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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
735 S.M.Mousavi et al.
© 2018 IAU, Arak Branch
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)
One-Dimensional Transient Thermal and Mechanical Stresses …. 736
© 2018 IAU, Arak Branch
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
737 S.M.Mousavi et al.
© 2018 IAU, Arak Branch
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)
One-Dimensional Transient Thermal and Mechanical Stresses …. 738
© 2018 IAU, Arak Branch
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)
739 S.M.Mousavi et al.
© 2018 IAU, Arak Branch
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)
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.
741 S.M.Mousavi et al.
© 2018 IAU, Arak Branch
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:
One-Dimensional Transient Thermal and Mechanical Stresses …. 742
© 2018 IAU, Arak Branch
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)
743 S.M.Mousavi et al.
© 2018 IAU, Arak Branch
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)
One-Dimensional Transient Thermal and Mechanical Stresses …. 744
© 2018 IAU, Arak Branch
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:
745 S.M.Mousavi et al.
© 2018 IAU, Arak Branch
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.
One-Dimensional Transient Thermal and Mechanical Stresses …. 746
© 2018 IAU, Arak Branch
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.
747 S.M.Mousavi et al.
© 2018 IAU, Arak Branch
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.
One-Dimensional Transient Thermal and Mechanical Stresses …. 748
© 2018 IAU, Arak Branch
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.
749 S.M.Mousavi et al.
© 2018 IAU, Arak Branch
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)
One-Dimensional Transient Thermal and Mechanical Stresses …. 750
© 2018 IAU, Arak Branch
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)
751 S.M.Mousavi et al.
© 2018 IAU, Arak Branch
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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