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The thermal effect on vibration of zigzag single walled carbon nanotubes using nonlocal Timoshenko beam theory
Computational Materials Science 51 (2012) 252–260
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
The thermal effect on vibration of zigzag single walled carbon nanotubes using nonlocal Timoshenko beam theory Mohamed Zidour a,b, Kouider Halim Benrahou a, Abdelwahed Semmah c, Mokhtar Naceri c, Hichem Abdesselem Belhadj a, Karima Bakhti a, Abdelouahed Tounsi a,⇑ a b c
Laboratoire des Matériaux et Hydrologie, Université de Sidi Bel Abbés, BP 89 Cité Ben M’hidi, 22000 Sidi Bel Abbés, Algeria Université Ibn Khaldoun, BP 78 Zaaroura, 14000 Tiaret, Algeria Université de Sidi Bel Abbés, Département de Physique, BP 89 Cité Ben M ’hidi, 22000 Sidi Bel Abbés, Algeria
a r t i c l e
i n f o
Article history: Received 18 May 2011 Received in revised form 10 July 2011 Accepted 12 July 2011 Available online 28 August 2011 Keywords: Single-walled carbon nanotubes Nonlocal elasticity Chirality Thermal effect
a b s t r a c t Based on nonlocal theory of thermal elasticity mechanics, a nonlocal elastic Timoshenko beam model is developed for free vibration analysis of zigzag single-walled carbon nanotube (SWCNT) considering thermal effect. The nonlocal constitutive equations of Eringen are used in the formulations. The equivalent Young’s modulus and shear modulus for zigzag SWCNT are derived using an energy-equivalent model. Results indicate significant dependence of natural frequencies on the temperature change as well as the chirality of zigzag carbon nanotube. These findings are important in mechanical design considerations of devices that use carbon nanotubes. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Carbon nanotubes (CNTs) are promising materials for the creation of novel nanodevices [1,2]. CNTs are cylindrical macromolecules composed of carbon atoms in a periodic hexagonal arrangement. As the nanotubes are found to have remarkable mechanical, physical, and chemical properties, they hold exciting promise as structural elements in nanoscale devices or reinforcing elements in superstrong nanocomposites [3,4]. In addition, CNTs are well known for their excellent rigidity, higher than that of steel and any other metal. Such superior properties are suitable for use in fabricating nanometre-scale electromechanical systems (NEMS). As an intimate understanding of the mechanical responses of individual carbon nanotubes is of great importance for their potential applications, the study of vibration behavior of carbon nanotubes is of practical interest. For the reasons of difficulties in the experimental characterization of nanotubes and the time-consuming and computationally expensive atomistic simulations, the mechanics of solids with continuum elastic models have been widely and successfully used to study the mechanical behavior of CNTs — the static deflection [5], buckling [6–8], thermal vibration [9,10], and resonant frequencies and modes [11–15]. However, the direct application
⇑ Corresponding author. Tel./fax: +213 48 564100. E-mail address: [email protected] (A. Tounsi). 0927-0256/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2011.07.021
of classical elastic models for CNTs may lead to inaccurate solutions because the influence of nanoscale size effects on the mechanical properties of CNTs cannot be captured by the classical models [16]. For these reasons, nonlocal continuum elastic models have been established to study the mechanical behaviors of CNTs. The nonlocal continuum or nonlocal elasticity theory was introduced by Eringen [17] to study screw dislocation and surface waves in solids. In the nonlocal elasticity theory, the size effects are captured by assuming the stress components at a point x are dependent not only on the strain components at the same point x but also on all other points in the domain [17]. It has been found that results from continuum mechanics approach are in good agreement with those of the atomistic approaches in many applications (i.e. [18,19]). Recently, the continuum mechanics approach has been widely and successfully used to study the mechanical behavior of CNTs, such as the static [20–24], the buckling [25–29], free vibration [30–35], wave propagation [36–42], thermo-mechanical analysis of CNTs [43,44]. More recently, Murmu and Adhikari [45] have analyzed the longitudinal vibration of double nanorod systems using the nonlocal elasticity. Murmu and Adhikari [46,47] have developed a nonlocal double-elastic beam model, and applied it to investigate the small scale effect on the free vibration and the axial instability of the double-nanobeam system. Sßimsßek [48] studied the dynamic behavior of a singlewalled carbon nanotube subjected to a moving harmonic load based on Eringen’s nonlocal elasticity theory. S ß imsßek [49] performed the dynamic analysis of an embedded single-walled carbon
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nanotube traversed by a moving nanoparticle based on the nonlocal Timoshenko beam theory, including transverse shear deformation and rotary inertia. Recently, S ß imsßek [50] presents a unique simple method of obtaining the exact solution based on the nonlocal Euler–Bernoulli beam theory for the forced vibration of an elastically connected double-carbon nanotube system (DCNTS) carrying a moving nanoparticle. In this paper, the dynamic behavior of zigzag SWCNT is investigated based on nonlocal Timoshenko beam model including both the temperature change and the chirality of zigzag carbon nanotube. The equivalent Young’s modulus and shear modulus for zigzag SWCNT are derived using an energy-equivalent model developed by Wu et al. [51]. The obtained results in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the thermal vibration properties of zigzag single-walled carbon nanotubes. 2. Atomic structure of carbon nanotube Carbon nanotubes are considered to be tubes formed by rolling a graphene sheet a bout the ~ T vector. A vector perpendicular to the ~ T is the chiral vector denoted by ~ Ch. 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 ~ a1 and ~ a2 as under
~ C h ¼ n~ a1 þ m~ a2
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, ~ a1 and ~ a2 . If the indices of translation are such that m = 0 and n = m 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 [52]
R¼
na pffiffiffi 3; 2p
ð2Þ
where a is the length of the carbon–carbon bond which is 1:42 A . Based on the link between molecular mechanics and solid mechanics, Wu et al. [51] developed an energy-equivalent model for studying the mechanical properties of SWCNTs. Using the same method, the equivalent Young’s modulus and shear modulus of zigzag nanotube are expressed as
Ez ¼
pffiffiffi 4 3KC 9Ct þ 4Ka2 tðk2z1 þ 2k2z2 Þ
; Gz ¼
pffiffiffi 2 3KC 6Ct þ 3Ka2 k2z t
ð3Þ
where K and C are the force constants, t is the thickness of the nanotube and the parameters kz1 , kz2 and kz are given by
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4 3 cos2 ðp=2nÞ cosðp=2nÞ pffiffiffi pffiffiffi ; kz2 8 3 2 3 cos2 ðp=2nÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 9 cos2 ðp=2nÞ p pffiffiffi ¼ pffiffiffi 1 ; and kz ¼ 4=3 cos2 2n 16 3 4 3 cos2 ðp=2nÞ
kz1 ¼
Letting n ? 1, the expressions of Young’s modulus and shear modulus of a graphite sheet is given by
Eg ¼
pffiffiffi 8 3KC 18Ct þ Ka2 t
; Gg ¼
pffiffiffi 2 3KC ð6C þ Ka2 Þt
ð5Þ
3. The nonlocal Timoshenko beam model of a SWCNT The elastodynamics differential equation that governs the thermo-mechanical vibration of the SWCNT based on the nonlocal Timoshenko beam model is [35]
" # 2 @2w @w @2 2@ w w ¼ qI 2 w ðe0 aÞ EI 2 þ bGA @x @x @x2 @t " # @ @w @2 @2w w þ Nt 2 w ðe0 aÞ2 2 @x @x @x @x " # 2 2 @ @ w ¼ qA 2 w ðe0 aÞ2 2 @x @t
ð6Þ
bGA
ð7Þ
where x is the axial coordinate, w the flexible transverse displacement of the neutral surface of the SWCNT, w is the rotation angle of cross section of the beam, q is the mass density of the material, A is the area of the cross section, E the modulus of elasticity, G the shear modulus, I the second moment of inertia of the cross-section of the SWCNT, b is the form factor of shear depending on the shape of the cross section. The recommended value of b, the adjustment coefficient, is 10/9 for a circular shape of the cross area [53]. Nt is the axial force arising from the thermal effect. The product term e0a, which has the dimension of length, is the small scale coefficient parameter that accounts for the extremely small size of the CNT. By equating the scale coefficient term to zero, one arrives at the vibration equation of the classical Timoshenko beam. The scale coefficient is the result of the formulation of the nonlocal theory of elasticity advanced by Eringen [17]. In this pioneering work, the small size effect is taken into account by specifying the stress state at a given point in the continuum to be a function of the strain states at all points in the continuum. Such a nonlocal stress–strain relationship accounts for the forces between atoms and their internal length scale. Some very recent developments of the theory can be found in the works of Reddy and Pang [54], Lu et al. [38] and Benzair et al. [35]. Zhang et al. [55], gave a predicted value of e0 and was approximated as 0.82. In the present result, this estimated value is used. a is an internal characteristic length (e.g. length of C–C bond, lattice spacing, granular distance). The thermally induced axial force (Nt) is derived from the constitutive relationship between the thermal strain and thermal stress, and it is defined as [35,56]
Nt ¼ aEAh
Fig. 1. Hexagonal lattice of graphene sheet including base vectors.
ð4Þ
ð8Þ
where h the temperature change during the thermal loading and a the thermal expansion coefficient. Since finding an analytical solution is possible for simply supported boundary conditions for the present problem, the SWNT beam is assumed simply supported. As a result, the boundary conditions have the following form [35]:
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ixt cosðkxÞ and kk ¼ kp=L wðx; tÞ ¼ Weixt sinðkxÞ; wðx; tÞ ¼ we
ð9Þ
is the where W is the amplitude of deflection of the beam, and w amplitude of the slope of the beam due to bending deformation alone. In addition, x is the frequency. Substitution of Eq. (9) into Eqs. (6) and (7) gives two branches of wave dispersion relation and the correspondent frequencies via nonlocal Timoshenko beam model are as follows;
ðx2k ÞNT ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ak a2k 4bk 2
ð10Þ
where
ak ¼
bk ¼
N t k2k bIGk2k þ bAG þ EIk2k 1 þ qA qI ð1 þ ðe0 aÞ2 k2k Þ
ð11Þ
EbGk4k
ðEIk2k þ bAGÞ 2 þ N k t k q2 ð1 þ ðe0 aÞ2 k2k Þ2 q2 IAð1 þ ðe0 aÞ2 k2k Þ
ð12Þ
It is seen that there are two wave modes for the Timoshenko beam model [38,41]. The wave mode with lower phase velocity is related to flexural wave, while the wave mode with higher phase velocity is related to shear wave out of interest.
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. As the structure characteristic of nanotube n increases, the effect of curvature diminishes gradually, and all of Young’s modulus and shear modulus approach to the values of graphite sheet. This result has been also obtained and discussed by an earlier MD simulation [61,62]. As indicated by Jiang et al. [58], the coefficients of thermal expansion for CNTs are negative at lower temperature and become positive at higher temperature. Consequently, two cases of low temperature and high temperature are considered. For the case of room or low temperature, we suppose a = 1.6 106 K1 [35,59,60] and for the case of high temperature, we suppose a = 1.1 106 K1 [35,59,60]. To investigate the effect of scale parameter and temperature change on vibrations of zigzag SWCNTs, the results including and excluding the thermal effect and the nonlocal parameter are compared. In addition, the vibration characteristics of different zigzag SWCNTs are compared in order to explore the effect of chirality. It follows that the ratios of the results with temperature change and nonlocal parameter to those without temperature change or nonlocal parameter are respectively given by:
vN ¼ 4. Results and discussions Based on the formulations obtained above with the nonlocal Timoshenko beam model, the effect of the lattice indices of translation n on the thermal vibration properties of zigzag single-walled nanotubes are discussed here. The parameters used in calculations for the zigzag SWCNTs are given as follows: the effective thickness of CNTs taken to be 0.258 nm [51], the force constants K /2 = 46,900 kcal/mol/nm2 and C /2 = 63 kcal/mol/rad2 [57], the mass density q = 2.3 g cm3 [35,41]. Fig. 2 shows the variation of Young’s modulus with the structure characteristic of nanotube n. It can be seen that Young’s modulus of zigzag nanotubes increases with increasing the value n. For the shear modulus of carbon nanotubes, the same variation trend is present in Fig. 3. From Figs. 2 and 3, it can be clearly observed that for carbon nanotubes with lattice index of translation n, Young’s modulus and shear modulus exhibit 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
ðxk ÞNT ðxk ÞNT ;v ¼ ðxk ÞLT th ðxk Þ0NT
ð13Þ
where (xk)LT is the frequency based on the local Timoshenko beam model including thermal effect and ðxk Þ0NT is the frequency based on the nonlocal Timoshenko beam model without thermal effect (h = 0). Figs. 4–7 illustrate the dependence of the frequency ratios (vN) on the chirality of zigzag carbon nanotube (n) for both cases of low and high temperatures. The frequency ratio vN serves as an index to assess quantitatively the scale effect on CNT vibration solution. It is clearly seen from Figs. 4–7 that for both cases of low and high temperatures, the frequency ratios (vN) are less than unity. This means that the application of the local Timoshenko 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 neglected. The frequency ratios (vN) 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.
1,1
Young's Moduli (Tpa)
1,0
0,9
Zigzag nanotube Graphite
0,8
0,7
0,6
0,5 0
2
4
6
8
10
12
14
n Fig. 2. The variation of Young’s modulus.
16
18
20
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M. Zidour et al. / Computational Materials Science 51 (2012) 252–260
0,5
Shear Moduli (Tpa)
0,4
Zigzag nanotube Graphite
0,3
0,2
0,1
0,0 0
2
4
6
8
10
12
14
16
18
20
18
20
n Fig. 3. The variation of Shear modulus.
1,00
N
0,98
k=1 k=2 k=4 k=6
0,96
0,94
0,92 2
4
6
8
10
n
12
14
16
Fig. 4. Relationship between the values of ratio vN, chirality of zigzag carbon nanotube n and the vibrational mode number k in the case of low or room temperature (h = 40 K) and L/d = 40.
1,00 0,99
k=1 k=2 k=4 k=6
N
0,98 0,97 0,96 0,95 0,94 2
4
6
8
10
n
12
14
16
18
20
Fig. 5. Relationship between the values of ratio vN, chirality of zigzag carbon nanotube n and the vibrational mode number k in the case of high temperature (h = 40 K) and L/d = 40.
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1,000
0,995
N
0,990
L/d =10 L/d =20 L/d =40
0,985
0,980
0,975
0,970 2
4
6
8
10
12
14
16
18
20
n Fig. 6. Relationship between the values of ratio vN, chirality of zigzag carbon nanotube n and the aspect ratio L/d in the case of low or room temperature (h = 40 K) and k = 1.
1,000
0,995
L/d =10 L/d =20 L/d =40
N
0,990
0,985
0,980
0,975
0,970 2
4
6
8
10
12
14
16
18
20
n Fig. 7. Relationship between the values of ratio vN, chirality of zigzag carbon nanotube n and the aspect ratio L/d in the case of high temperature (h = 40 K) and k = 1.
Furthermore, the chirality of nanotube has not obvious effect on the ratio vN when the fundamental mode is considered. It can be seen from Figs. 4 and 5 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 vibrational mode k. However, the scale effect becomes less significant with the increase of the length-to-diameter ratio (L/d) as is shown in Figs. 6 and 7. Therefore, it is clear that the small scale effect is significant for short CNTs. The effects of temperature change on the vibration frequencies for both cases of low and high temperatures are shown in Figs. 8 and 9 with the aspect ratio L/d = 40 and the vibrational mode k = 1. It can be seen that the frequency ratios (vth) vary linearly with the temperature change. In the case of room or low temperature (Fig. 8), the ratio vth increases monotonically as the temperature h increases, indicating that the effect of temperature change leads to an increase of the fundamental frequency and especially for the zigzag nanotubes with higher index of translation (n). Contrary to the case of room or low temperature, it can be seen from Fig. 9 that the frequency ratios (vth) are less than unity. This means that the values of (xk)NT considering the thermal effect are smaller than those excluding the influence of temperature change.
For the case of high temperature, the thermal effects on the vibration frequencies are shown in Fig. 9. Contrary to the case of room or low temperature, it can be seen from Fig. 9 that the frequency ratios (vth) are less than unity. This means that the fundamental frequency diminishes with increasing the temperature change. In addition, a zigzag nanotube with higher index of translation (n) will have the smallest fundamental frequency. Figs. 10–13 show the dependence of the frequency ratios (vth) on the chirality of zigzag carbon nanotube (n) for both cases of low and high temperatures. With the aspect ratio L/d = 40 and the temperature change h = 40 K, the relationship among the ratio vth, the index of translation n and the vibrational mode k is indicated in Figs. 10 and 11 for both cases of low and high temperatures, respectively. It can be seen that the chirality of nanotube has not obvious effect on the ratio vth for higher vibration modes. However, this effect becomes more significant for the fundamental mode. In addition, it can be observed that at low or room temperature the frequency ratio vth increases with the increase of the index of translation n, while at high temperature the ratio vth decreases with the increase of n. With the vibrational mode number k = 1 and the temperature change h = 40 K, the relationship among the ratio vth, the index
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1,06 1,05
th
1,04 1,03 1,02
(0,5) (0,10)
1,01
(0,15)
1,00
0
10
20
30
40
50
60
(K) Fig. 8. Thermal effects on vibration frequencies for different zigzag SWCNTs with the vibrational mode number k = 1 and L/d = 40 in the case of low or room temperature.
1,00
0,99
th
0,98
(0,5) 0,97
(0,10) (0,15)
0,96
0
10
20
30
40
50
60
(K) Fig. 9. Thermal effects on vibration frequencies for different zigzag SWCNTs with the vibrational mode number k = 1 and L/d = 40 in the case of high temperature.
1,040 1,035 1,030
k=1 k=2
1,025 th
k=4 1,020
k=6
1,015 1,010 1,005 1,000 2
4
6
8
10
12
14
16
18
20
n Fig. 10. Relationship between the values of ratio vth, chirality of zigzag carbon nanotube n and the vibrational mode number k in the case of low or room temperature (h = 40 K) and L/d = 40.
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1,000
0,995
0,990 th
k=1 k=2
0,985
k=4 k=6
0,980
0,975
0,970 2
4
6
8
10
12
14
16
18
20
n Fig. 11. Relationship between the values of ratio vth, chirality of zigzag carbon nanotube n and the vibrational mode number k in the case of the case of high temperature (h = 40 K) and L/d = 40.
1,040 1,035 1,030
L/d =10 L/d =20
1,025 th
L/d =40 1,020 1,015 1,010 1,005 1,000 2
4
6
8
10
12
14
16
18
20
n Fig. 12. Relationship between the values of ratio vth, chirality of zigzag carbon nanotube n and the aspect ratio L/d in the case of low or room temperature (h = 40 K) and k = 1.
1,000
0,995
0,990 th
L/d =10 L/d =20
0,985
L/d =40 0,980
0,975
0,970 2
4
6
8
10
12
14
16
18
20
n Fig. 13. Relationship between the values of ratio vth, chirality of zigzag carbon nanotube n and the aspect ratio L/d in the case of high temperature (h = 40 K) and k = 1.
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M. Zidour et al. / Computational Materials Science 51 (2012) 252–260 Table 1 The values of frequency ratios (vth) of zigzag carbon nanotubes using nonlocal Timoshenko beam model for different temperatures. Zigzag nanotube
Aspect ratio L/d
Temperature h (K)
High temperature
Low temperature
High temperature
Low temperature
(0, 5)
10
20 40 60 20 40 60 20 40 60
0.99934 0.99868 0.99803 0.99747 0.99494 0.99241 0.98995 0.97984 0.96960
1.00095 1.00191 1.00286 1.00366 1.00731 1.01094 1.01444 1.02863 1.04263
0.99994 0.99988 0.99982 0.99989 0.99979 0.99968 0.99969 0.99938 0.99906
1.00009 1.00017 1.00026 1.00015 1.00031 1.00046 1.00045 1.00091 1.00136
20 40 60 20 40 60 20 40 60
0.99917 0.99833 0.99750 0.99674 0.99349 0.99021 0.98689 0.97375 0.96043
1.00121 1.00242 1.00362 1.00470 1.00939 1.01406 1.01842 1.03665 1.05457
0.99995 0.99990 0.99984 0.99988 0.99976 0.99965 0.99961 0.99923 0.99884
1.00007 1.00015 1.00022 1.00017 1.00034 1.00051 1.00056 1.00112 1.00168
20 40 60 20 40 60 20 40 60
0.99912 0.99824 0.99736 0.99656 0.99311 0.98966 0.98621 0.97240 0.95807
1.00127 1.00255 1.00382 1.00496 1.00992 1.01484 1.01961 1.03887 1.05790
0.99995 0.99990 0.99985 0.99988 0.99976 0.99964 0.99959 0.99919 0.99878
1.00007 1.00015 1.00022 1.00018 1.00035 1.00053 1.00059 1.00118 1.00176
20
40
(0, 10)
10
20
40
(0, 15)
10
20
40
k=1
Table 2 The values of frequency ratios (vN) of zigzag carbon nanotubes using nonlocal Timoshenko beam model (h = 40 K). Zigzag nanotube
Aspect ratio L/d
(0, 5)
k=1
k=6
High temperature
Low temperature
High temperature
Low temperature
10 20 40
0.99569 0.99892 0.99952
0.99569 0.99892 0.99952
0.87303 0.96314 0.98310
0.87303 0.96314 0.98310
(0, 10)
10 20 40
0.99973 0.99892 0.99973
0.99973 0.99892 0.99973
0.99039 0.96314 0.99039
0.99039 0.96314 0.99039
(0, 15)
10 20 40
0.99988 0.99995 0.99997
0.99988 0.99995 0.99997
0.99569 0.99808 0.99892
0.99569 0.99808 0.99892
k=6
the temperature change and the chirality of zigzag carbon nanotube. According to the study, the results showed the dependence of the vibration characteristics on the chirality of zigzag carbon nanotube. It is shown that the dynamical properties of the nanotubes based on the classical Timoshenko beam theory are over estimated. Hence, the work in the manuscript not only reveals the significance of the small-scale effect and the chirality effect on CNTs mechanical response, but also points out the limitation of the applicability and feasibility of local continuum models in analysis of CNTs mechanical behaviors. The thermal effect on the frequencies decreases with the increase of the vibrational mode number k and increase with increasing the aspect ratio L/d, the index of translation n and temperature change h. Acknowledgments
of translation n and the aspect ratio L/d is indicated in Figs. 12 and 13 for both cases of low and high temperatures, respectively. As can be seen, at low or room temperature the ratio vth increases with increasing the index of translation (n) and becomes more significant with the increase of the aspect ratio L/d, while at high temperature the ratio vth diminishes with the increase of n and the aspect ratio L/d. Hence, the chirality effect on the ratio vth is significant for slender CNTs. The results of zigzag CNTs with different length-to-diameter ratios and different temperatures for the first and the sixth modes based on the nonlocal Timoshenko beam model are listed in Tables 1 and 2, respectively.
5. Conclusions This paper investigates the thermal vibration of zigzag SWCNTs based on Timoshenko beam theory and Eringen’s nonlocal elasticity theory. Theoretical formulations include the small scale effect,
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The thermal effect on vibration of zigzag single walled carbon nanotubes using nonlocal Timoshenko beam theory Mohamed Zidour a,b, Kouider Halim Benrahou a, Abdelwahed Semmah c, Mokhtar Naceri c, Hichem Abdesselem Belhadj a, Karima Bakhti a, Abdelouahed Tounsi a,⇑ a b c
Laboratoire des Matériaux et Hydrologie, Université de Sidi Bel Abbés, BP 89 Cité Ben M’hidi, 22000 Sidi Bel Abbés, Algeria Université Ibn Khaldoun, BP 78 Zaaroura, 14000 Tiaret, Algeria Université de Sidi Bel Abbés, Département de Physique, BP 89 Cité Ben M ’hidi, 22000 Sidi Bel Abbés, Algeria
a r t i c l e
i n f o
Article history: Received 18 May 2011 Received in revised form 10 July 2011 Accepted 12 July 2011 Available online 28 August 2011 Keywords: Single-walled carbon nanotubes Nonlocal elasticity Chirality Thermal effect
a b s t r a c t Based on nonlocal theory of thermal elasticity mechanics, a nonlocal elastic Timoshenko beam model is developed for free vibration analysis of zigzag single-walled carbon nanotube (SWCNT) considering thermal effect. The nonlocal constitutive equations of Eringen are used in the formulations. The equivalent Young’s modulus and shear modulus for zigzag SWCNT are derived using an energy-equivalent model. Results indicate significant dependence of natural frequencies on the temperature change as well as the chirality of zigzag carbon nanotube. These findings are important in mechanical design considerations of devices that use carbon nanotubes. Ó 2011 Elsevier B.V. All rights reserved.
1. Introduction Carbon nanotubes (CNTs) are promising materials for the creation of novel nanodevices [1,2]. CNTs are cylindrical macromolecules composed of carbon atoms in a periodic hexagonal arrangement. As the nanotubes are found to have remarkable mechanical, physical, and chemical properties, they hold exciting promise as structural elements in nanoscale devices or reinforcing elements in superstrong nanocomposites [3,4]. In addition, CNTs are well known for their excellent rigidity, higher than that of steel and any other metal. Such superior properties are suitable for use in fabricating nanometre-scale electromechanical systems (NEMS). As an intimate understanding of the mechanical responses of individual carbon nanotubes is of great importance for their potential applications, the study of vibration behavior of carbon nanotubes is of practical interest. For the reasons of difficulties in the experimental characterization of nanotubes and the time-consuming and computationally expensive atomistic simulations, the mechanics of solids with continuum elastic models have been widely and successfully used to study the mechanical behavior of CNTs — the static deflection [5], buckling [6–8], thermal vibration [9,10], and resonant frequencies and modes [11–15]. However, the direct application
⇑ Corresponding author. Tel./fax: +213 48 564100. E-mail address: [email protected] (A. Tounsi). 0927-0256/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2011.07.021
of classical elastic models for CNTs may lead to inaccurate solutions because the influence of nanoscale size effects on the mechanical properties of CNTs cannot be captured by the classical models [16]. For these reasons, nonlocal continuum elastic models have been established to study the mechanical behaviors of CNTs. The nonlocal continuum or nonlocal elasticity theory was introduced by Eringen [17] to study screw dislocation and surface waves in solids. In the nonlocal elasticity theory, the size effects are captured by assuming the stress components at a point x are dependent not only on the strain components at the same point x but also on all other points in the domain [17]. It has been found that results from continuum mechanics approach are in good agreement with those of the atomistic approaches in many applications (i.e. [18,19]). Recently, the continuum mechanics approach has been widely and successfully used to study the mechanical behavior of CNTs, such as the static [20–24], the buckling [25–29], free vibration [30–35], wave propagation [36–42], thermo-mechanical analysis of CNTs [43,44]. More recently, Murmu and Adhikari [45] have analyzed the longitudinal vibration of double nanorod systems using the nonlocal elasticity. Murmu and Adhikari [46,47] have developed a nonlocal double-elastic beam model, and applied it to investigate the small scale effect on the free vibration and the axial instability of the double-nanobeam system. Sßimsßek [48] studied the dynamic behavior of a singlewalled carbon nanotube subjected to a moving harmonic load based on Eringen’s nonlocal elasticity theory. S ß imsßek [49] performed the dynamic analysis of an embedded single-walled carbon
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nanotube traversed by a moving nanoparticle based on the nonlocal Timoshenko beam theory, including transverse shear deformation and rotary inertia. Recently, S ß imsßek [50] presents a unique simple method of obtaining the exact solution based on the nonlocal Euler–Bernoulli beam theory for the forced vibration of an elastically connected double-carbon nanotube system (DCNTS) carrying a moving nanoparticle. In this paper, the dynamic behavior of zigzag SWCNT is investigated based on nonlocal Timoshenko beam model including both the temperature change and the chirality of zigzag carbon nanotube. The equivalent Young’s modulus and shear modulus for zigzag SWCNT are derived using an energy-equivalent model developed by Wu et al. [51]. The obtained results in this paper can provide useful guidance for the study and design of the next generation of nanodevices that make use of the thermal vibration properties of zigzag single-walled carbon nanotubes. 2. Atomic structure of carbon nanotube Carbon nanotubes are considered to be tubes formed by rolling a graphene sheet a bout the ~ T vector. A vector perpendicular to the ~ T is the chiral vector denoted by ~ Ch. 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 ~ a1 and ~ a2 as under
~ C h ¼ n~ a1 þ m~ a2
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, ~ a1 and ~ a2 . If the indices of translation are such that m = 0 and n = m 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 [52]
R¼
na pffiffiffi 3; 2p
ð2Þ
where a is the length of the carbon–carbon bond which is 1:42 A . Based on the link between molecular mechanics and solid mechanics, Wu et al. [51] developed an energy-equivalent model for studying the mechanical properties of SWCNTs. Using the same method, the equivalent Young’s modulus and shear modulus of zigzag nanotube are expressed as
Ez ¼
pffiffiffi 4 3KC 9Ct þ 4Ka2 tðk2z1 þ 2k2z2 Þ
; Gz ¼
pffiffiffi 2 3KC 6Ct þ 3Ka2 k2z t
ð3Þ
where K and C are the force constants, t is the thickness of the nanotube and the parameters kz1 , kz2 and kz are given by
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4 3 cos2 ðp=2nÞ cosðp=2nÞ pffiffiffi pffiffiffi ; kz2 8 3 2 3 cos2 ðp=2nÞ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12 9 cos2 ðp=2nÞ p pffiffiffi ¼ pffiffiffi 1 ; and kz ¼ 4=3 cos2 2n 16 3 4 3 cos2 ðp=2nÞ
kz1 ¼
Letting n ? 1, the expressions of Young’s modulus and shear modulus of a graphite sheet is given by
Eg ¼
pffiffiffi 8 3KC 18Ct þ Ka2 t
; Gg ¼
pffiffiffi 2 3KC ð6C þ Ka2 Þt
ð5Þ
3. The nonlocal Timoshenko beam model of a SWCNT The elastodynamics differential equation that governs the thermo-mechanical vibration of the SWCNT based on the nonlocal Timoshenko beam model is [35]
" # 2 @2w @w @2 2@ w w ¼ qI 2 w ðe0 aÞ EI 2 þ bGA @x @x @x2 @t " # @ @w @2 @2w w þ Nt 2 w ðe0 aÞ2 2 @x @x @x @x " # 2 2 @ @ w ¼ qA 2 w ðe0 aÞ2 2 @x @t
ð6Þ
bGA
ð7Þ
where x is the axial coordinate, w the flexible transverse displacement of the neutral surface of the SWCNT, w is the rotation angle of cross section of the beam, q is the mass density of the material, A is the area of the cross section, E the modulus of elasticity, G the shear modulus, I the second moment of inertia of the cross-section of the SWCNT, b is the form factor of shear depending on the shape of the cross section. The recommended value of b, the adjustment coefficient, is 10/9 for a circular shape of the cross area [53]. Nt is the axial force arising from the thermal effect. The product term e0a, which has the dimension of length, is the small scale coefficient parameter that accounts for the extremely small size of the CNT. By equating the scale coefficient term to zero, one arrives at the vibration equation of the classical Timoshenko beam. The scale coefficient is the result of the formulation of the nonlocal theory of elasticity advanced by Eringen [17]. In this pioneering work, the small size effect is taken into account by specifying the stress state at a given point in the continuum to be a function of the strain states at all points in the continuum. Such a nonlocal stress–strain relationship accounts for the forces between atoms and their internal length scale. Some very recent developments of the theory can be found in the works of Reddy and Pang [54], Lu et al. [38] and Benzair et al. [35]. Zhang et al. [55], gave a predicted value of e0 and was approximated as 0.82. In the present result, this estimated value is used. a is an internal characteristic length (e.g. length of C–C bond, lattice spacing, granular distance). The thermally induced axial force (Nt) is derived from the constitutive relationship between the thermal strain and thermal stress, and it is defined as [35,56]
Nt ¼ aEAh
Fig. 1. Hexagonal lattice of graphene sheet including base vectors.
ð4Þ
ð8Þ
where h the temperature change during the thermal loading and a the thermal expansion coefficient. Since finding an analytical solution is possible for simply supported boundary conditions for the present problem, the SWNT beam is assumed simply supported. As a result, the boundary conditions have the following form [35]:
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ixt cosðkxÞ and kk ¼ kp=L wðx; tÞ ¼ Weixt sinðkxÞ; wðx; tÞ ¼ we
ð9Þ
is the where W is the amplitude of deflection of the beam, and w amplitude of the slope of the beam due to bending deformation alone. In addition, x is the frequency. Substitution of Eq. (9) into Eqs. (6) and (7) gives two branches of wave dispersion relation and the correspondent frequencies via nonlocal Timoshenko beam model are as follows;
ðx2k ÞNT ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ak a2k 4bk 2
ð10Þ
where
ak ¼
bk ¼
N t k2k bIGk2k þ bAG þ EIk2k 1 þ qA qI ð1 þ ðe0 aÞ2 k2k Þ
ð11Þ
EbGk4k
ðEIk2k þ bAGÞ 2 þ N k t k q2 ð1 þ ðe0 aÞ2 k2k Þ2 q2 IAð1 þ ðe0 aÞ2 k2k Þ
ð12Þ
It is seen that there are two wave modes for the Timoshenko beam model [38,41]. The wave mode with lower phase velocity is related to flexural wave, while the wave mode with higher phase velocity is related to shear wave out of interest.
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. As the structure characteristic of nanotube n increases, the effect of curvature diminishes gradually, and all of Young’s modulus and shear modulus approach to the values of graphite sheet. This result has been also obtained and discussed by an earlier MD simulation [61,62]. As indicated by Jiang et al. [58], the coefficients of thermal expansion for CNTs are negative at lower temperature and become positive at higher temperature. Consequently, two cases of low temperature and high temperature are considered. For the case of room or low temperature, we suppose a = 1.6 106 K1 [35,59,60] and for the case of high temperature, we suppose a = 1.1 106 K1 [35,59,60]. To investigate the effect of scale parameter and temperature change on vibrations of zigzag SWCNTs, the results including and excluding the thermal effect and the nonlocal parameter are compared. In addition, the vibration characteristics of different zigzag SWCNTs are compared in order to explore the effect of chirality. It follows that the ratios of the results with temperature change and nonlocal parameter to those without temperature change or nonlocal parameter are respectively given by:
vN ¼ 4. Results and discussions Based on the formulations obtained above with the nonlocal Timoshenko beam model, the effect of the lattice indices of translation n on the thermal vibration properties of zigzag single-walled nanotubes are discussed here. The parameters used in calculations for the zigzag SWCNTs are given as follows: the effective thickness of CNTs taken to be 0.258 nm [51], the force constants K /2 = 46,900 kcal/mol/nm2 and C /2 = 63 kcal/mol/rad2 [57], the mass density q = 2.3 g cm3 [35,41]. Fig. 2 shows the variation of Young’s modulus with the structure characteristic of nanotube n. It can be seen that Young’s modulus of zigzag nanotubes increases with increasing the value n. For the shear modulus of carbon nanotubes, the same variation trend is present in Fig. 3. From Figs. 2 and 3, it can be clearly observed that for carbon nanotubes with lattice index of translation n, Young’s modulus and shear modulus exhibit 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
ðxk ÞNT ðxk ÞNT ;v ¼ ðxk ÞLT th ðxk Þ0NT
ð13Þ
where (xk)LT is the frequency based on the local Timoshenko beam model including thermal effect and ðxk Þ0NT is the frequency based on the nonlocal Timoshenko beam model without thermal effect (h = 0). Figs. 4–7 illustrate the dependence of the frequency ratios (vN) on the chirality of zigzag carbon nanotube (n) for both cases of low and high temperatures. The frequency ratio vN serves as an index to assess quantitatively the scale effect on CNT vibration solution. It is clearly seen from Figs. 4–7 that for both cases of low and high temperatures, the frequency ratios (vN) are less than unity. This means that the application of the local Timoshenko 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 neglected. The frequency ratios (vN) 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.
1,1
Young's Moduli (Tpa)
1,0
0,9
Zigzag nanotube Graphite
0,8
0,7
0,6
0,5 0
2
4
6
8
10
12
14
n Fig. 2. The variation of Young’s modulus.
16
18
20
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0,5
Shear Moduli (Tpa)
0,4
Zigzag nanotube Graphite
0,3
0,2
0,1
0,0 0
2
4
6
8
10
12
14
16
18
20
18
20
n Fig. 3. The variation of Shear modulus.
1,00
N
0,98
k=1 k=2 k=4 k=6
0,96
0,94
0,92 2
4
6
8
10
n
12
14
16
Fig. 4. Relationship between the values of ratio vN, chirality of zigzag carbon nanotube n and the vibrational mode number k in the case of low or room temperature (h = 40 K) and L/d = 40.
1,00 0,99
k=1 k=2 k=4 k=6
N
0,98 0,97 0,96 0,95 0,94 2
4
6
8
10
n
12
14
16
18
20
Fig. 5. Relationship between the values of ratio vN, chirality of zigzag carbon nanotube n and the vibrational mode number k in the case of high temperature (h = 40 K) and L/d = 40.
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1,000
0,995
N
0,990
L/d =10 L/d =20 L/d =40
0,985
0,980
0,975
0,970 2
4
6
8
10
12
14
16
18
20
n Fig. 6. Relationship between the values of ratio vN, chirality of zigzag carbon nanotube n and the aspect ratio L/d in the case of low or room temperature (h = 40 K) and k = 1.
1,000
0,995
L/d =10 L/d =20 L/d =40
N
0,990
0,985
0,980
0,975
0,970 2
4
6
8
10
12
14
16
18
20
n Fig. 7. Relationship between the values of ratio vN, chirality of zigzag carbon nanotube n and the aspect ratio L/d in the case of high temperature (h = 40 K) and k = 1.
Furthermore, the chirality of nanotube has not obvious effect on the ratio vN when the fundamental mode is considered. It can be seen from Figs. 4 and 5 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 vibrational mode k. However, the scale effect becomes less significant with the increase of the length-to-diameter ratio (L/d) as is shown in Figs. 6 and 7. Therefore, it is clear that the small scale effect is significant for short CNTs. The effects of temperature change on the vibration frequencies for both cases of low and high temperatures are shown in Figs. 8 and 9 with the aspect ratio L/d = 40 and the vibrational mode k = 1. It can be seen that the frequency ratios (vth) vary linearly with the temperature change. In the case of room or low temperature (Fig. 8), the ratio vth increases monotonically as the temperature h increases, indicating that the effect of temperature change leads to an increase of the fundamental frequency and especially for the zigzag nanotubes with higher index of translation (n). Contrary to the case of room or low temperature, it can be seen from Fig. 9 that the frequency ratios (vth) are less than unity. This means that the values of (xk)NT considering the thermal effect are smaller than those excluding the influence of temperature change.
For the case of high temperature, the thermal effects on the vibration frequencies are shown in Fig. 9. Contrary to the case of room or low temperature, it can be seen from Fig. 9 that the frequency ratios (vth) are less than unity. This means that the fundamental frequency diminishes with increasing the temperature change. In addition, a zigzag nanotube with higher index of translation (n) will have the smallest fundamental frequency. Figs. 10–13 show the dependence of the frequency ratios (vth) on the chirality of zigzag carbon nanotube (n) for both cases of low and high temperatures. With the aspect ratio L/d = 40 and the temperature change h = 40 K, the relationship among the ratio vth, the index of translation n and the vibrational mode k is indicated in Figs. 10 and 11 for both cases of low and high temperatures, respectively. It can be seen that the chirality of nanotube has not obvious effect on the ratio vth for higher vibration modes. However, this effect becomes more significant for the fundamental mode. In addition, it can be observed that at low or room temperature the frequency ratio vth increases with the increase of the index of translation n, while at high temperature the ratio vth decreases with the increase of n. With the vibrational mode number k = 1 and the temperature change h = 40 K, the relationship among the ratio vth, the index
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1,06 1,05
th
1,04 1,03 1,02
(0,5) (0,10)
1,01
(0,15)
1,00
0
10
20
30
40
50
60
(K) Fig. 8. Thermal effects on vibration frequencies for different zigzag SWCNTs with the vibrational mode number k = 1 and L/d = 40 in the case of low or room temperature.
1,00
0,99
th
0,98
(0,5) 0,97
(0,10) (0,15)
0,96
0
10
20
30
40
50
60
(K) Fig. 9. Thermal effects on vibration frequencies for different zigzag SWCNTs with the vibrational mode number k = 1 and L/d = 40 in the case of high temperature.
1,040 1,035 1,030
k=1 k=2
1,025 th
k=4 1,020
k=6
1,015 1,010 1,005 1,000 2
4
6
8
10
12
14
16
18
20
n Fig. 10. Relationship between the values of ratio vth, chirality of zigzag carbon nanotube n and the vibrational mode number k in the case of low or room temperature (h = 40 K) and L/d = 40.
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1,000
0,995
0,990 th
k=1 k=2
0,985
k=4 k=6
0,980
0,975
0,970 2
4
6
8
10
12
14
16
18
20
n Fig. 11. Relationship between the values of ratio vth, chirality of zigzag carbon nanotube n and the vibrational mode number k in the case of the case of high temperature (h = 40 K) and L/d = 40.
1,040 1,035 1,030
L/d =10 L/d =20
1,025 th
L/d =40 1,020 1,015 1,010 1,005 1,000 2
4
6
8
10
12
14
16
18
20
n Fig. 12. Relationship between the values of ratio vth, chirality of zigzag carbon nanotube n and the aspect ratio L/d in the case of low or room temperature (h = 40 K) and k = 1.
1,000
0,995
0,990 th
L/d =10 L/d =20
0,985
L/d =40 0,980
0,975
0,970 2
4
6
8
10
12
14
16
18
20
n Fig. 13. Relationship between the values of ratio vth, chirality of zigzag carbon nanotube n and the aspect ratio L/d in the case of high temperature (h = 40 K) and k = 1.
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M. Zidour et al. / Computational Materials Science 51 (2012) 252–260 Table 1 The values of frequency ratios (vth) of zigzag carbon nanotubes using nonlocal Timoshenko beam model for different temperatures. Zigzag nanotube
Aspect ratio L/d
Temperature h (K)
High temperature
Low temperature
High temperature
Low temperature
(0, 5)
10
20 40 60 20 40 60 20 40 60
0.99934 0.99868 0.99803 0.99747 0.99494 0.99241 0.98995 0.97984 0.96960
1.00095 1.00191 1.00286 1.00366 1.00731 1.01094 1.01444 1.02863 1.04263
0.99994 0.99988 0.99982 0.99989 0.99979 0.99968 0.99969 0.99938 0.99906
1.00009 1.00017 1.00026 1.00015 1.00031 1.00046 1.00045 1.00091 1.00136
20 40 60 20 40 60 20 40 60
0.99917 0.99833 0.99750 0.99674 0.99349 0.99021 0.98689 0.97375 0.96043
1.00121 1.00242 1.00362 1.00470 1.00939 1.01406 1.01842 1.03665 1.05457
0.99995 0.99990 0.99984 0.99988 0.99976 0.99965 0.99961 0.99923 0.99884
1.00007 1.00015 1.00022 1.00017 1.00034 1.00051 1.00056 1.00112 1.00168
20 40 60 20 40 60 20 40 60
0.99912 0.99824 0.99736 0.99656 0.99311 0.98966 0.98621 0.97240 0.95807
1.00127 1.00255 1.00382 1.00496 1.00992 1.01484 1.01961 1.03887 1.05790
0.99995 0.99990 0.99985 0.99988 0.99976 0.99964 0.99959 0.99919 0.99878
1.00007 1.00015 1.00022 1.00018 1.00035 1.00053 1.00059 1.00118 1.00176
20
40
(0, 10)
10
20
40
(0, 15)
10
20
40
k=1
Table 2 The values of frequency ratios (vN) of zigzag carbon nanotubes using nonlocal Timoshenko beam model (h = 40 K). Zigzag nanotube
Aspect ratio L/d
(0, 5)
k=1
k=6
High temperature
Low temperature
High temperature
Low temperature
10 20 40
0.99569 0.99892 0.99952
0.99569 0.99892 0.99952
0.87303 0.96314 0.98310
0.87303 0.96314 0.98310
(0, 10)
10 20 40
0.99973 0.99892 0.99973
0.99973 0.99892 0.99973
0.99039 0.96314 0.99039
0.99039 0.96314 0.99039
(0, 15)
10 20 40
0.99988 0.99995 0.99997
0.99988 0.99995 0.99997
0.99569 0.99808 0.99892
0.99569 0.99808 0.99892
k=6
the temperature change and the chirality of zigzag carbon nanotube. According to the study, the results showed the dependence of the vibration characteristics on the chirality of zigzag carbon nanotube. It is shown that the dynamical properties of the nanotubes based on the classical Timoshenko beam theory are over estimated. Hence, the work in the manuscript not only reveals the significance of the small-scale effect and the chirality effect on CNTs mechanical response, but also points out the limitation of the applicability and feasibility of local continuum models in analysis of CNTs mechanical behaviors. The thermal effect on the frequencies decreases with the increase of the vibrational mode number k and increase with increasing the aspect ratio L/d, the index of translation n and temperature change h. Acknowledgments
of translation n and the aspect ratio L/d is indicated in Figs. 12 and 13 for both cases of low and high temperatures, respectively. As can be seen, at low or room temperature the ratio vth increases with increasing the index of translation (n) and becomes more significant with the increase of the aspect ratio L/d, while at high temperature the ratio vth diminishes with the increase of n and the aspect ratio L/d. Hence, the chirality effect on the ratio vth is significant for slender CNTs. The results of zigzag CNTs with different length-to-diameter ratios and different temperatures for the first and the sixth modes based on the nonlocal Timoshenko beam model are listed in Tables 1 and 2, respectively.
5. Conclusions This paper investigates the thermal vibration of zigzag SWCNTs based on Timoshenko beam theory and Eringen’s nonlocal elasticity theory. Theoretical formulations include the small scale effect,
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