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World Journal of Mechanics, 2013, 3, 6-12
http://dx.doi.org/10.4236/wjm.2013.35A002 Published Online August 2013 (http://www.scirp.org/journal/wjm)
Thermoelastic Problem of a Long Annular Multilayered
Cylinder
Yi Hsien Wu1*, Kuo-Chang Jane2
1Department of Information Management, Oriental Institute of Technology, Taipei, Chinese Taipei
2Department of Applied Mathematics, National Chung Hsing University, Taichung, Chinese Taipei
Email: *[email protected]
Received May 2, 2013; revised June 2, 2013; accepted June 9, 2013
Copyright © 2013 Yi Hsien Wu, Kuo-Chang Jane. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
Thermoelastic transient response of multilayered annu lar cylinde rs of infinite leng ths subjected to known inner pressure
and outer surfaces cooling are considered. A method based on the Laplace transformation and finite difference method
has been developed to analyze the thermoelasticity problem. Using the Laplace transform with respect to time, the gen-
eral solutions of the governing equations are obtained in transform domain. The solution is obtained by using the matrix
similarity transformation and inverse Laplace transform. Solutions for the temperature and thermal stress distributions
in a transient state were obtained. It was found that the temperature distribution, the displacement and the thermal
stresses change slightly as time increases.
Keywords: Thermoelastic; Multilayered Annular Cylinders; Laplace Transformation; Finite Difference Method
1. Introduction
A thermal problem arises when the composed materials
are generated by a sudden change in temperature. Shell
structures are widely used in contemporary industries, so
we must take care of the thermal problem. The shell
structures may be affected due to the pressure change or
the various temperature distributions. It is necessary to
solve for temperature or pressure at first.
The dynamic thermoelastic response of circular shell
rapidly change of thermal environments is important for
the design of many engineering structures. Due to the
complexity of the governing equations and the mathe-
matical difficulties associated with the solution, several
simplifications have been used. For example, Sherief and
Anwar [1] discussed the problem of an annular infinitely
long elastic circular. They have neglected both the in ertia
terms and the relaxation effects of the problem. Sherief
and Anwar [2] considered the thermoelasticity problem
of an infinitely long annular cylinder composed of two
different materials with axial symmetry. The solution
was obtained in the Laplace transform domain by using
the potential functio n approach.
The present work deals with the one-dimensional qua-
sistatic coupled thermoelastic problems of an infinitely
long annular multilayered cylinder composed of multi-
layered different materials. The medium has a pressure at
the inner layer, the temperature to be heated at the outer
layer, without body forces and internal heat generation.
Derivatives are approximated by central differences re-
sulting in an algebraic representation of the partial dif-
ferential equation. By taking the Laplace transform with
respect to time, the general solutions in the transform
domain are first obtained. The final solutions in the real
domain can be obtained by inverting the Laplace trans-
form.
2. Formulation
This work deals with the one-dimensional, quasi-static
coupled, thermoelastic problems of an infinitely long
annular cylinder composed of multilayered laminated
materials with axial symmetry under the following as-
sumptions: 1) Materials of each layer are assumed to be
non-homogeneous; 2) Deformation and strain satisfy the
Hooke’s law and small strain theory; 3) The composite
cylinder is constructed of multilayered laminates bonded
together p erfectly; 4) Th e medium is initiall y undisturb ed,
and without body forces and internal heat sources; 5) The
medium is applied by a force, which is the function of
time; 6) The temperature at inner layer and outer layer
are the functions of time.
We now consider an infinitely long annular cylinder
*Corresponding a uthor.
C
opyright © 2013 SciRes. WJM
Y. H. WU, K.-C. JANE 7
made of multiple layers of different materials. The inner
and outer radii of the cylinder are denoted by i and o,
respectively. The multilayered composite is assumed to
be heated suddenly at the inner and outer surface under
temperatures
r r
1
f
and 2
f
respectively.
The transient heat conduction equation for the
layer in dimensional form can be written as (see Equation
(1) below)
thi
where
0


1r
rr
rr
Er




and

1r
rr
rr
E


r



in which is the radial component of displacement,
is radius, and
U
rv
C
are specific heat and density
of material, r
and r
are the Poisson’s ratio ,
r
k
k
are radial, circumferential thermal conductivity, r
,
are radial and circumferential thermal expansion
coefficient, r, E E
are radial and circumferential
Young’s modulus, , 0 are the temperature, refer-
ence temperature, and
is time, respectively.
If the body forces are absent, the equation of equilib-
rium for a cylinder along the radial direction can be writ-
ten as


22
211
1
rr
rr
rrrr rr
r
EE
UU
U
EErr
rr
E
E
rr


 







 



 
 


(2)
The stress-displacement relations are
0
r
r
ri ri
ii
E
EU U
rr






 
 
 
(3)
0
rr
ii
ii
EE
UU
rr





 

 
 

(4)
where
, r
,
are Lame’s constant, radial and
circumferential stresses respectively.
Let the boundary conditions of multilayered cylinder
be at
0t 0U

at
1
rR

1
0
,e
ct
rrt P
 10
at
o
rR
,0
rrt

2
02 1e
ct
out r
f
  
where 1
f
, 2
f
, 0, r
P
are inner and outer surround-
ing temperatures, initial inner pressure, the initial tem-
perature at the outer layer respectively.
The non-dimensional variables are defined as fol-
lows:
00
T0

22 22
1
cos sincos sin
rr
ivv
i
kk kk
aCC







22 22
1
sincoscos sin
rr
ivv
i
kk kk
bCC







22 22
sincoscos sin
rr
ivv
ii
wCC

 





22
2
1
1
cos sin
r
v
kk
tR
C




1
rrR
22
1
cos sin
r
vi
uU R
C




1
irr r
i
eEE
 



ir
i
f
EE

22
0
cos sin
r
ir
i
vi
gC




r




22
0
cos sin
r
irr
vr i
i
E
hCE

 
  


rr





22
11
cos sin
r
r
ir
ivi
E
QC


0






22
21
cos sin
rr
ir
ivi
E
QC
 


0






31irir
Q
22
11
cos sin
rr r
iivi
E
RC



0






22
21
cos sin
r
r
iivi
E
RC


0






1
f

 

2
2
22 2222
0
22
0
1
cos sinsincoscossin
1
sin cos
rr vr
r
U
kk kkC
rr r
r
U
r

 
 
 









 

(1)
Copyright © 2013 SciRes. WJM
Y. H. WU, K.-C. JANE
8
31ii
R

10riri r



10ii



where , ,
Tt
r
, , ri
u
, i
are non-dimensional
temperature, time, radius, displacement, radial stress and
circumferential stress for the layer respectively.
ith
Substituting the nondimensional quantities into the
governing Equations (1)-(4), the transient heat conduc-
tion equation and stress-displacement relations have the
following nondimensional form:
2
2i
ib
u
aT
rrtr trtr


 

 
 

 
  



2
22
iiii
e
uuu TT
f
gh
rrrrrr
 
 
 (6)
123ri iii
uu
QQQ
rr
T

(7)
123iii i
uu
RRR
rr
T

(8)
i
w
u
(5)
3. Computational Procedures
Applying central difference in Equations (5)-(8), we ar-
rive at the following discretized equations:

1111 1
2
21
22
jjjjjjjjj
i
ii
jj jj
j
uu
TTTTTTu tt
w
ab
rr trtr
r
 

 
 
  
 

 
1
(9)

1111 11
22
211
22
j
jjjjj j
iijij
i
j
jj
j
j
uuuuuTT
efug
rr r
r
r
 
 


j
T
h
r
(10)
11
12
2
jjj
riiii j
jj
uu u
QQ
rr


3
QT
(11)
11
12
2
jj j
iii i
jj
uu u
RR
rr


3j
RT
(12)
where

11
j
rN 
and 1, 2,,jN
.
The Laplace transform of a function are de-
fined by

t
 
0ed
st
s
Lt t
 

t
Take the Laplace transform for Equations (9)-(12), we
obtain the following equations:
 

1111
,,
2
1,1 1,1
21()
2
1
2
jjjjji
iijin jjin j
jj j
j
jinj jinj
j
TTTTTw
abTsTusu
rr r
r
usuusu
r


 


(13)

1111 11
22
211
22
j
jjjjj jj
iijii
j
jj
j
j
uuuuuTT T
efug
rr r
r
r
 
 


j
h
r
(14)
Let the surface of the cylindrical inner surface be
stress free and subject to a time-dependent temperature.
After taking Laplace transformation, the boundary condi-
tions in transformed domain become

0
10 1
1
,
rr
P
rs
s
c


11
Tfs
0
at ;
1
rr

,
rrs
0 12
11
r
T
s
sc

 


at .
out
rr
and the interface conditions are as follows:
1
,
ii
ursu rs
,
1i
rr
1
,
ri ri
rs rs

, 1i
rr
1
,
ii
qrsqrs
, 1i
rr
1
,
ii
Trs T rs
,
1i
rr
where
1, 2,,1im
.
Substituting the boundary conditions and the interface
conditions into Equations (13), (14), we obtain the fol-
Copyright © 2013 SciRes. WJM
Y. H. WU, K.-C. JANE 9
lowing equation in matrix form (see Equations (15) be-
low)
where

1
31 1
12
11 1
2
1Qa
BQr


 





2
2i
j
i
a
Br

1
31 21 1
111111 1
1
1QQw
EQrQr

 



2
1
2
ii
j
j
i
i
ab
Crr
r

111
0
NN N
FD X YZG 1
2
ji
Dr

1
32
1
1
12
mmm
Nmk
m
Qab
AQr
r

 




m
r
i
j
j
w
Er


1
32
1
2
1mm
Nmm
Qa
BQr


 




1
2
ji
Fr

1
32
11
1
1mmm
Nmkmk
r
1
31 0
11111
1QP
XQQ




0
j
Y

1
32 2
12
1
12
mm
N
r
km
m
Qab
YQrr
r








QwQ
EQrQ
 
 


0
j
X
0Z
j

1
32 2
12
1
12
mm
N
r
km
m
Qab
ZQrr
r








0G
j

1
31 0
11
11
2
111 111
1
1
12
QP
ba
GT
QrrQ
r


 






where denotes the last layer, the last point, and
denotes it layer for
mk
ih
2,3, ,1jN
,
(see Equation (16) below)
where:
2i
ji
g
Hr
i
j
j
h
Ir
2
i
ji
g
Jr


2
11
2
i
j
j
i
i
e
Krr
r


22
2i
jj
i
f
Lr
r
 

2
11
2
i
j
j
i
i
e
Mrr
r

2,3, ,1jN

11 111
22 2222
12
111 111
11
22
111
NNN NNN
NN NNN
BC TXY
AB CTXY
sI scsc s
ABC TXY
AB TXY
EF
DE F
s
 
 1
2
1
N
N
Z
Z
Z
Z











 














 
11
22
2
11111
NNNN N
NNN N
uG
uG
DEFuG
DEu G

















 
(15)
111 11
1
2222 222
2
11 11 111
1
0
0
0
0
NN NN NNN
N
NNN NN
N
IJLM u
T
HIJKL Mu
T
HIJKLM u
T
HIK Lu
T
  

 


 


 


 



 


 


 


 

 








(16)
Copyright © 2013 SciRes. WJM
Y. H. WU, K.-C. JANE
10
Equations (15) and (16) can be rewritten in the following
matrix forms
 



 
 
1
2
1
11
j
jj
jjj
M
sITsN uX
sc
YZG
sc s


(17)


0
jj
RT Qu (18)
where the matrix
M
,
N,
R and
Q are the
corresponding matrix in Equations (15) and (16). Substi-
tuting Equation (17) into (18), we have
 

 
 
1
2
1
11
j
j
j
jj
A
sI TB
sc
CDF
sc s


(19)
where
 

 
1
11 1
A
NQRNM



 



1
11 1
j
j
BNQRNX
 


 



1
11 1
j
j
CNQRNY



 



1
11 1
j
j
DNQRNZ
 


 



1
11 1
j
j
F
NQRNG
 

Since the matrix
NN
is a nonsingular real
matrix, the matrix
possesses a set of linearly
independent eigenvectors, hence the matrix
N
is di-
agonalizable. There exist a nonsingular transition matrix
P such that

, that is, the ma-
trices
 
P gA
1
PA
dia
and
diag A are similar, where the ma-
trix
diag
1, 2jA,N
is a diagonal matrix with elements
j, where
,
j
is the eigenvalue of ma-
trix
.
The equation ca n be o bt ai n ed as












11
1
111
2
1
11
1
j
j
jj
PAPsIPT PB
sc
PCPD PF
sc s




j
(20)
Equation (20) can be rewritten as
 

 
 
1
2
1
11
j
j
jjj
diagAs ITB
sc
CDF
sc s




(21)
where

1
j
j
TPT
,



1
j
j
BPB

1
j
j
CPC
,



1
j
j
DPD
and


1
j
j
F
PF
From Equation (21), the following solutions can be
obtained immediately.




12
jj j
jii i
FBC D
Tj
i
s
scsscs ss
 

 
 
(22)
By applying the inverse Laplace transforms to Equa-
tion (22), we get the solution
j
T. The eigenvalue, ei-
genvector and inverse Laplace transform of matrix
can be solved by applying the IMSL MATH/LIBRARY
subroutines.
][A
After we have obtained
j
T, then we can use Equa-
tions (23) and (24) to obtain the solutions
j
T and
j
u

j
j
TPT
(23)


1
j
j
uQRT
 (24)
Substituting
j
T and
j
u into Equations (11) and (12),
we obtain the radial and circumferential stresses.
4. Numerical Results and Discussions
In this section, we present some numerical results of the
temperature distribution in a long multilayered co mposite
hollow cylinder, and displacement and thermal stresses
under temperature changes.
The inner and outer radii of the cylinder are assumed
to be 1.0 and 4.5 respectively. For an infinitely long an-
nular multilayered cylinder, the geometry and material
quantities of the cylinder (in the case of three layers,
layer 1:E = 58E6, k = 22,
= 0.2,
= 2.8E – 6,
= 0.095,
Cv = 0.31 and layer 2 :E = 30E6, k = 21,
= 0.35,
=
2.3E – 6,
= 0.053, Cv = 0.25 and layer 3 : E = 22E6, k =
17,
= 0.2,
= 2.8E – 6,
= 0.09, Cv = 0.17 ; in the case
of five layers, layer 1 : E = 58E6, k = 22,
= 0.2,
=
2.8E – 6,
= 0.095, Cv = 0.31 and layer 2 : E = 30E6, k =
21,
= 0.35,
= 2.3E – 6,
= 0.053, Cv = 0.25 and layer
3:E = 22E6, k = 17,
= 0.2,
= 2.8E – 6,
= 0.09, Cv =
0.17 and layer 4:E = 30E6, k = 21,
= 0.35,
= 2.3E – 6,
= 0.053, Cv = 0.25 and layer 5 : E = 22E6, k = 17,
=
0.2,
= 2.8E – 6,
= 0.09, Cv = 0.17). Each layer is as-
sumed to have a different thickness (in the case of three
layers, r1 = 1.5, r2 = 0.5 and r3 = 1.5; in the case of five
layers, r1 = 1.0, r2 = 0.5, r3 = 1.0, r4 = 0.5 and r5 = 0.5).
The pressure of the inner surface is assumed to be P0 =
1.5E6. The constant coefficient c1 = c2 = 1.0. The tem-
perature at inner surface is assumed to be 300, at outer
Copyright © 2013 SciRes. WJM
Y. H. WU, K.-C. JANE 11
surface which is a function of time is assumed to be 0 to
100. Figures 1-4 show some numerical results of three
and five layered cylinders at time step t = 0.5, 1, 2, 5 and
10.
Figures 1 and 2 show the temperature distributions
along radial direction for 3 and 5 layers case. Because of
the difference in thermal conductivity and the effect of
the outer layer is to be heated. As time is small,
say t = 0.5, the outer layer temperature which is to be
heated is not so more, so the distribution decreasing at
first and then increasing. Figures 3 and 4 show the dis-
placement along the radial direction. The maximum dis-
placement occurred at the interface of first and second
layers. Figures 5 and 6 show the radial stress distribution
r
along the radial direction. Figures 7 and 8 show the
circumferential stress
along the circumferential di-
rection.
Figure 1. Temperature distribution along radial direction
for 3 layers case.
Figure 2. Temperature distribution along radial direction
for 5 layers case.
Figure 3. Radial displacement distribution along radial
direction for 3 layers case.
Figure 4. Radial displacement distribution along radial
direction for 5 layers case.
Figure 5. Radial stress distribution along radial direction
for 3 layers case.
Copyright © 2013 SciRes. WJM
Y. H. WU, K.-C. JANE
Copyright © 2013 SciRes. WJM
12
Figure 8. Circumferential stress distribution along radial
direction for 5 layers case.
Figure 6. Radial stress distribution along radial direction
for 5 layers case.
distributions have been obtained, all of which can be
used to design useful structures or machines for engi-
neering applications. There is no limit to the number
of annular layers in a cylinder. Exemplifying numerical
results from three- and five- layered cylinders at different
time steps have been presented. The discontinuity in cir-
cumferential stress at each interface was found. It was
found that the temperature distribution, the displacement
and the thermal stresses vary slightly as the time in-
creases.
REFERENCES
[1] H. H. Sherief and M. N. Anwar, “Problem in Gene ralized
Thermoelasticity,” Journal of Thermal Stresses, Vol. 9,
No. 2, 1986, pp. 165-181.
doi:10.1080/01495738608961895
[2] H. H. Sherief and M. N. Anwar, “A Problem in General-
ized Thermoelasticity for an Infinitely Long Annular
Cylinder Composed of Two Different Materials,” Acta
Mechanica, Vol. 80, No. 1-2, 1989, pp. 137-149.
doi:10.1007/BF01178185
Figure 7. Circumferential stress distribution along radial
direction for 3 layers case.
A method based on the finite difference and Laplace
transformation has been developed to obtain numerical
results. The temperature, displacement and thermal stress