Free Vibration of Fgm Cylindrical Shell Reinforced With Single-Walled Carbon Nanotubes Using High Order Shear Deformation Theory

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


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,


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)


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)


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.


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


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


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.


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


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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