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Appl. Appl. Math.
ISSN: 1932-9466
Vol. 11, Issue 1 (June 2016), pp. 127 - 151
Applications and
Applied Mathematics: An International Journal
(AAM)
Thermal Vibration of Magnetostrictive Functionally Graded Material
Shells with the Transverse Shear Deformation Effects
C.C. Hong Department of Mechanical Engineering
Hsiuping University of Science and Technology
Taiwan, ROC
Received: June 10, 2014; Accepted: March 28, 2016
Abstract
The transverse shear deformation effect on the functionally graded material (FGM) circular
cylindrical shells with mounted magnetostrictive layer under thermal vibration is investigated by
using the generalized differential quadrature (GDQ) method. In the time dependent of
displacement field, the first-order shear deformation theory (FSDT) is used. The dynamic
equilibrium differential equations with displacements and shear rotations of FGM shells under
the magnetostrictive load and thermal load are normalized into the dynamic discrete equations.
The computational solutions for thermal stresses and center deflections of magnetostrictive FGM
circular cylindrical shells with four edges in simply supported boundary conditions are obtained.
Some parametric effects on the FGM shells are also investigated. They are thickness of mounted
magnetostrictive layer, control gain values, temperature of environment, and power law index of
FGM shells.
Keywords: shear deformation, FGM, shell, magnetostrictive, thermal vibration, GDQ, FSDT
AMS-MSC 2010 No.: 74B20, 74F05, 74H45, 74K25, 74S30
1. Introduction
There are many literatures describing in detail numerical investigations of functionally graded
material (FGM) shells. Kugler et al. (2013) presented the numerical enhanced finite elements
analysis for FGM shell-like and beam structures. Qu et al. (2013) examined the numerical effects
of the first-order shear deformation shell theory on the FGM cylindrical, conical and spherical
shells subjected to arbitrary boundary conditions. Torabi et al. (2013) presented the numerical
thermal buckling results for piezoelectric FGM conical shell. Alibeigloo et al. (2012)
128 C.C. Hong
investigated the numerical free vibration of a piezoelectric FGM cylindrical shell. Zahedinejad et
al. (2010) studied the numerical free vibration of FGM curved panels. Sepiani et al. (2010)
investigated the free vibration and buckling of FGM cylindrical shells with the transverse shear
and rotary inertia effects. There are some advanced researches on the dynamic vibrations
analyses of composite shells. Qatu et al. (2010) reviewed the articles of dynamic results on
laminated composite shells in 2000-2009. Lee et al. (2006) presented the finite element analysis
of transverse deflection damping on laminated composite shells with Terfenol-D material. Jafari
et al. (2005) studied the effect of first-order shear deformation shell theory on the free and forced
transient dynamic vibrations of composite circular cylindrical shells. Bhangale and Ganesan
(2005) investigated the frequency results of free vibration on FGM cylindrical shell with
piezoelectric and magnetostrictive materials.
Some of the application topics of DQM (differential quadrature method) are fundamental on
composite shells as can be seen in the references. Tornabene et al. (2014) studied the static
behaviors of doubly-curved anisotropic shells by using the DQM. Abediokhchi et al. (2013)
presented the bending analysis of moderately thick FGM conical panels by using the generalized
differential quadrature (GDQ) method. Alibeigloo and Nouri (2010) provided the static analysis
of FGM cylindrical shell with piezoelectric layers by using the DQM. Haftchenari et al. (2007)
presented the dynamic analysis of composite cylindrical shells by using the DQM. The author
has some GDQ experiences in the study of composite material shells and plates. Hong (2014)
presented the thermal vibration and transient response of Terfenol-D FGM Plates. The effects of
modified shear correction coefficient values on the center displacement were considered. Hong
(2013a) investigated the rapid heating transient response of Terfenol-D FGM square plates,
including the effects of temperature of environment and applied heat flux values.
Hong (2013b) studied the rapid heating of induced vibration of Terfenol-D FGM circular
cylindrical shells. Hong (2013c) investigated the thermal vibration of Terfenol-D FGM shells,
without considering the effects of transverse shear deformation. Hong (2010) presented the
computational approach of piezoelectric shells, with the effects of first-order shear deformation
theory. In this transverse shear deformation effect of GDQ study of magnetostrictive FGM shells
with four edges in simply supported boundary conditions, it is interesting to obtain the results of
thermal stresses and center deflections under uncontrolled/controlled gain values. Some
parametric effects on the magnetostrictive FGM shells are also analyzed. They are thickness of
mounted Terfenol-D layer, control gain values, temperature of environment and power-law index
of FGM. The novelty of this study is to provide a fundamental thermal vibration result for thick
Terfenol-D FGM shells with the first order shear deformation by using the GDQ method. It is
new for the calculations of not negligible shear strains, the constant value used simply for shear
correction coefficient and free vibration frequency with the not symmetric layers.
2. Formulation
2.1. Functionally graded material
Materials of FGM can be expressed in series form as follows (Pradhan (2005)),
AAM: Intern. J., Vol. 11, Issue 1 (June 2016) 129
i
n
i
ifgm VPPm
1
, (1)
where fgmP is the material properties of FGM, mn is the number of materials mixed to form the
FGM, iV is the volume fractions,
11
mn
i
iV ,
for all constituent materials, and iP is the individual constituent material properties, usually with
the form as follows,
)1( 3
3
2
21
1
10 TPTPTPTPPPi
, (2)
in which 2110 ,,, PPPP and 3P are the temperature coefficients, T is the temperature of
environment.
For a two-material ( mn = 2) FGM circular cylindrical shell as shown in Figure 1, the sum of
volume fractions is expressed in the form 121 VV . The variation form of
1
/ 2n
z hV R
h
used in the power-law function, where z is the thickness coordinate, h is the thickness of FGMs
shell, nR is the power law index (Chi and Chung, 2006).
Figure 1. Two-material FGM circular cylindrical shell with magnetostrictive layer
130 C.C. Hong
The material properties for Young’s modulus E (in units of G Pa) of power-law function two-
material FGM circular cylindrical shell, Equation (1) can be expressed as follows,
112 )
2/)(( E
h
hzEEE nR
fgm
, (3a)
and the others (values are much less than E ) are reasonably assumed in the simple algebraic
average form between the two components as follows (Delale and Erdogan, 1983):
2/)( 12 fgm , (3b)
2/)( 12 fgm , (3c)
2/)( 12 fgm , (3d)
2/)( 12 fgm , (3e)
2/)(12 vvfgmv CCC , (3f)
where fgmE , 1E and 2E are the Young’s modulus of the FGM shell, the constituent material 1
and 2, respectively, fgm , 1 and 2 are the Poisson’s ratios of the FGM shell, the constituent
material 1 and 2, respectively. fgm , 1 and 2 are the densities of the FGM shell, the
constituent material 1 and 2, respectively; fgm , 1 and 2 are the thermal expansion
coefficients of the FGM shell, the constituent material 1 and 2, respectively; fgm , 1 and 2 are
the thermal conductivities of the FGM shell, the constituent material 1 and 2, respectively, and
fgmvC , 1vC and
2vC are the specific heats of the FGM shell, the constituent material 1 and 2,
respectively. The terms of 1E , 2E , 1 , 2 , 1 , 2 , 1 , 2 , 1 , 2 , 1vC and
2vC can be
expressed in terms corresponding to iP in equation (2).
2.2. Displacement field
The time dependent displacements u , v and w of circular cylindrical shells are assumed in the
first order shear deformation theory (FSDT) form (Qatu et al., 2010) as in the following linear
equations,
),,(),,(0 txztxuu x , (4a)
),,(),,(0 txztxvv , (4b)
),,( txww , (4c)
where 0u and
0v are tangential displacements, w is transverse displacement of the middle-
surface of the shell, x and
are middle-surface shear rotations, x and are in-surface
coordinates of the shell, z is out of surface coordinates of the shell, and t is time.
AAM: Intern. J., Vol. 11, Issue 1 (June 2016) 131
2.3. GDQ method
In 1992, Shu and Richards presented the GDQ method (Shu and Richards, 1992). The GDQ
method can be restated as follows: the derivative of a smooth function at a discrete point in a
domain can be discrete by using an approximated weighting linear sum of the function values at
all the discrete points in the direction (Hong, 2014; Bert et al., 1989; Shu and Du, 1997). The
GDQ method is used to approximate the derivative of function.
2.4. Thermo-elastic stress-strain relations with magnetostrictive effect
For a preliminary data of study, it is reasonable to consider the FGM circular cylindrical shells as
a homogeneous material, mounted with magnetostrictive layers under uniformly distributed
loading and thermal effect, which is shown in Figure 1, where L is the axial length of shell,
is the meridian (arc angle) of shell panel, h* is the total thickness of magnetostrictive layer and
FGM shell, 3h is the thickness of magnetostrictive layer, 1h and 2h are the thickness of FGM
material 1 and FGM material 2, respectively,
),,(),,( 1*0 txTh
ztxTT
is the temperature difference between the FGM shell and curing area (a reference state).
For the plane stresses in the kth
layer of the magnetostrictive FGM circular cylindrical shell, the
in-plane stresses constitute the membrane stresses, bending stresses and thermal stresses are
given in the following equations (Hong, 2014; Lee and Reddy, 2005),
)()(36
32
31
)()(662616
262212
161211
)(
~0
0
~00
~00
~00
kzkkxx
xx
kkx
x
He
e
e
T
T
T
QQQ
QQQ
QQQ
, (5a)
and the shear stresses are given as follows,
)(
)(2515
2414
)()(5545
4544
)(~0
0
0~~0~~
kzkkxz
z
kkxz
z
Hee
ee
, (5b)
where x and
are the coefficients of thermal expansion, x
is the coefficient of thermal
shear, Qij is the stiffness of FGM shell,
x , and
xare in-plane strains, not negligible
z
and xz are shear strains,
xk , k and
xk are the curvatures, the strains and curvatures can be
expressed respectively in terms of displacement components and shear rotation as follows [Qu et
al. (2013); Qatu et al. (2010 )],
132 C.C. Hong
x
zx
u xx
0 , (6a)
)(1 0 wz
v
R
, (6b)
)(1 00
x
x zu
Rxz
x
v, (6c)
R
vw
Rz
01
, (6d)
R
u
x
wxxz
0
, (6e)
x
k xx
, (6f)
Rk
1, (6g)
x
xRx
k1
, (6h)
in which R is the middle-surface radius of shell, ije~ is the transformed magnetostrictive coupling
modulus, zH
~ is the magnetic field intensity and can be expressed in the following equations with
the term t
w
in the velocity feedback control system [(Krishna Murty et al. (1997)],
t
wtcktyxH cz
)(),,(
~, (7)
where ck is the coil constant value, )(tc is the control gain.
For the preliminary stresses calculation, simpler forms of Qij are used for FGM circular
cylindrical shells and given as follows [Sepiani et al. (2010)]:
22211
1 fgm
fgmEQQ
, (8a)
)1)(/1(
2112
fgm
fgmfgm
Rz
EQQ
, (8b)
)1)(/1(2
66
fgm
fgm
Rz
EQ
, (8c)
AAM: Intern. J., Vol. 11, Issue 1 (June 2016) 133
)1(2
44
fgm
fgmEQ
, (8d)
)1)(/1(2
55
fgm
fgm
Rz
EQ
, (8e)
0452616 QQQ . (8f)
For the layer of magnetostrictive material, the simple forms of Qij with Poisson’s ratios 0
are used and expressed as follows (Lee and Reddy, 2005; Hong, 2007),
112211 EQQ , (8g)
2/1166 EQ , (8h)
0261612 QQQ , (8i)
in which 11E is the Young’s modulus of magnetostrictive material.
2.5. Dynamic equilibrium differential equations
The dynamic equations of motion for a circular cylindrical shell introduced by Jafari et al. (2005)
are given in the dynamic equilibrium differential equations as follows,
2
2
2
21
tH
t
uN
Rx
N xxx
, (9a)
2
2
2
2
2
211
tH
t
v
x
vNQ
R
N
Rx
Na
x
, (9b)
2
2
2
211
t
wq
x
wNN
R
Q
Rx
Qa
x
, (9c)
2
2
2
21
tI
t
uHQ
M
Rx
M xx
xx
, (9d)
2
2
2
21
tI
t
vHQ
M
Rx
M x
, (9e)
where
dzzzIH
h
h),,1(),,( 22
2
0
*
* ,
0 is the density of ply, aN is the pulsating axial load, q is the applied external pressure load,
xN , N , xN , xM , M , xM , xQ and Q are stress resultants.
134 C.C. Hong
The behaviors of a multilayered circular cylindrical shell constructed of orthotropic ( A26=
2616 BB = 2616 DD = 45A = x 0), not symmetric ( ijB values are not ignored) layers are
considered. The constitutive relations including thermal and magnetostrictive loads effect
introduced by Lee et al. (2006) are given as follows,
6666
22122212
12111211
6666
22122212
12111211
0000
00
00
0000
00
00
DB
DDBB
DDBB
BA
BBAA
BBAA
M
M
M
N
N
N
x
x
x
x
x
x
x
x
k
k
k
x
x
x
x
x
x
x
x
M
M
M
N
N
N
M
M
M
N
N
N
~
~
~
~
~
~
, (10a)
z
xzx
A
A
Q
Q
44
55
0
0. (10b)
The dynamic equilibrium differential equations of FGM circular cylindrical shells in terms of
displacements and shear rotations included the magnetostrictive loads are expressed in the
following matrix forms,
000/00/)(0
0000/)(0/0
/0000000
000/00/)(0
0000/)(0/0
2
22666612
6612
2
6611
2
4455
22666612
6612
2
6611
RBBRBB
RBBRBB
RANA
RANARAA
RAARAA
a
a
2 2 2 2 2 2 2 2 2
0 0 0 0 0 0
2 2 2 2 2 2
t
u u u v v v w w w
x x x x x x
2
22666612
6612
2
6611
2
22666612
6612
2
6611
/2020)/2(0
0)/2(0/202
000000
/2020/)(20
0/)(20/202
RDDRDD
RDDRDD
RBMBRBB
RBBRBB
a
2 2 2 2 2 2
2 2 2 2
t
x x x
x x x x
AAM: Intern. J., Vol. 11, Issue 1 (June 2016) 135
0000//000
00000/00
/0000//
0000)/(000
00000/00
44
2
22
5512
4455
2
4455
2
4422
12
RARB
ARB
RAARARA
RAA
RA
0 0
t
x xu v w w
x x x x
4444
5555
44
2
44
00/0
000/
00000
/00/0
00000
ARA
ARA
RARA
t
xwvu 00
5
4
3
2
1
f
f
f
f
f
2
2
2
0
2
2
0
2
000
000
100
010
001
t
wt
vt
u
2
2
2
2
2
0
2
2
0
2
0010
0001
0000
2000
0200
t
t
t
vt
u
Hx
2
2
2
2
20
02
00
00
00
t
tI
x
, (11)
where 51 ,, ff are in the expressions of thermal loads ),( MN , external pressure load )(q and
magnetostrictive loads )~
,~
( MN as follows,
xxxx N
Rx
NN
Rx
Nf
~1
~1
1 ,
N
Rx
NN
Rx
Nf xx
~1
~1
2 ,
R
N
R
Nqf
~
3 ,
xxxx M
Rx
MM
Rx
Mf
~1
~1
4 ,
M
Rx
MM
Rx
Mf xx
~1
~1
5 ,
136 C.C. Hong
dzzTQQQMN x
h
h xxx ),1()(),( 16122
2
11
*
*
,
dzzTQQQMN xx
h
h),1()(),( 2622
2
2
12
*
* ,
dzzTQQQMN xx
h
hxx ),1()(),( 66262
2
16
*
* ,
dzzHeMN z
h
hxx ),1(~~)
~,
~( 22
2
31
*
* ,
dzzHeMN z
h
h),1(
~~)~
,~
( 22
2
32
*
* ,
dzzHeMN z
h
hxx ),1(~~)
~,
~( 22
2
36
*
* ,
dzzzQDBA
h
h ijijijij ),,1(),,( 22
2
*
* , ( , , , )i j 1 2 6 ,
dzQkkA
h
h jiji 2
2
*
* **** , )6,6;5,4,( **** jiji ,
dzzTQQQMN x
h
h xaa ),1()(),( 016122
2
11
*
*
,
in which k and k are the shear correction coefficients, aN and aM are the pulsating axial
load and moment in function of 0T .
For the 11A calculation with simpler forms of Qij stiffness integrations of magnetostrictive FGM
shells, the following equation can be obtained,
311
21
221
11 )1
(
)2
(1
hER
EERhA
n
n
. (12)
2.6. Dynamic discrete equations
The weighting coefficients of discrete equations in the two-dimensional GDQ method can be
applied to the discrete differential equations (11) under the vibrations of time sinusoidal
displacement and temperature as follows,
)],(),([ 0 xzxuu x )sin( tmn , (13a)
AAM: Intern. J., Vol. 11, Issue 1 (June 2016) 137
)],(),([ 0 xzxvv )sin( tmn , (13b)
),( xww )sin( tmn , (13c)
),( xxx )sin( tmn , (13d)
),( x )sin( tmn , (13e)
)],(),([ 1*0 xTh
zxTT )sin( t . (14a)
and with the simple vibration of temperature parameter
0),(0 xT ( aN = aM =0), (14b)
11 ),( TxT )/sin()/sin( Lx , (14c)
where mn is the natural frequency, is the frequency of applied heat flux, 1T is the amplitude
of temperature.
The following non-dimensional parameters are introduced to complete the discrete processes of
GDQ approach,
LxX / , (15a)
LuU /0 , (15b)
RvV /0 , (15c)
*/ hwW . (15d)
By considering four sides simply supported, not symmetric, orthotropic of laminated
magnetostrictive FGM shells under temperature loading, the following dynamic discrete
equations in matrix notation can be obtained.
FUVWSIFQSWSIKESSIXYBMSUVWAM , (16)
where
N
l
N
l
M
m
N
l
jlli
M
m
mimjmlmjlijlli VAUBUBAUASUVW1 1 1 1
,
)2(
,
1
,
)2(
,,
)1(
,
)1(
,,
)2(
,{
N
l
M
m
tM
m
mimjmlmjli
N
l
M
m
N
l
jlli
M
m
mimjmlmjli WBWBAWAVBVBA1 1 1
,
)2(
,,
)1(
,
)1(
,
1 1 1
,
)2(
,
1
,
)2(
,,
)1(
,
)1(
, } ,
N
l
N
l
M
m
M
mmi
xmjml
xmjlijl
xli BBAASSIXY1 1 1 1
,
)2(
,,
)1(
,
)1(
,,
)2(
,{
t
N
l
N
l
M
m
M
mmi
mjml
mjlijl
li BBAA }1 1 1 1
,
)2(
,,
)1(
,
)1(
,,
)2(
,
,
N
l
M
m
mimjjlli VBUASWSI1 1
,
)1(
,,
)1(
,{
N
l
N
ljl
xli
M
m
mimjjlli AWBWA1 1
,
)1(
,
1
,
)1(
,,
)1(
,
138 C.C. Hong
t
N
l
M
mmi
mjjl
li
M
mmi
xmj BAB }1 1
,
)1(
,,
)1(
,
1,
)1(
,
,
t
jijixjijiji WVUUVWSI }{
,,,,, ,
tFFFFFF }{ 54321 ,
in which )(
,
o
liA and )(
,
o
mjB are the weighting coefficients of GDQ approach related to the tho order
derivative of the function at coordinate of grid point ),( jix , Ni ,,1 and Mj ,,1 . The
elements of 95 matrix AM , 65 matrix BM , 85 matrix KE and 55 matrix FQ are
as follows,
)sin()/( 1111 tLAAM mn , 012 AM , )sin()/( 2
6613 tRLAAM mn ,
014 AM , )sin()]/()[( 661215 tRLRAAAM mn ,
016 AM , 0191817 AMAMAM , 021 AM ,
)sin()]/)[( 661222 tRAAAM mn , 023 AM ,
)sin()/( 2
6624 tLRAAM mn , 025 AM ,
)sin()/( 2
2226 tRRAAM mn , 0292827 AMAMAM ,
0363534333231 AMAMAMAMAMAM ,
)sin()/( 2*
5537 tLhAAM mn , 038 AM ,
)sin()/( *2
4439 thRAAM mn , )sin()/( 1141 tLBAM mn , 042 AM ,
)sin()/( 2
6643 tRLBAM mn ,
044 AM , )sin()]/()[( 661245 tRLRBBAM mn ,
046 AM , 0494847 AMAMAM , 051 AM ,
)sin()/1)(( 661252 tRBBAM mn , 053 AM ,
)sin()/( 2
6654 tLRBAM mn , 055 AM ,
)sin()/( 2
2256 tRRBAM mn , 0595857 AMAMAM ,
)sin()/2( 2
1111 tLBBM mn , 012 BM ,
)sin()/2( 2
6613 tRBBM mn , 014 BM ,
)sin()]/()(2[ 661215 tRLBBBM mn , 016 BM ,
021 BM , )sin(]/)(2[ 661222 tLBBBM mn ,
023 BM , )sin()/2( 2
6624 tLBBM mn ,
025 BM , )sin()/2( 2
2226 tRBBM mn ,
0363534333231 BMBMBMBMBMBM ,
)sin()/2( 2
1141 tLDBM mn , 042 BM ,
)sin()/2( 2
6643 tRDBM mn , 044 BM ,
)sin()]/()(2[ 661245 tRLDDBM mn , 046 BM ,
AAM: Intern. J., Vol. 11, Issue 1 (June 2016) 139
051 BM , )sin()]/()(2[ 661252 tRLDDBM mn ,
053 BM , )sin()/2( 2
6654 tLDBM mn ,
055 BM , )sin()/2( 2
2256 tRDBM mn ,
)cos()](~)(1
[)sin(1
1
1
31
**1213 tzzetck
Lht
Lh
R
AKE mnmnkk
N
k
cmn
k
,
)cos()](~)([1
1
1
36
*
14 tzzetckhR
KE mnmnkk
N
k
c
k
,
0181716151211 KEKEKEKEKEKE ,
)cos()](~)(1
[ 1
1
36
*
23 tzzetckL
hKE mnmnkk
N
k
c
k
,
)cos()](~)([1
)sin( 1
1
32
**
2
442224 tzzetckh
Rth
R
AAKE mnmnkk
N
k
cmn
k
,
0282726252221 KEKEKEKEKEKE ,
)sin()/( 5531 tRAKE mn , )sin()/( 2
4432 tRRAKE mn , )sin()/( 5535 tLAKE mn ,
037363433 KEKEKEKE , )sin()/( 4438 tRAKE mn ,
)sin(1
)( 5512*
43 tL
AR
BhKE mn )cos(]~
3
1)(
1[
*
3
1
3
1
31
* th
zzetck
Lh mnmn
kk
N
k
c
k
,
44KE )cos(]~
3
1)([
1*
3
1
3
1
36
* th
zzetckh
Rmnmn
kk
N
k
c
k
,
0484746454241 KEKEKEKEKEKE ,
53KE )cos(]~
3
1)(
1[
*
3
1
3
1
36
* th
zzetck
Lh mnmn
kk
N
k
c
k
,
)sin()( 44
2
22*
54 tR
A
R
BhKE mn )cos(]~
3
1)([
1*
3
1
3
1
32
* th
zzetckh
Rmnmn
kk
N
k
c
k
,
0585756555251 KEKEKEKEKEKE ,
)sin(2
11 tLFQ mnmn , 0151312 FQFQFQ , )sin(2 2
14 tHFQ mnmn ,
)sin()( 2
2
4422 tR
R
AFQ mnmn
, )sin()2( 244
25 tHR
AFQ mnmn ,
0242321 FQFQFQ ,
)sin(*2
33 thFQ mnmn )cos()](~)([1
1
1
32
* tzzetckhR
mnmnkk
N
k
c
k
,
035343231 FQFQFQFQ ,
)sin()( 25541 tLH
R
AFQ mnmn
, 44FQ )sin()2( 55
2 tAI mnmn ,
0454342 FQFQFQ ,
)sin()( 24452 tRH
R
AFQ mnmn
, )sin()2( 44
2
55 tAIFQ mnmn ,
140 C.C. Hong
0545351 FQFQFQ ,
in which F F1 5,..., are represented in the following discrete equation,
)sin()11
(,,
1
)1(
,
1
)1(
,1 tNBR
NAL
Fmijl x
M
m
mjx
N
l
li
,
)sin()11
(,,
1
)1(
,
1
)1(
,2 tNBR
NAL
Fmijl
M
m
mjx
N
l
li
,
jiqF ,3 ,
)sin()11
(,,
1
)1(
,
1
)1(
,4 tMBR
MAL
Fmijl x
M
m
mjx
N
l
li
,
)sin()11
(,,
1
)1(
,
1
)1(
,5 tMBR
MAL
Fmijl
M
m
mjx
N
l
li
.
And the following frequency parameter *f can be used,
11
* /4 AIRf mn . (17)
3. Some numerical results and discussions
The FGM material 1 is SUS304 (Stainless Steel), the FGM material 2 is 43 NSi (Silicon Nitride)
used to obtain the preliminary numerical data. The values of 2110 ,,, PPPP and 3P in Table 1 are
used to calculate material properties of FGM shells.
Table 1. Values of temperature-dependent coefficients of constituent FGM
Materials iP 0P
1P 1P 2P 3P
SUS304 1E ( aP ) 201.04E09 0 3.079E-04 -6.534E-07 0
1 0.3262 0 -2.002E-04 3.797E-07 0
1 (
3/ mKg ) 8166 0 0 0 0
1 (
1K ) 12.33E-06 0 8.086E-04 0 0
1 ( KmW / ) 15.379 0 0 0 0
1vC ( KKgJ / ) 496.56 0 -1.151E-03 1.636E-06 -5.863E-10
43 NSi 2E ( aP ) 348.43E09 0 -3.70E-04 2.16E-07 -8.946E-11
2 0.24 0 0 0 0
2 (
3/ mKg ) 2370 0 0 0 0
2 (
1K ) 5.8723E-06 0 9.095E-04 0 0
2 ( KmW / ) 13.723 0 0 0 0
2vC ( KKgJ / ) 555.11 0 1.016E-03 2.92E-07 -1.67E-10
AAM: Intern. J., Vol. 11, Issue 1 (June 2016) 141
The Terfenol-D of magnetostrictive material made in USA is used. The elastic modules and
)/10(5/912 6 Kx
of the Terfenol-D are listed as in the papers (Reddy and Chin,
1998; Shariyat, 2008). Three-layer )0/0/0( m Terfenol-D FGM circular cylindrical shell
including shear deformation is considered under four sides simply supported with aN = aM = 0
and with FGM material 1 of thickness 1h equal to FGM material 2 of thickness 2h . The
superscript of m denotes magnetostrictive layer. The following coordinates for the grid points
are used:
LN
ixi )]
1
1cos(1[5.0
, Ni ,,2,1 , (18a)
)]
1
1cos(1[5.0
M
jj , Mj ,,2,1 . (18b)
Usually, varied values of shear correction coefficient are functions of T , 3h and nR in the
Terfenol-D FGM calculation, and the constant value are used simply for shear correction
coefficient 6/5 kk . For the simplification, no heat generation, linear and uncouple case of
the heat conduction equation in the cylindrical coordinates is considered as follows (Hong,
2014),
t
T
z
T
R
T
x
TK
)(
2
2
22
2
2
2
. (19a)
With the forms of the parameter of )/( vCK , the host material conductivity fgm , specific
heatfgmvv CC , density fgm .
The heat conduction equation can be reduced and simplified to the following equation,
0)sin()cos(])/(1[ 22
2
ttRLK
L
. (19b)
Before the process of thermal vibrations of shell, it is needed to obtain the calculation values of
vibration frequency mn . It is reasonable to assumed that 0u 0v w x and are expressed in
the following time sinusoidal form of free vibration for Terfenol-D FGM shell (usually the
values of ijB are not zero) under four sides simply supported.
)cos()/cos(0 nLxmeau
ti
mnmn , (20a)
)sin()/sin(0 nLxmebv
ti
mnmn , (20b)
)cos()/sin( nLxmecw
ti
mnmn , (20c)
)cos()/cos( nLxmed
ti
mnxmn , (20d)
)sin()/sin(
nLxmeeti
mnmn , (20e)
142 C.C. Hong
where m is the number of axial half-waves, n is the number of circumferential waves mna mnb
mnc mnd and mne are the amplitudes of time sinusoidal form and expressed in the following
matrix equation,
0
0
0
0
0
2
2
2
2
5554535251
4544434241
3534333231
2524232221
1514131211
mn
mn
mn
mn
mn
e
d
c
b
a
IHHHHHH
HIHHHHH
HHHHH
HHHHHH
HHHHHH
, (21)
where
22
66
2
1111 )/()/( nRALmAH , nLmAARHH )/)()(/1( 66122112 ,
)/)(/( 1213 LmRAH , 22
66
2
1114 )/2()/(2 nRBLmBH ,
nLmBBRH )/)()(/2( 661215 ,
2
44
2
22
2
6622 /)/()/( RAnRALmAH , nRAAH ]/)[( 2
442223 ,
nLmBBRH )/)()(/2( 661224 , )/()/2()/(2 44
22
22
2
6625 RAnRBLmBH ,
)/)(/( 5531 LmRAH , nRAH )/( 2
4432 , 22
44
2
5533 )/()/( nRALmAH ,
)/(5534 LmAH , nRAH )/( 4435 , RAnRBLmBH /)/()/( 55
22
66
2
1141 ,
nLmBBRH )/)()(/1( 661242 , )/)(/( 551243 LmARBH ,
55
22
66
2
1144 )/2()/(2 AnRDLmDH , nLmDDRH )/)()(/2( 661245 ,
nLmBBRH )/)()(/1( 661251 , RAnRBLmBH /)/()/( 44
22
22
2
6652 ,
nRARBH )//( 44
2
2253 , nLmDDRH )/)()(/2( 661254 ,
44
22
22
2
6655 )/2()/(2 AnRDLmDH and 2
mn .
Thus, some of the lowest frequency of applied heat flux and vibration frequency 11 of shells
are found, at time t = 0.01s, 1s, 2s,…,5s, for Terfenol-D FGM circular cylindrical shells with h*
= 1.2 mm, 13 h mm, 1h = 2h = 0.1 mm, 05.0/ RL , 0q , 1 nm , 1nR , KT 653 ,
KT 11 as shown in Table 2 First, the computational results of the amplitude of center
deflection )2/,2/( Lw (unit mm) dynamic convergence study in Terfenol-D FGM circular
cylindrical shells are obtained and listed in Table 3 with )(tckc = 0, st 3 , h*= 1.2 mm, 13 h
mm, 1h = 2h = 0.1 mm, 0q , 1 nm , 1nR , KT 653 , KT 11 . The 19×19 grid point
is found in the good convergence result and can be used further in the GDQ thermal
computations of time responses for deflection and stress of Terfenol-D FGM circular cylindrical
shells. Secondly, the suitable )(tckc controlled gain values are used to control and reduce the
amplitude of center deflection )2/,2/( Lw (unit mm) for the Terfenol-D FGM circular
cylindrical shell. Table 4 shows the suitable )(tckc values versus time t for 1nR , thick shell
5/ * hL , 10 and thin shell 20/ * hL at KT 653 .
AAM: Intern. J., Vol. 11, Issue 1 (June 2016) 143
Figure 2a shows the frequency parameter *f versus n results investigated by using the GDQ
method for four sides simply supported Terfenol-D FGM circular cylindrical shells, Rh /* =
0.01, RL / = 0.05, stacking sequence )0/0/0( m at t = 6s, )(tckc = 0, 1nR . Figure 2b
shows the re-plotted, compared frequency parameter * versus n by Jafari et al. (2005) with
equation
* 11
* / AhLsmn ,
in which RLs 2 , for the clamped-free composite cylindrical shell, Rh /* = 0.002, RL / = 20,
stacking sequence )90/0/90( , Young’s modulus: 11E = 19GPa, 22E = 7.6GPa, 12G =4.1GPa,
13G = 4.1GPa, 23G =1.3GPa, Poisson’s ratios 12 = 0.26, density = 1643 3/ mkg . The very good
tendency is obtained between the two curves of frequency parameter versus n .
Figure 3 shows the response values of center deflection amplitude )2/,2/( Lw (unit mm)
versus time t in Terfenol-D FGM circular cylindrical shell with and without )(tckc values for
5/ * hL , 10 and 20, respectively, 1/ RL , h*= 1.2 mm, 13 h mm, 1h = 2h = 0.1 mm, 1nR ,
KT 653 , KT 11 , t = 0.1s-3.0s, time step is 0.1s. The controlled values of the center
deflection )2/,2/( Lw with )(tckc are smaller than the uncontrolled values of )2/,2/( Lw
without )(tckc , especially in the case of thick shell 5/ * hL . The amplitude )2/,2/( Lw can be
controlled and adjusted to a desired smaller value by using the suitable )(tckc value in Terfenol-
D FGM shell.
Table 2 and 11 of Terfenol-D FGM cylindrical shells
*/ hL
At t = 0.01 s At t = 1 s At t = 2 s
11
11 11
20 785.398 0.762939E-05 1.57080 0.762939E-05 0.785398 0.972637E-05
10 785.398 0.381470E-05 1.57080 0.470307E-04 0.785398 0.107896E-04
5 785.398 0.762939E-05 1.57080 0.381470E-05 0.785398 0.915527E-04
*/ hL
At t = 3 s At t = 4 s At t = 5 s
11
11 11
20 0.523599 0.190735E-05 0.392699 0.999928E-05 0.314159 0.999987E-05
10 0.523599 0.381470E-05 0.392699 0.381470E-05 0.314159 0.269740E-05
5 0.523599 0.410855E-04 0.392699 0.222433E-04 0.314159 0.152588E-04
144 C.C. Hong
The normal stress x and shear stress z are three-dimensional components and in functions of
x , and z . Their values vary through the shell thickness. Figure 4a shows the normal stress x
(unit G Pa) versus z and Figure 4b shows the shear stress z (unit G Pa) versus z at center
position ( 2/,2/ Lx ) of shells without )(tckc values, respectively at t = 0.1s, 5/ * hL ,
1/ RL . The absolute value (0.76E-05 G Pa) of normal stress x at *5.0 hz is found in the
greater value than the value (0.23E-09 G Pa) of shear stress z at *5.0 hz , thus the normal
stress x can be treated as the dominated stress. Figure 4c and Figure 4d show the time
responses of the dominated normal stress x (unit G Pa) and shear stress z (unit G Pa) at
center position of upper surface *5.0 hz with and without )(tckc values as the analyses of
deflection )2/,2/( Lw at 5/ * hL . Figure 4e and Figure 4f show the time response of the
dominated stress x (unit G Pa) at center position of upper surface *5.0 hz with and without
)(tckc values as the analyses of deflection )2/,2/( Lw at 10/ * hL and 20/ * hL ,
respectively. The thick ( 5/ * hL ) Terfenol-D FGM circular cylindrical shell under controlled
gain values has the larger absolute stresses x than uncontrolled gain values case.
Figure 5 shows the center deflection amplitude )2/,2/( Lw (unit mm) versus thickness 3h
(unit mm) of magnetostrictive layer in Terfenol-D FGM circular cylindrical shell with gain value
)(tckc = 910 for KT 100 , KT 653 respectively 5/ * hL and 1/ RL at 3t s. The
center deflection )2/,2/( Lw decreasing with 3h from 0.36 to 0.48mm, then increasing with
3h for all different values of nR andT .
Figure 6 shows the dominated stress x (unit G Pa) at center position of upper surface *5.0 hz
versus thickness 3h (unit mm) of magnetostrictive layer in Terfenol-D FGM circular cylindrical
shell with gain value )(tckc = 910 for KT 100 , KT 653 , respectively, 5/ * hL ,
1/ RL at 3t s. The absolute x value little decreasing with respect to 3h for all different
values of nR and increasing withT .
Figure 7a shows the center deflection amplitude )2/,2/( Lw (unit mm) versus T of Terfenol-
D FGM circular cylindrical shell with constant gain values )(tckc = 910 for 5/ * hL , 1/ RL
and h*= 1.2 mm, 13 h mm, 1h = 2h = 0.1 mm, KT 1001 at 1.0t s. The absolute value of
amplitude )2/,2/( Lw of Terfenol-D FGM shell can be controlled and adjusted to a desired
smaller value under more higher temperature KT 400 , for nR = 0.1 only, for the others nR
the amplitude )2/,2/( Lw are little increasing with T . Figure 7b shows the center deflection
amplitude )2/,2/( Lw (unit mm) versus 1T of Terfenol-D FGM shell with constant gain values
)(tckc = 910 for 5/ * hL , 1/ RL and h*= 1.2 mm, 13 h mm, 1h = 2h = 0.1 mm, KT 100 at
AAM: Intern. J., Vol. 11, Issue 1 (June 2016) 145
1.0t s. The absolute value of amplitude )2/,2/( Lw of Terfenol-D FGM shell increasing
with 1T for all different values of nR .
4. Conclusions
The GDQ solution provides us with a method to calculate the deflections and stresses in the
thermal vibration of Terfenol-D FGM circular cylindrical shells. The main point is clear from the
GDQ results to reduce the displacement and vibration by using the Terfenol-D FGM materials
and results show: (a) the amplitude of center deflection can be controlled and adjusted to a
desired smaller value by using the suitable gain value in Terfenol-D FGM shell. (b) The thick
Terfenol-D FGM shell under controlled gain values has the larger absolute stresses. (c) The
center deflection )2/,2/( Lw decreasing with 3h from 0.36 to 0.48mm, then increasing with
3h for all different values of nR . (d) The absolute x value little decreasing with respect to 3h
for all different values of nR and increasing withT . (e) The absolute value of amplitude
)2/,2/( Lw of Terfenol-D FGM shell increasing with 1T for all different values of nR . The
physical thermal vibration application of Terfenol-D FGM circular cylindrical shells might be
applied in the noise reducing relational fields of missile, airplane, ship and car.
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Table 3. Dynamic convergence of Terfenol-D FGM cylindrical shells
*/ hL
GDQ method Deflection )2/,2/( Lw at st 3
MN
05.0/ RL
1/ RL 2/ RL
100 1515 0.230012E-02 0.154303E-05 0.243478E-05
1717 0.239481E-02 0.144605E-05 0.222776E-05
1919 0.239483E-02 0.144604E-05 0.222779E-05
20 1515 0.142550E-01 0.578279E-02 0.252847E-02
1717 0.143640E-01 0.671408E-02 0.327779E-02
1919 0.143634E-01 0.671413E-02 0.327781E-02
10 1515 0.253981E-01 0.639501E-02 0.346163E-02
1717 0.254824E-01 0.677707E-02 0.378427E-02
1919 0.254700E-01 0.677719E-02 0.378427E-02
5 1515 0.651654E-02 0.357075E-02 0.234609E-02
1717 0.650966E-02 0.363410E-02 0.240864E-02
1919 0.652132E-02 0.363406E-02 0.240869E-02
Table 4. )(tckc versus t for 1nR
*/ hL t = 0.1s~0.6s t = 0.7s~ 3.0s
)(tckc )(tckc
5 910 910
*/ hL t = 0.1s~ 0.7s t = 0.8s~ 3.0s
)(tckc )(tckc
10 910 910
*/ hL t = 0.1s~ 0.3s t = 0.4s~ 3.0s
)(tckc )(tckc
20 910 810
148 C.C. Hong
Figure 2a. f
* versus n results by the GDQ method Figure 2b. Compared
versus n results
Figure 2. Frequency parameter f *versus n and compared
versus n
Figure 3a. )2/,2/( Lw versus t for 5/ * hL Figure 3b. )2/,2/( Lw versus t for 10/ * hL
Figure 3c. )2/,2/( Lw versus t for 20/ * hL
AAM: Intern. J., Vol. 11, Issue 1 (June 2016) 149
Figure 3. )2/,2/( Lw versus t for 5/ * hL , 10 and 20
Figure 4a. x versus z for 5/ * hL Figure 4b. z versus z for 5/ * hL
Figure 4c. x versus t for 5/ * hL Figure 4d. z versus t for 5/ * ha
Figure 4e. x versus t for 10/ * hL Figure 4f. x versus t for 20/ * hL
150 C.C. Hong
Figure 4. The stresses versus z and t for 5/ * hL , 10 and 20
Figure 5a. )2/,2/( Lw versus 3h for KT 100 Figure 5b. )2/,2/( Lw versus 3h for KT 653
Figure 5. )2/,2/( Lw versus 3h for KT 100 and K653
Figure 6a. x versus 3h for KT 100 Figure 6b. x versus 3h for KT 653
Figure 6. x versus 3h for KT 100 and K653
AAM: Intern. J., Vol. 11, Issue 1 (June 2016) 151
Figure 7a. )2/,2/( Lw versus T with KT 1001 Figure 7b. )2/,2/( Lw versus 1T with KT 100
Figure 7. )2/,2/( Lw versus T and 1T
