The electro-magneto-thermo-mechanical analysis of a thick walled sphere made of Functionaly Graded Piezoelectric Material

Abstract

The electro-magneto-thermo-mechanical behavior of a smart sphere made of functionally graded piezoelectric materiel (FGPM) is investigated. The vessel is subjected to an internal pressure, a uniform temperature field, an electric potential and a uniform magnetic field. Under such a loading condition initial elastic stresses are locked in the vessel. All of mechanical, thermal and piezoelectric properties except the Poisson ratio have been expressed as a power function of the sphere radious. It has been concluded from the present study that, by applying the propper material inhomogeneity parameter in the functionally graded piezoelectric material (FGPM) vessel, the distributions of stresses, electric potential and radial displacment in the FGPM can be controlled and the best material can be identified for use as sensors and actuators.

Full text

21st Annual International Conference on Mechanical Engineering-ISME2013
7-9 May, 2013, School of Mechanical Eng., K.N.Toosi University, Tehran, Iran
1
ISME2013-988
The electro-magneto-thermo-mechanical analysis of a thick walled sphere made of
Functionaly Graded Piezoelectric Material
A. Loghman 1,
M. Moradi 2
1Faculty of Mechanical Engineering, University of Kashan, Kashan, I. R. Iran, aloghman@kashanu.ac.ir
2Faculty of Mechanical Engineering, University of Kashan, Kashan, I. R. Iran, mehdimoradik@gmail.com
Abstract
The electro-magneto-thermo-mechanical behavior of a
smart sphere made of functionally graded piezoelectric
materiel (FGPM) is investigated. The vessel is subjected
to an internal pressure, a uniform temperature field, an
electric potential and a uniform magnetic field. Under
such a loading condition initial elastic stresses are
locked in the vessel. All of mechanical, thermal and
piezoelectric properties except the Poisson ratio have
been expressed as a power function of the sphere
radious. It has been concluded from the present study
that, by applying the propper material inhomogeneity
parameter in the functionally graded piezoelectric
material (FGPM) vessel, the distributions of stresses,
electric potential and radial displacment in the FGPM
can be controlled and the best material can be identified
for use as sensors and actuators.
Keywords: Functionally graded piezoelectric materiel
(FGPM) spheres, Magneto-electro-thermo-mechanical
loading, Stress and electric potential field
Introduction
Mechanical response of smart structures made of
functionally graded piezoelectric materials (FGPM) has
been an active area of research in the past decade.
Prominent feature of these materials is that their
properties are variable in the space in one or more
directions. Piezoelectric materials are deformed when
an electric potential is imposed, this feature of the
material can be used to make actuators. When
piezoelectric materials are deformed they will produce
electric potential which can be used as sensors. The
FGPM are used widely in aerospace industries, nuclear
reactors, and pressure vessels and etc. These structures
are subject to high temperature at constant loading
condition for long times. Time-dependent creep analysis
is essential to predict the performance of these elements
in the long terms.
Thermo-elastic stress analysis in functionally graded
piezoelectric cylinders, have been made by Khoshgoftar
et al. [1]. An Electro-thermo-mechanical behavior of
functionally graded piezoelectric rotating shafts was
presented by Ghorbanpour et al. [2]. Dai and Fu [3]
investigated the influence of magneto-thermo-elastic
loading on functionally graded hollow spheres and
cylinders. Wang and Zhu [4] imposed electrical and
mechanical loading on a FGPM sphere and the
constitutive equations were solved by using Frobenius
series. Dai et al. [5] considered a magnetic field in
addition to the electrical charge, temperature and
internal pressure on a hollow cylinder and showed that
selecting an appropriate power index for the
functionally graded material leads to have the optimized
stress distribution.
None of the above referred works have considered
the analysis of a FGPM spherical vessel under
combination of electric, magnetic, thermal and
mechanical loadings. In the present work the properties
of the FGPM sphere is assumed to be a power function
of radius. To find out the optimal power index, elastic
stress analysis has been conducted in the FGPM sphere.
Distribution of stresses, displacement and magnetic
field in thick walled FGPM sphere has been
investigated.
Mathematical modeling and governing equations
- General formulation for a thick walled FGPM
sphere
We supposed that the sphere is located in an
environment with constant temperature field T and
uniform magnetic field
(0,0, )HH

and subjected to an
internal pressure and a potential difference
()r
in
radial direction.
Fig (1). A sketch of the functionally graded piezoelectric
sphere under electrical, magnetic, thermal and internal &
external pressure loadings.
Due to the spherical symmetry circumferential and
tangential stresses and strains are the same
,
 


εε.


(1)
Considering Eq. (1) the constitutive relation for stress,
strain and electric displacement is expressed as follows:

2
ISME2013, 7-9 May, 2013
11 12 11 11rr r d
c c e T
dr


   
   
(2a)
21 22 12 21rd
c c e T
dr
 

   
   
(2b)
11 12
2.
rr d
D e e g pT
dr



    
(2c)
Here
r

,

are the radial and tangential total strains
respectively ,
d
dr

is the gradient of potential difference
in radial direction, T is the uniform temperature
distribution of the sphere.
rr

,


and
r
D
are radial
and tangential stresses, and electric displacement in
radial direction respectively.
The coefficients in Eq. (2) are in the power function
form and are defined as below:
0 0 0
11
00
00
11
, , ( 1,2) ,
( , 1,2) , ,
( 1,2) , ,
jj
ij ij
ii
E E r r e j e r
c i j c r p p r
i r g g r
  




   
  
  
(3)
0 0 0 0 0 0 0 0 0 0 0 0
11 12 11 11 12 21 21 22
( ) , ( ) , ( )P e e c c c c
    
     
(4)
In which
, , , , , ,g P c e E

are dielectric coefficient,
thermal modulus, the pyroelectric coefficient, elastic
coefficient, piezoelectric coefficient, conduction heat
transfer coefficient and elastic modulus respectively.
Radial dependent material properties are defined by the
power index β. In this study we have considered β
between -1.5 to 1.5.
When we do not have free electric displacement the
Maxwell equation is written as below [6]:
20,
rr
dD D
dr r

(5)
Solving Eq. (5) yields:
1
2
rq
Dr

(6)
And consequently from Eq. (6) and (2c) the gradient of
electric potential is obtained as:
111 12
2
1( . )
r
q
de e pT
dr g r


    
(7)
The strain - displacement relations considering spherical
symmetry are written as:
,
rdu
dr


(8a)
.
u
r



(8b)
Also considering the incompressibility condition of
material we have:
2.
cc
r


  
(9)
Substituting Eqs. (7), (8) and (9) into Eqs. (2a) (2b),
leads to the following equations for radial and
circumferential stresses in terms of displacement and
total creep strains:
1 2 1
1 2 3 5 ,
rr du
c r c r u c r T c r
dr
  


       
(10a)
1 2 1
6 7 8 10 .
du
c r c r u c r T c r
dr
  



       
(10b)
The constants of the above relations are as follows:
0 2 0 0
00
11 11 12
1 11 2 12
00
0 0 0
0
11 11 11 1
3 11 5
00
( ) 2
,
,.
e e e
c c c c
gg
e P e q
cc
gg


   
  
   
0 0 0 2
00
11 12 12
6 21 7 22
00
0 0 0
0
12 11 12 1
8 21 10
00
2 ( )
,
,.
e e e
c c c c
gg
e P e q
cc
gg


   
  
   
For the considered sphere the equilibrium equation in
the presence of magnetic force and absence of other
body forces can be written as follows:
2( ) 0.
2
rr
rr r
df
dr




  
(11)
That Lorentz force
r
f
defined as follows [7]:
22
( ) ( ).
rd du u
f r H dr dr r



   
(12)
In this relation
,H


are the permeability and the
intensity of magnetic field. The permeability varies
according to power law as:
0.r



(13)
Substituting Eqs. (10b), (10a) and (12) into equilibrium
equation leads the following differential equation:
2
21
1 2 3 4 5
2.
d u du
k r k r k u k r k r
dr
dr


          
(14)
The above constants in Eq. (14) are defined as follows:
02
11
02
2 1 2 6
02
3 2 7
4 3 8
5 10
,
(2 ) 2 2 ,
(1 ) 2 2 ,
( 2 (1 ) 2 ) ,
2.
k c H
k c c c H
k c c H
k c c T
kc








      
     
       

Putting the right side of differential Eq. (14) equal to
zero the general solution can be obtained. For this
purpose we assumed that the general solution of this
equation has the following form:
.
m
g
u Cr
(15(
Inserting Eq. (15) in the left hand side of Eq. (14) leads
to form and solve the characteristic equation, so we
have:
2
1 2 2 1 3 1
11
2
1 2 2 1 3 1
21
( ) 4
1,
2
( ) 4
1.
2
k k k k k k
mk
k k k k k k
mk
   

   

(16)
If m1 and m2 are real numbers, the general answer will
be as follows
12
12
.
mm
g
u C r C r
(17)
Those constant coefficients C2, C1 are achieved by using
the boundary conditions. In the concern problem using β
for elastic solution leads that
12
,mm
take only real
values.

3
ISME2013, 7-9 May, 2013
To obtain the inhomogeneous solution of Eq. (14), we
assume
p
u
as the following form:
1
12p
u w r w r



(18)
Putting Eq. (18) into Eq. (14) causes to have the
coefficients w2 and w1 as:
5
121 1 2 2 1 3
,
( (3 ) 2 )
k
wk k k k k k

     
(19a)
4
221 1 2 3 2
.
( ( ) )
k
wk k k k k

    
(19b)
Finally the complete solution to the radial direction of
displacement will be as follows:
12 1
1 2 1 2
mm
gp
u u u C r C r w r w r


     
(20)
With insertion Eqs. (20) and (3) in Eq. (7) , we have:
12
0 0 2
1 11 1 12 1
11
0 0 0 0
1 11 1 12 2 11 2 12
0 0 0
11 2 12 2
( ( 1) 2 )
( 2 ) ( 2 )
( ( 1) 2 )
mm
q e w e w r
d
dr g
C e m e r C e m e r
gg
e w e w p T r
g







   



  

(21)
Integrating Eq. (21) leads:
12
0 0 1
1 11 1 12 1
0 0 0 0
1 11 1 12 2 11 2 12
12
0 0 0 1
11 2 12 2 1
( ( 1) 2 )
() ( 1)
( 2 ) ( 2 )
( ( 1) 2 ) ,
( 1)
mm
q e w e w r
rg
C e m e r C e m e r
g m g m
e w e w p T r B
g








   

   

   


(22)
Substituting Eq. (20) into Eqs. (10a) and (10b) the
elastic solution is obtained as
12
12
11
1 1 1 2 2 1 2
1 1 2
2 1 2 1 2 3
0
11 1
02
( ( 1))
()
mm
rr
mm
c r C r m C r m w r w r
c r C r C r w r w r c r T
eq
gr
  
   
  
  
  
     
      

(23a)
12
12
11
6 1 1 2 2 1 2
1 1 2
7 1 2 1 2 8
0
12 1
02
( ( 1))
()
mm
mm
c r C r m C r m w r w r
c r C r C r w r w r c r T
eq
gr
  

   
  
  
  
     
      

(23b)
The unknown constants B1, q1 C2, C1are obtained using
the boundary conditions, we establish following system
of equations:
1 1 2 1 3 1 4 2 5
6 1 7 1 8 1 9 2 10
11 1 12 1 13 1 14 2 15
16 1 17 1 18 1 19 2 20
( )| ,
( )| ,
|,
|,
r a a
r b b
rr r a a
rr r b b
r S q S B S C S C S
r S q S B S C S C S
S q S B S C S C S p
S q S B S C S C S p








         
         
         
         
(24)
With solving system of Eq. (24) we reach to the
unknown coefficients. The unknown coefficients in
terms of system coefficients si (i = 1 ... 20) are listed in
Appendix.
Numerical results and discussion
The results presented in this paper are based on the
material properties and specifications introduced in
previous published work [1], [5], [10]. The mechanical
and electrical boundary conditions are also introduced.
0 0 6 0
11 2
2
0 0 9
12 22
0 7 9
1
22 , 1.2 , 0.3, 15.78 ,
5.35 , 7.4 , 873 , 1.4
4* , 2, 1,0,1,2 , 2.23
b
a
c
E Gpa e e
km
r
CC
e g e T K r
m N m
HA
e H e
mm


  



   
    

     
Creep specifications:
36
0 1 0
0.11 , 5 , 3 .b e b n

    
Boundary conditions:
100 ( ) , 0 , 100 ( ) , 0 .
a b a b
kW
P MPa P A

   
Fig (2). Radial stress in the thick wall sphere for β= 1.5, 0.5,
0,-0.5, -1.5
Fig (3). Circumferential stress in the thick wall sphere for β=
1.5, 0.5, 0,-0.5, -1.5
Fig (4). Radial displacement in the thick-walled sphere for β=
1.5, 0.5, 0,-0.5, -1.5

4
ISME2013, 7-9 May, 2013
Fig (5). Dimensionless electric field in the thick wall sphere
for β= 1.5, 0.5, 0,-0.5, -1.5
With the above material properties and boundary
conditions electro-magneto-thermo-elastic radial
stresses are shown in Figure (2). As it is clear in this
figure boundary conditions are satisfied for all
materials. Minimum absolute value of radial stress is for
β = -1.5, and the maximum absolute value of which
belongs to β = 1.5.
The electro-magneto-thermo-elastic circumferential
stresses for the FGPM sphere are depicted in figure (3).
The material in-homogeneity parameter β has a
significant effect on circumferential stresses so that
minimum absolute value of circumferential stress is for
β = -1.5, and the maximum absolute value of which
belongs to β = 1.5 and almost a uniform distribution of
circumferential stress can be reported for β = -1.5.
Figure (4) shows the radial displacement of spherical
vessel. In general radial displacements are decreasing
from the inner surface to outer surface of the FGPM
sphere. However the maximum positive displacements
belong to β=1.5 and minimum displacement is related to
β=-1.5. Figure (5) shows the dimensionless electric
potential distribution along thickness of the sphere. The
boundary conditions of the imposed electric potential
are satisfied and the maximum and minimum values
belong to β=1.5 and β=-1.5 respectively. Since the
sphere is radially polarized then the radial stress and
displacement are dependent on the electric field, so for
the material identified by β=1.5, electric potential field
has the greatest impact on the displacement and radial
stress.
Conclusion
In this study we presented an analytical method for
electro-magneto-thermo-mechanical analysis of a sphere
made of functionally graded piezoelectric material
(FGPM). Loading is composed of an electric potential
field, a uniform magnetic and thermal field and an
applied inner pressure. Distribution of stresses, radial
displacements, electric potential and perturbation of
magnetic field are investigated. The following
conclusions are made as:
1 - In the elastic state, the material inhomogeneity
parameter has a little effect on the radial stress
distribution, but it has a great effect on the
circumferential stress. The best material has been
identified by which the minimum shear stress
distribution will occur along the thickness of the FGPM
sphere.
2 –It is concluded that for the material identified by
β=1.5, electric potential field has the greatest impact on
the displacement and radial stress .
References
[1] M.J. Khoshgoftar, A. Ghorbanpour Arani, M. Arefi,
2009, “Thermo-elastic analysis of a thick walled
cylinder made of functionally graded piezoelectric
material”, SMART MATERIALS AND
STRUCTURES 18, 115007
[2] Ghorbanpour Arani, R. Kolahchi and A.A.
Mosallaie Barzoki, 2010, “Effect of material
inhomogeneity on electro-thermo-mechanical behaviors
of functionally graded piezoelectric rotating shaft”,
Applied Mathematical Modeling 36, 2771–2789
[3] H.L. Dai, Y.M. Fu, 2007, “Magneto-thermo-elastic
interactions in hollow structures of functionally graded
material subjected to mechanical loads”, International
Journal of Pressure Vessels and Piping, 84, 132–138
[4] H.M. Wang, Z.X. Xu, 2010, “Effect of material
inhomogeneity on electromechanical behaviors of
functionally graded piezoelectric spherical structures”,
Computational Materials Science, 48, 440–445
[5] H.L. Dai, Li Hong, Yi-Ming Fu, Xia Xiao, 2009,
“Analytical solution for electro-magneto-thermo-elastic
behaviors of a functionally graded piezoelectric hollow
cylinder”, Applied Mathematical Modeling, 34, 343-357
[6] P. Heyliger, 1996, “A note on the static behavior of
simply-supported laminated piezoelectric cylinders”,
Int. J. Solids Structure. 34, 3781–3794.
[7] J.D. Kraus, 1984, Electro-magnetic, McGraw Hill,
Inc., USA.

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ISME2013, 7-9 May, 2013
Appendix
q1=(s10 . s13 . s19 . s2 - s10 . s14 . s18 . s2 - s10 . s12 . s19 . s3 + s10 . s14 . s17 . s3 + s10 . s12 . s18 . s4 - s10 . s13 . s17 . s4 - s15 . s19 . s2 .
s8 + s15 . s19 . s3 . s7 + s14 . s2 . s20 . s8 - s14 . s20 . s3 . s7 + s15 . s18 . s2 . s9 - s15 . s18 . s4 . s7 - s13 . s2 . s20 . s9 + s13 . s20 . s4 . s7
- s13 . s19 . s5 . s7 + s14 . s18 . s5 . s7 - s15 . s17 . s3 . s9 + s15 . s17 . s4 . s8 + s12 . s20 . s3 . s9 - s12 . s20 . s4 . s8 + s12 . s19 . s5 . s8 -
s14 . s17 . s5 . s8 - s12 . s18 . s5 . s9 + s13 . s17 . s5 . s9)/ s21
B1= - (s1 . s10 . s13 . s19 - s1 . s10 . s14 . s18 - s10 . s11 . s19 . s3 + s10 . s14 . s16 . s3 + s10 . s11 . s18 . s4 - s10 . s13 . s16 . s4 - s1 . s15
. s19 . s8 + s15 . s19 . s3 . s6 + s1 . s14 . s20 . s8 - s14 . s20 . s3 . s6 + s1 . s15 . s18 . s9 - s15 . s18 . s4 . s6 - s1 . s13 . s20 . s9 + s13 . s20 .
s4 . s6 - s13 . s19 . s5 . s6 + s14 . s18 . s5 . s6 - s15 . s16 . s3 . s9 + s15 . s16 . s4 . s8 + s11 . s20 . s3 . s9 - s11 . s20 . s4 . s8 + s11 . s19 . s5
. s8 - s14 . s16 . s5 . s8 - s11 . s18 . s5 . s9 + s13 . s16 . s5 . s9)/ s21
C1=(s1 . s10 . s12 . s19 - s1 . s10 . s14 . s17 - s10 . s11 . s19 . s2 + s10 . s14 . s16 . s2 + s10 . s11 . s17 . s4 - s10 . s12 . s16 . s4 - s1 . s15 . s19 .
s7 + s15 . s19 . s2 . s6 + s1 . s14 . s20 . s7 - s14 . s2 . s20 . s6 + s1 . s15 . s17 . s9 - s15 . s17 . s4 . s6 - s1 . s12 . s20 . s9 + s12 . s20 . s4 . s6
- s12 . s19 . s5 . s6 + s14 . s17 . s5 . s6 - s15 . s16 . s2 . s9 + s15 . s16 . s4 . s7 + s11 . s2 . s20 . s9 - s11 . s20 . s4 . s7 + s11 . s19 . s5 . s7 -
s14 . s16 . s5 . s7 - s11 . s17 . s5 . s9 + s12 . s16 . s5 . s9)/ s21
C2= - (s1 . s10 . s12 . s18 - s1 . s10 . s13 . s17 - s10 . s11 . s18 . s2 + s10 . s13 . s16 . s2 + s10 . s11 . s17 . s3 - s10 . s12 . s16 . s3 - s1 . s15 . s18
. s7 + s15 . s18 . s2 . s6 + s1 . s13 . s20 . s7 - s13 . s2 . s20 . s6 + s1 . s15 . s17 . s8 - s15 . s17 . s3 . s6 - s1 . s12 . s20 . s8 + s12 . s20 . s3 .
s6 - s15 . s16 . s2 . s8 + s15 . s16 . s3 . s7 + s11 . s2 . s20 . s8 - s11 . s20 . s3 . s7 - s12 . s18 . s5 . s6 + s13 . s17 . s5 . s6 + s11 . s18 . s5 . s7
- s13 . s16 . s5 . s7 - s11 . s17 . s5 . s8 + s12 . s16 . s5 . s8)/ s21
s21= (s1 . s13 . s19 . s7 - s1 . s14 . s18 . s7 - s13 . s19 . s2 . s6 + s14 . s18 . s2 . s6 - s1 . s12 . s19 . s8 + s1 . s14 . s17 . s8 + s12 . s19 . s3 . s6 -
s14 . s17 . s3 . s6 + s1 . s12 . s18 . s9 - s1 . s13 . s17 . s9 + s11 . s19 . s2 . s8 - s11 . s19 . s3 . s7 - s12 . s18 . s4 . s6 + s13 . s17 . s4 . s6 - s14
. s16 . s2 . s8 + s14 . s16 . s3 . s7 - s11 . s18 . s2 . s9 + s11 . s18 . s4 . s7 + s13 . s16 . s2 . s9 - s13 . s16 . s4 . s7 + s11 . s17 . s3 . s9 - s11 .
s17 . s4 . s8 - s12 . s16 . s3 . s9 + s12 . s16 . s4 . s8)