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Vol.2, No.11, 1312-1317 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.211159
Copyright © 2010 SciRes. OPEN ACCESS
A finite difference scheme for magneto-thermo analysis
of an infinite cylinder
Daoud S. Mashat
Department of Mathematics, Faculty of Science, King AbdulAziz University, Jeddah, Saudi Arabia; [email protected]
Received 12 July 2010; revised 15 August 2010; accepted 20 August 2010.
ABSTRACT
A finite different scheme as well as least-square
method is presented for the magneto-thermo
analysis of an infinite functionally graded hollow
cylinder. The radial displacement, mechanical
stresses and temperature as well as the elec-
tromagnetic stress are investigated along the
radial direction of the cylinder. Material proper-
ties are assumed to be graded in the radial di-
rection according to a novel exponential -law
distribution in terms of the volume fractions of
the metal and ceramic constituents. The gov-
erning second -order differential equations are
derived from the equations of motion and the
heat-conduction equation. The system of differ-
ential equations is solved numerically and some
plots for displacement, radial stress, and tem-
perature are presented.
Keywords: Infinite Cylinder;
Electro-Magneto-Thermoelastic; Finite Difference
Method
1. INTRODUCTION
Magneto-thermal deformations of a cylinder can occur
when the cylinder is placed in a constant primary mag-
netic field or due to heat exchange with the external or
internal environments, or they can appear as the result of
the deformations themselves, when part of the mechani-
cal energy changes into heat [1-5]. Due to the complex-
ity of the governing equations and the mathematical dif-
ficulties associated with the solution, several simplifica-
tions have been used. Yang and Chen [6] discussed the
transient response of one-dimensional quasi -static
coupled thermo-elasticity problems of an infinitely long
annular cylinder composed of two different materials.
They applied the Laplace transform with respect to time
and used the Fourier series and matrix operations to ob-
tain the solution. Jane and Lee [7] considered the same
problem by using the Laplace transform and the finite
difference method. The cylinder is composed of multi-
layers with different materials. There is no limit of
number of annular layers of the cylinder in the computa-
tional procedures. Lee [8] presented axisymmetric
quasi-static-coupled thermoelastic problems for time-
dependent boundary condition. Laplace transform and
finite difference methods are used to analyze problems.
Using finite difference method, Awaji and Sivakuman
[9] studied the transient thermal stresses of a FGM hol-
low circular cylinder, which is cooled by surrounding
medium. A hybrid numerical method of the Laplace
transformation and the finite difference is applied by
Yang et al. [10] to solve the transient hygrothermal
problem of an infinitely long annular cylinder, in which
the temperature and moisture coupling at the inner and
outer surfaces is taken into account in the boundary con-
ditions. Chen [11] evaluated the tress intensity factors in
a cylinder with a circumferential crack by using the fi-
nite difference method. Jane and Lee [12] considered the
thermoelastic transient response of multilayered annular
cylinders of infinite lengths subjected to known tem-
peratures at traction -free inner and outer surfaces. A
method based on the Laplace transformation and finite
difference method has been developed to analyze the
thermo-elasticity problem.
The primary objective of this investigation is to
generate displacement, stresses, temperature, and
magnetic field in an infinite FGM hollow cylinder.
The present FGM cylinder is placed in a constant
primary magnetic field. It is made of an isotropic ma-
terial with material properties varying in the radial
thickness direction only according to a novel power
law form. The governing partial differential equations
are obtained in conservation forms and solved nu-
merically using finite difference method. Numerical
results for the variation of temperature, displacement
and stresses are presented for a metal-ceramic FG
cylinder. To make the study reasonably, temperature,
displacement, and stresses are distributed along the
D. S. Mashat / Natural Science 2 (2010) 1312-1317
Copyright © 2010 SciRes. OPEN ACCESS
1313
radial direction of the cylinder.
2. MATHEMATICAL MODEL
Let us consider a long cylinder of outer radius r = b,
inner radius r = a, and made of an exponentially graded
material. The cylindrical coordinates system (,, )rz
is
used with z-axis coinciding with the axis of the cylinder.
The strain axis is considered to be symmetric about the
z-axis. We have only the radial displacement r
u which
is independent of
and z. 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 the
time t only. The cylinder is placed in a constant primary
magnetic field 0
H
. The medium is assumed to be
non-ferromagnetic and ferroelectric. Neglecting the
Thompson effect, the simplified Maxwell’s equations of
electro -dynamics for perfectly conducting elastic me-
dium are:
,,0, 0,
h
hj EhE
t
  


(1)
where
,(),
u
EHhuH
t

 


 
(2)
in which
H
is the magnetic field, E
the electric
field, j
the current density, u
the mechanical dis-
placement, and h
is the perturbed magnetic.
The material properties of the FGM cylinder are as-
sumed to be function of the volume fraction of the con-
stituent materials. The functionally graded between the
physical properties and the radial direction r for ceramic
and metal FG cylinder is given by
() e,ln,
P
ra
Nbc
mP
m
P
b
Pr PNba P


 



(3)
where c
P and m
P are the corresponding properties of
ceramic (outer surface) and metal (inner surface), re-
spectively.
The equations of motion in the absence of the body
force are
,,
() ,
ij jij ji
ru
 (4)
where
is the material density of the cylinder and it is
also considered to be a function of r. The symbol ,
()
means differentiation with respect to
x
. The mechani-
cal stress tensor ij
and Maxwell’s electromagnetic
stress tensor ij
are given, respectively, by

1
()2,
,
ij iiijij
ijijjikkij
eT e
hHh HhH

 
 
 (5)
where ij
is Kronecker’s delta, 1
T the absolute tem-
perature,
and
Lamé’s coefficients,
the
magnetic permeability,(32 )
the stress tem-
perature modulus, in which
is the linear thermal ex-
pansion, and ij
e is the stain tensor,
1
,,
2.
ijijji
euu
(6)
For the present problem, considering the radial vibra-
tion of the medium, the only non-zero displacement is
(,)
r
urt, so that
,,0.
rr
rr zz
uu
eee
rr


(7)
Applying an initial magnetic field vector
0
0, 0,
H
H
in cylindrical polar coordinate (,, )rz
to Eqs.1 and 2, the field components in the medium are
then obtained as
00
0,,0 ,0,0,,
0,, 0.
rrr
z
uuu
EHhH
ttr
h
jr


 










(8)
The magneto -elasto-dynamic equation, Eq.4, in the
radial direction of the FG hollow cylinder is given by
2
2
1()(),
rr r
rr r
u
fr
rr t

 


(9)
where
,
rr
r
fr
(10)
is defined as Lorentz’s force, and
1
()2 ()(),
rr r
rr
uu u
rrrT
rrr
 





 (11)
1
()2 ()(),
rr r
uu u
rrrT
rrr

 



 (12)
2
0
() .
rr
rr
uu
rH rr





(13)
The heat conduction equation in the presence of heat
sources can be written in the form
2
1
1
2
112
d
111 ()
,
()d() ()
Qr
T
rrrr rtr
r











(14)
where 1
is the thermal diffusivity, 2
the thermal
conductivity, and Q is the intensity of the applied heat
source.
Generally, this study assumes that
,
,
,
,
1
, 2
,
and
of the FG cylinder change continu-
ously through the radial direction of the cylinder and
obey the gradation relation given in Eq.3.
D. S. Mashat / Natural Science 2 (2010) 1312-1317
Copyright © 2010 SciRes. OPEN ACCESS
1314
3. SOLUTION OF THE PROBLEM
Introducing the following dimensionless variables
may be simplifying the solving process:


1
00
11 22112
000
(,) (,)
,, ,,,
e
e
,,,,
mr
mm m
NA
NA
rr
rr
mm mm
Trtu rt
rat
RAT U
bbbT bT
THT

 
 
 
 

(15)
where 0
Tis the reference temperature. In what follows
we assume that the intensity of the applied heat source is
given by the following form
e,
R
QR

(16)
where
being a non -negative constant,
is
dimensionless time and
is a constant. The effect of
material properties variation of the FG cylinder can be
taken into account in Eqs.9 and 14. The substitution of
Eqs.10-12 into Eqs.9 and 14 with the aid of the dimen-
sionless variables given in Eq.15 produces the governing
equations for the FG cylinder as follows:

2
22
02
1e2e
NR NR
mm
UU
H
RRU
R
R
 


 

22
0e2 e
NR NR
mm
U
RN NHNR
 
 

 

22
0ee
NR NR
mm
T
RNNHURNT
R
 
 





2
()
2
2
e,
NRA U
R

(17)
12
1
2()
12
2
1ee,
N
RNR
NT
RR R
R






 





(18)
where
12
2
12
102
,,
e
ee
,.
m
mNA
m
N
ANA
m
mm mm
NNN
bb
T


 

 


(19)
Note that, m
, m
, m
, 1m
, 2m
, m
and m
are
Lamé’s constants, thermal modulus, thermal diffusivity,
thermal conductivity, magnetic permeability, and density
of the homogeneous metal material, respectively.
The dimensionless stresses induced by the tempera-
ture T and the electromagnetic stress are related to the
dimensionless radial displacement U by
11 e2ee,
NR NR
NR
mm
UUU T
RR R




 


 (20)
22 e2ee,
NR NR
NR
mm
UUU T
RR R



 


(21)
11 e.
NR UU
RR

 


(22)
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.
U
TU
 
(23)
The boundary conditions at the inner and outer radii
of the FG cylinder may be expressed as
0,0 at,
0,0 at1.
TU RA
TUR
R


(24)
4. NUMERICAL SCHEME
A finite element scheme is used here to get the tem-
perature and radial displacement. The finite difference
grids with spatial intervals h in the radial direction and k
as the time step, and use the subscripts i and n to denote
the ith discrete point in the R direction and the nth dis-
crete time. A mesh is defined by
,0, 1,2,...,,
,0, 1,2,3,....
i
n
RAih iJ
kn n

 (25)
The displacement and temperature may be given at
any nodal location by
(, ),(, ).
nn nn
ii ii
URU TRT

 (26)
The equation of motion and the heat conduction equa-
tion, given in Eqs.17 and 18, may be expressed in the
finite difference as

1
112 31
1
41 1
2(1 )
2,
nn nn
ii ii
nn nn
ii ii
UfU fUfU
UfThNTT


 
  (27)
1
11231 4
(1 4),
nn nn
ii ii
TgT gTgTg

 
(28)
where

2
2
1231 3
22
2
42
,,e,, ,
2( )
ee ,
2
Nih
NANih
m
f
k
ff fff
Aih
Vh A ih
k
fVh

 




(29)
1
12
()
2
1
2()()
2
4
1
e,
2
e,1,2,3,
()
NAih
ss
NN Aih
k
gg
h
kn
gs
Aih

 

(30)
D. S. Mashat / Natural Science 2 (2010) 1312-1317
Copyright © 2010 SciRes. OPEN ACCESS
1315
in which
1
()
()(2 )
2e[ ()(2)]
NAih
m
fhAih hN
hAih hN

 
 
()
2
0e[ ()(2)],
NAih
mHhAihhN

 (31)
22
2
() 22
2( )[1( )]
2e[2( ) ]
NAih
m
fAihhNAih
Aihh

 

()
222
0e{2()[1 ()]},
NAih
mHAihhNAih


(32)
3
()
()(2)
2e[ ()(2)]
NAih
m
fhAihhN
hAih hN

 
 
()
2
0e[()(2)],
NAih
mHhAihhN

 (33)
1
1
12
3
1
2,1,
1
2.
ghN g
Aih
ghN
Aih

 






(34)
5. NUMERICAL RESULTS
The above finite element scheme is used here to get
the temperature and radial displacement through the ra-
dial direction of the FGM hollow cylinder. The least
square method is used also to get the appropriate stresses
in the FGM hollow cylinder. The results are presented in
the non -dimensional form:
4171911
*11
10 ,10,10,10.TT 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
0410N/A

 and the relative permeability for
both aluminum m
K
and alumina c
K
. The material
properties are assumed to be as:
metal (aluminum):
3
70GPa,0.35,2700Kg/ m,2.3,
mmmm
EK


662
1
2
23.110/K,84.1810m/ s,
237W/(m K).
mm
m


 
ceramic (alumina):
3
116GPa,0.33,3000Kg/ m,1.0,
ccc c
EK

 
662
1
2
8.710/ K,1.0610m/s,
1.78W/(m K).
cm
m


 
Note that the properties of
,
, and
for metal
or ceramic are graded through the radial direction ac-
cording to the following relations:
,,
(1)(12)2(1)
(32),(, ).
pp p
pp
pp p
pppp
EE
pmc

 


 
 
Results are presented in Figures 1-6 for temperature,
radial displacement and redial stress according to the
fixed constants
6
00
0.25,0.2,20 K,10Oersted.AT H
 
The sensitivity of the time parameter *
and the
Figure 1. Variation of temperature T through the radial di-
rection of the FGM hollow cylinder at different times for
= 0.125.
Figure 2. Variation of radial displacement U through the ra-
dial direction of the FGM hollow cylinder at different times
for
= 0.125.
D. S. Mashat / Natural Science 2 (2010) 1312-1317
Copyright © 2010 SciRes. OPEN ACCESS
1316
Figure 3. Variation of radial stress
through the radial di-
rection of the FGM hollow cylinder at different times for
= 0:125.
Figure 4. Variation of temperature T through the radial direc-
tion of the FGM hollow cylinder at *
= 3.84 for different
values of
.
exponential factor
given in c (16) for the heat source,
are discussed here. The values of *
and
will be
chosen through the illustration figures.
Figures 1-3 represent the variations of the dimen-
sionless temperature T, radial displacement u, and radial
stress
through the radial direction of the FGM hollow
cylinder. Three values of the time parameter *
are
Figure 5. Variation of radial displacement U through the radial
direction of the FGM hollow cylinder at *
= 3.84 for differ-
ent values of
.
Figure 6. Variation of radial stress
through the radial
direction of the FGM hollow cylinder at *
= 3:84 for dif-
ferent values of
.
used. Figure 1 shows that the temperature is sharply
increasing to get its maximum at a neighborhood point
of the inner metal surface. After that, it slightly de-
creases to reach a fixed value at the outer ceramic sur-
face. The temperature decreases as *
decreases. Fig-
D. S. Mashat / Natural Science 2 (2010) 1312-1317
Copyright © 2010 SciRes. OPEN ACCESS
1317
ure 2 shows the displacement distribution through the
radial direction of the FGM cylinder. It is seen that the
displacement decreases dramatically within a very small
range of radial direction at first, and then it increases
gradually to a local higher value. Once again, it de-
creases gradually to a minimum value near the outer
surface. The displacement u decreases as *
increases.
The distribution of radial stress
through the radial
direction of the FGM cylinder is plotted in Figure 3 for
different values of *
. It is seen that
decreases
dramatically within a very small range of radial direction
at first, and then it decreases gradually to its minimum
value at the ceramic outer surface of the cylinder. Also,
decreases as *
increases.
The effects of the heat source intensity coefficient
on the temperature, displacement and stress at fixed time
parameter *3.84
are plotted in Figures 4-6. Figure
4 shows that for different
, the change tendencies of
temperature appear in same obviously. The temperature
increases as
decreases. Figures 5 and 6 show that
the higher values of heat source intensity coefficient
have only a little effect on u and
. It is seen that the
radial displacement and radial stress under larger
are evidently different from that under small
.
6. CONCLUSIONS
The main contribution in this paper is to describe the
effects of time parameter and heat source intensity of
exponentially graded material cylinder on temperature,
displacement and stresses. The results are very sensitive
to the change of time and heat source through the radial
direction of the cylinder. The solution method in this
article may be used as a useful reference to investigate
the temperature, radial displacement, radial and circum-
ferential stresses, and electromagnetic stress in the cyl-
inder. The results carried out can be used to predict the
electro-magneto-thermoelastic response at different
times and for different heat source intensities according
to the engineering requirements.
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