功能梯度材料板受溫度載重作用之理論推導及數值分析(III)

行政院國家科學委員會專題研究計畫 成果報告

功能梯度材料板受溫度載重作用之理論推導及數值分析 (III)

研究成果報告(精簡版)

計 畫 類 別 ： 個別型

計 畫 編 號 ： NSC 99-2221-E-011-040-

執 行 期 間 ： 99 年 08 月 01 日至 100 年 07 月 31 日 執 行 單 位 ： 國立臺灣科技大學營建工程系

計 畫 主 持 人 ： 張燕玲

公 開 資 訊 ： 本計畫可公開查詢

中 華 民 國 100 年 12 月 18 日

中 文 摘 要 ： 本研究推導簡支 FGM 矩形板在溫度載重下之撓曲、應力、軸 力、力矩等力學行為，並以 MARC 軟體進行有限元素法之數值 分析。溫度分佈為 方向呈線性變化、溫度分佈為 方向呈線 性變化，而且 FGM 矩形板內之材料分佈為：指數函數型、冪 次方函數型、及 S 型分佈時，探討其熱應力行為，以了解各 種材料分佈的優缺點，進而獲得最佳之材料分佈。

結果顯示，本文已推導出 FGM 矩形板在溫度載重下之撓曲、

應力、軸力、力矩等之 closed-form solution. 此 closed- form solution 與有限元素法之數值分析結果很一至，也驗 證此結果之正確性。此外，功能梯度材料的使用，能將板內 之最大熱應力由板之上緣（或下緣）移到板之內部，且熱應 力之最大值也明顯降低。

中文關鍵詞： 功能梯度材料板，理論推導，有限元素法，熱應力

英 文 摘 要 ： This study uses Fourier series expansion to analyze simply supported FGM plates with the Young’s modulus varying along the thickness of the plate under linear temperature change in both x- and z-axes. The

material properties of the FGM plate, Young’s modulus, Poisson’s ratio, and the linear

coefficients of the thermal expansion, on the upper and lower surfaces are different but are preassigned according to the performance demands. However, the material properties of the FGM plates vary

continuously in the thickness direction (z-axis) only, such that the Young’s modulus, Poisson’s ratio, and the linear coefficients of the thermal expansion are functions of z only.

Results show that the theoretical solutions of the deflection, strain, stress, and moment of an FGM plate with the Young’s modulus varying along the thickness, , have been derived based on the medium- thick plate assumption, and are functions of the quantities . Theoretical solutions are confirmed by finite element analysis. The use of FGM moves the maximum stress form the top or bottom surface to the inner portion of the FGM plate concerned, and

significantly reduces the maximum stress of the plates.

英文關鍵詞： Plate of Functionally Graded Material, Theroetical Derivation, Finite Element Method, Thermal Stress

行政院國家科學委員會專題研究計畫成果報告

功能梯度材料板受溫度載重作用之理論推導及數值分析 (I II )

計畫編號：NS C 9 9 - 22 2 1 - E- 0 1 1 - 0 4 0

執行期限：2010 年 8 月 1 日至 2011 年 7 月 31 日

Theoretical derivation and numerical analysis of functionally graded material plates subjected to thermal loadings (III)

Yen-Ling Chung, Professor of National Taiwan University of Science and Technology Chi-rong Chang and Xiu-Juan Yu, Research Assistant

1. Introduction

Composite media have been widely used because of the high performance demands of engineering devices. However in the interfaces of the composite medium, there exists stress concentration occurred by the mismatch of material properties. Specially, in the environment of high-temperature change, such as supersonic transport, coating process, nuclear fusion reactors and so on, high residual stresses due to the mismatch in the material properties will cause the cracking or debonding of the structure. Therefore, the concept of Functional Graded Material (FGM) was introduced [1,2] to decrease the mismatch in the material properties and to reduce the residual and thermal stresses.

FGMs are one kind of composite materials accomplished by continuously varying the volume fractions in thickness direction to obtain a predetermined profile [3-7]. FGMs for high-temperature application are widely compound by ceramics and metals. Ceramics in the ceramic/metal FGM are toughened and strengthened by metallic composition, but the ceramic provides thermal barrier property and protects

modulus and Poisson’s ratio are are approximately assumed as constant, cited by Paulino[8], such as MoSi2/Al2O3 system, TiC/SiC system. The Young’s modulus in zirconia/nickel FGM doen not change significantly because nickel alloys and partially stablized zirconia have similar Young’s modulus. The thermal stress problems of such kind of FGMs subjected to thermal loading will become much simplifier.

One of the most important issues in FGM study is how to reduce the thermal stresses and to determine the optimally compositional profile of FGM. Studies showed that adding a FGM in ceramic/metal composite material will significantly relax the residual stresses[ 9-12]. Moreover, using the FGM coating will decrease the damage of the structure when subjected to thermal shock. Lee and Erodgan [13]studied the exponentially metal-rich, ceramic-rich, and linear FGMs under uniform thermal loading and showed that the metal-rich has the lowest stress singularity. Chung and Ji [ 14-16] evaluated the thermal stresses of S-type FGM by the finite element method. Yang and Munz [17 ] analytically calculated the stresses in a joint with FGM by using the plate theory.

In this paper, the problem of simply supported rectangular FGM plates subjected to temperature distribution linearly change in x- and z-directions. The FGM plate is assumed to have constant coefficients of thermal expansion and Poisson’s ratio; however, the Young’s modulus of the FGM plates varys continuously throughout the plate thickness, according to the volume fraction of the constituent of the multi-phase materials based on power-law, sigmoid, and exponential functions.

The closed-form solution to the problem concerned that is not found in the literature is obtained using the Fourier series expansion and proved by finite element calculation in this study.

2. Governing equations of FGM plates under transverse and thermal Loading

Consider an elastic rectangular FGM plate of uniform thickness which is subjected to a distributed transverse load q x y and is exposed to a temperature _z( , ) change T x y z . The deformations and the stresses of the FGM plate are based ( , , ) upon the following assumptions:

(1) The material properties of the FGM plate, Young’s modulus, Poisson’s ratio, and the linear coefficients of the thermal expansion, on the upper and lower surfaces are different but are preassigned according to the performance demands. However, the material properties of the FGM plates vary continuously in the thickness direction (z-axis) only, such that the Young’s modulus, Poisson’s ratio, and the linear coefficients of the thermal expansion are functions of z only.

(2) The deflections of the FGM plate is small in comparison with its thickness h, such that the linear strain-displacement relations are valid.

(3) The FGM plate has medium thickness in which the plate is thick enough to carry transverse load but the plate is not so thick that transverse shear deformation becomes important. Consequently the stress in the thickness direction is negligible.

(4) A line element of the FGM plate normal to the middle surface of the plate remains straight and normal to the deformed middle surface, implying that the transverse strain components are negligibly small.

For a non-homogeneous elastic FGM plate in which the Young’s modulus E, Poisson’s ratio n and the coefficient of the thermal expansion a of the FGM plate are function of the spatial coordinate z, the stress-strain relation under thermal loading

( , , )

T x y z based on above assumptions is:

11 12

12 12

66

( ) ( ) 0 1

( ) ( ) ( , , )

( ) ( ) 0 1

1 ( )

0 0 ( ) 0

x x

y y

xy xy

E z E z

E z z T x y z

E z E z

E z z

s e

s e a

t g n

ì ü é ùì ü ì ü

ï ï=ê úï ï- ï ï

í ý ê úí ý - í ý

ï ï êë úûï ï ï ïî þ

î þ î þ

(1)

where

11 12

12 12 2

66

( ) ( ) 0 1 ( ) 0

( ) ( ) 0 ( ) ( ) 1 0

1 ( )

0 0 ( ) 1 ( )

0 0

2

E z E z z

E z E z E z z

E z z z

n n n

n

é ù

ê ú

é ù

ê ú

ê ú = ê ú

ê ú - ê ú

ê ú -

ë û ê ú

ë û

respect to the thickness coordinate z . The inplane axial forces N ，_x N_y，N_xy are defined by ^{/ 2}

/ 2 h

x h x

N =

ò

_- s dz^{, / 2}/ 2 h

y h y

N =

ò

_- s dz^{, and / 2}/ 2 h

xy h xy

N =

ò

_- t dz, and the bending momentsM ，_x M_y，M_xy are defined by ^{/ 2}

/ 2 h

x h x

M =

ò

_- z dzs ^{, / 2}/ 2 h

y h y

M =

ò

_- z dzs ^{, and}

/ 2 / 2 h

xy h xy

M =

ò

_- z dzt . Consequently, the axial forces and the bending moments in the matrix forms are as follows [ ]:

0

0

0

2 2

11 12 11 12 2

12 11 12 11 2

66 66 2

0 0

0 0

0 0 0 0 0

2

x T x

T

y y

xy xy

w

N A A B B x N

N A A B B w N

N A B y

w x y e

e g

ì -¶ ü

ï ï

ï ¶ ï

ì ü ì ü

ì ü é ù é ù

ï ï

ï ï ¶ ï ï

ï ï=ê úï ï+ê úï - ï-

í ý ê úí ý ê úí ¶ ý í ý

ï ï êë úûï ï êë úûï ï ï ï

î þ ïî ïþ ïï- ¶ ïï î þ

ï ¶ ¶ ï

î þ

(2)

0

0

0

2 2

11 12 11 12 2

12 11 12 11 2

66 66 2

0 0

0 0

0 0 0 0 0

2

x T x

T

y y

xy xy

w

M B B C C x M

M B B C C w M

M B C y

w x y e

e g

ì -¶ ü

ï ï

ï ¶ ï

ì ü ì ü

ì ü é ù é ù

ï ï

ï ï ¶ ï ï

ï ï=ê úï ï+ê úï - ï-

í ý ê úí ý ê úí ¶ ý í ý

ï ï êë úûï ï êë úûï ï ï ï

î þ ïî ïþ ïï- ¶ ïï î þ

ï ¶ ¶ ï

î þ

(3)

where

0, 0, 0

x y xy

e e g are the strains at the middle surface; w is the deflection of the plate; and

/ 2 / 2

( ) ( )

( , , ) 1 ( )

T h h

E z z

N T x y z dz

z a n

= -

ò

- ⁽⁴⁾

/ 2 / 2

( ) ( )

( , , ) 1 ( )

T h h

zE z z

M T x y z dz

z a n

= -

ò

- ⁽⁵⁾

The coefficients of A_ij，B_ij，C_ij in Eqs. (2) and (3) are the integration of the material properties of the FGM plate and they are:

/ 2 2

( _ij, _ij, _ij) ^h/ 2(1, , ) _ij( ) A B C h z z E z dz

=

ò

- ⁽⁶⁾

The equilibrium equations of the FGM plates are (Chi and Chung, 2006):

yx 0

x N

N

x y

¶ ¶

+ =

¶ ¶ (7)

yx y 0

N N

x y

¶ ¶

+ =

¶ ¶ (8)

2 2

2

2^x 2 M^xy M2^y 0

M

x x y y

¶ ¶

¶ + + =

¶ ¶ ¶ ¶ (9)

To satisfy the inplane equations of Eqs. (7) and (8), a stress function ^f( )^{x y, is}

introduced such that

2 x 2

N y

f

=¶

¶ ；

2 y 2

N x

f

=¶

¶ ；

2

Nxy

x y f

= - ¶

¶ ¶ (10)

Then by the use of Eqs. (2), (3) and (10), the strains at middle surface and the bending moment can be rearranged and expressed in terms of the stress function^f( )^{x y, and}

the deflection w . Consequently, the equilibrium equation, Eq. (9), becomes:

( )

( )

4 4 4

12 4 11 66 2 2 12 4

4 4 4

11 4 12 66 2 2 11 4

2 2 2 2

11 12 2 2 2 2

2 2

2 4

( )

T T T T

Q Q Q Q

x x y y

w w w

S S S S

x x y y

N N M M

Q Q

x y x y

f f f

¶ + - ¶ + ¶

¶ ¶ ¶ ¶

¶ ¶ ¶

+ + + +

¶ ¶ ¶ ¶

æ¶ ¶ ö æ¶ ¶ ö

= - + çè ¶ + ¶ ÷ çø è- ¶ + ¶ ÷ø

(11)

The definitions of the quantities Q_ij and S_ij can be found in Chi and Chung (2006).

The stress function ^f( )^{x y,} and the deflection w in Eq. (11) are unknowns, therefore one more equation is needed. A compatibility equation is then used to provide the governing equation for^f( )^{x y,} . The compatibility equation of a two-dimensional plate is

2 2

2

2 2

y xy

x

y x x y

e g

e ^{¶ ¶}

¶ + =

¶ ¶ ¶ ¶ . After some manipulations, the compatibility equation of an FGM plates can be rewritten in terms of stress function

( )^{x y,}

f and the deflection w as (Chi and Chung, 2006):

( )

( )

4 4 4

11 4 12 66 2 2 11 4

4 4 4 2 2

12 4 11 66 2 2 12 4 11 12 2 2

2

2 ( )

T T

P P P P

x x y y

w w w N N

Q Q Q Q P P

x x y y x y

f f f

¶ + - ¶ + ¶

¶ ¶ ¶ ¶

æ ö

¶ ¶ ¶ ¶ ¶

- ¶ - - ¶ ¶ - ¶ = - + çè ¶ + ¶ ÷ø

(12)

The equilibrium equations, Eq.(11), and the compatibility equation, Eq. (12), provide the simultaneous equations to solve for the stress function ^f( )^{x y,} and the deflection

w for an FGM plate subjected to thermal loads.

3. Series Solution of Simply Supported FGM plates under thermal loading

Consider a simply supported rectangular FGM plate with length a , width b, and uniform thickness h, as shown in Fig. 1. It is assumed that no transverse load acts on the plate, but the plate is subjected to a temperature change

( , , ) _V( ) _H( , )

T x y z =T z T x y where T z and _V( ) T x y are the temperature change in _H( , ) the z-direction and x-y plane, respectively. For the simply supported FGM plates, the transverse and tangential components of displacements are restricted to move but the normal components are allowed on all four edges. Therefore, the boundary conditions of the simply supported rectangular FGM plate are:

0 0

x x 0

v w at x and x a

N M

ì = =

= =

í = =

î (13a)

0 0

y y 0

u w at y and y b

N M

ì = =

= =

í = =

î (13b)

Expanding the thermal load T x y z into Fourier series: ( , , )

( , , ) _V( ) _H( , ) _V( ) _mnsinm xsinn y T x y z T z T x y T z T

a b

p p

= =

åå

^(14a)

where mn ⁴ H( )^{, sin sin}

m x n y

T T x y dxdy

ab a a

p p

=

ò ò

^(14b)

Substituting Eq. (22a) into Eqs. (9) and (10), we obtain that

* sin sin

T

mn m n

m x n y

N N T

a b

p p

=

åå

^(15a)

* sin sin

T

mn m n

m x n y

M M T

a b

p p

=

åå

^(15b)

Where

* / 2 / 2

( ) ( ) ( ) 1 ( )

h V

h

E z z T z

N dz

z a

n

= -

ò

- ^(16a)

* / 2 / 2

( ) ( ) ( ) 1 ( )

h V

h

zE z z T z

M dz

z a

n

= -

ò

- ^(16b)

To satisfy the loading condition in Eq. (14a) and the boundary conditions in Eqs. (13), the displacement w and the stress function ^f( )^{x y,} of the FGM plate can be in the form of:

( , ) _mnsin sin

m n

m x n y

w x y w

a b

p p

=

åå

^(17a)

( , ) _mnsin sin

m n

m x n y

x y a b

p p

f =

åå

f ^(17b)

where w and _mn f_mn are an unknown constants and can be determined by substituting Eq. (17) into the equilibrium and compatibility equations .

where w_mn and f_mn are unknown constants and can be determined from equilibrium and compatibility equations . By substituting Eq. (17) into the equilibrium and the compatibility equations in Eqs. (11) and (12), and then solving the simultaneous equations, the coefficients of w and _mn f_mn are obtained as:

2 2

mn mn mn mn

J K K H

w T T

K HJ K HJ

x ^- h f x ⁺ h

æ ö æ ö

=çè + ÷ø =çè + ÷ø (18)

where

( )

4 2 2 4

11 m 2 12 2 66 m n 11 n

H S S S S

a a b b

p p p p

æ ö æ ö æ ö æ ö

= ç ÷ + + ç ÷ ç ÷ + ç ÷

è ø è ø è ø è ø (18c)

( )

4 2 2 4

11 m 2 12 66 m n 11 n

J P P P P

a a b b

p p p p

æ ö æ ö æ ö æ ö

= ç ÷ + - ç ÷ ç ÷ + ç ÷

è ø è ø è ø è ø (18d)

( )

4 2 2 4

12 m 2 11 66 m n 12 n

K Q Q Q Q

a a b b

p p p p

æ ö æ ö æ ö æ ö

= ç ÷ + - ç ÷ ç ÷ + ç ÷

è ø è ø è ø è ø (18e)

( 11 12) ^{2 2}

m n

Q Q N M

a b

p p

x =^éë + ^*+ ^*ùû è^éêêë^æç ^ö÷ø +^æçè ^ö÷ø ^ùúúû (18f)

2 2

11 12

( ) m n

N P P

a b

p p

h= ^* + ^éêêë^æçè ^ö÷ø +^æçè ^ö÷ø ^ùúúû (18g)

Consequently, the strains at middle surface are:

( ) ( )

( )

( ) ( )

( )

( )

0

2

12 11

2

2

11 12

11 12

sin sin

sin sin

x mn m n

mn m n

T m

P K H Q J K

K HJ a

n m x n y

P K H Q J K

b a b

m x n y

P P N T

a b

e x h x h p

p p p

x h x h

p p

*

ìï æ ö

= + íïî - + + - çè ÷ø

æ ö üï

+ - + + - çè ÷ø ïþý

+ +

åå

åå

(19a)

( ) ( )

( )

( ) ( )

( )

( )

0

2

11 12

2

2

12 11

11 12

sin sin

sin sin

y mn m n

mn m n

T m

P K H Q J K

K HJ a

n m x n y

P K H Q J K

b a b

m x n y

P P N T

a b

e x h x h p

p p p

x h x h

p p

*

ìï æ ö

= + íïî - + + - çè ÷ø

æ ö üï

+ - + + - çè ÷ø ïþý

+ +

åå

åå

(19b)

0

cos cos

gxy = x + h - x - h

And the strain and stress fields of the FGM plate under thermal loads are found as:

( ) ( )( )

( ) ( )

( )

2

12 11

2

2

11 12

11 12

sin sin

sin sin

x mn m n

mn m n

T m

P K H Q z J K

K HJ a

n m x n y

P K H Q J K

b a b

m x n y

P P N T

a b

e x h x h p

p p p

x h x h

p p

*

ìï æ ö

= + íïîéë- + + + - ùûçè ÷ø

æ ö üï + -éë + + - ùû è ø ïþç ÷ ý

+ +

åå

åå

(19d)

( ) ( )

( )

^{( )( )}

( )

2

11 12

2

2

12 11

11 12

sin sin

sin sin

mn y

m n

mn m n

T m

P K H Q J K

K HJ a

n m x n y

P K H Q z J K

b a b

m x n y

P P N T

a b

e x h x h p

p p p

x h x h

p p

*

*

ìï æ ö

= + íïîéë- + + - ùûçè ÷ø

üï

æ ö

é ù

+ -ë + + + - ûè ø ïþç ÷ ý

+ +

åå

åå

(19e)

( ) ( )( )

66 66

2 2

cos cos

xy mn

m n

T P K H Q z J K

K HJ

m n m x n y

a b a b

g x h x h

p p p p

= + éë + - + - ùû

æ öæ ö

´çè ÷çøè ÷ø

åå

(19f)

and

( ) ( )( )

( ) ( ) ( )

2 2

12 11

2 2

2 2

11 12

11 12

( ) ( )

1 ( )

( ) sin sin

( ) ( )

1 ( )

mn x

m n

T

E z m n

P K H Q z J K z

z K HJ a b

n m m x n y

P K H Q J K z

b a a b

E z z

P P N z

p p

s x h x h n

n

p p p p

x h x h n

a n

*

ì é ù

ï æ ö æ ö

= - + íïîéë- + + + - ù êûêëçè ÷ø + çè ÷ø úúû éæ ö æ ö ùïü

+ -éë + + - ù êû è øêëç ÷ + çè ÷ø úýúïûþ

+ +

-

åå

sin sin ( ) ( ) ( )

sin sin

1 ( )

mn m n V

mn m n

m x n y

T a b

E z z T z m x n y

z T a b

p p

a p p

- n -

åå åå

(19g)

( ) ( )

( ) ( )( ) ( )

2 2

11 12

2 2

2 2

12 11

11 12

( ) ( )

1 ( )

( ) sin sin

( )

1 ( )

mn y

m n

mn

T

E z m n

P K H Q J K z

z K HJ a b

n m m x n y

P K H Q z J K z

b a a b

E z P P N T

z

p p

s x h x h n

n

p p p p

x h x h n

n ^*

ì é ù

ï æ ö æ ö

= - + íïîéë- + + - ù êûêëçè ÷ø + çè ÷ø úúû éæ ö æ ö ùïü

+ -éë + + + - ù êû è øêëç ÷ + çè ÷ø úýúïûþ

+ +

-

åå

sin sin ( ) ( ) ( )

sin sin

1 ( )

m n V

mn m n

m x n y

a b

E z z T z m x n y

z T a b

p p

a p p

- n -

åå åå

(19h)

( ) 2 66( ) ( 66 )( )

( ) 2

2 1 ( )

cos cos

xy mn

m n

E z T

P K H Q z J K

z K HJ

m n m x n y

a b a b

s x h x h

n

p p p p

= + + éë + - + - ùû

æ öæ ö

´çè ÷çøè ÷ø

åå

(19i)

The inplane axial forces and the bending moments of the FGM plate subjected to thermal loads are also obtained:

2 2

2 2 sin sin

x mn

m n

K H n m x n y

N T

y K HJ b a b

f x h p p p

¶ æ + öæ ö

= ¶ = -

åå

çè + ÷çøè ÷ø ^(19j)

2 2

2 2 sin sin

y mn

m n

K H m m x n y

N T

x K HJ a a b

f x h p p p

¶ æ + öæ ö

= ¶ = -

åå

çè + ÷çøè ÷ø ^(19k)

2

2 cos cos

xy mn

m n

K H m n m x n y

N T

x y K HJ a b a b

f x h p p p p

¶ æ + öæ öæ ö

= -¶ ¶ = -

åå

çè + ÷çøè ÷çøè ÷ø ^(19l)

and

( ) ( )

( ) ( )

( )

2

12 11

2

2

11 12

11 12

sin sin

sin sin

x mn m n

mn m n

T m

M Q K H S J K

K HJ a

n m x n y

Q K H S J K

b a b

m x n y

Q Q N M T

a b

x h x h p

p p p

x h x h

p p

* *

ìï æ ö

= + íïîéë + + - ùûçè ÷ø

æ ö üï +éë + + - ùû è ø ïþç ÷ ý

é ù

-ë + + û

åå

åå

(19m)

( ) ( )

( ) ( )

( )

2

11 12

2

2

12 11

11 12

sin sin

sin sin

y mn m n

mn m n

T m

M Q K H S J K

K HJ a

n m x n y

Q K H S J K

b a b

m x n y

Q Q N M T

a b

x h x h p

p p p

x h x h

p p

* *

ìï æ ö

= + íïîéë + + - ùûçè ÷ø

æ ö üï +éë + + - ùû è ø ïþç ÷ ý

é ù

-ë + + û

åå

åå

(19n)

( ) ( )

66 66

2 2

cos cos

xy mn

m n

M T Q K H S J K

K HJ

m n m x n y

a b a b

x h x h

p p p p

= -+ éë + + - ùû

æ öæ ö

´çè ÷çøè ÷ø

åå

(19o)

4. Solution to the FGM plate with E E z= ( ) only

If both the Young’s modulus and the Poisson’s ratio are considered when calculating the coefficients A_ij, B_ij, and C_ij in Eq. (6), the integration will turn out very complicate. Delale and Erdogan (1983) indicated that the influence of the Poisson’s ratio on the deformation of the FGM plates will be much less than that of the Young’s modulus. The same conclusion also obtained by Chi and Chung

(2006b).

Therefore, this paragraph will derive the solutions for the FGM plates havine constant Poisson’s ratio and the coefficient of thermal expansion but the Young’s modulus varying in the thickness direction. For the material with a and n = constant butE=E z( ), it can be found that:

12 11

A =nA , A₆₆ = -(1 n)A₁₁/ 2

12 11

B =nB , B₆₆ = -(1 n)B₁₁/ 2

12 11

C =nC , C₆₆= -(1 n)C₁₁/ 2

11 2

11

1 (1 )

P = v A

- , P₁₂ = -nP₁₁, P₆₆ = -2(1+v P) ₁₁

11 11

11

Q B

= - A , Q₁₂ = , 0 Q₆₆ =Q₁₁,

11 11 11 11

S =B Q +C , S₁₂ =vS₁₁, S₆₆ = -(1 n)S₁₁/ 2 (20)

4.1 Material gradation

Further assume that the Young’s modulus E z varies continuously in the ( ) thickness direction (z-axis) based on power-law function (simply called P-FGM in this study), sigmoid function (S-FGM), or exponential function (E-FGM).

(A) The P-FGM plates

For the P-FGM plates, the Young’s modulus is defined as:

( ) ( ) 1

[

1 ( )

]

2

E z =g z E + -g z E with ^{g z}( ) ^{z h/ 2 p}

h

æ + ö

= ç ÷

è ø (21)

where E and ₁ E are the Young’s moduli of the lowest (₂ z h= / 2) and top surfaces (z= -h/ 2) of the FGM plate, respectively. The variation of the Young’s modulus of a P-FGM plate (p= ) in the thickness direction with different steep of Young’s 2 modulus is plotted in Fig. 2.

(B) The S-FGM plates

The Young’s modulus of S-FGM plates based on two power-law functions is defined as:

1 1 1 2

( ) ( ) [1 ( )]

E z =g z E + -g z E for 0£ £z h/ 2 (22a)

2 1 2 2

( ) ( ) [1 ( )]

E z =g z E + -g z E for -h/ 2£ £z 0 (22b) with 1( )

1 / 2

1 2 / 2

h z p

g z h

æ - ö

= - çè ÷ø for 0£ £z h/ 2 (22c)

( )

2

1 / 2

2 / 2

h z p

g z h

æ + ö

= çè ÷ø for -h/ 2£ £z 0 (22d)

The variation of the Young’s modulus of an S-FGM plate (p= ) in the thickness 2 direction with different steep of Young’s modulus is plotted in Fig. 3

(C) The E-FGM plates

The Young’s modulus of the E-FGM plates is based on:

( / 2)

2 1 2

( ) ^{B z h} , where , 1ln( / )

E z Ae A E B E E

h

= + = = (23)

The variation of the Young’s modulus of an E-FGM plate in the thickness direction with different steep of Young’s modulus is plotted in Fig. 4.

By substituting the gradation of the Young’s moduli of P-, S-, or E-FGM plates in Eq. (21~23) into the definition of coefficients in Eqs. (6), coefficients A , ₁₁ B , ₁₁ C , 11 P , ₁₁ Q , and ₁₁ S of P-, S-, or E-FGM plates can be obtained, which are the ₁₁ same as those listed in Chi and Chung (2006a).

The thermal axial force N and thermal bending^* M defined in Eqs. (16) depend ^* on the temperature change in the z-direction T z . Therefore two kinds of thermal _V( ) loads are assumed. One is linear temperature change in the x-direction in whichT z =1 ; another is linear temperature change in the z-direction. _V( )

4.2 linear temperature change in the x-direction

The temperature distribution in the Fig. 1 problem linearly changes from T at ₀