Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 4, pp. 1005-1018, Warsaw 2015
DOI: 10.15632/jtam-pl.53.4.1005

FREE VIBRATION ANALYSIS OF THICK DISKS WITH VARIABLE
THICKNESS CONTAINING ORTHOTROPIC-NONHOMOGENEOUS

MATERIAL USING FINITE ELEMENT METHOD

Hassan Zafarmand, Mehran Kadkhodayan
Ferdowsi University of Mashhad, Department of Mechanical Engineering, Mashhad, Iran

e-mail: kadkhoda@um.ac.ir

Free vibration analysis of thick functionally graded nanocomposite annular and solid disks
with variable thickness reinforced by single-walled carbon nanotubes (SWCNTs) is presen-
ted. Four types of distribution of uniaxial aligned SWCNTs are considered: uniform and
three kinds of functionally graded (FG) distribution through radial direction of the disk.
The effective material properties of the nanocomposite disk are estimated by a micro me-
chanical model. The axisymmetric conditions are assumed and employing the graded finite
element method (GFEM), the equations are solved. The solution is considered for four dif-
ferent thickness profiles, namely constant, linear, concave and convex. The achieved results
show that the type of distribution and volume fraction of CNTs and thickness profile have
a great effect on normalized natural frequencies.

Keywords: free vibration, carbonnanotube, functionally gradedmaterial, thickdisk, variable
thickness, graded finite elementmethod

1. Introduction

In the recent years, nano-structured materials such as nanocomposites, have generated consi-
derable interest in the material research community and became an attractive new subject in
material science due to their potentially impressive mechanical properties. Carbon nanotubes
(CNTs) have illustrated remarkablemechanical, thermal and electrical properties. For example,
they could potentially have a Young’s modulus as high as 1TPa and a tensile strength appro-
aching 100GPa (Odegard et al., 2002). These enormous advantages make them highly desirable
candidates for the reinforcement of polymer composites, provided that good interfacial bonding
between CNTs and polymer and proper dispersion of the individual CNTs in the polymeric
matrix can be assured (Fiedler et al., 2006).
Themajority of researches performed on carbon nanotube reinforced composites (CNTRCs)

are focused on their material properties (Esawi and Farag, 2007; Thostenson et al., 2001; Dai,
2006;Kang et al., 2006; Lau et al., 2006).HanandElliott (2007), by theuse ofmolecular dynamic
simulation (MD), obtained the elastic modulus of composite structures reinforced byCNTs and
studied the effect of volume fraction of SWCNTs on mechanical properties of nanocomposites.
Hu et al. (2005), by analyzing the elastic deformation of a representative volume element (RVE)
under various loading conditions, evaluated themacroscopic elastic properties of CNTRCs. Zhu
et al. (2007) studied the effect of CNTs on the mechanical properties of polymeric composi-
tes. Their results showed that adding CNTs could greatly improve Young’s modulus. Due to
dependency of the interaction at the polymer and nanotube interface on the local molecular
structure and bonding, Odegard et al. (2003) a constitutive model for CNTRCs by utilizing an
equivalent-continuum modeling method proposed.
Functionally graded materials (FGMs) are special composite materials, microscopically in-

homogeneous, in whichmechanical properties vary smoothly and continuously from one surface



1006 H. Zafarmand,M. Kadkhodayan

to the other. This idea was used for the first time by Japanese researchers (Koizumi, 1993)
and led to the concept of FGMs. A wide range of researches have been carried out on FGMs
in various fields of mechanics. Motivated by the concept of FGMs, Shen (2009) presented a
kind of CNTRCs in which the volume fraction of CNTs is graded with certain rules through
desired directions and demonstrated that usingFG-CNTRCs improve themechanical properties
of the structures. Zhu et al. (2012) studied bending and free vibration analyses of composite
plates reinforced by SWCNTs using the finite element method based on the first order shear
deformation plate theory. The effective material properties of the FG-CNTRC are graded in the
thickness direction and are estimated according to the rule ofmixture. Zafarmand andKadkho-
dayan (2015) investigated nonlinear behaviour of FG-CNTRC rotating thick disks with variable
thickness where the nonlinear axisymmetric theory of elasticity is employed. The three dimen-
sional free vibration analysis of FC-CNTRC panels was investigated by Yas et al. (2013). The
boundary conditions were assumed to be simply supported and the equations were solved by
a generalized differential quadrature (GDQ) method. Ke et al. (2010) performed nonlinear free
vibration analysis of FG-CNTRC beams based on Timoshenko beam theory and Von-Karman
geometric nonlinearity. TheRitzmethodwas applied to derive the governing eigenvalue equation
whichwas then solved by a direct iterativemethod to obtain the nonlinear vibration frequencies
of FG-CNTRC beams with different end supports. Sobhani Aragh and Yas (2010) investigated
the static and free vibration characteristics of continuously graded fiber-reinforced (CGFR) cy-
lindrical shells based on three dimensional elasticity. The boundary conditions were assumed
to be simply supported and the equations were solved by a GDQ method. Moreover, several
researches were carried out about free vibration analysis of FG disks with variable thickness
(Alipour et al., 2010; Gupta et al., 2007; Efraim and Eisenberger, 2007; Tajeddini and Ohadi,
2011).
The purpose of this paper is to investigate free vibration analysis of thick FG-CNTRCannu-

lar and solid diskswith variable thickness.Material properties are assumed to vary continuously
through radial direction. The effective material properties of FG-CNTRC disks are estimated
using a micro-mechanical model and the normalized natural frequencies of FG-CNTRC annu-
lar and solid disks for various types of distributions and volume fractions of CNTs, boundary
conditions and thickness to radius ratios as well as different kinds of thickness profiles are com-
puted and compared. The difficulty in obtaining analytical solutions for the response of graded
material systems is due to the dispersive nature of heterogeneous material systems. Therefore,
analytical or semi-analytical solutions are available only through a number of problems with
simple boundary conditions. Thus, in order to find the solution for a thick FG-CNTRC disk of
variable thickness with various boundary conditions, powerful numerical methods such as the
graded finite element method (GFEM) are needed. The graded finite element incorporates the
gradient ofmaterial properties at the element scale in the framework of a generalized isoparame-
tric formulation. Some studies can be found in the literature on themodeling of non-homogenous
structures by using GFEM (Kim and Paulino, 2002; Zafarmand and Hassani, 2014; Ashrafi et
al., 2013).

2. Problem formulation

In this Section, different types of CNTs distributions through radial direction of the disk is
introduced. The axisymmetric governing equations ofmotion are obtained and the graded finite
element method is used for modeling the non-homogeneity of the material.

2.1. Material properties in FG-CNTRC disks

A thick FG-CNTRC disk of inner radius a, outer radius b and variable thickness h(r) is
considered. The geometry and coordinate system of the disk is shown in Fig. 1.



Free vibration analysis of thick disks with variable thickness... 1007

Fig. 1. Axisymmetric thick FG-CNTRC disk

This FG-CNTRC disk consists of SWCNTs (along the radial direction) and an isotropic
matrix. UD-CNTRC represents the uniform distribution and FG V, FG O and FG X-CNTRC,
the functionally graded distribution of CNTs in the radial direction of the nanocomposite disk.
Several studies have been published each with different focuses on mechanical properties of
CNTRCs. However, due to the simplicity and convenience, in the present study, the rule of
mixture is employed, and thus the effective material properties of CNTRCdisk can be obtained
as (Shen, 2009)

E1 = η1VCNTE
CNT
1 +VmE

m η2
Ei
=
VCNT
ECNTi

+
Vm
Em

(i =2,3)

η3
Gij
=
VCNT
GCNTij

+
Vm
Gm

νij = VCNTν
CNT
ij +Vmν

m (i 6= j)

ρ = VCNTρ
CNT +Vmρ

m

(2.1)

whereECNTi , G
CNT
ij , ν

CNT
ij and ρ

CNT are elasticitymodulus, shearmodulus,Poisson’s ratio and
density, respectively, of the CNTs, and Em, Gm, νm and ρm are the corresponding properties
of the matrix. ηj (j = 1,2,3) is the CNTs’ efficiency parameter which can be computed by
matching the elastic modulus of CNTRCs observed from theMD simulation results with those
obtained from the rule of mixture. VCNT and Vm are volume fractions of the CNTs andmatrix,
respectively, which are related by VCNT +Vm =1. The type of distribution and volume fraction
of the CNTs has serious effects on the disk behavior. As has been mentioned previously, four
types of distribution are utilized in this study; that is either uniformly distributed (UD) or
functionally graded (FG) in the radial direction of the disk. These types with the distribution
for a section in the r-z plane of the disk with a constant thickness profile are depicted in Fig. 2
(the dash lines are CNTs in a huge scale). Distributions of CNTs through the radial direction
of these four types of CNTRC disks are assumed to be as (Shen, 2009)

VCNT = V ∗CNT type UD

VCNT =2
r−a
b−a

V ∗CNT type FG V

VCNT =2
b−r
b−a

V ∗CNT type FG O

VCNT =4
∣∣∣
r−a
b−a

∣∣∣V ∗CNT type FG X

(2.2)

in which

rm =
a+ b
2

V ∗CNT =
wCNT

wCNT +
ρCNT
ρm
(1−wCNT)

(2.3)



1008 H. Zafarmand,M. Kadkhodayan

wherewCNT is themass fraction ofCNTswhich does not dependon the variable r and is unique
for all types of distribution. ρCNT and ρm are the densities of CNTs andmatrix, respectively.

Fig. 2. Different types of distribution: (a) UD, (b) FG V, (c) FG O, (d) FG X

2.2. Governing equations

The governing equations may be obtained based on Hamilton’s principle

t2∫

t1

δI dt =

t2∫

t1

δ(Π −T) dt =0 (2.4)

whereΠ andT arepotential andkinetic energy, respectively.These functionsandtheir variations
are

Π =
1
2

∫∫∫

Ω

εT ·σ dΩ δΠ = 1
2

∫∫∫

Ω

δεT ·σ dΩ

T =
1
2

∫∫∫

Ω

ρU̇T ·U̇ dΩ δT =
1
2

∫∫∫

Ω

ρU̇T ·δU̇ dΩ
(2.5)

whereΩ is the volume of the domain under consideration andρ is themass density that depends
on the r coordinate.
Substituting Eqs. (2.5) into Hamilton’s principle (Eq. (2.4)), applying the side conditions

δU|t1,t2 =0 and using part integration, one obtains
∫∫∫

Ω

δεT ·σ dΩ +
∫∫∫

Ω

ρÜT · δU̇ dΩ =0 (2.6)

The stress-strain relations fromHook’s law in matrix form are as

σ =Dε (2.7)

where the stress and strain components and the coefficients of elasticityD are as in the following
relations



Free vibration analysis of thick disks with variable thickness... 1009

σ =
{
σr σθ σz σrz

}T
ε=

{
εr εθ εz γrz

}T

D=




D11 D12 D13 0
D12 D22 D23 0
D13 D23 D33 0
0 0 0 D55




(2.8)

in which

D11 =
1−ν23ν32
E2E3∆

D22 =
1−ν13ν31
E1E3∆

D33 =
1−ν12ν21
E1E2∆

D12 =
ν21+ν31ν23
E2E3∆

D13 =
ν31+ν21ν32
E2E3∆

D23 =
ν32+ν12ν31
E1E3∆

D55 = G13 ∆ =
1−ν12ν21−ν23ν32−ν13ν31−2ν21ν32ν13

E1E2E3

(2.9)

where Ei, Gij and νij are found from (2.1). It is obvious that thematrixD is dependent on the
spatial variable r.
The strain-displacement equations based on the theory of linear theory of elasticity in cylin-

drical coordinates with the axisymmetric assumption are

εr =
∂u

∂r
εθ =

u

r
εz =

∂w

∂z
γrz =

∂u

∂z
+
∂w

∂r
(2.10)

where u and w are the radial and axial components of the displacement, respectively. Equation
(2.10) can be formulated in the matrix form as

ε=LU (2.11)

in which U is the displacements vector and L is a matrix containing partial differentiating
equations as

U=
{
u v

}T
L=

[
∂r 1/r 0 ∂z
0 0 ∂z ∂r

]T
(2.12)

Moreover, the boundary conditions used in this study are defined as follows:
Solid disk:
• Clamped r = b → u = w =0
• Simply supported r = b → σr = w =0
• Free r = b → σr = σrz =0

Annular disk:
• Clamped r = a,b → u = w =0
• Simply supported r = a,b → σr = w =0
• Free r = a,b → σr = σrz =0

2.3. Graded finite element modeling

In order to solve the governing equations, the isoparametric finite element method with
graded element properties is employed. For this purpose, the variational formulation is consi-
dered. In conventional finite element formulations, a predetermined set of material properties
are used for each element such that the property field is constant within an individual element.
Formodeling a continuously nonhomogeneousmaterial, thematerial property functionmust be



1010 H. Zafarmand,M. Kadkhodayan

discretized according to the size of elements mesh. This approximation can provide significant
discontinuities. Based on these facts, the graded finite element is strongly preferable for the
modeling of the present problem. Using the graded elements for the modeling of gradation of
the material leads to more accurate results than dividing the solution domain into homogenous
elements.
The finite element approximation of the domain is in the r-z plane, which is the plane of

revolution. The section of the cylinder in the r-z plane is considered and divided into a number
of simplex linear quadrilateral elements. For convenience, we use the local coordinate with its
variables (ξ,η) between −1 to 1, as shown in Fig. 3.

Fig. 3. Local coordinate

For element (e), the displacements are approximated as (Zienkiewicz, 2005)

U(e) =ΦΛ(e) (2.13)

where Φ is the matrix of linear shape functions in the local coordinate and Λ(e) is the nodal
displacement vector of the element, which is

Φ=

[
Φ1 0 Φ2 0 Φ3 0 Φ4 0
0 Φ1 0 Φ2 0 Φ3 0 Φ4

]

Λ(e) =
{
U1 V1 U2 V2 U3 V3 U4 V4

}T
(2.14)

in which

Φi =
1
4
(1+ ξiξ)(1+ηiη) (2.15)

To treat the material inhomogeneity by using the GFEM, it may be written

Ψ(e) =
4∑

i=1

ΨiΦi (2.16)

where Ψ(e) is the material property of the element.
Substituting Eq. (2.13) in Eq. (2.11) gives the strain matrix of element (e) as

ε(e) =BΛ(e) (2.17)

where

B=LΦ(e) (2.18)

By imposing Eqs. (2.7), (2.13), (2.17) into Eq. (2.6), it can be achieved as

δΛ(e)
T
(∫∫∫

Ω

BTDB dΩ

)
Λ(e)+ δΛ(e)

T
(∫∫∫

Ω

ρΦTΦ dΩ

)
Λ̈(e) =0 (2.19)



Free vibration analysis of thick disks with variable thickness... 1011

Since δΛ(e)
T
is the variation of the nodal displacements and is arbitrary, it can be omitted from

(2.19), thus this equation can be written as

M(e)Λ̈(e)+K(e)Λ(e) =0 (2.20)

where the mass and stiffness matrices are defined as

M(e) =
∫∫∫

Ω

ρΦTΦ dΩ K(e) =
∫∫∫

Ω

BTDB dΩ (2.21)

For finding the components of mass and stiffness matrices, the integral must be taken over the
elements volume. As D and ρ are not constant in the case of FG types distributions, these
matrices are evaluated by numerical integration for each element using the Gauss-Legendre
technique (Zienkiewicz, 2005). Now by assembling the element matrices, the global equations of
motion for the FG-CNTRC disks can be obtained as

MΛ̈+KΛ=0 (2.22)

Once the finite element equations are estimated, substituting Λ=Λ0eiωt (ω is the natural
frequency) into Eq. (2.22) leads to an eigenvalue problem that can be solved using standard
eigenvalue extraction procedures.
In this regard, for calculating the elemental characteristic matrices (2.21) with the use of

Gauss-Legendre technique, obtaining the global ones and solve the eigenvalue problem after
imposing the boundary conditions, a FE code is prepared by the authors.

3. Numerical results and discussion

3.1. Validation

Tovalidate the currentwork, thedataof anon-homogeneous solid circular platewithvariable
thickness can be used (Gupta et al., 2007). The outer radius of the disk is roand the variable
thickness is defined as

h(r)= h0(1+αr+βr
2) (3.1)

where h0, α and β are the thickness at the middle and taper parameters, respectively. The
elasticity modulus and density vary in the r direction as below

E(r)= E0e
µr ρ(r)= ρ0e

ηr (3.2)

in which µ and η are non-homogeneity parameters.
The comparison of the first three frequency parameter (Ω = ω

√
ρ0r
2
0(1−ν2)/E0) for a

circular plate of η =−0.5, µ =0.1, α =−0.5 and β =0.5 with the published data is shown in
Table 1 and a good agreement between these results is observed.

3.2. Numerical results

In this Section, the free vibrational response of FG-CNTRC disks with variable thickness
is presented. The disk is made of Polymethyl-methacrylate (PMMA) as the matrix, where
SWCNTs act as fibers aligned in the radial direction. The properties of basic materials are
(Han and Elliott, 2007; Shen, 2009)

Em =2.5GPa νm =0.34 ρm =1150kg/m3

ECNT1 =5.6466TPa E
CNT
2 =7.08TPa ν

CNT =0.175

ρCNT =1150kg/m3
(3.3)



1012 H. Zafarmand,M. Kadkhodayan

Table 1. Frequency parameter compared with the result by Gupta (2007)

Boundary
condition

h0 =0.1 h0 =0.2

GFEM
Gupta Difference

GFEM
Gupta Difference

(2007) [%] (2007) [%]

Ω1 0.4121 0.4083 0.9 0.7765 0.7673 1.2
Clamped Ω2 1.4657 1.4471 1.2 2.5050 2.4604 1.8

Ω3 3.0403 2.9897 1.6 4.7195 4.6123 2.3

Simply
supported

Ω1 0.1857 0.1852 0.3 0.3650 0.3641 0.2
Ω2 1.0800 1.0715 0.8 1.9362 1.9258 0.5
Ω3 2.5635 2.5349 1.1 4.1489 4.1205 0.7
Ω1 0.3358 0.3348 0.3 0.6503 0.6480 0.3

Free Ω2 1.3553 1.3436 0.9 2.3823 2.3572 1
Ω3 2.9541 2.9159 1.3 4.7125 4.6299 1.8

The key issue for successful application of the extended rule of mixture to CNTRCs is to
determine theCNT efficiency parameter ηj (j =1,2,3). However, there are no experiments con-
ducted to determine the value of ηj for CNTRCs (Shen, 2009). Han andElliott (2007), with the
use of MD simulation and energy minimization, obtained the elastic moduli of polymer/CNT
composites. In the conventional rule of mixture, the whole system is assumed to be continuum
and the interfaces between thematrix and fibers remain fully intact, thus the general macrosco-
pic rule of mixtures cannot be applied straightforwardly to composites with strong interfacial
interactions. Besides, micromechanics equations cannot capture the scale difference between the
nano andmicro levels. For this purpose, CNT efficiency parameters ηj (j =1,2,3) are obtained
by comparingYoung’s moduli ECNT1 and E

CNT
2 of CNTRCs achieved from the extended rule of

mixture to those fromMD simulation given byHan and Elliot (2007). It should be noticed that
there are noMD results available for shear modulus G12 in Han and Elliott (2007). The results
are shown in Table 2 and will be used in the present study, in which it is assumed η3 = 0.7η2
(Yas et al., 2013).

Table 2. Comparison of Young’s moduli for polymer/CNTRC at T0 =300 (Yas et al., 2013)

V ∗CNT

MD (Hanet al., 2007) Extended rule of mixture
E1 [GPa] E2 [GPa] E1 [GPa] η1 [–] E2 [GPa] η2 [–]

0.12 94.6 2.9 94.78 0.137 2.9 1.022
0.17 138.9 4.9 138.68 0.142 4.9 1.626
0.28 224.2 5.5 224.5 0.141 5.5 1.585

Furthermore, the thickness profile of FG-CNTRC disk is in the form of

h(r)=h0
(
1−q

(r
b

)m)
(3.4)

where h0, q and m are geometric parameters that 0¬ q < 1 and m > 0. By changing the values
of q and m, four different thickness profiles, namely constant, linear, concave and convex are
introduced in Table 3 and, in the case of a =0, b =0.5m, are shown in Fig. 4.

Table 3. Different kinds of thickness profiles

Constant Linear Concave Convex

q =0
q =0.7 q =0.7 q =0.7
m =1 m =0.5 m =2



Free vibration analysis of thick disks with variable thickness... 1013

Fig. 4. Different kinds of thickness profiles

It should be also stated that the plane of revolution of the disk (r-zplane) is devided into
1200 linear quadrilateral elements with mesh density of 60× 20 regularly placed along the r
and z direction. In the case of linear thickness profile, a schematic of finite element mesh is
illustrated in Fig. 5.

Fig. 5. Finite element mesh

Now, by introducing the normalized natural frequency as

Ω = ωb
√
ρm
Gm

(3.5)

the first five normalized natural frequencies for FG-CNTRCannular and solid disks for different
types of distribution andvolume fractions ofCNTs, boundary conditions and thickness to radius
ratios (h0/b) as well as different thickness profiles are presented.
The effect of volume fraction of CNTs (V ∗CNT) on the first five normalized natural frequen-

cies (Ω) of the solid UD andFG-CNTRC disk for different types of CNTs distribution is shown
inTable 4. In this case, theboundaryconditions are assumed tobe clamped, the thickness profile
to be constant and h0/b = 0.5. As it can be seen; the normalized natural frequencies increase
when V ∗CNT raises due to growth of the structure stiffness.Moreover, changing the type of CNTs
distribution, affects the magnitude of natural frequencies. The FG V type of distribution has
the highest natural frequencies and the FG O has the lowest ones.
The comparison of first five normalized natural frequencies for UD and FG-CNTRC solid

disks for different boundary conditions with the linear thickness profile, V ∗CNT = 0.12 and
h0/b =0.5 is presented inTable 5.According to the results, boundaryconditions have noticeable
effects on natural frequencies. Thus, by changing the boundary conditions, the free vibration
response of the disk can be controlled.



1014 H. Zafarmand,M. Kadkhodayan

Table 4. Comparison of the first five normalized natural frequencies (Ω) for clamped UD and
FG-CNTRC solid disks for different V ∗CNT with constant thickness and h0/b =0.5

V ∗CNT =0.12 V
∗
CNT =0.17 V

∗
CNT =0.28

UD FG V FG O FG X UD FG V FG O FG X UD FG V FG O FG X

Ω1 1.977 2.059 1.885 1.953 2.537 2.686 2.385 2.507 2.688 2.981 2.435 2.672
Ω2 4.669 4.699 4.631 4.576 6.009 6.075 5.943 5.882 6.339 6.536 6.219 6.231
Ω3 7.381 7.394 7.349 7.32 9.503 9.547 9.455 9.432 10.01 10.18 9.953 9.981
Ω4 10.13 10.14 10.11 10.06 13.06 13.09 13.02 12.98 13.74 13.91 13.73 13.71
Ω5 12.94 12.94 12.67 12.42 16.54 16.54 16.03 15.64 17.02 16.70 16.49 16.25

Table5.Comparisonof thefirstfivenormalizednatural frequencies (Ω) forUDandFG-CNTRC
solid disks for different boundary conditions with the linear thickness profile, V ∗CNT =0.12 and
h0/b =0.5

Clamped Simply supported Free
UD FG V FG O FG X UD FG V FG O FG X UD FG V FG O FG X

Ω1 1.44 1.53 1.323 1.426 1.317 1.427 1.125 1.335 2.942 3.009 2.775 2.723
Ω2 4.262 4.291 4.21 4.109 4.055 4.109 3.938 3.908 5.399 5.508 5.074 5.37
Ω3 6.936 6.957 6.884 6.818 6.669 6.709 6.581 6.579 7.945 8.085 7.658 7.905
Ω4 9.638 9.653 9.582 9.522 9.323 9.348 9.259 9.208 10.65 10.75 10.38 10.64
Ω5 12.4 12.41 12.35 12.33 12.03 12.04 11.98 11.96 13.41 13.49 10.79 13.39

Table 6 demonstrates the effect of thickness-to-radius ratio on the normalized natural fre-
quencies of UD and FG-CNTRC solid disks for different types of CNTs distribution with the
concave thickness profile and V ∗CNT =0.17, and the boundary conditions assumed to be simply
supported. It is obvious that enlarging the thickness-to-radius ratio leads to growth of natural
frequencies, and this growth is larger in the lower natural frequencies.

Table 6. Comparison of the first five normalized natural frequencies (Ω) for simply supported
UD and FG-CNTRC solid disks for different thickness-to-radius ratios (h0/b) with the concave
thickness profile and V ∗CNT =0.17

h0/b =0.1 h0/b =0.2 h0/b =0.3
UD FG V FG O FG X UD FG V FG O FG X UD FG V FG O FG X

Ω1 0.473 0.562 0.337 0.475 0.881 1.032 0.638 0.889 1.205 1.382 0.905 1.215
Ω2 2.125 2.203 1.971 1.732 3.493 3.596 3.278 3.01 4.271 4.372 4.057 3.867
Ω3 4.175 4.154 4.065 3.866 6.284 6.313 6.124 6.053 7.349 7.407 7.184 7.182
Ω4 6.604 6.467 6.555 6.304 9.35 9.312 9.257 9.022 10.64 10.65 10.55 10.36
Ω5 9.355 9.113 9.366 9.059 12.65 12.56 12.63 12.41 14.11 14.07 14.08 13.95

The influence of different thickness profiles on the normalized natural frequencies of free
UD and FG-CNTRC solid disks with V ∗CNT =0.28 and h0/b =0.5 for different types of CNTs
distribution is illustrated in Table 7. The achieved results show that the natural frequencies are
highly dependent on the thickness profile. Moreover, the highest natural frequencies occur in
the convex thickness profile and the smallest ones occur in the concave thickness profile.
Now, the free vibration response of FG-CNTRCannular disks of variable thickness is discus-

sed. As before, the effect of volume fraction of CNTs, boundary condition, thickness to radius
ratio and thickness profile on the first five normalized natural frequencies of annular UD and
FG-CNTRC disks are studied, in this case of a/b =0.2. These results are illustrated in Tables
8-11. As it can be concluded from these tables, the normalized natural frequncies increase when



Free vibration analysis of thick disks with variable thickness... 1015

Table 7. Comparison of the first five normalized natural frequencies (Ω) for free UD and
FG-CNTRC solid disks for different thickness profiles with h0/b =0.5 and V ∗CNT =0.28

Linear Concave Convex
UD FG V FG O FG X UD FG V FG O FG X UD FG V FG O FG X

Ω1 4.027 4.162 3.762 3.585 3.375 3.886 3.401 3.291 4.349 4.459 4.158 3.884
Ω2 7.38 7.566 6.942 7.257 7.014 7.211 6.479 6.896 7.793 7.95 7.482 7.65
Ω3 10.87 11.06 10.46 10.71 10.45 10.65 9.991 10.28 11.32 11.49 10.99 11.16
Ω4 14.49 14.67 14.17 14.42 14.04 14.23 13.71 13.97 14.95 15.13 14.64 14.89
Ω5 18.21 18.39 15.02 18.18 17.75 17.93 14.76 17.72 18.14 17.31 15.12 17.82

Table 8. Comparison of the first five normalized natural frequencies (Ω) for clamped UD and
FG-CNTRC annular disks for different V ∗CNT with constant thickness and h0/b =0.5

V ∗CNT =0.12 V
∗
CNT =0.17 V

∗
CNT =0.28

UD FG V FG O FG X UD FG V FG O FG X UD FG V FG O FG X

Ω1 3.203 3.237 3.167 3.301 4.122 4.194 4.076 4.314 4.348 4.541 4.321 4.757
Ω2 6.598 6.609 6.589 6.513 8.493 8.535 8.485 8.419 8.951 9.108 8.977 9.015
Ω3 10.03 10.04 10.02 10.03 12.93 12.96 12.92 12.98 13.61 13.77 13.66 13.83
Ω4 13.09 13.13 12.86 12.55 16.66 16.67 16.34 15.82 17.21 17.06 16.81 16.47
Ω5 13.55 13.56 13.54 13.1 17.47 17.51 17.29 16.61 18.31 18.49 18.09 17.29

Table 9. Comparison of first five normalized natural frequency (Ω) for UD and FG-CNTRC
annular disks for different boundary conditions with linear thickness profile, V ∗CNT = 0.12 and
h0/b =0.5

Clamped Simply supported Free
UD FG V FG O FG X UD FG V FG O FG X UD FG V FG O FG X

Ω1 2.829 2.838 2.801 2.928 2.228 2.264 2.124 2.337 3.205 3.234 3.066 2.945
Ω2 6.155 6.154 6.144 5.984 5.181 5.191 5.125 5.045 6.097 6.129 5.797 6.093
Ω3 9.478 9.477 9.456 9.422 8.178 8.14 8.152 8.129 9.232 9.215 8.944 9.196
Ω4 12.88 12.87 12.86 12.83 11.28 11.16 11.27 11.23 12.53 12.46 11.43 12.59
Ω5 16.37 16.36 16.35 15.49 14.47 14.26 13.39 14.49 15.93 15.09 12.28 15.16

Table 10. Comparison of first five normalized natural frequency (Ω) for simply supported
UD and FG-CNTRC annular disks for different thickness to radius ratio (h0/b) with concave
thickness profile and V ∗CNT =0.17

h0 =0.1 h0 =0.2 h0 =0.3
UD FG V FG O FG X UD FG V FG O FG X UD FG V FG O FG X

Ω1 1.429 1.193 1.418 1.432 2.212 2.023 2.156 2.329 2.593 2.515 2.488 2.781
Ω2 3.271 2.927 3.271 3.208 5.011 4.826 4.932 4.776 5.904 5.849 5.812 5.62
Ω3 5.712 5.279 5.767 5.592 8.421 8.209 8.418 8.285 9.626 9.561 9.609 9.559
Ω4 8.692 8.223 8.809 8.632 12.25 12.05 12.31 12.15 13.62 13.53 13.67 13.54
Ω5 12.14 11.69 12.31 12.12 16.37 16.19 16.48 16.34 17.81 17.64 17.76 17.83



1016 H. Zafarmand,M. Kadkhodayan

Table 11. Comparison of first five normalized natural frequency (Ω) for free UD and
FG-CNTRC annular disks for different thickness profile with h0/b =0.5 and V ∗CNT =0.28

Linear Concave Convex
UD FG V FG O FG X UD FG V FG O FG X UD FG V FG O FG X

Ω1 4.392 4.448 4.187 3.881 4.077 4.149 3.798 3.56 4.752 4.775 4.628 4.229
Ω2 8.336 8.399 7.978 8.281 7.928 7.986 7.472 7.887 8.814 8.858 8.585 8.731
Ω3 12.58 12.59 12.27 12.48 12.11 12.09 11.75 12.01 13.11 13.13 12.86 13.01
Ω4 17.03 16.98 15.97 17.09 16.54 16.44 15.69 16.61 17.56 16.91 16.07 17.63
Ω5 21.58 19.22 16.79 21.35 21.13 20.97 16.31 21.22 18.63 17.57 17.31 18.79

the volume fraction of CNTs or thickness-to-radius ratio raises. Moreover, the boundary condi-
tion and the type of CNTs distribution as well as the thickness profile have a great influence
on the normalized natural frequencies. Besides, in comparison to solid disks, annular disks of
the same conditions have larger normalized natural frequencies and, unlike the solid disks, the
highest normalized natural frequencies of annular disks occur in the case of FG X type of CNTs
distribution.
Thus, the normalized natural frequencies can be controlled and altered by changing the

distribution and volume fraction of CNTs, boundary condition, thickness-to-radius ratio and
the thickness profile.

4. Conclusions

In this study, free vibration analysis of thick FG-CNTRC annular and solid disks with variable
thickness is presented. The axisymmetric conditions are assumed, volume fraction of CNTs
is considered to be graded continuously along the radial direction and material properties are
estimated through the extended rule ofmixture. By employing the graded finite elementmethod
and Hamilton’s principle, the free vibrational response of FG-CNTRC disks is investigated.
Detailed parametric studies are performedto illustrate the effects of several parameters including
the type of distribution and volume fraction of CNTs, boundary condition, thickness-to-radius
ratio and the thickness profile on the normalized natural frequencies of FG-CNTRCdisks. From
this analysis, some typical conclusions can bemade:
• The normalized natural frequencies increase when theCNTs volume fraction or thickness-
-to-radius ratio rises.

• In solid FG-CNTRC disks, the FG V type of CNTS distribution has higher normalized
natural frequencies than the other types, and in the case of annular FG-CNTRCdisks, the
higher normalized frequencies occur in FG X type of CNTs distribution.

• By using the variable thickness profile, the normalized natural frequencies can be control-
led. Convexing the thickness profile generates larger normalized natural frequencies and
concaving the thickness profile, generates smaller ones.

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Manuscript received November 1, 2013; accepted for print June 1, 2015