Sound wave propagation in zigzag double-walled carbon nanotubes embedded in an elastic medium using nonlocal elasticity theory

Physica E 48 (2013) 118–123

Contents lists available at SciVerse ScienceDirect

Physica E journal homepage: www.elsevier.com/locate/physe

Sound wave propagation in zigzag double-walled carbon nanotubes embedded in an elastic medium using nonlocal elasticity theory Youcef Gafour a, Mohamed Zidour b,c, Abdelouahed Tounsi b,d,n, Houari Heireche a, Abdelwahed Semmah a a

De´partement de physique, Universite´ de Sidi Bel Abbe´s, BP 89 Cite´ Ben M0 hidi, 22000 Sidi Bel Abbe´s, Algeria Laboratoire des Mate´riaux et Hydrologie, Universite´ de Sidi Bel Abbe´s, BP 89 Cite´ Ben M0 hidi, 22000 Sidi Bel Abbe´s, Algeria c Universite´ Ibn Khaldoun, BP 78 Zaaroura, 14000 Tiaret, Algeria d De´partement de Ge´nie Civil, Faculte´ de Technologie, Universite´ de Sidi Bel Abbe´s, BP 89 Cite´ Ben M0 hidi, 22000 Sidi Bel Abbe´s, Algeria b

H I G H L I G H T S c

c

c

c

Wave propagation in zigzag DWCNT embedded in an elastic medium is studied. The influence of nonlocality on the vibration characteristics of DCWNTs is examined. The dependence of the frequencies on the chirality of zigzag carbon nanotube is shown. The equivalent Young’s modulus and shear modulus for zigzag SWCNT are derived using an energy-equivalent model.

G R A P H I C A L

A B S T R A C T

The influence of the chiral vector on the frequency of zigzag DWCNT embedded in an elastic medium is of concern. The problem is studied by using nonlocal elasticity theory.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 27 August 2012 Received in revised form 30 October 2012 Accepted 13 November 2012 Available online 2 December 2012

In the present work, nonlocal Euler–Bernoulli beam theory is used to investigate the wave propagation in zigzag double-walled carbon nanotube (DWCNT) embedded in an elastic medium. Winkler-type foundation model is employed to simulate the interaction of the DWCNT with the surrounding elastic medium. The DWCNTs are considered as two nanotube shells coupled through the van der Waals interaction between them. It is noticed in the presented study that the equivalent Young’s modulus for zigzag DWCNT is derived using an energy-equivalent model. Influences of nonlocal effects, the chirality of zigzag DWCNT, Winkler modulus parameter, and aspect ratio on the frequency of DWCNT are analyzed and discussed. The new features of the vibration behavior of zigzag DWCNTs embedded in an elastic medium and some meaningful results in this paper are helpful for the application and the design of nanostructures in which zigzag DWCNTs act as basic elements. & 2012 Elsevier B.V. All rights reserved.

1. Introduction Carbon nanotubes are tubular structures with nanometer diameter and micrometer length. Since the single-walled carbon nanotube (SWCNT) and multi-walled carbon nanotube (MWCNT) are found by Iijima [1], there have been extensive researches on n Corresponding author at: Laboratoire des Mate´riaux et Hydrologie, Universite´ de Sidi Bel Abbe´s, BP 89 Cite´ Ben M0 hidi, 22000 Sidi Bel Abbe´s, Algeria. E-mail address: [email protected] (A. Tounsi).

1386-9477/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.physe.2012.11.006

these nanomaterials. It has been found that carbon nanotubes possess many interesting and exceptional mechanical and electronic properties Therefore, it is expected that they can be used as promising materials for applications in nanoengineering. In order to make good use of these nanomaterials, it is important to have a good knowledge of their mechanical properties. Experiments at the nanoscale are extremely difficult and atomistic modeling remains prohibitively expensive for largesized atomic system. Consequently continuum models continue to play an essential role in the study of carbon nanotubes.

Y. Gafour et al. / Physica E 48 (2013) 118–123

Thereby size-dependent continuum based methods [2–4] are becoming popular in modeling small sized structures as it offers much faster solutions than molecular dynamics simulations for various engineering problems. Furthermore, the size dependent continuum mechanics are used because at small length-scales, the material microstructures (such as lattice spacing between individual atoms) become increasingly significant and its influence can no longer be ignored. The most widely reported theory for analyzing small scale structures is the nonlocal elasticity theory initiated by Eringen [5,6]. In nonlocal elasticity theory the small scale effects are captured by assuming that the stress at a point as a function not only of the strain at that point but also a function of the strains at all other points of the domain. The importance of nonlocal elasticity theory motivated the scientific community to explore the behavior of the micro/nanostructures more accurately and easily. The feasibility of nonlocal continuum theory in the field of nanotechnology was first reported by Peddieson et al. [7]. Various works related to nonlocal elasticity theory are found in several references [8–22]. Recently, considerable attention has been turned to the mechanical behavior of single and multi-walled carbon nanotubes embedded in polymer or metal matrix [23–28]. Vibration and buckling analyses [29–32] of CNTs have shown the employment of Winkler-type elastic foundation for modeling continuous surrounding elastic medium. Although several studies on the vibration behavior of CNTs have been carried out based on nonlocal Euler–Bernoulli beam theory, no studies can be found for the vibration behavior of zigzag CNTs embedded in an elastic medium. Therefore, in this paper, nonlocal Euler–Bernoulli beam theory has been implemented to investigate the vibration response of zigzag double-walled carbon nanotubes (DWCNTs) embedded in an elastic medium. Winkler-type model is employed to simulate the interaction of the DWCNTs with a surrounding elastic medium. The influence of the scale parameter, the van der Waals forces, Winkler modulus parameter and the effect of the chirality of zigzag DWCNT is taken into consideration in this study. It is hoped that the present analysis will be useful to researchers and engineers working on carbon nanotubes and CNT based composites.

Fig. 1. Hexagonal lattice of graphene sheet including base vectors.

Based on the link between molecular mechanics and solid mechanics, Wu et al. [34] developed an energy-equivalent model for studying the mechanical properties of SWCNTs. Using the same method, the equivalent Young’s modulus of zigzag SWCNT are expressed as pffiffiffi 4 3KC  , ESWNT ¼ ð3Þ 2 2 9Ct þ 4Ka2 t lz1 þ2lz2 where K and C are the force constants, t is the thickness of the nanotube and the parameters lz1 and lz2 are given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi   3 43cos2 p=2n cos p=2n pffiffiffi pffiffiffi , and kz1 ¼   8 32 3cos2 p=2n   129 cos2 p=2n ð4Þ kz2 ¼ pffiffiffi pffiffiffi 2  , 16 34 3cos p=2n Letting n-1, the expressions of Young’s modulus of a graphite sheet is given by pffiffiffi 8 3KC , ð5Þ Eg ¼ 18Ct þ Ka2 t Tu and Ou-Yang [35] indicated that the relation between Young’s modulus of multi-walled carbon nanotubes (MWCNTs) and the layer number N can be expressed as

2. Atomic structure of carbon nanotube EMWNT ¼ Carbon nanotubes are considered to be tubes formed by rolling ! a graphene sheet about the T vector. A vector perpendicular to ! ! the T is the chiral vector denoted by C h . The chiral vector and the corresponding chiral angle define the ! type of CNT, i.e. zigzag, armchair, chiral C h can be expressed with ! ! respect to two base vectors a 1 and a 2 as under: ! ! ! C h ¼ n a 1 þm a 2

na pffiffiffi 3, 2p

N t ESWNT N1 þ t=h h

ð6Þ

where EMWNT , ESWNT , t, Nand h are respectively the Young’s modulus of multi-walled nanotubes, Young’s modulus of singlewalled nanotubes, effective wall thickness of single-walled nanotubes, number of layers and layer distance. EMWNT ¼ ESWNT if N ¼ 1, which corresponds to the case of single-walled carbon nanotubes.

ð1Þ

where n and m are the indices of translation, which decide the structure around the circumference. Fig. 1 depicts the lattice indices ! ! of translation (n, m) along with the base vectors, a 1 and a 2 . If the indices of translation are such that m¼0 and n¼m then the corresponding CNTs are categorized as zigzag and armchair, respectively. Considering the chirality, the diameter and the chiral angle of the CNT can be calculated by the chiral vector for each nanostructure. The radius of the zigzag nanotube in terms of the chiral vector components can be obtained from the relation [33] R¼

119

ð2Þ

where a is the length of the carbon–carbon bond which is 1:42 A1.

3. Nonlocal double-elastic beam model The equations of motion for transversely vibrating Euler beam can be obtained as [36–38] @Q @2 w þ f ðxÞ þ pðxÞ ¼ qA 2 @x @t

ð7Þ

where pðxÞ is the distributed transverse force along axisx, wðx,t Þ is the transverse deflection, r is the density, A is the area of the cross section of the beam, f ðxÞ is the interaction pressure per unit axial length between the nanotube and the surrounding elastic medium, and Q is the resultant shear force on the cross section, which

120

Y. Gafour et al. / Physica E 48 (2013) 118–123

we have

satisfies the moment equilibrium condition @M Q¼ @x

ð8Þ

The resultant bending moment M in Eq. (8) is defined by Z M ¼ yrdA ð9Þ A

where y is the transverse coordinate measured positive in the direction of deflection and s is the nonlocal axial stress of the nonlocal continuum theory [6,7]. The one-dimensional nonlocal constitutive relation for the beam can be written as [6–8,16]

rðe0 aÞ2

@2 r ¼ Ee @x2

ð10Þ

where a is an internal characteristic length and e0 is a constant for adjusting the model in matching some reliable results by experiments or other models. E is the Young’s modulus of CNT. The nonlocal parameter ðe0 aÞ depends on the boundary conditions, chirality, mode shapes, number of walls, and type of motion [39]. So far, there is no rigorous study made on estimating the value of the nonlocal parameter. It is suggested that the value of nonlocal parameter can be determined by conducting a comparison of dispersion curves from the nonlocal continuum mechanics and molecular dynamics simulation [11,40]. In general, a conservative estimate of the nonlocal parameter is e0 a o 2:0 nmnm for a single wall carbon nanotube [11]. Using Bernoulli–Euler hypothesis for small deflection, the axial strain is given by [36] @2 w e ¼ y 2 @x

ð11Þ

According to Eqs. (10) and (11), Eq. (9) thus can be expressed as @2 M @2 w EI 2 ð12Þ 2 @x @x R 2 where I ¼ A y dA is the moment of inertia. By substituting Eqs. (7) and (8) into Eq. (12), the nonlocal bending moment M and shear force Q can be obtained as " # q2 w q2 w M ¼ EI 2 þ ðe0 aÞ2 qA 2 f ðxÞpðxÞ ð13Þ qx qt M ¼ ðe0 aÞ2

and Q ¼ EI

" # @3 w @3 w @f ðxÞ @pðxÞ 2  þ ð e a Þ q A  0 @x @x @x3 @x@t 2

p12 ¼ EI 1

@ 4 w1 @2 w1 @ 4 w1 @2 p12 þ q A1 ðe0 aÞ2 q A1  2 2 2 @x4 @x2 @t @x @t

f p12 ¼ EI 2

This is the general equation for transverse vibrations of an elastic beam under distributed transverse pressure with the surrounding elastic medium on the basis of nonlocal elasticity. It is noted that when the small scale parameter a vanishes, the above equation reduces to the classical Euler–Bernoulli expression (Eq. (7)). It is known that double-walled carbon nanotubes are distinguished from traditional elastic beam by their hollow two-layer structures and associated intertube van der Waals forces. Eq. (15) can be used to each of the inner and outer tubes of the doublewalled carbon nanotubes. Assuming that the inner and outer tubes have the same thickness and effective material constants,

!

ð16bÞ where subscripts 1 and 2 are used to denote the quantities associated with the inner and outer tubes, respectively,p12 denotes the van der Waals pressure per unit axial length exerted on the inner tube by the outer tube. The deflection of two tubes is coupled through the van der Waals force [41]. The van der Waals interaction potential, as a function of the interlayer spacing between two adjacent tubes, can be estimated by the Lennard–Jones model. The interlayer interaction potential between two adjacent tubes can be simply approximated by the potential obtained for two flat graphite monolayers, denoted bygðDÞ, where D is the interlayer spacing [42,43]. Since the interlayer spacing is equal or very close to an initial equilibrium spacing, the initial van der Waals force is zero for each of the tubes provided they deform coaxially. Thus, for small-amplitude sound waves, the van der Waals pressure should be a linear function of the difference of the deflections of the two adjacent layers at the point as follows: p12 ¼ cðw2 w1 Þ

ð17Þ

where c is the intertube interaction coefficient per unit length between two tubes, which can be estimated by [13] c¼

320ð2R1 Þerg=cm2 0:16d

2

ðd ¼ 0:142nmÞ

ð18Þ

where R1 is the radius of the inner tube. In addition the pressure per unit axial length, acting on the outermost tube due to the surrounding elastic medium, can be described by a Winkler type model [13] f ¼ kW w2

ð19Þ

where the negative sign indicates that the pressure f is opposite to the deflection of the outermost tube, and kW is spring constant of the surrounding elastic medium. Introduction of Eqs. (17) and (19) into Eqs. (16a) and (16b) yields cðw2 w1 Þ ¼ EI 1

ð15Þ

ð16aÞ

@ 4 w2 @2 w2 @ 4 w2 @2 f @2 p12 þ q A2 ðe0 aÞ2 q A2  2þ 4 2 2 2 @x @x @x2 @t @x @t

ð14Þ

The equation of motion (Eq. (7)) thus can be expressed by the transverse deflection as ! @4 w @2 w @4 w @2 f ðxÞ @2 p   f ðxÞ þ pðxÞ ¼ EI 4 þ q A 2 ðe0 aÞ2 q A @x @x2 @x2 @t @x2 @t 2

!

@ 4 w1 @2 w1 þ q A1 ðe0 aÞ2 @x4 @t 2

 q A1

@4 w1 @2 c 2 ðw2 w1 Þ 2 2 @x @x @t

kW w2 c ðw2 w1 Þ ¼ EI 2

! ð20aÞ

@ 4 w2 @2 w2 þ q A2 ðe0 aÞ2 4 @x @t 2

! @ 4 w2 @ 2 w2 @2  q A2 þ kW þ c 2 ðw2 w1 Þ @x2 @x @x2 @t 2 ð20bÞ With the effect of small length scale included, these two differential equations describe the free transverse vibrations of double-walled carbon nanotubes with the surrounding elastic medium modeled as an elastic foundation, and they are coupled together by the van der Waals interaction. Let us consider a double-walled carbon nanotube of length L in which the two ends are simply supported, so vibrational modes of

Y. Gafour et al. / Physica E 48 (2013) 118–123

the DWCNT are of the form [16] w1 ¼ a1 eio t sinkk x, w2 ¼ a2 eio t sinkk x,

and

kp kk ¼ ðk ¼ 1,2,:::::Þ L ð21Þ

where a1 and a2 are the amplitudes of deflections of the inner and outer tubes. Thus, the two k order resonant frequencies of the DWCNT with thermal effect can be obtained via nonlocal model by substituting Eq. (21) into Eqs. (20a) and (20b), which yields   qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 ð22Þ x2kI ¼ ak  a2k 4bk , x2kII ¼ ak þ a2k 4bk 2 2 with

an ¼

c ðA1 þ A2 Þ kW Ek4k ðA1 I 2 þ A2 I 1 Þ   þ þ qA1 A2 qA2 qA1 A2 1 þ ðe0 aÞ2 k2 k

ð23Þ

1

2

0

is that a carbon nanotube with smaller lattice indices of translation n has a larger curvature, which results in a more significant distortion of C–C bonds. As the structure characteristic of nanotube n increases, the effect of curvature diminishes gradually, and values of the Young’s modulus approach to the values of graphite sheet. This result has been also obtained and discussed by an earlier MD simulation [45,46]. To investigate the effect of scale parameter on vibrations of zigzag DWCNTs embedded in an elastic medium, the results including and excluding the nonlocal parameter are compared. In addition, the vibration characteristics of different zigzag DWCNTs are compared in order to explore the effect of chirality. It follows that the ratios of the results with the nonlocal parameter to those without nonlocal parameter are respectively given by:

vNI ¼

EI 1 þ EI 2 EI 1 EI 2   þ k8k bk ¼ c k4k  2 2 2 2 q A1 A2 1 þ ðe0 aÞ kk q2 A1 A2 1 þ ðe0 aÞ2 k2k 0 1 4 EI k c 1 k Ak  þ 2 þ@ q A1 A2 W q2 A A 1 þ ðe aÞ2 k2

ð24Þ

k

where okI is the lower natural frequency, okII is the higher natural frequency.

4. Results and discussions Based on the formulations obtained above with the nonlocal Euler–Bernoulli beam theory, the effect of both chirality and Winkler modulus parameter on vibration properties of zigzag double-walled nanotubes are discussed here. The parameters used in calculations for the zigzag DWCNTs are given as follows: the effective thickness of CNTs taken to be 0.258 nm [34], the force constants K/2¼46,900 kcal/mol/nm2 and C/2¼63 kcal/mol/ rad2 [44], the mass density r ¼2.3 g cm  3 [15] and layer distance h¼0.34 nm [35]. Fig. 2 shows the variation of Young’s modulus of the single and double-walled zigzag nanotubes with the structure characteristic of nanotube n. It can be seen that Young’s modulus of both zigzag SWCNT and DWCNT increases with increasing the value n. From Fig. 2, it can be clearly observed that for carbon nanotubes with lattice index of translation n, Young’s modulus exhibits a strong dependence on the structure characteristic of nanotube n. However for those with larger values of n, this dependence becomes very weak. The reason for this phenomenon

121

ðxkI ÞN , ðxkI ÞL

vNII ¼

ðxkII ÞN ðxkII ÞL

ð25Þ

where ðokI ÞL and ðokII ÞL are lower natural frequency and higher natural frequency, respectively, based on the local Euler–Bernoulli beam model and ðok ÞN is the frequency based on the nonlocal Euler–Bernoulli beam model. Figs. 3 and 4 respectively show the effect of small scale parameter on the lower and higher vibration response of embedded zigzag DWCNT with elastic medium modeled as Winkler-type foundation. For the present study, the nonlocal parameter (e0 a) values of DWCNT were taken in the range of 0– 2 nm as described in Ref. [11]. The aspect ratio L=d is taken as 40. The Winkler modulus ratio parameter (kW =c) values were taken in the range of 0–80. From Figs. 3 and 4, it is observed that there is a significant influence of small scale parameter on the vibration response of embedded zigzag DWCNT. Both lower frequency ratios (wNI ) and the higher frequency ratios (wNII ) considering nonlocal model are always smaller than the local (classical) model. This implies that the employment of the local Euler– Bernoulli beam model for DWCNT analysis would lead to an overprediction of the frequency if the small length scale effects between the individual carbon atoms are neglected. Further, with increase in e0 a values, the frequencies obtained by nonlocal Euler–Bernoulli theory become smaller compared to local model. Furthermore, it is seen that as the Winkler modulus ratio parameter increases, the frequency ratio increases. This increasing trend is attributed to the stiffness of the elastic medium. With higher values of Winkler modulus ratio parameter, the rate of increase of frequency ratio reduces. This implies that nonlocal or small-scale effect in vibration response of zigzag DWCNT looses its significance as the Winkler modulus ratio values increase. This

1.2

1,000

1.0

0,999

e0a=0 nm e0a=0.5 nm e0a=1 nm e0a=1.5 nm e0a=2 nm

0.9

Xnl

Young's Moduli (Tpa)

1.1

0.8

Graphite Zigzag SWCNT Zigzag DWCNT

0.7 0.6

0,998

zigzag nanotube (0,15)

0,997

0.5 0,996

0.4 2

4

6

8

10

12

14

16

18

20

n Fig. 2. The variation of Young’s modulus of zigzag single and double-walled carbon nanotubes.

10

20

30

40

50

60

k/c Fig. 3. Effect of Winkler modulus ratio parameter on the lower frequency ratio of zigzag DWCNT for various small-scale coefficients with (L=d ¼ 40 and k ¼ 6).

122

Y. Gafour et al. / Physica E 48 (2013) 118–123

1,00000

1,00000

0,99950

0,99925

N=1 N=2 N=3 N=4

0,99998

e0a=0 nm e0a=0.5 nm e0a=1 nm e0a=1.5 nm e0a=2 nm

Xnll

Xnll

0,99975

0,99996

zigzag nanotube (0,15) 0,99994

zigzag nanotube (0,15)

0,99992

0,99900 10

20

30

40

50

10

60

20

30

k/c Fig. 4. Effect of Winkler modulus ratio parameter on the higher frequency ratio of zigzag DWCNT for various small-scale coefficients with (L=d ¼ 40 and k ¼ 6).

40

50

60

k/c Fig. 6. Effect of Winkler modulus ratio parameter on the higher frequency ratio of zigzag DWCNT for various mode numbers with (e0 a ¼ 2nm and L=d ¼ 40). 1,000

1,00000

0,998

Xnl

k/c=5 k/c=10 k/c=15

0,996

0,99995

Xnl

N=1 N=2 N=3 N=4

0,99990

0,992 0,990

zigzag nanotube (0,15)

0,99985

0,994

0,988 2

4

6

8

10

0,99980 10

20

30

40

50

60

k/c Fig. 5. Effect of Winkler modulus ratio parameter on the lower frequency ratio of zigzag DWCNT for various mode numbers with (e0 a ¼ 2nm and L=d ¼ 40).

12

14

16

18

20

n Fig. 7. Relationship between the lower frequency ratio of DWCNT wNI , chirality of zigzag carbon nanotube n and the Winkler modulus ratio parameter (e0 a ¼ 2nm, k ¼ 6, L=d ¼ 40 ). 1,000 0,999

k/c=5 k/c=10 k/c=15

0,998

Xnll

is interpreted as follows: although the small-scale effect makes the CNTs more flexible as CNT being assumed as atoms linked by springs, the external elastic medium ‘‘grips’’ the CNTs and forces it to be stiffer. Hence the nonlocal effect is found to be more significant without the presence of elastic medium. Figs. 5 and 6 show the variation of frequency ratio with Winkler modulus ratio parameter for various modes of vibration. In the present study computation until four modes (k) of vibration are considered. It is observed that the nonlocal effects on vibration response are more significant for higher modes of vibration. This is interpreted from the fact that frequency ratio values for higher modes (k ¼ 3,4) are quite less than k ¼ 1. This significance of nonlocal effects in higher modes is attributed to the influence of small wavelength for higher modes. For smaller wavelengths, interactions between atoms are increasing and this leads to an increase in the nonlocal effects. Furthermore, as the Winkler modulus parameter increases, the frequency ratios increase for higher modes except for first mode of vibration. This implies that there is comparatively less effect of elastic medium on higher mode frequency of zigzag DWCNT. Figs. 7 and 8 show the dependence of the lower and higher frequency ratios on the chirality of zigzag carbon nanotube (n) for various Winkler modulus ratio parameter. The frequency ratios exhibit a dependence on the structure characteristic of zigzag carbon nanotube n. However for zigzag CNTs with larger values of n, this dependence becomes very weak. The reason for this phenomenon is that a carbon nanotube with smaller lattice indices of translation n has a larger curvature, which results in a more significant distortion of C–C bonds.

0,997 0,996 0,995 0,994 2

4

6

8

10

12

14

16

18

20

n Fig. 8. Relationship between the higher frequency ratio of DWCNT wNII , chirality of zigzag carbon nanotube n and the Winkler modulus ratio parameter (e0 a ¼ 2nm, k ¼ 6, L=d ¼ 40 ).

Figs. 9 and 10 illustrate the dependence of both lower frequency ratios (vNI ) and the higher frequency ratios (vNII ) on the chirality of zigzag carbon nanotube (n) for different values of the nonlocal parameter (e0 a). The frequency ratio serves as an index to assess quantitatively the scale effect on CNT vibration solution. It is clearly seen from Figs. 9 and 10 that for both cases of lower frequency ratios (vNI ) and the higher frequency ratios (vNII ), the frequency ratios are less than unity. This means that the application of the local Euler–Bernoulli beam model for CNT analysis would lead to an overprediction of the frequency if the scale effect between the individual carbon atoms in CNTs is

Y. Gafour et al. / Physica E 48 (2013) 118–123

the influences of long-range interatomic and intermolecular forces on the dynamic properties tend to be significant and cannot be neglected. The investigation presented may be helpful in the application of CNTs, such as ultrahigh-frequency resonators, electron emission devices, high-frequency oscillators and mechanical sensors.

1,000

e0a=0 nm e0a=0.5 nm e0a=1 nm e0a=1.5 nm e0a=2 nm

Xnl

0,996

0,992

123

Acknowledgments 0,988

2

4

6

8

10

12

14

16

18

20

This research was supported by the Algerian National Agency for Development of University Research (ANDRU) and University of Sidi Bel Abbes (UDL SBA) in Algeria.

n Fig. 9. Relationship between the lower frequency ratio of DWCNT wNI , chirality of zigzag carbon nanotube n and the nonlocal parameter (kW =c ¼ 5, k ¼ 6, L=d ¼ 40 ). 1,000

e0a=0 nm e0a=0.5 nm e0a=1 nm e0a=1.5 nm e0a=2 nm

0,998

Xnll

0,996

0,994

0,992

0,990 2

4

6

8

10

12

14

16

18

20

n Fig. 10. Relationship between the higher frequency ratio of DWCNT wNII , chirality of zigzag carbon nanotube n and the nonlocal parameter (kW =c ¼ 5, k ¼ 6, L=d ¼ 40 ).

neglected. The frequency ratios exhibit a dependence on the structure characteristic of zigzag carbon nanotube n. However for zigzag CNTs with larger values of n, this dependence becomes very weak. The reason for this phenomenon is that a carbon nanotube with smaller lattice indices of translation n has a larger curvature, which results in a more significant distortion of C–C bonds. Furthermore, the chirality of nanotube has not obvious effect on the ratios vNI or vNII when the local Euler–Bernoulli beam model is considered (e0 a ¼ 0). It can be seen also from Figs. 9 and 10 that the scale effect on the frequency ratios (vN ) diminishes with increasing the index of translation (n) and becomes more significant with the increase of the nonlocal parameter (e0 a). 5. Conclusions This paper studies the vibration of zigzag DWCNTs embedded in elastic medium based on Eringen’s nonlocal elasticity theory and the Euler–Bernoulli beam theory. Influence of the stiffness of the surrounding elastic medium on the lower and higher frequency of the zigzag CNTs is shown. According to the study, the results showed the dependence of the vibration characteristics on the chirality of zigzag DWCNTs and the nonlocal parameter. With the results, the dynamic properties of the DWCNT beam have been discussed in detail; they are shown to be very different from those predicted by classic elasticity when nonlocal effects become considerable. This means that as the length scales are reduced,

References [1] S. Iijima, Nature 354 (1991) 56. [2] S.J. Zhou, Z.Q. Li, Journal of Shandong University of Technology 31 (5) (2001) 401. [3] N.A. Fleck, J.W. Hutchinson, Advances in Applied Mechanics 33 (1997) 296. [4] F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, International Journal of Solids and Structures 39 (10) (2002) 2731. [5] A.C. Eringen, International Journal of Engineering Science 10 (1972) 1. [6] A.C. Eringen, Journal of Applied Physics 54 (1983) 4703. [7] J. Peddieson, G.G. Buchanan, R.P. McNitt, International Journal of Engineering Science 41 (2003) 305. [8] Y.Q. Zhang, G.R. Liu, X.Y. Xie, Physical Review 71 (2005) 195404-1. [9] P. Lu, H.P. Lee, C. Lu, P.Q. Zhang, International Journal of Solids and Structures 44 (2007) 5289. [10] P. Lu, H.P. Lee, C. Lu, P.Q. Zhang, Journal of Applied Physics 99 (2006) 510. [11] Q. Wang, C.M. Wang, Nanotechnology 18 (2007) 075702-1. [12] J.N. Reddy, S.D. Pang, Journal of Applied Physics 103 (2008) 023511. [13] L.J. Sudak, Journal of Applied Physics 94 (2003) 7281. [14] A. Benzair, A. Tounsi, A. Besseghier, H. Heireche, N. Moulay, L. Boumia, Journal of Physics D 41 (2008) 225404. [15] H. Heireche, A. Tounsi, A. Benzair, E.A. Adda Bedia, Physica E 40 (2008) 2791. [16] H. Heireche, A. Tounsi, A. Benzair, Nanotechnology 19 (2008) 185703. [17] T. Murmu, S.C. Pradhan, Physica E 41 (2009) 1232. [18] S.C. Pradhan, Physics Letters A 373 (2009) 4182. [19] S.C. Pradhan, T. Murmu, Computational Materials Science 47 (2009) 268. [20] M. S- ims-ek, Computational Materials Science 50 (2011) 2112. [21] M. S- ims- ek, Physica E 43 (2010) 182. [22] M. S- ims- ek, Steel and Composite Structures 11 (2011) 59. [23] C.Q. Ru, Journal of the Mechanics and Physics of Solids 49 (2001) 1265. [24] T. Kuzumaki, K. Miyazawa, H. Ichinose, K. Ito, Journal of Materials Research 13 (1998) 2445. [25] L.S. Schadler, S.C. Giannaris, P.M. Ajayan, Applied Physics Letters 73 (1998) 3842. [26] H.D. Wagner, O. Lourie, Y. Feldman, R. Tenne, Applied Physics Letters 72 (1998) 188. [27] C. Bower, R. Rosen, L. Jin, J. Han, O. Zhou, Applied Physics Letters 74 (1999) 3317. [28] D. Qian, E.C. Dickey, R. Andrews, T. Rantell, Applied Physics Letters 76 (2000) 2868. [29] A.R. Ranjbartoreh, A. Ghorbanpour, B. Soltani, Physica E 39 (2007) 230. [30] J. Yoon, C.Q. Ru, A. Mioduchowski, Composites Science and Technology 65 (2005) 1326. [32] J. Yoon, C.Q. Ru, A. Mioduchowski, International Journal of Solids and Structures 43 (2006) 3337. [33] Y. Tokio, Synthetic Metals 70 (1995) 1511. [34] Y. Wu, X. Zhang, A.Y.T. Leung, W. Zhong, Thin-Walled Structures 44 (2006) 667. [35] Z.C. Tu, Z.C. Ou-Yang, Physical Review B 65 (2002) 233407. [36] S.P. Timoshenko, Philosophical Magazine 41 (1921) 744. [37] W. Weaver, S.P. Timoshenko, D.H. Young, Vibration Problems in Engineering, Wiley, New York, 1990. [38] J.F. Doyle, Wave Propagation in Structures, 2nd edition, Springer, New York, 1997. [39] B. Arash, Q. Wang, Computational Materials Science 51 (2012) 303. [40] B. Arash, R. Ansari, Physica E 42 (2010) 2058. [41] B. Reulet, A.Yu. Kasumov, M. Kociak, R. Deblock, I.I. Khodos, B. Gorbatov, V.T. Volkov, C. Journet, H. Bouchiat, Physical Review Letters 85 (2000) 2829. [42] L.A. Girifalco, R.A. Lad, Journal of Chemical Physics 25 (1956) 693. [43] L.A. Girifalco, Journal of Chemical Physics 95 (1991) 5370. [44] W.D. Cornell, P. Cieplak, C.I. Bayly, et al., Journal of the American Chemical Society 117 (1995) 5179. [45] B.I. Yakobson, P.H. Avour, in: M.S. Dresselhaus, P.H. Avouris (Eds.), Carbon Nanotubes, Springer Verlag, Berlin-Heidelberg, 2001, Chapter 9, p. 287. [46] V.N. Popov, V.E. Van Doren, M. Balkanski, Physical Review B 61 (2000) 3078.