Modeling and Numerical Simulation of Material Science, 2013, *, **
doi:10.4236/mnsms.2013.***** Published Online ** 2013 (http://www.scirp.org/journal/mnsms)
Elasticity solution of functionally graded carbon
nanotube-reinforced composite cylindrical panel
A. Alibeigloo, M. Shaban
Department of mechanical engineering, Faculty of Engineering, Tarbiat modares University, Tehran, Iran
Email: [email protected], [email protected]
Received **** 2011
Abstract
Based on three-dimensional theory of elasticity, static analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical panel subjected to mechanical uniform load with simply supported boundary conditions
is carried out. By using Fourier series expansion along the axial and circumferential directions and state space technique
across the radial direction, closed form solution is derived. The effects of CNT distribution cases as well as the volume
fraction of CNT and span angle of the panel on the bending behavior of the cylindrical panel are examined.
Keywords: Carbon nanotube ; cylindrical panel; Static ; Elasticity; Analytical
1. Introduction
Study on the mechanical and thermal properties of
CNTRC structures has increased by many researchers in
recent years. Thostenson et al. [1] presented a review on
the researches and application of CNT and CNTRC.
Shen [2] used higher order shear deformation theory as
well as a von Kármán-type of kinematic nonlinearity to
investigate the postbuckling behavior of nanocomposite
cylindrical shells reinforced by SWCNTs and subjected
to axial environments. Nonlinear vibration of
FG-CNTRC cylindrical shell was investigated by Shen
and Xiang [3 using the equation of the motion base on
higher-order shear deformation theory with aVon Karman-type of kinematic nonlinearity. Moradi-Dastjerdi et
al. [4] analyzed the dynamic behavior of FG-CNTRC
cylindrical shell subjected to impact load by making the
use of mesh free method. Recently the author [5] presented an analytical solution for bending behavior of
FG-CNT composite plate integrated with piezoelectric
actuator and sensor under an applied electric field and
mechanical load. In the present work bending behavior
of FG-CNTRC cylindrical panel subjected to uniform
internal pressure is investigated.
2. Basic equations
A CNTRC cylindrical panel with geometry and dimensions according to the Fig.1 is considered.
The
Copyright © 2013 SciRes.
SWCNT reinforcement is either uniformly distributed
(UD) or functionally graded (FG) in four cases, FG-V,
FG-Λ, FG-X and FG-O in the thickness direction. Displacements component along the r, θ and z directions are
denoted by ur, uθ and uz, respectively. According to the
rule of mixture and considering the CNT efficiency parameters, the effective mechanical properties of mixture
of CNTs and matrix isotropic polymer can be written as
the follow
CN
E11  1VCN E11
+Vm E m
(1.1)
2 VCN Vm

 m
E 22 E CN
E
22
(1.2)
3
V
V
 CN
 mm
CN
G12 G12 G
(1.3)
Relation between the CNT and matrix volume fractions
is stated as
VCN  Vm  1
(2)
The volume fraction of CNT for five cases UD, FG-V,
FG   , FG  X and FG  O distribution along
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A. Alibeigloo, M. Shaban
the thickness according to the Fig.1 has the following
relations, respectively

VCNT  VCNT
(3.2)
 rR
 
VCNT  2  
 0.5  VCNT
h


(3.3)
rR
h

VCNT

rR  
VCNT  4  0.5 
 VCNT
h 

(3.4)
(3.5)
where

VCNT


r
   z

r
 Q11
Q
 12
Q
Q   13
 0
 0

 0
Q12
Q13
0
0
Q 22
Q 23
0
0
Q 23
Q33
0
0
0
0
Q 44
0
0
0
0
Q55
0
0
0
0
WCN
,
 CN   CN 
WCN  

 WCN
 m   m 
12  V

 Vm 
m
  VCNT CNT  Vmm
G12  G13  G 23 13  12
(4.2)
31  21
E
12  22 12
E11
The constitutive equations for CNTRC panel layer are
written as
  Q
 zr
 z 
T
(5 )
 r
T
0 
0 
0 

0 
0 

Q 66 
r z 1  2r



 0,
r
z r 
r
(6)
rz  z 1 z rz



 0,
r
z r 
r
(4.1)
of CNTs and matrix isotropic polymer are
32   23   21
z 
r zr 1 r 1


  r     0 ,
r
z r  r
(3.6)
and the other effective mechanical properties of mixture
E33  E 22
zr
and the relation between the stiffness elements, Qij and
engineering constants, Eij, Gij and νij are described in
appendix. In the absence of body forces, the governing
equilibrium equations in three dimensions are
where R is mid-radius of the panel . The Poisson’s ratio,ν12 and the density of the nanocomposite panel is assumed as

CNT
CNT 12
r
(3.1)
rR
 
VCNT  2 
 0.5  VCNT
 h

VCNT  4 
  z
The linear relations between the strain and displacements
are
u r 1 u 

r r 
z 
u z
z
 r 
u z u r
 u  u  1 u r



,  zr 
r
z
r
r r 
 z 
u  1 u z

,
z r 
, 

, r 
u r
,
r
,
(7)
3. Solution procedure
Following displacement and stress components satisfy
simply supported boundary condition
u r  ur sin  pn z  sin  pm
where
Copyright © 2013 SciRes.
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A. Alibeigloo, M. Shaban
u z  uz cos  pn z  sin  pm
[
  1.5
m
gr
cm3
]
1  0.149 , 2  0.934 ,
,
2  3
u   u sin  pn z  cos  pm
To show the effect of CNT in the bending behavior of
nanocomposite, numerical illustration is made. Numerical investigation were carried out and presented in Table
1 and Figs.2-3. According to the table, by increasing the
span angle of the panel, radial stress decreases whereas
the axial stress and transverse shear stress increase. From
this figure also it can be concluded that when the CNT
volume fraction increase, axial normal and transverse
shear stresses increase and radial stress as well as circumferential displacement decrease. Figs. 2a and 2b depict the effect of five cases of CNT distribution, UD, FGV, FG-Λ, FG-X and FG-O on the stress and displacement field for the CNTRC cylindrical panel. According
to the Figures, the transverse normal,  r and axial dis-
r   sin  pn z  sin  pm

r
   sin  pn z  sin  pm
z  z sin  pn z  sin  pm
z  z cos  pn z  cos  pm
zr  zr cos  pn z  sin  pm
placement, U z has minimum value in FG-Λ case and
r  r sin  pn z  cos  pm
(8 )
By using Eqs. (4)-(8), the following state-space equations for the FG-CNTRC layer is derived
d
  G ,
dr
where  


r
uz
u
ur
(9)
Table 1. Effect of CNT volume fraction on the stress
r  .
and displacement field at mid radius of cylindrical
T
rz
panel with various span and S = 10,
mn 5.
4. Numerical results and discussion
In this section a simply-supported FG-CNT cylindrical
panel with the following material properties for the CNT
and matrix polymer is considered to illustrate the foregoing analysis
CN
11
E
 5.6466
TPa
, E
CN
22
E
maximum value in FG-V case. Effect of CNT volume
fraction on stress and displacement fields is presented in
Figs.3a and 3b. According to Fig.3a increase the CNT
volume fraction causes to increase the axial stress nonlinearly with remaining almost constant at the outer radius. As the Fig. 3b depict, increase the CNT volume fraction causes to decrease axial displacements.
CN
33
 7.0800
TPa
,

VCNT
0.11
CN
CN
CN
G12
 G13
 1.9445TPa , 12
 0.19 ,

CNT
[
 1.4
gr
cm3
]
,
E  2.1
m
GPa
,
  0.34 ,
m
0.14
R
2

r
z
r
U

4
-0.390
10.091
-0.986
-1.328

3
-0.389
25.461
-1.251
-5.442
0.381
31.995
-1.571
-34.718
-0.381
11.545
-0.994
-1.292
-0.373
28.649
-1.256
-5.299
0.382
22.977
-1.559
-33.721
-0.376
12.951
-1.002
-1.253

2

4

3

2

4
Copyright © 2013 SciRes.
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MNSMS
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A. Alibeigloo, M. Shaban
0.17

3

2
-0.365
31.891
-1.264
-5.152
0.382
23.846
-1.556
-32.833
(a) Radial normal stress
(a) UD
(b) Axial displacement
Fig. 2. Distribution of mechanical entities along the
thickness for various cases of CNT distribution
for the cylindrical panel with

VCNT
=0.17, m =
n=25.
(b) FG- Λ
(c) FG-V
(d) FG-X
(a) Axial normal stress
(b) Axial displacement
Fig. 3. Effect of CNT volume fraction on through the
thickness stresses and displacements for the
FG   CNTRC cylindrical panel , with
L/h = 50, V  0.17
.

CNT
5. Conclusion
(e) FG-O
Fig.1. Geometry of CNTRC.
Copyright © 2013 SciRes.
Bending behavior of FG-CNTRC cylindrical panel with
simply supported edges and various cases of CNT distribution was examined. From numerical illustration the
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A. Alibeigloo, M. Shaban
following conclusion are derived;
 Radial normal and axial displacement in the case of
FG-Λ, at a point is always smaller in magnitude than
those at the corresponding points in the other two
cases of CNT distribution.
 Existence of CNT in cylindrical panel decreases the
axial displacement component and increases the axial stress.
6. Appendix
 a1
r

0


0
G = 
a6


 b6


 b9
b1
b2
0
0
0
1
r
b4
b5
b7
b8
b10
b11
a3
r2


z


z
a4
b3
0
a 8
r
a 2
r
0

b12
1
r
0

 

0 


a5 


0 


0 

2
 
r 

where
b9 
a 8 
r 
b8  
b 7  a 9
a10  2
r z
b10 
2 a3 2
b11  Q66 2  2 2
z
r 
 2 Q66  2

z 2 r 2 2
a10  2
r z
b1 2  
a3 
r 2 
Q11 
E11
E
1   2332  Q22  22 1  3113 


Q33 
E33
1  12 21 

Q66  G12 ,
Q1 2 
Q4 4  G
23
Q5 5  G
E1 1
  2 1  3 1 2 3

Q13 
E11
 31   2132 

,
Q 23 
E 22
 32  1231 

Q


Q Q 
a1   23  1 a 2   Q12  13 23 
Q33 
 Q33 

  1  12 21   2332  3113  2123213

Q2 
a 3   Q22  23 
Q33 

7. References
a6 
1
Q33
a7 
1
a4 
Q55
Q13
Q33
a2 
r z
b 4  a 7

z
b2 
Q 23
Q33

Q Q 
a1 0   Q 1 2 Q 66 2 3 1 3
Q33 

2


Q13
a 9   Q11 

Q33 

b1 
a8 
1
a5 
Q 44
a3 
r 2 
b5 
Copyright © 2013 SciRes.
b3  
a 8 
r 
1 
r 
b 6  a 7

z
13
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Part I: Axially-loaded shells”, Composite Structures.
Vol 93, pp.2096–2108, 2011.
[3] HS Shen, Y Xiang. “Nonlinear vibration of nanotube-reinforced composite cylindrical shells in
thermal invironments” . Computer Methods in Applied Mechanic Engineering, Vol. 213–216 pp.
196–205, 2012.
[4] R Moradi-Dastjerdi, M Foroutan, “A Pourasgha.
Dynamic analysis of functionally graded nanocomposite cylinders reinforced by carbon nanotube by a
mesh-free method”. Materials and Design Vol. 44,
MNSMS
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A. Alibeigloo, M. Shaban
pp. 256–266, 2013.
[5] Alibeigloo A. “Static analysis of functionally graded
carbon nanotube-reinforced composite plate embedded in piezoelectric layers by using theory of
elasticity”. Composite Structures Vol. 95, pp.
612-622, 2013.
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