hollow piezoelectric cylinder under transient...

Keywords: Transient, Thermoelasticity, Hollow Cylinder, Piezoelectric

1 Introduction

Piezoelectric materials are widely used due to their direct and inverse effects. The use of

piezoelectric layers as distributed sensors and actuators in structures to control noise and

deformations and suppress vibrations is quite common. Several research works have been

contributed to model and investigate the basic structural responses of piezoelectric materials

i.e. in the pioneering researches of Tiersten [10]. Alashti et al. [1] carried out three-dimensional

thermo-elastic analysis of a functionally graded cylindrical shell with piezoelectric layers by

differential quadrature method. Alibeigloo and Chen [2] obtained the elasticity solution for an

FGM cylindrical panel integrated with piezoelectric layers.

Chu and Tzou [3] 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. Fesharaki et al. [4] 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.

* M.Sc., Mechanical Engineering Department, South Tehran Branch, Azad University, Iran

[email protected] †Assistant Professor, Mechanical Engineering Department, South Tehran Branch, Azad University, Iran

[email protected] ‡M.Sc., Mechanical Engineering Department, South Tehran Branch, Azad University, Iran

[email protected]

S.M. Mousavi* M.Sc.

M. Jabbari† Assistant Professor

M.A. Kiani‡ M.Sc.

Hollow Piezoelectric Cylinder

under Transient Loads In this paper, transient solution of two dimensional

asymmetric thermal and mechanical stresses for a hollow

cylinder made of piezoelectric material is developed.

Transient temperature distribution, as function of radial

and circumferential directions and time with general

thermal boundary-conditions, is analytically obtained,

using the method of separation of variables and

generalized Bessel function. The results are the sum of

transient and steady state solutions that depend upon the

initial condition for temperature and heat source,

respectively. The general form of thermal and mechanical

boundary conditions is considered on the piezoelectric

cylinder. Material properties of piezoelectric cylinder are

the same along the thickness. A direct method is used to

solve the Navier equations, using the Euler equation and

complex Fourier series.

Iranian Journal of Mechanical Engineering Vol. 18, No. 2, Sep. 2017

24

He et al. [5] derived the active control of FGM plates with integrated piezoelectric sensors and

actuators. Mohazzab [6] presents the analytical solution of one-dimensional mechanical and

thermal stresses for a hollow cylinder made of functionally graded material. Vaghari et al. [7]

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.

Khoshgoftar et al. [8] presented the thermoelastic analysis of a thick walled cylinder made of

functionally graded piezoelectric material by using the separation of variables. Poultangari et

al. [9] presented a solution for the functionally graded hollow spheres under nonaxisymmetric

thermomechanical loads.

This paper present an analytical method to obtain the transient thermal and mechanical stresses

in a piezoelectric hollow cylinder subjected to the two-dimensional asymmetric loads.

Temperature distribution is assumed to be a function of radial and circumferential directions

and time. The Navier equations are solved analytically using a direct method of series

expansion.

2 Governing equation

2.1 Stress distribution

Consider a piezoelectric hollow cylinder of inner radius and outer radius b. Asymmetric

cylindrical coordinates (r, 𝜃) are considered along the radial and circumferential directions,

respectively. The governing two-dimensional strain-displacement relations in cylindrical

coordinates and electric field-electric potential relations are

1 1 1

2

1

rr r

r

u v u u v v

r r r r r r

E Er r

(1)

in which u, v, and 𝜓 as displacement components along the radial and circumferential

directions, and the electric potential, respectively. The constitutive relations describing the

electrical and the mechanical interaction for the piezoelectric material are

11 21 11

24 22

11 12 11

12 22 21

44 24

( , , )

( , , )

( , , )

( , , )

r rr r r

r

rr rr r r

rr r

r r

D

D e e E P T

e E P T

C C e E T r t

C C e E T r t

C e E

r t

r t

(2)

Where σij, εij(i, j = r, θ) and T(r, θ, t) are the stress, strain tensors and the temperature

distribution Cij and eij, ηij, Di, Pi and αi are elastic and piezoelectric coefficients, dielectric

constants, electric displacements, pyroelectric constant and thermal modulus respectively for

the piezoelectric material. The equilibrium equations in the radial and circumferential

directions, disregarding the body forces and inertia terms and equation of electrostatic are

Hollow Piezoelectric Cylinder under...

25

1 1( ) 0

1 20

1 10

rrrrr

rr

rr

rr

r r r

r r r

DDD

r r r

(3)

2.2 Heat Conduction Problem

The heat conduction equation in two-dimensional problem for piezoelectric cylinder leads to

, , ,2

1 1 ( , , )rr r

k R r tT T T T

c r r c

(4)

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

denotes derivative with respect to time. The initial and mixed boundary conditions are

11 12 , 1, , , , ,rX T a t X T a t g t (5-a)

21 22 , 2( , , ) ( , , ) ( , )rX T b t X T b t g t (5-b)

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

where 𝑋𝑖𝑗(𝑖, 𝑗 = 1, 2) are Robin-type constants related to the thermal boundary condition

parameters, and 𝑔3(𝑟, 𝜃) is the known initial condition. The solution of the heat conduction equations for temperature distribution in piezoelectric cylinder may be assumed to be of the

form

, , , , ( , , )T r t W r t Y r t (6)

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

Thus 𝑊(𝑟, 𝜃, 𝑡) is assumed a second order polynomial as

2( , , ) ( , ) ( , )W r t A t r B t r (7)

Substituting Eq. (7) into Eqs. (5-a), (5-b) yields

2

11 12 11 12 1( 2 ) ( ) ( , )A X a X a B X a X g t (8) 2

21 22 21 22 2( 2 ) ( ) ( , )A X b X b B X b X g t (9)

Substituting Eq. (7) into Eqs. (5-a), (5-b) yields

1 21 22 2 11 12

2 2

11 12 21 22 11 12 21 22

2 2

2 11 12 1 21 22

2 2

11 12 21 22 11 12 21 22

, ,( , )

2 2

, 2 , 2( , )

2 2

g t X b X g t X a XA t

X a aX X b X X a X X b bX

g t X a aX g t X b bXB t

X a aX X b X X a X X b bX

(10)

and substituting Eqs. (6), (7) into the heat conduction equation Eq. (4) yield

, , , 12

1 1rr r

kY Y Y Y R

c r r

(11)


26

Where

2

1 , , , ,

, , 14 ( )t t

R r t kR A r B r A A B B

c c r

(12)

The solution of Eqs. (11) may be obtained by the method of separation of variables. For the

general solution, Eq. (13) is substituted into Eq. (11) and using the generalized Bessel function

leads to

1

( , , ) ( ) ( )n

in

mn mn

n m

Y r t F r G t e

(13)

( ) ( )mn mn mnF r C r (14)

Where 𝐹𝑚𝑛(𝑟) is derived from the general solution of energy equation without heat source and Substituting Eq. (13) into the Eq. (11) yields

2

( )

2 || ( ) ||

dt dt

mn mn

mn mn

R tG t e b e dt

C r

(15)

where 𝜏 =𝑘

𝜌𝑐𝜆𝑚𝑛

2 and ‖𝐶 𝑚𝑛(𝜆 𝑚𝑛𝑟)‖ is the norm of the cylindrical function as

22|| ( ) || ( )

b

mn mn mn mna

C r C r r dr (16)

2

10

( ) ( )b

in

mn mna

R t r R C r e dr d

(17)

and 𝑏𝑚𝑛 is derived from the initial thermal boundary condition defined by Eq. (5-c) as

2

32 0

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

2 || ( ) ||

bin

mn mn mna

mn mn

b r g r W r C r e dr d GC r

(18)

2

( )( )

2 || ( ) ||

dt

mn mn

R tG t e dt

C r

(19)

where 𝐶 𝑚𝑛(𝜆 𝑚𝑛𝑟) is the mathematical Cylindrical Function given by

( ) ( ) ( )mn mn n mn mn n mnC r J r c Y r (20)

11 12

11 12

( ) ( )

( ) ( )

n mn n mnmn

n mn n mn

X J a X J ac

X Y a X Y a

(21)

Here, 𝐽𝑛 is the Bessel function of the first kind of order n, 𝑌𝑛 is the Bessel function of the second

kind of order n, the symbol (') denotes derivative with respect to r, and the eigenvalue 𝜆𝑚𝑛 is the positive roots of

11 12 11 12( ) ( ) ( ) ( ) ( ) ( ) 0

1,2,3,....

n mn n mn n mn n mn n mn n mnX J a X J a Y b X Y a X Y a J b

m

(22)

3 Solution of the problem

Using the relations (1), (2) and (3) the Navier equations in term of the displacements are


27

22 44 12 44 44 22 11, , , , , ,2 2 2

11 11 11 11 11

11 21 24, , ,

22 12 22

2

11 11 11 1

, , , ,2 2

44 4 4

1

4 4

1 1 1 1 1( )

1 1

1 1 1 11 1

1

1

rr r r rr

rrr r

rr r r

C C C C C C eu u u u v v

r C r C r C r C r C

e e eT T

C r C r

C C Cv v v v u

r r C r C r C

C C r

r

24 21, ,2

44

24, ,2

44 44

21 24 21 24 24 11 11, , , , , , ,2 2

11 11 11 11 11 11

22, , ,2

11 11 11 11

1

1 1

1 1 1 1 11

1 1 1

r

rr r r rr r

r rr

e eu

C r

eT

C r C r

e e e e eu u u v v

e r e r e r e r e e r

PP PTT T T

e r e r e r e r

(23)

To solve the Navier equations (23) consider the complex Fourier series for displacements

𝑢(𝑟, 𝜃, 𝑡) and 𝑣(𝑟, 𝜃, 𝑡) and electric potential 𝜓(𝑟, 𝜃) as

1 1 1

( , , ) ( , ) ( , , ) ( , ) ( , , ) ( , )n n n

in in in

mn mn mn

n m n m n m

u r t u r t e v r t v r t e r t r t e

(24)

Substituting Eqs. (24) into Eqs. (23) yield

212 44 44 2222 44

2 21 11 11 11

2

11 11 21 24,2

11 11 11 11 11

1 1 1 1

1 1 1

n

mn mn mn mn mn

n m

in rrmn mn mn r

in C C in C CC n Cu u u v v

r C r C r C r

e e e n ee T T

C C r C r C C r

2

22 12 22

2 21 44 44 44

24 21 24,2

44 44 44

2

21 24

2

11 11

1 1 1 11 1 1

1 1 1

1 11

n

mn mn mn mn mn

n m

in

mn mn

mn mn mn

n C C Cv v v in u in u

r C r C r C r

e e ein in e T

C r C r C r

e n eu u u

e r e r

21 24 24 11

21 11 11 11

2

11 22, ,2

11 11 11 11 11

1 1

1 1 1 1

n

mn mn mn

n m

in r rmn mn r

in e e ev in v

e r e r e

Pn P Pe T T T

e r e r e e r e r

(25)

Eqs. (25) are a system of ordinary differential equations with non-constant coefficients having

general and particular solutions. The general solutions for 𝑛 ≠ 0 are assumed as

( ) ( ) ( )g g g

n n nu r Rr v r Sr r Wr (26)

Where 𝑅, 𝑆, 𝑊 are the unknown constants and by using the specified boundary conditions are

determined. Substituting Eqs. (26) into Eqs. (25) yield


28

2 22 222 44 12 22 11 21 24

11 11 11 11 11

2212 22 44 22 24 21 24

44 44 44 44 44

22 21 24

11 11

0

1 1 0

C n C C C e e n eR i nS W

C C C C C

C C C n C e e ei nR S i nW

C C C C C

e n e

e e

2221 24 24 22 11

11 11 11 11

( )0

e e e nR i nS W

e e e e

(27)

Eqs. (27) 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 six roots η𝑛1

to η𝑛6

for the

equations are achieved and the general solution is

6 6 6

1 1 1

( ) , ( ) , ( )nk nk nkg g g

n nk n nk nk n nk nk

k k k

u r R r v r M R r r N R r

(28)

Where 𝑀𝑛𝑘 is the relation between constants 𝑅𝑛𝑘 and 𝑆𝑛𝑘 and 𝑁𝑛𝑘 is the relation between

constants 𝑅𝑛𝑘 and 𝑊𝑛𝑘 respectively and are obtained from Eq. (38) as

21,...,6 0nk nk nk nk nk nk nk nk nk nk nk

nk nk

nk nk nk nk nk nk nk nk nk nk

a c a c a a b c a b cM i N k n

b c b c c b c b c c

(29)

Where 𝑎𝑛𝑘 , 𝑏𝑛𝑘 , 𝑐𝑛𝑘 and 𝑎ˊ𝑛𝑘 , 𝑏ˊ𝑛𝑘 , 𝑐ˊ𝑛𝑘 are given in the appendix. For 𝑛 = 0 Eqs. (25) are independent of each other, which yield

22 11 11 210 0 0 0 0 ,2

1 11 11 11 11 11

1 1 1 1rrm m m m m r

m

C e e eu u u T T

r C r C C r C C r

(30)

0 0 0 ,21 44

1 1 1m m m

m

v v v Tr r C r

(31)

21 11 110 0 0 0 , ,

1 11 11 11 11 11 11

1 1 1 11 r r

m m m m r

m

Pe P Pu u T T T

e r e e r e e r e r

(32)

Two equations (30) and (32) are a system of ordinary differential equations. The general

solution of this case is considered as

0 0 0

4 6 4

0 0 0 0 0 0 0

1 5 1

( , ) ( , ) ( , )k k kg g g

m k m k m k k

k k k

u r t R r v r t R r r t Z R r

(33)

Where 22

11 0 21 011 22 2101,2 03,4 05,6 02 2

11 11 11 11 0

, 0, 1, 1,...,4k kk

k

e eC eZ k

C e

(34)

The particular solutions 𝑢𝑛𝑝 , 𝑣𝑛

𝑝 and 𝜓𝑛

𝑝 of Eqs. (25) for 𝑛 ≠ 0 are assumed as


29

2 3

1 2 3 4

0

2 3

5 6 7 8

0

2 3

9 10 11 12

0

( , ) ( ) ( ) ( )

( , ) ( ) ( ) ( )

( , ) ( ) ( ) ( )

p

mn mnk n mn mnk n mn mn mn mnk

k

p

mn mnk n mn mnk n mn mn mn mn

k

p

mn mnk n mn mnk n mn mn mn mn

k

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

v r t r D J r D Y r G t D r D r

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

(35)

we consider the definition of Bessel function of the first and second type as

2

0

11

cos 12)

n! 1(

si

k nk

nmnn mn n mn

n mn n mn

k

rJ r n J r

J r Y rk n nk

(36)

Substituting Eqs. (35) and (36) and using heat distribution in piezoelectric layers into Eqs. (25)

yield

1 1 5 2 9 3 4 2 1 6 2 10 3 4

1 11 5 12 9 13 14 2 11 6 12 10 13 14

1 21 5 22 9 23 24 2 21 6 22 10 23 24

mnk mnk mnk mnk mnk mnk mn



D x D x D x x D x D x D x c x



(37)

3 5 7 6 11 7 4 8 8 9 12 10

3 15 7 16 11 17 4 18 8 19 12 20

3 25 7 26 11 27 4 28 8 29 12 30

0 0

0 0

0 0

mn mn mn mn mn mn

mn mn mn mn mn mn

mn mn mn mn mn mn

D x D x D x D x D x D x



(38)

Eqs. (37) and (38) are four systems of algebraic equations. The determinant of coefficients Eqs.

(38) are zero, so the obvious answer is the only possible. 𝑥1 to 𝑥30 and 𝐷𝑚𝑛𝑘1 to 𝐷𝑚𝑛12 are

given in the Appendix. The particular solutions of Eqs. (30)-(32) for 𝑛 = 0 are 𝑢𝑚0𝑝 (𝑟, 𝑡),

𝑣𝑚0𝑝 (𝑟, 𝑡), 𝜓𝑚0

𝑝 (𝑟, 𝑡) are given in the appendix and substituting these equations into Eqs. (30) -

(32) lead to algebraic equation systems. 𝑥01 to 𝑥030 are 𝑥1 to 𝑥30 for 𝑛 = 0 and 𝑥031 to 𝑥036

and 𝐷𝑚0𝑘1 to 𝐷𝑚012 are given in the appendix. The complete solutions u(r, θ, t), v(r, θ, t) and

ψ(r, θ, t) are the sum of the general and particular solutions and are

0

0

6 4

0 1 2

1 1 1 0, 0

2 3

03 04

6 6

0 5 6

1 5 1 0, 0 , 0

, , ( ) ( ) ( )

, , ( ) ( )

knk

knk

n

nk k mnk n mn mnk n mn mn

n k k m kn

in

m m

nk nk k mnk n mn mnk n mn

k k m kn n

u r t R r R r r D J r D Y r G t

D r D r e

v r t M R r R r r D J r D Y r

2 3

07 08( )

n

n

in

mn m mG t D r D r e


30

0

6 4

0 0 9 10

1 1 1 0, 0

2 3

011 012

, , ( ) ( )

( )

knk

n

nk nk k k mnk n mn mnk n mn

n k k m kn

in

mn m m

r t N R r Z R r r D J r D Y r

G t D r D r e

(39)

Substituting Eqs. (39) into Eq. (1), (2) the stress are obtained as

0

6 41

1 11 12 11 11 0 12 11 0 0

1 1, 0

1

0 11 12 1 11 12

1 0

2

2 11 12 03 11 12 04

1

2 1 2 1

( ) 2 3

nk

k

n

rr nk nk nk nk nk k k k

n k kn

k mnk n mn

m k

mnk n mn mn m m

C C in M e N R r C C e Z

R r C k n C D J r C k n C

D Y r G t C C D r C C D r

0

6

12 0

5

1 2

12 5 6 07 08

1 0, 0

11 9 10 011

1 0

2

012

1

( ) ( ) ( )

2 1 ( ) 2 1 ( ) ( ) 2

3 ( )

k

k

k

mnk n mn mnk n mn mn m m

m kn

mnk n mn mnk n mn mn m

m k

nin

m r mn mn

n m

inC R

r inC D J r D Y r G t D r D r

e D k n J r D k n Y r G t D r

D r e C r

2

2

( )

2 || ( ) ||

dt dt in

n

mn mn

r r

R te b e dt e

C r

Ar Br

0

0

6 411

44 24 44 24 0 0

1 1, 0

2

44 1 2 03 04

1 0

61

44 0 0 44 5

5

( )

1 (2 ) ( )

knk

k

n

r nk nk nk nk nk k k

n k kn

mnk n j mn mnk n j mn j mn m m

m k

k k mnk n j mn

k

C in M M in e N R r C e Z in R r

inC D J r D Y r G t D r D r

C R r C D k n J r

6

1 0, 0

2

07 08 24 9 10

1 0

2

011 012

2 ( )

( ) 2 ( ) ( ) ( )

mnk n j mn

m kn

j mn m m mnk n mn mnk n j mn j mn

m k

in

m m

D k n Y r

G t D r D r in e D J r D Y r G t

D r D r e


31

0

6 41

12 22 21 12 0 22 21 0 0

1 1, 0

1

0 12 22 1 12 22

1 0

2

12 22 03 12 22 04 22

1

2 1 2 1

( ) 2 3

nk

k

n

nk nk nk nk nk k k k

n k kn

k mnk n mn n mn

m k

mn m m

C C in M e N R r C C e Z

R r C k n C D J r C k n C Y r

G t C C D r C C D r inC

0

61

0 22 5

5 1 0, 0

2

6 07 08 21 9

1 0

2

10 011 012

1

( ) ( ) ( ) 2 1 ( )

2 1 ( ) ( ) 2 3 ( )

k

k mnk

k m kn

n mn mnk n mn mn m m mnk n mn

m k

nin

mnk n mn mn m m mn mn

n m

R r inC D

J r D Y r G t D r D r e D k n J r

D k n Y r G t D r D r e C r

e

2

2

( )

2 || ( ) ||

dt dt in

n j j

mn mn

R tb e dt e A r B r

C r

(40)

It is recalled that 𝑅𝑛𝑘(𝑘 = 1, … ,6) are six unknown constants and therefore to determine these constants, six boundary conditions that may be either the given displacements or stresses, or

combinations are required.

1

2

3

( , ) ( )

( , ) ( )

( , ) ( )

u a f

v a f

a f

4

5

6

( , ) ( )

( , ) ( )

( , ) ( )

u b f

v b f

b f

7

8

( , ) ( )

( , ) ( )

rr

r

a f

a f

9

10

( , ) ( )

( , ) ( )

rr

r

b f

b f

(41)

Expanding these boundary conditions into complex Fourier series and select from the list of

Eqs. (41) and using the continuity conditions between layers lead to a system of six linear

equations to be solved for the constants 𝑅𝑛𝑘(𝑘 = 1, … ,6).

4 Numerical results and discussion

The analytical solutions obtained in the previous section are checked. Assume piezoelectric

material properties PZT-4 from following table

Table 1 Material properties of piezoelectric

Material Elastic constants, Gpa

𝐂𝟒𝟒 𝐂𝟑𝟑 𝐂𝟐𝟑 𝐂𝟐𝟐 𝐂𝟏𝟑 𝐂𝟏𝟐 𝐂𝟏𝟏 PZT-4

5.6 115 74 139 74 78 139

Piezoelectric constants, C/𝑚2 Permittivity,10−9C/𝑁𝑚2

Pyroelectric constants, 10−5C/𝐾𝑚2 Coefficient of thermal expansion, 1 𝐾⁄

αr = αθ

= α0

Pr = Pθ

= P0

η22 η11 e22 e12

e11 PZT-4

2.62 5.4 6.5 6.5 12.7 15.1 -5.2


32

Figure 1 Transient temperature distribution in hollow cylinde

Let us consider a thick hollow cylinder of radii 𝑎 = 0.64 𝑚 and 𝑏 = 0.68 𝑚 and thermal

conductivity, density and specific heat capacity are 𝑘 = 1.5 𝑊/𝑚 𝐾, 𝜌 = 7500 𝑘𝑔 𝑚3⁄ and

𝑐 = 350 𝐽 𝑘𝑔 𝐾⁄ respectively. As the example, Consider a hollow cylinder where the inside boundary is fixed with zero temperature and the outside boundary is assumed to be traction-

free with given temperature distribution 𝑔2(𝜃, 𝑡) = 20 sin(𝑡) cos(2𝜃) 𝐾.

Therefore, the assumed boundary conditions result in 𝑢(𝑎, 𝜃) = 0, 𝑣(𝑎, 𝜃) = 0, 𝜎𝑟𝑟(𝑏, 𝜃) =0, 𝜎𝑟𝜃(𝑏, 𝜃) = 0 and 𝑔1(𝜃, 𝑡) = 0. In this example the electrical boundary conditions

are 𝜓(𝑎, 𝜃) = 0 and 𝜓(𝑏, 𝜃) = 30 cos (2𝜃) 𝑣 and. The initial temperature is zero. The

cylinder is heated by the rate of energy generation per unit time and unit volume of (𝑟, 𝜃, 𝑡) =

6 × 106 ×1

𝑟sin(t)cos(2θ)

𝑊

𝑚3 .

r

Figure. (1) illustrates the cylinder temperature at 𝜃 = −𝜋 over the course of 10 seconds. The

temperature on the vertical axis is plotted against the time in seconds on the horizontal axis.

The results are the sum of transient and steady state solutions that depend upon the initial

condition for temperature and heat source, respectively. With small range of radius, the

temperature distribution in the piezoelectric layers is close, but as may be seen temperature

increases as radius increase.

Figure. (2) shows temperature distribution for different radius r and time t at 𝜃 = −𝜋 . Radial and shear stresses distribution along the thickness and for a specific angle over the course of 10

seconds are show in the Figures. (3),(4), can easily be seen from these figures which radial and

circumferential stresses decreases when the radius increases. To verify the proposed method,

consider the piezoelectric cylinder with similar boundary conditions and inner radius a =0.64 m and outer radius and b = 0.68 m. Figure (5) illustrates radial stress in piezoelectric

cylinder along the thickness by present and finite element method. The results are in good

agreement with obtained results from finite element method.


33

Figure 2 temperature distribution at 𝜃 = −𝜋 Figure 3 Radial stress

Figure 4 Hoop stress Figure 5 The comparison of Hoop stress along the

thickness by present and the finite element method

5 Conclusions

This paper presents a direct method of solution to obtain the transient mechanical and thermal

stresses in a piezoelectric cylinder with heat source. The advantage of this method, compared

to the conventional potential function method, is its mathematical strength to handle more

general types of the mechanical and thermal boundary conditions. More complicated

mechanical and thermal boundary conditions may be handled using the proposed method.

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

heat source. Temperature distribution are zero at t=0 due to the initial temperature. According

to the given mechanical boundary conditions, stresses at the outside and displacements at the

inside surface are zero and temperature increases as radius increase.

References

[1] Alashti, A.R., and Khorsand, M., “Three-dimensional Thermo-elastic Analysis of a

Functionally Graded Cylindrical Shell with Piezoelectric Layers by Differential

Quadrature Method”, Int. J. Pressure Vessels and Piping, Vol. 88, No. 5-7, pp. 167-180,

(2011).


34

[2] Alibeigloo, A., and Chen, W.Q., “Elasticity Solution for an FGM Cylindrical Panel

Integrated with Piezoelectric Layers”, Eur. J. Mech. A/Solids, Vol. 29, No. 4, pp. 714-723,

(2010).

[3] Chu, H.S., and Tzou, J.H., “Transient Response of a Composite Hollow Cylinder Heated

by a Moving Line Source”, ASME, Vol. 3, pp. 677-682, (1987).

[4] Fesharaki, J.J., Fesharaki, J.V., Yazdipoor, M., and Razavian, B., “Two-dimensional

Solution for Electro-mechanical Behavior of Functionally Graded Piezoelectric Hollow

Cylinder”, Applied Mathematical Modeling, Vol. 36, No. 11, pp. 5521–5533, (2012).

[5] He, X.Q., Ng, T.Y., Sivashanker, S., and Liew, K.M., “Active Control of FGM Plates with

Integrated Piezoelectric Sensors and Actuators”, Int. J. Solid Struct, Vol. 38, No. 9, pp.

1641-1655, (2001).

[6] Jabbari, M., Mohazzab, A.H., and Bahtui, A., “One-dimensional Moving Heat Source in a

Hollow FGM Cylinder”, ASME J. Appl. Mech, Vol. 131, No. 2, pp. 12021-12027, (2009).

[7] Jabbari, M., Vaghari, A.R., Bahtui, A., and Eslami, M.R., “Exact Solution for Asymmetric

Transient Thermal and Mechanical Stresses in FGM Hollow Cylinders with Heat Source”,

Str. Mech. & Eng, Vol. 29, No. 5, pp. 551–565, (2008).

[8] Khoshgoftar, M.J., Ghorbanpour Aranl, A., and Arefi, M., “Thermoelastic Analysis of a

Thick Walled Cylinder Made of Functionally Graded Piezoelectric Material”, Smart Mater.

Struct, Vol. 18, No. 11, pp. 5007, (2009).

[9] Poultangari, R., Jabbari, M., and Eslami, M.R, “Functionally Graded Hollow Spheres under

Non-axisymmetric Thermo-mechanical Loads”, Int. J. Pressure Vessels Piping, Vol. 85,

No. 5, pp. 295-305, (2008).

[10] Tiersten, H.F, "Linear Piezoelectric Plate Vibrations", New York, Plenum Press, (1969).

Nomenclature

a, b inner and outer radius of the hollow cylinder

iT temperature distribution

ik thermal conductivity

i mass density

ic specific heat capacity

R energy source

r radial coordinate


35

circumferential coordinate

t time

, 1,2ijX i j Robine-type constants

1 ,g t ambient temperature inner the cylinder

2 ,g t ambient temperature outer the cylinder

3 ,g r t initial temperature for the cylinder

mn eigenvalues for the cylinder

,n nJ Y Bessel functions for the first and the second kinds of order n

,u v Displacement components

C Elastic coefficient

e Piezoelectric coefficient

Dielectric constant

P Pyroelectric constant

Thermal modulus

Poisson's ratio

ij Strain tensor ( , ) ( , )i j r

ij Stress tensor ( , ) ( , )i j r


36

Appendix

22 22 44 12 22

11 11

2 22 211 21 24 12 22 44 22

11 11 11 44 44 44

24 21 243 4 7 8 11

44 44

,

, (1 ) (1 ) ,

( ) ,

nk nk nk nk

nk nk nk nk nk nk nk

nk nk mn mn mn mn mn

C n C C Ca b n

C C

e e n e C C C n Cc a n b

C C C C C C

e e ec n D D D D D

C C

12

212 44 12 2222 44

1 2

11 11 11

2

11 11 21 11 21 243 4

11 11 11 11 11

212 44 222 44

5 6

11

0

2 2 1 3 2 1 , 2

32 2 1 2 , 2

24 ,

mn

rr

D

in C C in C CC n Cx k n k n k n x k n

C C C

e e e e e n ex k n k n k n x k n

C C C C C

in C C CC n Cx x

C

22 11 21 24

7

11 11

2 212 44 2222 44 11 21 24

8 9 10

11 11 11

12 44 12 2211

44 44

2

2212

44

24 2113

44

4 2,

3 2 9 39 , ,

21 2

2 2 1 3 2

2

e e n ex

C C

in C C CC n C e e n ex x x

C C C

C C C Cx in k n in

C C

n Cx k n k n k n

C

e ex in k n in

C

24 21 44 12 2214 15

44 44 44

2 2

22 24 21 44 12 22 2216 17 18 19

44 44 44 44

24 21 2120 21

44 11

2 3 2, ,

3 2 4 33 , , 8

4 3, 2 2 1 3 2

ine e C C Cx x in

C C C

n C e e C C C n Cx x in x in x

C C C C

e e e ex in x k n k n k n

C e

2

11 21 24

11

21 24 2122

11 11

2

11 11 22 1123

11 11 11

2

11 21 24 21 2424 25 26

11 11 11 11 11

27

2

2 2 1 3 2

4 2 22 , ,r r

e n e

e

e e ex in k n in

e e

nx k n k n k n

e e e

PP P e e n e e ex k n in x x in

e e e e e

nx

2 2 2

22 11 11 21 24 21 24 22 1128 29 30

11 11 11 11

031 1 032 1 033 1, 034 1,

11 11 44 44

035 1 1, 036

11 11

4 9 3 3 2 9, , ,

3 3, , ,

2 3,

r r

r r r

e e n e e e nx x in x

e e e e

x B x A x B x AC C C C

P P Px B B x

e e e

1 1,

11 11

rPA A

e


37

4 2 3 1 4 3 1 2 4

14 12 13 11 14 13 11 12 14

24 22 23 21 24 23 21 22 24

1 5 9

1 2 3 1 2 3 1 2 3

11 12 13 11 12 13 11 12 13

21 22 23 21 22 23 21 22 23

4 2 3

14 12 13

24 22

2

, ,mnk mnk mnk

mn

mn

mn

mnk

x x x x x x x x x

x x x x x x x x x

x x x x x x x x xD D D

x x x x x x x x x

x x x x x x x x x

x x x x x x x x x

c x x x

c x x x

c x xD

1 4 3 1 2 4

11 14 13 11 12 14

23 21 24 23 21 22 24

6 10

1 2 3 1 2 3 1 2 3

11 12 13 11 12 13 11 12 13

21 22 23 21 22 23 21 22 23

, ,

mn mn

mn mn

mn mn

mnk mnk

x c x x x x c x

x c x x x x c x

x x c x x x x c xD D

x x x x x x x x x

x x x x x x x x x

x x x x x x x x x

2 3

0 0 1 0 0 0 2 0 0 0 03 04

0

2 3

0 0 5 0 0 0 6 0 0 0 07 08

0

2 3

0 0 9 0 0 0 10 0 0 0 011 012

0

0

0 1

( , ) ( ) ( ) ( )

( , ) ( ) ( ) ( )

( , ) ( ) ( ) ( )

p

m m k m m k m m m m

k

p

m m k m m k m m m m

k

p

m m k m m k m m m m

k

m k

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

v r t r D J r D Y r G t D r D r

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

x

D

4 03 04 03 031 07

024 023 024 023 035 027

0 2 03

01 03 01 03 05 07

021 023 021 023 025 027

032 010 01 04 01 04

036 030 021 024 021 024

04 0 9 0 10

08 010 01 03

028 030 021 023

, ,

,

mn

mn

m k m

mn

mn

m m k m k

x c x x x x

x x c x x x xD D

x x x x x x

x x x x x x

x x x x x c x

x x x x x c xD D D

x x x x

x x x x

01 03

021 023

05 031 08 032

025 035 028 036

011 012

05 07 08 010

025 027 028 030

033 03405 06 07 08

, ,

,

0 03 8

m m

m m m m

x x

x x

x x x x

x x x xD D

x x x x

x x x x

x xD D D D


38

چکیده

ده از ساخته ش یتوخال یلندرس ینامتقارن برا یکیو مکان حرارتیتنش یگذرابعدی تحلیل دومقاله یندر ا

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یعموم بعو تا یرهامتغ یبا استفاده از روش جداساز ی،کل یحرارت یمرز یطو زمان با شرا محیطیو یشعاع

یتگبس یباست که به ترت یدارحالت گذرا و پاهای حلمجموع آن حاصل از یجنتاکه بسل، به دست آمده است

دما و منبع حرارت دارند. یهاول یطبه شرا

ه شده است. در نظر گرفت یزوالکتریکپ یلندرس یبر رو یکیو مکان یحرارت یمرز یطشرا یشکل کلدر این مقاله

روش یرحل معادلات ناو یبرادر این مقاله است. یکسانطول ضخامت رد یزوالکتریکپ یلندرخواص مواد س

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