A Green’s function approach to the deflection of a FGMplate under transient thermal loading

K.-S. Kim, N. Noda

Summary Green’s function approach is adopted for analyzing the deflection and the transienttemperature distribution of a plate made of functionally graded materials (FGMs). Thegoverning equations for the deflection and the transient temperature are formulated intoeigenvalue problems by using the eigenfunction expansion theory. Green’s functions forsolving the deflection and the transient temperature are obtained by using the Galerkin methodand the laminate theory, respectively. The eigenfunctions of Green’s function for the deflectionare approximated in terms of a series of admissible functions that satisfy the homogeneousboundary conditions of the plate. The eigenfunctions of Green’s function for the temperatureare determined from the continuity conditions of the temperature and the heat flux at inter-faces.

Keywords Green’s Function, Functionally Graded Material, Plate Deflection, TransientThermal Loading

1IntroductionA functionally graded material (FGM) is characterized by continuously changing materialproperties due to a graded composition from one surface to the other. For heterogeneousmaterials such as FGMs, the heat conduction equation and the governing equation fordeflection in a FGM plate become complex forms with position-dependent coefficients becauseof the non-uniform material properties. Theoretical treatments for those equations are verydifficult.

There are few reports, [1–5], on the solution of transient thermal stress problems in an FGMplate. Almost all papers, which treated transient problems, have used the Laplace or the Fouriertransformations to solve the governing equation. On the other hand, a Green’s functionapproach was used [6] for two-dimensional homogeneous elastic problems. Green’s functionwas used to analyze steady thermal stresses in a two-dimensional FGM plate [7] .

In this paper, we discuss the deflection of an FGM plate under transient thermal loading.Since the compositions of the FGM are functions of the position between a metal surface and aceramic surface, it is assumed that the thermal properties of FGM are dependent on theunidirectional position. The transient temperature distribution for a three-dimensional FGMplate with unidirectionally dependent properties is formulated by a Green’s function approachbased on the laminate theory. An approximate solution for each layer by the eigenfunctionexpansion method is substituted into the governing equation to yield an eigenvalue problem.The eigenvalues and the corresponding eigenfunctions of each layer constitute a Green’sfunction solution for obtaining the three-dimensional transient temperature distribution.

Archive of Applied Mechanics 72 (2002) 127–137 � Springer-Verlag 2002

DOI 10.1007/s00419-002-0172-6

127

Received 9 October 2000; accepted for publication 3 April 2001

K.-S. KimDepartment of Aeronautical Engineering,Inha Technical College,253 Yonghyun-dong Nam-ku, Incheon, 402-752, Korea

N. Noda (&)Department Mechanical Engineering,Shizuoka University, 3-5-1 Johoku Hamamatsu,432-8561, Japane-mail: tmnnoda@ipc.shizuoka.ac.jp

The solution to the governing equation of a FGM plate derived by the classical (Kirchhoff)theory is formulated by a Green’s function approach based on the Galerkin method, [6].Green’s function for the deflection of a FGM plate is expressed by eigenfunctions, which areapproximated in terms of a series of admissible functions that satisfy the homogeneousboundary conditions to which the plate is subjected.The coefficients of each eigenfunction aredetermined by the Galerkin method.

As an example, a FGM plate made of zirconium oxide and titanium alloy under transientthermal loading is selected. Numerical results, such as the temperature distribution and thedeflection of a FGM plate, are shown in the figures.

2Analysis of the temperature field

2.1Three-dimensional heat conduction equationAssuming that the thermal properties are dependent on a z-directional position, we considerthe heat conduction equation of the cartesian coordinate system ðx; y; zÞ in the absence of aheat source as

o

oxkðzÞ oT

ox

� �þ o

oykðzÞ oT

oy

� �þ o

ozkðzÞ oT

oz

� �¼ qðzÞcpðzÞ

oT

ot; ð1Þ

where T ¼ Tðx; y; z; tÞ is temperature, and t is time. The thermomechanical parameters q, cp

and k are the position-dependent density, specific heat and thermal conductivity, respectively.For convenience in the analysis, we consider Tðx; y; z; tÞ constructed by the superposition of

the simpler problem as

Tðx; y; z; tÞ ¼ Tsðx; y; zÞ þ hðx; y; z; tÞ ; ð2Þ

where Ts and h are solutions of the steady-state problem with non-homogeneous boundaryconditions and the unsteady-state problem with homogeneous boundary conditions,respectively. Thus, we can obtain the steady state solution Tsðx; y; zÞ by using the standardGalerkin-based Green function, [8].

2.2Unsteady heat conduction equationWe consider a laminated medium consisting of L layers in the z direction. Assuming that thenumber of laminae becomes sufficiently large, and the thermal properties of each layer areconstants, the governing equations without internal heat generation for each layer and theinitial conditions are given as

o2hi

ox2þ o2hi

oy2þ o2hi

oz2¼ 1

ki

ohi

ot; ð3Þ

hiðx; y; z; 0Þ ¼ Fiðx; y; zÞ in zi�1 � z � zi; i ¼ 1; 2; . . . ; L at t ¼ 0 ; ð4Þ

where hi and ki are the temperature change and the thermal diffusivity of the i-th layer,respectively.

The solutions of Eq. (3), [9], are assumed to be

hiðx; y; z; tÞ ¼X1k¼1

X1m¼1

X1n¼1

ckmn/ikðxÞuimðyÞwinðzÞe�ða2kþb2

mþc2nÞt

in zi�1 � z � zi; i ¼ 1; 2; . . . ; L for t > 0 ; ð5Þ

where ckmn are constants to be evaluated. Substitution of Eq. (5) into Eq. (3) yields eigenvalueproblems as

o2/ik

ox2þ a2

k

ki/ik ¼ 0 in 0 � x � a ; ð6Þ

128

o2uim

oy2þ b2

m

kiuim ¼ 0 in 0 � y � b ; ð7Þ

o2win

oz2þ c2

n

kiwin ¼ 0 in zi�1 � z � zi; i ¼ 1; 2; . . . ; L for t > 0 ; ð8Þ

where a and b denote the x-directional and y-directional length of the plate, ak; bm and cn arek-th, m-th, and n-th eigenvalues, and /ikðxÞ, uimðyÞ and winðzÞ are eigenfunctions corre-sponding to the eigenvalues, respectively.

Applying the initial conditions of Eq. (4) to Eq. (5), coefficients ckmn yield

ckmn ¼ 1

Nkmn

XL

j¼1

kj

kj

Za

0

Zb

0

Zzj

z¼zj�1

Fjðx0; y0; z0Þ/jkðx0Þujmðy0Þwjnðz0Þdz0 dy0 dx0 ; ð9Þ

where the norm Nkmn is defined as

Nkmn ¼XL

j¼1

kj

kj

Za

0

Zb

0

Zzj

z¼zj�1

/2jkðx0Þu2

jmðy0Þw2jnðz0Þdz0 dy0 dx0 : ð10Þ

Substitution of Eq. (9) into Eq. (5) and introduction of Green’s function yield

hiðx; y; z; tÞ ¼XL

j¼1

Za

0

Zb

0

Zzj

z¼zj�1

Gijðx; y; z; tjx0; y0; z0; t0Þjt¼0Fjðx0; y0; z0Þdz0 dy0 dx0 ; ð11Þ

where Gijðx; y; z; tjx0; y0; z0; t0Þjt0¼0 is defined as

Gijðx; y; z; tjx0; y0; z0; t0Þjt0¼0

¼X1k¼1

X1m¼1

X1n¼1

1

Nkmn

kj

kj/ikðxÞ/jkðx0ÞuimðyÞujmðy0ÞwinðzÞwjnðz0Þe�ða2

kþb2

mþc2nÞt : ð12Þ

2.3Determination of eigenfunctions and eigenvaluesThe general solutions /ikðxÞ, uimðyÞ and winðzÞ of Eqs. (6)–(8) can be written as

/ikðxÞ ¼ Aik sinakffiffiffiffiki

p x

� �þ Bik cos

akffiffiffiffiki

p x

� �in 0 � x � a ; ð13Þ

uimðyÞ ¼ Cim sinbmffiffiffiffiki

p y

� �þ Dim cos

bmffiffiffiffiki

p y

� �in 0 � y � b ; ð14Þ

winðzÞ ¼Ein sincnffiffiffiffiki

p ðz � ziÞ� �

þ Fin coscnffiffiffiffiki

p ðz � ziÞ� �

in zi�1 � z � zi; i ¼ 1; 2; . . . ; L at t > 0 ð15Þ

where Ail, Bil, Cim, Dim, Ein and Fin are coefficients to be evaluated.The functions /ikðxÞ and uimðyÞ of Eqs. (13) and (14) are determined from the homogeneous

boundary conditions, and the coefficients Ein and Fin of function winðzÞ are obtained in thematrix form as

Ein

Fin

� �¼ P Q

R S

� Eiþ1;n

Fiþ1;n

� �in zi�1 � z � zi; i ¼ 1; 2; . . . ; L at t > 0 ; ð16Þ

129

where

P ¼ kiþ1

ki

ffiffiffiffiffiffiffiffiki

kiþ1

scos

cnffiffiffiffiffiffiffiffikiþ1

p ðzi � ziþ1Þ� �

;

Q ¼ � kiþ1

ki

ffiffiffiffiffiffiffiffiki

kiþ1

ssin

cnffiffiffiffiffiffiffiffikiþ1

p ðzi � ziþ1Þ� �

;

R ¼ sincnffiffiffiffiffiffiffiffikiþ1

p ðzi � ziþ1Þ� �

;

S ¼ coscnffiffiffiffiffiffiffiffikiþ1

p ðzi � ziþ1Þ� �

:

ð17Þ

Therefore, the eigenvalues cn are determined from Eq. (16) and the homogeneous boundaryconditions at the outer surfaces, and the corresponding eigenfunctions for all layers are de-termined from Eq. (16).

3Analysis of deflectionAssuming that the thermal properties are dependent on the z-directional position, we considerthe deflection of a FGM plate subjected to the three-dimensional unsteady temperature changegiven in Eq. (2). According to the classical (Kirchhoff) theory, the governing equation for aFGM plate without external loads is given as

Lwðx; yÞ ¼ �qtðx; yÞ ; ð18Þ

where wðx; yÞ is the deflection of the plate, qtðx; yÞ is a distributed thermal load and L is definedas the following operator:

L ¼ D11o4w

ox4þ D33

o4w

ox2oy2þ D11

o4w

oy4; ð19Þ

in which

D11 ¼Zh=2

�h=2

EðzÞð1 � m2ðzÞÞ z2 dz; D22 ¼

Zh=2

�h=2

EðzÞmðzÞð1 � m2ðzÞÞ z2 dz;

D66 ¼Zh=2

�h=2

EðzÞð1 þ mðzÞÞ z2 dz; D33 ¼ 2ðD22 þ D66Þ; qtðx; yÞ ¼ o2Mt

ox2þ o2Mt

oy2;

ð20Þ

and where

Mt ¼Zh=2

�h=2

aðzÞEðzÞð1 � mðzÞÞ �ssz dz ;

and a, E and m are the coefficients of linear thermal expansion, Young’s modulus and Poisson’sratio, respectively; h and �ss ¼ Tðx; y; z; tÞ � T0 denote the thickness of a FGM plate and thetemperature difference from the initial state, respectively.

The unknown wðx; yÞ is expressed as

wðx; yÞ ¼ �ZZ

R

Gðn; g; x; yÞqtðn; gÞdn dg ; ð21Þ

130

where R is the middle plate surface. The function Gðn; g; x; yÞ is Green’s function, whichsatisfies

L�Gðn; g; x; yÞ ¼ dðx � n; y � gÞ ; ð22Þ

where dðx � n; y � gÞ is the two-dimensional Dirac delta-function and L� is the adjoint oper-ator of L; however, in this case, L ¼ L� (self-adjoint).

It can be shown that Green’s function Gðn; g; x; yÞ can be expressed as, [6],

Gðn; g; x; yÞ ¼X1p¼1

/pðx; yÞ/pðn; gÞkp

; ð23Þ

where /pðx; yÞ and kp are the p-th eigenfunction and eigenvalue, respectively, for the followingeigenvalue problem:

L/pðn; gÞ ¼ kp/pðn; gÞ : ð24Þ

The eigenfunctions are assumed as a series of admissible functions as

/pðx; yÞ ¼XN

q¼1

cpqfqðx; yÞ ; ð25Þ

where cpq are the undetermined coefficients and the admissible functions fqðx; yÞ satisfy theprescribed homogeneous boundary conditions. If the region is expressed by a set of curvedlines ðg1ðx; yÞ; g2ðx; yÞ; . . .Þ, their forms, [6], are given by

fqðx; yÞ ¼ g1ðx; yÞ � g2ðx; yÞ � � ðF00 þ F10x þ F01y þ F20x2 þ F11xy þ F02y2 þ Þ ;

ð26Þ

where the unknown coefficients Fij are determined so as to satisfy the boundary conditions.Substituting Eq. (25) into Eq. (24) and using the Galerkin method, Eq. (24) yields the fol-

lowing matrix form:

XN

q¼1

Arqcpq ¼XN

q¼1

kpBrqcpq ; ð27Þ

where

Arq ¼ZZ

R

frðx; yÞLfqðx; yÞdx dy; Brq ¼ZZ

R

frðx; yÞfqðx; yÞdx dy : ð28Þ

4Numerical results and discussionNumerical calculations are carried out for an FGM plate with all edges clamped, which is madeof zirconium oxide and titanium alloy. For the numerical calculations, we used the dimen-sionless quantities as follows:

ð�xx; �yy; �zzÞ ¼ x

a;y

b;

z

h

� �; s ¼ km

t

a2; �wwð�xx; �yy; sÞ ¼ wðx; y; tÞ

ðamTmaÞ ;

where �xx, �yy and �zz denote, respectively, the dimensionless position in the x, y and z-direction ofan FGM plate; a, b and h denote, respectively, the length of x and y-direction and thicknessof the FGM plate; �ww and s denote, respectively, the dimensionless deflection and the dimen-sionless time; and km and am are the thermal diffusivity and the linear expansion coefficient ofmetal, respectively.

131

As an illustrative example, we consider the following temperature boundary conditions:

Tð�xx; �yyÞ ¼ T0 þ 16Tm�xxð1 � �xxÞ�yyð1 � �yyÞ at 0 < �xx < 1; 0 < �yy < 1; �zz ¼ 1;

Tð�xx; �yyÞ ¼ T0 at 0 < �xx < 1; 0 < �yy < 1; �zz ¼ 0;

Tð�xx; �yyÞ ¼ T0 at �xx ¼ 0 and �xx ¼ 1; 0 < �yy < 1; 0 < �zz < 1;

Tð�xx; �yyÞ ¼ T0 at 0 < �xx < 1; �yy ¼ 0 and �yy ¼ 1; 0 < �zz < 1 ;

where T0 and Tm denote the initial temperature and an arbitrary parameter, respectively. Thethermal boundary condition on the upper face of the FGM plate is shown in Fig. 1. We assumedthat the volumetric ratio of metal, the porosity and the z-direction material properties as afunction of position are expressed as Vm ¼ ð1 � �zzÞM , P ¼ Ap�zzð1 � �zzÞ and those in Ref. [1],respectively.

Figures 2 and 3 show the dimensionless transient temperature

�TTð�xx; �yy; �zz; sÞ ¼ Tðx; y; z; tÞ � T0

Tm

at �zz ¼ 0:8 and �zz ¼ 0:5 in the three-dimensional FGM plate. After checking the convergence ofseries in the temperature solution, the truncated numbers of series are selected as k ¼ 20,m ¼ 20, n ¼ 50, and the number of layers L ¼ 50 are used in Eq. (11).

Fig. 1. Thermal boundary conditionon the upper face of an FGM plateða ¼ bÞ

Fig. 2. Dimensionless transient tem-perature with dimensionless positionat �zz ¼ 0:8 and s ¼ 0:01 (Vm ¼ 50%,P ¼ 0:0, a ¼ b, a ¼ 5h)

132

Fig. 3. Dimensionless transient tem-perature with dimensionless positionat �zz ¼ 0:5 and s ¼ 0:01 (Vm ¼ 50%,P ¼ 0:0, a ¼ b, a ¼ 5h)

Fig. 4. Green’s function Gð�nn, �gg; x, y)with a point load applied at (0.5, 0.5),Vm ¼ 50%, P ¼ 0:0, a ¼ b, a ¼ 5h

Fig. 5. Green’s function Gð�nn, �gg; x, y)with a point load applied at (0.25,0.25), Vm ¼ 50%, P ¼ 0:0, a ¼ b,a ¼ 5h

133

Figures 4 and 5 show examples of Green’s function Gð�nn; �gg; �xx; �yyÞ given in Eq. (23), where apoint load is applied at (0.5, 0.5) and (0.25, 0.25) in an FGM plate with all clamped edges,respectively.

Table 1 shows the values for dimensionless deflection �ww at ð�xx; �yyÞ ¼ ð0:5; 0:5Þ for the trun-cated numbers of series and the number of layers in Eq. (11). When the number of admissiblefunctions N is fixed, the convergence of the dimensionless deflection depends on the number ofz-directional eigenvalue cn and layers L. When the truncated numbers of series and the numberof layers are selected as k ¼ 20, m ¼ 20, n ¼ 50 and L ¼ 50, the sufficient convergence for thesolution is achieved.

Figures 6–8 show the dimensionless deflection �ww of a FGM and a homogeneous plate at threekinds of position with variation of the number of admissible functions fqðx; yÞ given in Eq. (25).As shown in Figs. 6–8, the dimensionless deflection values converge around the number ofadmissible functions N ¼ 10 independently of the dimensionless time and the dimensionlesspositions, and the convergence of the dimensionless deflection is very fast within a shortdimensionless time ðs ¼ 0:01Þ. As the volumetric ratio of metal Vm increases, the dimensionlessdeflection �ww increases.

Figures 9 and 10 show the dimensionless deflection �ww of a FGM plate at �xx ¼ 0:5, whichcorresponds to different values of the dimension ratio ðDR ¼ A=hÞ in steady and unsteadystates, respectively.

Figures 11 and 12 show the dimensionless deflection distribution �ww over a FGM plate with allclamped edges at s ¼ 0:01 and s ¼ 1, respectively.

Table 1. Comparison of the dimensionless deflection �wwð�xx ¼ 0:5; �yy ¼ 0:5; sÞ in an FGM plate with allclamped edges (Vm = 50%, P = 0.0, a = b, a = 5h)

N k m n L �wwð�xx ¼ 0:5; �yy ¼ 0:5; sÞ

s ¼ 0:01 s ¼ 1

15 10 10 20 20 0.439776E-1 0.539336E-115 20 20 20 30 0.437514E-1 0.538116E-115 20 20 30 40 0.437370E-1 0.538116E-115 20 20 40 50 0.437341E-1 0.538116E-115 20 20 50 50 0.437341E-1 0.538116E-1

Fig. 6. Dimensionless deflection�wwð�xx; �yy; sÞ of an FGM and a homo-geneous plate at �xx ¼ 0:25 and �yy ¼ 0:25ðP ¼ 0:0; a ¼ b; a ¼ 5hÞ

134

5ConclusionsGreen’s function approach to the analysis of the three-dimensional heat conduction equationand the deflection of a FGM plate with unidirectionally dependent properties is proposed.Green’s functions are formulated by the use of the laminate theory and the Galerkin method,

Fig. 8. Dimensionless deflection�wwð�xx; �yy; sÞ of an FGM and a homo-geneous plate at �xx ¼ 0:50 and �yy ¼ 0:50ðP ¼ 0:0; a ¼ b; a ¼ 5hÞ

Fig. 7. Dimensionless deflection�wwð�xx; �yy; sÞ of an FGM and a homo-geneous plate at �xx ¼ 0:25 and �yy ¼ 0:50ðP ¼ 0:0; a ¼ b; a ¼ 5hÞ

135

respectively, and are expressed by eigenvalues and corresponding eigenfunctions. With theGreen’s functions for a temperature field and the deflection of an FGM plate, it is possible toexpress the transient temperature and the deflection as a convolution-type integral betweenGreen’s function and thermal loadings; thus redundant calculations can be avoided when thethermal boundary conditions are altered. By comparison of numerical results with the numberof admissible functions, we show that the convergences of the dimensionless deflection are veryfast in steady and unsteady states. Applications of the proposed method to the analysis of the

Fig. 9. Dimensionless deflection�wwð�xx; �yy; sÞ at �xx ¼ 0:5 with variousdimension ratios in an FGM plateðVm ¼ 50%; P ¼ 0:0; a ¼ b; s ¼ 0:01Þ

Fig. 10. Dimensionless deflection�wwð�xx; �yy; sÞ at �xx ¼ 0:5 with variousdimension ratios in an FGM plateðVm ¼ 50%; P ¼ 0:0; a ¼ b; s ¼ 1Þ

Fig. 11. Dimensionless deflection�wwð�xx; �yy; sÞ of an FGM plate with allclamped edges at s ¼ 0:01ðVm ¼ 50%; P ¼ 0:0; a ¼ b; a ¼ 5hÞ

136

transient temperature and the deflection of a practical FGM plate subjected to transient thermalloadings are possible with sufficient accuracy.

References1. Obata, Y.; Noda, N.: Unsteady thermal stresses in a functionally gradient material plate (Analysis

of one-dimensional unsteady heat transfer problem). Trans JSME, Series A 59 560 (1993) 1090–1096(in Japanese)

2. Obata, Y.; Noda, N.: Steady thermal stresses in a hollow circular cylinder and a hollow sphere of afunctionally gradient material. J Thermal Stresses 17(4) (1994) 471–487

3. Tanigawa, Y.: Some basic thermoelastic problems for nonhomogeneous structural materials. TransASME J Appl Mech 48(6) (1995) 287–300

4. Ootao, Y.; Tanigawa, Y.: Three dimensional transient thermal stress analysis of a nonhomogeneoushollow sphere with respect to rotating heat source. Trans JSME, Series A 60 578 (1994) 2273–2279(in Japanese)

5. Tanigawa, Y.; Akai, T.; Kawamura, R.; Oka, N.: Transient heat conduction and thermal stress problemsof a nonhomogeneous plate with temperature-dependent material properties. J Thermal Stresses 19(1996) 77–102

6. Diaz, R.; Nomura, S.: Numerical Green’s function approach to finite-sized plate analysis. Int J SolidsStruct 33 (1996) 4215–4222

7. Nomura, S.; Sheahen, D.M.: Green’s function approach to the analysis of functionally graded materials.ASME MD-80 (1997) 19–23

8. Beck, J.V.; Cole, K.D.; Haji-Sheikh, A.; Litkouki, B.: Heat conduction using Green’s functions.Washington D.C., Hemisphere Publishing Co., 1992

9. Bytovski, A.G.: Characteristics of distributed parameter systems. Moscow, Nauka 1979

Fig. 12. Dimensionless deflection�wwð�xx; �yy; sÞ of an FGM plate with allclamped edges at s ¼ 1ðVm ¼ 50%; P ¼ 0:0; a ¼ b; a ¼ 5hÞ

137