Applied Mathematics, 2011, 2, 123-130
doi:10.4236/am.2011.21014 Published Online January 2011 (http://www.SciRP.org/journal/am)
Copyright © 2011 SciRes. AM
Numerical Solution for a FG Cylinder Problem Using
Finite-Difference Method
Daoud S. Mashat
Department of Mathematics, Faculty of Science, King AbdulAziz University, Jeddah, Saudi Arabia
E-mail: dmashat@kau.edu.sa
Received August 28, 2010; revised November 24, 2010; accepted Nove mber 28, 2010
Abstract
A refined finite-difference approach is presented to solve the thermoelastic problem of functionally graded
cylinders. Material properties of the present cylinder are assu med to be graded in the radial direction accord-
ing to a power-law distribution in terms of the volume fractions of the metal and ceramic constituents. The
governing second-order differential equations are derived from the motion and the heat-conduction equations.
Numerical results for dimensionless temperature, radial displacement, mechanical stresses and electromag-
netic stress are distributed along the radial directions. The effects of time parameter and the functionally
graded coefficient are investigated.
Keywords: Functionally Graded, Annular Cylinder, Electro-Magneto-Thermoelastic Field
1. Introduction
Magneto-thermoelastic functionally graded material is a
kind of magneto-thermoelastic material with material
composition and properties varying continuously along
certain directions. It is the composite material intention-
ally designed so that they possess desirable properties for
some specific applications. The advantage of this new
kind of materials can improve the reliability of life span
of magnetic devices. Recently, there has been growing
interest in materials deliberately fabricated so that their
electric, magnetic, thermal and mechanical properties
vary continuously in space on the macroscopic scale.
This research subject is so new that only a few results
can be found in the literatures.
In recent years the theory of magneto -thermo -elasticity
that deals with the interactions among strain, temperature
and electromagnetic fields has drawn the attention of
many researchers. This theory is extensive uses in d iverse
field such as geophysics for understanding the effect of
the Earth’s magnetic field on seismic waves, damping of
acoustic waves in a magnetic field, and emissions of
electromagnetic radiations from nuclear devices. It is
also developed a highly sensitive superconducting mag-
netometer, electrical power engineering and optics.
In the field of magneto-elasticity or magneto-thermo-
elasticity, many studies have been conducted on an ana-
lytical treatment of an interaction between elastic, elec-
tromagnetic, and temperature fields (e.g., Paria [1], Wang
et al. [2], Wang and Dai [3], Wang and Dong [4], Ba-
nerjee and Roychoudhuri [5]).
The magneto-thermoelastic problem of functionally
graded material (FGM) hollow structures subjected to
mechanical loads is considered by Dai and Fu [6]. Hou
and Leung [7] analyzed the plane strain dynamic prob-
lem of a magneto -electro -elastic hollow cylinder by virtue
of the separation of var iab les, orthog on al exp ansion tech-
nique and the interpolation method. Buchanan [8] consi-
dered the free vibration problem of an infinite magne-
to-electro-elastic cylinder. The frequency behaviour of
clamped-clamped magneto -electro -el astic cylin drica l shel ls
is analysed by Annigeri et al. [9] using the semi-ana-
lytical finite element approach. Hou et al. [10] studied
the transient responses of a special non-homogeneous
magneto-electro-elastic hollow cylinder for axisymme-
tric plane strain problem. The magnetother-moelastic
problem in non-homogeneous isotropic cylinder in a
primary magnetic field is discussed by Abd-El-Salam et
al. [11]. The dynamic and quasi-static behaviors of mag-
neto -thermo -elastic stresses in a conducting hollow circu-
lar cylinder subjected to an arbitrary variation of mag-
netic field are investigated by Higuchi et al. [12].
Bhangale and Ganesan [13] studied the free vibration
on FGM magneto-electro-elastic cylindrical shells using
a series solution. Kong et al. [14] presented an analytical
method to investigate thermo -magneto-elastic stresses
D. S. MASHAT
Copyright © 2011 SciRes. AM
124
and perturbation of the magnetic field vector in a con-
ducting non-homogeneous hollow cylinder under thermal
shock. Wang and Dai [15] presented magneto-thermo-
dynamic stresses and perturbation of magnetic field vec-
tor in an orthotropic thermoelastic cylinder. Dai and
Wang [16] presented an analytical solution for magne-
to-thermo-electro-elastic problems of a piezoelectric hol-
low cylinder placed in an axial magnetic field subjected
to arbitrary thermal shock, mechanical load and transient
electric excitation.
The objective of this investigation is to generate dis-
placement, stresses, temperature, and magnetic field in a
FGM annular cylinder. Material prop erties of the present
cylinder are assumed to be graded in the radial direction
according to a power-law distribution in terms of the
volume fractions of the metal and ceramic constituents.
The governing motion and the heat-conduction equations
are obtained in conservation forms and solved numeri-
cally using finite difference method. A refined finite-
difference approach is presented to solve the present
problem. Numerical results for the variation of tempera-
ture, displacement and stresses are presented for a metal-
ceramic FG cylinder. To make the study reasonably,
temperature, displacement, and stresses are given for
different homogenization schemes and exponents in the
power-law that d escribes along- the- thick ness variation of
the constituents. The effects of many parameters are in-
vestigated.
2. Mathematical Model
Let us consider an annular cylinder of outer radius r = b,
inner radius r = a, and made of a functionally graded
material. The cylindrical coordinates system
,,rz
for the axially symmetric problem is used with z-axis
coinciding with the axis of the cylinder. We have only
the radial displacement r
u which is independent of
and z. The cylinder is placed in a constant primary mag-
netic field 0
H
as shown in Figure 1. In a generalized
plane strain, we suppose that the planes perpendicular to
the z-axis and r
u is a function of the radial direction r
and time t only. The Cauchy’s relations are
,, 0,
rr
rrzz zr rz
uu
eeeeee
rr
 

(1)
where ij
e are the strain components. The mechanical
stress components rr
and
, and Maxwell’s elec-
tromagnetic stress component rr
are given, respec-
tively, by
 


 


2
0
2,
2,
.
rr rrrr
rr
rr rr
re ererT
re ererT
rH ee

 

 
 


 

(2)
(a) (b)
Figure 1. Temperature distribution through the radial direction of the FGM annular cylinder at (a) different times, and (b)
for different values the gradation exponent.
D. S. MASHAT
Copyright © 2011 SciRes. AM
125
where T is the absolute temperature,
and
Lamé’s coefficients,
the magnetic permeability,

32

 the stress-temperature modulus, in
which
is the linear thermal expansion, and 0
H
is
the primary magnetic field.
The magneto-elasto-dynamic equation in the radial
direction of the FG annular cylinder is given by


2
2
1,
rrrr r
rr
u
r
rrr t


 



(3)
where
is the material density of the cylinder. The
heat conduction equation in the absence of heat source
can be written in the form [17]
 
22
10
22
11
,
r
rCrTrTu
rrtr r
rt



 

 

 





(4)
where 0
T is the reference temperature, Cc
the
specific heat at constant volume, and
is the thermal
conductivity.
The material properties like
,
,
,
,
, c,
and
of the FGM cylinder are assumed to be function
of the volume fraction of the constituent materials. The
functionally graded between the physical properties and
the radial direction r for ceramic and metal FG cylinder
is given by


1,
j
mc c
ra
PrP PP
ba

 


(5)
where c
P and m
P are the corresponding properties of
ceramic and metal, respectively, and j is the volume frac-
tion exponent which takes values greater than or equal to
zero. The value of j equal to zero represents a fully metal
annular cylinder. The above power-law assumption re-
flects a simple rule of mixtures used to obtain the effec-
tive properties of the ceramic-metal FGM annular cy-
linder. The rule of mixtures applies only to the radial
direction. Note that the volume fraction of the metal is
high near the inner surface of the cylinder, and that of
ceramic high near the outer surface. In addition, Equa-
tion (3) indicates that the inner surface of the annular
cylinder is metal-rich whereas its outer surface is ceram-
ic-rich. The density and other physical components of the
annular cylinder according to the power law, and the
power-law exponent may be varied to obtain different
distributions of the component materials through the
radial direction of the cylinder.
3. Solution of the Problem
Introducing the following dimensionless variables may
be simplifying the solving process:



00
11 222
000
,,
,,, ,,
1
,,,,
r
rr
rr
Trtu rt
rat
RAT U
bbb TbT
THT

 
 

(6)
where
is dimensionless time and 0
is the linear
thermal expansion constant. The effect of material prop-
erties variation of the FG cylinder can be taken into ac-
count in Equations (3) and (4) using Equation (5). The
substitution of Equations (1) and (2) into Equations (3)
and (4) with the aid of the dimensionless variables given
in Equation (6) produces the governing equations for the
FG cylinder as follows:





2
*2 2
2
*2 2
2
*2
2,
mc
mc
UU U
RRU RVR
RR
R
T
RV RURV RRT
R
U
R









(7)
22
12
22
11
,TU
RRR R
R



 

 

 




 (8)
where


22
00
2
0
*
*0
12
**
2, 2,
1
,,
1
,,
mm mmcc cc
j
mc mc
HH
R
HVR
A
T
CC
 
 
 

 

 



(9)
where the letter
in the expression

*.
mc c
VR

 (10)
is given instead of the parameters
,
,
, C or
.
Note that, the prime ()
in Equation (7) denotes diffe-
rentiation with respect to R. Also,
p
,
p
,
p
,
p
,
p
C,
p
and
p
(p = m or c) are Lamé’s constants,
thermal modulus, thermal conductivity, specific heat,
magnetic permeability, and density of the homogeneous
metal or ceramic material, respectively, and the parame-
ter
in Equation (10) may be used to represent one of
these constants.
The dimensionless stresses induced by the temperature
T and the electromagnetic stress are related to the dimen-
sionless radial displacement U by
***
11 2,
UUUT
RR R






 (11)
***
22 2,
UUUT
RR R




 (12)
D. S. MASHAT
Copyright © 2011 SciRes. AM
126
*.
UU
RR

 


(13)
Once again the letter
in Equation (10) is given in-
stead of the parameters
,
,
, or
.
The elastic solution for the FG hollow cylinder is
completed by the application of the initial and boundary
conditions. The initial conditions can be expressed as
0,0 at0.
TU
TU


 
 (14)
The boundary conditions at the inner and outer radii of
the FG cylinder may be expressed as
11
11
e,0at ,
0, 0at1,
TRA
TR



(15)
where is an exponent of the decayed heat flux.
4. Numerical Scheme
A finite element scheme is used here to get the tempera-
ture and radial displacement. The finite difference grids
with spatial intervals h (mesh width) in the radial direc-
tion and k as the time step for the maximum time max
,
and use the subscripts i and n to denote the ith discrete
point in the R direction and the nth discrete time. A mesh
is defined by
max
1
,0,1,2,,, .
,0,1,2,3,,, .
i
n
A
RAihiNh N
knnK k
K
 
 
(16)
The displacement and temperature may be given for
positive integers N and K at any nodal location by

,,,.
nn nn
iiii
URU TRT


(17)
The equation of motion and the heat conduction equa-
tion, given in Equations (7) and (8), may be expressed in
the finite difference as


11
11231
411 5
2
,
nn nnn
ii iii
nn n
ii i
UfU fUfUU
fT TfT




 (18)



11
112 31
11
2
1111
21111
21
2
,
2
nn nnn
ii iii
nnn
iii
nnnn
iiii
TgT gTgTT
UUU
Aih
UUUU
h





 


(19)
where

2*
1*
21 ,
2imci
i
k
f
V
hAih
h








(20a)
2*
2*2 2
21 ,
()
ii
i
k
f
V
Aih
hAih




(20b)

2*
3*
21 ,
2imci
i
k
f
V
hAih
h





(20c)
2
2*
45
**
,,
2
mc
ii
ii
k
k
f
fV
h


 (20d)

 
21
123 2
,,1 ,1,1.
22
khh
ggg
A
ihAih
h
 


(20e)
Note that


*
1
0
1
,,0,1,2,, ,
1
1,0, 0.
11
j
imcici
j
iN
Aih
VVi N
A
jAih
ViiNVV
AA



 









(21)
The mechanical stresses and electromagnetic stress are
given, accordingly, by

*** *
111 1
2,
2
nnn n
ii i
ii iii
UUU T
kAih
 



(22)


** **
221 1
2,
2
nnnn
ii
iii iii
UUUT
kA ih




(23)
*11 .
2
nn n
ii i
i
UU U
kAih


 


(24)
The initial conditions in Equation (14) may be written
as
0110 11
0, , 0, .
iiiiii
TTTUUU

 
(25)
Putting n = 0 in Equations (18) and (19), one obtains

10000
11 31 411
1,
2
iiiii
UfUfUfTT
 

(26)


1001
2
11 31
11 11
21111
1
2
,
4
iii i
iiii
TgTgT U
Aih
UUUU
h





(27)
where 1, 2, 3,,1.iN
The boundary conditions
given in Equation (15) at R = A may be written as
011 00
2
e, ,
2
nknnnn n
mm
mm
h
TUU UT
A



 


(28)
and at R = 1, we get

11
2
0, .
2
nnnnn
NNN cNcN
cc
h
TUUUT



 
(29)
D. S. MASHAT
Copyright © 2011 SciRes. AM
127
Putting i = 0 in Equation (18) and using the forward
difference approximation for the temperature’s derivative
with the aid of Equation (28), one obtains



100 001
01312300
000
42 134
2
2
2
43e,
2
nn nn
mm
nn kn
m
mm
h
UffUffUU
A
h
fT Tff


 


 




 


(30)
where
2
0
121
,
2m
m
k
fhhA




(31a)
2
0
222
21
2,
m
m
k
fhA

 


(31b)
2
0
321
,
2m
m
k
fhhA




(31c)
2
0
4.
2m
m
k
fh
(31d)
Putting i = N in Equatio n (18) and using the b ackward
difference approximation for the temperature’s derivative
with the aid of Equation (29), one obtains


121131
1421
2
2
4,
nN NnNNn
c
NNN
cc
nNnn
NNN
h
Uf fUffU
UfT T



 


 
(32)
where

2
12
2,
2
c
N
c
kh
fh
(33a)
22
22
2
2,
c
N
c
kh
fh
 (33b)

22
34
2
2,.
2
2
c
NN
c
c
c
kh k
ff
h
h

(33c)
5. Numerical Results
The temperature, displacement and stresses for the
present cylinder are obtained using the above finite ele-
ment scheme. The results are presented in the non -di-
mensional form:
 
68
1 211 22
10,,,10 .uU
 


All results of this article are for aluminum as inner
metal surface and alumina as outer ceramic surface.
Generally, the magnetic permeability 0
p
p
K

(p = m
or c) is given in terms of the permeability of space
72
0410NA

 and the relative permeability for
both aluminum m
K
and alumina c
K
. The material
properties are assumed to be as:
Metal (aluminum):
3
70 GPa,0.35,2700 Kgm,
mmm
E


6
23.110K,2.3,237 WmK.
mmm
K

 
Ceramic (alumina):
3
116 GPa,0.33,3000 Kgm,
ccc
E


6
8.710K,1.0,1.78 WmK.
ccm
K


Note that the properties of
,
, and
for metal
or ceramic are graded through the radial direction ac-
cording to the following relations:



,,
112 21
32 ,,.
pp p
pp
pp p
pppp
EE
pmc

 


 
 
Results are presented in Figures 1-5 for temperature,
radial displacement, radial stress, circumferential stress
and electromagnetic stress according to the fixed con-
stants
5
00
1,0.25,27 K,210Oersted,2.5.bA TH 
The sensitivity of the ti me parameter
and the FGM
exponent j are discussed through the figures. Figure 1
represents the variation of the dimensionless temperature
T through the radial direction of the FGM annular cy-
linder. Four values of the time parameter
with j = 3
are used in Figure 1(a), while four values of the FGM
exponent j with 0.3
are used in Figure 1(b). Simi-
lar results for the dimensionless displacement u, the radi-
al stress 1
, the circumferential stress 2
and the elec-
tromagnetic stress
are plotted in Figures 2-5, re-
spectively.
Figure 1 shows that the absolute value of the temper-
ature decreases as the time parameter
increases and
the FGM exponent j decreases. It is to be noted that the
temperature is maximum for the homogeneous metal
cylinders. The same behavior occurs for displacement
and stress quantities. The solution satisfied the boundary
conditions (see Figures 1 and 3) and the difference be-
tween homogeneous and FGM cases is shown. The vari-
ation of temperature, displacement, and stresses are due
to the effect of inertia and magnetic field. It is seen that,
the influence of the FGM on temperature, displacement
and stresses is very pronounced. Finally, it is interested
to see that all quantities my by vanished near and at the
external ceramic surface of the annular cylinder.
D. S. MASHAT
Copyright © 2011 SciRes. AM
128
(a) (b)
Figure 2. Displace ment distribution through the radial direction of the FGM annular cylinder at (a) different times, and (b)
for different values the gradation exponent.
(a) (b)
Figure 3. Radial stress distribution through the radial direction of the FGM annular cylinder at (a) different times, and (b)
for different values the gradation exponent.
D. S. MASHAT
Copyright © 2011 SciRes. AM
129
(a) (b)
Figure 4. Circumferential stress distribution through the radial direction of the FGM annular cylinder at (a) different times,
and (b) for different values the gradation exponent.
(a) (b)
Figure 5. Elec tromagnetic stress distribution thro ugh the radial direction of the FGM annular cylinde r at (a) different times,
and (b) for different values the gradation exponent.
D. S. MASHAT
Copyright © 2011 SciRes. AM
130
6. Conclusions
The main contribution in this paper is to describe second-
order explicit finite-difference scheme. This scheme helps
us to solve the coupled hyperbolic equations on a uni-
form grid, and is quite efficient for computation thermal
stresses. The results obtained show that behavior of the
temperature, displacement and stresses may change sig-
nificantly by reason of influence of exponent heat flux
and primary magnetic field in homogeneous and func-
tionally graded cases. These results are specific for the
example considered, but the example may have different
trends because of the dependence of the results on the
magnetic and thermal constants of the metal-ceramic
functionally graded material.
7. References
[1] G. Paria, “Magneto-Elasticity and Magneto-Thermo-Ela-
sticity,” Advances in Applied Mechanics, Vol. 10, 1966,
pp. 73-112. doi:10.1016/S0065-2156(08)70394-6
[2] X. Wang, G. Lu and S. R. Guillow, “Magnetothermody-
namic Stress and Perturbation of Magnetic Field Vector
in a Solid Cylinder,” Journal of Thermal Stresses, Vol.
25, No. 10, 2002, pp. 909-926. doi:10.1080/0149573029
0074397
[3] X. Wang and H. L. Dai, “Magnetothermodynamic Stress
and Perturbation of Magnetic Field Vector in an Ortho-
tropic Thermoelastic Cylinder,” International Journal of
Engineering Science, Vol. 42, No. 5-6, 2004, pp. 539-
556. doi:10.1016/j.ijengsci.2003.08.002
[4] X. Wang and K. Dong, “Magnetothermodynamic Stress
and Perturbation of Magnetic Field Vector in a Non-
Homogeneous Thermoelastic Cylinder,” European Jour-
nal of Mechanics-A/Solids, Vol. 25, No. 1, 2006, pp. 98-
109. doi:10.1016/j.euromechsol.2005.07.003
[5] S. Banerjee and S. K. Roychoudhuri, “Magneto-Thermo-
Elastic Interactions in an Infinite Isotropic Elastic Cy-
linder Subjected to a Periodic Loading,” International
Journal of Engineering Science, Vol. 35, No. 4, 1997, pp.
437- 444. doi:10.1016/S0020-7225(96)00070-5
[6] H. L. Dai and Y. M. Fu, “Magnetothermoelastic Inter-
actions in Hollow Structures of Functionally Graded Ma-
terial Subjected to Mechanical Loads,” International
Journal of Pressure Vessels and Piping, Vol. 84, No. 3,
2007, pp. 132-138. doi:10.1016/j.ijpvp.2006.10.001
[7] P. F. Hou and A. Y. T. Leung, “The Transient Responses
of Magneto -Electro-Elastic Hollow Cylinders,” Smart
Materials and Structures, Vol. 13, No. 4, 2004, pp. 762-
776. doi:10.1088/0964-1726/13/4/014
[8] G. R. Buchanan, “Free Vibration of an Infinite Magne-
to-Electroelastic Cylinder,” Journal of Sound and Vibra-
tion, Vol. 268, No. 2, 2003, pp. 413-426. doi:10.
1016/S0022-460X(03)00357-2
[9] A. R. Annigeri, N. Ganesan and S. Swarnamani, “Free
Vibrations of Clamped-Clamped Magneto-Electro-Elastic
Cylindrical Shells,” Journal of Sound and Vibration, Vol.
292, No. 1-2, 2006, pp. 300-314. doi:10.1016/j.jsv.2005.
07.043
[10] P. F. Hou, H. J. Ding and A. Y. T. Leung, “The Transient
Responses of a Special Non-Homogeneous Magneto-
Electro-Elastic Hollow Cylinder for Axisymmetric Plane
Strain Problem,” Journal of Sound and Vibration, Vol.
291, No. 1-2, 2006, pp. 19-47. doi:10.1016/j.jsv.2005.
05.022
[11] M. R. Abd-El-Salam, A. M. Abd-Alla and H. A. Hosham,
“A Numerical Solution of Magneto -Thermoelastic Problem
in Non-Homogeneous Isotropic Cylinder by the Finite-
Difference Method,” Applied Mathematical Modelling,
Vol. 31, No. 8, 2007, pp. 1662-1670. doi:10.1016/j.apm.
2006.05.009
[12] M. Higuchi, R. Kawamura and Y. Tanigawa, “Dynamic
and Quasi-Static Behaviors of Magneto-Thermo-Elastic
Stresses in a Conducting Hollow Circular Cylinder Sub-
jected to an Arbitrary Variation of Magnetic Field,” In-
ternational Journal of Mechanical Sciences, Vol. 50, No.
3, 2008, pp. 365-379.
[13] R. K. Bhangale and N. Ganesan, “Free Vibration Studies
of Simply Supported Non-Homogeneous Functionally
Graded Magneto -Electro - Elastic Finite Cylindrical Shell s,”
Journal of Sound and Vibration, Vol. 288, No. 1-2, 2005,
pp. 412-422. doi:10.1016/j.jsv.2005.04.008
[14] T. Kong, D. X. Li and X. Wang, “Thermo-Magneto -
Dynamic Stresses and Perturbation of Magnetic Field
Vector in a Non-Homogeneous Hollow Cylinder,” Ap-
plied Mathematical Modelling, Vol. 33, No. 7, 2009, pp.
2939-2950. doi:10.1016/j.apm.2008.10.003
[15] X. Wang and H. L. Dai, “Magneto -Thermo-Dynamic
Stress and Perturbation of Magnetic Field Vector in a
Hollow Cylinder,” Journal of Thermal Stresses, Vol. 3,
2004, pp. 269-288. doi:10.1080/01495730490423900
[16] H. L. Dai and X. Wang, “Magneto-Thermo-Electro-
Elastic Transient Response in a Piezoelectric Hollow Cy-
linder Subjected to Complex Loadings,” International
Journal of Solids and Structures, Vol. 43, No. 18-19,
2006, pp. 5628-5646. doi:10.1016/j.ijsolstr.2005.06.092
[17] S. B. Mukhopadhyay, “Thermoelastic Interactions with-
out Energy Dissipation in an Unbounded Body with a
Spherical Cavity Subjected to Harmonically Varying
Temperature,” Mechanics Re search Communications, Vol.
31, No. 1, 2004, pp. 81-89. doi:10.1016/S0093-6413(03)
00082-X