higher-order shear deformation beam theory reinforced composite face sheets based...

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Mechanics of Advanced Materials and Structures

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Thermo-mechanical vibration analysis of sandwichbeams with functionally graded carbon nanotube-reinforced composite face sheets based on ahigher-order shear deformation beam theory

Farzad Ebrahimi & Navid Farzamand nia

To cite this article: Farzad Ebrahimi & Navid Farzamand nia (2016): Thermo-mechanicalvibration analysis of sandwich beams with functionally graded carbon nanotube-reinforcedcomposite face sheets based on a higher-order shear deformation beam theory, Mechanics ofAdvanced Materials and Structures, DOI: 10.1080/15376494.2016.1196786

To link to this article: http://dx.doi.org/10.1080/15376494.2016.1196786

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ACCEPTED MANUSCRIPT

ACCEPTED MANUSCRIPT 1

Thermo-mechanical vibration analysis of sandwich beams with functionally graded carbon

nanotube-reinforced composite face sheets based on a higher-order shear deformation

beam theory

Farzad Ebrahimi and Navid Farzamand nia

Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International

University, Qazvin, Iran

Received: 2015-12-12

Accepted: 2016-05-05

Abstract

This paper proposes a higher-order shear deformation beam theory (HSBT) for free vibration analysis of

functionally graded carbon nanotube-reinforced composite (FG-CNTRC) sandwich beams in thermal

environment. The temperature- dependent material properties of FG-CNTRC beam are supposed to vary

continuously in the thickness direction and are estimated through the rule of mixture. The governing

equations and boundary conditions are derived by using Hamilton's principle and the Navier solution

procedure is used to achieve the natural frequencies of the sandwich beam in thermal environment. A

parametric study is led to carry out the effects of carbon nanotube volume fractions, slenderness ratio and

core-to-face sheet thickness ratio on free vibration behavior of sandwich beams with FG-CNTRC face

sheets. Numerical results are also presented in order to compare the behavior of sandwich beams

including uniformly distributed carbon nanotube-reinforced composite (UD-CNTRC) face sheets to those

including FG-CNTRC face sheets.

Keywords: Free Vibration; FG-CNTRC; New HSBT; Sandwich Beam; Thermal Environments

--------------------------------------------------------------------

CONTACT Farzad Ebrahimi

[email protected]

Department of Mechanical Engineering, Faculty of Engineering, Imam Khomeini International

University, Qazvin, Iran.

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1. Introduction

The use of sandwich structures is growing so rapidly all over the world and has attracted

increasing attention due to their super fantastic characteristics. The need for higher performances

and lower weight of the structures makes sandwich construction one of the best choices to be

applied in aircrafts, space vehicles and transportation systems. Functionally graded materials

(FGMs) are composite materials with inhomogeneous micromechanical structures in which the

material properties of the structure change smoothly between two surfaces. The advantages of

this combination lead to novel structures which can withstand in large mechanical loadings under

high temperature environments [1]. Presenting novel properties, FGMs have also attracted

intensive research interests, which were mainly focused on their static, dynamic and vibration

characteristics of functionally graded (FG) structures [2, 3]. Also many studies have been

conducted to investigate vibration, buckling and post-buckling behavior of sandwich structures

with FGM face sheets [4-8].

Carbon nanotubes (CNTs) have extraordinary mechanical properties. Due to their outstanding

properties such as superior mechanical, electrical and thermal properties, they have attracted

growing interest and are considered to be the most promising materials applied in nano-

engineering applications in recent years [9-11]. The CNT-based nanocomposite devices can

withstand high temperature during manufacturing and operation. Thus it is important to have a

good knowledge of the thermal properties of nanotubes. Some related studies show that the

physical properties of CNTs depend strongly on temperature, from which it is believed that the

elastic constants of nanotubes, such as Young’s modulus and shear modulus, are also

temperature dependent [12, 13]. However, it is remarkably difficult to directly measure the

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mechanical properties of individual CNTs in the experiment due to their extremely small size. In

1994, Ajayan et al. [14] studied polymer composites reinforced by aligned CNT arrays. Since

then, many researchers inspected the material properties of CNTRCs [15-19]. Ashrafi and Hubert

[20] modeled the elastic properties of CNTRCs through a finite element analysis. Xu et al. [21]

examined the thermal behavior of single-walled carbon nanotubes (SWCNTs) polymer–matrix

composites. Han and Elliott [22] employed molecular dynamics (MD) to simulate the elastic

properties of CNTRCs. These studies proved that adding a small amount of CNTs can

significantly improve the mechanical, electrical, and thermal properties of polymeric composites.

The results were also helpful for prediction of the global response of CNTRCs as an actual

structural constituent.

Studies on CNTRCs have revealed that distributing the CNTs in a uniform way as the

reinforcements in matrix can lead to only intermediate improvement of the mechanical

characteristics [23, 24]. This is principally because of the weak interfacial bonding strength

between the CNTs and the matrix. Shen [25] used the idea of functionally graded distribution of

CNTS in composites. Ke et al. [26, 27] examined the effect of FG-CNT volume fraction on the

nonlinear vibration and dynamic stability of composite beams. Wang and Shen [28] studied the

vibration behavior of CNTRC plates in thermal environments. They mentioned that generally the

CNTRC plates with symmetrical distribution of CNTs have higher natural frequencies, but lower

linear to nonlinear frequency ratios than ones with unsymmetrical or uniform distribution of

CNTs. Wang and Shen [29] studied the nonlinear bending and vibration of sandwich plates with

CNTRC face sheets in sandwich structures with FG-CNTRC face sheets. The effects of nanotube

volume fraction, foundation stiffness, core-to-facing thickness ratio, temperature change, and in-

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plane boundary conditions on the nonlinear vibration and bending behaviors of sandwich plates

with CNTRC facings sheets were considered. Yang et al. [30] examined the dynamic buckling

FG nanocomposite beams reinforced by CNT as a core and integrated with two surface bonded

piezoelectric layers. Wu et al. [31] investigated free vibration and buckling of sandwich beams

which are reinforced with FG-CNTRCs face sheets based on a Timoshenko beam theory (TBT).

It is well known that the classic beam theory (CBT) is only appropriate to slender beams and

lower modes of vibration and can not describe higher modes of vibration, especially for short

beams. For moderate beams, it undervalues deflection and overrates buckling load and natural

frequencies because of ignoring the shear deformation effect. Thus, the first-order beam theory

(FOBT) is proposed to overcome the limitations of the CBT by accounting for the transverse

shear effects. However to including this effect, a shear correction factor for FOBT (but not for

HSBT) is required to account for the discrepancy between the actual stress state and the assumed

constant stress state since the FOBT can't predict the zero shear stress conditions on the top and

bottom surfaces of the beam. The efficiency of the HSBT strongly relies on the suitable selection

of displacement field which is an attracting subject interested by many researchers. The HSBT

can be formulated based on the assumption of the higher-order variation of axial displacement or

both axial and transverse displacements through the beam depth. Although there are many

references available on static and vibration, the research on the displacements and natural

frequencies of FG beams with HSBT is limited.

Kadoli et al. [32] suggested a displacement field based on HSBT to study the static behavior

of FGM beams under ambient temperature. Simsek [33] analyzed natural frequency of various

higher order FG beams under different boundary conditions. Mahi et al. [34] presented a unified

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formulation to study the free vibration of a higher order graded beam. Larbi et al. [35] presented

an efficient shear deformation beam theory depended on neutral surface position for static and

vibrational analysis of FG beams. Vo et al. [36] investigated static and vibration behavior of FG

beams developing a novel refined theory. Nguyen et al. [37] studied vibration and stability of FG

sandwich beams via a new hyperbolic shear deformation theory. In another study, Vo et al. [38]

provided finite element vibration and buckling analysis of FG sandwich beams through a quasi-

3D theory in which both shear deformation and thickness stretching effects are included. Also,

Atmane et al. [39] applied an efficient beam theory to investigate the effects of thickness

stretching and porosity on mechanical responses of FGM beams resting on elastic foundation.

It should be noted that several aforementioned studies ([40-42]) neglected the thickness

stretching effect, which becomes very important for thick plates [43]. In order to include shear

deformation and thickness stretching effects, the quasi-3D theories, which are based on a higher-

order variation through the thickness of the in-plane and transverse displacements, are used. By

using these theories, although a lot of work has been done [44-46], the research on FG beams is

still limited. Carrera et al. [47] developed Carrera Unified Formulation (CUF) using various

refined beam theories (polynomial, trigonometric, exponential and zig-zag), in which non-

classical effects including the stretching effect were automatically taken into account. Recently,

he and his co-workers also used CUF to investigate the free vibration of laminated beam [48] and

FG layered beams [49]. To the best of the authors’ knowledge, there is no paper assigned for free

vibration of FG-CNTRC sandwich beam in a thermal environment using refined shear

deformation theory in the open literature. This complicated problem is not well-investigated and

there is a need for further studies.

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In this paper vibration analysis of sandwich beams with a stiff core and FG-CNTRC face

sheets reinforced by SWCNTs in thermal environment is presented within the framework of a

HSBT. The in-plane displacements use a combination of hyperbolic tangent function and

polynomial ones. The thickness stretching effect is ignored. This new theory has five degrees of

freedom, provides parabolic transverse shear strains across the thickness direction and hence,

does not need shear correction factor. Moreover, zero-traction boundary conditions on the top

and bottom surfaces of the plate are satisfied rigorously. The material characteristics of CNTs are

supposed to change in the thickness direction in a FG form. They are also can be calculated

through a micromechanical model where the CNT efficiency parameter ( ) is determined by

matching the elastic modulus of CNTRCs calculated from the rule of mixture with those gained

from the molecular dynamics (MDs) simulations. Navier-type analytical solution is obtained for

sandwich beams with simply supported boundary conditions. A parametric study is directed to

show the effects of CNT volume fraction, slenderness ratio and core-to-face sheet thickness ratio

in a uniform thermal environment on the free vibration properties of sandwich beams with FG-

CNTRC face sheets. Numerical results for sandwich beams with UD-CNTRC are also presented

for validation and comparison.

2. Sandwich Beam with CNTRC Face Sheets

Assume a symmetric sandwich beam with length L, width b and total thickness h subjected to

an axial load as shown in Figure 1. The sandwich beam is made of two CNTRC face sheets with

equal thickness of fh and have a stiff core layer of thickness ch .

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Moreover, in this study two distribution of CNTs, i.e. FG and uniform, are considered. The

material properties can be determined through the rule of mixture [50]:

11 1 11cn

cn m mE V E V E (1a)

*2

22 22 22

cn mcn m

V V

E E E (1b)

3

12 12

cn m

cn

m

V V

G G G

(1c)

where 11cnE , 22

cnE and 12cnG are Young’s moduli and shear modulus of CNTs, respectively and mE

and mG are the properties for the matrix. i ( 1,2,3i ) is the CNT efficiency parameter

accounting for the scale-dependent material properties and can be obtained by matching the

elastic modulus of CNTRCs achieved from molecular dynamics simulation and those extracted

from the rule of mixture. mV and cnV are the volume fraction of matrix and the CNTs with

following relation:

1cn mV V (2)

The material properties of FG ceramic–metal composites, most commonly vary along

thickness direction in two ways; either a power law or an exponential distribution. However, in

nanocomposites, the manufacturing method for such a graded distribution is so expensive and

difficult. It is supposed that cnV for the FG-CNTRC face sheets changes linearly across the

thickness because a better way to manufacture CNTRC is to align CNTs functionally graded in a

polymer matrix and only linear distribution can be obtained in engineering practice, so the

distribution for the top face sheet can be expressed by:

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*

2 ccn cn

f

z hV V

h (3a)

and also for the bottom face sheet:

*

2 ccn cn

f

z hV V

h (3b)

where *cnV can be described as:

w

w w

* cncn

cn cncn cn

m m

V . (4)

In which w cn is the mass fraction of CNT, whereas m and cn are the densities of matrix and

CNT, respectively. There is a simple relation for *cnV in UD-CNTRCs which can be given by

*cn cnV V , so it is obvious that the mass fraction for UD-CNTRC and FG-CNTRC face sheets are

equal.

The density and Poisson's ratio of the CNTRC face sheets can be described as:

cn cn m mV V (5)

cn cn m mV V (6)

in which m and cn are Poisson's ratio of the matrix and CNT, respectively. As sandwich beams

are used mostly in high temperature environments, significant changes in mechanical properties

(such as Young's modulus and thermal expansion coefficient) of the ingredient materials are

expected. Thus it is necessary to take into account this temperature-dependency of properties for

precise prediction of the mechanical behavior of the structure. The behavior of FG materials

subjected to high temperatures can be predicted more precisely with considering the temperature

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dependency of material properties. The nonlinear equation of thermo-elastic material properties

as a function of temperature ( )T K can be expressed as follows [51]:

1 2 3

0 1 1 2 3( 1 )P P P T P T P T P T (7)

where 0 1 1 2

, , ,P P P P and 3P are the temperature dependent coefficients which can be seen in Table

1 for Ti-6Al-4V .

As there are not specific temperature dependency expressions for material properties of CNTs,

one should estimate CNT efficiency parameters 1 and 2 by matching the Young’s modulus 11E

and 22E of CNTRCs obtained by the rule of mixture to those from the MD simulations given by

Han and Elliott [22]. It should be noted that only should be used in beam theories. The

results are presented in Table 2.

3. Theoretical Formulations

3.1. Governing equations

The equations of motion are derived based on a HSBT firstly used by Reddy. The displacement

field at any point of the beam can be written as:

1 , , , ( )b sdw dwu x z t u x t z f z

dx dx (8.a)

2 , , 0u x z t (8.b)

3( , , ) ( , ) ( , )b su x z t w x t w x t (8.c)

In which u is the axial displacement of mid-plane along x -axis, ,b sw w are the bending and shear

components of displacement along the mid-plane of the beam and t is the time. f(z) represents the

distribution of the transverse shear strains and shear stress through the beam depth. This function

is considered to satisfy the stress-free boundary conditions on the top and bottom surfaces of the

11E

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beam. Thus it is not required to use any shear correction factor. The displacement relation of the

new hyperbolic shear deformation theory based on Mahi et al. [53] can be obtained by using the

following hyperbolic shape function:

Then the strains displacement relation of shear deformation beam theory can be expressed as:

2

2

2

2

bx

sx

u wz

x

wf

xx

(10.a)

(1 ) s sxzf w w

gx x x

(10.b)

where xx , xz are the normal and shear strains and ( ) (1 )df

g zdz

is the shape function of the

transverse shear strains as following:

The Euler Lagrange equations has been used to derive the equation of motion by using a

Hamilton’s principle, which can be stated as:

2

1

( ) 0t

t

U T V dt (12)

where 1t , 2t are the initial and end time, u is the virtual variation of strain energy, V is the

virtual variation of work done by external loads and T is the virtual variation of kinetic energy.

Strain energy, kinetic energy and potential energy (external loading) have been calculated and

the equation of motion has been obtained by using rules of calculus of variations and Hamilton's

principle.

3

2 2

4( ) tanh(2 ) ( )

2 3cosh (1)

h z zf z

h h

(9)

22

2 2

2 4( ) 1 1 tanh ( ) ( )

cosh (1)

z zg z

h h

(11)

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0

( )L

ij ij xx xx xz xzv A

U dV dxdA (13)

Where is the variational symbol, A is the cross-section area of the uniform beam, xx the axial

stress and xz is the shear stress, by substituting the expressions for xx and xz into Eq. (13) as:

2

20

0

2

2

LL

b sxx xz

A

s

A

u w wu z f dAdx g dAdx

x x x

w

x

(14)

where N , bM sM ,Q as the stress resultants are defined as following and by replacing these

resultants into Eq(14), get to Eq(16).:

( , , ) (1, , ) , ( ) ( )b s xx xzA A

N M M z f dA Q g dA (15)

2 2

2 20

b sb s s

L N M M Qu w w w dx

x xx xu

(16)

The kinetic energy expression be expressed as:

2 2 2

1 2 3

0

1,

2

L

A

u u uT z T dAdx

t t t

(17)

u

t

the velocity along x,y and z-axes has been obtained by derivative of the coordinates with

respect to time( t ), the first variation of the virtual kinetic energy can be written in the form:

2 2 22 2 2 2 2

0

2 2 22 2

1( , )[( ) ( ) ( ) 2

2

2 ( ) 2 ( ) ( ) ( ) 2 ]

Lb s b

A

s b s b s b s

u w w u wT z t z f z

t t x t x t t x

u w w w w w w wf zf dAdx

t t x t x t x t t t t

(18)

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2 3 3 2 3

0 2 2 12 2 2 2 20

2 3 3 3

1 22 2 2 2

2 2 2 2

0 2 2 2 2

1[ [ ]

2

[ ] [ ]

[ ]]

Lb s b

b s b

s b ss s b

b s b sb s s b

u w w u wT I u I w k w I w u

t t x t x t t x

u w w wJ w u J w w

t t x t x t x

w w w wI w w w w dx

t t t t

(19)

where 0 1 2 1 2 2( , , , , , )I I I J J k are the mass moment of inertias that can be defined as:

20 1 2( , , ) ( , )(1, , )

A

I I I z T z z dA (20)

1 2 2( , , ) ( , )(1, , )A

J J K f z T z f dA

also the first variation of potential energy can be written in the form:

0

( ) ( )[ ( ) ( ) ( ) ]dx

Ls b s b

s b

w w w wV f x u q x w w N

x x

(21)

In this study for analyzing vibration of porous FG beam in thermal environment, the first

variation of external loadings due to a temperature change can be obtained as:

0

( ) ( )dx

LT s b s bw w w wV N

x x

(22)

where TN is defined as following:

exp( , ) ( , )TN E z T z T Tdz (23)

In which exp is the coefficient of thermal dilatation that is typically positive and very small

(0 1) .

Applying the principle of virtual displacements which states if a system is in equilibrium, then

the total virtual work done is zero (Eq. (12)) thus we obtain:

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2

1

2 3 3 2 2 2 3

0 1 1 0 12 2 2 2 2 2 2

0

4 4 2 2 4 4

2 2 2 22 2 2 2 2 2 2 2 2 2

3 2 2

1 02 2

[( ) ( ( )

) (

(

t L

b s b b s

t

Tb s s b s

b

s b

N u w w M w w uI I J u I I

x t t x t x x t t t x

w w N M Q w wI J w J k

xt x t x x x t x t x

u w wJ I

t x t t

2

2 2) ) ] 0

T

s

Nw

x

(24)

By collecting the coefficients of u , bw and sw and setting to zero, the governing equations of

motion of porous FG Reddy beam theory in thermal environment can be obtained as:

2 3 3

0 1 12 2 2( : 0), b s

N u w wu I I J

x t t x t x

(25)

2 2 2 2 4

0 22 2 2 2 2 2

3 4

1 22 2 2

( ): 0 , ( )Ts b s s b ss

b

M Q w w w w ww N I k

xx x t t t x

u wJ J

t x t x

(26)

2 2 2 2 3 4 4

0 1 2 22 2 2 2 2 2 2 2 2

( ): 0 , ( )Tb b s b s b sb

M w w w w u w ww N I I I J

x x t t t x t x t x

(27)

For a material that is linearly elastic and obeys the 1D Hooke’s law, the relation between stress-

strain can be described as;

xx xxE z (28.a)

xz xzG z (28.b)

where G is the shear modulus and E is the Young’s modulus, by substituting the Eqs. (10.a),

(10.b) into Eqs. (28.a) and (28.a) and subsequent results into Eq.(15) and integrating over the

beam’s cross-section, stress resultants can be derived as following:

2 2 2 2

2 2 2 2,b s b sxx xx xx b xx xx xx

u w w u w wN A B C M B D E

x xx x x x

(29)

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2 2

2 2,b s ss xx xx xx xz

u w w wM C E F Q A

x xx x

In which the cross-section stiffness are defined as:

2 2( , , , , , ) (1, , , , , ) ,xx xx xx xx xx xxA

A B C D E F z f z f z f E z T dA (30)

2 ,xzA

A g G z T dA

(31)

And the last form of Euler-Lagrange equations for FG based on new hyperbolic shear

deformation beam theory in thermal environment in terms of displacementu , bw and sw can be

derived as:

2 3 3 2 3 3

0 1 12 3 3 2 2 2

b s b sxx xx xx

u w w u w wA B C I I J

x x x t t x t x

(32)

3 4 4 2 2 3

0 13 4 4 2 2 2

4 4

2 22 2 2 2

( ) ( )Tb s b s s bxx xx xx

b s

u w w w w w w uB D E N I I

x x x x t t x

w wI J

t x t x

(33)

3 4 4 2 2 2

03 4 4 2 2 2

4 3 4

2 1 22 2 2 2 2

( ) ( )Tb s s b s s bxx xx xx xz

s b

u w w w w w w wC E F A N I

x x x x x t

w u wk J J

t x t x t x

(34)

3. Solution method

3.1. Analytical solution

In this section, an analytical solution of the Euler-Lagrange equations for free vibration of

simply-supported porous FG beam based on Navier type method is presented. The displacement

functions are expressed as combinations of non-significant coefficients and known trigonometric

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functions to satisfy Lagrange equations and boundary conditions at 0 ,x x L the following

displacements functions are assumed to be formed:

1

( , ) cos( ) miw tmm

mu x t u x e

L

(35.a)

1

( , ) sin( ) miw tb bmm

mw x t w x e

L

(35.b)

1

( , ) sin( ) miw ts smm

mw x t w x e

L

(35.c)

In which ( , , )m b m s mu w w are the unknown Fourier coefficients which may be calculated for each

value of m. Boundary conditions for a simply-supported beam are as follows:

(0) 0 , ( ) 0u

u Lx

(36.a)

(0) ( ) 0 , (0) ( ) 0w w L Lx x

(36.b)

Substituting Eqs. (35.a) - (35.c) into Eqs. (32) - (34) respectively, leads to the following

relations:

2 3 3 2 20 1

21

( ) ( ) ( ) ( )

( )

xx m xx bm xx sm m bm

sm

m m m mA u B w C w I u I w

L L L L

mJ w

L

(37)

3 4 4 2

2 2 2 2 2 20 1 2 2

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

Txx m xx bm xx sm bm sm

sm bm m bm sm

m m m mB u D w E w N w w

L L L L

m m mI w w I u I w J w

L L L

(38)

3 4 4 2 2

2 2 2 2 2 20 2 1 2

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

Txx m xx bm xx sm xz sm bm sm

sm bm sm m bm

m m m m mC u E w F w A w N w w

L L L L L

m m mI w w k w J u J w

L L L

(39)

The analytical solutions can be obtained from the following equation:

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2 2 3 2 3 20 1 1[ ( ) ] [ ( ) ( )] [ ( ) ( )] 0xx m xx bm xx sm

m m m m mA I u B I w C J w

L L L L L

(40)

3 2 4 2 2 21 0 2

4 2 2 20 2

[ ( ) ( )] [ ( ) ( ) ( ( ) )]

[ ( ) ( ) ( ( ) )] 0

Txx m xx bm

Txx sm

m m m m mB I u D N I I w

L L L L L

m m mE N I J w

L L L

(41)

3 2 4 2 2 2 21 0 2

4 2 2 2 22 0

[ ( ) ( )] [ ( ) ( ) ( ) ]

[ ( ) ( ) ( ) ( ( ) )] 0

Txx m xx bm

Txx xz sm

m m m m mC J u E N I J w

L L L L L

m m m mF A N k I w

L L L L

(42)

11 12 13

21 22 23

31 32 33

0

m

sm

bm

a a a u

a a a w

a a a w

(43)

For the non-trivial solutions of Eq. (43), it is necessary that the determinant of the following

coefficient matrix is equal to zero:

11 12 13

21 22 23

31 32 33

0

a a a

det a a a

a a a

(44)

By setting the determinant of the coefficient matrix to zero, one can find the natural frequencies

n of the structure.

4. Uniform temperature rise (UTR)

The initial temperature of the sandwich beam is assumed to be (0

300T K ), which is a stress

free state, and uniformly changes to final temperature and the temperature rise is given by:

0T T T (45)

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6. Results and Discussion

6.1 Comparison studies

To evaluate accuracy of the present study, the first three dimensionless natural frequencies of

sandwich beams with FG-CNTRC facing sheets previously analyzed by Wu [31] are reexamined

as presented in Tables 6. One may clearly notice here that the Non-dimensional fundamental

frequency parameters obtained in the present investigation are in excellent agreement to the

results presented by Wu [31] and validates the proposed method of solution.

6.2 Free vibration analysis

In this study, Poly-methyl methacrylate (PMMA) as the matrix with following material

properties: 0.34m , 1150m kg/m3 645(1 0.0005 ) 10m T /K and (3.52 0.0034 )mE T GPa, in

which 0T T T and 300T K (room temperature) and this leads to 645.0 10m /K and 2.5mE

GPa at 300T [30]. Also the armchair (10, 10) SWCNTs, with material properties of 11 5.6466cnE

TPa, 22 7.08cnE TPa, 12 1.9445

cnG TPa, 611 3.4584 10cn /K, 622 5.1682 10

cn /K 1400cn kg/m3 and

0.175cn at room temperature, [54] are selected as the reinforcement for CNTRCs. The CNT

efficiency parameter j is obtained by matching the Young's modulus 11E and 22E and shear

modulus 12G of CNTRCs determined from the rule of mixture against those from the MD

simulations given by Han and Elliott [22]. Shen and Zhang [54] presented the following values

1 0.137 , 2 1.022 , 3 0.715 for the case of * 0.12cnV and 1 0.142 , 2 1.626 , 3 1.138 for

* 0.17cnV ; and 1 0.141 , 2 1.585 , 3 1.109 for * 0.28cnV . In this study a Titanium alloy (Ti-6Al-

4V) with the following properties is considered for core material: 113.8cE GPa, 4430c kg/m3

and 0.342c . Also the total thickness of the sandwich beam is assumed chosen to be 10 mm.

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However, the thickness of core layer and face sheets change arbitrarily as the core-to-face sheet

thickness ratio changes: hc/hf = 8, 6, 4. The natural frequencies of the sandwich beams with FG-

CNTRC face sheets with various temperature rises are tabulated in Tables 7-9.

Table 7 and Figure 2 present the first three natural frequencies of S-S sandwich beams with

CNTRC face sheets with different CNT volume fraction *cnV . The core-to-face sheet thickness

ratio and the slenderness ratio are kept unchanged at hc/hf=8 and L/h=20, respectively. It is

observed that the natural frequency of the sandwich beam increases with an increase in the CNT

volume fraction *cnV but decreases as the temperature increases.

Also it can be found out that the results of present theory are in excellent agreement with those

predicted by TBT for all values of small scale coefficient and length-to-depth ratio even for short

beams at the higher vibration modes. This is where the effects of transverse shear deformation

and rotary inertia are significant. It is worth noting that the TBT requires a shear correction

factor to satisfy the free transverse shear stress conditions on the top and bottom surfaces of the

beam, while the present theory satisfies the free transverse shear stress conditions on the top and

bottom surfaces of the beam without using any shear correction factors.

Furthermore, it is observed that the natural frequencies of the sandwich beam with UD-

CNTRC face sheets is also lower than those of the beam with FG-CNTRC face sheets. This is

because the sandwich beam with UD-CNTRC face sheets has a lower stiffness than the beam

with FG-CNTRC face sheets.

Table 8 and Figure 3, present the first three natural frequencies of S-S sandwich beams with

CNTRC face sheets but with different slenderness ratio L/h. The core-to-face sheet thickness

ratio and the CNT volume fraction are kept unchanged at hc/hf=8 and *

cnV =0.17, respectively. It is

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observed that the natural frequency of the sandwich beam decreases with an increase in the

slenderness ratio but decreases as the temperature increases.

Also it can be found out that the results of present theory are in excellent agreement with those

predicted by TBT for all values of small scale coefficient and length-to-depth ratio even for short

beams at the higher vibration modes. This is where the effects of transverse shear deformation

and rotary inertia are significant. It is worth mentioning that the TBT requires a shear correction

factor to satisfy the free transverse shear stress conditions on the top and bottom surfaces of the

beam, while the present theory satisfies the free transverse shear stress conditions on the top and

bottom surfaces of the beam without using any shear correction factors.

Furthermore, it is observed that the natural frequencies of the sandwich beam with UD-

CNTRC face sheets is also lower than that of the beam with FG-CNTRC face sheets. This is

because the sandwich beam with UD-CNTRC face sheets has a lower stiffness than the beam

with FG-CNTRC face sheets.

Table 9 and Figure 4, present the first three natural frequencies of S-S sandwich beams with

CNTRC face sheets but with different core-to-face thickness ratio hc/hf. The slenderness ratio

and the CNT volume fraction are kept unchanged at L/h=20 and *cnV =0.17, respectively. It is

observed that the natural frequency of the sandwich beam increases with an increase in the core-

to-face thickness ratio but decreases as the temperature increases. Furthermore, it is observed

that the natural frequencies of the sandwich beam with UD-CNTRC face sheets is also lower

than that of the beam with FG-CNTRC face sheets. This is because the sandwich beam with UD-

CNTRC face sheets has a lower stiffness than the beam with FG-CNTRC face sheets.

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7. Conclusions

A higher-order shear deformation beam theory is presented for free vibration analysis of

isotropic and FG-CNTRC sandwich beams in a uniform thermal environment. The obtained

solutions are in excellent agreement with those derived from earlier works. The proposed theory

is accurate and efficient in investigating free vibration and buckling behaviors of FG-CNTRC

sandwich beams. The effects of CNT volume fraction, core-to-face sheet thickness ratio and

slenderness ratio, on vibration behavior of stiff-cored sandwich beams with CNTRC face sheets

are evaluated with respect to uniform temperature changes through a parametric study.

Numerical results show that CNT volume fraction, and slenderness ratios have significant

influences on the natural frequencies of the structure whereas the effects of temperature change

and core-to-face sheet thickness ratio is less pronounced. It is found that the natural frequencies

of the sandwich beam decrease with an increase in temperature change, core-to-face and

slenderness ratio, but they would increase with an increase in CNT volume fraction. The

numerical results also pointed out that the sandwich beam with UD-CNTRC face sheets has

lower vibration performances compared to beam with FG-CNTRC face sheets.

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Fig. 1. A simple scheme of Sandwich beam with CNTRC face sheets.

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Figure 2- First dimensionless natural frequency of S-S sandwich beams with FG-CNTRC face

sheets with different CNT volume fraction

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.1 0.15 0.2 0.25 0.3Firs

t N

atu

ral F

req

ue

ncy

S-S ΔT=0 S-S ΔT=100 S-S ΔT=200

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Figure 3-First dimensionless natural frequencies of S-S sandwich beams with FG-CNTRC face

sheets with different slenderness ratios

0.05

0.1

0.15

0.2

0.25

0.3

0.35

5 10 15 20 25 30 35Fir

st N

atu

ral F

req

uen

cy

L/h


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Figure 4- first dimensionless natural frequencies of S-S sandwich beams with FG-CNTRC face

sheets with different core-to-face thickness ratio

0.15

0.155

0.16

0.165

0.17

0.175

0.18

3.5 4.5 5.5 6.5 7.5 8.5

Firs

t N

atu

ral F

req

uen

cy


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Table 1. Temperature- dependent properties for .

Material Properties

122.56e+9 0 -4.586e-4 0 0

7.5788e-6 0 6.638e-4 -3.147e-6 0

Ti-6Al-4V

0P

1P

1P

2P

3P

Ti-6Al-4V

( )E Pa

1( )K

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Table 2. Temperature- dependent properties for CNTs

Temperature (oK)

300 5.6466 7.0800 1.9445 3.4584

400 5.5679 6.9814 1.9703 4.1496

500 5.5308 6.9348 1.9643 4.5361

11 ( )cnE TPa 22 ( )

cnE TPa 12 ( )cnG TPa 1( )cn K

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Table 6. Comparison of the first three dimensionless natural frequencies of sandwich beams with

FG-CNTRC face sheets

( / 20L h , / 8c fh h ).

Mode * 0.12

cnV * 0.17

cnV * 0.28

cnV

Present [31] Present [31] Present [31]

1 FG 0.1453 0.1453 0.1588 0.1588 0.1825 0.1825

UD 0.1432 0.1432 0.1560 0.1560 0.1785 0.1785

2 FG 0.5730 0.5730 0.6247 0.6247 0.7174 0.7174

UD 0.5650 0.5650 0.6140 0.6140 0.6997 0.6997

3 FG 1.2599 1.2599 1.3689 1.3689 1.5554 1.5554

UD 1.2429 1.2429 1.3465 1.3465 1.5247 1.5246

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Table 7. Effect of nanotube volume fraction on first three dimensionless natural frequencies of

sandwich beams with FG-CNTRC face sheets

( / 20L h , / 8c fh h ).

0T 100T 200T

Mod

e

B.S

.

*cnV *

cnV *

cnV

0.12 0.17 0.28 0.12 0.17 0.28 0.12 0.17 0.28

1 S-S FG 0.1453 0.158

8

0.182

5

0.142

4

0.155

9

0.179

7

0.139

6

0.1531 0.177

0

S-S UD 0.1432 0.156

0

0.178

5

0.140

2

0.153

1

0.175

7

0.137

3

0.1503 0.172

9

2 S-S FG 0.5730 0.624

7

0.714

7

0.562

5

0.614

5

0.704

9

0.552

1

0.6044 0.695

1

S-S UD 0.5650 0.614

0

0.699

7

0.554

4

0.603

8

0.689

8

0.543

9

0.5936 0.680

0

3 S-S FG 1.2599 1.368

9

1.555

4

1.239

4

1.346

7

1.534

0

1.219

0

1.3246 1.512

6

S-S UD 1.2429 1.346

5

1.524

6

1.221

4

1.324

2

1.503

1

1.199

9

1.3019 1.481

6

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Table 8. Dimensionless first three natural frequencies of sandwich beams with FG-CNTRC face

sheets and different values of /L h

( / 8c fh h , * 0.17cnV ).

0T 100T 200T

Mod

e

B.S. /L h /L h /L h

10 20 30 10 20 30 10 20 30

1 S-S FG 0.312

4

0.158

8

0.106

2

0.307

3

0.155

9

0.103

9

0.302

3

0.1531 0.101

7

S-S UD 0.307

0

0.156

0

0.104

3

0.301

9

0.153

1

0.102

0

0.296

9

0.1503 0.099

7

2 S-S FG 1.175

6

0.624

7

0.421

7

1.156

2

0.614

5

0.414

5

1.136

9

0.6044 0.407

4

S-S UD 1.157

4

0.614

0

0.414

3

1.137

9

0.603

8

0.407

1

1.118

5

0.5936 0.400

0

3 S-S FG 2.428

5

1.368

9

0.937

1

2.386

6

1.346

7

0.921

9

2.344

8

1.3246 0.906

7

S-S UD 2.395

8

1.346

5

0.921

0

2.353

8

1.324

2

0.905

7

2.311

8

1.3019 0.890

4

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ACCEPTED MANUSCRIPT


Table 9. Dimensionless first three natural frequencies of sandwich beams with FG-CNTRC face

sheets and various values of /c fh h

( / 20L h , * 0.17cnV ).

0T 100T 200T

Mod

e

B.S

.

/c fh h /c fh h /c fh h

8 6 4 8 6 4

1 S-S FG 0.158

8

0.164

2

0.174

3

0.155

9

0.161

5

0.171

8

0.153

1

0.1588 0.1693

S-S UD 0.156

0

0.159

9

0.166

8

0.153

1

0.157

1

0.164

2

0.150

3

0.1544 0.1616

2 S-S FG 0.624

7

0.645

2

0.683

1

0.614

5

0.635

8

0.674

7

0.604

4

0.6265 0.6664

S-S UD 0.614

0

0.628

8

0.654

8

0.603

8

0.619

3

0.646

4

0.593

6

0.6099 0.6380

3 S-S FG 1.368

9

1.411

4

1.488

7

1.346

7

1.391

0

1.470

6

1.324

6

1.3706 1.4525

S-S UD 1.346

5

1.377

1

1.430

3

1.324

2

1.356

5

1.412

1

1.301

9

1.3360 1.3940

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higher-order shear deformation beam theory reinforced composite face sheets based...

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