Copyright © 2013, American-Eurasian Network for Scientific Information publisher
JOURNAL OF APPLIED SCIENCES RESEARCH
JOURNAL home page: http://www.aensiweb.com/jasr.html 2013 December; 9(13): pages 6301-6315.
Published Online: 15 January 2014. Research Article
Corresponding Author: Reza Azarafza, Department of Mechanical Engineering, Malek ashtar University of Technology,
Tehran, Iran.
E-mail: [email protected], Phone: +98-21-777-45103)
Free Vibration Of Fgm Cylindrical Shell Reinforced With Single-Walled Carbon
Nanotubes Using High Order Shear Deformation Theory
1Hossein Farahani,
2Reza Azarafza and
3Farzan Barati
1Department of Mechanics, College of Engineering, Hamedan Science and Research Branch, Islamic Azad University, Hamedan, Iran. 2Department of Mechanical Engineering, Malek ashtar University of Technology, Tehran, Iran.
3Department of Mechanical Engineering, Islamic Azad University, Hamedan Brunch, Hamedan, Iran.
Received: 12 November 2013; Revised: 14 December, 2013; Accepted: 20 December 2013.
© 2013 AENSI PUBLISHER All rights reserved
ABSTRACT
In this paper study of the free vibration of FGM thin cylindrical shell reinforced with single- walled carbon nanotubes with non-
uniform distribution is presented. Fundamental relations, the equilibrium and stability equations are derived using the third order shear
deformation theory. simply supported boundary condition is considered for both edge of the shell and wave method is used to solve the
problem . The effects of different materials and different volume ratio on the natural frequencies are described. The analytical results are compared and validated using the results obtained from the papers is demonstrator the certify equation. The results show that changes in
shell geometry, the type of polymer and varying the volume fraction of the carbon nanotube significantly affect on the natural frequencies.
Key words: Cylindrical shell, carbon nanotubes, functional graded materials, third order shear deformation theory
INTRODUCTION
Functional graded material for the first time in
Japan was built in 1984 [1] and first were used in
heat resistant material to cover the space shuttle and
the nuclear reactors that are affected by in high
temperatures and very of thermoelastic analysis is
done in this area [2,3]. In recent decades a dramatic
increase in demand for structures with high
resistance to high temperature, energy absorption and
light weight, there have been many studies on
behavior of FGM cylindrical shells. Buckling of
structures made from FGM are subjected to
mechanical loads by Brush and Almorth [4] have
been investigated. Golterman [5] a relative method
for predicting buckling of thin-walled cylinder using
a simple and well-known theories presented. He
critical elastic buckling of a cylindrical shell
completely consistent with the classical theory for
the two- mode factor loading estimates and destroyed
in accordance with a code of classical stability theory
Kuiter be calculated according to [6]. Winterstetter
and Schmidt [7] in the context of a comprehensive
review and analysis of the axial buckling of
cylindrical steel shells under combined loads are
being carried out. Vodenticharova and Ansourian [8]
analysed the buckling of cylindrical shells under
uniform lateral pressure are paid. Pelletier [9], the
steady-state response of a thick cylindrical shell
subjected to mechanical loading and thermal FGM
has been graded, were reviewed and analysed.
Boundary conditions considered for the backrest
shell is simple and it is assumed that the material
properties in the radial direction is changed. Rahimi
and colleagues [10] vibrations of a cylindrical shell
with functional graded ring intermediate amplifier
based on the theory Sanders looked. Using
Hamilton's equations of motion and apply the Ritz
method is obtained. And material properties vary
along the radius. Akhlaghi and Asghari [11] The
natural frequencies of a thick -walled cylinder with
finite length consisting of a functional graded
material properties in the longitudinal and radial
directions are changed based on the three-
dimensional elasticity equations were investigated.
Pradham et al. [12] the vibrational parameters of
FGM cylindrical shells with different boundary
conditions began. The effect of different boundary
conditions and different volume ratios on the natural
frequency of the shell. They used Rail methods to
solve the problem. Reddy et al. were examined of
FGM cylindrical shells under axial harmonic balance
dynamics. Their stability, Mathieu-Hill equations to
obtain the equations of motion and Bolotin 's method
to solve it began. Sofiyev [14] the balance consisting
of ceramic and metal composite cylindrical shell
support an axial impact loading was investigated.,
His love relationships can be used to obtain the
equations of Lagrangian Galerkin method for solving
the governing equations chose.Ritz method is used to
6302 Hossein Farahani et al, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6301-6315
obtain equations for the effect of volume fraction on
natural frequencies can be investigated. Shen
nonlinear behavior of composite plates reinforced
with nanotubes were investigated under thermal load.
Li Zhang and Hui-Shen on the thermal buckling
analysis of composite plates reinforced with carbon
nanotubes temperature dependent material properties
were considered in molecular dynamics method to
obtain these properties.
We analyse the free vibration of cylindrical
shells reinforced with single-walled carbon
nanotubes (SWCNT) in the non-uniform distribution
explains, the fact that the reinforcing properties page
as graded (FGM) is be. Finally, to investigate the free
vibration of cylindrical shell nanocomposite with
PMMA and PMPV matrix reinforced with single-
walled carbon nanotubes with simply supported
boundary condition, explains. Carbon nanotubes
properties considered in this study, extracted of paper
No.20 For to validation the equations, the results
obtained from the numerical solution for a
homogeneous cylindrical shell are compared with the
results presented in Paper No. 27.
Modelling of problem
The figure below shows the micromechanical
model and the coordinate system of the page.
Figure.1 show two layers of FGM cylindrical shell is
symmetrical relative to the middle level. The length ,
radius and thickness, respectively is determined with
L, R and h.
Fig. 1: Coordinate system and the geometry of the cylindrical shell [21].
In this study, the modified law of mixtures is used to determine the properties of the FGM shell.
Cylindrical shell is orthoteropic properties. volume fraction of nanotubes on the upper and lower plate is
considered [19].
*4CN CN
zV V
h (1)
Such that:
* ( / ) ( / )
CNCN
CN CN m CN m CN
wV
w w
(2)
is the mass fraction of nanotubes.
(3)
In the above equations, the index represents the single-walled nanotubes and index is the matrix.
Young's modulus and shear modulus is defined as follows [14].
(4)
(5)
(6)
1 CN mV V
11 1 11 CN mCN mE V E V E
2
22 22
CN m
CN m
V V
E E E
3
12 12
CN m
CN m
V V
G G G
6303 Hossein Farahani et al, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6301-6315
In the above equations E, G, and V are represents the Young's modulus, shear modulus and volume
fraction.
4 3, , , , , , , 1 1 123
WU x y z t U x y t z x y t z
xh
(7)
4 1 3, , , , , , , 2 2 223
W VU x y z t V x y t z x y t z
R Rh
(8)
, , , , ,3U x y z t W x y t (9)
FGM cylindrical shell equations of motion based on third order shear deformation theory:
Where, U, V and W, are displacements of arbitrary point through the cylindrical shell along coordinate (x,
y, z). In the above equations U, V and W, are to change the location of a point on z=0. also 1 2, are normal
rotation around the transverse axis (y,x).
Strain-displacement relationships:
The strain-displacement relationships for a thin shell are (Najafizadeh and Isvandzibaei, 2009).
1 1 2 1 1 11 31 2 2 13 11
1
U U A AU
A RA
R
(10)
2 1 2 222 3
2 1 1 232
2
1
1
U U A AU
A RA
R
(11)
333
3
U
(12)
3 31 2
1 21 212
2 13 3 3 32 1 1 2
2 1 1 2
1 1
1 1 1 1
A AR RU U
A A A AR R R R
(13)
3 3113 1
1 3 13 31 1
1 1
1 1
1 1
UUA
RA A
R R
(14)
3 3223 2
2 3 23 32 2
2 2
11
1 1
UUA
RA A
R R
(15)
1 21 2
, r r
A A
(16)
That and are the fundamental form parameters or Lame parameters, are the
displacement at any point ( ), R1 and R2 are the radius of curvature related to
respectively.
The third-order theory of Reddy [24] used in the present study is based on the following displacement field,
6304 Hossein Farahani et al, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6301-6315
2 3, , , ,1 1 1 2 3 1 1 2 3 1 1 2 3 1 1 2U u (17)
2 3, , , ,2 2 1 2 3 2 1 2 3 2 1 2 3 2 1 2U u (18)
,3 3 1 2U u (19)
These equations can be reduced by satisfying the stress-free conditions on the top and bottom faces of the
laminates, which are equivalent to:
13 23 0 : 2
hat z (20)
The displacement field can be written as follows:
3 311 1 1 2 3 1 1 2 1 3 1
1 1 1
1, ,
uuU u c
R A
(21)
3 322 2 1 2 3 2 1 2 1 3 2
2 2 2
1, ,
uuU u c
R A
(22)
3 3 1 2,U u (23)
Necessary parameters for the cylindrical shell are defined as:
1 2 3
2 1
1 2
, ,
,
1 ,
x z
R R R
A A R
(24)
Where 1 2
4
3 C
h [25].
Stress-strain relationships:
Relationship stress - strain as [23].
11 12
21 22
44
55
66
0 0 0
0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
xx xx
z z
xz xz
x x
Q Q
Q Q
Q
Q
Q
(25)
For an orthotropic cylindrical shell the reduced
stiffness Qij ( i , j=1, 2 and 6) is defined as
1111
12 211
EQ
v v
(26)
12 22 21 1112
12 21 12 211 1
v E v EQ
v v v v
(27)
2222
12 21
1
EQ
v v
(28)
44 23 2 12Q G k G (29)
55 13 1 12Q G k G (30)
66 12Q G (31)
Coefficients 12 13 23, ,G G G , 12 21 11 22, , ,v v E E is
calculated from the modified mixtures.
The stress resultants:
For a thin cylindrical shell the force and moment
results are defined as [23],
2
2
h
xx xx
hx x
N
N dz
N
(32)
2
2
h
xx xx
hx x
M
M z dz
M
(33)
6305 Hossein Farahani et al, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6301-6315
23
2
h
xx xx
hx x
P
P z dz
P
(34)
2
2
h
xz xz
z zh
Qdz
Q
(35)
22
2
h
xz xz
z zh
Rz dz
R
(36)
23
2
h
xz xz
z zh
Pz dz
P
(37)
For Young's modulus, shear modulus, density
and moments of inertia of the equation (38) is used:
2
23
4
25
6
1
ij
ijh
ij
ij
hij
ij
ij
A
zB
zD
zE dz
F z
G z
H z
(38)
Whit Placement in the relationship, the resultant forces and moments will be obtained as:
0 0 1 11 1 21 1 1 21
3 312 211 1 21
1
1
a g xx b h xx
xx
d j xx
E E v E E k v k
Nv v E E k k
(39)
0 0 1 12 2 12 2 2 12
3 312 212 2 12
1
1
a g xx b h xx
d j xx
E E v E E k v k
Nv v E E k k
(40)
0 1 312 12 12 12 12 12 x a g xy b h x d j xN G G G G k G G k (41)
0 0 1 11 1 21 1 1 21
3 312 211 1 21
1
1
b h xx c i xx
xx
e k xx
E E v E E k v k
Mv v E E k k
(42)
0 0 1 12 2 12 2 2 12
3 312 212 2 12
1
1
b h xx c i xx
e k xx
E E v E E k v k
Mv v E E k k
(43)
0 1 312 12 12 12 12 12 x b h xy c i x e k xM G G G G k G G k (44)
0 0 1 11 1 21 1 1 21
3 312 211 1 21
1
1
d j xx e k xx
xx
f l xx
E E v E E k v k
Pv v E E k k
(45)
0 0 1 12 2 12 2 2 12
3 312 212 2 12
1
1
d j xx e k xx
f l xx
E E v E E k v k
Pv v E E k k
(46)
0 1 312 12 12 12 12 12 x d j xy e k x f l xP G G G G k G G k (47)
0 21 12 12 12 12 xz a g xz c i xzQ k G G G G
(48)
212 12
0 32 12 12 12 12 c i zz a g z d j zG GQ k G G G G
(49)
0 21 12 12 12 12 xz c i xz e k xzR k G G G G
(50)
0 2 32 12 12 12 12 12 12 z c i z e k z m n zR k G G G G G G
(51)
6306 Hossein Farahani et al, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6301-6315
0 21 12 12 12 12 xz d j xz m n xzP k G G G G
(52)
0 2 32 12 12 12 12 12 12 z d j z m n z f l zP k G G G G G G
(53)
The equations of motion for vibration of a generic shell:
The equations of motion for vibration of a generic shell can be derived by using Hamilton's principle which
is described by
0
0
tK U V dt (54)
δU is the potential energy, δV virtual work done by external forces and δK is kinetic energy allowed.
Potential energy is obtained as follows:
2
2
(0) (1) (3)(0) (1) (3)
0 1 3 0 2(0) (2)
(3)
[
]
h
ij ij
hx
xx xx xx xx xx xx
x
x x x xz xz xz xz z zx x x z z
z z
U R dx d dz
U N M k P k N M k P k
N M k P k Q R Q R
P R
dx d
(55)
Kinetic energy is calculated from the following equation:
2 ˙ ˙ ˙ ˙ ˙ ˙
1 1 2 2 3 3
2
2 ˙ ˙ ˙ ˙3 3
2 2
2
˙ ˙3
2
( )
4 4 {(
3 3
4 1
3
h
hx
h
x x x x
hx
K U U U U U U
wu z z u z z w
x xh h
w vv z z
R Rh
˙ ˙3
2
4 1 1
3
}
v z z w vR Rh
w w R dx d dz
(56)
Due to the absence of external forces δV (virtual work done by external forces) is zero.
wave method:
Method for solving the wave displacement components are defined as [26].
, ,
, ,
, ,
, ,
, ,
m
m
m
m
m
i n t k x
i n t k x
i n t k x
i n t k xx
i n t k x
u x t A e
v x t B e
w x t C e
x t D e
x t E e
(57)
Where, m and n are wave numbers and the simply supported boundary condition in both edges is satisfied. By substituting Eq. 57 in Eqs. (39-53) and obtaining coefficient matrix then setting the determinant of coefficient matrix to zero, the natural frequencies may be determined with respect to m and n.
6307 Hossein Farahani et al, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6301-6315
Results And Discussion
Material properties:
Single-walled carbon nanotubes (10, 10) have been used to reinforce the cylindrical shell. Temperature and
size-dependent properties of nanotubes have been considered and studied by molecular dynamics and the results of which are presented below.
Table 1: Properties of Single-walled carbon nanotubes (10, 10) (L=9.26 nm , R=0.68 nm , h=0.067 nm , ) [20].
5.6466 7.0800 1.9445
Properties of composite materials used in the shell in Tables.2 and Tables.3 have been identified.
Table 2: The elastic modulus PMPV / CNT reinforced by nano-tubes (10,10) [19].
0.934 2.2 0.149 94.57 0.11
0.941 2.3 0.150 120.09 0.14
1.381 3.5 0.149 145.08 0.17
Table 3: The elastic modulus of PMMA / CNT reinforced by nano-tubes (10,10)[20]
1.022 2.9 0.137 94.78 0.12
1.626 4.9 0.142 138.689 0.17
1.585 5.5 0.141 224.50 0.28
Numerical analysis and compare the results of
different shell materials with different volume than
intended and the natural frequency changes after
every m, n, h/R is obtained. N and m are respectively
the wave number of environmental and longitudinal
wave number.
Numerical results for the nanocomposites
(10,10) with regard to the sex of the shell is
presented. Ω Is the dimensionless natural vibration
frequency to evaluate the results, the following are
considered: [27].
accuracy of the results:
Table 4: verify the equations and compared with the reference
n (Shah et al., 2009). Present
1 0.140641 0.141123
2 0.054323 0.05434
3 0.027074 0.027093
4 0.017776 0.017784
5 0.017088 0.017092
Table 4 shows the results, there is little error
indicates that the equation is true.
Results:
Natural frequencies of FGM cylindrical shells with
different volume ratios of nanotubes and polymer
PMPV:
a) First natural frequency of nanotubes with
various volume ratios and polymer PMPV according
to changes in the longitudinal direction of the wave
number are obtained. Here n is equal to 1 in all cases.
The results in Table.5 and Figure.2 can be seen.
Table 5: The values of the dimensionless natural frequencies of FGM cylindrical shell in terms of the variation m
m 0.11 0.14 0.17
1 0.269967 0.2784 0.338424
2 0.392924 0.401725 0.493565
3 0.474443 0.484241 0.596208
4 0.53664 0.547427 0.664457
5 0.586783 0.598449 0.737516
6308 Hossein Farahani et al, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6301-6315
Fig. 2: Nondimensional natural frequency Ω for different volume ratios PMPV in m
B) Natural frequencies with different volume
ratios of nanotubes and polymer PMPV according to
environmental changes in wave number are obtained.
In this case m = 1 in all cases we have considered.
The results in Table.6 and Figure.3 can be seen.
Table 6: The values of the dimensionless natural frequencies of FGM cylindrical shell in terms of the variation of n
n
0.11 0.14 0.17
1 0.269967 0.2784 0.338424
2 0.0340121 0.0366588 0.0422179
3 0.0160027 0.0175364 0.0198064
4 0.0101621 0.0111105 0.0126367
5 0.00916664 0.0097687 0.0115664
Fig. 3: Nondimensional natural frequency Ω for different volume ratio of n times PMPV
C) natural frequencies with different volume
ratios of nanotubes and polymer PMPV according to
changes in h / R to obtain. In this case, m and n = 1
in all cases we have considered. The results in
Table.7 and Figure.4 is observed.
Table 7: The values of the dimensionless natural frequencies of FGM cylindrical shell in terms of the variation of h / R
h/R 0.11 0.14 0.17
0.01 0.269972 0.278405 0.33843
0.02 0.269986 0.278422 0.338449
0.03 0.270011 0.278451 0.338479
0.04 0.270045 0.27849 0.338522
0.05 0.270089 0.278541 0.338577
6309 Hossein Farahani et al, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6301-6315
Fig 4: Nondimensional natural frequency Ω for different volume ratios PMPV terms of h / R
Natural frequencies of FGM cylindrical shells with
different volume ratios of polymer nanotubes and
PMMA:
A) Natural frequencies with different volume
ratios of nanotubes and PMMA polymer according
obtain m changes. In this case n = 1 in all cases we
have considered. The results in Table.8 and Figure .5
can be seen.
Table 8: The values of the dimensionless natural frequencies of FGM cylindrical shell in terms of the change m
m 0.12 0.17 0.28
1 0.304015 0.389414 0.426979
2 0.448784 0.580311 0.62429
3 0.543492 0.704236 0.754489
4 0.615318 0.797838 0.853631
5 0.673066 0.87295 0.933489
Fig. 5: Natural frequency Ω for the different volume ratio of PMMA in m
B) Natural frequencies with different volume
ratios of nanotubes and PMMA polymer according
obtain environmental changes in wave number. In
this case m = 1 in all cases we have considered. The
results in Table .9 and Figure .6 is observed.
6310 Hossein Farahani et al, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6301-6315
Table 9: The values of the dimensionless natural frequencies of FGM cylindrical shell in terms of the variation of n
n 0.12 0.17 0.28
1 0.304015 0.389414 0.426979
2 0.0358979 0.0442234 0.0525729
3 0.0165577 0.0202071 0.0245915
4 0.0106329 0.0131462 0.0158583
5 0.0100424 0.0128542 0.0149211
Fig. 6: Natural frequency Ω for the different volume ratio PMMA based on the
C) natural frequencies with different volume
ratios of PMMA and nanotubes and polymers based
on changes in h / R to obtain. In this case, m and are
equal to (1) in all cases we have considered. The
results in Table .10 as well as Figure .7 can be seen.
Table 10: values of the dimensionless natural frequencies of FGM cylindrical shell in terms of the variation of h / R
h/R 0.12 0.17 0.28
0.01 0.30402 0.38942 0.426987
0.02 0.304034 0.389438 0.42701
0.03 0.304059 0.389468 0.427049
0.04 0.304093 0.38951 0.427104
0.05 0.304137 0.389564 0.427174
Fig. 7: Natural frequency Ω for the different volume ratio of PMMA versus h / R
6311 Hossein Farahani et al, 2013 /Journal Of Applied Sciences Research 9(13), December, Pages: 6301-6315
Conclusion:
A study on the vibration of functionally graded
cylindrical shell with carbon nanotube support has
been presented.
Dimensionless frequencies for each of the
polymers with carbon nanotube support in three
states has obtained the following results .
a. The first mode frequency n terms
b. The second mode frequency in terms of m
c. The third mode frequency versus h / R
The study showed that carbon nanotube support
has significant influence on the frequencies. this is
because the functionally graded cylindrical shells
exhibit interesting frequency characteristics when the
constituent volume fractions are varied. In addition,
sometimes the frequency of the polymer has been
effective. So that the polymer PMMA has better
performance .
Finally we can say that increasing the volume
percent carbon nanotube (as far as frequency
increases are stopped) to increase the frequency that
this increase in frequency is good for strengthening
the shell.
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