hi

1. Static and free vibration analyses of carbon nanotube-reinforced composite plates
using finite element method with first order shear deformation plate theory
Ping Zhu a
, Z.X. Lei a,b,c
, K.M. Liew a,b,⇑
a
Department of Civil and Architectural Engineering, City University of Hong Kong, Kowloon, Hong Kong
b
USTC–CityU Joint Advanced Research Centre, Suzhou, PR China
c
CAS Key Laboratory of Mechanical Behavior and Design of Materials, University of Science and Technology of China, PR China
a r t i c l e i n f o
Article history:
Available online 15 November 2011
Keywords:
Bending
Free vibration
Carbon nanotube
Composite
First order shear deformation theory
Finite element method
a b s t r a c t
This paper mainly presents bending and free vibration analyses of thin-to-moderately thick composite
plates reinforced by single-walled carbon nanotubes using the finite element method based on the first
order shear deformation plate theory. Four types of distributions of the uniaxially aligned reinforcement
material are considered, that is, uniform and three kinds of functionally graded distributions of carbon
nanotubes along the thickness direction of plates. The effective material properties of the nanocomposite
plates are estimated according to the rule of mixture. Detailed parametric studies have been carried out
to reveal the influences of the volume fractions of carbon nanotubes and the edge-to-thickness ratios on
the bending responses, natural frequencies and mode shapes of the plates. In addition, the effects of dif-
ferent boundary conditions are also examined. Numerical examples are computed by an in-house finite
element code and the results show good agreement with the solutions obtained by the FE commercial
package ANSYS.
Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Carbon nanotubes (CNTs) have been widely accepted owing to
their remarkable mechanical, electrical and thermal properties
and the applications of CNTs are thus drawing much attention cur-
rently. Conventional fiber-reinforced composite materials are nor-
mally made of stiff and strong fillers with microscale diameters
embedded into various matrix phases. The discovery of CNTs
may lead to a new way to improve the properties of resulting com-
posites by changing reinforcement phases to nano-scaled fillers
[1]. Carbon nanotubes are considered as a potential candidate for
the reinforcement of polymer composites, provided that good
interfacial bondinged between CNTs and polymer and proper dis-
persion of the individual CNTs in the polymeric matrix can be guar-
anteed [2]. The CNT–polymer interfacial shear strength was
determined according to a series of pull-out tests of individual car-
bon nanotubes embedded within polymer matrix by Wagner et al.
[3,4], which demonstrated that carbon nanotubes are effective in
reinforcing a polymer due to remarkably high separation stress.
During pullout, it can be expected that yielding of surrounding
polymer would occurs before interfacial failure between the nano-
tube and polymer matrix. Molecular dynamics (MD) simulations
were performed to predict the interfacial bonding by considering
three-dimensional cross-links and stronger interfacial adhesion
can be achieved through functionalization of the nanotube surface
to form chemical bonding to the chains of polymer matrix [5,6].
Another fundament issue is the dispersion of carbon nanotubes
in the matrix, since CNTs are tend to agglomerate and entangle
because of their enormous surface area and high aspect ratio.
Amino-functionalized CNTs are therefore developed to improve
their dispersion in polymer resins [7].
The constitutive models and mechanical properties of carbon
nanotube polymer composites have been studied analytically,
experimentally, and numerically. A review and comparisons of
mechanical properties of single- and multi-walled carbon nano-
tube reinforced composites fabricated by various processes were
given by Coleman et al. [8], in which the composites based on
chemically modified nanotubes showed the best results since the
functionalization should significantly enhance both dispersion
and stress transfer. Tensile tests of multi-walled carbon nano-
tube-reinforced composites have demonstrated that an addition
of small amounts of the nano-scaled reinforcement results in con-
siderable increases in the elastic modulus and break stress [9,10].
According to the equilibrium molecular structure obtained with
the MD simulation, an equivalent continuum modeling was devel-
oped for prediction of bulk mechanical properties of single-walled
carbon nanotubes (SWCNTs)/polymer composites [11,12]. The
macroscopic elastic properties of were CNTs reinforced composites
were evaluated through a layered cylindrical representative
volume element (RVE) with an embedded nanotube by a
0263-8223/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved.
doi:10.1016/j.compstruct.2011.11.010
⇑ Corresponding author at: Department of Civil and Architectural Engineering,
City University of Hong Kong, Kowloon, Hong Kong.
E-mail address: kmliew@cityu.edu.hk (K.M. Liew).
Composite Structures 94 (2012) 1450–1460
Contents lists available at SciVerse ScienceDirect
Composite Structures
journal homepage: www.elsevier.com/locate/compstruct

2. three-dimensional finite element model [13,14]. Different elastic
moduli and constants can be calculated based on the stress–strain
curves of CNTs reinforced composites derived from MD simula-
tions [15,16]. Nonlinear responses of nano-reinforced polymers
were predicted via an atomistic-based continuum multiscale mod-
el by taking nonlinearity of the interatomic potentials into account
[17].
In actual structural applications, carbon nanotube-reinforced
composites (CNTRC), as a type of advanced material, may be incor-
porated in the form of beams, plates or shells as structural compo-
nents. It is thus of importance to explore mechanical responses of
the structures made of CNTRC. Wuite and Adali firstly used the
classical laminated beam theory to analyze symmetric cross-ply
and angle-ply laminated beams stacked with multiple transversely
isotropic layers reinforced by CNTs in different aligned directions
and isotropic beams composed of randomly oriented CNTs dis-
persed in a polymer matrix, based on micromechanical constitu-
tive models developed according to the Mori–Tanaka method
[18]. With the decomposition into two superposable stages, Vode-
nitcharova and Zhang [19] developed a continuum model for pure
bending of a straight nanocomposite beam with a circular cross
section reinforced by a single-walled carbon nanotube, by which
the critical stress of local buckling of SWCNT in longitudinal bend-
ing was derived for the assessment of the onset of a possible fail-
ure. A type of novel actuator comprised of a piezoceramic matrix
reinforced by SWCNTs was proposed to apply for active controls
of smart structures by Ray and Batra [20]. The effective properties
of the resulting piezoelectric composites were determined by a
micromechanical analysis and the active control system exhibited
better performance in the controllability of vibrations. The same
authors [21] then proposed another new hybrid piezoelectric com-
posite containing SWCNTs and piezoelectric fibers as reinforce-
ments embedded in a conventional polymer matrix. The value of
the effective piezoelectric coefficient related to the in-plane actua-
tion was found to increase significantly due to the addition of
CNTs. With the finite element method, Formica et al. [22] studied
vibration behaviors of cantilevered CNTRC plates by employing
an equivalent continuum model according to the Mori–Tanaka
scheme. They affirmed that the CNTRC plates can be tailored so
as to respond with a desired mode to a certain external excitation.
The buckling behaviors of laminated composite plates reinforced
by SWCNTs were investigated analytically and numerically based
on the classical laminated plate theory and third-order shear defor-
mation theory, respectively [23], in which the optimized orienta-
tion of CNTs to achieve the highest critical load and
corresponding mode shape were calculated for different kinds of
boundary conditions as well as aspect ratios of the plates.
Motivated by the concept of functionally graded materials
(FGM), Shen [24] suggested that for CNT-based composite struc-
tures the distributions of CNTs within an isotropic matrix were de-
signed purposefully to grade with certain rules along desired
directions for the improvement of the mechanical properties of
the structures and the nonlinear bending behaviors of the resulting
functionally graded CNT-reinforced composite (FG-CNTRC) plates
in thermal environments were presented. The nonlinear free vibra-
tions of FG-CNTRC Timoshenko beams with symmetric and unsym-
metrical distributions of CNTs along the thickness direction were
analyzed with the Ritz method and direct iterative technique
[25]. Then thermo-mechanical nonlinear dynamic characteristics
of uniformly distributed CNT-reinforced composite (UD-CNTRC)
plates were studied by Shooshtari et al. [26]. The critical buckling
temperature and thermal postbuckling behaviors of bi-layer com-
posite plates reinforced by symmetrically distributed CNTs with
respect to mid-plane were reported by Shen and Zhang [27]. The
results showed that the plate with intermediate nanotube volume
fraction may not lead to intermediate buckling temperature and
initial thermal postbuckling strength in some cases. Then Shen
and Zhu [28] presented the buckling loads and postbuckling equi-
librium paths of the same bi-layer nanocomposite plates subjected
to uniaxial compressions in thermal environments since the prop-
erties of CNTs are temperature-dependent. For CNTRC cylindrical
shells [29,30], postbuckling analyses, respectively, subject to the
axial compression and lateral pressure in thermal environments
were performed, and the results revealed that the CNT volume
fraction has a significant effect on the buckling loading and post-
buckling behavior. Wang and Shen [31] studied the large ampli-
tude vibration of CNTRC plates resting on an elastic foundation in
thermal environments. They found that the natural frequencies
are reduced while the ratios of nonlinear to linear frequencies
are increased with the elevated temperature or the decreasing
foundation stiffness. Recently, a new type of sandwich plate that
consists of two thin composite face sheets reinforced by CNTs
and a thick homogeneous core layer was proposed [32], and non-
linear vibration and bending analyses in thermal environments of
the sandwich structure were conducted.
For the most analyses of CNTRC plates and shells available in
the literatures [24,27–32], an analytical method involving a two-
step perturbation technique was used. However, it is necessary
to develop efficient numerical models for dealing with complicated
analyses of the novel advanced structures with complex geometri-
cal shapes and boundary conditions. In the present work, a finite
element model is developed to carry out bending and free vibra-
tion analyses of CNTRC plates based on the first-order shear defor-
mation plate theory. The CNTs are assumed to be uniformly
distributed or functionally graded in the thickness direction. The
material properties of SWCNT are determined according to molec-
ular dynamics simulations and then the effective material proper-
ties of CNTRCs are estimated through the rule of mixture in which
the CNT efficiency parameters are introduced to account for the
scale-dependence of the resulting nanostructures. The influences
of CNT volume fraction, boundary conditions and width-to-thick-
ness ratio on the bending responses and natural frequencies are
discussed for the CNTRC plates. The numerical results are com-
pared with those that are given by using the FE commercial pack-
age ANSYS. A good agreement is observed between the two groups
of solutions.
2. Carbon nanotube reinforced composite plates
As shown in Fig. 1, in this paper, four types of CNTRC plates with
length a, width b and thickness h were considered. UD-CNTRC rep-
resents the uniform distribution and FG-V, FG-O and FG-X CNTRC
the functionally graded distributions of carbon nanotubes in the
thickness direction of the composite plates. The effective material
properties of the two-phase nanocomposites, mixture of CNTs
and an isotropic polymer, can be estimated according to the
Mori–Tanaka scheme [33] or the rule of mixture [34,35]. Due to
the simplicity and convenience, in the present study the rule of
mixture was employed by introducing the CNT efficiency parame-
ters and the effective material properties of CNTRC plates can thus
be written as [24]
E11 ¼ g1VCNTECNT
11 þ VmEm
ð1aÞ
g2
E22
¼
VCNT
ECNT
22
þ
Vm
Em ð1bÞ
g3
G12
¼
VCNT
GCNT
12
þ
Vm
Gm ð1cÞ
where ECNT
11 , ECNT
22 and GCNT
12 indicate the Young’s moduli and shear
modulus of SWCNTs, respectively, and Em
and Gm
represent the
P. Zhu et al. / Composite Structures 94 (2012) 1450–1460 1451

3. corresponding properties of the isotropic matrix. To account for the
scale-dependent material properties, gj (j = 1, 2, 3), the CNT effi-
ciency parameters, were introduced and can be calculated by
matching the effective properties of CNTRC obtained from the MD
simulations with those from the rule of mixture. VCNT and Vm are
the volume fractions of the carbon nanotubes and matrix, respec-
tively, and it is noticeable that the sum of the volume fractions of
the two constituents equals to unity.
The uniform and three types of functionally graded distribu-
tions of the carbon nanotubes along the thickness direction of
the nanocomposite plates depicted in Fig. 1 are assumed to be as
following:
VCNT ¼ VÃ
CNT ðUD CNTRCÞ
VCNTðzÞ ¼ 1 þ 2z
h
À Á
VÃ
CNT ðFG-V CNTRCÞ
VCNTðzÞ ¼ 2 1 À 2jzj
h
VÃ
CNT FG-O CNTRCð Þ
VCNTðzÞ ¼ 2 2jzj
h
VÃ
CNT ðFG-X CNTRCÞ
8
:
ð2aÞ
where
VÃ
CNT ¼
wCNT
wCNT þ ðqCNT=qmÞ À ðqCNT=qmÞwCNT
ð2bÞ
where wCNT is the mass fraction of the carbon nanotube in the com-
posite plate, and qm
and qCNT
are the densities of the matrix and
CNT. Note that the UD-CNTRC and the other three FG-CNTRC plates
possess the same overall mass fraction (wCNT) of the carbon
nanotubes.
Similarly, the thermal expansion coefficients, a11 and a22,
respectively, in the longitudinal and transverse directions, Pois-
son’s ratio m12 and the density q of the nanocomposite plates can
be determined in the same way as
m12 ¼ VÃ
CNTmCNT
12 þ Vmmm
ð3aÞ
q ¼ VCNTqCNT
þ Vmqm
ð3bÞ
a11 ¼ VCNTaCNT
11 þ Vmam
ð3cÞ
a22 ¼ 1 þ mCNT
12
À Á
VCNTaCNT
22 þ ð1 þ mm
ÞVmam
À m12a11 ð3dÞ
where mCNT
12 and mm
are Poisson’s ratios, and aCNT
11 , aCNT
22 and am
are the
thermal expansion coefficients of the CNT and matrix, respectively.
Note that m12 is considered as constant over the thickness of the
functionally graded CNTRC plates.
3. Finite element formulation
3.1. Displacement filed and strains of CNTRC plates
Considering moderately thick CNTRC plates (Fig. 1), the first or-
der shear deformation plate theory (FSDT) was employed to ac-
count for the displacement field {u, v, w}T
within a plate domain,
according to displacements and rotations of the mid-plane of the
plate [36].
uðx; y; z; tÞ
vðx; y; z; tÞ
wðx; y; z; tÞ
8
:
9
=
;
¼
u0ðx; y; tÞ
v0ðx; y; tÞ
w0ðx; y; tÞ
8
:
9
=
;
þ z Á
uxðx; y; tÞ
uyðx; y; tÞ
0
8
:
9
=
;
ð4Þ
where u0, v0 and w0 represent the respective translation displace-
ments of a point at the mid-plane of the plate in x, y, and z direc-
tions; ux and uy denote rotations of a transverse normal about
the positive y and negative x axes, respectively.
The linear in-plane and transverse shear strains are given by
exx
eyy
cxy
8
:
9
=
;
¼ e0 þ zj;
cyz
cxz
'
¼ c0 ð5aÞ
where
e0 ¼
@u0
@x
@v0
@y
@u0
@y
þ @v0
@x
8
:
9
=
;
; j ¼
@ux
@x
@uy
@y
@ux
@y
þ
@uy
@x
8
:
9
=
;
; c0 ¼
uy þ @w0
@y
ux þ @w0
@x
( )
By employing the Voigt’s notation, the constitutive equations
are written in the form
z
yh/2
h/2
z
xx
yh/2
h/2
a a
b b
z
yh/2
h/2
z
xx
yh/2
h/2
a a
b b
(a) UD (b) FG-V
(c) FG-O (d) FG-X
Fig. 1. Configurations of the carbon nanotube-reinforced composite plates. (a) UD CNTRC plate; (b) FG-V CNTRC plate; (c) FG-O CNTRC plate; (d) FG-X CNTRC plate.
1452 P. Zhu et al. / Composite Structures 94 (2012) 1450–1460

4. rxx
ryy
rxy
ryz
rxz
8
:
9
=
;
¼
Q11ðz;TÞ Q12ðz;TÞ 0 0 0
Q12ðz;TÞ Q22ðz;TÞ 0 0 0
0 0 Q66ðz;TÞ 0 0
0 0 0 Q44ðz;TÞ 0
0 0 0 0 Q55ðz;TÞ
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
exx
eyy
exy
eyz
exz
8
:
9
=
;
À
a11
a22
0
0
0
8
:
9
=
;
DT
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
ð6aÞ
where
Q11 ¼
E11
1 À m12m21
; Q22 ¼
E22
1 À m12m21
; Q12 ¼
m21E11
1 À m12m21
; Q66
¼ G12; Q44 ¼ G23; Q55 ¼ G13
and DT is the temperature change with respect to a reference state.
E11 and E22 are the effective Young’s moduli of CNTRC plates in the
principle material coordinates G12, G13, and G23 are the shear mod-
uli; and m12 and m21 are Poisson’s ratios.
3.2. Bending of CNTRC plates
The generalized displacements at any point within an element
Xe
are approximated as
uh
0ðx; tÞ ¼
Xm
i¼1
we
i ðxÞd
e
i ðtÞ ð7Þ
where we
i are the Lagrangian interpolation functions and d
e
collect
nodal displacements of the element Xe
. Following the standard pro-
cedure, the finite element equations of CNTRC plates subjected to a
uniformly distributed transverse load can be expressed as
Kd ¼ F ð8Þ
where K and F are the global stiffness matrix and load vector which
are computed by assembling the element stiffness matrix Ke
and
element load vector Fe
. Ke
is given by
Table 1
Temperature-dependent material properties of (10,10) SWCNT (L = 9.26 nm, R = 0.68 nm, h = 0.067 nm, mCNT
12 ¼ 0:175).
Temperature (K) ECNT
11 (TPa) ECNT
22 (TPa) GCNT
12 (TPa) aCNT
11 (10À6
/K) aCNT
22 (10À6
/K)
300 5.6466 7.0800 1.9445 3.4584 5.1682
500 5.5308 6.9348 1.9643 4.5361 5.0189
700 5.4744 6.8641 1.9644 4.6677 4.8943
Table 2
Effects of CNT volume fraction VÃ
CNT and width-to-thickness ratio (b/h) on the non-dimensional central deflection w ¼ Àw0=h for CNTRC plates under a uniformly distributed load
q0 = À1.0 Â 105
N/m2
.
VÃ
CNT b/h SSSS CCCC SCSC SFSF
Present ANSYS Present ANSYS Present ANSYS Present ANSYS
0.11 10 UD 3.739 Â 10À3
3.739 Â 10À3
2.228 Â 10À3
2.227 Â 10À3
3.325 Â 10À3
3.325 Â 10À3
3.444 Â 10À3
3.445 Â 10À3
FG-V 4.466 Â 10À3
4.461 Â 10À3
2.351 Â 10À3
2.351 Â 10À3
3.853 Â 10À3
3.848 Â 10À3
4.186 Â 10À3
4.188 Â 10À3
FG-O 5.230 Â 10À3
5.216 Â 10À3
2.512 Â 10À3
2.506 Â 10À3
4.433 Â 10À3
4.422 Â 10À3
4.967 Â 10À3
4.960 Â 10À3
FG-X 3.177 Â 10À3
3.176 Â 10À3
2.109 Â 10À3
2.104 Â 10À3
2.867 Â 10À3
2.864 Â 10À3
2.905 Â 10À3
2.909 Â 10À3
20 UD 3.628 Â 10À2
3.629 Â 10À2
1.339 Â 10À2
1.338 Â 10À2
3.393 Â 10À2
3395 Â 10À2
3.341 Â 10À2
3.341 Â 10À2
FG-V 4.879 Â 10À2
4.876 Â 10À2
1.593 Â 10À2
1.591 Â 10À2
4.381 Â 10À2
4.375 Â 10À2
4.544 Â 10À2
4.547 Â 10À2
FG-O 6.155 Â 10À2
6.136 Â 10À2
1.860 Â 10À2
1.856 Â 10À2
5.389 Â 10À2
5.372 Â 10À2
5.797 Â 10À2
5.786 Â 10À2
FG-X 2.701 Â 10À2
2.703 Â 10À2
1.150 Â 10À2
1.150 Â 10À2
2.587 Â 10À2
2.586 Â 10À2
2.484 Â 10À2
2.490 Â 10À2
50 UD 1.155 1.155 0.2618 0.2618 1.099 1.100 1.068 1.068
FG-V 1.653 1.652 0.3649 0.3653 1.504 1.502 1.540 1.541
FG-O 2.157 2.150 0.4719 0.4705 1.909 1.903 2.030 2.025
FG-X 0.7900 0.7915 0.1894 0.1900 0.7728 0.7732 0.7338 0.7361
0.14 10 UD 3.306 Â 10À3
3.305 Â 10À3
2.087 Â 10À3
2.086 Â 10À3
2.974 Â 10À3
2.974 Â 10À3
3.029 Â 10À3
3.029 Â 10À3
FG-V 3.894 Â 10À3
3.889 Â 10À3
2.182 Â 10À3
2.177 Â 10À3
3.412 Â 10À3
3.407 Â 10À3
3.615 Â 10À3
3.616 Â 10À3
FG-O 4.525 Â 10À3
4.512 Â 10À3
2.313 Â 10À3
2.307 Â 10À3
3.916 Â 10À3
3.905 Â 10À3
4.236 Â 10À3
4.230 Â 10À3
FG-X 2.844 Â 10À3
2.842 Â 10À3
1.979 Â 10À3
1.975 Â 10À3
2.583 Â 10À3
2.580 Â 10À3
2.593 Â 10À3
2.596 Â 10À3
20 UD 3.001 Â 10À2
3.002 Â 10À2
1.188 Â 10À2
1.188 Â 10À2
2.852 Â 10À2
2.854 Â 10À2
2.760 Â 10À2
2.760 Â 10À2
FG-V 4.025 Â 10À2
4.021 Â 10À2
1.390 Â 10À2
1.388 Â 10À2
3.689 Â 10À2
3.683 Â 10À2
3.722 Â 10À2
3.724 Â 10À2
FG-O 5.070 Â 10À2
5.053 Â 10À2
1.604 Â 10À2
1.600 Â 10À2
4.557 Â 10À2
4.541 Â 10À2
4.719 Â 10À2
4.709 Â 10À2
FG-X 2.256 Â 10À2
2.258 Â 10À2
1.036 Â 10À2
1.035 Â 10À2
2.184 Â 10À2
2.183 Â 10À2
2.078 Â 10À2
2.083 Â 10À2
50 UD 0.9175 0.9182 0.2131 0.2131 0.8890 0.8897 0.8505 0.8503
FG-V 1.326 1.325 0.2955 0.2958 1.234 1.232 1.229 1.229
FG-O 1.738 1.732 0.3805 0.3797 1.583 1.577 1.619 1.615
FG-X 0.6271 0.6284 0.1560 0.1566 0.6206 0.6209 0.5854 0.5872
0.17 10 UD 2.394 Â 10À3
2.394 Â 10À3
1.412 Â 10À3
1.411 Â 10À3
2.124 Â 10À3
2.124 Â 10À3
2.207 Â 10À3
2.207 Â 10À3
FG-V 2.864 Â 10À3
2.861 Â 10À3
1.486 Â 10À3
1.483 Â 10À3
2.461 Â 10À3
2.458 Â 10À3
2.691 Â 10À3
2.693 Â 10À3
FG-O 3.378 Â 10À3
3.368 Â 10À3
1.595 Â 10À3
1.591 Â 10À3
2.865 Â 10À3
2.858 Â 10À3
3.206 Â 10À3
3.202 Â 10À3
FG-X 2.012 Â 10À3
2.011 Â 10À3
1.318 Â 10À3
1.316 Â 10À3
1.804 Â 10À3
1.802 Â 10À3
1.842 Â 10À3
1.845 Â 10À3
20 UD 2.348 Â 10À2
2.349 Â 10À2
8.561 Â 10À3
8.560 Â 10À3
2.190 Â 10À2
2.191 Â 10À2
2.162 Â 10À2
2.162 Â 10À2
FG-V 3.174 Â 10À2
3.171 Â 10À2
1.021 Â 10À2
1.020 Â 10À2
2.834 Â 10À2
2.831 Â 10À2
2.961 Â 10À2
2.963 Â 10À2
FG-O 4.020 Â 10À2
4.007 Â 10À2
1.198 Â 10À2
1.195 Â 10À2
3.518 Â 10À2
3.507 Â 10À2
3.787 Â 10À2
3.779 Â 10À2
FG-X 1.737 Â 10À2
1.738 Â 10À2
7.290 Â 10À3
7.285 Â 10À3
1.653 Â 10À2
1.653 Â 10À2
1.596 Â 10À2
1.600 Â 10À2
50 UD 0.7515 0.7521 0.1698 0.1699 0.7135 0.7140 0.6950 0.6951
FG-V 1.082 1.081 0.2384 0.2386 0.9790 0.9780 1.010 1.010
FG-O 1.416 1.411 0.3085 0.3079 1.252 1.248 1.333 1.329
FG-X 0.5132 0.5141 0.1223 0.1227 0.4990 0.4993 0.4758 0.4772
P. Zhu et al. / Composite Structures 94 (2012) 1450–1460 1453

5. Ke
¼ Ke
m þ Ke
b þ Ke
s ð9aÞ
where
Ke
m ¼
Z
Xe
ðBmÞT
ABm dX þ
Z
Xe
ðBmÞT
BBb dX þ
Z
Xe
ðBbÞT
BBm dX ð9bÞ
Ke
b ¼
Z
Xe
ðBbÞT
DBb dX ð9cÞ
Ke
s ¼
Z
Xe
ðBsÞT
AsBs dX ð9dÞ
Bm ¼
wi;x 0 0 0 0
0 wi;y 0 0 0
wi;y wi;x 0 0 0
2
6
4
3
7
5; Bb ¼
0 0 0 wi;x 0
0 0 0 0 wi;y
0 0 0 wi;y wi;x
2
6
4
3
7
5;
Bs ¼
0 0 wi;y 0 wi
0 0 wi;x wi 0
#
ð9eÞ
in which the components of the extensional stiffness A, bending-
extensional coupling stiffness B, bending stiffness D, and transverse
shear stiffness As are defined by
ðAij; Bij; DijÞ ¼
Z h=2
Àh=2
Qijð1; z; z2
Þdz; As
ij ¼ K
Z h=2
Àh=2
Qij dz ð10aÞ
where Aij, Bij and Dij are valid for i, j = 1, 2, 6, and As
ij for i, j = 4, 5
according to the Voigt notation. K denotes the transverse shear cor-
rection coefficient for FGM, given by [37]
K ¼
5
6 À ðmcVc þ mmVmÞ
ð10bÞ
The load vector of the element Xe
, Fe
is written as
Fe
¼
Z
Xe
½We
ŠT
qdX þ
Z
Xe
ðBmÞT
ðBbÞT
 à NTh
MTh
#
dX ð11aÞ
wiwhere q is the transverse load intensity over the element, and NTh
and MTh
denote the thermal force and thermal moment resultants,
respectively, and are given by
Fig. 2. Non-dimensional axial stress rxx ¼ rxxh
2
=ðjq0ja2
Þ in the CNTRC plates under
a uniform load q0 = À1.0 Â 105
N/m2
(SSSS).
Fig. 3. Non-dimensional axial stress rxx ¼ rxxh
2
=ðjq0ja2
Þ in the CNTRC plates under
a uniform load q0 = À1.0 Â 105
N/m2
(CCCC).
Fig. 4. Non-dimensional axial stress rxx ¼ rxxh
2
=ðjq0ja2
Þ in the CNTRC plates under
a uniform load q0 = À1.0 Â 105
N/m2
(SCSC).
Fig. 5. Non-dimensional axial stress rxx ¼ rxxh
2
=ðjq0ja2
Þ in the CNTRC plates under
a uniform load q0 = À1.0 Â 105
N/m2
(SFSF).
1454 P. Zhu et al. / Composite Structures 94 (2012) 1450–1460

6. NTh
¼
Z h=2
Àh=2
Q11a11 þ Q12a22
Q12a11 þ Q22a22
0
2
6
4
3
7
5DT dz; MTh
¼
Z h=2
Àh=2
Q11a11 þ Q12a22
Q12a11 þ Q22a22
0
2
6
4
3
7
5DTzdz ð11bÞ
3.3. Free vibration of CNTRC plates
In a free vibration analysis, the CNTRC plate is assumed to un-
dergo a harmonic motion and the obtained eigenvalue equations
are
ðK À x2
MÞd ¼ 0 ð12Þ
Table 3
Effects of width-to-thickness ratio and CNT volume fraction on the non–dimensional natural frequency x ¼ xða2
=hÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qm=Em
p
for CNTRC square plates (SSSS).
VÃ
CNT b/h Modes (m,n) UD FG-V FG-O FG-X
Present ANSYS Present ANSYS Present ANSYS Present ANSYS
0.11 10 (1,1) 13.532 13.521 12.452 12.495 11.550 11.600 14.616 14.659
(1,2) 17.700 17.658 17.060 17.210 16.265 16.409 18.646 18.824
(1,3) 19.449 19.447 19.499 19.479 19.499 19.479 19.499 19.479
(1,4) 19.449 19.447 19.499 19.479 19.499 19.479 19.499 19.479
(2,1) 27.569 27.325 27.340 27.524 26.513 26.687 28.519 28.742
(2,2) 32.563 32.488 31.417 31.398 30.280 30.274 33.598 33.584
20 (1,1) 17.355 17.328 15.110 15.103 13.523 13.531 19.939 19.916
(1,2) 21.511 21.439 19.903 19.879 18.486 18.464 23.776 23.773
(1,3) 32.399 32.063 31.561 31.233 30.166 29.830 34.389 34.123
(1,4) 38.898 38.894 38.998 38.958 38.998 38.958 38.998 38.958
(2,1) 38.898 38.894 38.998 38.958 38.998 38.958 38.998 38.958
(2,2) 50.199 49.122 47.739 47.470 43.948 43.732 52.268 51.078
50 (1,1) 19.223 19.184 16.252 16.216 14.302 14.290 22.984 22.910
(1,2) 23.408 23.310 21.142 21.030 19.373 19.274 26.784 26.660
(1,3) 34.669 34.272 33.350 32.740 31.615 31.013 37.591 37.016
(1,4) 54.043 52.770 53.430 51.354 51.370 49.326 56.946 54.912
(2,1) 70.811 70.363 60.188 59.584 53.035 52.569 83.150 79.630
(2,2) 72.900 72.395 62.780 62.198 55.823 55.362 84.896 82.297
0.14 10 (1,1) 14.306 14.296 13.256 13.300 12.338 12.391 15.368 15.413
(1,2) 18.362 18.325 17.734 17.894 16.848 17.003 19.385 19.573
(1,3) 19.791 19.790 19.879 19.858 19.879 19.859 19.879 19.859
(1,4) 19.791 19.790 19.879 19.858 19.879 19.859 19.879 19.859
(2,1) 28.230 27.989 28.021 28.218 27.003 27.193 29.398 29.632
(2,2) 33.646 33.573 32.678 32.660 31.633 31.624 34.631 34.621
20 (1,1) 18.921 18.893 16.510 16.503 14.784 14.794 21.64247 21.620
(1,2) 22.867 22.796 21.087 21.069 19.462 19.449 25.35981 25.365
(1,3) 33.570 33.237 32.617 32.299 30.906 30.584 35.93751 35.684
(1,4) 39.583 39.579 39.759 39.717 39.759 39.717 39.75897 39.717
(2,1) 39.583 39.579 39.759 39.717 39.759 39.717 39.75897 39.717
(2,2) 51.422 50.340 51.078 49.813 47.341 47.113 54.06184 52.866
50 (1,1) 21.354 21.311 17.995 17.956 15.801 15.788 25.555 25.474
(1,2) 25.295 25.192 22.643 22.531 20.563 20.469 29.192 29.065
(1,3) 36.267 35.866 34.660 34.052 32.509 31.918 39.833 39.257
(1,4) 55.608 54.320 54.833 52.739 52.184 50.145 59.333 57.272
(2,1) 78.110 77.629 66.552 65.893 58.748 58.237 87.814 82.437
(2,2) 80.015 79.482 68.940 68.315 61.277 60.782 91.299 90.389
0.17 10 (1,1) 16.815 16.801 15.461 15.514 14.282 14.345 18.278 18.330
(1,2) 22.063 22.011 21.307 21.490 20.091 20.269 23.541 23.760
(1,3) 24.337 24.335 24.511 24.485 24.512 24.486 24.512 24.486
(1,4) 24.337 24.335 24.511 24.485 24.512 24.486 24.512 24.486
(2,1) 34.448 34.141 34.273 34.494 32.766 32.983 36.245 36.511
(2,2) 40.630 40.535 39.263 39.237 37.763 37.755 42.150 42.131
20 (1,1) 21.456 21.422 18.638 18.628 16.628 16.640 24.764 24.735
(1,2) 26.706 26.615 24.734 24.699 22.739 22.710 29.819 29.809
(1,3) 40.401 39.977 39.471 39.048 37.139 36.719 43.612 43.257
(1,4) 48.674 48.669 49.023 48.972 49.024 48.972 49.024 48.972
(2,1) 48.674 48.669 49.023 48.972 49.024 48.972 49.024 48.972
(2,2) 62.723 61.372 59.191 58.855 54.367 54.099 66.616 65.074
50 (1,1) 23.697 23.649 19.982 19.938 17.544 17.529 28.413 28.322
(1,2) 28.987 28.865 26.204 26.061 23.783 23.659 33.434 33.274
(1,3) 43.165 42.667 41.646 40.871 38.855 38.109 47.547 46.797
(1,4) 67.475 65.880 66.943 64.318 63.179 60.652 72.570 69.940
(2,1) 87.385 86.830 74.030 73.285 65.154 64.581 102.939 101.739
(2,2) 90.031 89.403 77.343 76.622 68.579 68.011 105.334 101.877
P. Zhu et al. / Composite Structures 94 (2012) 1450–1460 1455

7. where the global stiffness matrix K is the same as that in Eq. (8) and
the global mass matrix is computed by assembling the element
mass matrix given by
Me
¼
Z
Xe
½We
ŠT
½qŠ½We
ŠdX ð13aÞ
where
½qŠ ¼
I0 0 0 I1 0
0 I0 0 0 I1
0 0 I0 0 0
I1 0 0 I2 0
0 I1 0 0 I2
2
6
6
6
6
6
6
4
3
7
7
7
7
7
7
5
ð13bÞ
in which I0, I1 and I2 are the normal, coupled normal-rotary and ro-
tary inertial coefficients, respectively and defined by
Table 4
Effects of width-to-thickness ratio and CNT volume fraction on the non-dimensional natural frequency x ¼ xða2
=hÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
qm=Em
p
for CNTRC square plates (CCCC).
VÃ
CNT b/h Modes (m,n) UD FG-V FG-O FG-X
Present ANSYS Present ANSYS Present ANSYS Present ANSYS
0.11 10 (1,1) 17.625 17.616 17.211 17.247 16.707 16.744 18.083 18.129
(1,2) 23.041 22.968 22.818 22.922 22.253 22.353 23.606 23.728
(1,3) 33.592 33.244 33.070 33.060 32.378 32.372 34.338 34.338
(1,4) 33.729 33.659 33.552 33.534 32.857 32.837 34.467 34.464
(2,1) 37.011 36.970 36.528 36.804 35.809 36.083 37.447 37.400
(2,2) 37.317 37.319 37.437 37.394 37.447 37.400 37.786 38.084
20 (1,1) 28.400 28.354 26.304 26.248 24.486 24.443 30.421 30.380
(1,2) 33.114 33.012 31.496 31.405 29.795 29.699 35.036 35.003
(1,3) 44.559 44.070 43.589 42.991 41.895 41.279 46.480 45.984
(1,4) 59.198 58.978 56.249 55.894 53.557 53.204 61.980 61.692
(2,1) 61.851 61.556 59.221 59.090 56.617 56.464 64.562 63.321
(2,2) 63.043 61.709 62.608 60.643 60.719 58.742 65.174 64.551
50 (1,1) 39.730 39.580 34.165 33.967 30.303 30.161 46.166 45.896
(1,2) 43.876 43.633 39.043 38.727 35.444 35.163 49.934 49.609
(1,3) 54.768 54.076 51.204 50.129 47.878 46.811 60.225 59.271
(1,4) 74.488 72.523 72.202 68.973 68.842 65.631 79.534 76.535
(2,1) 98.291 97.473 86.291 84.997 77.468 76.338 108.694 101.276
(2,2) 100.537 98.942 89.054 87.839 80.460 79.376 110.921 109.493
0.14 10 (1,1) 18.127 18.119 17.791 17.832 17.311 17.351 18.593 18.642
(1,2) 23.572 23.501 23.413 23.525 22.782 22.893 24.243 24.369
(1,3) 34.252 33.901 34.101 34.094 33.411 33.404 35.224 35.224
(1,4) 34.650 34.582 34.275 34.262 33.441 33.438 35.411 35.410
(2,1) 37.921 37.885 37.538 37.827 36.788 37.080 38.169 38.121
(2,2) 37.972 37.975 38.159 38.114 38.169 38.121 38.789 39.095
20 (1,1) 29.911 29.870 27.926 27.875 26.127 26.087 31.857 31.824
(1,2) 34.516 34.426 32.976 32.900 31.186 31.107 36.487 36.469
(1,3) 45.898 45.426 44.989 44.411 43.034 42.448 48.087 47.602
(1,4) 61.628 61.420 58.951 58.610 56.403 56.060 64.334 64.059
(2,1) 64.199 63.013 61.816 61.717 59.277 59.155 66.912 65.275
(2,2) 64.496 64.082 64.135 62.170 61.793 59.839 67.148 66.928
50 (1,1) 43.583 43.426 37.568 37.357 33.369 33.217 50.403 50.125
(1,2) 47.479 47.237 42.175 41.857 38.145 37.869 54.025 53.706
(1,3) 57.968 57.288 53.963 52.906 50.055 49.020 64.112 63.184
(1,4) 77.395 75.445 74.785 71.567 70.646 67.477 83.394 80.410
(2,1) 106.371 101.841 94.022 92.669 84.799 83.601 112.896 105.403
(2,2) 106.487 105.539 96.573 95.324 87.511 86.383 119.134 117.701
0.17 10 (1,1) 22.011 21.998 21.544 21.587 20.833 20.879 22.748 22.804
(1,2) 28.801 28.708 28.613 28.738 27.651 27.776 29.878 30.023
(1,3) 42.015 41.578 41.431 41.418 40.501 40.493 43.293 43.287
(1,4) 42.132 42.044 42.119 42.087 40.781 40.759 43.588 43.572
(2,1) 46.250 46.197 45.796 46.136 44.699 45.046 47.071 47.009
(2,2) 46.694 46.697 47.055 46.996 47.071 47.009 47.606 47.969
20 (1,1) 35.316 35.257 32.686 32.613 30.325 30.270 38.062 38.008
(1,2) 41.253 41.122 39.279 39.156 36.848 36.726 44.105 44.050
(1,3) 55.627 55.011 54.560 53.791 51.757 50.989 58.927 58.269
(1,4) 73.769 73.490 70.149 69.696 66.657 66.208 77.640 77.271
(2,1) 77.109 76.926 73.926 73.748 70.401 70.200 81.040 80.539
(2,2) 78.801 76.932 78.522 76.025 75.018 72.558 82.932 81.012
50 (1,1) 49.074 48.888 42.078 41.833 37.247 37.073 57.245 56.907
(1,2) 54.324 54.019 48.309 47.908 43.577 43.228 62.236 61.817
(1,3) 68.069 67.198 63.755 62.387 58.890 57.568 75.746 74.496
(1,4) 92.868 90.400 90.293 86.200 84.717 80.745 100.850 96.950
(2,1) 121.669 120.647 106.513 104.900 95.462 94.059 137.913 128.908
(2,2) 124.518 123.489 110.055 108.533 99.142 97.793 138.485 136.114
1456 P. Zhu et al. / Composite Structures 94 (2012) 1450–1460

8. ðI0; I1; I2Þ ¼
Z h=2
Àh=2
qðzÞð1; z; z2
Þdz ð13cÞ
4. Numerical results
In this section, the bending and free vibration analyses are pre-
sented to investigate the mechanical characteristics of the CNTRC
plates by several numerical examples. PmPV [38] is considered as
the matrix, the material properties of which are assumed to be
mm
= 0.34, qm
= 1.15 g/cm3
, and Em
= 2.1 GPa at room temperature
(300 K). In the present study, the armchair (10,10) SWCNTs are
chosen as the reinforcements. According to the MD simulation re-
sults of Zhang and Shen [39], the material properties of the SWCNT
are listed in Table 1. The CNT efficiency parameters gj introduced in
Eq. (1a) through Eq. (1c) can be determined according to the effec-
tive properties of CNTRCs available (e.g. in [38]) by matching the
Young’s moduli E11 and E22 with the counterparts computed by
the rule of mixture. For example, g1 = 0.149 and g2 = 0.934 for
the case of VÃ
CNT ¼ 0:11, and g1 = 0.150 and g2 = 0.941 for the case
of VÃ
CNT ¼ 0:14, and g1 = 0.149 and g2 = 1.381 for the case of
VÃ
CNT ¼ 0:17. In addition, we assume that g3 = g2 and
G23 = G13 = G12. These values will be used in the following numeri-
cal examples.
4.1. Bending of CNTRC plates
Several numerical examples are provided to investigate the
bending analysis of CNTRC plates under mechanical loading, in
which the width-to-thickness ratios (b/h) of plates are set to 10,
20 and 50, and the thickness is taken to be 2.0 mm. Four types of
square plates (a/b = 1), UD-CNTRC, FG-V, FG-O and FG-X CNTRC,
are considered with four kinds of boundary conditions – all edges
simply supported (SSSS) or clamped (CCCC), and two opposite
edges simply supported and the other two clamped (SCSC) or free
(SFSF). The boundary conditions are defined as following:
us ¼ w0 ¼ us ¼ 0 ðSimply supported edgeÞ
un ¼ us ¼ w0 ¼ un ¼ us ¼ 0 ðClamped edgeÞ
ð14Þ
(a) UD CNTRC plate (b) FG-V CNTRC plate
(c) FG-O CNTRC plate (d) FG-X CNTRC plate
Fig. 6. Variation of the fundamental frequency parameter x with the CNT volume fraction VÃ
CNT for CNTRC plates. (a) UD CNTRC plate; (b) FG-V CNTRC plate; (c) FG-O CNTRC
plate; (d) FG-X CNTRC plate.
P. Zhu et al. / Composite Structures 94 (2012) 1450–1460 1457

9. where the subscripts n and s denote the normal and tangential
directions, respectively, on the boundaries. The applied uniformly
distributed load q0 is À0.1 MPa. The various non-dimensional
parameters are defined as
Central deflection : w ¼
w0
h
ð15aÞ
Central axial stress : rxx ¼
rxxh
2
jq0ja2
ð15bÞ
(a) (b)
(c) (d)
(e) (f)
Fig. 7. The free vibration mode shapes of a SSSS square CNTRC plate. (a) 1st Mode (m = 1, n = 1); (b) 2nd Mode (m = 1, n = 2); (c) 3rd Mode (m = 1, n = 3); (d) 4th Mode (m = 1,
n = 4); (e) 5th Mode (m = 2, n = 1); (f) 6th Mode (m = 2, n = 2).
1458 P. Zhu et al. / Composite Structures 94 (2012) 1450–1460

10. Thickness coordinate : z ¼
z
h
ð15cÞ
Table 2 shows the non-dimensional central deflection w for the
UD and three types of FG CNTRC square plates subject to a uniform
transverse load q0 with different values of CNT volume fraction
VÃ
CNT (=0.11, 0.14 and 0.17), various values of width-to-thickness
ratio (b/h) and different boundary conditions. Note that we assume
that the UD-CNTRC and FG-CNTRC plates have the same overall
mass fraction wCNT of the carbon nanotube for the purpose of com-
parisons. The results alongside given by ANSYS (SHELL181 element
for UD-CNTRC plates and SHELL99 element for FG-CNTRC plates)
are provided to demonstrate the accuracy of the FE model used
in the present study. It can be found that the two groups of results
agree very well. It can be observed that the volume fraction of the
CNT has so much influence on the central deflection of the plates
that only 6% increase in the volume fraction of CNT may lead to
more than 30% decrease in the central deflection. It is noticeable
that the central deflections of FG-V and FG-O CNTRC plates are lar-
ger than deflections of UD-CNTRC plates while those of FG-X
CNTRC plate are smaller though these four types of plates have
the same mass fraction of the CNT. This is because that the form
of distribution of reinforcements can affect the stiffness of plates
and it is thus expected that the desired stiffness can be achieved
by adjusting the distribution of CNTs along the thickness direction
of plates. It is concluded that reinforcements distributed close to
top and bottom are more efficient than those distributed nearby
the mid-plane for increasing the stiffness of plates.
According to the results, the CNT volume fraction VÃ
CNT has little
effect on the central axial stresses of CNTRC plates. Fig. 2 depicts
the non-dimensional central axial stresses rxx distributed along
the thickness direction for various CNTRC plates subjected to a uni-
form transverse load q0 with the volume fraction VÃ
CNT ¼ 0:17. The
central axial stress distributions in UD, FG-O and FG-X CNTRC
plates possess anti-symmetry due to the reinforcements being
symmetric with respect to the mid-plane. For FG-V CNTRC plates,
the axial stress is close to zero at the bottom where the concentra-
tion of CNT vanishes and the same case exists at the top and bot-
tom for FG-O CNTRC plates. Similar trends can also be observed
from Figs. 3–5 for the plates with CCCC, SCSC and SFSF boundary
conditions. The values of the axial stresses of CCCC CNTRC plates
are much smaller than those of the plates with the other three
boundary conditions since the CCCC plates have the smallest cen-
tral deflections. The distributions and values of the central axial
stresses rxx are very close for the SSSS, SCSC and SFSF CNTRC plates
because of the identical boundary condition along x direction.
4.2. Free vibration of CNTRC plates
The dynamic characteristics of various CNTRC plates are pre-
sented in this section. The material and geometric properties are
the same as those for bending analyses. The non-dimensional fre-
quency parameter x is defined as
x ¼ xmn
a2
h
ffiffiffiffiffiffiffi
qm
Em
r
ð16Þ
where xmn is the natural frequency of the CNTRC plates. The sub-
scripts m and n are the number of half-waves in the mode shapes
in the x and y directions, respectively. The frequency parameters
of various types of CNTRC plates with different width-to-thickness
ratios and CNT volume fractions are listed in Tables 3 and 4 com-
pared with those that are calculated by ANSYS (SHELL181 element
for UD-CNTRC plates and SHELL99 element for FG-CNTRC plates).
The natural frequencies of the CNTRC plates are subject to the sim-
ilar influence from width-to-thickness ratio and volume fraction of
CNT observed in the previous bending analyses, however the
variations of natural frequency exhibit opposite trends relative to
the changes of central deflections.
Fig. 6 illustrates the effect of the CNT volume fraction on the
fundamental frequency parameters of SSSS and CCCC CNTRC
plates. For different types of distribution of the CNT, the effect of
the volume fraction is nearly linear on the variation of fundamental
frequency parameter. When the width-to-thickness ratio increases,
the incremental changes of frequency parameters of CCCC plates
are much larger than those for SSSS plates, which may attribute
to CCCC CNTRC plates possessing stronger constraints and thus
being sensitive to the change of b/h. And with the increase in the
value of b/h, SSSS CNTRC plates obviously become less sensitive
to the change of the width-to-thickness ratio.
Fig. 7 gives the first six mode shapes of a SSSS FG-V CNTRC plate
(b/h = 50). Since the reinforcement only aligns in x direction, the
mechanical properties of transverse direction (in y axis) of CNTRC
plates are weaker and then the mode sequence is dissimilar to that
for an isotropic plate, compared with the modes shapes given in
[40]. The mode shapes (m = 2, n = 1) and (m = 2, n = 2) become
higher order over (m = 1, n = 3) and (m = 1, n = 4) because the dif-
ference of mechanical properties between the longitudinal and
transverse directions of CNTRC plates exists.
5. Conclusion
In this paper, bending and free vibration analyses of carbon
nanotube-reinforced composite plates are investigated by a finite
element modal. The effective material properties of the CNTRC
are estimated by the rule of mixture with introducing CNT effi-
ciency parameters gj accounting for the size-dependence. Numer-
ical examples are provided to present the static and dynamic
characteristics of CNTRC plates with different distributions of rein-
forcement, width-to-thickness ratios and boundary conditions. The
results by the present finite element model agree well with those
obtained by using the commercial package ANSYS. It is concluded
that for the bending analysis, the CNT volume fraction, the width-
to-thickness ratio and the boundary condition significantly influ-
ence the bending deflection, while the effect of the CNT volume
fraction on the central axial stress can be neglected. For the free
vibration analysis, it is found that both the CNT volume fraction
and the width-to-thickness ratio have pronounced effect on the
natural frequencies and vibration mode shapes of the CNTRC plate.
Acknowledgements
The work described in this paper was fully supported by Grants
from the Research Grants Council of the Hong Kong Special Admin-
istrative Region, China [Project No. 9041674, CityU 118411] and
National Science Foundation of China [Grant No. 11172253].
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