Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory

Author’s Accepted Manuscript Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory Yugang Tang, Ying Liu, Dong Zhao www.elsevier.com/locate/physe

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S1386-9477(16)30293-4 http://dx.doi.org/10.1016/j.physe.2016.06.007 PHYSE12482

To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 25 April 2016 Revised date: 30 May 2016 Accepted date: 7 June 2016 Cite this article as: Yugang Tang, Ying Liu and Dong Zhao, Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2016.06.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory Yugang Tang, Ying Liu*, Dong Zhao Department of Mechanics, School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, PR China *

E-mail address: [email protected]

In this paper, the viscoelastic wave propagation in an embedded viscoelastic single-walled carbon nanotube (SWCNT) is studied based on the nonlocal strain gradient theory. The characteristic equation for the viscoelastic wave in SWCNTs is derived. The emphasis is placed on the influence of the tube diameter on the viscoelastic wave dispersion. A blocking diameter is observed, above which the wave could not propagate in SWCNTs. The results show that the blocking diameter is greatly dependent on the damping coefficient, the nonlocal and the strain gradient length scale parameters, as well as the Winkler modulus of the surrounding elastic medium. These findings may provide a prospective application of SWCNTs in nanodevices and nanocomposites. Keywords: Viscoelastic single walled nanotube; Viscoelastic wave; Blocking diameter; Nonlocal strain gradient theory 1.

Introduction Since the discovery of carbon nanotubes (CNTs) by Iijima [1], there are amounts of researches on properties of CNTs by

adopting theoretical analysis, experimental investigation and atomistic simulations [2-8]. Along with the continuous finding of the distinguished mechanical, chemical, thermal and electrical properties, CNTs have been widely used in nanodevices and nanocomposities [9, 10]. In recent years, with the growing interest in terahertz physics of nanoscale materials and devices, the studies of vibration and wave propagation in CNTs become more and more important for CNT-based nanodevices and nano- composites [11-17]. Lately, viscoelastic properties of CNTs

have been reported by Xu et al. [18] for an operational temperature range of -196oC

to 1000 oC. It is found that this temperature invariant viscoelasticity can be affected by the van de Waals interaction and the thermal stability of CNTs. Moreover, as the size of the CNT is close to atom scale, the size effect becomes very obvious and must be taken into account for a better prediction of the mechanical properties of CNTs [19-21]. So the nonlocal elasticity theory developed by Eringen [22, 23] is adopted to analyze the size-dependent vibration and wave propagation behaviors of viscoelastic CNTs. In the nonlocal elasticity theory, the size effect is characterized by assuming that the stress at a reference point is a function of the strain field at every point of the body. Based on the nonlocal elasticity theory, Chang and Lee [24] explored the thermal and foundation effects on the vibrational frequency of a viscoelastic CNT. Ghavanloo and Fazelzadeh [25] studied the flexural vibration of viscoelastic CNTs conveying fluid and embedded in viscous fluid. The viscoelastic property of the CNT, as well as the coupled effects of temperature change, nonlocal parameter and the external viscous fluid on the frequencies and the structural instabilities were analyzed and discussed. Ghorbanpour Arani et al.[26] investigated the combined effects of the magnetic field and visco-Pasternak foundation on the vibrational behavior of double bonded viscoelastic CNTs conveying viscous fluid by adopting the visco-surface nonlocal theory. For the wave in CNTs, Pang et al. [27] used the nonlocal elasticity theory to discuss the viscoelastic damping and surface effects on the transverse wave propagation in viscoelastic CNTs. However, it is pointed out by some investigators [28-33] that there are some limited situations in the capability of the nonlocal elasticity theory to identify the size-dependent stiffness. Specifically speaking, the nonlocal elasticity theory cannot characterize the stiffness enhancement effects, which are observed in experiments, as well as the modified couple stress theory prediction [34]. In addition, it is questionable that there needs no extra boundary conditions 1

in the nonlocal elastic model since some extra boundary conditions are expected to be added in the higher-order elasticity models [28, 35, 36]. More recently, Lim et al. [31] proposed the nonlocal strain gradient theory, introducing two independent small length-scale parameters, namely, the nonlocal parameter and the strain gradient length scale parameter into a single theory. In the nonlocal strain gradient theory, the stress takes into account not only the non-gradient nonlocal elastic stress field [22, 23], but also the nonlocality of higher-order strain gradient stress field [37], which reasonably explains the size-dependent wave propagation behavior of CNTs and results in an excellent matching with the results given by molecular dynamics (MD) simulations. Li et al. [32] investigated the influences of longitudinal magnetic fields and surface effects on wave propagation in viscoelastic SWCNTs by use of the nonlocal strain gradient theory. The results showed that the effects of the nonlocal parameter and the material length scale parameter on the wave dispersion relation are significant. Li and Hu [33] applied the nonlocal strain gradient theory to study the wave propagation in fluid-conveying viscoelastic SWCNTs. The effects of the damping coefficient, internal moving fluid, the nonlocal parameter and the material length scale parameter on the phase velocity and damping ratio were analyzed. Unfortunately, although the effect of the tube size on the wave propagation in CNTs has been realized, to the best of authors’ knowledge, the distinguish propagation characteristic due to the viscoelasticity of CNTs and the variation of the tube diameter has not yet been fully considered, especially when the effect of the material length scale is included. In this paper, the viscoelastic wave propagation in an embedded viscoelastic SWCNT is investigated based on the nonlocal strain gradient theory. The distinct propagation phenomenon for waves in visco-CNTs caused by the tube diameter variation is clarified. The influence of the foundation stiffness, the damping coefficient, as well as the nonlocal and the material length scale parameters on the wave dispersion in visco-CNTs is discussed in detail. 2.

Theoretical formulation The geometry of a SWCNT is considered to be a tube constructed by rolling up a graphite sheet, as shown in Fig. 1. Then

the chiral vector Ch and the chiral angle θ are with the form [38]

Ch  na1  ma2 ,

  arccos

2n  m 2 n  nm  m2 2

(1)

,

(2)

where a1 and a2 are the two unit directional vectors. The chirality indices (n, m) decide the structure around the circumference.

Fig. 1. Schematic representation of a SWCNT.

The diameter of a CNT is then determined as [38]

D

3a



n 2  nm  m2 ,

(3)

where a is the length of the carbon-carbon bond with the value a=0.142 nm. Many experiments have been carried out to measure Young’s modulus of CNTs [39-42]. Considering the diversity in the experimental values reported, the theoretical analysis and atomistic simulations on the Young’s modulus of CNTs have been made to verify the experimental findings and offer some required information. Poncharal et al. [42] determined the Young's 2

modulus of multiwall carbon nanotubes (MWCNTs) using a transmission electron microscope. Their results showed that the Young’s modulus dropped sharply with an increase of tube diameter. Yao and Lordi [43] calculated the Young’s modulus of SWCNTs via MD simulations and the same results have been obtained. Based on these experimental and atomistic simulation results, Sakharova et al. [44] evaluated the tensile and bending rigidities and, subsequently, the Young's modulus of CNTs by using an equivalent beam approach. They gave the approximated relations among the tensile rigidity EA, bending rigidity EI, and the tube diameter, that is

EA    D  D0  ,

(4)

EI    D  D0  ,

(5)

3

where α, β, and D0 are the fitting parameters, E is the Young’s modulus. The cross-sectional area, A, and the moment of inertia, I, are expressed as

A

 2 2  D  h    D  h     Dh,

(6)

I

  4 4  D  h   D  h  ,

(7)

4



64 



where h is the wall thickness of CNTs. Using Eqs. (4) to (7), the tube diameter D is given as

D 8

EI  h2 . EA

(8)

Substitution of Eq. (8) into Eq. (4) leads the expression for the Young's modulus of SWCNTs, that is,

E

EA  A

  D  D0    D  D0  h 8  h2  2

.

(9)

To consider both the effects of the nonlocal parameter and the material length scale parameter, the nonlocal strain gradient theory is adopted in our discussion. The constitutive relation for the normal stress and strain is written as [31]

 xx   e0 a 

2

 2 xx  2 xx  E ( xx  l 2 ), 2 x  x2

(10)

where x is the axial coordinate. σxx and εxx are the normal stress and strain, respectively. e0 is a constant appropriated to each material and a is an internal characteristic length (e.g. the length of the carbon-carbon bond, lattice spacing, granular distance). In the present study, a is the length of the carbon-carbon bond. e0a is the nonlocal parameter introduced to consider the significance of nonlocal elastic stress field. According to the molecular mechanics and MD simulations, many researchers proposed a conservative estimation of the nonlocal parameter for CNTs, that is, 0
 xx   e0 a 

2

 2 xx  E xx . x 2

(11)

When e0a=0, it is degenerated to the one based on the pure strain gradient model [50, 51]

 xx  E ( xx  l 2

 2 xx ). x 2

(12)

Considering the viscoelastic damping effect of CNTs, the Young’s modulus E can be replaced with an operator

E 1  g   / t  based on the Kelvin-Voigt viscoelastic damping model [25–27]. Then, the previous constitutive relation Eq. 3

(10) is rewritten as 2  2   xx    e a    xx  0 x 2    2 xx   E (1  g )( xx  l 2 ), t  x2

(13)

where g is damping coefficient. When the flexural wave propagates in an embedded SWCNT, the general equation of wave motion for the Rayleigh beam model is expressed as [32, 33]

V 2 w 4 w  q ( x)   A 2   I 2 2 , x t x t

(14)

M V, x

(15)

where w denotes the transverse displacements. ρ and V are the mass density and the shear force, respectively. q(x) is the external force which is induced by the normal pressure of the elastic medium, that is,

q( x)  kw w,

(16)

where kw is the Winkler modulus of the surrounding elastic medium. M is the bending moment, which is defined as

M   z xx dA .

(17)

2 w . x 2

(18)

A

The normal strain εxx for the Rayleigh beam model is given as

 xx   z Using Eqs. (13), (17) and (18), we have

2 M x 2  2 w 4 w   EI (1  g ) ( 2  l 2 4 ).  t x x

M   e0 a 

2

(19)

By substituting Eqs. (16) and (19) into Eqs. (14) and (15), we can obtain the governing differential equation of wave motion based on the nonlocal strain gradient theory and the Rayleigh beam model as

EI (1  l 2

2  4 w )(1  g ) 4 2  t x x

2   w 4 w 2    1   e0 a  (  A 2   I 2 2  k w w)  0. 2  x  t x t  2

(20) It is seen that if the effects of the surrounding elastic medium and rotary inertia are ignored, Eq. (20) degenerates to the equation of wave motion of the nonlocal strain gradient theory derived by Lim et al. [31]

2  w ) x 2 x 4 2 2   w 2    1   e0 a  A 2  0 . x 2  t  4

EI (1  l 2

(21)

Furthermore, when the small scale and the strain gradient length scale effects are ignored, Eq. (21) degenerates to the local Euler–Bernoulli beam model for wave motion 4

 w  w  A 2  0 . x 4 t 4

EI

3.

2

(22)

Wave propagation in a SWCNT The solution of the transverse displacement for the flexural wave in an embedded SWCNT can be assumed as the

following form

w  x, t   We 

i kx t 

(23)

,

where i  1 , W denotes the wave amplitude, k and ω are the wave number and the frequency of the wave motion, respectively. Inserting Eq. (23) into Eq. (20), the complex-valued dispersion relation can be given as

1   e0 a 2 k 2    A   Ik 2   2  iEI 1  l 2 k 2  k 4 g   2 2 2  EI 1  l k  k 4  kw 1   e0 a  k 2   0.   (24) And then we can obtain the complex circular frequency of the viscoelastic flexural wave



a2i  4a1a3  a22 2a1

,

(25)

where 2 a1  1   e0 a  k 2    A   Ik 2   

a2  gEI 1  l 2 k 2  k 4

(26)

2 a3  EI 1  l 2 k 2  k 4  kw 1   e0 a  k 2  .  

Eq. (25) shows that the complex frequency can be separated into real and imaginary parts [52]. The real and imaginary parts denote the natural frequency and decay rate (equivalent viscous damping or Neperian frequency in the units of Neper) of the CNT for a constant wave number, respectively [27,32,33,53]. According to Eq. (25), the complex phase velocity of the viscoelastic flexural wave in CNTs is given as

c

 k



a2i  4a1a3  a22 2a1k

.

(27)

The real and imaginary parts of c, marked as Re(c) and Im(c), are the phase velocity and a measure of the decay rate of waves in CNTs, respectively [32,33]. The damping ratio δ is then defined as [32, 33]

 

Im  

Re  



Im  c 

Re  c 

.

(28)

It should be noted that the value of the phase velocity must be positive if the wave can propagate in viscoelastic CNTs. Then we can obtain the critical condition for the viscoelastic wave propagation in visco-SWCNT, that is

4a1a3  a22  0. For an elastic beam (g=0), the left term of Eq. (29) is reduced to

5

(29)

2 4a1a3  a22  4 1   e0 a  k 2    A   Ik 2    

EI 1  l k  k 2

2

4



2  kw 1   e0 a  k 2  .  

(30)

It is seen that the right term of Eq. (30) is always positive, indicating that the wave can always propagate in elastic CNTs. 4.

Results and discussions Seen as Eq. (9), the Young’s modulus of the CNTs is directly related to the tube diameter. In this section, the influence of

the tube diameter on the wave dispersion in viscoelastic SWCNTs is investigated. The thickness of SWCNT h is 0.34 nm [32, 33]. The mass density ρ is 2300 kg/m3 [14, 15]. Following Sakharova et al. [44], the fitting parameters α, β and D0 are set to be 1131.66nN nm-1, 143.48nN nm-1 and 2.8×10-7nm, respectively. Firstly, the variation of the complex phase velocities with respect to the tube diameter at different damping coefficients is considered. Following Pang et al. [27], the damping coefficients are given as: g=0.5×10-12s, 0.75×10-12s, 1×10-12s, 1.25×10-12s and 1.5×10-12s, respectively. Moreover, no special explanation, we have the strain gradient length scale parameter l=1nm, the nonlocal parameter e0a=1nm, wave number k=2×108m-1 and the Winkler modulus kw=1×107N/m2 [54]. Seen as Fig. 2(a), when the damping coefficient is small, see, g=0.5×10-12s, the phase velocity is firstly decreased and then increased along with the increase of the tube diameter and tends to a constant value. However, along with the increase of the damping coefficient, see, g>0.75×10-12s in the present discussion, the phase velocity is firstly increased to a maximum value and then dropped to zero along with the further increase of the tube diameter. Correspondingly, seen as Fig. 2(b), the damping ratio is gradually increased along with the increase of the tube diameter, and tends to infinite at the critical diameter when Re(c)=0. The results indicate that there exists a blocking diameter, above which the phase velocity of the transverse wave in viscoelastic CNTs equals to zero, and the energy is rapidly dissipated. Fig. 2 indicates that the blocking diameter is greatly damping coefficient dependent. Along with the increase of the damping coefficient, the blocking diameter is decreased. When g=1.0×10-12s, the blocking diameter is D=7.47nm. When g=1.25×10-12s, it is reduced to 5.69 nm, and dropped to 4.63 nm when g=1.5×10-12s.

6

Fig. 2. The phase velocity and the damping ratio as a function of the tube diameter at different damping coefficients: (a) phase velocity; (b) damping ratio.

Eq. (25) gives the complex frequency of the viscoelastic wave motion in CNTs for a real wave number. In Fig. 3, the variation of the phase velocity and the damping ratio with respect to the tube diameter at different wave numbers are plotted. In the calculation, the damping coefficient is g=1×10-12s, the strain gradient length scale parameter l=1nm, the nonlocal parameter e0a=1nm and the Winkler modulus kw=1×107 N/m2. It can be seen from Fig. 3(a) that the phase velocity is gradually increased to a constant value along with the increase of the tube diameter when the wave number is small, see, k=1.5×108m-1 in the present discussion. Meanwhile, the damping ratio is small. Along with the increase of the wave number, the wave blocking appears, and the larger the wave number is, the smaller the blocking diameter is. But along with the increase of the wave number, the sensitivity of the blocking diameter to the wave number k is weakened. It is seen that the damping ratio tends to infinite at the blocking diameter (Fig. 3(b)). Then we consider the limiting situation of Eq. (29). If k=0, Eq. (29) is degenerated to

4a1a3  a22  4kw  A  0,

(31)

which means that the viscoelastic wave could always propagate in the visco-CNTs when k is small. If k tends to infinite, Eq. (29) is reduced to

4a1a3  a22    gEI  l 4 k 12  0, 2

(32)

It means that when the wave number k is larger enough, the wave could not propagate in viscoelastic CNTs any more, which shows a good agreement with results of previous work [32, 33].

7

Fig. 3. The phase velocity and the damping ratio as a function of the tube diameter with different wave numbers: (a) phase velocity; (b) damping ratio.

Small scale effect plays an important role in the dynamic performance description of CNTs. Fig. 4 displays the influence of the nonlocal parameter on the blocking diameters for visco-CNTs. The damping coefficient g=1×10-12s, the strain gradient length scale parameter l=1nm, wave number k=2×108m-1 and the Winkler modulus kw=1×107 N/m2 are taken. Fig. 4(a) indicates that the blocking diameter depends upon the value of the nonlocal parameter. Along with the increase of the nonlocal parameter, the blocking diameter is increased. Let us consider the limiting situation if e0a tends to infinite.

At this time, Eq.

(29) is degenerated to

4a1a3  a22  4  e0 a  k 4  A  Ik 2  kw  0 , 4

which means that when the nonlocal effect is strengthened, the wave blocking may not form.

8

(33)

Fig. 4. The phase velocity and the damping ratio as a function of the tube diameter with different nonlocal parameters: (a) phase velocity; (b) damping ratio.

The influence of the strain gradient length scale parameter on the blocking diameter for visco-CNTs is shown in Fig. 5. In the calculation, we have the nonlocal parameter e0a=1nm, the damping coefficient g=1×10-12s wave number k=2×108m-1 and the Winkler modulus kw=1×107 N/m2. Seen as Fig. 5(a), the blocking diameter becomes smaller with the increase of the strain gradient length scale parameter. According to Eq. (29), if let l tends to infinite, Eq. (29) is degenerated to the following form, that is,

4a1a3  a22    gEIl 2 k 6   0, 2

(34)

which implies that the wave may not propagate in viscoelastic CNTs when the material length scale parameter is larger enough.

Fig. 5. The phase velocity and the damping ratio as a function of the tube diameter with different strain gradient length scale parameters: (a) phase velocity; (b) damping ratio.

The influence of the Winkler modulus on the blocking diameters for visco-CNTs is illustrated in Fig. 6. The strain gradient length scale parameter l=1nm, the damping coefficient g=1×10-12s, the nonlocal parameter e0a=1nm and the wave number k=2×108m-1. It is observed that the blocking diameter is insensitive to the Winkler modulus when the Winkler modulus kw<1×109N/m2, indicating that the influence of the softer foundation on the blocking diameter of visco-CNTs is limited. Whilst for the stiff foundation, that is, foundation with larger Winkler modulus, the blocking diameter is increased along with the increase of the Winkler modulus. This could be explained to let kw in Eq. (29) tends to infinite, which leads to 2 4a1a3  a22  4kw 1   e0 a  k 2   

9

2

  A   Ik   0. 2

(35)

Eq. (35) means that the critical condition for the wave propagation in visco-CNTs will be always fulfilled when the foundation is stiff enough.

Fig. 6. The phase velocity and the damping ratio as a function of the tube diameter with different Winkler modulus: (a) phase velocity; (b) damping ratio. 5.

Conclusion In this paper, the viscoelastic wave propagation in an embedded viscoelastic SWCNT is investigated based on the

nonlocal strain gradient theory. The critical propagation condition for the viscoelastic wave in visco-CNTs is obtained by analyzing the closed-form expression of the complex phase velocity. When the critical condition for the viscoelastic wave is not satisfied, wave blocking is found during the wave propagation in visco-CNTs, which will not form in elastic CNTs. Summarizing the numerical results above we can conclude that

(1) Along with the increase of the wave number, the blocking diameter is decreased. But the wave blocking will not form when the wave number is small.

(2) The blocking diameter is damping coefficient dependent. Larger damping coefficient corresponds to a smaller blocking diameter. For a certain damping coefficient, the enhancement of the nonlocal effect and foundation stiffness or the decrease of the material length scale parameter may increase the blocking diameter. It is expected that these results will be useful for the perspective application of visco-CNTs in nanotechnology.

Acknowledgment The second author thanks the support from the Fundamental Research Funds for the Central Universities of China 10

(2014JBZ014). Supports from the National Natural Science Foundation of China (11272046), and National Basic Research Program of China (973Program) (2015CB057800), are acknowledged.

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Highlights



The characteristic equation for the viscoelastic wave in embedded viscoelastic SWCNTs is derived based on the nonlocal strain gradient theory.

  

The critical propagation condition for the viscoelastic wave in visco-CNTs is obtained. A blocking diameter is observed, above which the wave could not propagate in SWCNTs. The enhancement of the nonlocal effect and foundation stiffness or the decrease of the damping coefficient and material length scale parameter increase the blocking diameter.

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