along the negative z direction (b = 1).

Fig. 8 Variation of the dimensionless transverse deflection (W(a/2, h, z)) of the radially graded annular plate with the mechanical load

(Q) for different values of a volume fraction index (n) and b inner ceramic rich plate-edge temperature (k = 1, b = 2)


123

If this intrinsic deflection is considered along the

positive z direction (b = 2), then the corresponding

thermo-elastic equilibrium paths (Q vs. W) are illus-

trated in Fig. 8a, b. It may be observed from these

figures that the graded annular plate is in stable

equilibrium under any applied mechanical load. This

may be due to the fact that the direction of the applied

mechanical load is the same as that of the initial

(Q = 0) thermal bending deflection. Thus, the results

as illustrated in Figs. 6, 7 and 8 imply that the

direction of applied mechanical load with respect to

that of the intrinsic deflection of the plate is an

important factor for the snap-through equilibrium of

the imperfect radially graded annular plate with a

heated edge. If the direction of intrinsic deflection of

the annular plate can be known prior to the application

of the transverse mechanical load, then its unstable

equilibrium can be removed easily.

In the preceding results (Figs. 6, 8), the inner edge

of the imperfect radially graded annular plate is

considered as the heated ceramic rich edge (k = 1).

However, similar studies are also carried out

Fig. 9 a Variation of the dimensionless transverse deflection (W(a/2, h, z)) of the radially graded annular plate with the mechanical

load (Q); variations of b the upper and c the lower critical mechanical loads with the volume fraction index (n, k = 2, b = 1)


123

considering the outer edge of the graded annular plate

as the heated ceramic rich edge (k = 2). In the present

study, the graded annular plate with the outer ceramic

rich edge is designated as the reversely graded annular

plate (k = 2). For an applied temperature

(Tc = 400 K), Fig. 9a illustrates thermo-elastic equi-

librium paths (Q vs. W) of the imperfect reversely

graded annular plate for two different values of volume

fraction index (n = 1 and 2). The direction of intrinsic

deflection of the plate is considered as negative

z direction and the mechanical load acts along the

positive z direction. In comparison to the previous

results (Fig. 6a), the present result (Fig. 9a) exhibits

increased thermal bending deflection (Q = 0) of the

graded annular plate although volume fraction index

(n) and temperature (Tc) remain constant. In fact, the

temperature in the previously graded annular plate

(k = 1) is significantly less than that in the reversely

Fig. 10 a Variation of the dimensionless transverse deflection

(W(a/2, h, z)) of the radially graded annular plate with the

mechanical load (Q); variations of b the upper and c the lower

critical mechanical loads with the outer ceramic rich plate-edge

temperature (Tc, k = 2, b = 1)


123

graded annular plate (k = 2). Note that, Fig. 3a, b also

show this difference of temperature in the graded

annular plate. Figure 9b, c illustrate the variations of

upper and lower critical mechanical loads with the

volume fraction index (n) of the reversely graded

annular plate (k = 2). It may be observed from these

figures that the nature of variation of a critical

mechanical load (upper or lower) is almost the same

as that in the previous case (Fig. 6b, c). The only

difference is in the magnitude of a critical mechanical

load due to the difference of initial thermal bending

deflection.

For a particular volume fraction index (n = 1) of

the imperfect (b = 1) reversely graded (k = 2) annu-

lar plate, Fig. 10a illustrates thermo-elastic equilib-

rium paths (Q vs. W) for two different temperatures

(Tc). It may be observed from Figs. 10a and 7a that the

deflection of the graded annular plate during the snap

of equilibrium significantly increases when the loca-

tion of its ceramic rich edge is shifted from inner edge

to outer edge without altering applied temperature (Tc)

and volume fraction index (n). However, the varia-

tions of the corresponding upper and lower critical

mechanical loads with the temperature (Tc) are

illustrated in Fig. 10b, c for two different values of

the volume fraction index. From Figs. 10 and 7, it may

be observed that the nature of variation of a critical

mechanical load is not much dependent on the location

(inner or outer edge) of the heated ceramic rich edge.

Figure 11a, b illustrate thermo-elastic equilibrium

paths (Q vs. W) of the reversely graded (k = 2)

annular plate when its intrinsic deflection is along the

positive z direction (b = 2). It may be observed from

Figs. 11 and 8 that the nonlinearity in the thermo-

elastic response of the graded annular plate reduces

when the location of the ceramic rich edge is shifted

from inner edge to outer edge for constant values of

applied temperature (Tc) and volume fraction index

(n). This may be due to the fact that the overall

stiffness of the reversely graded (k = 2) annular plate

is significantly more than that of the previously graded

(k = 1) annular plate. However, it may be noticed

from the foregoing results that the thermo-elastic

deformation characteristics of the imperfect radially

graded annular plate do not vary significantly for two

different locations (inner and outer edges) of the

heated ceramic rich edge.

Figure 12a, b demonstrate the distributions of

dimensionless radial ð�rrÞ and circumferential ð�rhÞstresses across the radial span of the graded annular

plate (k = 1). The direction of intrinsic deflection of

the annular plate is considered as the negative

z direction (b = 1) and the mechanical load acts along

the positive z direction. It may be observed from

Fig. 12a, b that the maximum stress concentration of

the radial stress ð�rrÞ occurs at the stiffer (ceramic rich)

Fig. 11 Variation of dimensionless transverse deflection (W(a/2, h, z)) of the radially graded annular plate with the mechanical load

(Q) for different values of a volume fraction index (n) and b outer ceramic rich plate-edge temperature (k = 2, b = 2)


123

edge of the annular plate but the maximum circum-

ferential stress ð�rhÞ may not happen at the same

location. Also, the effect of the volume fraction index

(n) on the maximum value of a stress ð�rr or �rhÞ is

indicatively lesser than that of the temperature (Tc) on

the same quantity. However, the variations of dimen-

sionless stresses ð�rr and �rhÞ across the thickness of

the graded annular plate are illustrated in Fig. 12c, d

corresponding to the radial locations of their

ð�rr and �rhÞ maximum values. It may be observed

from these figures that the circumferential stress

significantly varies along the thickness direction while

the radial stress weakly varies along the same direc-

tion. The foregoing distributions of radial and cir-

cumferential stresses are illustrated considering the

direction of intrinsic deflection of the plate as the

negative z direction (b = 1). For the opposite direc-

tion of intrinsic deflection of the same plate, similar

distributions of stresses are also demonstrated in

Fig. 13a–d. It may be observed from these figures that

Fig. 12 Distributions of a radial and b circumferential stresses across the radial span of the radially graded annular plate; c, d the

corresponding through-thickness stress distributions [k = 1, b = 1, p = 500 N/m2 (Q = 3.5)]


123

the nature of variation of a stress ð�rr or �rhÞ does not

change for the alteration of the direction of intrinsic

deflection of the graded annular plate.

For an imperfect (b = 1) reversely graded (k = 2)

annular plate, the variations of dimensionless stresses

ð�rr and �rhÞ along the radial direction are illustrated in

Fig. 14a, b. Unlike the previously graded annular plate

(k = 1, Fig. 12a), the maximum radial stress ð�rrÞ does

not occur at the ceramic rich edge of the reversely

graded annular plate (Fig. 14a). It may be observed


corresponding through-thickness stress distributions [k = 1, b = 1, p = 7.15 9 103 N/m2 (Q = 50)]


123

from Figs. 12c and 14c that the radial stress ð�rrÞsignificantly varies across the thickness of the

reversely graded (k = 2) annular plate while the same

stress insignificantly varies across the thickness of the

previously graded (k = 1) annular plate. It may also

be observed from Figs. 12d and 14d that the nature of

variation of the circumferential stress across the

thickness of the reversely graded (k = 2) annular

plate is similar to that for the previously graded

(k = 1) annular plate. Figure 15a–d demonstrate the

distributions of dimensionless stresses ð�rr and �rhÞ

when the direction of intrinsic deflection of the

reversely graded (k = 2) annular plate is considered

as the positive z direction (b = 2). It may be observed

from Figs. 14 and 15 that the direction of intrinsic

deflection of the reversely graded annular plate does

not have much effect on the radial ð�rrÞ and circum-

ferential ð�rhÞ stresses. However, Figs. 12, 13, 14 and

15 demonstrate an important fact that the radial

location corresponding to the maximum value of a

stress ð�rr or �rhÞ insignificantly depends on applied

temperature (Tc) and volume fraction index (n). Also,


corresponding through-thickness stress distributions [k = 2, b = 1, p = 500 N/m2 (Q = 3.5)]


123

the maximum radial stress concentration always

occurs at the stiffer (ceramic rich) edge when it is

located at the inner edge of the graded annular plate. In

case of a graded annular plate with outer ceramic rich

(stiffer) edge, the stiffer edge may not be a location of

maximum radial stress concentration.


corresponding through-thickness stress distributions [k = 2, b = 2, p = 7.15 9 103 N/m2 (Q = 50)]


123

5 Conclusions

A geometrically nonlinear finite element analysis of an

imperfect radially graded annular plate with a heated

edge is presented. The geometrical imperfection of the

graded annular plate is assumed in aspect of its little

intrinsic transverse deflection. One of the edges of the

graded annular plate is exposed to a high temperature.

The effect of this temperature on the nonlinear flexural

behaviour of the imperfect graded annular plate under

a transverse mechanical load is investigated. The

temperature is uniformly distributed across the thick-

ness of the annular plate while it varies along the radial

direction only. The temperature-dependent material

properties of the annular plate are graded along the

radial direction according to a simple power-law that

is formed by power-law exponent and material

properties of constituent materials (ceramic and

metal). The kinematics of deformation of the graded

annular plate is defined according to the FSDT. Based

on the Von Karman nonlinear strain–displacement

relations for an imperfect annular plate, the nonlinear

finite element equations of equilibrium of the graded

annular plate are derived employing the principle of

minimum potential energy. A single nodal displace-

ment-control solution strategy is described for numer-

ical solutions of the nonlinear finite element equations

of equilibrium. The numerical results are presented

either for the inner ceramic rich edge or for the outer

ceramic rich edge of the imperfect radially graded

annular plate. The analysis reveals the following

important findings,

(1) A geometrically imperfect radially graded annu-

lar plate with a heated edge may undergo snap-

through equilibrium when the direction of the

applied transverse mechanical load is in opposite

to that of the intrinsic deflection of the plate due

to its geometrical imperfection.

(2) The thermal bending deflection of the imperfect

radially graded annular plate due to its heated

edge is the main reason for its unstable equilib-

rium under thermo-mechanical load. The volume

fraction index of the same graded plate is the

main parameter for alleviating this unstable

equilibrium.

(3) The critical mechanical load corresponding to

the snap of equilibrium of the graded annular

plate significantly increases with the increase of

its (plate) temperature. The same (critical load)

also increases with the increase of the volume

fraction index even though the temperature

remains constant.

(4) The radial location of the maximum value of a

stress (radial or circumferential) insignificantly

depends on temperature and volume fraction

index of the radially graded annular plate.

(5) The maximum radial stress concentration always

occurs at the stiffer (ceramic rich) edge when it is

located at the inner edge of the graded annular

plate.

(6) The thermo-elastic deformation characteristics

of the radially graded annular plate do not change

significantly due to different locations (inner and

outer edges) of its ceramic rich edge.

References

Alibeigloo, A.: Three-dimensional semi-analytical thermo-

elasticity solution for a functionally graded solid and an

annular circular plate. J. Therm. Stresses 35, 653–676

(2012)

Allahverdizadeh, A., Naei, M.H., Rastgo, A.: The effects of

large vibration amplitudes on the stresses of thin circular

functionally graded plates. Int. J. Mech. Mater. Des. 3,

161–174 (2006)

Alwar, R.S., Reddy, S.: Large deflection static and dynamic

analysis of isotropic and orthotropic annular plate. Int.

J. Nonlinear Mech. 14, 347–359 (1978)

Bakhshi, M., Bagri, A., Eslami, M.R.: Coupled thermoelasticity

of functionally graded disk. Mech. Adv. Mater. Struct. 13,

219–225 (2006)

Bayat, M., Saleem, M., Sahari, B.B., Hamouda, A.M.S., Mahdi,

E.: Thermo elastic analysis of a functionally graded rotat-

ing disk with small and large deflections. Thin-Walled

Struct. 45, 667–691 (2007)

Bayat, M., Sahari, B.B., Saleem, M., Hamouda, A.M.S., Reddy,

J.N.: Thermo elastic analysis of functionally graded rotat-

ing disks with temperature-dependent material properties:

uniform and variable thickness. Int. J. Mech. Mater. Des. 5,

263–279 (2009)

Chia, C.Y.: Nonlinear Analysis of Plates. McGraw-Hill, Inc.,

New York (1980)

Chiba, R.: Stochastic heat conduction analysis of a functionally

graded annular disc with spatially random heat transfer

coefficients. Appl. Math. Model. 33, 507–523 (2009)

Fallah, F., Nosier, A.: Nonlinear behaviour of functionally

graded circular plates with various boundary supports

under asymmetric thermo-mechanical loading. Compos.

Struct. 94, 2834–2850 (2012)

Golmakani, M.E., Kadkhodayan, M.: An investigation into the

thermoelastic analysis of circular and annular functionally

graded material plates. Mech. Adv. Mater. Struct. 21, 1–13

(2014)


123

Jagtap, K.R., Lal, A., Singh, B.N.: Stochastic nonlinear bending

response of functionally graded material plate with random

system properties in thermal environment. Int. J. Mech.

Mater. Des. 8, 149–167 (2012)

Jalali, S.K., Naei, M.H., Poorsoljouy, A.: Thermal stability

analysis of circular functionally graded sandwich plates of

variable thickness using pseudospectral method. Mater.

Des. 31, 4755–4763 (2010)

Koizumi, M.: Concept of FGM. Ceram. Trans. 34, 3–10 (1993)

Lanhe, W.: Thermal buckling of a simply supported moderately

thick rectangular FGM plate. Compos. Struct. 64, 211–218

(2004)

Lee, Y.Y., Zhao, X., Reddy, J.N.: Postbuckling analysis of

functionally graded plates subject to compressive and

thermal loads. Comput. Methods Appl. Mech. Eng. 199,

1645–1653 (2010)

Ma, L.S., Wang, T.J.: Nonlinear bending and post-buckling of a

functionally graded circular plate under mechanical and

thermal loadings. Int. J. Solids Struct. 40, 3311–3330

(2003)

Najafizadeh, M.M., Heydari, H.R.: Thermal buckling of func-

tionally graded circular plates based on higher order shear

deformation plate theory. Eur. J. Mech. A 23, 1085–1100

(2004)

Nie, G.J., Batra, R.C.: Stress analysis and material tailoring in

isotropic linear thermoelastic incompressible functionally

graded rotating disks of variable thickness. Compos.

Struct. 92, 720–729 (2010)

Noda, N.: Thermal stresses in functionally graded materials.

J. Therm. Stresses 22, 477–512 (1999)

Peng, X.L., Li, X.F.: Thermal stress in rotating functionally

graded hollow circular disks. Compos. Struct. 92,

1896–1904 (2010)

Prakash, T., Ganapathi, M.: Asymmetric flexural vibration and

thermoelastic stability of FGM circular plates using finite

element method. Composites B 37, 642–649 (2006)

Praveen, G.N., Reddy, J.N.: Nonlinear transient thermo-elastic

analysis of functionally graded ceramic metal plates. Int.

J. Solids Struct. 35, 4457–4476 (1998)

Reddy, J.N., Berry, J.: Nonlinear theories of axisymmetric

bending of functionally graded circular plates with modi-

fied couple stress. Compos. Struct. 94, 3664–3668 (2012)

Reddy, J.N., Chin, C.D.: Thermo mechanical analysis of func-

tionally graded cylinders and plates. J. Therm. Stresses 21,

593–626 (1998)

Sepahi, O., Forouzan, M.R., Malekzadeh, P.: Large deflection

analysis of thermo-mechanical loaded annular FGM plates

on nonlinear elastic foundation via DQM. Compos. Struct.

92, 2369–2378 (2010)

Sepahi, O., Forouzan, M.R., Malekzadeh, P.: Thermal buckling

and postbuckling analysis of functionally graded annular

plates with temperature-dependent material properties.

Mater. Des. 32, 4030–4041 (2011)

Shariat, B.A.S., Eslami, M.R.: Buckling of thick functionally

graded plates under mechanical and thermal loads. Com-

pos. Struct. 78, 433–439 (2007)

Shen, H.S.: Nonlinear bending response of functionally graded

plates subjected to transverse loads and in thermal envi-

ronments. Int. J. Mech. Sci. 44, 561–584 (2002)

Woo, J., Meguid, S.A.: Nonlinear analysis of functionally gra-

ded plates and shallow shells. Int. J. Solids Struct. 38,

7409–7421 (2001)

Wu, T.L., Shukla, K.K., Huang, J.H.: Post-buckling analysis of

functionally graded rectangular plates. Compos. Struct. 81,

1–10 (2007)

Yang, J., Shen, H.S.: Nonlinear bending analysis of shear

deformable functionally graded plates subjected to thermo-

mechanical loads under various boundary conditions.

Composites B 34, 103–115 (2003)

Zhao, X., Lee, Y.Y., Liew, K.M.: Mechanical and thermal

buckling analysis of functionally graded plates. Compos.

Struct. 90, 161–171 (2009)


123

Date post:	22-Jan-2017
Category:	Documents
Upload:	satyajit
View:	212 times
Download:	0 times

Nonlinear analysis of an imperfect radially graded annular plate with a heated edge

Documents