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© 2018 IAU, Arak Branch. All rights reserved. Journal of Solid Mechanics Vol. 10, No. 4 (2018) pp. 734-752 One-Dimensional Transient Thermal and Mechanical Stresses in FGM Hollow Cylinder with Piezoelectric Layers S.M. Mousavi * , M. Jabbari, M.A. Kiani Mechanical Engineering Department, South Tehran Branch, Islamic Azad University, Iran Received 7 July 2018; accepted 12 September 2018 ABSTRACT In this paper, an analytical method is developed to obtain the solution for the one dimensional transient thermal and mechanical stresses in a hollow cylinder made of functionally graded material (FGM) and piezoelectric layers. The FGM properties are assumed to depend on the variable r and they are expressed as power functions of r but the Poisson’s ratio is assumed to be constant. Transient temperature distribution, as a function of radial direction and time with general thermal boundary conditions on the inside and outside surfaces, is analytically obtained for different layers, using the method of separation of variables and generalized Bessel function. A direct method is used to solve the Navier equations, using the Euler equation and complex Fourier series. This method of solution does not have the limitations of the potential function or numerical methods as to handle more general types of the mechanical and thermal boundary conditions. © 2018 IAU, Arak Branch. All rights reserved. Keywords: Transient; Symmetric thermal stress; Hollow cylinder; Functionally graded material; Piezoelectric. 1 INTRODUCTION UNCTIONALLY graded materials (FGMs) are composites with material properties varying smoothly in one or more directions which exhibit preferred structural responses. These materials are useful to withstand high thermal stresses where high heat fluxes and large temperature gradients exist. Therefore, these materials are chosen to use in structure components of aircraft, aerospace vehicles, nuclear plants as well as various temperature shielding structures widely used in industries [1]. On the other hand, piezoelectric materials are widely used in modern engineering due to its direct and inverse effects. The usage of piezoelectric layer as distributed sensors and actuators in active structure control as noise attenuation, deformation control, and vibration suppression have attracted serious attention. Piezoelectric materials are, in fact, capable of altering the response of the structures through sensing and actuation [2], by integrating the surface bonded and embedded actuators in structural systems, the desired localized strains may be induced in the structures thanks to the application of an appropriate voltage to the actuators. Such an electromechanical coupling allows closed-loop control systems to be built up, in which piezoelectric materials play the role of both the actuators and the sensors. The theoretical analysis of a three-dimensional transient thermal stress problem for a nonhomogeneous hollow circular cylinder due to a moving heat source in the axial direction from the inner and /or outer surfaces is developed by Ootao and Tanigawa [3]. Using perturbation techniques, Obata and ______ * Corresponding author. Tel:. +98 9124213078. E-mail address: [email protected] (S.M. Mousavi). F

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Page 1: One-Dimensional Transient Thermal and Mechanical Stresses in FGM Hollow Cylinder …jsm.iau-arak.ac.ir/article_545716_2fec8bf49c3416c8383be... · 2021. 1. 10. · mechanical and thermal

© 2018 IAU, Arak Branch. All rights reserved.

Journal of Solid Mechanics Vol. 10, No. 4 (2018) pp. 734-752

One-Dimensional Transient Thermal and Mechanical Stresses in FGM Hollow Cylinder with Piezoelectric Layers

S.M. Mousavi *, M. Jabbari, M.A. Kiani

Mechanical Engineering Department, South Tehran Branch, Islamic Azad University, Iran

Received 7 July 2018; accepted 12 September 2018

ABSTRACT

In this paper, an analytical method is developed to obtain the solution for the

one dimensional transient thermal and mechanical stresses in a hollow

cylinder made of functionally graded material (FGM) and piezoelectric

layers. The FGM properties are assumed to depend on the variable r and they

are expressed as power functions of r but the Poisson’s ratio is assumed to be

constant. Transient temperature distribution, as a function of radial direction

and time with general thermal boundary conditions on the inside and outside

surfaces, is analytically obtained for different layers, using the method of

separation of variables and generalized Bessel function. A direct method is

used to solve the Navier equations, using the Euler equation and complex

Fourier series. This method of solution does not have the limitations of the

potential function or numerical methods as to handle more general types of

the mechanical and thermal boundary conditions.

© 2018 IAU, Arak Branch. All rights reserved.

Keywords: Transient; Symmetric thermal stress; Hollow cylinder;

Functionally graded material; Piezoelectric.

1 INTRODUCTION

UNCTIONALLY graded materials (FGMs) are composites with material properties varying smoothly in one

or more directions which exhibit preferred structural responses. These materials are useful to withstand high

thermal stresses where high heat fluxes and large temperature gradients exist. Therefore, these materials are chosen

to use in structure components of aircraft, aerospace vehicles, nuclear plants as well as various temperature shielding

structures widely used in industries [1]. On the other hand, piezoelectric materials are widely used in modern

engineering due to its direct and inverse effects. The usage of piezoelectric layer as distributed sensors and actuators

in active structure control as noise attenuation, deformation control, and vibration suppression have attracted serious

attention. Piezoelectric materials are, in fact, capable of altering the response of the structures through sensing and

actuation [2], by integrating the surface bonded and embedded actuators in structural systems, the desired localized

strains may be induced in the structures thanks to the application of an appropriate voltage to the actuators. Such an

electromechanical coupling allows closed-loop control systems to be built up, in which piezoelectric materials play

the role of both the actuators and the sensors. The theoretical analysis of a three-dimensional transient thermal stress

problem for a nonhomogeneous hollow circular cylinder due to a moving heat source in the axial direction from the

inner and /or outer surfaces is developed by Ootao and Tanigawa [3]. Using perturbation techniques, Obata and

______ *Corresponding author. Tel:. +98 9124213078.

E-mail address: [email protected] (S.M. Mousavi).

F

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735 S.M.Mousavi et al.

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Noda presented a solution for the transient thermal stresses in a plate made of FGM [4]. Jabbari et al. [5] studied a

general solution for mechanical and thermal stresses in a functionally graded hollow cylinder due to non-

axisymmetric steady-state load. They applied separation of variables and the complex Fourier series to solve the heat

conduction and Navier equations. These authors [6] also studied the mechanical and thermal stresses in functionally

graded hollow cylinder due to radially symmetric loads. Poultangari et al. [7] presented a solution for the

functionally graded hollow spheres under non-axisymmetric thermo mechanical loads. He et al. [8] derived the

active control of FGM plates with integrated piezoelectric sensors and actuators. Fesharaki et al. [9] presented 2D

solution for electro-mechanical behavior of functionally graded piezoelectric hollow cylinder by using the separation

of variables method and complex Fourier series; the Navier equations in term of displacements are derived and

solved. Hosseini and Akhlaghi [10] presented transient heat conduction in a cylindrical shell of functionally graded

material by using analytical method. Chu and Tzou [11] presented the transient response of a composite finite

hollow cylinder heated by a moving line source on its inner boundary and cooled convectively on the exterior

boundary using Eigen function expansion method and the Fourier series. Vaghari et al. [12] presented an analytical

method to obtain the transient thermal and mechanical stresses in a functionally graded hollow cylinder subjected to

the two-dimensional asymmetric loads. Mohazzab [13] presents the analytical solution of one-dimensional

mechanical and thermal stresses for a hollow cylinder made of functionally graded material.

In this paper, an analytical method is presented to obtain the transient thermal and mechanical stresses in a

functionally graded hollow cylinder with piezoelectric internal and external layers subjected to the one-dimensional

transient symmetric loads. Temperature distribution is considered in transient symmetric case and mechanical and

thermal boundary conditions are considered in general forms.

2 GOVERNING EQUATION

2.1 Stress distribution

Consider a functionally graded hollow cylinder of inner radius b and outer radius c, with external and internal

piezoelectric layers of radius d, a respectively as shown in Fig.1. Symmetric cylindrical coordinate (r) is considered

along the thickness. The FGM layer is graded through the r-direction thus the material properties are function of r.

Fig.1

Geometry of the problem studied.

The governing one-dimensional strain-displacement relations in cylindrical coordinate and electric field-electric

potential relations are

rr r

u uE

r r r

(1)

In which ,u as displacement component along the radial direction, and the electric potential, respectively. The

constitutive relations describing the mechanical and electrical interaction for a piezoelectric material are

11 21 11

11 12 11

12 22 21

( , )

( , )

( , )r rr r r

rr rr r r

rr r

D e e E P T

C C e E T r t

C C e E T r t

r t

(2)

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One-Dimensional Transient Thermal and Mechanical Stresses …. 736

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and the symmetric stress-strain relations for the FGM layer are

2 2 2

2 2

2 2 2

2 2

2 2 2

2 2

( ) ( ) ( )1 ( , )

1 1 2 1 2

( ) ( ) ( )1 ( , )

1 1 2 1 2

( ) ( ) ( )( , )

1 1 2 1 2

rr rr

rr

zz rr

E r E r rT r t

E r E r rT r t

E r E r rT r t

(3)

where ,ij ij and ( , )T r t are the stress and strain tensors and the temperature distribution, , , , ,ij ij ij i iC e D P and

i are elastic and piezoelectric coefficients, dielectric constants, electric displacements, pyroelectric constant and

thermal modulus respectively for the piezoelectric layers and ( ), ( )r E r are the coefficient of thermal expansion

and the Young’s modules respectively, v is Poisson’s ratio that assumed to be a constant for the FGM layer and

1 2

2 2( ) ( )m m

E r E r r r (4)

Here 1 2

0 0( ) / , ( ) /m m

E E b b b b , where ( )E b and ( )b are the modulus of elasticity and the coefficient

of thermal expansion of the inner FG material at r b , 1m and

2m are the material constants. The equilibrium

equations in the radial direction, disregarding the body forces and inertia terms and equation of electrostatic are

1 1( ) 0 0rr rr

rrrr

DD

r r r r

(5)

2.2 Heat conduction problem

The heat conduction equation for the functionally graded cylinder is

2,2

2 2, 2,

2 2 2 2 2

( ) 1 ( , )

( ) ( ) ( ) ( )

r

rr r

kk r R r tT T T

r c r k r r c r

(6)

where ( )k r is the thermal conductivity, ( )c r is specific heat capacity, ( )r is mass density, and ( , )R r t is the

energy source. A comma denotes partial differentiation with respect to the space variable. The symbol dot (·)

denotes derivative with respect to time. The initial and boundary conditions are assumed as:

2 1

2 1( ) | |r c r c

T Tk r k

r r

(7a)

2 3( , ) ( , )T b t T b t (7b)

2 3( ,0) ( )T r g r (7c)

The thermal FGM properties are assumed to be described with the power law functions as:

3 54

2 0 2 0 2 0( ) ( ) ( )m mm

k r k r r r c r c r (8)

Here, 31 2

0 0 0( ) / , ( ) / , ( ) /mm m

k k b b b b c c b b where ( ), ( )k b b and ( )c b are the coefficient of thermal

conduction, specific heat capacity and mass density of the inner FG material at r b and 3 4,m m and 5m are the

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737 S.M.Mousavi et al.

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material constants. As material properties in piezoelectric material is constant the heat conduction equation in one-

dimensional problem for the outside piezoelectric layer ( 1)j and the inside piezoelectric layer ( 3)j leads to

, ,

1 ( , )1,3

j

j j rr j r

j j j j

k R r tT T T j

c r c

(9)

and initial and boundary conditions for the 1st and 3

rd piezoelectric layers are assumed as:

11 1 12 1, 1, ,rX T d t X T d t g t (10a)

1 2, ,T c t T c t (10b)

1 2,0T r g r (10c)

21 3 22 3, 4( , ) ( , ) ( )rX T a t X T a t g t (11a)

32

2 3( ) | |r b r b

TTk r k

r r

(11b)

3 5( ,0) ( )T r g r (11c)

where ( , 1,2)ijX i j are Robin-type constants related to the thermal boundary condition parameters, and

2 3 5( ), ( ), ( )g r g r g r are the known initial condition. The solution of the heat conduction equations for temperature

distribution in FGM and piezoelectric layers may be assumed to be of the form

, , ( , )T r t W r t Y r t (12)

where ( , )W r t is considered in a way that the boundary conditions of ( , )Y r t become zero. Thus ( , )W r t is

assumed a second order polynomial as:

2( , ) ( ) ( )W r t A t r B t r (13)

Substituting Eq. (13) into Eqs. (7a), (10a) and (11a) for FGM, 1st and 3

rd layers respectively yields

2 2 2 1 1, 1 1( ) 2 , 2rk c A c B k Y c t A c B (14)

2

1 11 12 1 11 12 1( 2 ) ( ) ( )A X d X d B X d X g t (15)

2

3 21 22 3 21 22 4( 2 ) ( ) ( )A X a X a B X a X g t (16)

and substituting Eqs. (12), (13) into the heat conduction equation Eq. (9) for piezoelectric layers and Eq. (6) for the

FGM layer yield

, ,

11,3

j

j j rr j r j

j j

kY Y Y R j

c r

(17a)

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2

2 2, 3 2, 2

2 2

1( 1)rr r

kY Y m Y R

c r

(17b)

where

2

, ,

( , )4 1, 3

j j

j j t j t j

j j j j

k BR r tR A r B r A j

c c r

(18a)

2 2 2

2 2, 2, 2 3 3

2 2 2 2

( , )2 (2 ) ( 1)t t

k BR r tR A r B r A m m

c c r

(18b)

The solution of Eqs. (17) may be obtained by the method of separation of variables, generalized Bessel function

as:

( , ) ( ) ( )Y r t F r G t (19)

( ) ( ) 1,3j j jF r C r j (20a)

3

22 ( ) ( )

m f

P n

rF r r C

f

(20b)

where ( )nF r is derived from the general solution of energy equation without heat source and substituting Eqs. (20)

into the Eq. (19) yield

( , ) ( ) ( )j j j jY r t C r G t (21a)

3

22

0

( , ) ( )

m f

P n n

n

rY r t r C G t

f

(21b)

and substituting Eq. (21) into the Eqs. (17) yield

2

( )1,3

|| ( ) ||

j jdt dtj

j j

j j

R tG t e b e dt j

C r

(22a)

2 2

*

2

2 2 2

( )( )

( )

dt dt

nf

P n

R tG t e b e dt

rC

f

(22b)

where 2j

j j

j j

k

c

and ( )j jC r is the norm of the cylindrical function for piezoelectric layers as:

2

2

1 1 1 1|| ( ) || ( )d

cC r C r r dr

(23a)

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739 S.M.Mousavi et al.

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22

3 3 3 3|| ( ) || ( )b

aC r C r r dr (23b)

*

1 1 1 1( ) ( , ) ( )d

cR t r R r t C r dr (24a)

*

3 3 3 3( ) ( , ) ( )b

aR t r R r t C r dr (24b)

and 20

2

0 0

.n

k

c

f

P n

rC

f

is the norm of the cylindrical function of the FGM layer as:

2 2

f fc

P n P nb

r rC C r dr

f f

(25a)

3 1

22 2( ) ( , )

m fc

P nb

rR t r R r t C dr

f

(25b)

and nb is derived from the initial thermal boundary condition defined by Eq. (7c), (10c), (11c) for the first, FGM,

and third layer respectively as:

*

1 2 1 1 1 12

1 1

1( ) ( ,0) ( ) (0)

|| ( ) ||

d

cb r g r W r C r dr G

C r

(26a)

3 1

*22 3 2 22

1( ) ( ,0) (0)

m fc

n P nbf

P n

rb r g r W r C dr G

frC

f

(26b)

*

3 5 3 3 3 32

3 3

1( ) ( ,0) ( ) (0)

|| ( ) ||

b

ab r g r W r C r dr G

C r

(26c)

*

*

2

( )( ) 1,3

|| ( ) ||

j dtj

j

j j

R tG t e dt j

C r

(27a)

2

*

* 2

2 2

( )( )

( )

dt

f

P n

R tG t e dt

rC

f

(27b)

where ( ), 1,3,f

j j P n

rC r j C

f

are the mathematical Cylindrical Function given by

0 0( ) ( ) ( ) 1,3j j j j jC r J r c Y r j

(28a)

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One-Dimensional Transient Thermal and Mechanical Stresses …. 740

© 2018 IAU, Arak Branch

2

f f f

P n P n n P n

r r rC J c J

f f f

(28b)

11 0 1 12 0 1

1

11 0 1 12 0 1

( ) ( )

( ) ( )

X J d X J dc

X Y d X Y d

(29a)

21 0 3 22 0 3

3

21 0 3 22 0 3

( ) ( )

( ) ( )

X J a X J ac

X Y a X Y a

(29b)

2

( )

( )

f

P n

n f

P n

bJ

fcb

Jf

(29c)

3

5 4 3

1( 2)

2 2

mf m m m P

f

(30)

Here, 0J is the Bessel function of the first kind of order zero,

0Y is the Bessel function of the second kind of

order zero, the symbol (') denotes derivative with respect to r, and the eigenvalues 1 3, and

n are respectively the

positive roots of

11 0 1 12 0 1 0 1 11 0 1 12 0 1 0 1( ) ( ) ( ) ( ) ( ) ( ) 0X J d X J d Y c X Y d X Y d J c (31a)

21 0 3 22 0 3 0 3 21 0 3 22 0 3 0 3( ) ( ) ( ) ( ) ( ) ( ) 0X J a X J a Y b X Y a X Y a J b (31b)

3 3

3 3

13 2 2

13 2 2

2

02

m mf f f f

P n P n P n P n

m mf f

P n P n

mb c c bJ c J c J J

f f f f

m c cc J c J

f f

(31c)

where 1 2 3 1 2 3, , , , ,A A A B B B are six unknowns to be obtained from Eqs. (14) -(16) and (32) -(34) by system of linear

equations

2 2

1 1 2 2 2( , )A c B c Y c t A c B c (32)

2 2

2 2 3 3 3( , )A b B b Y b t A b B b (33)

2 2, 2 2 3 3 3( ) ( , ) 2 2rk b Y b t A b B k A b B (34)

The solution of this system of equation is given in the Appendix.

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3 SOLUTION OF THE PROBLEM

3.1 Piezoelectric layer

Using the relations (1), (2) and (5) the Navier equations in term of the displacements are

22 11 11 21

, , , , ,2

11 11 11 11 11

21 11 11

, , , , ,

11 11 11 11 11

1 1 1 1

1 1 11

r

rrrr r rr r r

r r

rr r rr r

C e e eu u u T T

r C C C r C C rr

e P Pu u T T

e r e e r e e r

(35)

Eqs. (35) are a system of differential equations having general and particular solutions. The general solutions are

assumed as:

1 1

3 3

( ) ( )

( ) ( )

g g

g g

u r rR Wr r

R Wu r r

(36)

where R,W and ,R W are the unknown constants for the 1st and 3

rd layers respectively and by using the specified

boundary conditions are determined. Substituting Eqs. (36) into Eqs. (35) yield

2 222 11 21

11 11 11

2 221 11

11 11

0

0

R WC e e

R WC C C

R We

R We e

(37)

Eqs. (37) are a system of algebraic equations. For obtaining the nontrivial solution of the equations, the

determinant of system should be equal to zero. So the four roots 1 to

4 for the equations are achieved which

include repeated roots of zero, hence a solution of the form of lnr must be considered in the general solutions as:

2 21 1 3 44

1 1 3 443 3

( ) ( ) ln

ln( ) ( )

k k

g g

k k

kg gk kk k

u r rW W W W rWN r N r

W W rWW Wu r r

(38)

where kN is the relation between constants ,k kW W and ,k kR R respectively and are obtained from Eq. (37) as:

21

22

1,2k

k

k

b eN k N

a C

(39)

where , , ,k k k ka b a b are given in the Appendix. The particular solutions pu and p of Eqs. (35) for piezoelectric

layers ( 1,3)j are assumed as:

1 1 2 3 42 3

0 0

0 9 10 12113

1 5 76 2

0 0

13 15143

( , )( ) ( ) ( )

( , )

( , )( ) ( ) ( )

( , )

p

j j jpk

p

j j jp

u r t D D D Dr J r Y r G t r r

D D DDu r t

r t D D DDr J r Y r G t r

D DDr t

8 3

0 16k

rD

(40)

We consider the definition of Bessel function of the first and second type as:

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One-Dimensional Transient Thermal and Mechanical Stresses …. 742

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2

0 ! 1

11

2

cos cos 1( )

sin sin

k nk

n

k

n

n n n n

n

r

J rk

J r n J r J r n J rY r

n

k n

n

(41)

Substituting Eqs. (40) and (41) and using heat distribution in piezoelectric layers into Eqs. (35) yield

51 2 6 1

1 2 3 1 2 3

9 13 10 314

5 2 6 11

8 9 108 9 10

10 39 13 14

DD D D cx x x x x x

D D D cD

D D D cDx x xx x x

D cD D D

(42)

7 83 4

4 5 6 7

15 12 1611

7 83 4

11 12 13 14

15 12 1611

0 0

0 0

D DD Dx x x x

D D DD

D DD Dx x x x

D D DD

(43)

Eqs. (42) and (43) are four systems of algebraic equations. The determinant of coefficients Eqs. (43) are zero, so

the obvious answer is the only possible. 1x to

14x and 1D to

16D are given in the Appendix. The complete

solutions ( , )u r t and ( , )r t for piezoelectric layers ( 1,3)j are the sum of the general and particular solutions

and are

21 1 24

0 0

1 04 9 103

21 5 63 4

0

1 3 4 13 13

,( ) ( ) ( )

,

, ln( )

ln,

k

k

k

k j j j

k kk

k

j

k k

u r t D DW WN r N r J r Y r G t

W D DWu r t

r t D DW W W rr r J r

W W rW D Dr t

0

0 4

( ) ( )j j

k

Y r G t

(44)

Substituting Eqs. (44) into Eq. (1), the strains for piezoelectric layers are obtained as:

2

1 1 21

0 0

1 0 9 103

2 1 ( ) 2 1 ( ) ( )krr k

k k j j j

k krr k

D DWN r k J r k Y r G t

D DW

21 1 241

0 0

1 04 9 103

( ) ( ) ( )kk

k j j j

k kk

D DW WNN r J r Y r G t

W D DW r

2

51 641

0 0

1 043 13 14

12 1 ( ) 2 1 ( ) ( )k

r k

k j j j

k kr k

DE DW Wr k J r k Y r G t

WE W D Dr

21 41

11 12 11 12 11 11 12

1 03 4

51 2

0 11 12 0 11 0

09 10 13

6

14

12 1

2 1 ( ) 2 1 ( )

2

krr k

k k k

k krr k

j j j j

k

W WC C N e r C N e C k C

W Wr

DD DJ r C k C Y r G t e k J r

D D D

Dk

D

*

2

0 2

( )1 ( ) ( ) ( )

|| ( ) ||

j jdt dtj

j j r j j j r j r j

j j

R tY r G t C r e b e dt A r B r

C r

(45)

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21 41

12 22 21 22 21 12 22

1 03 4

51 2

0 12 22 0 21 0

09 10 13

6

14

12 1

2 1 ( ) 2 1 ( )

2

kk

k k k

k kk

j j j j

k

W WC C N e r C N e C k C

W Wr

DD DJ r C k C Y r G t e k J r

D D D

Dk

D

*

2

0 2

( )1 ( ) ( ) ( )

|| ( ) ||

j jdt dtj

j j j j j j j

j j

R tY r G t C r e b e dt A r B r

C r

21 41

11 21 11 21 11 11 21

1 03 4

51 2

0 11 21 0 11 0

09 10 13

6

14

12 1

( ) 2 1 ( ) ( ) 2 1 ( )

krr k

k k k

k krr k

j j j j

k

D W We e N r e N e k e

D W Wr

DD DJ r e k e Y r G t k J r

D D D

D

D

*

2

0 2

( )2 1 ( ) ( ) ( )

|| ( ) ||

j jdt dtj

j j r j j j r j r j

j j

R tk Y r G t P C r e b e dt P A r P B r

C r

(45)

3.2 FGM layer

Using the relations (1), (3), (4) and (5), the Navier equations in term of the displacements are

2 211

, 1 , 0 1 2 ,2

1 1 1( 1) 1 ( )

1 1

m m

rr r r

mu m u u m m r T r T

r r

(46)

Eqs. (46) is a differential equation having general and particular solutions. The general solutions are assumed as:

( )gu r E r (47)

where E is the unknown constant and by using the specified boundary conditions is determined. Substituting Eq.

(47) into Eq. (46) yields

2 1

1 1 01

mm

(48)

Eq. (48) has two roots 1 2, as:

2

1 1 1

1,2 12 4 1

m m m

(49)

The particular solution pu of Eq. (46) for FGM layer is assumed as:

2 2 21 2 3

17 18 2 19 20

0 0

( , ) ( )f f

m m mp

P n P n n

n k

r ru r t r D J D J G t D r D r

f f

(50)

where 3

2

m . Substituting Eq. (50) and (41) and using heat distribution in FGM layers into Eq. (46) yield

17 1 2 18 3 4D y y D y y (51)

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19 5 6 20 7 8D y y D y y

(52)

where constants 1y to

3y and 17D to

20D are given in the Appendix. The complete solutions ( , , )u r t for FGM

layer is

2 2 2

21 2 3

17 18 2 19 20

1 0 0

, ( ) ( ) ( )j

f fm m m

j P n P n n

j n k

r ru r t E r r D J D J G t D r D r

f f

(53)

Substituting Eq. (53) into Eq. (1), the strains for FGM layer are obtained as:

2

2 2

21

2 17 2 18

1 0 0

1 2

2 2 19 2 20

1 2 1 2

( ) 2 3

j

fm

rr j j P n

j n k

fm m

P n n

rE r r m k P D J m k P D

f

rJ G t m D r m D r

f

2 2 2

21 1 2

17 18 2 19 20

1 0 0

( )j

f fm m m

j P n P n n

j n k

r rE r r D J D J G t D r D r

f f

1 1 2

11 2 1 2

210

2

1 0 0

17 2 18 2 2

1 2

19 2 20 0

1 1 1 21 1 2

1 1 2 ( ) 1 2

1 3 1

jm m m

rr j j

j n k

f f

P n P n n

m mm m m m

EE r r m k P

r rD J m k P D J G t m

f f

D r m D r r

32

22 1 2 1 2

2

0

*2 12

2 0 2 0 2

2

( )1 1

|| ( ) ||

m fn

P n

n

dt dt m m m m

n f

P n

rC

f

R te b e dt A r B r

rC

f

1 1 2

11 2 1 2

210

2

1 0 0

17 2 18 2 2

1 2

19 2 20 0

1 1 2 11 1 2

1 2 1 ( ) 2 1

3 1 1

jm m m

j j

j n k

f f

P n P n n

m mm m m m

EE r r m k P

r rD J m k P D J G t m

f f

D r m D r r

32

22 1 2 1 2

2

0

*2 12

2 0 2 0 2

2

( )1 1

|| ( ) ||

m fn

P n

n

dt dt m m m m

n f

P n

rC

f

R te b e dt A r B r

rC

f

(54)

It is recalled that ( 1,2), ( 1,...,4)j kE j W k and ( 1,...,4)kW k are ten unknown constants for the FGM shell

and the outer and the inner piezoelectric layers respectively and can be evaluated by satisfying the boundary

conditions and continuity requirements on the interface. Therefore, displacements, electric potential, stress and other

responses can be evaluated. The corresponding boundary conditions can be written as:

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1 1 1 3

3 2 3

0 0

0

rr

rr r

r d p r c r b

r a p D r d r a v

(55)

where 1p and

2p are the outer and inner pressure, respectively. In addition, the continuity requirements for the

stresses and displacements on the interfaces must be satisfied, therefore we have

1 2 1 2

2 3 2 3

rr rr

rr rr

u r c u r c r c r c

u r b u r b r b r b

(56)

4 NUMERICAL RESULTS AND DISCUSSION

Assume material properties for piezoelectric on first layer as an actuator and for third layer piezoelectric as a sensor

PZT-4 from following table

Table1

Material properties of piezoelectric.

Material Elastic constants, Gpa

PZT-4 11C

12C 13C

22C 23C

33C 44C

139 78 74 139 74 115 25.6

Piezoelectric constants, 2/C m Permittivity, 9 210 /C Nm Pyroelectric constants, 5 210 /C Km Coefficient of thermal

expansion, 610 1/ K

PZT-4 11e

12e 22e

11 22

0rP P P 0r

-5.2 15.1 12.7 6.5 6.5 5.4 2.62

Let us consider a thick hollow cylinder of radii 0.96 , 1 , 1.24r m b m c m and 1.28d m .The Poisson's

ratio is assumed 0.3 and modulus of elasticity, coefficient of thermal expansion, thermal conductivity, density and

specific heat capacity for FGM layer are 6 3

0 0 0 0200 , 1.2 10 / , 2 / , 7800 /E GPa K k W mK kg m ,

0 420 /c J kgK and thermal conductivity, density and specific heat capacity for piezoelectric layers are

31.5 / , 7500 /k W mK kg m and 350 /c J kgK respectively. For simplicity of analysis, we consider the

power law of material properties be the same as 1 2 3 4 5m m m m m m . As the example Consider a hollow

cylinder subjected to thermo mechanical loads, and its mechanical, thermal and electrical boundary conditions are

respectively, taken as:

1 1 3 1

3 1 3 4

30 0 0 20sin

0 0 20 0

rr

rr r

r d MPa r c r b g t t K

r a D r d r a v g t

(57)

The initial temperature for the FGM shell and the inner and the outer piezoelectric layers are zero. The cylinder

is heated by the rate of energy generation per unit time and unit volume of 6

3

1( , ) 6 10 sin( )

WR r t t

r m . Figs. 2-8

illustrates the temperature profile, radial displacement, radial and circumferential stress for FGM layer and electric

potential and radial electrical displacement for piezoelectric layers at the middle radius of the layers over the course

of 10 seconds for different power law indices respectively. The value of 0m corresponds to pure metal. The

curves associated with the non-zero heat source follow the sine-form pattern of the assumed heat source.

Temperature distribution are zero at 0t due to the initial temperature. It can easily be seen as the graded index

increase the values of temperature, radial displacement, radial and circumferential stresses for FGM layer and

electric potential and radial electrical displacement for piezoelectric layers increase at the same time.

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Fig.2

Transient temperature distribution in the FGM layer.

Fig.3

Radial displacement in the FGM layer.

Fig.4

Radial stress in the FGM layer.

Fig.5

Circumferential stress in the FGM layer.

Fig.6

Radial electrical displacement in the actuator layer.

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Fig.7

Electric potential distribution in the actuator layer.

Fig.8

Electric potential distribution in the sensor layer.

For different values of m, the temperature profile for piezo FGM hollow cylinder, radial displacement, radial and

circumferential stresses for FGM layer, electric potential and radial electrical displacement for piezoelectric layers

along the radial direction at 2t seconds are plotted in Figs. 9-15. From Fig. 9, one can see the temperatures

satisfy the prescribed thermal boundary conditions at the internal and external boundaries, the temperatures decrease

along the thickness and increase as the graded index m increases at the same radial point. Fig. 10, shows that the

radial displacements for FGM layer increase gradually from the inner surface to the outer surface and the radial

displacements increase as the graded index m increases at the same radial point.

Fig.9

Temperature distribution in the piezo FGM hollow cylinder along

the thickness.

Fig.10

Radial displacement in the FGM hollow cylinder along the

thickness.

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It can easily be seen from Fig. 11 that the radial stresses increase with the increase of power law parameters m at

the same radial point. The distribution of circumferential stresses is shown in Fig. 12 and similar patterns can also

observe in this figure. Fig. 13 illustrates radial electrical displacement distributions with various for actuator layer.

It can easily be seen from this figure which radial electrical displacements decrease as the graded index decreases

at the same radial point. Fig. 14 indicates the distribution of electric potential in the actuator along the thickness. Fig.

15 shows the distribution of electric potential in the sensor layer that increases as the graded index decreases at

the same radial point. According to the given mechanical boundary conditions, radial electrical displacement at the

outside and electric potential at the inside surfaces of the actuator layer are zero and the electric potential in sensor

layer satisfies the prescribed electrical boundary conditions.

Fig.11

Radial stress in the FGM layer along the thickness.

Fig.12

Circumferential stress in the FGM layer along the thickness.

Fig.13

Radial electrical displacement in the actuator layer along the

thickness.

Fig.14

Electric potential distribution in the actuator along the thickness.

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749 S.M.Mousavi et al.

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Fig.15

Electric potential distribution in the sensor layer along the

thickness.

To verify the proposed method, assuming that the electric potential to the piezoelectric layers are zero consider

the functionally graded hollow cylinder with inner radius 0.02b m and outer radius 0.024c m . The comparison

of the present method and the method in Ref. [12] for 1 2 311.9839, 1.3103, 1.4937m m m and the initial

temperature of the cylinder ( ,0) 50 (100 )T r r C , where is the mathematical Gamma function such that the

cylinder is heated by 6

3

1( , ) 6 10 sin(5 )

WR r t t

r m , are illustrated in Fig. 16. The results are in good agreement

with obtained results from Ref. [12].

Fig.16

The comparison of present results and method in Ref. [12].

5 CONCLUSIONS

This paper presents an analytical study of piezo thermoelastic behavior of an FGM hollow cylinder with

piezoelectric layers subjected to radially symmetric loadings with heat source. The method of solution is based on

the direct method, rather than other methods. This method does not have the limitations of the potential function or

numerical methods and more various thermal boundary conditions may be handled using the proposed method. The

potential function method requires a wide experience of particular solutions and even then is not guaranteed to be

successful and requires examination of the assumed solutions with a view toward finding one that will satisfy the

governing equations and boundary conditions. In addition, analytical solution is the solution for multitude of

particular cases, while the numerical solution has to be obtained anew for each such case separately. Moreover,

numerical method always works with iteration, while analytical methods the final answer is straight forward. By using this method and considering the special boundary conditions and material properties for piezoelectric-

FGM-piezoelectric hollow cylinder, the mechanical and electrical displacements and stresses can be controlled and

optimized to design and use this kind of structures.

APPENDIX A

22

3 21 22 31 1 11 12

1 1 1

2 2 11 12 2 21 22 2

2 2

2 2 211 12 1 1 11 12

2 2 2

11 12 2 2 11 12

( 2 ) 22 ( 2 ), 2 2 ,

( ) ( ) ( ) ( ) ( ) ( )

( 2 ) 2 ( 2 ), , 0

( ) ( )

k X a X a kk k X d X dx c y b c z b

k c k c X d X k b X a X k b

c X d X d k k c X d X dx c c y c z

X d X k c k c X d X

(A.1)

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2 2

2 21 1 11 12 21 22

3 3 3

2 2 11 12 21 22

3 4 1 1 1

1 2, 1,

2 21 22 2 11 12 2

1 1 1

2 2 1,

11 12 2 2

2 ( 2 ) ( 2 ), 2

( ) ( ) ( )

k( ,t) ( , t)

( ) ( ) ( ) k ( )

g k( , t) ( , t)

( ) ( )

r r

r

k k b X d X d b X a X ax bc y b bc z b

k c k c X d X X a X

k g k gh Y b Y c

k b X a X k c X d X c

c k c c gh Y c Y c

X d X k c k c

1

11 12

4 1 1 1

3 3 1,

21 22 2 2 11 12

g k( , t) ( , t)

( ) ( ) ( )r

X d X

b k b b gh Y b Y c

X a X k c k c X d X

(A.1)

1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3

1 2 3

1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3

h y z x y h x y h

h y z x y h x y h

h y z x y h x y hA A A

x y z x y z x y z

x y z x y z x y z

x y z x y z x y z

(A.2)

2 2

1 1 11 12 1 1 1 11 12

1 2 1, 1 2

11 12 2 11 12

2

4 3 21 22

3

21 22

( 2 ) ( 2 ), ( , ) 2 2

( )

( 2 )

r

g A X d X d k g A X d X dB B Y c t A c A c

X d X k c X d X

g A X a X aB

X a X

(A.3)

2 222 11 21

11 11 11

2 221 11

11 11

,

,

k k k k k

k k k k k

C e ea b

C C C

ea b

e e

(A.4)

3 4 7 8

11 12 15 16

0

0

D D D D

D D D D

(A.5)

22 11 11 21 11 21

1 2

11 11 11 11

22 11 21 22

3 4 5 6

11 11 11 11 11

11 21

7

11

32 2 1 6 1 , 2 2 1 2

4 22 , 4 , , 9

9 3

rr

C e e e e ex k k k x k k k

C C C C

C e e Cx k x x x

C C C C C

e ex

C

21 11 21

8

11 11

, 2 2 1 2 3e e e

x k k ke e

11 11 11

9 10

11 11 11 11 11

11 21 11 11 21 11

11 12 13 14

11 11 11 11

2 2 1 6 , 2

4 2 4 9 3 9, , ,

r rP Px k k k x k

e e e e e

e e e ex x x x

e e e e

(A.6)

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751 S.M.Mousavi et al.

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1 1

3 2 1 3

3 3

1 13 2 1 3

10 9 8 10

3 351 2 610 9 8 10

1 2 1 2 1 2 1 29 10 13 14

8 9 8 9 8 9 8 9

1 2 1 2 2

, , ,

2 2 1 2 2 3 2 1

c cx x x x

c c

c cx x x xx x x x

c cDD D Dx x x x

x x x x x x x xD D D D

x x x x x x x x

y f k P f k P m m f k P m m

2

1

1 2

2 1 2

3 2 1 2 2

2

1

1 2

4 2 1 2

5 2 2 1 2

1 21 1 1

1 2 2

(1 )2

(1 )

2 2 1 2 2 3 2 1

1 21 1 1

1 2 2

(1 )2

(1 )

2 1 1

n

nmm m

y m m f k P

y f k P f k P m m f k P m m

nmm m

y c m m f k P

y m m m m

12 11

m

6 1 2 2

1

7 2 2 2 1

8 1 2 2

(1 )1

(1 )

3 2 3 1 11

(1 )2

(1 )

y m m B

my m m m m

y m m A

(A.7)

6 82 4

17 18 19 20

1 3 5 7

y yy yD D D D

y y y y

(A.8)

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