Wave characteristics in aligned forests of single-walled carbon nanotubes using nonlocal discrete and continuous theories

International Journal of Mechanical Sciences 90 (2015) 278–309

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Wave characteristics in aligned forests of single-walled carbon nanotubes using nonlocal discrete and continuous theories Keivan Kiani n Department of Civil Engineering, K.N. Toosi University of Technology, P.O. Box 15875-4416, Valiasr Ave., Tehran, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 18 July 2014 Received in revised form 25 October 2014 Accepted 13 November 2014 Available online 20 November 2014

Elastic waves within forests of single-walled carbon nanotubes (SWCNTs) have been of particular interest to researchers of nanotechnology and applied physics. To date, wave motion in individual single-, double-, and multi- walled carbon nanotubes has been extensively investigated, however, characteristics of the transverse waves in three-dimensional clusters of SWCNTs have not been revealed. In this paper, using nonlocal Rayleigh, Timoshenko, and higher-order beam theories, shear and flexural frequencies as well as their corresponding phase and group velocities of transverse waves within such nanostructures are studied via discrete and continuous models. Using continuous models, the explicit expressions of the above-mentioned characteristics of waves are obtained. The efficacy of the proposed continuous models is proved by comparing their results with those of the nonlocal discrete models. The roles of wavenumber, radius of the constitutive SWCNTs, slenderness ratio, small-scale parameter, intertube distance, and population of the ensemble on the characteristics of transverse waves are comprehensively examined. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Vertically aligned ensembles of SWCNT Elastic transverse waves Nonlocal beam theories Shear and flexural frequencies Phase and group velocities

1. Introduction The geometry of a single-walled carbon nanotube (SWCNT) can be visualized by rolling a graphene sheet into a seamless cylinder. Most SWCNTs have a diameter about 1 nm and their length-to-diameter ratio can be up to 132,000,000 [1]. These sheets are wrapped at specific directions, called chiral angles. A combination of chirality and radius can significantly affect the nanotube properties [2–4]. The exceptional electrical properties of SWCNTs [5,6], their extraordinary strength [7–10], and efficiency in heat conduction [11–13] offer them for a large body of applications such as nanotechnology, electronics, optics, materials science, and medicine. Generally, ensembles of SWCNTs (ESWCNTs) as well as SWCNT-based composites are efficiently exploited for the above-mentioned purposes. Hereinafter, we focus on the vertically aligned ESWCNTs to investigate their vibration behaviors for the considered functions. Chemical vapor deposition (CVD) is the most common technique used for commercial production of CNTs; however, the resulting nanotubes are often haphazardly oriented. If plasma is produced by the application of a strong electric field during growth of CNTs (i.e., plasma-enhanced CVD), the nanotubes will grow along the direction of the applied electric field [14–16]. Under particular reaction conditions, even in the absence of a plasma, closely spaced nanotubes will maintain a vertical growth direction resulting in a dense array of tubes resembling a carpet or forest. For example, Murakami et al. [17] synthesized vertically aligned SWCNTs on quartz substrates. To this end, low-temperature CVD from ethanol was performed by using densely mono-dispersed Co–Mo catalyst, and the thickness of the resulting film was a few micrometers. In brief, a vertically aligned ESWCNT is a group of parallel SWCNTs placed at the vicinity of each other. The deformation interactions occur because of the existing van der Waals (vdW) forces between the constitutive atoms of each tube and those of its neighboring ones. Additionally, the intertube distance, arrangement, and other geometrical properties of the ESWCNTs can influence on the vdW interactional forces and their vibration behaviors as well. Since the past decade, the potential application of ESWCNTs in electronics (as micro-electro-mechanical devices) [18–23], mechanical (as oscillator and actuator) [24–26], and medicine (as drug deliverer) [27–29] has intensified the need to realize their true mechanisms of vibrations and wave propagation. To date, vibrations and wave propagation within an individual SWCNT have been greatly paid attention to by many researchers. For instance, free vibrations and instabilities [30–34], forced vibrations due to moving loads and nanoparticles [35–38] and inside fluid flow [39–42], wave propagation within SWCNTs [43–46], and vibrations and instabilities of magnetically affected SWCNTs [47–50] are among

n

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http://dx.doi.org/10.1016/j.ijmecsci.2014.11.011 0020-7403/& 2014 Elsevier Ltd. All rights reserved.

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

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the important findings in the past ten years. However, vibrations of and waves propagation in ESWCNTs have not been thoroughly addressed yet. With respect to their potential applications and the importance of the subject, the present work is devoted to examine characteristics of transverse waves in vertically aligned ESWCNTs. Since only transverse vibration of the nanostructure is of concern, appropriate beam theories are employed. For this purpose, Rayleigh, Timoshenko, and higher-order beam theories are implemented for modeling of the problem at hand. Through using such a variety of models, we also seek the capabilities of the proposed model in capturing the shear deformation of the ESWCNTs. In other words, such studies are also aimed to explain that under what circumferences the results of Rayleigh or Timoshenko model would be reliable. At the nano-scale, the stress of each point of the nanostructure is also affected by the stresses of the neighboring points due to the existing inter-atomic bonds. It implies that the classical theory of elasticity cannot rationally predict deformation regime of nanostructures undergoing dynamic loads since it explains that the stress state of each point only depends on the strain state of that point. On the other hand, atomistic-based approaches have been frequently used for predicting elastic properties and mechanical behavior of SWCNTs and their composites [51–55]; however, such methods are enormously more expensive in time and labor costs compared with the continuumbased models. To conquer the above-mentioned drawbacks of the classical continuum theory and to reduce the costs of atomic models, several novel theories have been developed to incorporate the size-effect into the equations of motion of very small structures [56–62]. Among the proposed theories, the nonlocal continuum theory of Eringen [62–65] has gained much popularity among researchers because of two facts. The first is that it can be simply applied to the classical version of the equations of motion (i.e., an operator acts on the inertial and force terms of the governing equations). The second one is that there exists many studies display that through choosing an appropriate value for the small-scale parameter, a reasonably good agreement between the results of the nonlocal model and those of the atomistic-based approach can be achieved ([66–69]). As a result, the author employs the nonlocal continuum theory of Eringen in deriving the equations of motion of 3D ESWCNTs based on the above-mentioned beam models. As it will be explained, both nonlocal discrete and continuous versions of the governing equations for the nanostructure are established. In the present work, transverse wave characteristics within 3D ESWCNTs is aimed to be carefully studied. Using nonlocal continuum theory of Eringen and Hamilton's principle, nonlocal-discrete equations of motion of the problem based on the Rayleigh, Timoshenko, and higher-order beam theories are developed. For more convenient in studying the problem, some original continuous models are also developed based on the newly developed discrete models. By taking the opportunity of using these continuous models, exploring crucial characteristics of the transverse waves in the nanostructure could be carried out more efficiently and systematically. In a particular case, the obtained results by the continuous models are verified with those of discrete models and other works and a reasonably good agreement is reported. Subsequently, the influences of the crucial factors of the geometry, small-scale parameter, and population of the 3D ESWCNTs on the characteristics of the waves are examined. The capabilities of the proposed nonlocal continuous models in predicting the flexural and shear frequencies as well as their corresponding phase and group velocities are also investigated.

2. The details of the under study problem Consider a vertically aligned ESWCNTs whose intertube distance along the y and z axes and length are equal to d and lb, respectively, as shown in Fig. 1(a). The ensemble consists of Ny and Nz tubes along the y and z axes, respectively. For continuum-based modeling of the problem, each SWCNT of the ensemble is replaced by an equivalent continuum structure (ECS) whose most of its dominant frequencies are identical to those of the parent structure. The ECS is a hollow circular cylinder with mean radius rm and length lb. The elasticity modulus, density, Poisson's ratio, shear elastic modulus, cross-sectional area, and second moment inertia of the ECS are denoted by Eb, ρb, νb, Gb, Ab, and Ib, respectively. The SWCNTs of the ensemble interact with each other through the intertube vdW forces. Such forces are correspond to the tight attractions of the nanotubes within the ensemble. As it will be explained in the upcoming part, not only relative movement of two adjacent tubes in the plane passes through their revolutionary axes, but also relative displacement perpendicular to such a plane can cause extra vdW forces that play a crucial key in vibration behavior of the nanostructure. The vdW forces between two tubes are modeled by elastically continuous springs. For two nearest tubes, pffiffiffi the constants of springs are represented by C v ? and C v J , whereas the constants pertinent to the tubes with the intertube distance d 2 are denoted by C d ? and C d J (see Fig. 1(b)).

3. Assessing vdW forces between two adjacent SWCNTs According to the Lennard–Jones's potential function for two atoms [70]: "   6 # σ 12 σ Φij ðλÞ ¼ 4ϵ  ;

λ

λ

ð1Þ

pffiffiffi where λ is the distance between the ith and jth atoms, σ ¼ r a = 6 2, ϵ denotes the well depth, and ra represents the distance between two atoms at the equilibrium state. The vdW force between two atoms i and j, f ij , is evaluated as "    8 # ! dΦ 24ϵ σ 14 σ ð2Þ f ij ¼  eλ ¼ 2 2  λ; σ dλ λ λ ! ! ! where λ is the position vector of the atom j with respect to the atom i, eλ denotes its unit base vector (i.e., eλ ¼ λ = J λ J ). By representing the coordinates of the atom i of the first ! SWCNT and the atom j of the second SWCNT (as shown in Fig. 2) by ðx1 ; r m cos ϕ1 ; r m sin ϕ1 Þ and ðx2 ; r m cos ϕ2 ; d þ r m sin ϕ2 Þ, respectively, λ is readily expressed by !

λ ¼ ðx2  x1 Þex þ ðrm ð cos φ2  cos φ1 Þ  ΔV Þey þ ðr m ð sin φ2  sin φ1 Þ þ d  ΔWÞez ; 0 r x1 ; x2 r lb ; 0 r φ1 ; φ2 r 2π ;

ð3Þ

280

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

Fig. 1. (a) An oblique view of a 3D ESWCNTs; (b) The top view of the continuum-based model for the ESWCNTs with the intertube elastically continuous springs.

V (x,t)

V2(x,t)

1

W1(x,t) y

λ

rm

z

W (x,t) 2 r

m

π/2−φ

π/2−φ

O

2

1

O

1

2

d Fig. 2. The geometry of two adjacent tubes.

where ΔWðx; tÞ ¼ W 1 ðx; tÞ W 2 ðx; tÞ, ΔVðx; tÞ ¼ V 1 ðx; tÞ  V 2 ðx; tÞ, W 1 ðx; tÞ=V 1 ðx; tÞ and W 2 ðx; tÞ=V 2 ðx; tÞ represent the transverse displacement fields of the first and second SWCNTs along the z/y axis, respectively, ex , ey , and ez are the unit base vectors associated with the considered rectangular coordinate system. The components of the vdW force on the SWCNTs because of their relative displacements can be evaluated by Kiani [71]: 24ϵσ 2CNT Fx ¼ lb σ 2

Z

24ϵσ 2CNT lb σ 2

Z

24ϵσ 2CNT lb σ 2

Z

Fy ¼

Fz ¼

lb

Z

0

0 lb

Z

0

lb

Z

0

λ

λ

Z 2π Z 2π "  14  8 # σ σ 2  ðr m ð cos φ2  cos φ1 Þ  ΔVÞ dφ1 dφ2 dx1 dx2 ; 0

lb 0

Z 2π Z 2π "  14  8 # σ σ 2  ðx2  x1 Þ dφ1 dφ2 dx1 dx2 ; 0

0

lb 0

lb

0

λ

λ

 Z 2π Z 2π "  14  8 # r m ð sin φ2  sin φ1 Þ þ σ σ dφ1 dφ2 dx1 dx2 ; 2  λ λ d  ΔW 0 0

ð4Þ

pffiffiffi where σ CNT ¼ 4 3=9a2 is the surface density of the carbon atoms, and a is the length of the carbon–carbon bond. By estimating the components of the vdW force in Eq. (4) by the Taylor expansion up to the first-order about the equilibrium state, the only components of the resulted vdW force between two adjacent SWCNTs due to their relative transverse displacements are obtained as

ΔF y ¼ C v ? ΔV ; ΔF z ¼ C v J ΔW;

ð5Þ

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

281

where C v ? ðr m ; dÞ ¼ 

256ϵr 2m 9a4 lb Z

lb



Z

0

C v J ðr m ; dÞ ¼ 

lb 0

256ϵr 2m 9a4 lb Z

lb

 0

Z

lb 0

8 h   2 i 9 12 7 8 > r m cos φ2  cos φ1 > > Z 2π Z 2π > = < σ χ  14χ i dφ1 dφ2 dx1 dx2 ; 6h     σ > 0 0 > > χ  4  8χ  5 rm cos φ2  cos φ1 2 > ; : 2

ð6aÞ

8 h   2 i 9 12 7 8 > d þ r m sin φ2  sin φ1 > > Z 2π Z 2π > = < σ χ  14χ dφ1 dφ2 dx1 dx2 ; h i 6     σ 2 > > 0 0 > > χ  4  8χ  5 d þ rm sin φ2  sin φ1 ; : 2

ð6bÞ





χ ðx1 ; x2 ; φ1 ; φ2 ; rm ; dÞ ¼ ðx2  x1 Þ2 þ2r 2m 1  cos ðφ2  φ1 Þ þd2 þ 2rm dð sin φ2  sin φ1 Þ:

ð6cÞ

4. Establishment of discrete models based on nonlocal beam theories In this part, characteristics of the elastic waves within 3D ensembles of SWCNTs are studied using nonlocal classical and shear deformable beam models. To this end, the transverse equations of motion for each SWCNT within the ensemble are derived. Thereafter, through eigenvalue analysis of the set of governing equations, the natural frequencies and their corresponding phase velocities of the elastic waves are numerically calculated. 4.1. Elastic waves within 3D ESWCNTs via NRBT 4.1.1. Nonlocal discrete equations of motion using NRBT Based on the hypothesis of the Rayleigh's beam model, the kinetic energy, TR, and the elastic strain energy of the 3D ensemble of SWCNTs, UR, are stated in the context of the nonlocal continuum theory of Eringen as follows: 0 0 0 !2 !2 1 !2 !2 11 Z lb 1 Ny N z ∂V Rmn ∂W Rmn A ∂2 V Rmn ∂2 W Rmn AA R @ @ @ þ Ib dx; ð7aÞ ∑ ∑ ρ Ab þ þ T ¼ 2m¼1n¼1 0 b ∂t ∂t ∂t∂x ∂t∂x 0

1 ∂2 V Rmn  nl R ∂2 W Rmn  nl R M  M B C 2 bymn ∂x B C ∂x2 bzmn

C B     2 2     C B R R R R B þ C v J V mn  V ðm þ 1Þn 1  δmNy þ V mn  V ðm  1Þn 1  δ1m C B C B 

C B C 2   2    B C 1  δ1n þ V Rmn  V Rmðn þ 1Þ 1  δnNz B þ C v ? V Rmn  V Rmðn  1Þ C B C B C  2  B C    R R B þC C 1  X  X 1  δ δ 1m 1n dJ B C mn ðm  1Þðn  1Þ B C  2  B C    R R B þC C 1  δmNy 1  δnNz d J X mn  X ðm þ 1Þðn þ 1Þ B C Z C lb B Ny N z 1  2  B C R    ∑ ∑ U ¼ R R B Cdx; 1  δ1m 1  δnNz C 2 m ¼ 1 n ¼ 1 0 B þ C d J Y mn  Y ðm  1Þðn þ 1Þ B C   B C 2    B þC C R R 1  δmN y 1  δ1n B C d J Y mn  Y ðm þ 1Þðn  1Þ B C B C  2     B C R R 1  δ1m 1  δnNz B þ C d ? X mn  X ðm  1Þðn þ 1Þ C B C B C  2    B C R R 1  δmNy 1  δ1n B þ C d ? X mn  X ðm þ 1Þðn  1Þ C B C B C   2    B C B þ C d ? Y Rmn  Y Rðm  1Þðn  1Þ C 1  δ1m 1  δ1n B C B C   2     @ A R R 1  δmNy 1  δnNz þ C d ? Y mn  Y ðm þ 1Þðn þ 1Þ

ð7bÞ

 pffiffiffi pffiffiffi  nl R R where X Rmn ¼ 2=2ðW Rmn þ V Rmn Þ and Y Rmn ¼ 2=2  W Rmn þ V Rmn . Further, δmn, VRmn, WRmn, ðM nl bymn Þ , and ðM bzmn Þ in order are the Kronecker delta tensor, transverse displacements along the y and z axes, and the nonlocal bending moments of the (m,n)th SWCNT about the y and z axes based on the NRBT. In the context of the nonlocal continuum theory of Eringen, the nonlocal bending moment of the (m,n)th SWCNT of the ensemble based on the NRBT are provided by [72,73,44]: 



M nl bymn M nl bzmn

R

 R ∂2 W Rmn ðe0 aÞ2 M nl ¼  Eb I b ; bymn ;xx ∂x2

ð8aÞ

R

 R ∂2 V Rmn  ðe0 aÞ2 M nl ¼  Eb I b ; bzmn ;xx ∂x2

ð8bÞ

282

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

where e0 a denotes the small-scale parameter. By employing Hamilton's principle and Eqs. (7a), and (7b) and, the nonlocal transverse equations of motion of the nanostructure are obtained as follows: ( ! ∂4 V Rmn ∂2 V Rmn ∂4 V Rmn þ Ξ ρ A  I Eb I b b b 2 b ∂x4 ∂t 2 ∂t ∂x2 h     i þ C v J V Rmn  V Rðm þ 1Þn 1  δmNy þ V Rmn  V Rðm  1Þn 1  δ1m h     i þ C v ? V Rmn V Rmðn  1Þ 1  δ1n þ V Rmn  V Rmðn þ 1Þ 1  δnNz     þ 0:5C d J W Rmn þV Rmn  W Rðm  1Þðn  1Þ  V Rðm  1Þðn  1Þ 1  δ1n 1  δ1m     þ 0:5C d J W Rmn þV Rmn  W Rðm þ 1Þðn þ 1Þ  V Rðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy     þ 0:5C d J V Rmn  W Rmn  V Rðm þ 1Þðn  1Þ þ W Rðm þ 1Þðn  1Þ 1  δ1n 1  δmN y     þ 0:5C d J V Rmn  W Rmn  V Rðm  1Þðn þ 1Þ þ W Rðm  1Þðn þ 1Þ 1  δnNz 1  δ1m     þ 0:5C d ? V Rmn  W Rmn þ W Rðm  1Þðn  1Þ  V Rðm  1Þðn  1Þ 1  δ1n 1  δ1m     þ 0:5C d ? V Rmn  W Rmn þ W Rðm þ 1Þðn þ 1Þ  V Rðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy     þ 0:5C d ? V Rmn þ W Rmn  V Rðm  1Þðn þ 1Þ  W Rðm  1Þðn þ 1Þ 1  δnNz 1  δ1m )     þ 0:5C d ? V Rmn þ W Rmn  V Rðm þ 1Þðn  1Þ  W Rðm þ 1Þðn  1Þ 1  δ1n 1  δmNy ¼ 0; ð9aÞ

Eb I b

( ! ∂4 W Rmn ∂2 W Rmn ∂4 W Rmn þ Ξ ρ A  I b b b ∂x4 ∂t 2 ∂t 2 ∂x2   h   i þ C v J W Rmn  W Rmðn þ 1Þ 1  δnNz þ W Rmn  W Rmðn  1Þ 1  δ1n h     i þ C v ? W Rmn  W Rðm  1Þn 1  δ1m þ W Rmn  W Rðm þ 1Þn 1  δmNy     þ 0:5C d J W Rmn þV Rmn  W Rðm  1Þðn  1Þ  V Rðm  1Þðn  1Þ 1  δ1n 1  δ1m     þ 0:5C d J W Rmn þV Rmn  W Rðm þ 1Þðn þ 1Þ  V Rðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy     þ 0:5C d J W Rmn V Rmn  W Rðm  1Þðn þ 1Þ þ V Rðm  1Þðn þ 1Þ 1  δnNz 1  δ1m     þ 0:5C d J W Rmn V Rmn  W Rðm þ 1Þðn  1Þ þ V Rðm þ 1Þðn  1Þ 1  δ1n 1  δmN y     þ 0:5C d ? W Rmn  V Rmn  W Rðm  1Þðn  1Þ þ V Rðm  1Þðn  1Þ 1  δ1n 1  δ1m     þ 0:5C d ? W Rmn  V Rmn  W Rðm þ 1Þðn þ 1Þ þ V Rðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy     þ 0:5C d ? W Rmn þ V Rmn  W Rðm  1Þðn þ 1Þ  V Rðm  1Þðn þ 1Þ 1  δnNz 1  δ1m )     þ 0:5C d ? W Rmn þ V Rmn  W Rðm þ 1Þðn  1Þ  V Rðm þ 1Þðn  1Þ 1  δ1n 1  δmNy ¼ 0:

ð9bÞ

where Ξ ½: ¼ ½:  ðe0 aÞ2 ∂∂x½:2 : Eqs. (9a) and (9b) furnish us regarding the free transverse vibrations of the constitutive SWCNTs of a 3D ensemble in the context of the nonlocal continuum theory of Eringen using hypotheses of the Rayleigh beam model. These are called discrete equations since the displacements associated with each SWCNT of the ensemble are stated by exclusive functions. The displacement field of each SWCNT can be affected by the displacements of its neighboring tubes. To determine such unknown fields, a set of 2N y N z second-order partial differential equations (PDEs) should be appropriately solved. In order to study the free dynamic response of the nanostructure in a more general context, we define the following dimensionless parameters: sffiffiffiffiffiffiffiffiffiffi x V Rmn W Rmn z 1 Eb I b R R ξ ¼ ; V mn ¼ ; W mn ¼ ; γ¼ ; τ¼ 2 t; lb lz lb lb ρb Ab l 2

b

4

4

C l C l e a R l d R μ ¼ 0 ; C v½: ¼ v½: b ; C d½: ¼ d½: b ; λ ¼ b ; d ¼ ; ½: ¼ J or ? : lb lz Eb I b Eb I b rb

ð10Þ

By introducing Eq. (10) to Eqs. (9a) and (9b), the dimensionless nonlocal-discrete equations of motion of the 3D ensemble are derived as follows: ( R R 4 R ∂4 V mn ∂2 V mn  2 ∂ V mn þ Ξ  λ 4 2 ∂τ 2 ∂ξ ∂τ 2 ∂ξ   h R   R i R R R þ C v J V mn  V ðm þ 1Þn 1  δmN y þ V mn  V ðm  1Þn 1  δ1m h R     R i R R R þ C v ? V mn  V mðn  1Þ 1  δ1n þ V mn  V mðn þ 1Þ 1  δnNz  R    R R R R þ 0:5C d J W mn þ V mn  W ðm  1Þðn  1Þ  V ðm  1Þðn  1Þ 1  δ1n 1  δ1m

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

283

 R    R R R R þ 0:5C d J W mn þ V mn  W ðm þ 1Þðn þ 1Þ  V ðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy  R    R R R R þ 0:5C d J V mn  W mn  V ðm þ 1Þðn  1Þ þ W ðm þ 1Þðn  1Þ 1  δ1n 1  δmNy  R    R R R R þ 0:5C d J V mn  W mn  V ðm  1Þðn þ 1Þ þ W ðm  1Þðn þ 1Þ 1  δnNz 1  δ1m  R    R R R R þ 0:5C d ? V mn  W mn þ W ðm  1Þðn  1Þ  V ðm  1Þðn  1Þ 1  δ1n 1  δ1m  R    R R R R þ 0:5C d ? V mn  W mn þ W ðm þ 1Þðn þ 1Þ  V ðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy  R    R R R R þ 0:5C d ? V mn þ W mn  V ðm  1Þðn þ 1Þ  W ðm  1Þðn þ 1Þ 1  δnNz 1  δ1m  R    R R R R þ 0:5C d ? V mn þ W mn  V ðm þ 1Þðn  1Þ  W ðm þ 1Þðn  1Þ 1  δ1n 1  δmN y R

∂4 W mn ∂ξ

4

) ¼ 0;

( R R 4 ∂2 W mn  2 ∂ W mn λ 2 2 ∂τ ∂ τ 2 ∂ξ h       R i R R R R þ C v J W mn W mðn þ 1Þ 1  δnNz þ W mn  W mðn  1Þ 1  δ1n h R     R i R R R þ C v ? W mn W ðm  1Þn 1  δ1m þ W mn  W ðm þ 1Þn 1  δmNy  R    R R R R þ 0:5C d J W mn þ V mn W ðm  1Þðn  1Þ  V ðm  1Þðn  1Þ 1  δ1n 1  δ1m  R    R R R R þ 0:5C d J W mn þ V mn W ðm þ 1Þðn þ 1Þ  V ðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy  R    R R R R þ 0:5C d J W mn  V mn W ðm  1Þðn þ 1Þ þ V ðm  1Þðn þ 1Þ 1  δnNz 1  δ1m  R    R R R R þ 0:5C d J W mn  V mn W ðm þ 1Þðn  1Þ þ V ðm þ 1Þðn  1Þ 1  δ1n 1  δmNy  R    R R R R þ 0:5C d ? W mn  V mn  W ðm  1Þðn  1Þ þ V ðm  1Þðn  1Þ 1  δ1n 1  δ1m  R    R R R R þ 0:5C d ? W mn  V mn  W ðm þ 1Þðn þ 1Þ þ V ðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy  R    R R R R þ 0:5C d ? W mn þ V mn  W ðm  1Þðn þ 1Þ  V ðm  1Þðn þ 1Þ 1  δnNz 1  δ1m )  R    R R R R ¼ 0; þ 0:5C d ? W mn þ V mn  W ðm þ 1Þðn  1Þ  V ðm þ 1Þðn  1Þ 1  δ1n 1  δmNy

ð11aÞ

þΞ

ð11bÞ

where Ξ ½: ¼ ½:  μ2 ½:;ξξ . 4.1.2. Characteristics of the transverse waves based on the NRBT The elastic transverse waves within the (m,n)th tube of the 3D ESWCNTs based on the NRBT are considered in the following form:   R R R R R 〈V mn ; W mn 〉 ¼ 〈V mn0 ; W mn0 〉ei ϖ τ  k x ξ ; ð12Þ R

R

where V mn0 and W mn0 represent the dimensionless amplitudes of the transverse waves, ϖ R is the dimensionless frequency, and k x is the dimensionless wave number. Without loss of generality, it is assumed that the exterior tubes of the ensembleare prohibited from  any 2 R R movement. In view of this assumption and by substituting Eq. (12) into Eqs. (11a) and (11b), it is obtained:  ϖ R M þ K x R0 ¼ 0 R R where  M and RK can  be easily calculated. The if and only if condition for existence of nontrivial solution to the resulting equations is R R 2 det  ϖ M þ K ¼ 0. By solving these set of equations for ϖ R , the frequency of the wave with wavenumber k x is evaluated. The phase velocity is defined as the ratio of the frequency to the wavenumber. Therefore, the phase velocity of the transverse waves within the 3D ESWCNTs is calculated by sffiffiffiffiffi ϖ R Eb : vRp ¼ λk x ρb

4.2. Elastic waves within 3D ESWCNTs via NTBT 4.2.1. Nonlocal discrete equations of motion using NTBT Using the hypotheses of the Timoshenko beam theory [74] in the context of the nonlocal continuum theory of Eringen [63–65], the kinetic energy, TT, the elastic strain energy, UT, of the 3D ensemble of SWCNTs accounting for the intertube vdW forces are expressed as follows: 0 0 0 !2 1 !2 !2 11 T !2 Z lb T ∂Θymn ∂Θzmn 1 N y Nz ∂V Tmn ∂W Tmn AA T @ @ A @ T ¼ ∑ ∑ þ Ab dx; ð13aÞ ρ I þ þ 2m¼1n¼1 0 b b ∂t ∂t ∂t ∂t

284

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

0

1 ! T  T ∂Θzmn  nl T ∂V Tmn T nl B C M bzmn þ  Θzmn Q bymn B C ∂x ∂x B C B C ! T B C T     T T ∂ Θ ∂W mn B C ymn T nl nl B  C M bymn þ  Θymn Q bzmn B C ∂x ∂x B C 

B C    2 2     B C T T T T B þ C v J V mn  V ðm þ 1Þn C 1  δ  V 1  δ þ V mN 1m z mn ðm  1Þn B C B 

C B C 2    2   B C B þ C v ? V Tmn V Tmðn  1Þ 1  δ1n þ V Tmn  V Tmðn þ 1Þ 1  δnNz C B C B C  2  B C   T T B þC C 1  X  X 1  δ δ 1m 1n d J B C mn ðm  1Þðn  1Þ B C Z l Ny Nz b B  2  C 1   T T T B C dx; ∑ ∑ U ¼ 1  δmNy 1  δnNz þ C d J X mn  X ðm þ 1Þðn þ 1Þ B C 2m¼1n¼1 0 B C  2  B C    B þC C T T 1  δ1m 1  δnNz B C d J Y mn  Y ðm  1Þðn þ 1Þ B C B C  2     B C T T 1  δmNy 1  δ1n B þ C d J Y mn  Y ðm þ 1Þðn  1Þ C B C B C  2     B C 1  δ1m 1  δnNz B þ C d ? X Tmn X Tðm  1Þðn þ 1Þ C B C B C   2   B C B þ C d ? X Tmn X Tðm þ 1Þðn  1Þ C 1  δmNy 1  δ1n B C B C   2    B C T T B þ C d ? Y mn  Y ðm  1Þðn  1Þ C 1  δ1m 1  δ1n B C B C   2     @ A T T 1  δmNy 1  δnNz þ C d ? Y mn  Y ðm þ 1Þðn þ 1Þ

ð13bÞ

   T  T  T pffiffiffi  T pffiffiffi  T T nl nl T T T T T nl T T where  XTmn ¼ 2=2 W mn þV mn and Y mn ¼ 2=2  W mn þ V mn . Additionally, Vmn, Wmn, Θymn , Θzmn , Q bymn , Q bzmn , M bymn , and M nl represent the transverse displacements of the (m,n)th SWCNT along the y and z axes, angles of deformation about the y and z bzmn axes, nonlocal shear forces pertinent to the y and z axes, and nonlocal bending moments about the y and z axes, respectively. According to the nonlocal continuum theory of Eringen, the nonlocal shear forces and bending moments within the (m,n)th SWCNT are given by [75,76]: !  T  T ∂V Tmn T nl 2  Q nl  ðe aÞ Q ¼ k G A Θ ð14aÞ s b b 0 bymn bymn zmn ; ;xx ∂x 





Q nl bzmn

T

M nl bymn

T

!  T ∂W Tmn T   ðe0 aÞ2 Q nl ¼ k G A Θ s b b bzmn ymn ; ;xx ∂x T  T ∂Θymn ; ðe0 aÞ2 M nl ¼  E I b b bymn ;xx ∂x

ð14cÞ

T  T ∂Θzmn ; ð14dÞ  ðe0 aÞ2 M nl ¼  E I b b bzmn ;xx ∂x  Rt  T by employing Hamilton's principle, 0 δT  δU T dt ¼ 0, the equations of motion in terms of nonlocal forces are obtained. Through introducing (14a)–(14d) to the resulting equations, the nonlocal discrete equations of motion of 3D ensembles of SWCNTs accounting for intertube vdW interactional forces can be described in terms of deformation fields of the NTBT as ( ) ! T T ∂2 Θzmn ∂2 Θzmn ∂V Tmn T  Ξ ρb I b G A Θ I ¼ 0; ð15aÞ  k  E s b b b b z mn ∂x ∂x2 ∂t 2

M nl bzmn

T

ð14bÞ

( ! T ∂2 V Tmn ∂Θzmn ∂2 V Tmn ks Gb Ab  þ Ξ ρb Ab ∂x ∂x2 ∂t 2   h   T i T T þC v J V mn  V ðm þ 1Þn 1  δmNz þ V mn  V Tðm  1Þn 1  δ1m h     i þC v ? V Tmn  V Tmðn  1Þ 1  δ1n þ V Tmn  V Tmðn þ 1Þ 1  δnNz     þ0:5C d J W Tmn þ V Tmn W Tðm  1Þðn  1Þ V Tðm  1Þðn  1Þ 1  δ1n 1  δ1m     þ0:5C d J W Tmn þ V Tmn W Tðm þ 1Þðn þ 1Þ V Tðm þ 1Þðn þ 1Þ 1  δnNz 1  δmN y     þ0:5C d J V Tmn  W Tmn V Tðm þ 1Þðn  1Þ þ W Tðm þ 1Þðn  1Þ 1  δ1n 1  δmNy     þ0:5C d J V Tmn  W Tmn V Tðm  1Þðn þ 1Þ þ W Tðm  1Þðn þ 1Þ 1  δnNz 1  δ1m     þ0:5C d ? V Tmn  W Tmn þ W Tðm  1Þðn  1Þ  V Tðm  1Þðn  1Þ 1  δ1n 1  δ1m     þ0:5C d ? V Tmn  W Tmn þ W Tðm þ 1Þðn þ 1Þ  V Tðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

285

    þ0:5C d ? V Tmn þ W Tmn  V Tðm  1Þðn þ 1Þ  W Tðm  1Þðn þ 1Þ 1  δnNz 1  δ1m

)     T T T T þ0:5C d ? V mn þ W mn  V ðm þ 1Þðn  1Þ  W ðm þ 1Þðn  1Þ 1  δ1n 1  δmNy ¼ 0; (

∂2 Θymn T

Ξ ρb I b

∂t 2

)  k s Gb A b

∂W Tmn T  Θymn ∂x

!

∂2 Θymn

ð15bÞ

T

 Eb I b

∂x2

¼ 0;

ð15cÞ

( T ! ∂2 W Tmn ∂Θymn ∂2 W Tmn  þ Ξ ρb Ab ks Gb Ab ∂x ∂x2 ∂t 2 h     i þC v J W Tmn  W Tmðn þ 1Þ 1  δnNz þ W Tmn  W Tmðn  1Þ 1  δ1n h     i þC v ? W Tmn W Tðm  1Þn 1  δ1m þ W Tmn  W Tðm þ 1Þn 1  δmN y     þ0:5C d J W Tmn þ V Tmn  W Tðm  1Þðn  1Þ V Tðm  1Þðn  1Þ 1  δ1n 1  δ1m     þ0:5C d J W Tmn þ V Tmn  W Tðm þ 1Þðn þ 1Þ V Tðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy     þ0:5C d J W Tmn  V Tmn  W Tðm  1Þðn þ 1Þ þV Tðm  1Þðn þ 1Þ 1  δnNz 1  δ1m     þ0:5C d J W Tmn  V Tmn  W Tðm þ 1Þðn  1Þ þV Tðm þ 1Þðn  1Þ 1  δ1n 1  δmNy     þ0:5C d ? W Tmn  V Tmn  W Tðm  1Þðn  1Þ þ V Tðm  1Þðn  1Þ 1  δ1n 1  δ1m     þ0:5C d ? W Tmn  V Tmn  W Tðm þ 1Þðn þ 1Þ þ V Tðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy     þ0:5C d ? W Tmn þ V Tmn  W Tðm  1Þðn þ 1Þ  V Tðm  1Þðn þ 1Þ 1  δnNz 1  δ1m )     þ0:5C d ? W Tmn þ V Tmn  W Tðm þ 1Þðn  1Þ  V Tðm þ 1Þðn  1Þ 1  δ1n 1  δmNy ¼ 0:

ð15dÞ

In order to analyze the problem in a more general framework, the following dimensionless parameters are considered: sffiffiffiffiffiffiffiffiffiffi 2 2 C d½: lb T T C v½: lb V Tmn W Tmn 1 ks Gb Eb I b T T T T T T t; χ ¼ V mn ¼ ; W mn ¼ ; Θ ymn ¼ Θymn ; Θ zmn ¼ Θzmn ; τ ¼ ; C v½: ¼ ; C d½: ¼ ; ½: ¼ J or ? 2 lb lb ρb ks Gb Ab ks Gb Ab lb ks Gb Ab l

ð16Þ

b

by introducing Eq. (16) to Eqs. (15a)–(15d), the dimensionless discrete equations of motion of the ensembles of SWCNTs on the basis of the NTBT are derived as follows: 8 9 ! T T T < = 2 ∂2 Θ zmn T ∂V mn  2 ∂ Θ zmn Ξ λ  Θ χ ¼ 0; ð17aÞ   zmn 2 : ∂τ 2 ; ∂ξ ∂ξ 0

1 ( T   h T 2 T   T i ∂Θ zmn T T A þ Ξ ∂ V mn þ C T  V mn  V ðm þ 1Þn 1  δmNz þ V mn  V ðm  1Þn 1  δ1m  vJ 2 2 ∂ξ ∂τ ∂ξ   h T   T i T T T þC v ? V mn  V mðn  1Þ 1  δ1n þ V mn V mðn þ 1Þ 1  δnNz  T    T T T T þ0:5C d J W mn þ V mn  W ðm  1Þðn  1Þ  V ðm  1Þðn  1Þ 1  δ1n 1  δ1m  T    T T T T þ0:5C d J W mn þ V mn  W ðm þ 1Þðn þ 1Þ  V ðm þ 1Þðn þ 1Þ 1  δnNz 1  δmN y  T    T T T T þ0:5C d J V mn  W mn  V ðm þ 1Þðn  1Þ þ W ðm þ 1Þðn  1Þ 1  δ1n 1  δmNy  T    T T T T þ0:5C d J V mn  W mn  V ðm  1Þðn þ 1Þ þ W ðm  1Þðn þ 1Þ 1  δnNz 1  δ1m  T    T T T T þ0:5C d ? V mn W mn þ W ðm  1Þðn  1Þ  V ðm  1Þðn  1Þ 1  δ1n 1  δ1m  T    T T T T þ0:5C d ? V mn W mn þ W ðm þ 1Þðn þ 1Þ  V ðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy  T    T T T T þ0:5C d ? V mn þW mn  V ðm  1Þðn þ 1Þ W ðm  1Þðn þ 1Þ 1  δnNz 1  δ1m )  T    T T T T þ0:5C d ? V mn þW mn  V ðm þ 1Þðn  1Þ W ðm þ 1Þðn  1Þ 1  δ1n 1  δmNy ¼ 0; T

2 @∂ V mn

8 <

Ξ λ :

 2∂

T

9

!

T

T Θ ymn = ∂2 Θ ymn T ∂W mn  Θ χ ¼ 0;   y 2 mn ∂τ 2 ; ∂ξ ∂ξ

2

ð17bÞ

ð17cÞ

286

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

0 

∂ξ

2

T

þC v ?

h



1

( T 2 A þ Ξ ∂ W mn 2 ∂τ ∂ξ T

T

2 @∂ W mn

∂Θ ymn

T

T

W mn  W ðm  1Þn

T

þC v J

h

T

T

W mn  W mðn þ 1Þ

    T i T 1  δnNz þ W mn  W mðn  1Þ 1  δ1n

    T i T 1  δ1m þ W mn  W ðm þ 1Þn 1  δmNy

 T    T T T T þ0:5C d J W mn þ V mn  W ðm  1Þðn  1Þ  V ðm  1Þðn  1Þ 1  δ1n 1  δ1m  T    T T T T þ0:5C d J W mn þ V mn  W ðm þ 1Þðn þ 1Þ  V ðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy  T    T T T T þ0:5C d J W mn  V mn  W ðm  1Þðn þ 1Þ þ V ðm  1Þðn þ 1Þ 1  δnNz 1  δ1m  T    T T T T þ0:5C d J W mn  V mn  W ðm þ 1Þðn  1Þ þ V ðm þ 1Þðn  1Þ 1  δ1n 1  δmNy  T    T T T T þ0:5C d ? W mn  V mn  W ðm  1Þðn  1Þ þ V ðm  1Þðn  1Þ 1  δ1n 1  δ1m  T    T T T T þ0:5C d ? W mn  V mn  W ðm þ 1Þðn þ 1Þ þ V ðm þ 1Þðn þ 1Þ 1  δnNz 1  δmN y  T    T T T T þ0:5C d ? W mn þ V mn  W ðm  1Þðn þ 1Þ  V ðm  1Þðn þ 1Þ 1  δnNz 1  δ1m )  T    T T T T ¼ 0: þ0:5C d ? W mn þ V mn  W ðm þ 1Þðn  1Þ  V ðm þ 1Þðn  1Þ 1  δ1n 1  δmNy

ð17dÞ

Eqs. (17a)–(17d) represent a set of 4N y N z second-order coupled PDEs. In the following part, characteristics of the transverse waves in the 3D ensemble will be evaluated based on the proposed model in this part.

4.2.2. Characteristics of the transverse waves based on the NTBT The harmonic waves in the (m,n)th tube of the 3D ESWCNTs based on the NTBT are assumed as follows:   T T T T T T T R R 〈V mn ; Θ zmn ; W mn ; Θ ymn 〉 ¼ 〈V mn0 ; Θ zmn0 ; W mn0 ; Θ ymn0 〉ei ϖ τ  k x ξ ; T

T

T

ð18Þ

T

where V mn0 , Θ zmn0 , W mn0 , and Θ ymn0 denote the dimensionless amplitudes of the transverse waves, and ϖ T is the dimensionless frequency of the 3D ESWCNTs modeled based on the NTBT. The exterior SWCNTs of the 3D ESWCNTs are assumed to be prevented from any lateral movement. into account this condition and by introducing Eq. (18) to Eqs. (17a)–(17d), one can arrive at:    2 T By T taking T T T  ϖ T M þK x ¼ 0 where  M and K can be easily calculated. A nontrivial solution to the resulting equations would exist if and 0  T T 2 only if: det  ϖ T M þ K ¼ 0. By solving these equations for ϖ T , the frequency of the transverse wave of wavenumber k x can be readily calculated. Subsequently, the phase velocity of the transverse waves in the 3D ESWCNTs based on the NTBT is evaluated by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vTp ¼ ϖ T =k x ks Gb =ρb .

4.3. Elastic waves in 3D ESWCNTs via NHOBT 4.3.1. Nonlocal discrete equations of motion using NHOBT By employing the higher-order beam theory of Bickford-Reddy [77,78] in the context of the nonlocal continuum theory, the kinetic energy, T H , and the strain energy of the 3D ensemble of SWCNTs, UH, are stated as

0 0

1 !2 !2 1 H 2 H 2 ∂ V ∂ W mn mn A B I0 @ C þ B C ∂t∂x ∂t∂x B C B C B C ! ! ! 2 2 Z H H H H lb B Ny H H N 2 2 z ∂Ψ ymn ∂Ψ ymn ∂ W mn ∂Ψ ymn ∂Ψ ymn ∂ W mn C 1 B C ∑ ∑ TH ¼ B þ I2 C dx; þ þ þ α2 I 6  2α I 4 C 2m¼1n¼1 0 B ∂t ∂t ∂t∂x ∂t ∂t ∂t∂x B C B C ! ! ! 2 2 B C H H H H H H B C ∂Ψ zmn ∂Ψ zmn ∂2 V mn ∂Ψ zmn ∂Ψ zmn ∂2 V mn 2 @ þ I2 A þ þ þ α I6  2αI 4 ∂t ∂t ∂t∂x ∂t ∂t ∂t∂x

ð19aÞ

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

0 B B B B B B B B B B B B B B B B B B B B B B B Z l Ny Nz b B 1 H B ∑ ∑ U ¼ 2m¼1n¼1 0 B B B B B B B B B B B B B B B B B B B B B B B @

H ∂Ψ ymn 

H

þ Ψ

∂W H mn þ ∂x

! 

α

H



H 

1

C C C C !  H  C H H     H H ∂Ψ zmn ∂V mn C H nl nl nl C M bzmn þ Ψ zmn þ α P bymn þ Q bymn þ C ;x ∂x ∂x C 

C 2   2    C H H H H 1  δmNz þ V mn  V ðm  1Þn 1  δ1m C þ C v J V mn  V ðm þ 1Þn C 

C C 2   2    C H H þ Cv ? V H 1  δ1n þ V H 1  δnNz C mn V mðn  1Þ mn  V mðn þ 1Þ C C  2  C   H H C 1  δ1m 1  δ1n þ C d J X mn  X ðm  1Þðn  1Þ C C  2  C   H H C dx; 1  δmNy 1  δnNz þ C d J X mn  X ðm þ 1Þðn þ 1Þ C C  2  C    C H H 1  δ1m 1  δnNz þ C d J Y mn  Y ðm  1Þðn þ 1Þ C C C  2     C H H 1  δmNy 1  δ1n þ C d J Y mn  Y ðm þ 1Þðn  1Þ C C C  2     C H X 1  δ δ 1  þ Cd ? XH C nN z 1m mn ðm  1Þðn þ 1Þ C C  2    C H C X 1  δ δ 1  þ Cd ? XH mN y 1n mn ðm þ 1Þðn  1Þ C C  2    C H H C 1  δ1m 1  δ1n þ C d ? Y mn  Y ðm  1Þðn  1Þ C C  2     A H H 1  δmNy 1  δnNz þ C d ? Y mn  Y ðm þ 1Þðn þ 1Þ

∂x

M nl bymn

H ymn

P nl bzmn

;x

þ

Q nl bzmn

287

ð19bÞ

   H  H  H pffiffiffi  H pffiffiffi  H H H H nl H H 2=2 W mn þ V H 2=2  W H , M nl , Q nl , and where X H bymn mn ¼ mn and Y mn ¼ mn þ V mn . Furthermore, Vmn, Wmn, Ψ ymn , Ψ zmn , M bymn bzmn  H Q nl represent the deflection field associated with the y and z axes, deflection's angles about the y and z axes, nonlocal bending bzmn moments about the y and z axes, and nonlocal shear forces along the y and z axes of the (m,n)th SWCNT modeled based on the NHOBT, respectively. In the framework of the NHOBT, the nonlocal forces within the (m,n)th SWCNT are expressed in terms of deformation fields as [33,79]: ! H H ∂2 M H ∂Ψ ymn ∂Ψ ymn ∂2 W H bymn 2 mn  þ MH  ðe aÞ ¼ J α J ; ð20aÞ 0 2 4 bymn ∂x ∂x ∂x2 ∂x2 2 MH bzmn  ðe0 aÞ

QH bymn þ α

∂P H bymn

¼ κ Ψ zmn þ H

QH bzmn þ α ¼κ Ψ

H ymn

! H H ∂2 M H ∂Ψ zmn ∂Ψ zmn ∂2 V H bzmn mn  þ ¼ J α J ; 2 4 ∂x ∂x ∂x2 ∂x2 !

∂x

∂P H bzmn ∂x

Ab

!

!

H

ðe0 aÞ2

∂P bzmn ∂2 QH bzmn þ α ∂x ∂x2

κ¼

ρb zn dA;

Z Ab

ð20cÞ

!

! ! H H ∂2 Ψ ymn ∂2 Ψ ymn ∂3 W H ∂W H 2 mn mn þ  α J6 þ ; þ αJ 4 ∂x ∂x2 ∂x2 ∂x3

α ¼ 1=ð3r 2o Þ; Z

∂P bymn ∂2 QH bymn þ α ∂x ∂x2

! ! H H ∂2 Ψ zmn ∂2 Ψ zmn ∂3 V H ∂V H 2 mn mn  α J þ ; þ αJ 4 6 ∂x ∂x2 ∂x2 ∂x3

where

In ¼

H

 ðe0 aÞ2

ð20bÞ

ð20dÞ

Gb ð1  3αz2 ÞdA;

Jn ¼

Z Ab

Eb zn dA; n ¼ 0; 2; 4; 6:

ð21Þ

By implementation of the Hamilton's principle through using Eqs. (20a)–(20d), the nonlocal discrete equations of motion of 3D ensembles of vertically aligned SWCNTs based on the NHOBT are obtained as follows: ( )  ∂2 Ψ H ∂3 V H zmn 2 mn Ξ I2  2αI 4 þ α2 I6 þð α I  α I Þ 6 4 ∂t 2 ∂t 2 ∂x !  ∂2 Ψ H  ∂ 3 V H ∂V H H zmn mn þ α J 4  α2 J 6 ¼ 0; ð22aÞ þ κ Ψ zmn þ mn  J 2  2αJ 4 þ α2 J 6 2 ∂x ∂x ∂x3

288

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

! ! H H H ∂Ψ zmn ∂2 V H ∂3 Ψ zmn ∂2 Ψ zmn ∂3 V H 2 mn mn þ κ þ α J6 þ  αJ 4 ∂x ∂x2 ∂x3 ∂x2 ∂x3 ( H h    ∂3 Ψ zmn   i ∂2 V H ∂4 V H H H mn  α2 I 6 2 mn þ C v J V H þ Ξ I0  α2 I 6  αI 4 1  δmNz þ V H 1  δ1m mn V ðm þ 1Þn mn  V ðm  1Þn 2 2 2 ∂t ∂t ∂x ∂t ∂x h     i H H H þ C v ? V mn V mðn  1Þ 1  δ1n þ V H 1  δnNz mn  V mðn þ 1Þ     H H H þ 0:5C d J W H 1  δ1n 1  δ1m mn þV mn  W ðm  1Þðn  1Þ  V ðm  1Þðn  1Þ     H H H 1  δnNz 1  δmNy þ 0:5C d J W H mn þV mn  W ðm þ 1Þðn þ 1Þ  V ðm þ 1Þðn þ 1Þ     H H H 1  δ1n 1  δmN y þ 0:5C d J V H mn  W mn  V ðm þ 1Þðn  1Þ þW ðm þ 1Þðn  1Þ     H H H 1  δnNz 1  δ1m þ 0:5C d J V H mn  W mn  V ðm  1Þðn þ 1Þ þW ðm  1Þðn þ 1Þ     H H H 1  δ1n 1  δ1m þ 0:5C d ? V H mn  W mn þ W ðm  1Þðn  1Þ  V ðm  1Þðn  1Þ     H H H 1  δnNz 1  δmNy þ 0:5C d ? V H mn  W mn þ W ðm þ 1Þðn þ 1Þ  V ðm þ 1Þðn þ 1Þ     H H H 1  δnNz 1  δ1m þ 0:5C d ? V H mn þ W mn  V ðm  1Þðn þ 1Þ  W ðm  1Þðn þ 1Þ )     H H H 1  þ W  V  W δ δ 1  ¼ 0; þ 0:5C d ? V H mN 1n y mn mn ðm þ 1Þðn  1Þ ðm þ 1Þðn  1Þ

Ξ

( ) H  ∂2 Ψ ymn ∂3 W H 2 mn I 2  2αI 4 þ α2 I 6 þ ð α I  α I Þ 6 4 ∂t 2 ∂t 2 ∂x

þ κ Ψ ymn þ H

! H  ∂2 Ψ ymn  ∂3 W H ∂W H mn mn þ αJ 4  α2 J 6 ¼ 0;  J 2  2α J 4 þ α 2 J 6 2 ∂x ∂x ∂x3

! ! ( H H H ∂3 Ψ ymn ∂2 Ψ ymn ∂3 W H  ∂3 Ψ ymn ∂2 W H ∂2 W H ∂4 W H 2 mn mn mn þ  α2 I 6 2 mn þ α J6 þ  α2 I 6  αI 4 κ  αJ 4 þ Ξ I0 2 3 2 3 2 2 ∂x ∂x ∂x ∂x ∂x ∂t ∂t ∂x ∂t ∂x2 h     H i H H H þC v J W mn  W mðn þ 1Þ 1  δnNz þ W mn  W mðn  1Þ 1  δ1n h     i H H þC v ? W H 1  δ1m þ W H 1  δmNy mn  W ðm  1Þn mn  W ðm þ 1Þn     H H H þ0:5C d J W H 1  δ1n 1  δ1m mn þ V mn W ðm  1Þðn  1Þ V ðm  1Þðn  1Þ     H H H 1  δnNz 1  δmN y þ0:5C d J W H mn þ V mn W ðm þ 1Þðn þ 1Þ V ðm þ 1Þðn þ 1Þ     H H H 1  δnNz 1  δ1m þ0:5C d J W H mn  V mn W ðm  1Þðn þ 1Þ þV ðm  1Þðn þ 1Þ     H H H 1  δ1n 1  δmNy þ0:5C d J W H mn  V mn W ðm þ 1Þðn  1Þ þV ðm þ 1Þðn  1Þ     H H H 1  δ1n 1  δ1m þ0:5C d ? W H mn  V mn  W ðm  1Þðn  1Þ þ V ðm  1Þðn  1Þ     H H H 1  δnNz 1  δmNy þ0:5C d ? W H mn  V mn  W ðm þ 1Þðn þ 1Þ þ V ðm þ 1Þðn þ 1Þ     H H H þ0:5C d ? W H 1  δnNz 1  δ1m mn þ V mn  W ðm  1Þðn þ 1Þ  V ðm  1Þðn þ 1Þ )     H H H 1  1  ¼ 0: þ V  W  V δ δ þ0:5C d ? W H mN y 1n mn mn ðm þ 1Þðn  1Þ ðm þ 1Þðn  1Þ

ð22bÞ

ð22cÞ

∂Ψ ymn H

ð22dÞ

To analyze the problem in a more general context, the following dimensionless quantities are considered: sffiffiffiffi H H VH WH α J6 H H H H V mn ¼ mn ; W mn ¼ mn ; Ψ ymn ¼ Ψ ymn ; Ψ zmn ¼ Ψ zmn ; τ ¼ 2 t; lb lb I0 l b

γ ¼ 2 1

αI 4  α2 I 6 2 I 0 lb

;

γ ¼ 2 2

α2 I 6 2 I 0 lb

;

κ l2 γ ¼ 2b ; α J6 2 3

γ 27 ¼

κ I0 l4b ; ðI 2  2αI 4 þ α2 I 6 Þα2 J 6

γ 28 ¼

γ 29 ¼

ðαJ 4  α2 J 6 ÞI 0 lb ; ðI 2  2αI 4 þ α2 I 6 Þα2 J 6

C v½: ¼

2

H

αJ  α2 J γ ¼ 4 2 6; α J6 2 4

γ ¼ 2 6

αI 4  α2 I 6 ; I 2  2α I 4 þ α 2 I 6

ðJ 2  2αJ 4 þ α2 J 6 ÞI 0 lb ; ðI 2  2αI 4 þ α2 I 6 Þα2 J 6 2

4

C v½: lb ; α2 J 6

H

C d½: ¼

4

C d½: lb ; α2 J 6

½: ¼ J or ? :

ð23Þ

By introducing Eq. (23) to Eqs. (22a)–(22d), the dimensionless governing equations of the 3D ensemble of SWCNTs describing its transverse motion are derived as 8 9 ! H H <∂ 2 Ψ H 3 H = 3 H ∂2 Ψ zmn H ∂V mn zmn 2 ∂ V mn 2 2 2 ∂ V mn Ξ  γ γ Ψ þ γ þ γ ¼ 0; ð24aÞ þ  6 7 8 9 z mn 2 3 : ∂τ 2 ∂τ 2 ∂ξ ; ∂ξ ∂ξ ∂ξ

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

0 γ

2@ 3

289

1 H H H H ∂Ψ zmn ∂2 V mn ∂3 Ψ zmn ∂4 V mn A  γ2 þ þ 4 2 3 4 ∂ξ ∂ξ ∂ξ ∂ξ

( H H 4 H ∂3 Ψ zmn ∂2 V mn 2 2 ∂ V mn þ γ  γ 1 2 2 ∂τ 2 ∂ τ 2 ∂ξ ∂τ 2 ∂ ξ h    H    i H H H H þC v J V mn V ðm þ 1Þn 1  δmNz þ V mn V ðm  1Þn 1  δ1m   h H   H i H H H þC v ? V mn  V mðn  1Þ 1  δ1n þ V mn V mðn þ 1Þ 1  δnNz  H    H H H H þ0:5C d J W mn þ V mn  W ðm  1Þðn  1Þ  V ðm  1Þðn  1Þ 1  δ1n 1  δ1m  H    H H H H þ0:5C d J W mn þ V mn  W ðm þ 1Þðn þ 1Þ  V ðm þ 1Þðn þ 1Þ 1  δnNz 1  δmN y  H    H H H H þ0:5C d J V mn  W mn  V ðm þ 1Þðn  1Þ þ W ðm þ 1Þðn  1Þ 1  δ1n 1  δmNy  H    H H H H þ0:5C d J V mn  W mn  V ðm  1Þðn þ 1Þ þ W ðm  1Þðn þ 1Þ 1  δnNz 1  δ1m  H    H H H H þ0:5C d ? V mn W mn þ W ðm  1Þðn  1Þ  V ðm  1Þðn  1Þ 1  δ1n 1  δ1m  H    H H H H þ0:5C d ? V mn W mn þ W ðm þ 1Þðn þ 1Þ  V ðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy  H    H H H H þ0:5C d ? V mn þW mn  V ðm  1Þðn þ 1Þ W ðm  1Þðn þ 1Þ 1  δnNz 1  δ1m )  H    H H H H þ0:5C d ? V mn þW mn  V ðm þ 1Þðn  1Þ W ðm þ 1Þðn  1Þ 1  δ1n 1  δmNy ¼ 0; þΞ

ð24bÞ

8 9 ! H H = H H <∂2 Ψ H 3 ∂2 Ψ ymn H ∂W mn ∂3 W mn ymn 2 ∂ W mn 2 Ξ  γ6 2 þ γ 29 ¼ 0;  γ 28 þ γ 7 Ψ ymn þ 2 2 3 : ∂τ ∂τ ∂ξ ; ∂ξ ∂ξ ∂ξ 0 γ

2@ 3

H

∂Ψ yn

þ

1 H ∂2 W n A

H

γ

2 4

∂3 Ψ yn

H

þ

∂4 W n

þΞ

8 <∂ 2 W H

H

γ

mn þ 21 2

∂3 Ψ ymn

ð24cÞ

H

 γ 22

∂4 W mn

2 3 4 2 : ∂τ ∂ξ ∂τ 2 ∂ ξ ∂ξ ∂ξ ∂ξ ∂τ 2 ∂ξ h i         H H H H H þC v J W mn  W mðn þ 1Þ 1  δnNz þ W mn  W mðn  1Þ 1  δ1n h       i H H H H H þC v ? W mn  W ðm  1Þn 1  δ1m þ W mn  W ðm þ 1Þn 1  δmN y  H    H H H H þ0:5C d J W mn þ V mn  W ðm  1Þðn  1Þ  V ðm  1Þðn  1Þ 1  δ1n 1  δ1m  H    H H H H þ0:5C d J W mn þ V mn  W ðm þ 1Þðn þ 1Þ  V ðm þ 1Þðn þ 1Þ 1  δnNz 1  δmN y  H    H H H H þ0:5C d J W mn  V mn  W ðm  1Þðn þ 1Þ þ V ðm  1Þðn þ 1Þ 1  δnNz 1  δ1m  H    H H H H þ0:5C d J W mn  V mn  W ðm þ 1Þðn  1Þ þ V ðm þ 1Þðn  1Þ 1  δ1n 1  δmNy  H    H H H H þ0:5C d ? W mn  V mn  W ðm  1Þðn  1Þ þ V ðm  1Þðn  1Þ 1  δ1n 1  δ1m  H    H H H H þ0:5C d ? W mn  V mn  W ðm þ 1Þðn þ 1Þ þ V ðm þ 1Þðn þ 1Þ 1  δnNz 1  δmNy  H    H H H H þ0:5C d ? W mn þ V mn  W ðm  1Þðn þ 1Þ  V ðm  1Þðn þ 1Þ 1  δnNz 1  δ1m  H   o H H H H þ0:5C d ? W mn þ V mn  W ðm þ 1Þðn  1Þ  V ðm þ 1Þðn  1Þ 1  δ1n 1  δmNy ¼ 0:

4.3.2. Characteristics of the transverse waves based on the NHOBT The elastic transverse waves in the (m,n)th SWCNT of the ensemble based on the NHOBT is considered as:   H H H H H H R H R 〈V mn ; Ψ zmn ; W mn ; Ψ ymn 〉 ¼ 〈V mn0 ; Ψ zmn0 ; W mn0 ; Ψ ymn0 〉ei ϖ τ  k x ξ ; H

H

H

ð24dÞ

ð25Þ

H

where V mn0 , Ψ zmn0 , W mn0 , and Ψ ymn0 are the dimensionless amplitudes of the transverse waves, and ϖ H is the dimensionless frequency of the 3D ESWCNTs based on the NHOBT. Without loss of generality, the exterior SWCNTs are assumed to be prohibited from any lateral    2 H H H H movement. In view of this assumption and by substituting Eq. (25) into Eqs. (24a)–(24d):  ϖ H M þ K x H 0 ¼ 0 where M and K can be readily obtained. The if and only if condition for existence of a nontrivial solution to the these set of equations is    2 H H ¼ 0. By solving such a set of equations for ϖ H , the frequency of the transverse waves for each level of the det  ϖ H M þ K wavenumber k x is calculated. Furthermore, the phase velocity of the transverse waves within the 3D ESWCNTs based on the NHOBT is rffiffiffiffiffi J6 H . evaluated as: vH p ¼ αϖ =lb k x I0

290

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

5. Establishment of continuous models based on the nonlocal beam theories 5.1. Elastic waves within 3D ESWCNTs via continuous NRBT 5.1.1. Nonlocal continuous governing equations using NRBT Based on the discrete governing equations of the NRBT, Eqs. (9a) and (9b), the nonlocal equations of motion of the (m,n)th SWCNT of the ensemble can be expressed by ( ! ∂4 V Rmn ∂2 V Rmn ∂4 V Rmn Eb I b þ Ξ ρ A  I b b 2 b ∂x4 ∂t 2 ∂t ∂x2     þ C v J 2V Rmn V Rðm þ 1Þn  V Rðm  1Þn þ C v ? 2V Rmn  V Rmðn  1Þ  V Rmðn þ 1Þ   1 þ ðC d J  C d ? Þ W Rðm  1Þðn þ 1Þ þ W Rðm þ 1Þðn  1Þ  W Rðm þ 1Þðn þ 1Þ  W Rðm  1Þðn  1Þ 2 )   1 þ ðC d J þ C d ? Þ 4V Rmn  V Rðm  1Þðn þ 1Þ  V Rðm þ 1Þðn þ 1Þ V Rðm þ 1Þðn  1Þ  V Rðm  1Þðn  1Þ ¼ 0; ð26aÞ 2

Eb I b

( ! ∂4 W Rmn ∂2 W Rmn ∂4 W Rmn þ Ξ ρ A  I b b b ∂x4 ∂t 2 ∂t 2 ∂x2     þ C v J 2W Rmn  W Rmðn þ 1Þ  W Rmðn  1Þ þC v ? 2W Rmn  W Rðm  1Þn  W Rðm þ 1Þn   1 þ ðC d J  C d ? Þ V Rðm  1Þðn þ 1Þ þ V Rðm þ 1Þðn  1Þ V Rðm þ 1Þðn þ 1Þ  V Rðm  1Þðn  1Þ 2 )   1 þ ðC d J þ C d ? Þ 4W Rmn  W Rðm  1Þðn þ 1Þ  W Rðm þ 1Þðn þ 1Þ  W Rðm þ 1Þðn  1Þ  W Rðm  1Þðn  1Þ ¼ 0: 2

ð26bÞ

In order to construct a continuous model based on the discrete relation in Eqs. (26a) and (26b), we introduce two continuous displacement functions of the form v ¼ vðx; y; z; tÞ and w ¼ wðx; y; z; tÞ such that: ½mn ðx; tÞ  ½:ðx; ymn ; zmn ; tÞ; ½ðm  1Þðn  1Þ ðx; tÞ  ½:ðx; ymn  d; zmn  d; tÞ; ½ðm  1Þðn þ 1Þ ðx; tÞ  ½:ðx; ymn  d; zmn þ d; tÞ; ½ðm þ 1Þðn  1Þ ðx; tÞ  ½:ðx; ymn þ d; zmn  d; tÞ; ½ðm þ 1Þðn þ 1Þ ðx; tÞ  ½:ðx; ymn þ d; zmn þ d; tÞ; ð27Þ  ½○   ½○  ½○ ½○ where ½ð½:Þ ¼ V v or W w and ½○ ¼ R or T or H. The transverse displacements of the neighboring SWCNTs of the (m,n)th SWCNT are now approximated by sixth-order Taylor polynomials as in the following form: ! 6 i i ∂i ½:ðx; ymn ; zmn ; tÞ j ij ½:ðx; ymn 7 d; zmn 7 d; tÞ ¼ ∑ ∑ ð 7 dÞ ð 7dÞ ; ð28Þ ij ∂zj ∂yi  j i¼1j¼0 where ∂0 ½:ðx; y; z; tÞ=∂y0 ∂z0 ¼ ½:ðx; y; z; tÞ; ½: ¼ v½○ and w½○ . By substituting Eq. (28) into Eqs. (26a) and (26b) in view of Eq. (27), the nonlocal continuous version of the free vibration of SWCNTs’ ensembles based on the NRBT are derived as follows: 8 >  <  ∂ 2 vR ∂ 4 vR ∂ 4 vR E b I b 4 þ Ξ ρb A b 2  I b 2 > ∂x ∂t ∂t ∂x2 : ! 2 4 6 2 R d ∂ 4 vR d ∂ 6 vR d ∂ 8 vR 2 ∂ v þ þ þ  Cv J d ∂y2 12 ∂y4 360 ∂y6 20; 160 ∂y8 ! 2 4 6 2 R d ∂4 vR d ∂6 vR d ∂ 8 vR 2 ∂ v þ þ þ  Cv ? d ∂z2 12 ∂z4 360 ∂z6 20; 160 ∂z8   2 2 R 2 R 3 2 ∂ v ∂ v d ∂4 vR ∂4 vR ∂4 vR 2 þ ∂z2 þ 12 4 þ 6∂z2 ∂y2 þ ∂y4 ∂y ∂z 26 6 R 7  ðC d J þ C d ? Þd 4 5 4 6 R d ∂ v v ∂6 vR ∂6 vR þ 15∂z∂4 ∂y þ 360 2 þ15∂z2 ∂y4 þ ∂y6 ∂z6 9 "  #> 2 4  = ∂2 wR d ∂4 wR ∂4 wR d ∂6 wR ∂6 wR ∂6 wR 2 þ 3 5 þ10 3 3 þ 3 þ þ 3  ðC d J  C d ? Þd 2 ¼ 0; ð29aÞ 5 3 > ∂y∂z 3 ∂y∂z 180 ∂y ∂z ∂y∂z ∂y ∂z ∂y ∂z ; 8 >  <  ∂2 wR ∂ 4 wR ∂4 wR Eb I b 4 þ Ξ ρb Ab 2  I b 2 2 > ∂x ∂t ∂t ∂x : C v J d

2

2

4

6

∂ 2 wR d ∂4 wR d ∂ 6 wR d ∂8 wR þ þ þ 2 4 6 12 ∂z 360 ∂z 20; 160 ∂z8 ∂z

!

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

291

! 2 4 6 ∂2 wR d ∂4 wR d ∂6 wR d ∂8 wR  Cv ? d þ þ þ ∂y2 12 ∂y4 360 ∂y6 20; 160 ∂y8   2 2 R 2 R 3 4 R 4 R ∂ w ∂ w d2 ∂4 wR þ 6∂z∂ 2w∂y2 þ ∂∂yw4 2 þ ∂z2 þ 12 ∂z4 2 6 ∂y 6 R 7  ðC d J þ C d ? Þd 4 5 4 d ∂ w ∂6 wR ∂6 wR ∂6 w R þ 15 þ 15 þ þ 360 6 4 2 2 4 6 ∂z ∂z ∂y ∂z ∂y ∂y 2

9  6 R #> = 6 R 6 R ∂ v d ∂ v ∂ v d ∂ v ∂ v ∂ v þ 3 þ10 3 3 þ 3 þ 2 þ ¼ 0: ∂y∂z 3 ∂y∂z3 ∂y3 ∂z 180 ∂y5 ∂z ∂y∂z5 > ∂y ∂z ;

"  ðC d J  C d ? Þd

2

2

2 R

4 R

4 R



4

ð29bÞ

By introducing Eq. (10) to Eqs. (29a) and (29b), the dimensionless continuous equations of motion describing transverse motion of the ensemble's SWCNTs are stated as

∂4 v R ∂ξ

4

þΞ

8 > > > > > > <∂ 2 v R > ∂τ 2 > > > > > :



 κd

2

2

λ

∂4 v R

2

∂τ 2 ∂ξ

2

0



R B∂ CvJ @

2 R

v

∂η2

þ

κd

2

12

1  4  6 κ d ∂6 v R κ d ∂8 v R C ∂4 v R þ þ A 360 ∂η6 20; 160 ∂η8 ∂ η4

2

R

d C v ?

4

6

∂2 v R d ∂4 v R d ∂6 v R d ∂8 v R þ þ þ 2 4 6 12 ∂γ 360 ∂γ 20; 160 ∂γ 8 ∂γ

!

2

3 ! 2 4 R 4 R ∂2 v R ∂2 v R d ∂4 v R 2 ∂ v 4∂ v 6 κ2 7 þ 2 þ þ 6κ þκ 2 7 12 ∂γ 4 ∂γ ∂γ 2 ∂η2 ∂η4  R 6 6 ∂η 7 R !7  CdJ þCd? 6 4 6 7 6 R 6 R 6 R 6 R 6 7 d ∂ v 2 ∂ v 4 ∂ v 6∂ v 4 þ 5 þ 15 κ þ 15 κ þ κ 360 ∂γ 6 ∂ γ 4 ∂ η2 ∂γ 2 ∂η4 ∂η6 ! 39 4 R > ∂2 w R d 2 ∂4 w R > > 2∂ w > 6 2 ∂η∂γ þ 3 ∂η∂γ 3 þ κ ∂η3 ∂γ 7> > = 6 7  R  R 6 !7 ¼ 0; κ CdJ Cd? 6 7 4 6 R 6 R 6 R > 6 7 ∂ w ∂ w ∂ w > 4 þ d 5> > 3κ 4 5 þ 10κ 2 3 3 þ 3 > > 180 ∂ η ∂γ ∂η∂γ 5 ∂η ∂γ ; 2

ð30aÞ

8 > > > <∂2 w R 4 R ∂4 w R 2 ∂ w þ Ξ λ 4 2 2 > ∂ τ > ∂ξ ∂τ 2 ∂ξ > : ! 2 4 6 ∂2 w R d ∂4 w R d ∂ 6 w R d ∂8 w R d þ þ þ 360 ∂γ 6 20; 160 ∂γ 8 ∂γ 2 12 ∂γ 4 0 1  2  4  6  2 R 2 R κ d ∂4 w R κ d ∂6 w R κ d ∂8 w R C B∂ w þ þ  κd C v ? @ 2 þ A 12 ∂η4 360 ∂η6 20; 160 ∂η8 ∂η 2

R CvJ

2 3  2 R R R R R 2∂2 w ∂2 w d ∂4 w 2 ∂4 w 4∂4 w κ þ þ þ 6 κ þ κ  2 2 4 2 2 4 12 ∂γ ∂η ∂γ ∂γ ∂η ∂η 6 7 R R 7  CdJ þCd? 6 4  R R R R 4 5 d ∂6 w 2 ∂6 w 4 ∂6 w 6∂6 w þ 360 ∂γ 6 þ 15κ ∂γ 4 ∂η2 þ 15κ ∂γ 2 ∂η4 þ κ ∂η6 

2

39 > > > κ 7= 7 ¼ 0;  4  5> 6 R 6 R 6 R d > þ 180 3κ 4 ∂∂η5v∂γ þ10κ 2 ∂η∂ 3v∂γ 3 þ3∂∂η∂vγ 5 > ; R

2

2 6 2∂∂ηv∂γ þ d3

 R R κ CdJ Cd? 6 4



R

R

4 ∂4 v þ 2 ∂∂η3v∂γ ∂η∂γ 3



ð30bÞ

where y ly

l ly

R

η ¼ ; κ ¼ z ; C d½: ¼

2 4

C d½: d lb 2 E b I b lz

; vR ¼

vR wR ; wR ¼ ; lb lb

v R ¼ v R ðξ; η; γ ; τÞ; w R ¼ w R ðξ; η; γ ; τÞ; ½: ¼ J or ? :

ð31Þ

292

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

5.1.2. Characteristics of the transverse waves within 3D ESWCNTs using continuous NRBT The harmonic form of the continuous dimensionless displacements of the 3D ESWCNTs can be expressed as follows:   R 〈v R ; w R 〉 ¼ 〈v R0 ; w R0 〉ei ϖ τ  k :r ;

ð32Þ

where and represent the dimensionless amplitudes of the elastic transverse waves, ϖ is the dimensionless frequency, k ¼ k x ex þ k y ey þk z ez is the dimensionless wavenumber vector in which its components are related to the those of the wavenumber by k x ¼ lb kx , k y ¼ ly ky , k z ¼ lz kz , and r ¼ ξex þ ηey þ γ ez is the dimensionless position vector. By substituting Eq. (32) into Eqs. (30a) and (30b), one can arrive at #!( R ) 

" v0 Γ1 Γ2  2 1 0 0 þ  ϖR ¼ ; ð33Þ Γ2 Γ3 0 1 0 w R0 v R0

w R0

R

where

  2  2  4  6 3 R C v J κk y d 6 κk y d κk y d κk y d 7 þ  Γ1 ¼   5  2 41   2  þ 2 12 360 20; 160 1 1 1 þ λ kx 1 þ λ kx 1 þ μk x  2 2  2  4  6 3 R R R 2 C v ? kzd kzd kzd kzd 7 2 CdJ þCd?  6 þ  þ þ 1  4 5  2  2 κ k y þ k z 12 360 20; 160 1 1 1 þ λ kx 1 þ λ kx # 2 4    4 6 d d 2 2 2 4 4 2 4 2 4 2 4 6 6  k þ 6κ k y k z þ κ k y þ k þ 15κ k z k y þ 15κ k z k y þ κ k y ; 12 z 360 z 4

kx  



Γ2 ¼

R

R

κ k y k z C d J C d ?

"

 2 1 1 þ λ kx

2

2

d 3

#  2  d 4   4  2 2 2 4 3 κ k y þ 10 κ k y k z þ 3k z k z þ κk y þ ; 180

 2  2  4  6 3 R C v ? κk y d κk y d κk y d κk y d 7 6 þ  Γ3 ¼   5  2 41   2  þ 2 12 360 20; 160 1 1 1 þ λ kx 1 þ λ kx 1 þ μk x  2 2  2  4  6 3 " R R R  2 C v J kzd k d k d k d z z z 2 CdJ þCd? 6 7 þ  þ 5þ  2 41   2 κ k y þk z 12 360 20; 160 1 1 1 þ λ kx 1 þ λ kx # 2 4  d  6  4 2 2 4 4 2 2 4 6 d  k z þ 6κ 2 k y k z þ κ 4 k y þ k z þ 15κ 2 k z k y þ 15κ 4 k z k y þ κ 6 k y : 12 360 4

kx  

ð34Þ

A nontrivial solution to the set of linear equations in Eq. (34) would be obtained if the determinant of the coefficient matrix pertinent to the dimensionless amplitude vector is set equal to zero. By doing this, the dispersion relation of the 3D ESWCNTs on the basis of the NRBT is derived as follows:  R 4  R 2 ϖ  ϖ ðΓ 1 þ Γ 3 Þ þ Γ 1 Γ 3  Γ 22 ¼ 0; ð35Þ by solving Eq. (35) for ϖ R , two dimensionless frequencies for the transverse waves within the 3D vertically aligned ESWCNTs are calculated as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u u  2   u u Γ þ Γ Γ  Γ Γ1  Γ3 2 t 1 tΓ 1 þ Γ 3 3 1 3 2 2 R R  þ ϖ1 ¼ þ Γ2; ϖ2 ¼ þ Γ2 : ð36Þ 2 2 2 2 We define the phase velocity of the transverse waves by vRp ¼ ωR =kx where ωR and kxpare the ffiffiffiffiffiffiffiffiffiffiffiffi ffi frequency and the wavenumber of the transverse waves in the nanostructure on the basis of the NRBT. Since ωR =kx ¼ ϖ R =λk x Eb =ρb , in view of Eq. (36), the phase velocities pertinent to ϖR1 and ϖR2 are obtained as: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 u 0 u 0  2   u u 1 E Γ þ Γ Γ  Γ 1 Γ1  Γ3 2 u u E b @Γ 1 þ Γ 3 3 1 3 2A R 2 b@ 1 R t t ð37Þ  þ þ Γ 2 ; vp2 ¼ þ Γ 2 A: vp1 ¼ 2 2 2 2 k x λ ρb k x λ ρb The final crucial characteristic of the transverse waves of our interest is the group velocity. This parameter is defined by vRg ¼ ∂ωR =∂kx . Since ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂ωR =∂kx ¼ 1=λ ðEb =ρb Þ∂ϖ R =∂k x , in view of Eqs. (34) and (36), the group velocities of the transverse waves within the vertically aligned 3D ESWCNTs are derived as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 u 0 u 0 !  1=2 !  1=2    2 u u Γ1  Γ3 2 ∂ Γ ∂ Γ Γ  Γ u B ∂Γ 1 ∂Γ 4 u 1 4 1 3 2 2 C C u B u B þ  þ Γ2 þ þ þ Γ2 C B C 2 2 1 u 1 u ∂k x ∂k x ∂k x ∂k x C R C uEb B uEb B vRg1 ¼ ¼ ; v ; ð38Þ B C B     C g2    C uρ B uρ B R R C 4λϖ 1 u b 4 λϖ u 1 ∂ Γ ∂ Γ ∂ Γ b ∂ Γ ∂ Γ ∂ Γ 1 1 3 2 1 3 2 A @ A 1 t @  t  ðΓ 1  Γ 3 Þ þ 2Γ 2  2  ðΓ 1  Γ 3 Þ þ 2Γ 2 2 ∂k x ∂k x ∂k x ∂k x ∂k x ∂k x

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

293

where      2   2 2  2   2 2 3 5 3 5 2 2 4k x 1 þ μk x 1 þ μk x Γ1 4k x 1 þ μk x 1 þ μk x Γ3  2μ2 k x  2k x λ  2μ2 k x  2k x λ 2 ∂Γ 1 ∂Γ 2 2k x λ Γ 2 ∂Γ 3   ¼ ¼ ¼ ; :  2 ;  2   2   2   2  1 1 1 ∂k x ∂k x ∂k x 1 þ λ kx 1 þ λ kx 1 þ λ kx 1 þ μk x 1 þ μk x

ð39Þ

5.2. Elastic waves within 3D ESWCNTs via continuous NTBT 5.2.1. Nonlocal continuous governing equations using NTBT On the basis of the NTBT and using Eqs. (15a)– (15d), the equations that describe transverse motion of the (m,n)th SWCNT of the 3D ensemble are stated by ( ) ! T T ∂2 Θzmn ∂2 Θzmn ∂V Tmn T  Ξ ρb I b G A Θ ¼ 0; ð40aÞ  k s b b zmn  E b I b 2 2 ∂x ∂x ∂t ∂2 V Tmn ∂Θzmn  ∂x ∂x2 T

(

!

∂2 V Tmn ∂t 2     þC v J 2V Tmn  V Tðm þ 1Þn  V Tðm  1Þn þ C v ? 2V Tmn  V Tmðn  1Þ  V Tmðn þ 1Þ

ks Gb Ab

þ Ξ ρb A b

  1 þ ðC d J  C d ? Þ W Tðm  1Þðn þ 1Þ þ W Tðm þ 1Þðn  1Þ  W Tðm þ 1Þðn þ 1Þ  W Tðm  1Þðn  1Þ 2 )   1 þ ðC d J þ C d ? Þ 4V Tmn  V Tðm  1Þðn þ 1Þ  V Tðm þ 1Þðn þ 1Þ  V Tðm þ 1Þðn  1Þ  V Tðm  1Þðn  1Þ ¼ 0; 2 (

∂2 Θymn T

Ξ ρb I b

∂t

2

)  k s Gb A b

∂W Tmn T  Θymn ∂x

!

∂2 Θymn

ð40bÞ

T

 Eb I b

∂x2

¼ 0;

( T !     ∂2 W Tmn ∂Θymn ∂2 W Tmn  þ C v J 2W Tmn  W Tmðn þ 1Þ  W Tmðn  1Þ þ C v ? 2W Tmn  W Tðm  1Þn  W Tðm þ 1Þn ks Gb Ab þ Ξ ρb Ab 2 2 ∂x ∂x ∂t   1 þ ðC d J  C d ? Þ V Tðm  1Þðn þ 1Þ þV Tðm þ 1Þðn  1Þ  V Tðm þ 1Þðn þ 1Þ  V Tðm  1Þðn  1Þ 2 )   1 þ ðC d J þ C d ? Þ 4W Tmn W Tðm  1Þðn þ 1Þ W Tðm þ 1Þðn þ 1Þ  W Tðm þ 1Þðn  1Þ  W Tðm  1Þðn  1Þ ¼ 0: 2

ð40cÞ

ð40dÞ

Now the following new continuous angles of deflection are taken into account:

θTy ðx; ymn ; zmn ; tÞ  ΘTymn ðx; tÞ; θTz ðx; ymn ; zmn ; tÞ  ΘTzmn ðx; tÞ;

ð41Þ

by introducing Eqs. (27), (28) and (41) to Eqs. (40a)–(40d), the continuous version of the equations of motion associated with the free transverse vibration of 3D ensembles of SWCNTs based on the NTBT are obtained ( )  T  T T ∂2 θ z ∂v ∂2 θ T  θz  Eb I b 2z ¼ 0; Ξ ρ b I b 2  k s Gb A b ð42aÞ ∂x ∂x ∂t

ks Gb Ab

C v J d

2

∂ 2 vT ∂ θ  ∂x ∂x2

T z

2

!

8 > > > <

∂ 2 vT þ Ξ ρb Ab 2 þ > ∂t > > : 4

6

∂ 2 vT d ∂ 4 vT d ∂6 vT d ∂8 vT þ þ þ ∂y2 12 ∂y4 360 ∂y6 20; 160 ∂y8

!

! 2 4 6 ∂ 2 vT d ∂ 4 vT d ∂6 vT d ∂8 vT þ þ þ ∂z2 12 ∂z4 360 ∂z6 20; 160 ∂z8 2 3  2  4 T ∂2 vT ∂2 vT d ∂ v ∂4 vT ∂ 4 vT þ 6 þ 6 2 þ 2 þ 7 12 ∂z4 ∂z ∂z2 ∂y2 ∂y4 7 2 6 ∂y 7  ðC d J þC d ? Þd 6   4 6 d ∂ 6 vT ∂ 6 vT ∂6 vT ∂ 6 vT 7 4 5 þ þ 15 þ 15 þ 360 ∂z6 ∂z4 ∂y2 ∂z2 ∂y4 ∂y6 C v ? d

2

294

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

"  ðC d J C d ? Þd

(

∂2 θ y T

Ξ ρb I b

2

9 > >  6 T #> = 6 T 6 T ∂ w d ∂ w ∂ w d ∂ w ∂ w ∂ w ¼ 0; þ 3 5 þ 10 3 3 þ 3 þ 2 þ 3 5 3 > ∂y∂z 3 ∂y∂z 180 ∂y ∂z ∂y∂z ∂y ∂z ∂y ∂z > > ;

)  k s Gb A b

∂t 2

2

∂2 wT ∂θ y  ∂x ∂x2

C v J d

T

4

T



4

 T  T ∂2 θ y ∂w T  θy  Eb I b 2 ¼ 0; ∂x ∂x

! T

ks Gb Ab

2 4

T

ð42cÞ

8 > > > > <

∂ 2 wT þ Ξ ρb A b 2 þ > ∂t > > > :

2

4

6

∂2 wT d ∂4 wT d ∂6 wT d ∂8 wT þ þ þ 2 4 6 12 ∂z 360 ∂z 20; 160 ∂z8 ∂z

2

ð42bÞ

!

! 2 4 6 ∂ 2 wT d ∂4 wT d ∂6 wT d ∂8 wT þ þ þ ∂y2 12 ∂y4 360 ∂y6 20; 160 ∂y8 2 3  2  4 T ∂2 wT ∂2 wT d ∂ w ∂ 4 wT ∂4 wT þ þ þ6 2 2 þ 6 7 2 12 ∂z4 ∂z2 ∂z ∂y ∂y4 7 2 6 ∂y 7  ðC d J þC d ? Þd 6  4  6 T 6 d ∂ w ∂6 wT ∂6 wT ∂6 wT 7 4 5 þ þ 15 4 2 þ 15 2 4 þ 360 ∂z6 ∂z ∂y ∂z ∂y ∂y6 C v ? d

2

"  ðC d J C d ? Þd

2

2

2



4



∂2 vT d ∂4 vT ∂ 4 vT d ∂6 vT ∂6 vT ∂ 6 vT þ 3 þ 10 3 3 þ 3 þ þ ∂y∂z 3 ∂y∂z3 ∂y3 ∂z 180 ∂y5 ∂z ∂y∂z5 ∂y ∂z

9 > > > #> = > > > > ;

¼ 0:

In view of Eq. (16), Eqs. (42a)– (42d) can be rewritten in the dimensionless form as follows: ( ! T) T T ∂2 θ ∂v T ∂2 θ Ξ λ  2 2z   θ z  χ 2z ¼ 0; ∂τ ∂ξ ∂ξ



T!

∂θ  z 2 ∂ξ ∂ξ

∂2 v T

þΞ

8 > > > > > > <∂ 2 v T > ∂τ 2 > > > > > :



 κd

2

0



T B∂ CvJ @

2 T

v

∂η2

þ

κd

2

12

ð43aÞ

1  4  6 κ d ∂6 v T κ d ∂8 v T C ∂4 v T þ þ A 360 ∂η6 20; 160 ∂η8 ∂η4

! 2 4 6 ∂2 v T d ∂4 v T d ∂6 v T d ∂8 v T þ þ þ ∂γ 2 12 ∂γ 4 360 ∂γ 6 20; 160 ∂γ 8 2 3 ! 2 4 T 4 T ∂2 v T ∂2 v T d ∂4 v T 2 ∂ v 4∂ v 6 κ2 7 þ 2 þ þ6κ þκ 2 7 12 ∂γ 4 ∂γ ∂ γ 2 ∂ η2 ∂η4  T 6 6 ∂η 7 T 7 !  CdJ þCd? 6 4 6 7 6 T 6 T 6 T 6 7 d ∂6 v T ∂ v ∂ v ∂ v 2 4 6 4 þ 5 þ 15κ þ 15κ þκ 6 4 2 2 4 6 360 ∂γ ∂γ ∂η ∂γ ∂η ∂η 2 39 ! 2 > > ∂2 w T d ∂4 w T ∂4 w T 2 > 62 7> > 6 ∂η∂γ þ 3 ∂η∂γ 3 þ κ ∂η3 ∂γ 7>  T  6 7= T 6 7 ! 7 ¼ 0; κ CdJ Cd? 6 4 > 6 d ∂6 w T ∂6 w T ∂ 6 w T 7> > 4 þ 5> 3κ 4 5 þ 10κ 2 3 3 þ 3 > > ; 180 ∂ η ∂γ ∂ η∂ γ 5 ∂η ∂γ 2

ð42dÞ

T

d C v ?

8 <

Ξ λ

@

9

!

T

θy = ∂2 θ y T ∂w T  θ y  χ 2 ¼ 0;  2 ∂τ ; ∂ξ ∂ξ

 2∂

:

0

T

ð43bÞ

2

∂2 w T ∂ξ

2

1 T



∂θ y ∂ξ

AþΞ

8 > > > > > > <∂ 2 w T > ∂τ 2 > > > > > :

2

ð43cÞ

2

T

d CvJ

4

6

∂ 2 w T d ∂ 4 w T d ∂6 w T d ∂8 w T þ þ þ 2 4 6 12 ∂γ 360 ∂γ 20; 160 ∂γ 8 ∂γ

!

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309



 κd

2

0

 T

κd

2

T B∂ w Cv? @ 2 þ 12 ∂η 2



κd

T

4

 T

κd

295

1

6 T

∂ w ∂ w ∂ w C þ þ A 360 ∂η6 20; 160 ∂η8 ∂η4 4

6

8

2

3 ! 2 4 T 4 T ∂2 w T ∂2 w T d ∂4 w T 2 ∂ w 4∂ w 6 κ2 7 þ þ þ 6κ þκ 7 12 ∂γ 4 ∂η2 ∂γ 2 ∂γ 2 ∂η2 ∂ η4  T 6 6 7 T 7 !  CdJ þCd? 6 4 6 7 6 T 6 T 6 T 6 7 d ∂6 w T ∂ w ∂ w ∂ w 2 4 6 4 þ 5 þ 15κ þ 15κ þκ 6 4 2 2 4 6 360 ∂γ ∂γ ∂η ∂γ ∂η ∂η 2 39 ! 2 > 4 T > ∂2 v T d ∂4 v T 2 ∂ v > 62 7> > 6 ∂η∂γ þ 3 ∂η∂γ 3 þ κ ∂η3 ∂γ 7>  T  6 7= T 7 ¼ 0; ! κ CdJ Cd? 6 4 6 7> 6 7> d ∂6 v T ∂6 v T ∂6 v T > 4 þ 5> 3κ 4 5 þ 10κ 2 3 3 þ 3 > > ; 180 ∂η ∂ γ ∂η∂γ 5 ∂η ∂γ

ð43dÞ

where 2 2

C d½: d lb

T

C d½: ¼

θ

; vT ¼

2 k s Gb A b l z T T z ¼ zð ; ;

T vT T wR T T T ;w ¼ ; θ ¼ θy ; θ z ¼ θz ; ½: ¼ J or ?; lb lb y T

T

θ ξ η γ ; τÞ; v T ¼ v T ðξ; η; γ ; τÞ; θ y ¼ θ y ðξ; η; γ ; τÞ; w T ¼ w T ðξ; η; γ ; τÞ:

ð44Þ

5.2.2. Characteristics of the transverse waves within 3D ESWCNTs using continuous NTBT The elastic transverse waves within 3D vertically aligned ESWCNTs in which modeled based on the continuous NTBT are taken into account as follows:   T T T T T ð45Þ 〈v T ; θ z ; w T ; θ y 〉 ¼ 〈v T0 ; θ z0 ; w T0 ; θ y0 〉ei ϖ τ  k:r ; by substituting this form of the deformation field into Eqs. (43a)– (43d), the following set of algebraic equations are derived: 8 T 9 8 9 v0 > 0 2 3 2 > > ν1 ν2 ν3 0 31> > >0> > 1 0 0 0 > > > > > T > > > > = <0> = < B  6 7 C 6 7 0 7C θ z0  0 1 0 0 7 6 ν4 ν5 0 B T 26 ; ¼ B ϖ 6 7C 7þ6 T @ 4 0 0 1 0 5 4 ν6 0 ν7 ν8 5A> w0 > > >0> > > > > ; : > > > > T > > > > 0 0 ν9 ν10 0 0 0 0 1 ; :θ > y0

where

ν1 ¼



2 kx



1 þ μk x T þCv?







2



T 6  2 þ C v J κ k y d 41 

kzd

2

2



6 41 

kzd

2

12

κk y d 12

2

 þ

κk y d

4

360



6 3

κk y d 7  5 20; 160

 6 3 "  T   2 kzd 2 T 7 þ  κk y þ k z 5þ CdJ þCd? 360 20; 160 

kzd

4

# 2  d4  6  2 2 4 4 2 2 4 6 d  4 k z þ6κ 2 k y k z þ κ 4 k y þ k z þ15κ 2 k z k y þ 15κ 4 k z k y þ κ 6 k y ; 12 360

# 4   4  2 2 4 d 3 κ k y þ 10 κ k y k z þ3k z ν3 ¼ κ κk y þ ; 180   2 2 λ2 1 þ χ k x kx kxλ ν2 ¼  i  2 ; ν4 ¼ i  2 ; ν5 ¼  2 ; 1 þ μk x 1 þ μk x 1 þ μk x " # 2  T   2  d 4   4  2 2 2 4 d T 3 κ k y þ 10 κ k y k z þ3k z þ ν6 ¼ κ k y k z C d J  C d ? 2  k z þ κk y ; 3 180 

T kykz C d J

T Cd?



"

2

d 2 3

2 kz þ



2 

2  2  4  6 3 2   κ k d κ k d κ k d y y y kx T 6 7 þ  ν7 ¼ 5   2 þ C v ? κ k y d 41  12 360 20; 160 1 þ μk x 2  2  4  6 3  T   2  2 kzd kzd kzd 2 T T 6 7 þ C v J k z d 41  þ  κk y þ k z 5 þ C d J þC d ? 12 360 20; 160

ð46Þ

296



K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

# 2  d4  6  2 2 4 4 2 2 4 6 d  4 k z þ 6κ 2 k y k z þ κ 4 k y þ k z þ 15κ 2 k z k y þ 15κ 4 k z k y þ κ 6 k y ; 12 360

k ν8 ¼  i  x 2 ; 1 þ μk x

k λ ν9 ¼ i x 2 ; 1 þ μk x 2



ν10 ¼

2

λ2 1 þ χ k x



 2 : 1 þ μk x

ð47Þ

A non-trivial solution to Eq. (46) would exist if and only if the determinant of the dimensionless amplitude vector would be zero. By doing so, one can arrive at the dispersion relation of the 3D ESWCNTs based on the NTBT:  8  6  4  2 ð48Þ P T8 ϖ T þ P T6 ϖ T þ P T4 ϖ T þ P T2 ϖ T þ P T0 ¼ 0; where P T8 ¼ 1; P T4

P T6 ¼  ðν1 þ ν5 þ ν7 þ ν10 Þ;

¼  ν2 ν4 þ ν1 ν5  ν3 ν6 þ ν1 ν7 þ ν5 ν7  ν8 ν9 þ ν1 ν10 þ ν5 ν10 þ ν7 ν10 ;

P T2 ¼ ν3 ν5 ν6 þ ν2 ν4 ν7  ν1 ν5 ν7 þ ν1 ν8 ν9 þ ν5 ν8 ν9 þ ν2 ν4 ν10  ν1 ν5 ν10 þ ν3 ν6 ν10  ν1 ν7 ν10  ν5 ν7 ν10 ; P T0 ¼ ν2 ν4 ν8 ν9  ν1 ν5 ν8 ν9  ν3 ν5 ν6 ν10  ν2 ν4 ν7 ν10 þ ν1 ν5 ν7 ν10 :

ð49Þ

1; …; 4, A more detail study of Eq. (48) reveals that it generally has four positive and four negative roots. The positive ones, ϖ represent the frequencies of the transverse waves within the vertically aligned ESWCNTs on the basis of the NTBT. The phase and group velocities pertinent to these frequencies can be determined in terms of the dimensionless wavenumber and frequency as follows: sffiffiffiffiffiffiffiffiffiffi ωTi ks Gb ϖ Ti T vpi ¼ ¼ ; kx ρb k x sffiffiffiffiffiffiffiffiffiffi ∂ωTi ks Gb ∂ϖ Ti vTgi ¼ ¼ ; ð50Þ ∂k ρb ∂k x T i ;i¼

where ∂ϖ Ti ∂k x

∂P T8 

¼

∂k x

8

T



6

T



4

T



2

T

6 4 2 0 ϖ Ti þ ∂P ϖ Ti þ ∂P ϖ Ti þ ∂P ϖ Ti þ ∂P ∂k ∂k ∂k ∂k

x ;  7  5  3 8P T8 ϖ Ti þ 6P T6 ϖ Ti þ 4P T4 ϖ Ti þ 2P T2 ϖ Ti x

x

x

ð51Þ

by using Eqs. (47) and (49), the values of ð∂P T2i  2 Þ=∂k x ; i ¼ 1; …; 5 can be easily evaluated. 5.3. Elastic waves within 3D ESWCNTs via continuous NHOBT 5.3.1. Nonlocal continuous governing equations using NHOBT By employing Eqs. (22a)–(22d), the transverse equations of motion of the (m,n)th SWCNT in the 3D ensemble could be written in the following form: ( )  ∂2 Ψ H ∂3 V H zmn 2 mn Ξ I 2  2αI 4 þ α2 I6 þ ð α I  α I Þ 6 4 ∂t 2 ∂t 2 ∂x !  ∂2 Ψ H  ∂ 3 V H ∂V H H zmn mn þ αJ 4  α2 J 6 ¼ 0; ð52aÞ þ κ Ψ zmn þ mn  J 2  2αJ 4 þ α2 J 6 2 ∂x ∂x ∂x3 ! ! H H H ∂Ψ zmn ∂2 V H ∂3 Ψ zmn ∂2 Ψ zmn ∂3 V H 2 mn mn þ α J þ α J þ  4 6 ∂x ∂x2 ∂x3 ∂x2 ∂x3 ( H  ∂3 Ψ zmn ∂2 V H ∂4 V H mn  α2 I 6 2 mn  α2 I 6  αI 4 þ Ξ I0 ∂t 2 ∂t 2 ∂x ∂t ∂x2     H H H H H þ C v J 2V H mn  V ðm þ 1Þn  V ðm  1Þn þ C v ? 2V mn  V mðn  1Þ V mðn þ 1Þ   1 H H H þ ðC d J  C d ? Þ W H ðm  1Þðn þ 1Þ þW ðm þ 1Þðn  1Þ  W ðm þ 1Þðn þ 1Þ  W ðm  1Þðn  1Þ 2 )   1 H H H H  V  V  V V þ ðC d J þ C d ? Þ 4V H ¼ 0; mn ðm  1Þðn þ 1Þ ðm þ 1Þðn þ 1Þ ðm þ 1Þðn  1Þ ðm  1Þðn  1Þ 2 κ

Ξ

( ) H  ∂2 Ψ ymn ∂3 W H 2 mn I 2  2αI 4 þ α2 I 6 þ ð α I  α I Þ 6 4 ∂t 2 ∂t 2 ∂x ! H  ∂2 Ψ ymn  ∂ 3 W H ∂W H H mn mn þ αJ 4  α2 J 6 ¼ 0;  J 2 2αJ 4 þ α2 J 6 þ κ Ψ ymn þ 2 ∂x ∂x ∂x3

ð52bÞ

ð52cÞ

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

297

! ! H H ∂3 Ψ ymn ∂2 Ψ ymn ∂3 W H ∂2 W H 2 mn mn  αJ 4 þ þ α J6 þ ∂x ∂x2 ∂x3 ∂x2 ∂x3

∂Ψ ymn H

κ

(

 ∂ Ψ ymn ∂2 W H ∂4 W H mn  α2 I 6 2 mn  α2 I 6  αI 4 ∂t 2 ∂t 2 ∂x ∂t ∂x2     H H H H H þC v J 2W mn W mðn þ 1Þ  W mðn  1Þ þ C v ? 2W H mn  W ðm  1Þn  W ðm þ 1Þn   1 H H H þ ðC d J  C d ? Þ V H ðm  1Þðn þ 1Þ þ V ðm þ 1Þðn  1Þ  V ðm þ 1Þðn þ 1Þ V ðm  1Þðn  1Þ 2 )   1 H H H H W  W  W  W þ ðC d J þ C d ? Þ 4W H ¼ 0: mn ðm  1Þðn þ 1Þ ðm þ 1Þðn þ 1Þ ðm þ 1Þðn  1Þ ðm  1Þðn  1Þ 2 3

þ Ξ I0

H

ð52dÞ

Now the following continuous functions are considered for deflection's angles of the constitutive SWCNTs of the ensemble:

ψ

H y ðx; ymn ; zmn ; tÞ 

Ψ Hymn ðx; tÞ;

ψ Hz ðx; ymn ; zmn ; tÞ  Ψ Hzmn ðx; tÞ;

ð53Þ

by introducing Eqs. (27), (28), and (53) to Eqs. (52a)– (52d), the continuous equations of motion pertinent to the free transverse vibration of 3D SWCNTs’ ensembles in accordance with the NHOBT are derived as follows:   ∂ 2 ψ H ∂ 3 vH z Ξ I 2  2α I 4 þ α 2 I 6 þ ðα 2 I 6  α I 4 Þ 2 2 ∂t ∂t ∂x    ∂ 2 ψ H  ∂3 vH ∂vH H 2 z  J 2  2α J 4 þ α J 6 þκ ψz þ þ αJ 4  α2 J 6 ¼ 0; ð54aÞ ∂x ∂x2 ∂x3  H   2 H  ∂ ψ z ∂ 2 vH ∂3 ψ H ∂ ψ z ∂3 vH z þ 2  αJ 4 þ α2 J 6 þ 3 3 2 ∂x ∂x ∂x ∂x ∂x 8 > > > > > < ∂2 vH  ∂3 ψ H ∂4 vH  α2 I 6  αI 4 2 z  α2 I 6 2 þ þ Ξ I0 2 > ∂t ∂t ∂x ∂t ∂x2 > > > > :

κ

C v J d

2

2

4

6

∂2 vH d ∂4 vH d ∂6 vH d ∂ 8 vH þ þ þ 2 4 6 12 ∂y 360 ∂y 20; 160 ∂y8 ∂y

!

! 2 4 6 ∂2 vH d ∂4 vH d ∂6 vH d ∂8 vH þ þ þ ∂z2 12 ∂z4 360 ∂z6 20; 160 ∂z8 2 3  2  4 H ∂2 vH ∂2 vH d ∂ v ∂ 4 vH ∂ 4 vH þ þ þ 6 þ 6 7 2 12 ∂z4 ∂z2 ∂z2 ∂y2 ∂y4 7 2 6 ∂y 7  ðC d J þC d ? Þd 6   4 6 7 6 H 6 H 6 H 6 H d ∂ v ∂ v ∂ v ∂ v 4 5 þ þ 15 4 2 þ 15 2 4 þ 6 360 ∂z6 ∂z ∂y ∂z ∂y ∂y C v ? d

2

"  ðC d J C d ? Þd

Ξ

( 

I 2  2αI 4 þ α I 6 2



þκ ψH y þ κ

∂ψ H y 8 > > > > > <

∂x

þ

2

 2 4  ∂2 wH d ∂4 wH ∂4 wH d ∂6 wH ∂6 wH þ 3 5 þ 10 3 3 þ 3 þ þ 3 3 ∂y∂z 3 ∂y∂z 180 ∂y ∂z ∂y∂z5 ∂y ∂z ∂y ∂z

∂2 ψ H y

 H

∂w ∂x

2

∂t 2

∂ 3 wH þ ðα I 6  α I 4 Þ 2 ∂t ∂x

∂3 ψ H ∂ 2 wH  2 ∂ 4 wH y  α I 6  αI 4 2  α2 I 6 2 þ þ Ξ I0 2 > ∂t ∂t ∂x ∂t ∂x2 > > > > : C v J d

C v ? d

2

4

6

∂ 2 wH d ∂4 wH d ∂ 6 wH d ∂8 wH þ þ þ 2 4 6 12 ∂z 360 ∂z 20; 160 ∂z8 ∂z 2

¼ 0;

ð54bÞ

)

 ∂2 ψ H  ∂ 3 w H y  J 2  2α J 4 þ α 2 J 6 þ αJ 4  α2 J 6 ¼ 0; ∂x2 ∂x3

2

> > > > > ;

2

! ! ∂3 ψ H ∂2 ψ H ∂2 wH ∂3 wH y y 2 α J þ α J þ  4 6 ∂x2 ∂x3 ∂x2 ∂x3

2

9 > > > > #> = 6 H ∂ w

4

6

!

∂ 2 wH d ∂4 wH d ∂6 wH d ∂ 8 wH þ þ þ 2 4 6 12 ∂y 360 ∂y 20; 160 ∂y8 ∂y

!

ð54cÞ

298

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

2

3  2  4 H ∂2 wH ∂2 wH d ∂ w ∂4 wH ∂4 wH þ þ þ6 2 2 þ 6 7 2 12 ∂z4 ∂z2 ∂z ∂y ∂y4 7 2 6 ∂y 7  ðC d J þC d ? Þd 6  4  6 H 6 7 6 H 6 H 6 H d ∂ w ∂ w ∂ w ∂ w 4 5 þ þ 15 4 2 þ15 2 4 þ 6 6 360 ∂z ∂z ∂y ∂z ∂y ∂y "  ðC d J C d ? Þd

2

2

9 > > > > #> = 6 H ∂ v

 2 4  ∂2 vH d ∂4 vH ∂4 vH d ∂6 vH ∂ 6 vH þ 3 þ 10 3 3 þ 3 þ þ ∂y∂z 3 ∂y∂z3 ∂y3 ∂z 180 ∂y5 ∂z ∂y∂z5 ∂y ∂z

> > > > > ;

¼ 0:

ð54dÞ

By introducing Eq. (23) to Eqs. (54a)– (54d), the dimensionless continuous governing equations describing transverse vibration of 3D ensembles of SWCNTs on the basis of the NHOBT are derived as follows: ( ) ! 3 H ∂2 ψ H ∂v H ∂2 ψ H ∂3 v H H 2 ∂ v 2 z z Ξ  γ γ ψ þ þ γ 29 3 ¼ 0; ð55aÞ þ  γ 28 6 7 z 2 2 2 ∂τ ∂τ ∂ξ ∂ξ ∂ξ ∂ξ 8 > > > > > > <∂2 v H

! ∂ψ H ∂2 v H ∂3 ψ H ∂4 v H ∂3 ψ H ∂4 v H z z þ 2  γ 24 þ 4 þΞ þ γ 21 2 z  γ 22 þ 3 2 2 > ∂ τ ∂ξ ∂ τ ∂ ξ > ∂ξ ∂ξ ∂ξ ∂τ 2 ∂ξ > > > > :

 γ 23



 κd

2

0

1  2  4  6 2 H 4 H 6 H 8 H κ d κ d κ d ∂ v ∂ v ∂ v ∂ v H B C CvJ @ 2 þ þ þ A 12 ∂η4 360 ∂η6 20; 160 ∂η8 ∂η

! 2 4 6 ∂2 v H d ∂4 v H d ∂6 v H d ∂8 v H d þ þ þ ∂γ 2 12 ∂γ 4 360 ∂γ 6 20; 160 ∂γ 8 2 3 ! 2 ∂2 v H ∂2 v H d ∂4 v H ∂4 v H ∂4 v H 2 2 4 6κ 7 þ 2 þ þ6κ þκ 7 12 ∂γ 4 ∂η2 ∂γ ∂ γ 2 ∂ η2 ∂ η4  H 6 6 7 H 6 !7  C d J þC d ? 6 4 7 H H H H 6 6 6 6 6 7 d ∂ v 2 ∂ v 4 ∂ v 6∂ v 4 þ 5 þ 15 κ þ 15 κ þ κ 360 ∂γ 6 ∂γ 4 ∂η2 ∂ γ 2 ∂ η4 ∂η6 2

H Cv?

39 ! 2 > 4 H > ∂2 w H d ∂4 w H 2∂ w > 62 7> > 6 ∂η∂γ þ 3 ∂η∂γ 3 þ κ ∂η3 ∂γ 7>  H  6 7= H ¼ 0; !7 κ CdJ Cd? 6 4 6 7 6 H 6 H > 6 d ∂ 6 w H 7> > 4∂ w 2 ∂ w > 4 þ 5 3κ þ 10κ þ3 > > ; 180 ∂η5 ∂γ ∂η∂γ 5 ∂ η3 ∂ γ 3 2

Ξ

8 > > > > > > <∂ 2 ψ H > ∂τ > > > > > :

y 2

 γ 26

! ∂2 ψ H ∂3 w H ∂w H ∂3 w H y H 2 2 g þ γ ψ þ γ þ γ 29 ¼ 0;  7 8 y 2 3 ∂τ 2 ∂ξ ∂ξ ∂ξ ∂ξ

( ∂3 ψ H ∂2 w H ∂4 w H y þ þ þΞ þ γ 21 2  γ 22 þ γ γ 2 3 4 2 2 ∂τ ∂ξ ∂τ ∂ ξ ∂ξ ∂ξ ∂ξ ∂τ 2 ∂ξ ! 2 4 6 2 H ∂2 w H d ∂4 w H d ∂6 w H d ∂8 w H d C v J þ þ þ 12 ∂γ 4 360 ∂γ 6 20; 160 ∂γ 8 ∂γ 2 2 3



 κd

∂ψ H y

2

∂2 w H

!

2 4

0



H B∂ Cv? @

2

wH

∂η2

2

þ

∂3 ψ H y

κd

2

12

∂4 w H

1  4  6 κ d ∂6 w H κ d ∂8 w H C ∂4 w H þ þ A 360 ∂η6 20; 160 ∂η8 ∂η4

3 ! 2 4 H 4 H ∂2 w H ∂ 2 w H d ∂4 w H 2 ∂ w 4∂ w 6 κ2 7 þ þ þ 6κ þκ 7 12 ∂γ 4 ∂η2 ∂γ 2 ∂γ 2 ∂η2 ∂ η4  H 6 6 7 H 7 !  CdJ þCd? 6 4 6 7 6 H 6 H 6 H 6 7 d ∂6 w H ∂ w ∂ w ∂ w 2 4 6 4 þ 5 þ 15κ þ 15κ þκ 6 4 2 2 4 6 360 ∂γ ∂γ ∂η ∂γ ∂ η ∂η

ð55bÞ

ð55cÞ

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

299

39 ! 2 > > > ∂2 v H d ∂4 v H ∂4 v H 2 > 62 7> þ þ κ > > 3 3 6 7 ∂ η ∂ γ 3 ∂ η ∂ γ ∂ η ∂ γ  H 6 7= H 6 7 ! 7 ¼ 0; κ CdJ Cd? 6 4 > 6 d ∂6 v H ∂6 v H ∂ 6 v H 7> > 4 þ 5> 3κ 4 5 þ 10κ 2 3 3 þ 3 > > > 180 ∂ η ∂γ ∂η∂γ 5 ∂η ∂γ ; 2

ð55dÞ

where H

C d½: ¼

2 4

C d½: d lb

α

2 J l2 6 z

; vH ¼

vH wH H H H ; wH ¼ ; ψ y ¼ ψH y ; ψ z ¼ ψ z ; ½: ¼ J or ? ; lb lb

ψ ¼ ψ Hz ðξ; η; γ ; τÞ; v H ¼ v H ðξ; η; γ ; τÞ; ψ Hy ¼ ψ Hy ðξ; η; γ ; τÞ; w H ¼ w H ðξ; η; γ ; τÞ:

ð56Þ

5.3.2. Characteristics of the transverse waves within 3D ESWCNTs using continuous NHOBT The elastic transverse waves within the 3D ESWCNTs based on the NHOBT are considered as   H H H H H H i ϖ H τ  k :r ; 〈v H ; ψ H z ; w ; ψ y 〉 ¼ 〈v 0 ; ψ z0 ; w 0 ; ψ y0 〉e

ð57Þ

H z

H H H in which 〈v H 0 ; ψ z0 ; w 0 ; ψ y0 〉 denotes the dimensionless vector of the amplitudes associated with the dynamic deformation of the 3D ESWCNTs on the basis of the NHOBT, and ϖ H is the dimensionless frequency of the transverse waves. By introducing Eq. (57) to Eqs. (55a)– (55d), one can arrive at the following set of equations:

0

2

@

4 0 0

ζ1 ζ2 B 6 B  H 2 6 ζ 3 ζ 4 B ϖ 6 6 B 0

0

0 0

ζ5 ζ7

3 2 η1 0 7 6 0 7 6 η4 7þ6 6 ζ6 7 5 4 η3 0 ζ8

318

9

8 9

> > > v > > η2 η3 0 > > > 0> > > > > 0 > > > > > > > 7C> H = η5 0 0 7C< ψ z0 = < 0 > 7C ¼ C> w H > > 0 > ; 0 η6 η7 7 > > 5A> > > 0 > > > > > > > > > 0 η8 η9 > ; > :ψH > ; :0> y0 H

ð58Þ

where 2

ζ 2 ¼  γ 21 k x ;

ζ 3 ¼  γ 26 k x ;

ζ 4 ¼ 1;

50

50

40

40

30

30 ϖ

ϖ1

2



ζ1 ¼ 1 þ k x γ2 ;

20

20

10

10

0

1

2

3

4

5

k /π x

6

7

8

0

1

2

3

5

4

6

7

8

k /π x

Fig. 3. Comparison of the predicted dispersions curves by the proposed models with those of the model of Murmu and Adhikari ([80]) for a double-nanobeam-system ((...) R NRBT, (– –) NTBT, (—) NHOBT; (□) Model of Murmu and Adhikari [80]; μ¼ 0.5, C v J ¼ 10).

300

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

Table 1 Comparison of the predicted flexural wave frequency by the proposed nonlocal discrete models and that of the proposed nonlocal continuous models in the case of k x ¼ k y ¼ k z ¼ π and e0 a ¼ 1 nm.

Continuous models

NRBT

NTBT

NHOBT

Discrete models

NRBT

NTBT

NHOBT



2

ζ5 ¼ 1 þ k x γ2 ;

ζ 6 ¼  γ 21 k x ;

λ

Ny ¼ Nz ¼ 5

Ny ¼ Nz ¼ 7

Ny ¼ Nz ¼ 9

N y ¼ N z ¼ 11

10 20 40 10 20 40 10 20 40

2.746482 1.526161 1.435976 2.462735 1.515176 1.435889 2.565119 1.519497 1.435925

2.566852 1.142248 0.979348 2.250989 1.125657 0.979098 2.365483 1.132177 0.979200

2.498238 0.965246 0.746675 2.168732 0.944711 0.746274 2.288449 0.952790 0.746437

2.465344 0.869744 0.607336 2.128986 0.846448 0.606797 2.251345 0.855623 0.607016

10 20 40 10 20 40 10 20 40

2.746482 1.526159 1.435974 2.462735 1.515175 1.435888 2.565118 1.519496 1.435924

2.566852 1.142248 0.979348 2.250989 1.125657 0.979097 2.365483 1.132177 0.979200

2.498238 0.965246 0.746675 2.168732 0.944711 0.746274 2.288449 0.952790 0.746437

2.465344 0.869744 0.607336 2.128986 0.846448 0.606797 2.251345 0.855623 0.607016

ζ 7 ¼  γ 26 k x ;

ζ 8 ¼ 1;

ð59aÞ

2 2  2  4  6 3 4   kxγ3 þ kx κ k d κ k d κ k d y y y H 6 7 þ  η1 ¼ 5  2 þC v J κ k y d 41  12 360 20; 160 1 þ μk x 2  2  4  6 3 "  H   2   k d k d k d z z z 2 2 H H 6 7 þ  κk y þ k z þ C v ? k z d 41  5 þ C d J þC d ? 12 360 20; 160 

þ

# 2  d4  6  2 2 4 4 2 2 4 6 d  4 k z þ 6κ 2 k y k z þ κ 4 k y þ k z þ 15κ 2 k z k y þ 15κ 4 k z k y þ κ 6 k y ; 12 360 

T

T

η3 ¼ κ k y k z C d J  C d ?



"

#  2  d 4   4  2 2 2 4 3 κ k y þ 10 κ k y k z þ 3k z k z þ κk y þ ; 180

2

2

d 3

2 2  2  4  6 3 4   kxγ3 þ kx κk y d κk y d κk y d 7 H 6 þ  η6 ¼ 5  2 þC v ? κ k y d 41  12 360 20; 160 1 þ μk x 2  2  4  6 3 "  H   2  2 kzd kzd kzd 2 H H 6 7 þ C v J k z d 41  þ  þ C þ C κk y þ k z 5 dJ d? 12 360 20; 160 

# 2  d4  6  d  4 2 2 2 4 4 2 4 2 4 2 4 6 6  k þ 6κ k y k z þ κ k y þ k þ 15κ k z k y þ 15κ k z k y þ κ k y ; 12 z 360 z   2  k x γ 23  k x γ 4 η2 ¼  i ;  2 1 þ μk x   2  k x γ 23  k x γ 4 η7 ¼  i ;  2 1 þ μk x

η4 ¼ i

  2  k x γ 27  k x γ 9

;  2 1 þ μk x   2  k x γ 27  k x γ 9 η8 ¼ i ;  2 1 þ μk x



η5 ¼

γ 27 þ k x γ 8

 2 ; 1 þ μk x 

η9 ¼

2

γ 27 þ k x γ 8

2

 2 : 1 þ μk x

ð59bÞ

The if and only if condition for existence of a non-trivial solution to Eq. (58) is the vanishing of the determinant of the coefficient matrix associated with the dimensionless vector of the amplitudes. As a result, the characteristic equation of the 3D ESWCNTs in which simulated via the NHOBT is obtained as  H 8  H 6  H 4  H 2 PH þ PH þP H þ PH þ PH ð60Þ 8 ϖ 6 ϖ 4 ϖ 2 ϖ 0 ¼ 0; where PH 8 ¼ ζ 1 ζ 4 ðζ 5 ζ 8  ζ 6 ζ 7 Þ þ ζ 3 ζ 2 ðζ 6 ζ 7  ζ 5 ζ 8 Þ;

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

301

Fig. 4. Frequencies, phase and group velocities as a function of the dimensionless wavenumber for three levels of the small-scale parameter: ((...) NRBT, (– –) NTBT, (—) NHOBT; (○) e0 a ¼0, (□) e0 a ¼ 1, ( ) e0 a ¼ 2 nm; k y ¼ k z ¼ π).

▵

PH 6 ¼ ζ 1 ζ 4 ζ 6 η8  ζ 1 ζ 4 ζ 5 η9  ζ 3 ζ 2 η7 ζ 7 þ ζ 3 ζ 2 η6 ζ 8 þ η1 ζ 4 ζ 6 ζ 7 þ ζ 1 ζ 4 η7 ζ 7 þ ζ 3 ζ 2 ζ 5 η9  ζ 1 η5 ζ 5 ζ 8  η4 ζ 2 ζ 6 ζ 7 þ ζ 3 η2 ζ 5 ζ 8  ζ 3 ζ 2 ζ 6 η8 þ ζ 1 η5 ζ 6 ζ 7  ζ 3 η2 ζ 6 ζ 7  ζ 1 ζ 4 η6 ζ 8  η1 ζ 4 ζ 5 ζ 8 þ η4 ζ 2 ζ 5 ζ 8 ; 2 PH 4 ¼ ζ 1 ζ 4 η6 η9  η4 η2 ζ 5 ζ 8 þ η4 η2 ζ 6 ζ 7  η3 ζ 4 ζ 8

 ζ 1 ζ 4 η7 η8 þ ζ 1 η5 ζ 5 η9 þ ζ 1 η5 η6 ζ 8  ζ 1 η5 ζ 6 η8  ζ 1 η5 η7 ζ 7 þ η1 ζ 4 ζ 5 η9 þ η1 ζ 4 η6 ζ 8  η1 ζ 4 ζ 6 η8  η1 ζ 4 η7 ζ 7 þ η1 η5 ζ 5 ζ 8  η1 η5 ζ 6 ζ 7  ζ 3 ζ 2 η6 η9 þ ζ 3 ζ 2 η7 η8  ζ 3 η2 ζ 5 η9  ζ 3 η2 η6 ζ 8 þ ζ 3 η2 ζ 6 η8 þ ζ 3 η2 η7 ζ 7  η4 ζ 2 ζ 5 η9  η4 ζ 2 η6 ζ 8 þ η4 ζ 2 ζ 6 η8 þ η4 ζ 2 η7 ζ 7 ; 2 PH 2 ¼ η1 ζ 4 η7 η8 þ η3 η5 ζ 8 þ ζ 3 η2 η6 η9 þ η4 η2 ζ 5 η9

þ η23 ζ 4 η9 þ η1 η5 η7 ζ 7  ζ 3 η2 η7 η8  η4 η2 η7 ζ 7 þ ζ 1 η5 η7 η8  η1 ζ 4 η6 η9 þ η4 ζ 2 η6 η9 þ η1 η5 η6 ζ 8  ζ 1 η5 η6 η9  η4 ζ 2 η7 η8  η4 η2 ζ 6 η8 þ η1 η5 ζ 6 η8 þ η4 η2 η6 ζ 8  η1 η5 ζ 5 η9 ; 2 PH 0 ¼ η1 η5 η6 η9  η1 η5 η7 η8  η4 η2 η6 η9 þ η4 η2 η7 η8  η3 η5 η9 :

ð61Þ

In most of the cases, Eq. (60) has four positive roots as well as four negative roots. The positive ones, ϖ H i ; i ¼ 1; …; 4, are the dimensionless frequencies of the transverse waves within the nanostructure in accordance with the NHOBT. The phase and group velocities associated

302

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

Fig. 5. Frequencies, phase and group velocities as a function of mean radius of the ECS for three levels of the SWCNT's length: ((...) NRBT, (– –) NTBT, (—) NHOBT; (○) lb ¼10, (□) lb ¼ 15, ( ) lb ¼30 nm; k y ¼ k z ¼ π, k x ¼ 2π).

▵

with these frequencies are determined by sffiffiffiffi sffiffiffiffi ωHi α J 6 ϖ Hi H ∂ωHi α J 6 ∂ϖ Hi ¼ ¼ ; v ¼ ¼ ; vH pi kx lb I 0 k x g i ∂kx lb I 0 ∂k x

ð62Þ

where ∂ϖ H i ∂k x

 ∂P H 8 ¼

∂k x

8

ϖ Hi þ

 ∂P H 6

6

ϖ Hi þ

 ∂P H 4

4

ϖ Hi þ

 ∂P H 2

2

ϖ Hi þ

∂P H 0

∂k x ∂k x ∂k x ∂k x ;  H 7  H 5  H 3 H 8P H þ 6P H þ 4P H þ2P H 8 ϖi 6 ϖi 4 ϖi 2 ϖi

ð63Þ

in which the statements ð∂P H 2i  2 Þ=∂k x could be readily evaluated through using Eqs. (59a), (59b) and (61). 6. Results and discussion In this part, the predicted results by the proposed models are compared with those of other available works in particular cases. Further, the obtained results by the proposed nonlocal continuous models are verified with those of the nonlocal discrete models. After ensuring regarding the accuracy of the suggested models, the roles of the crucial parameters on the characteristics of the elastic transverse waves within the 3D ESWCNTs are carefully addressed. 6.1. Validation of the proposed discrete models To ensure regarding the accuracy of the proposed models, some comparison studies are carried out in particular cases. Consider a double-SWCNT-system with the given data in Ref. [80] (viz. Ny ¼1 and Nz ¼ 2). The under study system is not allowed to move along the y direction (namely, C v ? ¼ 0), however, each tube can freely vibrate and deflect along the z axis. Murmu and Adhikari [80] studied vibration behaviors of such a system by using nonlocal Euler–Bernoulli beam theory. The explicit expressions of the flexural frequencies associated with the in-phase sequence (i.e., the first vibration mode) and the out-of-phase sequence (i.e., the second vibration mode) were derived. In Fig. 3, the predicted dispersion curves of the first two frequencies (i.e., flexural frequencies) by the proposed discrete models based on the NRBT, NTBT, and NHOBT and those of the model of Murmu and Adhikari [80] are plotted. As it is seen in Fig. 3, there exists a fairly good agreement between the predicted results by the proposed models and those of Murmu and Adhikari [80] for low levels of the wavenumber (i.e., for low levels of the mode number). Generally, for most values of the wavenumber, the predicted results by the discrete

K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

303

Fig. 6. Frequencies, phase and group velocities as a function of slenderness ratio for three levels of the small-scale parameter: ((...) NRBT, (– –) NTBT, (—) NHOBT; (○) e0 a ¼ 0, (□) e0 a ¼ 1, ( ) e0 a ¼ 2 nm; k y ¼ k z ¼ π, k x ¼ 2π).

▵

NRBT and those of Murmu and Adhikari [80] are close to each other. The main reason of this fact is that the effects of shear deformation are excluded in the formulations of these models. By an increase of the wavenumber, the discrepancies between the above-mentioned models and those of nonlocal shear deformable beam theories (NSDBT) (i.e., NTBT and NHOBT) would magnify. The results of NSDBT are commonly overestimated by the NRBT. As it is obvious in Fig. 3, in most of the cases, the predicted results by the NTBT and those of the NHOBT are in line and close to each other.

6.2. Validation of the proposed continuous models A comparison study is performed to check the accuracy of the proposed nonlocal continuous models. In Table 1, the predicted frequencies of the transverse waves within the nanosystem based on the proposed nonlocal continuous and discrete models are given in the case of k x ¼ k y ¼ k z ¼ π and e0 a ¼ 1 nm. The results are provided for three levels of the slenderness ratio (i.e., λ ¼10, 20, and 40) and four populations of the ensemble of SWCNTs (i.e., Ny ¼Nz ¼ 5, 7, 9, and 11). The properties of the nanotubes are Eb ¼1 TPa,ρb ¼2500 kg/m3, νb ¼0.2, rm ¼ 1 nm, tb ¼0.34 nm, and d ¼ 2rm þ t b . A brief survey of the given results in Table 1 reveals that all of the proposed nonlocal continuous models can successfully capture the predicted results by their corresponding discrete models. Additionally, by an increase of the number of constitutive SWCNTs or slenderness ratio of the nanosystem, all proposed models display that the frequency of the transverse wave would decrease. In the following parts, the roles of both population and slenderness ratio of the nanosystem on the characteristics of the transverse waves within vertically aligned ESWCNTs will be investigated in some detail.

6.3. Parametric studies The 3D ESWCNTs under study consists of 7  7 vertically aligned tubes whose ECSs have the following properties: Eb ¼1 TPa,

ρb ¼2500 kg/m3, νb ¼0.2, rm ¼1 nm, tb ¼0.34 nm, and d ¼ 2rm þ t b . In the following, based on the proposed continuous models, the

influences of the wavenumber, radius of the constitutive SWCNTs of the ensemble, slenderness ratio, small-scale parameter, intertube distance, and population of the ensemble on the important characteristics of the transverse waves within such nanostructures are discussed and explained in some detail. Additionally, the capabilities of the NRBT and NTBT in capturing the characteristics of transverse waves are addressed by comparing their predicted results with those predicted by the NHOBT as a reference model.

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K. Kiani / International Journal of Mechanical Sciences 90 (2015) 278–309

Fig. 7. Frequencies, phase and group velocities as a function of intertube distance for three levels of the wavenumber: ((...) NRBT, (– –) NTBT, (—) NHOBT; (○) k x ¼ 2π, (□) k x ¼ 4π, ( ) k x ¼ 6π; k y ¼ k z ¼ π).

▵

6.3.1. Role of the wavenumber on the characteristics of the transverse waves Effect of the wavenumber on the characteristics of the flexural and shear waves within the 3D ESWCNTs is of particular interest. The predicted frequencies and their corresponding phase and group velocities by the proposed nonlocal continuous models as a function of the dimensionless wavenumber are plotted in Fig. 4. The results are provided for three levels of the small-scale parameter (i.e., e0 a ¼0, 1, and 2 nm) as well as k y ¼ k z ¼ π . Flexural waves. As it is seen in Fig. 4, the flexural frequencies commonly increase with the dimensionless wavenumber irrespective of the small-scale parameter. As the influence of the small-scale parameter increases, the flexural frequencies of all models decrease and the influence of the wavenumber on the flexural frequencies would lessen. The predicted flexural frequencies by the NHOBT are generally overestimated by the NRBT and underestimated by the NTBT. As the wavenumber increases, the discrepancies between the predicted flexural frequencies by the proposed models would magnify. Such discrepancies would be followed with a lower rate for higher levels of the wavenumber. Such facts also hold true for the phase velocities of the flexural frequencies. Concerning the phase velocities pertinent to the flexural frequencies, all proposed nonlocal continuous models predict that the phase velocities decrease with the wavenumber up to k x ¼ 4. Thereafter, phase velocities associated with the nonzero small-scale parameter would generally reduce as the wavenumber increases. However, the local models (i.e., e0a ¼0) display that the phase velocities of the flexural waves for k x 4 4 would magnify with the wavenumber. Concerning the group velocities associated with the flexural waves, the local models predict that the group velocities increase with the wavenumber up to a certain value. For wavenumbers greater than that value, variation of the wavenumber has a slight influence on the variation of the group velocities. However, for a nonzero small-scale parameter and a wavenumber greater than that specified value, the predicted group velocities would commonly reduce as the wavenumber increases. Shear waves. According to the plotted results in Fig. 4, for e0 a r 1, the predicted frequencies of shear waves magnify with the wavenumber. In the case of e0 a ¼ 0 nm, the frequencies monotonically magnify with the wavenumber, however, for e0 a ¼ 1 nm, the frequencies converge to specified values. In the case of e0 a ¼ 2 nm, the frequencies would lessen as the wavenumber increases. Further, their values would converge to low specified levels. Generally, the discrepancies between the frequencies of the NTBT and those of the NHOBT reduce as the wavenumber increases. The phase velocities pertinent to the shear frequencies decrease as the wavenumber increases. The predicted phase velocities by the NHOBT are commonly higher than those of the NTBT. The group velocities associated with lower shear frequencies (i.e., vg3 ) usually magnify with the wavenumber whereas those of higher shear frequencies (i.e., vg4 ) commonly decrease as the wavenumber increases.

6.3.2. Role of the radius of the SWCNT on the characteristics of the transverse waves An interesting parametric study has been conducted to determine the effect of the radius of the constitutive SWCNTs of the 3D ensemble on the characteristics of the elastic transverse waves. The predicted frequencies, phase and group velocities of both flexural and shear waves as a function of mean radius of the ECS pertinent to the SWCNTs of the ensemble are demonstrated in Fig. Fig. 5. The results of

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Fig. 8. Frequencies, phase and group velocities as a function of the small-scale parameter for three levels of the wavenumber: ((...) NRBT, (– –) NTBT, (—) NHOBT; (○) k x ¼ π, (□) k x ¼ 2π, ( ) k x ¼ 4π; k y ¼ k z ¼ π, λ¼ 10).

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the continuous models based on the NRBT, NTBT, and NHOBT are given for three levels of the length of SWCNTs, namely lb ¼ 10, 15, and 20 nm. Flexural waves. The frequencies of the flexural waves commonly magnify as the radius of the SWCNTs increases. A close scrutiny of the vdW forces as a function of the mean radius of the SWCNT shows that C v J , the spring constant associated with the most strong component of the intertube vdW force, magnify with the mean radius up to rm ¼ 3 nm. However, for r m 4 3, its magnitude would reduce as the mean radius increases. On the other hand, by increasing the mean radius, the bending rigidity of the SWCNTs increases. Such an evidence indicates that the bending stiffness of the nanostructure generally increases with the radius of the SWCNT. By this detail, the growing of the flexural frequencies with radius of the SWCNT can be interpreted. The corresponding phase and group velocities also magnify as the radius of the SWCNT increases. Generally, the discrepancies between the results of various nonlocal continuous models magnify with the radius of the SWCNT. Such discrepancies are more obvious for 3D ESWCNTs with lower lengths since the ratio of the shear strain energy to the total strain energy are higher for such nanostructures. As a result, the effect of shear deformation becomes larger and the discrepancies between the predicted results by the proposed models would be more obvious. Shear waves. According to Fig. 5, the shear frequencies as well as their corresponding phase velocities of the 3D ESWCNTs would decrease as the radius of the SWCNT increases. However, the group velocities generally magnify with the radius of the SWCNT. A more detail survey displays that the discrepancies between the results of the NTBT and those of the NHOBT commonly decrease as the radius of the SWCNT grows. Such discrepancies are more apparent for those ensembles with higher levels of length. 6.3.3. Role of the slenderness ratio on the characteristics of the transverse waves Another fascinating scrutiny has been carried out to determine the influence of the slenderness ratio on the characteristics of the transverse waves within the 3D ESWCNTs. The plots of the flexural and shear frequencies as well as their corresponding phase and group velocities are illustrated in Fig. 6. Based on the proposed nonlocal continuous models, the results are provided for three levels of the smallscale parameter (i.e., e0 a ¼ 0; 1, and 2 nm) in the case of k x ¼ 2π , and k y ¼ k z ¼ π . Flexural waves. As it is seen in Fig. 6, the flexural frequencies commonly decrease with the slenderness ratio. A more detail study reveals that C v J slightly increases with the slenderness ratio. On the other hand, the mass of the nanostructure linearly increases with its length. Further, based on the NRBT, the flexural stiffness of the nanostructure is inversely proportional to the third power of the length. Since the influence of the slenderness ratio on the stiffness is more apparent in compare to that of the vdW force, thereby, the flexural frequencies of the nanostructure would lessen with the slenderness ratio. It is worth mentioning that the effect of the variation of slenderness ratio on the flexural frequencies of 3D ESWCNTs with lower levels of the small-scale parameter is more obvious. As the slenderness ratio increases, the share of shear strain energy within the total strain energy of the nanostructure decreases. As a result, the discrepancies between the results of the proposed models would decrease by an increase of the slenderness ratio. In most of the cases, the predicted results by the

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Fig. 9. Frequencies, phase and group velocities as a function of the ensemble's population for three levels of the wavenumber: ((...) NRBT, (– –) NTBT, (—) NHOBT; (○) k x ¼ 2π, (□) k x ¼ 4π, ( ) k x ¼ 6π; k y ¼ k z ¼ π, lb ¼ 15 nm).

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NHOBT are between those of the NRBT and the NTBT such that the NRBT overestimates the results of both NTBT and NHOBT. The local models based on the NRBT, NTBT, and NHOBT (i.e., e0 a ¼ 0) predict that both phase and group velocities would decrease with the slenderness ratio. Nevertheless, the nonlocal models display that both phase and group velocities increase with the slenderness ratio up to a certain level. Thereafter, by increasing the slenderness ratio, their values would decrease. Additionally, by an increase of the slenderness ratio, the discrepancies between the phase and group velocities of various nonlocal continuous models would decrease. Shear waves. Depend on the magnitude of the small-scale parameter, shear frequencies decrease or increase with the slenderness ratio. For lower levels of this parameter (i.e., e0 a r 1), the frequencies associated with the shear waves decrease with the slenderness ratio, however, for a high level of the small-scale parameter (i.e., e0 a ¼ 2 nm), the predicted shear frequencies by the NTBT and NHOBT would magnify as the slenderness ratio increases. All proposed continuous models predict that the phase velocities of the shear frequencies would increase with the slenderness ratio. Furthermore, the choice of the small-scale parameter has fairly no influence on the trend of the plots of the phase velocities of the shear waves as a function of the slenderness ratio. The group velocities pertinent to the lower shear frequencies commonly decrease with the slenderness ratio, however, those associate with the higher shear frequencies usually increase as the slenderness ratio increases.

6.3.4. Role of the intertube distance on the characteristics of the transverse waves An important parametric study is provided to investigate the effect of the intertube distance on the characteristics of the transverse waves within the 3D ESWCNTs. The predicted results by various models are demonstrated in Fig. 7 for three levels of the wavenumber (i.e., k x ¼ 2π , 4π, and 6π) in the case of k y ¼ k z ¼ π and λ ¼ 10. Flexural waves. Based on the plotted results in Fig. 7, the predicted flexural frequencies by various models would slightly decrease with the intertube distance. Such a reduction is more obvious for lower levels of the intertube distance and higher wavenumbers. This fact also holds true for their corresponding phase velocities. The main reason of this fact is the lessening of the intertube vdW forces by an increase of the intertube distance. A close scrutiny of the demonstrated results reveals that the variation of the intertube distance has fairly no influence on the discrepancies between the predicted results by the proposed continuous models. However, by increasing the wavenumber, the relative discrepancies between various models would lessen. The plotted results also display that the group velocities of the flexural waves would trivially magnify as the intertube distance increases. Such a fact is more obvious for higher levels of the wavenumber and lower levels of the intertube distance. Furthermore, the NRBT overestimates the predicted results by the NTBT and NHOBT. Shear waves. The plotted results in Fig. 7 show that the variation of the intertube distance has a fairly trivial impact on the variation of the shear frequencies and their corresponding phase velocities. Generally, the predicted frequencies and phase velocities would magnify with the wavenumber, irrespective of the intertube distance. However, the group velocities of the lower shear frequencies would commonly decrease with the wavenumber. Further studies indicate that the discrepancies between the predicted results associated with

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the shear waves slightly affected by the intertube distance. Additionally, the discrepancies between the results of the NTBT and those of the NHOBT would commonly increase as the wavenumber increases. 6.3.5. Role of the small-scale parameter on the characteristics of the transverse waves A crucial investigation is performed to study the role of the small-scale parameter on the frequencies, phase and group velocities of the transverse waves in the 3D ESWCNTs. These parameters in terms of the small-scale parameter are plotted in Fig. 8 for three levels of the wavenumber (i.e., k x ¼ π, 2π , and 4π ) for a fairly stocky SWCNTs (i.e., λ ¼10). Flexural waves. As it is observed in Fig. 8, flexural frequencies as well as their corresponding phase velocities would lessen as the smallscale parameter increases. The rate of reduction is also more obvious for higher wavenumbers. A detail scrutiny of the plotted results displays that variation of the small-scale parameter has a very small influence on the discrepancies between the flexural frequencies as well as phase velocities of the proposed models. Furthermore, such discrepancies would commonly magnify as the wavenumber increases. For instance, the NTBT underestimates the flexural frequencies and the corresponding phase velocities of the NHOBT with relative error lower than 5, 12.5, and 19.5 percent for k x ¼ π , 2π , and 4π , respectively. Additionally, for these wavenumbers, the NRBT overestimates the results of the NHOBT with relative error lower than 8.5, 26, and 43 percent. According to the obtained results, the discrepancies between the predicted group velocities by the proposed continuous models generally magnify as the influence of the small-scale parameter becomes highlighted. As the wavenumber increases, the discrepancies between the group velocities of various models would also increase. Shear waves. Based on the plotted results in Fig. 8, the shear frequencies as well as their corresponding phase and group velocities would decrease as the small-scale parameter magnifies. In most of the cases, the predicted results by the NHOBT are underestimated by the NTBT. A close survey of the obtained results indicates that the variation of the small-scale parameter has a very trivial effect on the discrepancies between the predicted shear frequencies as well as corresponding group velocities by the NTBT and those of the NHOBT. Furthermore, by an increase of the wavenumber, such discrepancies would reduce. For example, for k x ¼ π , 2π , and 4π , the NTBT could reproduce the results of the NHOBT with relative error lower than 18, 12, and 6 percent, respectively. 6.3.6. Role of the population of the ESWCNTs on the characteristics of the transverse waves Finally, we are interested in understanding the influence of the number of SWCNTs on the characteristics of the elastic transverse waves within the nanostructure. To this end, the plots of the frequencies as well as their corresponding phase and group velocities in terms of the number of SWCNTs along the y axis, namely Ny, are provided in Fig. 9 (note that Nz ¼ Ny). The predicted results by the continuous models based on the NRBT, NTBT, and NHOBT are demonstrated for a 3D ESWCNTs with lb ¼15 nm. The nanostructure under study is aimed to transfer transverse waves with three levels of the wavenumber (i.e., k x ¼ 2π , 4π , and 6π ). Flexural waves. According to the plotted results in Fig. 9, the flexural frequencies slightly decrease as the population of the 3D ESWCNTs magnifies. For lower levels of the population and wavenumber, the rate of the variation of the flexural frequencies and corresponding phase velocity in terms of Ny is more apparent. Furthermore, the group velocities associated with the flexural frequencies commonly increase with the number of SWCNTs within the ensemble, particularly for lower levels of the wavenumber. A close assessment of the obtained results indicates that the discrepancies between the predicted flexural frequencies and phase velocities by the NRBT and those of the NHOBT would trivially increase with the population of the ensemble. However, the discrepancies between the group velocities of the NRBT and those of the NHOBT would lessen by an increase of the population of the 3D ESWCNTs. Such results display that the NRBT can capture the flexural frequencies and corresponding phase velocities of the NHOBT with relative error lower than 9, 25, and 36 percent for k x ¼ 2π , 4π , and 6π , respectively. Additionally, in the case of Ny ¼ 4ð14Þ, the discrepancies between the predicted group velocities by the NRBT and those of the NHOBT in order are 55(23), 72(61), and 94(88) percent. Further investigations of the obtained results show that the discrepancies between the predicted flexural frequencies and phase velocities by the NTBT and those of the NHOBT commonly magnify as the population of the ensemble grows. Such an issue is more obvious for higher levels of the wavenumber. For instance, for the abovementioned wavenumbers, the NTBT can reproduce the predicted flexural frequencies and phase velocities by the NHOBT with 4, 11.5, and 16 percent, respectively. Furthermore, for all levels of the population of the ensemble, the group velocities of the NHOBT in order are underestimated by the NTBT with relative error lower than 12, 26, and 38 percent for k x ¼ 2π , 4π , and 6π . Shear waves. According to the demonstrated results in Fig. 9, shear frequencies as well as corresponding phase velocities are basically not affected by the population of the ensemble. It implies that the discrepancies between the predicted shear frequencies as well as phase velocities by the NHOBT and those by the NTHOBT do not vary with the ensemble's population. However, such discrepancies are more obvious for lower levels of the wavenumber. For example, the NTBT can capture the shear frequencies and corresponding phase velocities of the NHOBT with relative error lower than 18.5, 12.5, and 9 percent for k x ¼ 2π , 4π , and 6π , respectively. In addition, no regular pattern for the effect of the population of the ensemble on the group velocities of the shear frequencies is detected.

7. Concluding remarks Study characteristics of the elastic transverse waves within 3D ESWCNTs is of interest in the context of the nonlocal continuum theory of Eringen. By exploiting Hamilton's principle, the discrete equations of motion associated with the free vibration of the nanostructure are derived based on the NRBT, NTBT, and NHOBT. For an ensemble with N y  N z SWCNTs, the equations of motion of the aforementioned models consist of 2Ny N z , 4N y N z , and 4N y N z unknown fields. In addition to the size problem of the governing equations, finding explicit statements for the characteristics of the transverse waves would not be an easy task. To conquest the above-mentioned problems associated with the discrete models, suitable continuous models are developed. The analytical expressions of shear and flexural frequencies as well as their corresponding phase and group velocities are extracted for the proposed nonlocal continuous models. The efficiency of the proposed continuous models is examined through various comparison studies and a reasonably good agreement has been reported. Subsequently, the influences of the geometric data of the ensemble, its population, and small-scale parameter on the characteristics of the transverse waves are addressed in some detail. The major obtained results are as follows:

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 The flexural frequencies generally magnify with the wavenumber. The influence of the wavenumber on the flexural frequencies would 

 

 

lessen as the effect of the small-scale parameter becomes highlighted. However, the trend of the plots of shear frequencies in terms of wavenumber strongly relies on the small-scale parameter. The flexural frequencies as well as their corresponding phase and group velocities generally magnify with the radius of the SWCNT. However, the shear frequencies and their corresponding phase velocities would reduce as the radius of the SWCNT grows. Further, the discrepancies between the results of the proposed models would grow with the radius of the SWCNTs, particularly for stockier 3D ESWCNTs. The flexural frequencies would lessen as the slenderness ratio increases. However, the variation of the shear frequencies as a function of the slenderness ratio also relies on the small-scale parameter. By an increase of the slenderness ratio, the ratio of the shear strain energy to the total strain energy would reduce; thereby, the predicted results by the proposed models become closer. The flexural frequencies as well as their corresponding phase velocities would trivially lessen as the intertube distance increases. However, their corresponding group velocities would slightly increase with the intertube distance. Such a fact is more apparent for higher levels of the wavenumber. Additionally, the intertube distance has fairly no influence on the discrepancies between the results of the proposed models. Generally, the predicted shear and flexural frequencies as well as their corresponding phase and group velocities would reduce as the effect of the small-scale parameter becomes highlighted. Furthermore, variation of the small-scale parameter has a very trivial effect on the discrepancies between the results of various models. The flexural frequencies and corresponding phase velocities commonly lessen as the population of the ensemble grows. However, the corresponding group velocities magnify with the ensemble's population. Such a fact is more apparent for lower levels of the wavenumber. Furthermore, shear frequencies as well as their phase velocities are slightly affected by the population of the 3D ESWCNTs.

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