Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory

Computational Materials Science 46 (2009) 854–859

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Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory T. Murmu, S.C. Pradhan * Department of Aerospace Engineering, Indian Institute of Technology, Kharagpur, West Bengal 721 302, India

a r t i c l e

i n f o

Article history: Received 16 November 2008 Received in revised form 18 January 2009 Accepted 15 April 2009 Available online 17 May 2009 PACS: 61.46.Fg 46.70.p 46.15.x Keywords: Nonlocal elasticity Thermal vibration Single-walled carbon nanotube Differential quadrature

a b s t r a c t A single-elastic beam model has been developed to analyze the thermal vibration of single-walled carbon nanotubes (SWCNT) based on thermal elasticity mechanics, and nonlocal elasticity theory. The nonlocal elasticity takes into account the effect of small size into the formulation. Further, the SWCNT is assumed to be embedded in an elastic medium. A Winkler-type elastic foundation is employed to model the interaction of the SWCNT and the surrounding elastic medium. Differential quadrature method is being utilized and numerical solutions for thermal-vibration response of SWCNT is obtained. Influence of nonlocal small scale effects, temperature change, Winkler constant and vibration modes of the CNT on the frequency are investigated. The present study shows that for low temperature changes, the difference between local frequency and nonlocal frequency is comparatively high. With embedded CNT, for soft elastic medium and larger scale coefficients ðe0 aÞ the nonlocal frequencies are comparatively lower. The nonlocal model-frequencies are always found smaller than the local model-frequencies at all temperature changes considered. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction Among several possible smart nanoscale materials, carbon nanotubes (CNTs) have aroused great interest in the scientific community because of their exceptional mechanical, electronic, electrochemical, and thermal properties [1–3]. Carbon nanotubes, discovered in 1991 by Iijima [4], are long and thin cylinders of macromolecules composed of carbon atoms in a periodic hexagonal arrangement. The CNTs hold exciting promise in useful potential applications, as electrodes in supercapacitors, as cable materials for space elevators [5], as structural elements in nanoscale devices and reinforcing element in superstrong and conducting nanocomposites [6]. Much effort has so far been devoted to the study of the various aspects of nanotubes such as buckling, mechanical, chemical, and electrical properties. For the sake of the difficulties in experimental characterization of nanotubes and time-consuming and computationally expensive atomistic simulations, elastic continuum models have been widely used to study the vibrational behavior of CNTs. Some works are cited herein. Yoon et al. [7] studied the vibration behavior of multiwall carbon nanotubes embedded in an elastic medium using multiple-elastic beam model. Further * Corresponding author. Tel.: +91 3222 283008; fax: +91 3222 282242. E-mail address: [email protected] (S.C. Pradhan). 0927-0256/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2009.04.019

Yoon et al. [8] studied the vibration of short CNTs using Timoshenko beam model. Wang et al. [9] used Timoshenko beam model and DQ method for free vibration analysis of multi-walled carbon nanotubes. Zhang et al. [10], developed a double-elastic beam model is developed for transverse vibrations of double-walled carbon nanotubes under compressive axial load using Euler–Bernoulli beam theory. Study of vibrational and buckling behavior of carbon nanotubes is of practical interest for better understanding of mechanical responses of CNTs [11,12]. However, only a limited portion of the literature is concerned with the vibration and buckling analysis of carbon nanotubes considering the thermal effects. Thermal vibration frequencies could be used to calculate Young’s modulus of various nanotubes. Zhang et al. [13] studied the thermal effect on the vibration of double-walled carbon nanotubes using thermal elasticity. Wang et al. [14] studied the thermal effects on the vibration and instability of conveying fluid CNTs based on thermal elasticity mechanics. Hsu et al. [15] analyzed the frequency of chiral SWCNT subjected to thermal vibration and using Timoshenko beam model. Ni et al. [16] conducted an analysis of buckling behavior of singlewalled CNTs subjected to axial compression under a thermal environment. Yao and Han [17] performed a buckling analysis of MWNTs subjected to torsional load under temperature field. It is concluded that at room or lower temperature the critical load for infinitesimal buckling of a multi-walled carbon nanotubes

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(MWNT) increases as the value of the temperature change increases, while at a temperature higher than room temperature theoretical load for infinitesimal buckling of a MWNT decrease as the value of the temperature change increases. Li and Kardomateas [18] investigated the thermal buckling phenomenon of multiwalled carbon nanotubes in an elastic medium using nonlocal theory. Based on thermal elasticity mechanics, Zhang et al. [19] developed elastic multiple column model for column buckling of MWNTs with large aspect ratios under axial compression coupling with temperature change. They concluded that at low or room temperature the buckling strain including thermal effect is larger than that excluding the thermal effect and increases with the increase of temperature change. In most of these references the size effects of the CNTs were not taken into consideration. However some thermal based works with molecular dynamics (MD) solution has been reported. Cao et al. [20] carried out molecular dynamics (MD) and continuum mechanics analyses to study the effects of axial tension and compression, bending, torsion on the vibrational characteristics of single-walled carbon nanotubes. Temperature effects were also considered. Further, Cao et al. [21] performed numerical investigation to explore the thermal vibration behaviors of SWCNTs using MD simulations. Yao and Lordi [22] used MD simulation and thermal vibration frequencies to calculate Young’s modulus of various nanotubes. Since these simulations are computationally expensive and time consuming there is a need to develop the size-dependent continuum models for the thermo-mechanical vibration of SWCNT. One widely used theory is the nonlocal elasticity theory first initiated by Eringen [23]. 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 nonlocal elasticity contains information about inter atomic forces; and the internal length scale is introduced into the constitutive equations simply as a material parameter. The importance of nonlocal elasticity theory motivated the scientific community to explore the behavior of the micro/nano structures much accurately and easily. The feasibility of nonlocal continuum theory in the field of nanotechnology was first reported by Peddieson et al. [24]. Various works related to nonlocal elasticity theory are found in several references (Zhang et al. [25], Sudak [26], Lu et al. [27], Zhang et al. [28], Wang et al. [29], Murmu and Pradhan [30] and Pradhan and Murmu [31]). In this present paper a thermal vibration model is proposed to analyze the natural frequency of single-walled carbon nanotubes with simply supported ends using the nonlocal elasticity theory and Euler–Bernoulli theory. In addition the SWCNT is considered to be embedded in an elastic medium. A Winkler-type foundation model is assumed for simulating the interaction of the CNT and the elastic medium (polymer matrix). A differential quadrature (DQ) approach is being utilized and numerical solutions for the frequencies are obtained. Influence of nonlocal small scale effects, Winkler modulus parameter, temperature and modes of vibration of the CNT on the frequency are investigated and discussed. Difference of frequency results is shown for models with classical elasticity theory and nonlocal elasticity theory.

persion. The scale effects are accounted in this theory by considering internal size as a material parameter. The most general form of the constitutive relation for nonlocal elasticity involves an integral over the whole body. The basic equations for a linear homogenous nonlocal elastic body neglecting the body force are given as

rij;j ¼ 0 rij ðxÞ ¼ eij

Z

kðjx  x0 j; sÞ C ijkl ekl ðx0 Þ dVðx0 Þ;

ð1Þ

1 ¼ ðui;j þ uj;i Þ 2

The terms rij , C ijkl and eij are the stress, fourth order elasticity and strain tensors, respectively. kðjx  x0 j; sÞ is the nonlocal modulus or attenuation function incorporating into constitutive equations the nonlocal effects at the reference point x produced by local strain at the source x0 jx  x0 j represents the distance in Euclidean form and s is a material constant that depends on the internal (e.g. lattice parameter, granular size, distance between C–C bonds) and external characteristics lengths (e.g. crack length, wave length). Material constant s is defined as e0 a=‘. e0 is a constant for adjusting the model in matching with experimental results and by other models. The parameter e0 is estimated such that the relations of the nonlocal elasticity model could provide satisfied approximation of atomic dispersion curves of plane waves with those of atomic lattice dynamics. A value of 0.39 was used by Eringen [23] for e0. Duan et al. [32] presented a calibration of the small scaling parameter e0 using nonlocal Timoshenko beam theory and MDS results at room temperature conditions. The terms a and ‘ denote the internal and external lengths, respectively. Sudak [26] used the length of C–C bond equal to 0.142 nm for CNT stability analysis. When the internal characteristic length (C–C bonds length) is negligible compared to external characteristic length, s approaches zero and hence nonlocal elasticity theory reduces to classical elasticity theory. However when the internal characteristic length is reasonably close to external characteristic length then s approaches unity. This makes the nonlocal elasticity theory to reduce to atomic lattice dynamics. Thus for nano size structures such as graphene sheets, carbon nanotubes, micro/nano rods where internal and external characteristic lengths are of similar orders, one should use nonlocal theory for the analysis. This will provide better prediction of mechanical behaviors of the nano-structures. As solving of integral constitutive Eq. (1) is difficult, a simplified equation of differential form is used as a basis of all nonlocal constitutive formulation

ð1  s2 ‘2 r2 Þr ¼ t;

s¼

e0 a ‘

ð2Þ

where t = C:e and ‘:’ represents the double dot product. It is noted that the parameter e0 is estimated such that the relations (2) of the model could provide satisfied approximation of atomic dispersion curves of plane waves with those of atomic lattice dynamics. The above nonlocal constitutive Eq. (2) has been recently employed widely for the study of micro and nanostructure elements. The nonlocal constitutive relation (Eq. (2)) can be approximated to a onedimensional form as

2. Formulations

rðxÞ  ðe0 aÞ2

@ 2 rðxÞ ¼ EeðxÞ @x2

ð3Þ

sðxÞ  ðe0 aÞ2

@ 2 sðxÞ ¼ GcðxÞ @x2

ð4Þ

2.1. Nonlocal elasticity The essence of the nonlocal elasticity theory is that the stress field at a reference point x in an elastic continuum depends not only on strain at that point but also on strains at all other points in the domain [23]. This is in accordance with the atomic theory of lattice dynamics and experimental observations on phonon dis-

8x 2 V

where E is the Young’s modulus and G is the shear modulus. Thus, the scale coefficient (eoa) in the modeling will lead to small scale effect on the response of structures in nano-size.

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2.2. Thermal-nonlocal beam model for embedded SWCNT Many studies showed that the classic Euler-elastic beam offers a simple and reliable model for an overall mechanical deformation of CNTs, provided the characteristic wavelength is much larger than the diameter of CNTs. Therefore, the present work studies the thermal effect on the vibration of SWCNTs described by the Eringen nonlocal elasticity theory [23] and classic Bernoulli–Euler beam model, as shown in Fig. 1. In the present theory the plane cross sections of the beam remain plane during flexure and that the radius of curvature of a bent beam is large compared to the beam’s depth. In addition, CNT is assumed to be simply-supported at both ends, with or without being embedded into an elastic medium such as polymer. The governing equation of motion for the free vibration of a SWCNT considering thermal effects based on nonlocal elastic theory can be derived as

½EI þ N temp ðe0 aÞ2  þ kw þ m

@4w @4w @2w  mðe0 aÞ2 2 2  ½kðe0 aÞ2 þ N temp  2 4 @x @x @x @t

@ 2 wð0; tÞ @ 2 wðL; tÞ ¼ ¼0 @x2 @x2

ð9Þ

The solution of the differential equation given in Eq. (5) can be assumed as

wðx; tÞ ¼ WðxÞ eixt

ð10Þ

where W is the nondimensional amplitude and x is the circular natural pffiffiffiffiffiffiffi frequency. The term i is the conventional imaginary number 1. For the sake of simplicity the following dimensionless variables are introduced: 2

@2w ¼0 @t 2

ð5Þ

where x is an axial coordinate, t is time, w(x,t) is deflection of the CNT, I is the moment of inertia of the cross-section of the CNT, A is the area of the innermost cross-section of the CNT. The term k denotes the Winkler constant of the surrounding elastic medium described as a Winkler-type elastic foundation [33]. The term ðe0 aÞ denotes the small scale coefficient accounting the small size effects. Ntemp is the additional axial force arising due to thermal effects. It should be noted that the Eq. (5) reduces to the conventional Euler–Bernoulli equation when scale coefficient ðe0 aÞ is reduced to zero. On the basis of the theory of thermal elasticity mechanics, the axial force N temp can be written as [19]

Ntemp ¼ 

It should be noted that M is the nonlocal bending moment and not the classical bending moment. However it is interesting to note that for simply supported boundary condition, the boundary equations for the classical beam model and nonlocal beam models are same [30]. This is in view of w ¼ 0 at the boundaries. Consequently the nonlocal effects are neglected. For other boundary conditions this criteria may not apply. The derivative nonlocal boundary condition for the present study is thus reduced to

EA ax T 1  2m

ð6Þ

where ax is the coefficient of thermal expansion in the direction of x axis, and m is the Poisson’s ratio, respectively. T denotes the change in temperature. In the present study, it is assumed that only axial load due to temperature change exists on the SWCNT. Here it should be noted that the Young’s modulus is assumed to independent of temperature. According Hsieh et al. [34] Young’s modulus of an SWCNT is insensitive to temperature change in the tube at temperatures of less than approximately 1100 K, but decreases at higher temperatures. Here changes for low temperature environment will be considered. Consider the SWCNT simply-supported at the two ends. The CNT is considered to be of length L. The simply supported boundary conditions are specified by

wð0; tÞ ¼ wðL; tÞ ¼ 0

ð7Þ

M nonlocal ¼ 0 at x ¼ 0 and L:

ð8Þ

 temp ¼ Ntemp L ; N EI

X ¼ x=L;

X2 ¼

mL4 2 x; EI

b¼

eo a ; L

4

K¼

kL EI ð11Þ

Substituting Eqs. (10) and (11) into Eq. (5), the governing differential equation of motion can be deduced into the following dimensionless form 4

2

 temp b2 Þ d W  ðX2 b2 þ Kb2 þ N  temp Þ d W þ KW þ X2 W ¼ 0 ð1 þ N 4 2 dX dX ð12Þ If the effect of thermal vibration and nonlocal parameter are not taken into account (i.e., N temp ¼ 0; b ¼ 0), the equation reduces to that of classical Euler–Bernoulli beam model. 2.3. Solution by differential quadrature method The essence of DQ method is that a derivative of a function F is approximated as a weighted linear sum of all functional values within the computational domain. The differential quadrature method is proved to be a good computational tool for various engineering problems [35–37]. The derivative of a function F is assumed as n  N X d F  ðnÞ ¼ C i j Fðxj Þ n dx x¼xi j¼1

ð13Þ

where ð1Þ

C ij ¼

Rðxi Þ ðxi  xj ÞRðxj Þ

i; j ¼ 1; 2; :::; N;

i–j

ð14Þ

and

Rðxi Þ ¼

N Y

ðxi  xj Þ;

i–j

ð15Þ

j¼1

L ð1Þ

ð1Þ

C ij ¼ C ii ¼ 

ð1Þ

C ik ;

i ¼ 1; 2; :::; N; i – k; i ¼ j

ð16Þ

k¼1

Elastic Medium CNT

N X

d

The higher order weighting coefficient matrices are obtained from matrix multiplication: ð2Þ

C ij ¼

N X

ð1Þ

ð1Þ

C ik C kj ;

ð3Þ

Cij ¼

N X

k¼1 ð4Þ

Fig. 1. Simply-supported single-walled carbon nanotubes embedded in an elastic medium. The elastic medium is characterized by Winkler constant, k.

Cij ¼

N X k¼1

k¼1 ð1Þ

ð3Þ

C ik C kj ¼

N X k¼1

ð3Þ

ð1Þ

C ik C kj

ð1Þ

ð2Þ

C ik C kj ¼

N X

ð2Þ

ð1Þ

C ik C kj ;

k¼1

ð17Þ

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T. Murmu, S.C. Pradhan / Computational Materials Science 46 (2009) 854–859

The grid point is chosen based on well established grid points, Chebyshev–Gauss–Lobatto points [37]:

0.20

  1 ði  1Þ Xi ¼ p ; 1  cos 2 ðN  1Þ

0.18

T=10 K

0.16

T=25 K

ð18Þ

Substitution of Eq. (13) into Eq. (12) and rearranging, one could obtain the following differential quadrature eigen-value equation

½KfWg ¼ X2 fWg

ð19Þ

Using Eq. (19), one can easily compute the eigen-values numerically and obtain the frequencies of the CNTs with various parameter values. The nonlocal boundary conditions are incorporated within the formulation during the determination of weighting coefficients (MWCM approach). Details of the procedure can be seen in Murmu and Pradhan [30]. The present article thus brings out the simplicity and generality of employing DQ approach in the field of nonlocal elasticity theory.

frequency deviation percent

i ¼ 1; 2; :::; N

T=0.0 K

T=50 K 0.14 0.12 0.10 0.08 0.06 0.04 0.02

3. Numerical results and discussion

0.00

Frequency deviation percent ¼

100  ðfrequencyLocal  frequencyNonlocal Þ frequencyLocal

ð20Þ

The parameter is used to give a better illustration of the nonlocal effects in thermo-mechanical vibration response of CNT. Four different set of temperature change is considered viz, T = 0 K, 10 K, 25 K and 50 K. It is considered that the temperature change T is uniformly distributed in the SWCNT. Fig. 2 shows that as the small scale coefficient ðe0 aÞ increases the frequency deviation percent also increases. This reduction in fundamental frequency values of CNTs with scale coefficient is accounted for the small size of CNT. The rise of frequency deviation percent with ðe0 aÞ is found to be significantly dependent on temperature change. For 0 K, the frequency deviation percent increases heavily for scale coefficient ðe0 aÞ more than 1 nm. In addition for large temperature changes,

0.0

0.5

1.0

1.5

2.0

Scale coefficient, eoa Fig. 2. Variation of frequency deviation percent with scale coefficient e0 a for different change in temperature for SWCNT.

the difference between local frequency and nonlocal frequency is comparatively reduced. This shows that the thermal effect on CNTs reduces the scale effects (nonlocal effects). Fig. 3 depicts the variation of frequency deviation percent of carbon nanotubes with different stiffness of elastic medium. A Winkler-type foundation model is assumed for simulating the interaction of the CNT and the elastic medium. The stiffness of elastic medium is described by Winkler constant. For this study we suppose a temperature change of T = 30 K. As the stiffness of elastic medium increases (Winkler constant) the frequency deviation percent of carbon nanotubes decreases. It is obvious from the definition of frequency deviation percent that for local frequency is independent of Winkler constant values. For larger values of scale coefficients, the frequency deviation percent decreases with

2.50E-05

Frequency deviation percent

Based on the formulations obtained above with the nonlocal Bernoulli–Euler beam model, the thermal vibration properties of SWCNT are investigated and discussed here. A large aspect ratio of SWCNT is assumed to neglect the shear deformation in the analysis. To obtain the natural frequencies for a single-walled carbon nanotube (SWCNT) which is embedded in an elastic medium (Fig. 1), Eq. (12) is transformed to a DQ analogous form. A computer code is developed based on the differential quadrature approach. The grid distributions were taken based on Chebyshev– Gauss–Lobatto points. Sufficient number of grid points (N = 20) were employed to predict accurate nonlocal frequency results in the analysis. The effective properties of SWCNT are taken as that of Reddy and Pang [38]. The Young’s modulus E = 1000 GPa, mass density q = 2300 kg/m3, Poisson’s ratio m = 0.19 are considered in the analysis. An aspect ratio (L/2R) of 100 is taken in the analysis. The diameter is assumed as 1.0 nm. The scale coefficients are taken as eoa = 0.0 nm, 0.5 nm, 1.0 nm, 1.5 nm and 2.0 nm. These values were adopted because eoa should be smaller than 2.0 nm for carbon nanotubes as described by Wang and Wang [39]. It is reported (Jiang et al. [40]) that all the coefficients of thermal expansion for SWCNT are negative at low and room temperature and are positive at high temperature. In the present study, temperature change at low or room temperatures is considered. The coefficient of thermal expansion for CNTs is taken as 1:6  106 K1 [17]. First, the effect of surrounding elastic medium is neglected here (K W ¼ 0). Fig. 2 illustrates the variation of frequency deviation percent of carbon nanotubes with different temperature changes. Frequency deviation percent is defined as

Local model eoa=0.5 nm eoa=1.0 nm eoa=1.5 nm eoa=2.0 nm

2.00E-05

1.50E-05

1.00E-05

5.00E-06

0.00E+00

-5.00E-06

0.5

1

1.5

2

Winkler constant (GPa) Fig. 3. Variation of frequency deviation percent with Winkler constant for different values of scale coefficients e0 a (T = 30 K).

T. Murmu, S.C. Pradhan / Computational Materials Science 46 (2009) 854–859

Different percent ¼

100  ðfrequencyT¼T K  frequencyT¼0 K Þ frequencyT¼0 K ð21Þ

As the Winkler constant increases the difference percent of nonlocal frequencies decreases. For larger temperature change (T = 50 K), the drop of difference percent is quite significant. This shows the significant influence of temperature on the vibration response of embedded CNT. Fig. 5 shows the variation of fundamental frequencies with temperature change for frequencies computed from local model and nonlocal model ðe0 a ¼ 5 nmÞ: As the change in temperature increases the frequencies also increase. Consistent results of larger frequency values with thermal effects for SWCNT are also reported in Wang et al. [14] and Zhang et al. [13]. These conclusions are for the case of low or room temperature. The present thermal frequency behavior could be attributed to the fact that the SWCNT is subjected to initial axial load due to applied thermal stress, which increases the frequency. In Hsu et al. [15] frequency decreases with increasing thermal effects. In this study positive value for thermal coefficient was considered indicating a temperature change in high temperature environment. The present authors however not necessarily agree with the results reported in Refs. [20–22]. In these papers, the frequency is considered independent of temperature. However it should be noted that in these papers [20–22], molecular dynamic simulation has been carried out in a ‘‘constant temperature bath”. The present paper however refers to a ‘‘temperature change”. Moreover the present research results dealt with a simply supported-simply supported CNT which is unlike the results of Cao et al. [21] and Yao and Lordi [22], with cantilever CNT.

9.5 Local model Nonlocal model

9.0

Fundamental ferquency (GHz)

Winkler modulus significantly. However this trend shows the reducing influence of nonlocal effects in hard elastic medium (such as polymer matrix). For soft elastic medium the nonlocal frequencies have smaller values for larger scale coefficients ðe0 aÞ: The effect of temperature change on the frequency of CNT embedded in elastic medium is investigated. The frequency is numerically obtained considering the nonlocal effects. A scale coefficient ðe0 aÞ value of 1 nm is employed in the analysis. Fig. 4 shows the plot for difference percent versus Winkler constant for various changes in temperature. Difference percent is defined as

8.5 8.0 7.5 7.0 6.5 6.0 5.5 5.0 0

10

20

40

30

50

Temperature change (K) Fig. 5. Variation of fundamental frequencies with temperature change of SWCNT considering the local and the nonlocal models.

60

50

Frequency (GHz)

858

m=1 m=2 m=3

40

30

20

10

0.011 T=0 K T=10 K

Difference percent (freq)

0.009

0

T=25 K

0

T=50 K

10

20

30

40

50

Temperature change (K)

0.007

Fig. 6. Variation of natural frequencies with temperature change of SWCNT for the first three modes of vibration.

0.005

0.003

0.001

-0.001 0.5

1

1.5

2

Winkler constant (GPa) Fig. 4. Variation of difference percent with Winkler constant for various changes in temperature of SWCNT.

As found earlier the nonlocal model-frequencies are always smaller than the local model-frequencies. These features clearly show the importance of nonlocal elasticity in thermo-mechanical analysis of CNT. Finally variation of frequencies with temperature is shown for first three modes of vibration (Fig. 6). The frequencies are computed considering nonlocal effects ðe0 a ¼ 1:5 nmÞ: For all modes of vibration considered, the nonlocal frequencies increase with increase in temperature. Not much difference in variation trend of frequency is observed for the modes considered. However the effect of temperature change for the case of low or room temperature change on the variation of frequency with low and high mode

T. Murmu, S.C. Pradhan / Computational Materials Science 46 (2009) 854–859

effects). With embedded CNT, for soft elastic medium and larger scale coefficients ðe0 aÞ the nonlocal frequencies are comparatively lower. The nonlocal model-frequencies are always found smaller than the local model-frequencies at all temperature considered. Also not much difference in variation trend of frequency is observed for the first three modes of vibration. Finally the present study clearly shows the importance of using nonlocal elasticity in thermo-mechanical analysis of CNT.

2.0 T=0K T=10K

1.8

frequency ratio

T=25K T=50K

1.6

859

1.4

References

1.2 1.0 0.8 0

1

2

3

4

5

6

7

n Fig. 7. Thermal effects on the frequency ratio in the case of low or room temperature change using nonlocal theory.

numbers is shown in Fig. 7. The figure plots frequency ratio against the mode numbers. Frequency ratio is defined as the ratio of frequency based on nonlocal Euler–Bernoulli theory including thermal effects to the frequency based on nonlocal Euler–Bernoulli theory without thermal effects. It is clearly observed from the figure that frequencies with thermal effects are larger than those ignoring the influence of temperature change. Moreover the thermal effect on the frequencies decreases with the increase in the vibrational mode number n. The present results are consistent with that of Zhang et al. [13]. However the present frequencies would be comparatively smaller than that of Zhang et al. [13] because of the introduction of small scale or nonlocal effects. 4. Conclusions A thermo-mechanical nonlocal model has been developed to analyze the thermal vibration of single-walled carbon nanotubes. A Winkler-type elastic foundation is employed to model the surrounding elastic medium. Differential quadrature method is used to obtain numerical solutions for thermal-vibration response of SWCNT. For low temperature changes, the difference between local frequency and nonlocal frequency is comparatively high. In addition the thermal effect on CNTs reduces the scale effects (nonlocal

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