Nonlocal discrete and continuous modeling of free vibration of stocky ensembles of vertically aligned single-walled carbon nanotubes

Current Applied Physics 14 (2014) 1116e1139

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Nonlocal discrete and continuous modeling of free vibration of stocky ensembles of vertically aligned single-walled carbon nanotubes Keivan Kiani* Department of Civil Engineering, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran

a r t i c l e i n f o

a b s t r a c t

Article history: Received 12 October 2013 Received in revised form 4 May 2014 Accepted 7 May 2014 Available online 11 June 2014

Free dynamic analysis of transverse motion of vertically aligned stocky ensembles of single-walled carbon nanotubes is of particular interest. A linear model is developed to take into account the van der Waals forces between adjacent SWCNTs because of their bidirectional transverse displacements. Using Hamilton's principle, the discrete equations of motion of free vibration of the nanostructure are obtained based on the nonlocal Rayleigh, Timoshenko, and higher-order beam theories. The application of such discrete models for frequency analysis of highly populated ensembles would be associated with so much computational effort. To overcome such a problem, some useful nonlocal continuous models are established. The obtained results reveal that the newly developed models can successfully capture the predicted fundamental frequencies of the discrete models. Through various numerical studies, the roles of slenderness ratio, radius of the SWCNT, small-scale parameter, population of the ensemble, and intertube distance on the fundamental flexural frequency of the nanostructure are examined and discussed. The capabilities of the proposed nonlocal continuous models in predicting flexural frequencies of the nanostructure are also addressed. © 2014 Elsevier B.V. All rights reserved.

Keywords: Ensemble of SWCNTs Frequency analysis Nonlocal discrete models Nonlocal continuous models Galerkin approach

1. Introduction A forest or an ensemble of single-walled carbon nanotubes (ESWCNTs) consists of multiple SWCNTs at the vicinity of each other in which interact with each other by the van der Waals (vdW) forces. Using chemical vapor deposition in a suitably controlled environment, growth of randomly oriented ESWCNTs [1,2] or even vertically aligned ESWCNTs [3e6] on an appropriate substrate would be possible. The orientation of the SWCNTs within the layered ensemble should be chosen such that the nanosystem could perform more appropriately and efficiently for the assigned jobs. For instance, from applied mechanics standpoint, the arrangement and directions of the SWCNTs within an ensemble network should be taken into account such that the exerted electro-mechanical forces cause lower stresses and displacements within the nanostructure. Herein, without dealing about the engineering design of such nanosystems, we are interested in examining free dynamic analysis of vertically aligned ESWCNTs. Carbon nanotubes (CNTs) have been under great investigation because of their brilliant physical and chemical properties [7,8].

* Tel.: þ98 21 88779473; fax: þ98 21 88779476. E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.cap.2014.05.018 1567-1739/© 2014 Elsevier B.V. All rights reserved.

Such excellent properties have been provided them for a broad range of applications such as drug delivery systems [9e12], nanosensors [13e16], nanocomposites [17e19], and nano-electromechanical systems [20e24]. For these potential applications, determination of the dynamic behavior of ESWCNTs is a basic step toward their optimal design. To date, various aspects of vibrations of individual CNTs have been studied and a sufficient knowledge regarding the roles of the influential parameters on their vibration behavior have been accumulated. Most of the undertaken works before 2008 had been performed in the context of the classical theory of elasticity. Several examples of such an evidence are the free transverse vibration of CNTs [25,26], their transverse vibrations and instabilities due to moving inside fluids flow [27e29], and their nonlinear frequency analysis [30,31]. Further experimental and theoretical studies [32e34] show that the common continuum mechanics cannot accurately capture the mechanical and vibration behavior of nanostructures. This fact becomes highlighted when the wavelength of the propagated wave would be comparable with the interatomic bonds. Under such a circumstance, irrespective of the influence of the propagated wave on the vibration of each point of the continuum, the state of stress at each point is also affected by those of its neighboring points. Such an important phenomenon cannot be interpreted by the models which are inherently established

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

based on the classical continuum mechanics. To remove such a deficiency, some advanced continuum mechanics have been developed during the past century including micropolar of Cosserat and Cosserat [35], couple stress of Toupin and Mindlin [36,37], gradient elasticity of Aifantis [38e40], and nonlocal continuum mechanics of Eringen [41e45]. The latter theory has been paid attention to by many researchers who study dynamic behavior of nanostructures. The main reason of this fact maybe its simplicity in application to the classical governing equations. Based on the nonlocal continuum theory, the stress of each point (i.e., nonlocal stress) does not rely on the state of stress at that point, but also is affected by the stress of its neighboring points. Such a dependency is formulated by using appropriate kernel functions for the under study spatial domains. The influence domain of the kernel functions is commonly specified by a so-called small-scale parameter. In limit, when the small-scale parameter vanishes, the nonlocal stress approaches the local stress (i.e., classical stress). Through choosing an appropriate value for the small-scale parameter, a reasonably good agreement between the predicted results by this theory and those of atomic models has been reported for various problems [46e51]. So far, the nonlocal continuum theory of Eringen has been employed for dynamic analysis of individual CNTs including free vibrations [52e57], dynamic instability [58,59], their interactions with inside fluid flow [60e64] and moving nanoparticles [65e71]. However, free vibrations of ensemble networks of vertically aligned SWCNTs and their related dynamical problems have not been addressed. In this paper, free transverse vibrations of ESWCNTs are of concern. For this purpose, the interactional vdW forces between two adjacent SWCNTs due to their transverse displacements are evaluated via Lennard-Jones potential function. Using Hamilton's principle, the discrete and continuous equations of motion of the nanostructure based on the nonlocal Rayleigh, Timoshenko, and higher-order beam theories are established. Using modal analysis as well as Galerkin approach, the frequencies of the nanostructure are extracted via proposed nonlocal discrete and continuous models. In a particular case, the predicted results by the nonlocal discrete models are compared with those of another work, and a reasonably good agreement is achieved. The efficiency of the proposed continuous models is successfully checked. The roles of influential factors on the fundamental frequencies of the ESWCNTs are then addressed in some details.

1117

(a)

(b) Fig. 1. (a) A schematic representation of an ensemble of vertically aligned SWCNTs with uniform distribution; (b) an elastic linear model to the considered ESWCNTs accounting for intertube vdW forces.

adjacent ones. Based on the developed spring-tube model for such forces (see Fig. 1(b)), the discrete equations of motion of ESWCNTs are then established via nonlocal Rayleigh, Timoshenko, and higher-order beam theories.

2. Description of the nanomechanical problem Consider an ensemble of vertically aligned SWCNTs whose intertube distances in both y and z directions are equal to d as demonstrated in Fig. 1(a). The ensemble consists of Ny and Nz tubes along the y and z axes, respectively. For nonlocal continuum-based modeling of the problem, each SWCNT is substituted by an equivalent continuum structure (ECS) whose most of dominant frequencies are identical to those of the SWCNT evaluated by an appropriate atomistic approach. The research works of Gupta and Batra [72] and Batra and Gupta [73] showed that such an ECS is a hollow cylindrical shell of length, lb, and mean radius, rm, similar to those of the SWCNT. The density, elastic moduli, shear elastic modulus, Poisson's ratio, cross-sectional area, moment of inertia of the ECS's cross-section in order are denoted by rb, Eb, Gb, nb, Ab, and Ib. The dynamic transverse displacement components of the (m,n) th SWCNT along the y and z axes are represented by Vmn ¼ Vmn (x,t) and Wmn ¼ Wmn (x,t), respectively (see Fig. 1(b)). Every SWCNT interacts with its neighboring tubes because of the existing vdW forces between their atoms. In the upcoming part, a linear model is developed to take into account the interactional vdW forces between each tube with its

3. Assessing vdW forces between two adjacent-deformed SWCNTs In this part, a simple model for evaluation of the interactional vdW forces between two adjacent tubes due to their transverse motions is proposed. This model explains that not only transverse displacements of the tubes along the plane passes their center-lines but also those displacements which are perpendicular to this plane would contribute to the vdW forces. Such forces are applied in the parallel and perpendicular directions to the plane's surface, respectively. According to the Lennard-Jones's potential function [74,75] for two atoms:

Fij ðlÞ ¼ 4ε


  s 12 s6 ;  l l

(1)

pffiffiffi where l is the distance between the ith and jth atoms, s ¼ ra = 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, fij, is evaluated as:

1118

K. Kiani / Current Applied Physics 14 (2014) 1116e1139


 dF 24ε s14 s8 ! el ¼ 2 2 l; f ij ¼   dl l l s

(2)

! where l is the position vector of the atom j with respect to the ! ! atom i, el denotes its unit base vector (i.e., el ¼ l =k l k). 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 ! (x1,rmcos ɸ1,rmsin ɸ1) and (x2,rmcos ɸ2,d þ rmsin ɸ2), respectively, l is readily expressed by:

! l ¼ðx2  x1 Þex þ ðrm ðcos 42  cos 41 Þ  DVÞey þ ðrm ðsin 42  sin 41 Þ þ d  DWÞez ;

(3)

0  x1 ; x2  lb ; 0  41 ; 42  2p; where rm denotes the mean radius of the ECS pertinent to the SWCNT, DW(x,t) ¼ W1(x,t)  W2(x,t), DV(x,t) ¼ V1(x,t)  V2(x,t), W1(x,t)/V1(x,t) and W2(x,t)/V2(x,t) 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 transverse components of the vdW force on the SWCNTs accounting for their relative displacements can be evaluated as:

Fy ¼

Fz ¼

24εs2CNT lb s 2 24εs2CNT lb s2

2 256εrm Cv⊥ ðrm ; dÞ ¼   9a4 lb

Zlb Zlb Z2p Z2p h i s12 c7  14c8 ðrm ðcos 42  cos 41 ÞÞ2 0

0

0

0



s6 h 4 c  8c5 ðrm ðcos42  cos41 ÞÞ2 2

(6a)

i

d41 d42 dx1 dx2 ;

Cvk ðrm ; dÞ ¼ 

2 256εrm  4 9a lb

Zlb Zlb Z2p Z2p h i s12 c7  14c8 ðd þ rm ðsin 42  sin 41 ÞÞ2 0

0

0

0



i s6 h 4 c  8c5 ðd þ rm ðsin 42  sin 41 ÞÞ2 2



d41 d42 dx1 dx2 ; (6b)

 Zlb Zlb Z2p Z2p
  s 14 s8 2 ðrm ðcos 42  cos 41 Þ  DVÞd41 d42 dx1 dx2 ;  l l 0

0

0

0

(4)

 Zlb Zlb Z2p Z2p
  s 14 s8 2 ðrm ðsin 42  sin 41 Þ þ d  DWÞd41 d42 dx1 dx2 ;  l l 0

0

0

0

pffiffiffi where sCNT ¼ 4 3=9a2 is the surface density of the carbon atoms, and a is the length of the carbonecarbon 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, and then evaluating the integrals, the transverse components of the linear vdW force between two adjacent SWCNTs because of their relative displacements are calculated by:

DFy ¼ Cv⊥ DV; DFz ¼ Cvk DW;

(5)

2 ð1  cosð42  41 ÞÞ cðx1 ; x2 ; 41 ; 42 ; rm ; dÞ ¼ðx2  x1 Þ2 þ 2rm

þ d2 þ 2rm dðsin 42  sin 41 Þ:

(6c)

4. Development of discrete models based on nonlocal beam theories In this part, free transverse vibrations of 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 of the nanostructure are numerically calculated.

where, 4.1. Free dynamic analysis of ESWCNTs via NRBT 4.1.1. Nonlocal 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 ESWCNTs, UR, are stated in the context of the nonlocal continuum theory of Eringen as follows: Ny X Nz Z 1 X T ¼ rb Ab 2 m¼1 n¼1 lb

R

0

þ Ib Fig. 2. The geometry of two deflected SWCNTs for evaluating the perpendicular and parallel components of the intertube vdW force.

R v2 Vmn vtvx

!2 þ

R vVmn vt

!2

R v2 Wmn vtvx

R vWmn vt

þ

!2 !

!2 !! dx;

(7a)

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

0 2

R R  Vmn Mbnlzmn 2

2

R R  Wmn Mbnlymn þ 2

1119

1

v C B v  C B C B vx vx C B h 2    2 i C B BC R R R R 1  dmN þ Vmn  Vðm1Þn ð1  d1m Þ þ C C B vk Vmn  Vðmþ1Þn y C B C B h 2 2  C B

i R R C B Cv⊥ V R  V R ð1 Þ þ  d þ V  V 1  d 1n mn mn C B mðn1Þ mðnþ1Þ nNz C B C B  2 C B R R C B Cdk Xmn  Xðm1Þðn1Þ ð1  d1m Þð1  d1n Þþ C B C B  2   C B

C B C XR  XR 1  dmN 1  dnN þ C B dk mn ðmþ1Þðnþ1Þ y z C lb B Z N N y C B z X X   1 C B 2

R U ¼ Cdx; B C YR  YR ð1  d1m Þ 1  dnN þ ðm1Þðnþ1Þ C 2 m¼1 n¼1 B dk mn z C B 0 B C     2 C B R R C B Cdk Ymn 1  dmN ð1  d1n Þþ  Yðmþ1Þðn1Þ C B y C B  2 C B

C BC R R X  X ð1  d Þ 1  d þ C B d⊥ mn 1m ðm1Þðnþ1Þ n N z C B C B  2   C B R R C B Cd⊥ X  X 1  d ð1  d Þþ 1n mn ðmþ1Þðn1Þ C B mNy C B C B   2 C B R R C B Cd⊥ Ymn  Yðm1Þðn1Þ ð1  d1m Þð1  d1n Þþ C B C B     C B 2

A @C R R 1  dmN 1  dnN d⊥ Ymn  Yðmþ1Þðnþ1Þ y z

pffiffiffi pffiffiffi R ¼ 2 ðW R þ V R Þ and Y R ¼ 2 ðW R þ V R Þ. Further, where Xmn mn mn mn mn mn 2 2 R R nl Þ , and ðM nl Þ in order are the Kronecker delta and the dmn, ðMby bzmn mn nonlocal bending moments of the (m,n)th SWCNT about the y and z axes based on the NRBT. In the framework of the nonlocal continuum theory of Eringen, the nonlocal bending moment of each tube based on the NRBT are given by Refs. [76,77]:





Mbnlymn

Mbnlzmn

R

 R R v2 Wmn  ðe0 aÞ2 Mbnlymn ¼ Eb Ib ; 2 ;xx vx

(8a)

R

 R R v2 Vmn  ðe0 aÞ2 Mbnlzmn ¼ Eb Ib ; 2 ;xx vx

(8b)

(

(7b)

where e0a denotes the small-scale parameter. This parameter is determined by comparing of the predicted dispersion curves by the nonlocal model and those of an atomic approach. For a SWCNT, boundary conditions, chirality, and length to diameter ratio are among the most crucial factors that influence on this parameter [46]. The magnitude of this parameter have been frequently considered in the range of 0e2 nm [46,52,78,79]. By employing Hamilton's principle and Eqs. (7a) and (7b), the nonlocal transverse equations of motion of the nanostructure are obtained as follows:

h     i R R R R V 1  d þ V ð1  d Þ þ C  V  V Þ þ 1m vk mn mn ðmþ1Þn ðm1Þn m N y vx4 vt 2 vt 2 vx2 i h   

R R R R ð1  d1n Þ þ Vmn Cv⊥ Vmn  Vmðn1Þ  Vmðnþ1Þ 1  dnN þ z    

R R R R R R R R 0:5Cdk Wmn ð1  d1n Þð1  d1m Þ þ 0:5Cdk Wmn þ Vmn  Wðm1Þðn1Þ  Vðm1Þðn1Þ þ Vmn  Wðmþ1Þðnþ1Þ  Vðmþ1Þðnþ1Þ 1  dnN z       R R R R  1  dmN þ 0:5Cdk Vmn ð1  d1n Þ 1  dmN þ  Wmn  Vðmþ1Þðn1Þ þ Wðmþ1Þðn1Þ y y    

R R R R R R R R ð1  d1n Þ  Wmn  Vðm1Þðnþ1Þ þ Wðm1Þðnþ1Þ  Wmn þ Wðm1Þðn1Þ  Vðm1Þðn1Þ 0:5Cdk Vmn 1  dnN ð1  d1m Þ þ 0:5Cd⊥ Vmn z    

R R R R 1  dmN þ  Wmn þ Wðmþ1Þðnþ1Þ  Vðmþ1Þðnþ1Þ  ð1  d1m Þ þ 0:5Cd⊥ Vmn 1  dnN z y     

R R R R R R R R 1  d1n 1  Vðmþ1Þðn1Þ  Wðmþ1Þðn1Þ 0:5Cd⊥ Vmn þ Wmn  Vðm1Þðnþ1Þ  Wðm1Þðnþ1Þ 1  dnN ð1  d1m Þ þ 0:5Cd⊥ Vmn þ Wmn z   dmN ¼ 0; Eb Ib

R v4 Vmn

þ X rb ðAb

R v2 Vmn

 Ib

R v4 Vmn

y

(9a)

1120

Eb Ib

K. Kiani / Current Applied Physics 14 (2014) 1116e1139 R v4 Wmn 4

R v2 Wmn



vt

2

 Ib

R v4 Wmn 2

2

Þ þ Cvk

h   i

 R R R R Wmn ð1  d1n Þ þ  Wmðnþ1Þ  Wmðn1Þ 1  dnN þ Wmn

z vt vx  i    R R R R R R R R Cv⊥ Wmn ð1  d1m Þ þ Wmn 1  dmN þ 0:5Cdk Wmn ð1  d1n Þ  Wðm1Þn  Wðmþ1Þn þ Vmn  Wðm1Þðn1Þ  Vðm1Þðn1Þ y    

R R R R 1  dmN þ þ Vmn  Wðmþ1Þðnþ1Þ  Vðmþ1Þðnþ1Þ  ð1  d1m Þ þ 0:5Cdk Wmn 1  dnN z y    

R R R R R R R R ð1  d1n Þ  Vmn  Wðm1Þðnþ1Þ þ Vðm1Þðnþ1Þ  Vmn  Wðmþ1Þðn1Þ þ Vðmþ1Þðn1Þ 0:5Cdk Wmn 1  dnN ð1  d1m Þ þ 0:5Cdk Wmn z      R R R R R R R ð1  d1n Þð1  d1m Þ þ 0:5Cd⊥ Wmn  Vmn  Wðm1Þðn1Þ þ Vðm1Þðn1Þ  Vmn  Wðmþ1Þðnþ1Þ  1  dmN þ 0:5Cd⊥ Wmn y    



R R R R R 1  dmN þ 0:5Cd⊥ Wmn þ Vðmþ1Þðnþ1Þ þ Vmn  Wðm1Þðnþ1Þ  Vðm1Þðnþ1Þ 1  dnN 1  dnN ð1  d1m Þþ z y z     R R R R 1  d1n 1  dmN ¼ 0: (9b) þ Vmn  Wðmþ1Þðn1Þ  Vðmþ1Þðn1Þ 0:5Cd⊥ Wmn

h

vx

( þ X rb ðAb

y

nanostructure in a more general context, we define the following dimensionless parameters: Eqs. (9a) and (9b) furnish us regarding the free transverse vibrations of the constitutive SWCNTs within a ESWCNTs 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 SWCNTs. To determine such unknown fields, a set of 2NyNz second-order partial differential equations (PDEs) should be appropriately solved. In order to study the dynamic response of the

(

R

v4 V mn 4

vx h R

þX

R

R

v2 V mn

 l2

2

vt

v4 V mn 2

2

vt vx

R

þ C vk

x R VR WR z 1 R x ¼ ;V mn ¼ mn ;W mn ¼ mn ;g ¼ ;t ¼ 2 lb lz lb lb l b

Cv½:: l4b

sffiffiffiffiffiffiffiffiffiffi Eb Ib t; rb Ab

(10)

Cd½:: l4b

e a R l d R ;C ¼ ;l ¼ b ;d ¼ ;½:: ¼k or ⊥; m ¼ 0 ;C v½:: ¼ lb lz Eb Ib d½:: Eb Ib rb where lz ¼ (Nz  1)d. By introducing Eq. (10) to Eqs. (9a) and (9b), the dimensionless nonlocal-discrete equations of motion of the ESWCNTs are derived as follows:

h R    R  i R R V mn  V ðmþ1Þn 1  dmN þ V mn  V ðm1Þn ð1  d1m Þ þ y

  R 

i R R R R R R R R þ 0:5C dk W mn þ V mn  W ðm1Þðn1Þ  V ðm1Þðn1Þ ð1  d1n Þð1  d1m Þ C v⊥ V mn  V mðn1Þ ð1  d1n Þ þ V mn  V mðnþ1Þ 1  dnN z  R    R 

 R R R R R R R R 1  dmN þ 0:5C dk V mn  W mn  V ðmþ1Þðn1Þ þ W ðmþ1Þðn1Þ þ 0:5C dk W mn þ V mn  W ðmþ1Þðnþ1Þ  V ðmþ1Þðnþ1Þ 1  dnN z y   R   R 

R R R R R R  ð1  d1n Þ 1  dmN þ 0:5C dk V mn  W mn  V ðm1Þðnþ1Þ þ W ðm1Þðnþ1Þ 1  dnN ð1  d1m Þ þ 0:5C d⊥ V mn  W mn y z   R  

 R R R R R R 1  dmN þ W ðm1Þðn1Þ  V ðm1Þðn1Þ ð1  d1n Þð1  d1m Þ þ 0:5C d⊥ V mn  W mn þ W ðmþ1Þðnþ1Þ  V ðmþ1Þðnþ1Þ 1  dnN z y  R   R 

R R R R R R R R þ 0:5C d⊥ V mn þ W mn  V ðm1Þðnþ1Þ  W ðm1Þðnþ1Þ 1  dnN ð1  d1m Þ þ 0:5C d⊥ V mn þ W mn  V ðmþ1Þðn1Þ  W ðmþ1Þðn1Þ 1 z   ¼ 0;  d1n 1  dmN 



y

(11a)

(

R

v4 W mn

þX

R

v2 W mn

R

 l2

v4 W mn

R

þ C vk

h

R

R

W mn  W mðnþ1Þ



1  dnN



 i  R R þ W mn  W mðn1Þ ð1  d1n Þ þ

z vx4 vt2 vt2 vx2 h   i  R   R R R R R R R R R þ 0:5C dk W mn þ V mn  W ðm1Þðn1Þ  V ðm1Þðn1Þ ð1  d1n Þ C v⊥ W mn  W ðm1Þn ð1  d1m Þ þ W mn  W ðmþ1Þn 1  dmN y  R  

 R R R R 1  dmN þ  ð1  d1m Þ þ 0:5C dk W mn þ V mn  W ðmþ1Þðnþ1Þ  V ðmþ1Þðnþ1Þ 1  dnN z y  R   R 

R R R R R R R R 0:5C dk W mn  V mn  W ðm1Þðnþ1Þ þ V ðm1Þðnþ1Þ 1  dnN ð1  d1m Þ þ 0:5C dk W mn  V mn  W ðmþ1Þðn1Þ þ V ðmþ1Þðn1Þ ð1  d1n Þ z   R   R R R R  1  dmN þ 0:5C d⊥ W mn  V mn  W ðm1Þðn1Þ þ V ðm1Þðn1Þ ð1  d1n Þð1  d1m Þþ y  R    R 

 R R R R R R R R 1  dmN þ 0:5C d⊥ W mn þ V mn  W ðm1Þðnþ1Þ  V ðm1Þðnþ1Þ 0:5C d⊥ W mn  V mn  W ðmþ1Þðnþ1Þ þ V ðmþ1Þðnþ1Þ 1  dnN z y  R   

R R R R ¼ 0;  1  dnN ð1  d1m Þ þ 0:5C d⊥ W mn þ V mn  W ðmþ1Þðn1Þ  V ðmþ1Þðn1Þ 1  d1n 1  dmN z

y

(11b)

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

1121

set of Eigenvalue equations for 6R, the natural frequencies of the nanostructure would be determined. where X½: ¼ ½:  m2 ½:;xx : 4.1.2. Frequency analysis of ESWCNTs modeled based on NRBT Assume that both ends of each SWCNT are simple supports, and the exterior ones are prevented from any transverse movement. Using assumed mode method, the following admissible deformation fields are considered for the SWCNTs of the ensemble: R

W mn ðx;tÞ ¼

Np X

R

R

W mnp ðtÞsinðppxÞ;V mn ðx;tÞ ¼

p¼1

Np X

R

V mnp ðtÞsin ðppxÞ;

p¼1

(12) by substituting such a form of deformation field into Eqs. (11a) and (11b) accounting for the conditions of the exterior SWCNTs of the ensemble, the following set of equations is obtained: Rv

M

2 R

x

vt

2

R

þ K xR ¼ 0;

4.2. Free dynamic analysis of ESWCNTs via NTBT 4.2.1. Nonlocal equations of motion using NTBT Using the hypotheses of the Timoshenko beam theory in the context of the nonlocal continuum theory of Eringen [41e43], the kinetic energy, TT, the elastic strain energy, UT, of the ESWCNTs accounting for the intertube vdW forces are expressed by:

0 0 !2 l Ny X Nz Z b vQTymn 1 X T ¼ rb @Ib @ þ 2 m¼1 n¼1 vt T

0

T vVmn vt

þ Ab

!2 þ

T vWmn vt

vQTzmn vt

!2 !1 Adx;

!2 1 A

(14a)

(13)

1 ! T T T   T vV C B vQzmn mn Mbnlzmn þ  QTzmn Qbnlymn þ C B C B vx vx C B C B ! C B vQT     T T T C B vW y mn mn C B Qbnlzmn þ Mbnlymn þ  QTymn C B vx vx C B C B B h 2 2 i C 

C B T T T T 1  dmN þ Vmn  Vðm1Þn ð1  d1m Þ þ C B Cvk Vmn  Vðmþ1Þn z C B B i C h 2 2  C B

T T T T BC 1  dnN þ C C B v⊥ Vmn  Vmðn1Þ ð1  d1n Þ þ Vmn  Vmðnþ1Þ z C B C B   2 C B T T C B Cdk Xmn  Xðm1Þðn1Þ ð1  d1m Þð1  d1n Þþ C B C B l bB     C Z N N 2 y

z C B T T 1 X X T X 1  d  X C þ 1  d Cdx; B dk mn ðmþ1Þðnþ1Þ U ¼ m N n N y z C 2 m¼1 n¼1 B C B  2 C 0 B

C B C YT  YT ð1 Þ  d 1  d þ 1m dk C B mn ðm1Þðnþ1Þ nNz C B C B  2   C B T T C B Cdk Ymn  Yðmþ1Þðn1Þ 1  d ð1  d Þþ 1n mNy C B C B C B  2

C B T T C B Cd⊥ Xmn  Xðm1Þðnþ1Þ ð1  d1m Þ 1  dnNz þ C B C B     C B 2 T T C BC X 1  d ð1  d  X Þþ 1n C B d⊥ mn ðmþ1Þðn1Þ mNy C B C B   2 C B T T C B Cd⊥ Ymn  Yðm1Þðn1Þ ð1  d1m Þð1  d1n Þþ C B C B C B  2  

A @ T T 1  dmN  Yðmþ1Þðnþ1Þ Cd⊥ Ymn 1  dnN y z 0

R

R

R

where xR ¼ < W mnp ; V mnp > T and the dimensionless matrices M R and K can be readily calculated. Eq. (13) represents a set of secondorder linear ODEs with 2(Ny  2)(Nz  2) unknowns for an arbitrary R mode number p. Now let xR ðtÞ ¼ xR0 ei6 t where xR0 is the dimenR sionless amplitude vector, and 6 is the dimensionless natural frequency of the ESWCNTs based on the NRBT. By substituting this form of the unknown vector into Eq. (13) and solving the resulting

(14b)

pffiffiffi pffiffiffi 2 T T T 2 T T T 2 ðWmn þ Vmn Þ, Ymn ¼ 2 ðWmn þ Vmn Þ. Also, Vmn , T T T T nl Þ , ðQ nl Þ , ðM nl Þ , and ðM nl Þ represent QTzmn , ðQby bzmn bymn bzmn mn

T ¼ where Xmn T , QT , Wmn ymn

the transverse displacements of the (m,n)th SWCNT along the y and z axes, angles of deformation about the y and z axes, nonlocal shear forces pertinent to the y and z axes, and nonlocal bending moment about the y and z axes, respectively. According to the nonlocal

1122

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

continuum theory of Eringen, the nonlocal shear forces and bending moments within the (m,n)th SWCNT are given by Refs. [78,80]:



Qbnlymn

) !  T v2 QTzmn v2 QTzmn vVmn T  Q X rb Ib G A ¼ 0;  k  Eb Ib s b b z 2 mn vx vt vx2

!  T T vVmn  QTzmn ;  ðe0 aÞ2 Qbnlymn ¼ ks Gb Ab ;xx vx

T

for the intertube vdW interactional forces are described in terms of deformation fields of the NTBT as:

(16a)

(15a)

( h   i T vQTzmn

 T v2 Vmn T T T V ð1  d Þ þ X r ks Gb Ab ð  A þ C  V  V Þ þ þ V 1  d 1m vk b b mn mn ðmþ1Þn ðm1Þn mNz vx vx2 vt 2 i h     

T T T T T T T T þ 0:5Cdk Wmn Cv⊥ Vmn ð1  d1n Þ þ Vmn ð1  d1n Þð1  d1m Þ  Vmðn1Þ  Vmðnþ1Þ þ Vmn  Wðm1Þðn1Þ  Vðm1Þðn1Þ 1  dnN z      

T T T T T T T T 1  dmN þ 0:5Cdk Vmn þ Vmn  Wðmþ1Þðnþ1Þ  Vðmþ1Þðnþ1Þ  Wmn  Vðmþ1Þðn1Þ þ Wðmþ1Þðn1Þ þ 0:5Cdk Wmn 1  dnN z y    

T T T T  ð1  d1n Þ 1  dmN þ 0:5Cdk Vmn  Wmn  Vðm1Þðnþ1Þ þ Wðm1Þðnþ1Þ 1  dnN ð1  d1m Þ y z    

T T T T T T T T ð1  d1n Þð1  d1m Þ þ 0:5Cd⊥ Vmn  Wmn þ Wðm1Þðn1Þ  Vðm1Þðn1Þ  Wmn þ Wðmþ1Þðnþ1Þ  Vðmþ1Þðnþ1Þ þ0:5Cd⊥ Vmn 1  dnN z    

T T T T  1  dmN þ 0:5Cd⊥ Vmn þ Wmn  Vðm1Þðnþ1Þ  Wðm1Þðnþ1Þ 1  dnN ð1  d1m Þþ y z     T T T T 1  d1n 1  dmN ¼ 0; (16b) þ Wmn  Vðmþ1Þðn1Þ  Wðmþ1Þðn1Þ 0:5Cd⊥ Vmn y T v2 Vmn



Qbnlzmn

T

!  T T vWmn  QTymn ;  ðe0 aÞ2 Qbnlzmn ¼ ks Gb Ab ;xx vx (15b)

) !  T v2 QTymn v2 QTymn vWmn T  Q G A I ¼ 0;  k  E X rb Ib s b b b b y mn vx vt 2 vx2 (16c)



Mbnlymn

T



 ðe0 aÞ2 Mbnlymn

T ;xx

vQTymn ; ¼ Eb Ib vx

(15c)

In order to analyze the problem in a more general framework, the following dimensionless parameters are introduced:

( h   i T vQTymn

 T v2 Wmn T T T W ð1  d Þ þ X r ks Gb Ab ð  A þ C  W  W Þ þ þ W 1  d 1n vk b b mn mn mðnþ1Þ mðn1Þ n N z vx vx2 vt 2 h   i    T T T T T T T T ð1  d1m Þ þ Wmn 1  dmN þ 0:5Cdk Wmn ð1  d1n Þ Cv⊥ Wmn  Wðm1Þn  Wðmþ1Þn þ Vmn  Wðm1Þðn1Þ  Vðm1Þðn1Þ y    

T T T T 1  dmN þ  ð1  d1m Þ þ 0:5Cdk Wmn þ Vmn  Wðmþ1Þðnþ1Þ  Vðmþ1Þðnþ1Þ 1  dnN z y    

T T T T T T T T ð1  d1n Þ  Wðmþ1Þðn1Þ þ Vðmþ1Þðn1Þ 0:5Cdk Wmn  Vmn  Wðm1Þðnþ1Þ þ Vðm1Þðnþ1Þ 1  dnN ð1  d1m Þ þ 0:5Cdk Wmn  Vmn z      T T T T T T T ð1  d1n Þð1  d1m Þ þ 0:5Cd⊥ Wmn  Vmn  Wðm1Þðn1Þ þ Vðm1Þðn1Þ  Vmn  Wðmþ1Þðnþ1Þ  1  dmN þ 0:5Cd⊥ Wmn y     



T T T T T 1  dmN þ 0:5Cd⊥ Wmn þ Vðmþ1Þðnþ1Þ þ Vmn  Wðm1Þðnþ1Þ  Vðm1Þðnþ1Þ 1  dnN 1  dnN ð1  d1m Þþ z y z     T T T T 1  d1n 1  dmN ¼ 0: (16d) þ Vmn  Wðmþ1Þðn1Þ  Vðmþ1Þðn1Þ 0:5Cd⊥ Wmn y T v2 Wmn



Mbnlzmn

T

 T vQTzmn ;  ðe0 aÞ2 Mbnlzmn ¼ Eb Ib ;xx vx Z

by employing Hamilton's principle, 0

t

(15d)

ðdT T  dU T Þdt ¼ 0, the

equations of motion in terms of nonlocal forces are obtained. Through introducing Eqs. (15a)e(15d) to the resulting equations, the nonlocal discrete equations of motion of ESWCNTs accounting

T Vmn WT T T T ;W mn ¼ mn ;Qymn ¼QTymn ;Qzmn ¼QTzmn ; lb lb sffiffiffiffiffiffiffiffiffiffi Cd½: l2b Cv½: l2b T 1 ks Gb Eb Ib T t;h¼ t¼ ;C v½: ¼ ;C d½: ¼ ;½:¼k or⊥; 2 lb rb ks Gb Ab ks Gb Ab ks Gb Ab l T

V mn ¼

b

(17)

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

by introducing Eq. (17) to Eqs. (16a)e(16d), the dimensionless discrete equations of motion of the ensembles of SWCNTs on the basis of the NTBT are derived as follows:

) T  v2 Qzmn 2 X l  vt2

T

vV mn T  Qzmn vx

!

T

v2 Qzmn vx2

¼ 0;

T

Np X

p¼1

p¼1

Np X

T

Np X

W mn ðx;tÞ ¼ Qymn ðx;tÞ ¼

T

h

T

Np X

1123 T

W mnp ðtÞsin ðppxÞ;V mn ðx;tÞ ¼ T

Qymnp ðtÞcos ðppxÞ;Qzmn ðx;tÞ ¼

p¼1

(18a)

T

V mnp ðtÞsin ðppxÞ; T

Qzmnp ðtÞcos ðppxÞ;

p¼1

(19)

( T T h T   i vQzmn

 T v2 V mn T T T Þ þ ð  X þ C vk V mn  V ðmþ1Þn 1  dmN þ V mn  V ðm1Þn ð1  d1m Þ þ 2 2 z vx vx vt h T    T   T

i T T T T T T T þ 0:5C dk W mn þ V mn  W ðm1Þðn1Þ  V ðm1Þðn1Þ ð1  d1n Þð1  d1m Þ C v⊥ V mn  V mðn1Þ ð1  d1n Þ þ V mn  V mðnþ1Þ 1  dnN z  T    T 

 T T T T T T T T 1  dmN þ 0:5C dk V mn  W mn  V ðmþ1Þðn1Þ þ W ðmþ1Þðn1Þ þ 0:5C dk W mn þ V mn  W ðmþ1Þðnþ1Þ  V ðmþ1Þðnþ1Þ 1  dnN z y   T  

T T T T  ð1  d1n Þ 1  dmN þ 0:5C dk V mn  W mn  V ðm1Þðnþ1Þ þ W ðm1Þðnþ1Þ 1  dnN ð1  d1m Þþ y z  T   T 

T T T T T T T T 0:5C d⊥ V mn  W mn þ W ðm1Þðn1Þ  V ðm1Þðn1Þ ð1  d1n Þð1  d1m Þ þ 0:5C d⊥ V mn  W mn þ W ðmþ1Þðnþ1Þ  V ðmþ1Þðnþ1Þ 1  dnN z   T  

T T T T  1  dmN þ 0:5C d⊥ V mn þ W mn  V ðm1Þðnþ1Þ  W ðm1Þðnþ1Þ 1  dnN ð1  d1m Þþ y z  T    T T T T ¼ 0; (18b) 0:5C d⊥ V mn þ W mn  V ðmþ1Þðn1Þ  W ðmþ1Þðn1Þ 1  d1n 1  dmN T

v2 V mn

y

) T  v2 Qymn X l2  vt2

T

vW mn T  Qymn vx

!

T

h

v2 Qymn 2

vx

¼ 0;

(18c)

by introducing Eq. (19) to Eqs. (18a)e(18d) through accounting for the boundary conditions of the exterior SWCNTs of the ensemble, one can arrive at the following set of equations:

!

( # " " T     T  T

 T v2 W mn T T T T T þX   þ C vk W mn  W mðnþ1Þ 1  dnNz þ W mn  W mðn1Þ ð1  d1n Þ þ C v⊥ W mn  W ðm1Þn 2 2 vx vt vx #    T   T T T T T T  ð1  d1m Þ þ W mn  W ðmþ1Þn 1  dmNy þ 0:5C dk W mn þ V mn  W ðm1Þðn1Þ  V ðm1Þðn1Þ ð1  d1n Þð1  d1m Þ T

v2 W mn

T

vQymn

 T    T 

 T T T T T T T T þ 0:5C dk W mn þ V mn  W ðmþ1Þðnþ1Þ  V ðmþ1Þðnþ1Þ 1  dnNz 1  dmNy þ 0:5C dk W mn  V mn  W ðm1Þðnþ1Þ þ V ðm1Þðnþ1Þ  T    T 

T T T T T T  1  dnNz ð1  d1m Þ þ 0:5C dk W mn  V mn  W ðmþ1Þðn1Þ þ V ðmþ1Þðn1Þ ð1  d1n Þ 1  dmNy þ 0:5C d⊥ W mn  V mn   T  

 T T T T T T  W ðm1Þðn1Þ þ V ðm1Þðn1Þ ð1  d1n Þð1  d1m Þ þ 0:5C d⊥ W mn  V mn  W ðmþ1Þðnþ1Þ þ V ðmþ1Þðnþ1Þ 1  dnNz 1  dmNy !  T 

T T T T T T T T T þ 0:5C d⊥ W mn þ V mn  W ðm1Þðnþ1Þ  V ðm1Þðnþ1Þ 1  dnNz ð1  d1m Þ þ 0:5C d⊥ W mn þ V mn  W ðmþ1Þðn1Þ  V ðmþ1Þðn1Þ !  1  d1n

!) 1  dmNy

¼ 0:

Eqs. (18a)e(18d) represent a set of 4NyNz second-order coupled PDEs that should be solved for the dimensionless deformation fields of the SWCNTs.

(18d)

2 T Tv x 2

M

vt

T

þ K xT ¼ 0; T

(20) T

T

T

where xT ¼ < W mnp ; V mnp ; Qymnp ; Qzmnp > T and the dimensionless T

4.2.2. Frequency analysis of ESWCNTs modeled based on NTBT All SWCNTs have simple supports at their ends. The exterior ones are also prohibited from any lateral movement. As a result, the following admissible deformation field could be considered:

T

matrices M and K can be easily evaluated. Eq. (20) denotes a set of 4(Ny  2)(Nz  2) ODEs for each mode number p. Let xT ðtÞ ¼ xT0 ei6

T

xT0

t

where is the dimensionless amplitude vector, and 6T is the dimensionless natural frequency of the ESWCNTs based on the

1124

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

moments about the y and z axes, and nonlocal shear force along 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 [83]:

NTBT. By substituting such a vector into Eq. (20) and solving the resulting set of Eigenvalue equations for 6T , the natural frequencies are readily calculated. 4.3. Free dynamic analysis of ESWCNTs via NHOBT 4.3.1. Nonlocal equations of motion using NHOBT By employing the higher-order beam theory of BickfordeReddy [81,82] in the context of the nonlocal continuum theory, the kinetic energy, T H, and the strain energy of the ESWCNTs, UH, are stated as:

T

H

0 l Ny X Nz Z b 1 X @I0 ¼ 2 m¼1 n¼1 0

þ I2

vJH zmn vt

H v2 Vmn vtvx

!2 þ a2 I6

!2 þ

H v2 Wmn vtvx

H JH v2 Vmn zmn þ vt vtvx

!2 ! þ I2

!2  2aI4

vJH ymn

vJH zmn vt

MbHymn  ðe0 aÞ2

vx2

¼ J2

! H vJH vJH ymn ymn v2 Wmn  aJ4 þ ; vx vx vx2 (22a)

!2

vt

v2 MbHymn

þ a2 I6

vJH zmn vt

JH ymn vt

þ

H v2 Wmn vtvx

!2  2aI4

!1 H v2 Vmn Adx; þ vtvx

vJH ymn

vJH ymn

vt

vt

þ

H v2 Wmn vtvx

!

(21a)

1 ! H       H vJ H H H vWmn C B ymn B Mbnlymn a Pbnlzmn þC þ JH þ Qbnlzmn ymn þ vx C B vx ;x C B C B !   C B H     H H H C B vJH vV z nl H nl nl mn mn B Mbzmn a Pbymn þ C þ Jzmn þ þ Qbymn C B vx ;x vx C B C B C B h 2 2 i 

C B H H H H 1  dmN þ Vmn  Vðm1Þn ð1  d1m Þ þ C B Cvk Vmn  Vðmþ1Þn z C B C B i h 2 2  C B

H H H H BC 1  dnN þ C C B v⊥ Vmn  Vmðn1Þ ð1  d1n Þ þ Vmn  Vmðnþ1Þ z C B C B   2 C B H H C B Cdk Xmn ð1 Þð1 Þþ  d  d  X 1m 1n ðm1Þðn1Þ C B C B l  2   C Ny X Nz Z b B

X C B H H 1 H X 1  d  X C þ 1  d Cdx; B dk mn ðmþ1Þðnþ1Þ U ¼ m N n N y z C B 2 m¼1 n¼1 B C  2 C 0 B

H H C BC Y  Y ð1 Þ  d 1  d þ 1m C B dk mn ðm1Þðnþ1Þ nNz C B C B     2 C B H H C B Cdk Ymn  Yðmþ1Þðn1Þ 1  d ð1  d Þþ 1n mNy C B C B C B  2

C B H H C B Cd⊥ Xmn  Xðm1Þðnþ1Þ ð1  d1m Þ 1  dnNz þ C B C B     C B 2 H H C BC 1  dmN ð1  d1n Þþ C B d⊥ Xmn  Xðmþ1Þðn1Þ y C B C B   2 C B H H C B Cd⊥ Ymn  Y ð1  d Þð1  d Þþ 1m 1n ðm1Þðn1Þ C B C B C B  2  

A @ H H 1  dmN Cd⊥ Ymn  Yðmþ1Þðnþ1Þ 1  dnN y z 0

H , W H , JH , JH , ðM nl ÞH , ðM nl ÞH , ðQ nl ÞH , and where Vmn mn ymn zmn bymn bzmn bymn nl ÞH represent the deflection field associated with the y and z ðQbz mn

axes, deflection's angles about the y and z axes, nonlocal bending

MbHzmn  ðe0 aÞ2

v2 MbHzmn vx2

(21b)

¼ J2

! 2 H vJH vJH zmn zmn v Vmn  aJ4 þ ; vx vx vx2 (22b)

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

! H 2 vPby 2 v mn H H  ðe0 aÞ Qbymn þ a Qbymn þ a vx vx vx2 ! ! H H v2 JH v2 JH vVmn v3 Vmn zmn zmn H 2 ¼ k Jzmn þ þ aJ4  a J6 þ ; vx vx2 vx2 vx3 H vPby mn

!

(22c) ! H vPbz v2 H mn  ðe0 aÞ þa Qbzmn þ a vx vx vx2 ! ! H H v2 J H v2 JH vWmn v3 Wmn ymn ymn 2 ¼ k JH þ aJ þ  a J þ ; 4 6 ymn vx vx2 vx2 vx3 H vPbz mn

H Qbz mn

!

2

(22d)



3ro2 ; k ¼

Z

  Gb 1  3az2 dA;

Z n

(23)

n

rb z dA; Jn ¼

Eb z dA; n ¼ 0; 2; 4; 6: Ab

By implementation of the Hamilton's principle through using Eqs. (22a)e(22d), the nonlocal discrete equations of motion of vertically aligned ESWCNTs based on the NHOBT are obtained as follows:

( X

  v2 JH  v3 V H  zmn mn I2  2aI4 þ a2 I6 þ a2 I6  aI4 2 vt vt 2 vx !  v2 JH  H vVmn zmn H  J2  2aJ4 þ a2 J6 þ k Jzmn þ vx vx2  v3 V H  mn þ aJ4  a2 J6 ¼ 0; vx3

)

(24a)

ymn vt 2

 v3 W H  mn þ a2 I6  aI4 vt 2 vx

)

!  v2 JH  H vWmn ymn (24c)  J2  2aJ4 þ a2 J6 þ þk vx vx2  v3 W H  mn þ aJ4  a2 J6 ¼ 0; vx3 ! H vJH v3 JH v2 JH v2 Wmn ymn ymn ymn þ þ a2 J6  aJ4 k 2 3 vx vx vx vx2 ! ( ! 3 H H v JH v3 Wmn v2 Wmn v4 W H ymn 2  a2 I6 2 mn þ  a I6  aI4 X I0 3 2 2 vx vt vt vx vt vx2 "   

 H H H H þ Cvk Wmn  Wmðnþ1Þ  Wmðn1Þ 1  dnNz þ Wmn



AZ b

 v2 JH

JH ymn

"

 ð1  d1n Þ þ Cv⊥

a¼1

Ab

I2  2aI4 þ a2 I6

#

where

In ¼

( 

X

1125

H Wðmþ1Þn



  H H H Wmn ð1  d1m Þ þ Wmn  Wðm1Þn

#    H H H 1  dmNy þ Vmn  Wðm1Þðn1Þ þ 0:5Cdk Wmn

  H H H ð1  d1n Þð1  d1m Þ þ 0:5Cdk Wmn  Vðm1Þðn1Þ þ Vmn   

H H  Wðmþ1Þðnþ1Þ  Vðmþ1Þðnþ1Þ 1  dnNz 1  dmNy  

H H H H  Vmn  Wðm1Þðnþ1Þ þ Vðm1Þðnþ1Þ þ 0:5Cdk Wmn 1  dnNz  H H H  ð1  d1m Þ þ 0:5Cdk Wmn  Vmn  Wðmþ1Þðn1Þ     H H H ð1  d1n Þ 1  dmNy þ 0:5Cd⊥ Wmn þ Vðmþ1Þðn1Þ  Vmn  H H ð1  d1n Þð1  d1m Þ  Wðm1Þðn1Þ þ Vðm1Þðn1Þ  

H H H H  Vmn  Wðmþ1Þðnþ1Þ þ Vðmþ1Þðnþ1Þ þ 0:5Cd⊥ Wmn 1  dnNz    H H H  1  dmNy þ 0:5Cd⊥ Wmn þ Vmn  Wðm1Þðnþ1Þ H  Vðm1Þðnþ1Þ



H H þ Vmn 1  dnNz ð1  d1m Þ þ 0:5Cd⊥ Wmn !



H Wðmþ1Þðn1Þ



H Vðmþ1Þðn1Þ

! 1  d1n

!) 1  dmNy

¼ 0: (24d)

! ! (  v3 JH  H H H vJH v3 JH v2 J H v2 Vmn v3 Vmn v2 Vmn v4 V H zmn zmn zmn zmn 2 þ k þ a J6 þ  a2 I6  aI4  aJ4 X I0  a2 I6 2 mn2 2 3 2 3 2 2 vx vx vx vx vx vt vt vx vt vx # " " #       

 H

H H H H H H H þCvk Vmn  Vðmþ1Þn 1  dmNz þ Vmn  Vðm1Þn ð1  d1m Þ þ Cv⊥ Vmn  Vmðn1Þ ð1  d1n Þ þ Vmn  Vmðnþ1Þ 1  dnNz    

H H H H H H H H ð1  d1n Þð1  d1m Þ þ 0:5Cdk Wmn þ Vmn  Wðm1Þðn1Þ  Vðm1Þðn1Þ þ Vmn  Wðmþ1Þðnþ1Þ  Vðmþ1Þðnþ1Þ þ 0:5Cdk Wmn 1  dnNz        H H H H H H H ð1  d1n Þ 1  dmNy þ 0:5Cdk Vmn  1  dmNy þ 0:5Cdk Vmn  Wmn  Vðmþ1Þðn1Þ þ Wðmþ1Þðn1Þ  Wmn  Vðm1Þðnþ1Þ    

H H H H H H H ð1  d1n Þð1  d1m Þ þ 0:5Cd⊥ Vmn þ Wðm1Þðnþ1Þ  Wmn þ Wðm1Þðn1Þ  Vðm1Þðn1Þ  Wmn 1  dnNz ð1  d1m Þ þ 0:5Cd⊥ Vmn    



H H H H H H þ Wðmþ1Þðnþ1Þ  Vðmþ1Þðnþ1Þ þ Wmn  Vðm1Þðnþ1Þ  Wðm1Þðnþ1Þ 1  dnNz 1  dmNy þ 0:5Cd⊥ Vmn 1  dnNz ð1  d1m Þ ! ! !) H H H H þ 0:5Cd⊥ Vmn þ Wmn  Vðmþ1Þðn1Þ  Wðmþ1Þðn1Þ

1  d1n

1  dmNy

¼ 0; (24b)

1126

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

To analyze the problem in a more general context, the following dimensionless quantities are considered:

H Vmn WH aI  a2 I H H H ; W mn ¼ mn ; Jymn ¼ JH ; Jzmn ¼ JH ; g21 ¼ 4 2 6 ; y z mn mn lb lb I0 l

H

V mn ¼

b

2

kl2b 2 ; g4 a2 J6

2

aI4  a I6 ; I2  2aI4 þ a2 I6  4 J2  2aJ4 þ a2 J6 I0 l2b kI l 0 b   ; g28 ¼ ; g27 ¼ I2  2aI4 þ a2 I6 a2 J6 I2  2aI4 þ a2 I6 a2 J6  aJ4  a2 J6 I0 l2b Cd½: l4 Cv½: l4 H H 2  ; C v½: ¼ 2 b ; C d½: ¼ 2 b ; ½: ¼k or ⊥: g9 ¼ a J6 a J6 I2  2aI4 þ a2 I6 a2 J6

g22 ¼

a I6 I0 l2b

; g23 ¼

¼

aJ4  a J6

2

a2 J6

; g26 ¼

(25)

By introducing Eq. (25) to Eqs. (24a)e(24d), the dimensionless governing equations of transverse motion of the ESWCNTs are derived as:

) !  2 H H H H 3 H v Jzmn v2 Jzmn vV mn v3 V mn H 2 v V mn 2 2 X  g J þ þ g29 ¼ 0; þ g  g zmn 7 6 8 2 2 2 3 vx vt vt vx vx vx H

g23

H

vJzmn v2 V mn þ vx vx2

! 

H v3 Jzmn g24 3

vx

H

þ

v4 V mn vx4

( þX

(26a)

" H 3 H  H  4 H

v2 V mn H H 2 v Jzmn 2 v V mn þ g þ C V  V  g 1  dmNz mn ðmþ1Þn vk 1 2 vt2 vt2 vx vt2 vx2

# " #  H     H  H  H

H H H H H H H þ V mn  V ðm1Þn ð1  d1m Þ þ C v⊥ V mn  V mðn1Þ ð1  d1n Þ þ V mn  V mðnþ1Þ 1  dnNz þ 0:5C dk W mn þ V mn  W ðm1Þðn1Þ   H    H

 H H H H H H H  V ðm1Þðn1Þ ð1  d1n Þð1  d1m Þ þ 0:5C dk W mn þ V mn  W ðmþ1Þðnþ1Þ  V ðmþ1Þðnþ1Þ 1  dnNz 1  dmNy þ 0:5C dk V mn  W mn     

H H H H H H H  V ðmþ1Þðn1Þ þ W ðmþ1Þðn1Þ ð1  d1n Þ 1  dmNy þ 0:5C dk V mn  W mn  V ðm1Þðnþ1Þ þ W ðm1Þðnþ1Þ 1  dnNz ð1  d1m Þ  H   H  H H H H H H H H þ 0:5C d⊥ V mn  W mn þ W ðm1Þðn1Þ  V ðm1Þðn1Þ ð1  d1n Þð1  d1m Þ þ 0:5C d⊥ V mn  W mn þ W ðmþ1Þðnþ1Þ  V ðmþ1Þðnþ1Þ   H 



H H H H H H H  1  dnNz 1  dmNy þ 0:5C d⊥ V mn þ W mn  V ðm1Þðnþ1Þ  W ðm1Þðnþ1Þ 1  dnNz ð1  d1m Þ þ 0:5C d⊥ V mn þ W mn ! H

H

 V ðmþ1Þðn1Þ  W ðmþ1Þðn1Þ

! 1  d1n

!) 1  dmNy

¼ 0; (26b)

! ) H  2 H H H H 3 3 v Jymn v2 Jymn vW mn H 2 v W mn 2 2 2 v W mn  g X  g J þ þ g ¼ 0; þ g y 7 6 8 9 mn vx vt2 vt2 vx vx2 vx3

(26c)

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

1127

( " H H H  H  4 v3 Jymn

 H v2 W mn H H 2 2 v W mn þ g þ C W mn  W mðnþ1Þ 1  dnNz þ W mn  g vk 1 2 2 3 4 2 2 2 2 vx vt vt vx vx vx vx vt vx # # "      H   H H H H H H H H H þ 0:5C dk W mn þ V mn  W ðm1Þðn1Þ  W mðn1Þ ð1  d1n Þ þ C v⊥ W mn  W ðm1Þn ð1  d1m Þ þ W mn  W ðmþ1Þn 1  dmNy H

vJyn

g23

H

þ

v2 W n

!

H

 g24

v3 Jyn

H

þ

v4 W n

þX

  H    H

 H H H H H H H  V ðm1Þðn1Þ ð1  d1n Þð1  d1m Þ þ 0:5C dk W mn þ V mn  W ðmþ1Þðnþ1Þ  V ðmþ1Þðnþ1Þ 1  dnNz 1  dmNy þ 0:5C dk W mn  V mn     

H H H H H H H  W ðm1Þðnþ1Þ þ V ðm1Þðnþ1Þ 1  dnNz ð1  d1m Þ þ 0:5C dk W mn  V mn  W ðmþ1Þðn1Þ þ V ðmþ1Þðn1Þ ð1  d1n Þ 1  dmNy  H   H  H H H H H H H H þ 0:5C d⊥ W mn  V mn  W ðm1Þðn1Þ þ V ðm1Þðn1Þ ð1  d1n Þð1  d1m Þ þ 0:5C d⊥ W mn  V mn  W ðmþ1Þðnþ1Þ þ V ðmþ1Þðnþ1Þ   H 



H H H H H H H  1  dnNz 1  dmNy þ 0:5C d⊥ W mn þ V mn  W ðm1Þðnþ1Þ  V ðm1Þðnþ1Þ 1  dnNz ð1  d1m Þ þ 0:5C d⊥ W mn þ V mn ! H

H

 W ðmþ1Þðn1Þ  V ðmþ1Þðn1Þ

! 1  d1n

!) 1  dmNy

¼ 0: (26d)

4.3.2. Frequency analysis of ESWCNTs modeled based on NHOBT Each vertically aligned SWCNT has simple supports at its both ends. Further, the exterior SWCNTs of the ensemble are not allowed to move in the transverse directions. Thereby, using assumed mode method, the transverse displacements and the angles of deformation of the (m,n)th tube are expressed by:

H W mn ðx; tÞ

¼

Np X

H W mnp ðtÞsin

H

Np X

ðppxÞ;

H

V mnp ðtÞsin ðppxÞ;

p¼1

H

Jymn ðx; tÞ ¼

Np X p¼1

5. Development of continuous models based on nonlocal beam theories As it was explained in the previous parts, the size of the set of discrete governing equations would magnify with the number of constitutive SWCNTs of the ensemble. It implies that dynamic analysis of largely populated ensembles based on the discrete models would associated with a huge computational effort and labor cost. Thereby, development of appropriate continuous models for vibration analysis of SWCNTs' ensembles would be of great importance, particularly for highly populated ESWCNTs. Based on the proposed discrete models, suitable continuous models are established in the following parts.

p¼1

V mn ðx; tÞ ¼

solving the resulting set of Eigenvalue equations for 6H , the natural frequencies are determined.

H

Jymnp ðtÞcos ðppxÞ; 5.1. Nonlocal continuous modeling of ESWCNTs via NRBT

H

Jzmn ðx; tÞ ¼

Np X p¼1

H

Jzmnp ðtÞcos ðppxÞ;

(27)

by introducing Eq. (27) to Eqs. (26a)e(26d) and through taking into account the boundary conditions of the exterior ensemble's SWCNTs, the following set of algebraic equations is obtained:

Hv

M

2 H

x

vt2

H

þ K xH ¼ 0; H

(28) H

H

H

where xH ¼ < W mnp ; V mnp ; Jymnp ; Jzmnp > T and the dimensionless H

H

matrices M and K are readily derived. For each mode number p, Eq. (28) is a set of 4(Ny  2)(Nz  2) ODEs. By considering xH ðtÞ ¼ i6 t where x H is the dimensionless amplitude vector, and 6H xH 0e 0 denotes the dimensionless natural frequency of the ESWCNTs according to the NHOBT. By substituting this vector into Eq. (28) and H

5.1.1. Nonlocal continuous governing equations using NRBT According to the discrete governing equations of the ESWCNTs on the basis of the NRBT, Eqs. (9a) and (9b), the nonlocal equations of motion of the (m,n)th interior SWCNT of the ensemble can be expressed by:

( !  R R R v4 Vmn v2 Vmn v4 Vmn R R 2Vmn Eb Ib þ X r  I  Vðmþ1Þn A þ C vk b b b vx4 vt 2 vt 2 vx2    1  R R R R Cdk  Cd⊥ þ Cv⊥ 2Vmn þ  Vðm1Þn  Vmðn1Þ  Vmðnþ1Þ 2   R R R R þ Wðmþ1Þðn1Þ  Wðmþ1Þðnþ1Þ  Wðm1Þðn1Þ  Wðm1Þðnþ1Þ  1 R R R Cdk þ Cd⊥ 4Vmn  Vðm1Þðnþ1Þ  Vðmþ1Þðnþ1Þ þ 2 )  R R  Vðmþ1Þðn1Þ  Vðm1Þðn1Þ ¼ 0; (29a)

1128

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

( !  R R R v4 Wmn v2 Wmn v4 Wmn R þ X r  I A þ Cvk 2Wmn b b b 2 2 4 2 vx vt vt vx    R R R R R þ Cv⊥ 2Wmn  Wmðnþ1Þ  Wmðn1Þ  Wðm1Þn  Wðmþ1Þn  1 R R R Cdk  Cd⊥ Vðm1Þðnþ1Þ þ þ Vðmþ1Þðn1Þ  Vðmþ1Þðnþ1Þ 2  1  R R R Cdk þ Cd⊥ 4Wmn þ  Vðm1Þðn1Þ  Wðm1Þðnþ1Þ 2 )  R R R ¼ 0:  Wðmþ1Þðnþ1Þ  Wðmþ1Þðn1Þ  Wðm1Þðn1Þ

Eb Ib

(29b) In order to construct a continuous model based on the discrete relation in Eqs. (29a) and (29b), we introduce two continuous displacement functions of the form v ¼ v(x,y,z,t) and w ¼ w(x,y,z,t) such that:

½  mn ðx; tÞz½:ðx; ymn ; zmn ; tÞ; ½  ðm1Þðn1Þ ðx; tÞz½:ðx; ymn  d; zmn  d; tÞ; ½  ðm1Þðnþ1Þ ðx; tÞz½:ðx; ymn þ d; zmn  d; tÞ; ½  ðmþ1Þðn1Þ ðx; tÞz½:ðx; ymn  d; zmn þ d; tÞ; ½  ðmþ1Þðnþ1Þ ðx; tÞz½:ðx; ymn þ d; zmn þ d; tÞ;

( ! v4 wR v2 wR v4 wR v2 wR Eb Ib 4 þ X rb Ab 2  Ib 2 2  Cvk d2 vx vt vt vx vz2 ! d2 v4 wR d4 v6 wR d6 v8 wR v2 wR þ þ þ  Cv⊥ d2 4 6 8 12 vz 360 vz 20160 vz vy2 !  d2 v4 wR d4 v6 wR d6 v8 wR þ þ þ  Cdk 12 vy4 360 vy6 20160 vy8 " !  2 R v2 wR d2 v4 wR v4 w R v4 wR 2 v w þ Cd⊥ d þ þ þ6 2 2þ 12 vz4 vy2 vz2 vz vy vy4 !#  d4 v6 wR v6 wR v6 wR v6 wR þ þ 15 4 2 þ 15 2 4 þ  Cdk 6 6 360 vz vz vy vz vy vy " !  v2 vR d2 v4 vR v4 vR d4 v6 vR þ 3 5  Cd⊥ d2 2 þ þ vyvz 3 vyvz3 vy3 vz 180 vy vz !#) v6 vR v6 v R þ 10 3 3 þ 3 ¼ 0: vyvz5 vy vz (32b)

(30)

where ½ð½:Þ ¼ V ½+ ðv½+ Þ or W ½+ ðw½+ Þ and [B] ¼ R or T or H, and (ymn,zmn) represents the coordinates of the revolutionary axis of the (m,n)th SWCNT in the y-z plane. The transverse displacements of the neighboring SWCNTs of the (m,n)th tube are now approximated by sixth-order Taylor polynomials as in the following form:

½:ðx; ymn ±d; zmn ±d; tÞ ¼

6 P i P

i

i¼1 j¼0

ij

j

ij

ð±dÞ ð±dÞ

!

vi ½:ðx; ymn ; zmn ; tÞ vzj vyij

(31)

By introducing Eq. (10) to Eqs. (32a) and (32b), the dimensionless continuous equations of motion describing transverse motion of the ESWCNTs ensemble are stated by:

;

0

where v ½:ðx;y;z;tÞ ¼ ½:ðx; y; z; tÞ; ½: ¼ v½+ and w½+ . By substituting Eq. vy0 vz0 (31) into Eqs. (29a) and (29b) in view of Eq. (30), the nonlocal continuous version for free transverse vibrations of SWCNTs' ensembles based on the NRBT are derived as follows:

( ! v4 v R v2 vR v4 vR v2 vR d2 v4 vR Eb Ib 4 þ X rb Ab 2  Ib 2 2  Cvk d2 þ 12 vy4 vx vt vt vx vy2 ! d4 v6 vR d6 v8 vR v2 vR d2 v4 vR þ þ þ  Cv⊥ d2 6 8 360 vy 20160 vy 12 vz4 vz2 ! "   2 R d4 v6 vR d6 v8 vR v2 vR 2 v v d þ þ þ C þ  C dk d⊥ 360 vz6 20160 vz8 vy2 vz2 ! d2 v4 vR v4 vR v4 vR d4 v6 vR v6 vR þ þ6 2 2þ 4 þ þ 15 4 2 4 6 12 vz 360 vz vz vy vy vz vy !# "   6 R 6 R 2 R v v v v v w d2 v4 wR þ þ 15 2 4 þ 6  Cdk  Cd⊥ d2 2 vyvz 3 vz vy vy vyvz3 !#) ! v4 wR d4 v6 wR v6 wR v6 wR 3 5 þ 10 3 3 þ 3 þ 3 ¼ 0; þ 180 vy vz vyvz5 vy vz vy vz (32a)

( v2 vR v4 vR v2 vR ðkdÞ2 v4 vR R þX  l2  ðkdÞ2 C vk þ 4 2 2 2 12 vh4 vt vh2 vx vt vx ! 2 2 R ðkdÞ4 v6 vR ðkdÞ6 v8 vR v2 vR d v4 vR þ þ d C þ  v⊥ 360 vh6 20160 vh8 vg2 12 vg4 ! " 4 6   R 2 R d v6 vR d v8 v R v2 vR R 2v v þ þ C þ C þ  k dk d⊥ 360 vg6 20160 vg8 vh2 vg2 ! 2 4 d v4 vR v4 vR v4 vR d v6 vR þ þ 6k2 2 2 þ k4 4 þ 4 12 vg 360 vg6 vg vh vh !#   R v6 vR v6 vR v6 vR R þ 15k2 4 2 þ 15k4 2 4 þ k6 6  k C dk  C d⊥ vg vh vg vh vh " ! 2 4 4 R v2 wR d v4 wR d v6 wR 2v w þ 3k4 5 þ k  2 þ 3 3 vhvg 3 vhvg 180 vh vg vh vg !#) R R v6 w v6 w þ 10k2 3 3 þ 3 ¼ 0; vhvg5 vh vg

v4 vR

(33a)

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

( 2 4 R 2 R v4 wR v2 w R v2 wR d v4 wR 2 v w þ X  l  d C þ vk 12 vg4 vt2 vg2 vx4 vt2 vx2 ! 4 6 d v6 wR d v8 wR v2 wR ðkdÞ2 v4 wR 2 R þ þ C þ  ðkdÞ v⊥ 360 vg6 20160 vg8 12 vh4 vh2 " !   R ðkdÞ4 v6 wR ðkdÞ6 v8 wR v2 wR v2 wR R þ þ þ  C dk þ C d⊥ k2 6 8 360 vh 20160 vh vh2 vg2 ! 2 4 4 R 4 R d v4 wR d v6 wR 2 v w 4v w þ þ 6k þ k þ 12 vg4 360 vg6 vg2 vh2 vh4 !#   R v6 wR v6 w R v6 w R R þ 15k2 4 2 þ 15k4 2 4 þ k6  k C dk  C d⊥ 6 vg vh vg vh vh " ! 2 4 4 R v2 vR d v4 vR d v6 vR 2 v v þ 3k4 5 þ k  2 þ vhvg 3 vhvg3 180 vh vg vh3 vg !#) v6 vR v6 vR þ 10k2 3 3 þ 3 ¼ 0: vhvg5 vh vg (33b) where

h¼ R

Cd½: d2 l4b R vR R wR y lz R ; k ¼ ; C d½: ¼ ;v ¼ ;w ¼ ; ly ly lb lb Eb Ib l2z R

R

(34)

R

v ¼ v ðx; h; g; tÞ; w ¼ w ðx; h; g; tÞ; ½: ¼k or ⊥:

5.1.2. Frequency analysis of ESWCNTs using continuous NRBT The dimensionless deflections in y and z directions are represented in the following harmonic form:

vR ðx; h; g; tÞ ¼

N pv mv N nv N X X X

vRmnp ðtÞfvmnp ðx; h; gÞ;

m¼1 n¼1 p¼1

wR ðx; h; g; tÞ ¼

pw NX mw N nw N X X

(35) wRmnp ðtÞfw mnp ðx; h; gÞ;

where fvmnp and fw mnp are the shape functions associated with the deflection fields along the y and z axes, respectively. For simply supported ensembles of SWCNTs with immovable exterior nanotubes, the following admissible shape functions are taken into account:

(36)

Additionally, vRmnp /wRmnp are their corresponding unknown coefficients, Nmw/Nmv, Nnw/Nnv, and Npw/Npv in order are the number of vibration modes associated with the deflection fields along the x, y, and z axes. By multiplying both sides of Eqs. (34a) and (34b) by dvR

2h 4h

i R vv Mb

i R wv Mb

8 2 R9 2 < v v2 = hKR ivv

h R ivw 3 Mb 5 h R iww Mb

vt

: v2 w R ; vt2

þ 4h

b

i R wv

Kb

and dwR , where d is the variational sign, and taking the required integration by parts, one can arrive at the following set of equations: where the submatrices associated with the mass and stiffness matrices could be readily calculated.

5.2. Nonlocal continuous modeling of ESWCNTs via NTBT 5.2.1. Nonlocal continuous governing equations using NTBT On the basis of the NTBT and using Eqs. (16a)e(16d), the equations of motion describing transverse motion of the (m,n)th interior SWCNT of the ESWCNTs are stated by:

) !  T v2 QTzmn v2 QTzmn vVmn T  Q X rb Ib G A ¼ 0;  k  Eb Ib s b b z 2 mn vx vt vx2 (38a) ! (  T T vQTzmn v2 Vmn v2 Vmn T ks Gb Ab þ X r b Ab  þ Cvk 2Vmn vx vx2 vt 2    T T T T T þ Cv⊥ 2Vmn  Vðmþ1Þn  Vðm1Þn  Vmðn1Þ  Vmðnþ1Þ  1 T T T Cdk  Cd⊥ Wðm1Þðnþ1Þ þ þ Wðmþ1Þðn1Þ  Wðmþ1Þðnþ1Þ 2  1  T T T C þ Cd⊥ 4Vmn þ  Wðm1Þðn1Þ  Vðm1Þðnþ1Þ 2 dk )  T T T  Vðmþ1Þðnþ1Þ  Vðmþ1Þðn1Þ  Vðm1Þðn1Þ ¼ 0; (38b) ) !  T v2 QTymn v2 QTymn vWmn T  Q G A I ¼ 0;  k  E X rb Ib s b b b b ymn vx vt 2 vx2 (38c) !

(

 T T vQTymn v2 Wmn v2 Wmn T þ X rb Ab  þ Cvk 2Wmn 2 2 vx vx vt    T T T T T þ Cv⊥ 2Wmn  Wmðnþ1Þ  Wmðn1Þ  Wðm1Þn  Wðmþ1Þn  1 T T T Cdk  Cd⊥ Vðm1Þðnþ1Þ þ þ Vðmþ1Þðn1Þ  Vðmþ1Þðnþ1Þ 2  1  T T T Cdk þ Cd⊥ 4Wmn þ  Vðm1Þðn1Þ  Wðm1Þðnþ1Þ 2 )  T T T ¼ 0:  Wðmþ1Þðnþ1Þ  Wðmþ1Þðn1Þ  Wðm1Þðn1Þ

ks Gb Ab

m¼1 n¼1 p¼1

fvmnp ðx; h; gÞ ¼ sin ðmpxÞsin ðnphÞsin ðppgÞ; fw mnp ðx; h; gÞ ¼ sin ðmpxÞsin ðnphÞsin ðppgÞ:

1129

(38d) Now the following new continuous angles of deflection functions are taken into account:

i 3 R vw ( R) ( ) Kb 0 v 5 ¼ ; h R iww R 0 w Kb h

(37)

1130

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

Fig. 3. Comparison of the predicted flexural frequencies by the proposed models with those of another work [84] for a double-nanobeam-system: ((…) NRBT, () NTBT, (d) R NHOBT, (D) Ref. [84]; C vk ¼ 10).

qTy ðx; ymn ; zmn ; tÞzQTymn ðx; tÞ;

(39)

qTz ðx; ymn ; zmn ; tÞzQTzmn ðx; tÞ;

by introducing Eqs. (30), (31), and (39) to Eqs. (38a)e(38d), the continuous version of the equations of motion associated with the free transverse vibration of ensembles of SWCNTs based on the NTBT are obtained:

n v2 qT X rb Ib 2z vt

ks Gb Ab

)

  ks Gb Ab

v2 vT vqTz  vx vx2

!

vvT  qTz vx



( þ X rb Ab

d4 v6 vT d6 v8 vT þ þ 360 vy6 20160 vy8

!

 Eb Ib

v2 qTz vx2

¼ 0;

(40a)

v2 v T v2 vT d2 v4 vT þ Cvk d2 þ 2 12 vy4 vt vy2

v2 vT d2 v4 vT þ 12 vz4 vz2 ! "   2 T d4 v6 vT d6 v8 vT v2 vT 2 v v þ þ þ 2  Cdk þ Cd⊥ d 2 360 vz6 20160 vz8 vy vz ! 4 T 4 T 4 T 6 T 6 T 2 4 d v v v v v v d v v v v þ þ6 2 2þ 4 þ þ 15 4 2 12 vz4 360 vz6 vz vy vy vz vy !# "   v6 vT v6 vT v2 wT d2 v4 wT 2 þ þ 15 2 4 þ 6  Cdk  Cd⊥ d 2 vyvz 3 vyvz3 vz vy vy !#) ! v4 wT d4 v6 wT v6 wT v6 wT þ 3 ¼ 0; þ 3 5 þ 10 3 3 þ 3 180 vy vz vyvz5 vy vz vy vz  Cv⊥ d2

(40b)

)  T  v2 qTy v2 qTy vw  qTy  Eb Ib 2 ¼ 0; X rb Ib 2  ks Gb Ab vx vt vx T v2 wT vqy  vx vx2

!

(40c)

(

v2 wT v2 wT þ Cvk d2 2 vt vz2 ! d2 v4 wT d4 v6 wT d6 v8 wT v2 wT þ þ þ  Cv⊥ d2 4 6 8 12 vz 360 vz 20160 vz vy2 !  d2 v4 wT d4 v6 wT d6 v8 wT þ þ þ  Cdk 12 vy4 360 vy6 20160 vy8 " !  2 T v2 wT d2 v4 wT v4 wT v4 wT 2 v w þ Cd⊥ d þ þ þ6 2 2þ 12 vz4 vy2 vz2 vz vy vy4 !#  d4 v6 wT v6 wT v6 wT v6 wT þ þ 15 4 2 þ 15 2 4 þ  Cdk 6 6 360 vz vz vy vz vy vy " !  v2 vT d2 v4 vT v4 vT d4 v6 vT þ 3 5  Cd⊥ d2 2 þ þ vyvz 3 vyvz3 vy3 vz 180 vy vz !#) v6 vT v6 vT þ 10 3 3 þ 3 ¼ 0: vyvz5 vy vz

ks Gb Ab

þ X rb Ab

(40d) In view of Eq. (17), Eqs. (40a)e(40d) can be rewritten in the dimensionless form as follows: T n v2 q X l2 2z vt

)

 

vvT T  qz vx



T

h

v2 qz 2

vx

¼ 0;

(41a)

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

Table 1 Comparison of the predicted results by the discrete models and those of nonlocal continuous ones for different levels of the ensemble's population and slenderness ratio.

!

( v2 vT v2 vT ðkdÞ2 v4 vT 2 T þX  þ ðkdÞ C þ vk 12 vh4 vt2 vh2 ! 2 2 T ðkdÞ4 v6 vT ðkdÞ6 v8 vT v2 vT d v4 vT þ þ þ  d C v⊥ 6 8 2 360 vh 20160 vh 12 vg4 vg " ! 4 6   T d v6 vT d v8 vT v2 vT v2 vT T k2 2 þ þ þ C þ C  dk d⊥ 6 8 360 vg 20160 vg vh vg2 ! 2 4 4 T 4 T d v4 vT d v6 vT 2 v v 4v v þ þ 6k þ k þ 12 vg4 360 vg6 vg2 vh2 vh4 !#   T v6 vT v6 vT v6 v T T þ 15k2 4 2 þ 15k4 2 4 þ k6 6  k C dk  C d⊥ vg vh vg vh vh " ! 2 4 4 T v2 wT d v4 wT d v6 wT 2v w þ 3k4 5 þ k  2 þ 3 vhvg 3 vhvg3 180 vh vg vh vg !#) 6 T 6 T v w v w þ 10k2 3 3 þ 3 ¼ 0; vhvg5 vh vg v2 vT

T

vq  z vx vx2

l Discrete models

10 15 30 NTBT 10 15 30 NHOBT 10 15 30 Continuous NRBT 10 models 15 30 NTBT 10 15 30 NHOBT 10 15 30

(41b) T) T   T v2 qy v2 qy vw T 2   h X l q ¼ 0;  y vx vt2 vx2

1131

NRBT

Ny ¼ Nz ¼ 5 Ny ¼ Nz ¼ 7 Ny ¼ Nz ¼ 9 Ny ¼ Nz ¼ 11 2.379006 1.709242 1.418869 2.165378 1.667933 1.418095 2.242269 1.683699 1.418408 2.379007 1.709243 1.418870 2.165379 1.667934 1.418096 2.242270 1.683700 1.418410

2.169156 1.384324 .983494 1.921212 1.328546 .981950 2.010895 1.349866 .982573 2.169156 1.384324 .983494 1.921212 1.328546 .981950 2.010895 1.349866 .982573

2.087511 1.245128 .765769 1.824158 1.180941 .763564 1.919694 1.205532 .764453 2.087511 1.245128 .765769 1.824158 1.180941 .763565 1.919694 1.205532 .764453

2.048031 1.174051 .638340 1.776725 1.104801 .635564 1.875312 1.131381 .636683 2.048031 1.174051 .638340 1.776725 1.104801 .635564 1.875312 1.131381 .636683

5.2.2. Frequency analysis of ESWCNTs using continuous NTBT The dimensionless deformation fields of ESWCNTs in which modeled based on the proposed continuous NTBT can be written as follows:

(41c)

T! vqy

( 2 2 T v2 wT v2 wT v2 wT d v4 wT þ  X þ d C þ  vk vx 12 vg4 vt2 vg2 vx2 ! 4 6 d v6 wT d v8 wT v2 wT ðkdÞ2 v4 wT T þ þ þ  ðkdÞ2 C v⊥ 360 vg6 20160 vg8 12 vh4 vh2 ! "   T ðkdÞ4 v6 wT ðkdÞ6 v8 wT v2 wT v2 wT T þ þ þ  C dk þ C d⊥ k2 6 8 360 vh 20160 vh vh2 vg2 ! 2 4 4 T 4 T d v4 wT d v6 wT 2 v w 4v w þ þ 6k þ k þ 4 2 2 4 12 vg 360 vg6 vg vh vh !#   T v6 wT v6 wT v6 w T T þ 15k2 4 2 þ 15k4 2 4 þ k6  k C dk  C d⊥ 6 vg vh vg vh vh " ! 2 4 4 T v2 vT d v4 vT d v6 vT 2 v v þ 3k4 5 þ k  2 þ 3 3 vhvg 3 vhvg 180 vh vg vh vg !#) 6 T 6 T v v v v þ 10k2 3 3 þ 3 ¼ 0; vhvg5 vh vg

T

qz ðx; h; g; tÞ ¼

T qz qzmnp ðtÞfmnp ðx; h; gÞ;

m¼1 n¼1 p¼1

vT ðx; h; g; tÞ ¼

N pv mv N nv N X X X

vTmnp ðtÞfvmnp ðx; h; gÞ;

m¼1 n¼1 p¼1 T qy ðx; h; g; tÞ

¼

N pv mv N nv N X X X m¼1 n¼1 p¼1

wT ðx; h; g; tÞ ¼

(43)

qy qymnp ðtÞfmnp ðx; h; gÞ;

pw NX mw N nw N X X

T

wTmnp ðtÞfw mnp ðx; h; gÞ;

m¼1 n¼1 p¼1

where the shape functions associated with the deformation fields of the nanostructure at hand are as:

fvmnp ðx; h; gÞ ¼ fw mnp ðx; h; gÞ ¼ sin ðmpxÞsin ðnphÞsin ðppgÞ; qy qz fmnp ðx; h; gÞ ¼ fmnp ðx; h; gÞ ¼ cos ðmpxÞsin ðnphÞsin ðppgÞ: (44)

(41d) where

pw NX mw N nw N X X

T

By multiplying both sides of Eqs. (41a)e(41d) in order by dqz , T

dv ,

T dqy ,

T

dw , and taking the required integration by parts, the

following set of algebraic equations is obtained:

T

C d½:: ¼ T qz

¼

Cd½:: d2 l2b ks Gb Ab l2z

T qz ðx; h; g; tÞ; vT

; vT ¼ T

¼v

vT T wR T T ;w ¼ ; q ¼ qTy ; qz ¼ qTz ; ½:: ¼k or ⊥; lb lb y

T ðx; h; g; tÞ; qy

¼

T qy ðx; h; g; tÞ; wT

T

¼ w ðx; h; g; tÞ:

(42)

1132

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

Fig. 4. Variation of the fundamental frequency as a function of the slenderness ratio for different levels of the small-scale parameters: ((…) NRBT, () NTBT, (d) NHOBT; (B) e0a ¼ 0, (,) e0a ¼ 1, (D) e0a ¼ 2 nm; Ny ¼ Nz ¼ 1000).

Fig. 5. Variation of the fundamental frequency as a function of the mean radius of the SWCNT for different values of length: ((…) NRBT, () NTBT, (d) NHOBT; (B) lb ¼ 10, (,) lb ¼ 15, (D) lb ¼ 30 nm; Ny ¼ Nz ¼ 1000).

Fig. 6. Variation of the fundamental frequency as a function of the small-scale parameter for different levels of the slenderness ratio: ((…) NRBT, () NTBT, (d) NHOBT; (B) l ¼ 10, (,) l ¼ 15, (D) l ¼ 30; Ny ¼ Nz ¼ 1000).

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

8 9 3> v Q > > h iq w > > vt > > > M > > 7> > h i 7< v v = T z

2

2h

i h T iqz v h T iqz qy T qz qz T z Mb M M b 6h i h b i h bi 6 T vqz T vv T vqy T vw 6 Mb 7 M M M 6h i h bi h bi h bi 7 6 T qy qz T qy v T qy qy T qy w 7 6 Mb 7 Mb Mb Mb 4 5 h T iwqz h T iwv h T iwqy h T iww Mb Mb Mb Mb

2h

2

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

2 T

vt2

> v Q > > > > > > vt > > > > : > ; T y

2

H jH y ðx; ymn ; zmn ; tÞzJymn ðx; tÞ;

2

v2 wT

H jH z ðx; ymn ; zmn ; tÞzJzmn ðx; tÞ;

(45)

3

by introducing Eqs. (30), (31), and (47) to Eqs. (46a)e(46d), the continuous equations of motion pertinent to the free transverse vibration of SWCNTs' ensembles in accordance with the NHOBT are derived as follows:

( )   v2 jH   v3 vH 2 2 z X I2  2aI4 þ a I6 þ a I6  aI4 vt 2 vt 2 vx   v2 jH  vvH z  J2  2aJ4 þ a2 J6 þ k jH z þ vx vx2  v3 vH  þ aJ4  a2 J6 ¼ 0; vx3

where the mass and stiffness matrices are easily determined. 5.3. Nonlocal continuous modeling of ESWCNTs via NHOBT 5.3.1. Nonlocal continuous governing equations using NHOBT By employing Eqs. (24a)e(24d), the equations of motion describing transverse motion of the (m,n)th interior SWCNT of the ensemble could be written in the following form:

X

( 

2

I2  2aI4 þ a I6

 v2 JH

zmn vt 2



2

þ a I6  aI4

)

 v3 V H

mn

vt 2 vx

þk

JH zmn

vV H þ mn vx

!

 v2 JH  v3 V H   zmn mn þ aJ4  a2 J6 ¼ 0;  J2  2aJ4 þ a2 J6 2 vx vx3

! ! (  v3 J H  H H H vJH v3 JH v2 JH v2 Vmn v3 Vmn v2 Vmn zmn zmn zmn zmn 2 2 k þ þ a J þ  a I  aI  aJ X I 4 6 0 6 4 vx vx2 vx3 vx2 vx3 vt 2 vt 2 vx     v4 V H H H H H H H þ Cv⊥ 2Vmn  Vðmþ1Þn  Vðm1Þn  Vmðn1Þ  Vmðnþ1Þ  a2 I6 2 mn2 þ Cvk 2Vmn vt vx   1 H H H H C  Cd⊥ Wðm1Þðnþ1Þ þ þ Wðmþ1Þðn1Þ  Wðmþ1Þðnþ1Þ  Wðm1Þðn1Þ 2 dk )   1 H H H H H C þ Cd⊥ 4Vmn  Vðm1Þðnþ1Þ  Vðmþ1Þðnþ1Þ  Vðmþ1Þðn1Þ  Vðm1Þðn1Þ þ ¼ 0; 2 dk

X

( 

2

I2  2aI4 þ a I6

 v2 JH

ymn vt 2



(47)

vt2

iqz qz h T iqz v h T iqz qy h T iqz w K K Kb T9 8 9 6h i hb i h bi 78 > Qz > 0 h i 6 T vqz T vv T vqy > T vw 7> > > > > > > 6 Kb 7> < T> = > <0> = Kb Kb Kb v 6 7 þ h i ¼ ; T 6 T qy qz h T iqy v h T iqy qy h T iqy w 7> > 0> Qy > > > > > > > > 6 Kb 7> Kb Kb Kb > > : ; 4 5: w T ; 0 h T iwqz h T iwv h T iwqy h T iww Kb Kb Kb Kb T Kb

1133

2

þ a I6  aI4

 v3 W H

)

mn

þk

vt 2 vx

JH ymn

H vWmn þ vx

!

(48a)

(46a)

(46b)

 v2 J H  v3 W H   ymn 2 mn  J2  2aJ4 þ a2 J6 þ aJ  a J ¼ 0; 4 6 vx2 vx3 (46c)

! (  v3 JH  H H v3 Wmn v2 Wmn v4 W H ymn  a2 I6 2 mn  a2 I6  aI4 X I0 3 2 3 2 2 vx vx vx vx vt vt vx vt vx2     1  H H H H H H H H C  Cd⊥ Vðm1Þðnþ1Þ þ Cv⊥ 2Wmn þ þ Cvk 2Wmn  Wmðnþ1Þ  Wmðn1Þ  Wðm1Þn  Wðmþ1Þn þ Vðmþ1Þðn1Þ 2 dk )  1   H H H H H H H C þ Cd⊥ 4Wmn  Wðm1Þðnþ1Þ  Wðmþ1Þðnþ1Þ  Wðmþ1Þðn1Þ  Wðm1Þðn1Þ ¼ 0:  Vðmþ1Þðnþ1Þ  Vðm1Þðn1Þ þ 2 dk k

vJH ymn

þ

H v2 Wmn 2 vx

!

 aJ4

v3 JH ymn

þ a2 J6

v2 JH ymn

þ

(46d)

1134

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

! (  v3 jH v 3 jH v2 jH v3 vH v2 vH  v4 vH 2 z z z þ  aJ4 3 þ a J6 X I0 2  a2 I6  aI4  a2 I6 2 2 2 3 2 vx vx vx vt vt vx vt vx ! ! 2 H 4 H 6 H 8 H 2 H 4 H 6 H 2 4 6 2 4 d v v d v v d v v d v v d v v d6 v8 vH 2 v v 2 v v þ Cvk d þ þ þ þ þ þ  Cv⊥ d 12 vy4 360 vy6 20160 vy8 12 vz4 360 vz6 20160 vz8 vy2 vz2 " ! !#   2 H v2 vH d2 v4 vH v4 vH v4 vH d4 v6 vH v6 vH v6 vH v6 vH 2 v v þ þ þ6 2 2þ þ 15 4 2 þ 15 2 4 þ  Cdk þ Cd⊥ d þ 12 vz4 360 vz6 vy2 vz2 vz vy vy4 vz vy vz vy vy6 " !#) !   v2 wH d2 v4 wH v4 wH d4 v6 w H v6 wH v6 wH þ 3 5 þ 10 3 3 þ 3 þ ¼ 0;  Cdk  Cd⊥ d2 2 þ vyvz 3 vyvz3 vy3 vz 180 vy vz vyvz5 vy vz vjH v2 vH z þ k vx vx2

!

(

 v 2 jH   v3 wH  y þ a2 I6  aI4 X I2  2aI4 þ a2 I6 2 vt vt 2 vx

   v 2 jH   v3 wH vwH y 2  J þ  2aJ þ a J þ aJ4  a2 J6 ¼ 0; þ k jH 2 4 6 y 2 vx vx vx3

! (  v3 jH v3 wH v2 wH  v4 wH y  a2 I6 2 2 X I0 2  a2 I6  aI4 3 2 3 2 vx vx vx vx vt vt vx vt vx ! ! 2 H 4 H 6 H 8 H 2 H 4 H 2 4 6 2 4 d v w d v w d v w d v w d v6 w H d6 v8 wH 2 v w 2 v w þ Cvk d þ þ þ þ þ þ  Cv⊥ d 12 vz4 360 vz6 20160 vz8 12 vy4 360 vy6 20160 vy8 vz2 vy2 " ! !#   2 H v2 wH d2 v4 wH v4 wH v4 wH d4 v6 wH v6 wH v6 wH v6 wH 2 v w þ þ þ6 2 2þ þ 15 4 2 þ 15 2 4 þ þ  Cdk þ Cd⊥ d 12 vz4 360 vz6 vy2 vz2 vz vy vy4 vz vy vz vy vy6 " !#) !   v2 vH d2 v4 vH v4 vH d4 v6 v H v6 vH v6 vH  Cdk  Cd⊥ d2 2 þ 3 5 þ 10 3 3 þ 3 þ ¼ 0: þ vyvz 3 vyvz3 vy3 vz 180 vy vz vyvz5 vy vz k

vjH y

þ

v2 wH vx2

!

)

 aJ4

v 3 jH y

þ a2 J6

v2 jH y

(48b)

(48c)

þ

(48d)

By introducing Eq. (25) to Eqs. (48a)e(48d), the dimensionless continuous governing equations describing transverse motion of ensembles of SWCNTs on the basis of the NHOBT are derived as follows:

n v 2 jH 3 H 2v v z X  g 6 vt2 vt2 vx

)

 H vvH v2 j v3 vH H  g28 2z þ g29 3 ¼ 0; þ g27 jz þ vx vx vx

(49a)

( ! 3 H 4 H 2 H v2 vH ðkdÞ2 v4 vH ðkdÞ4 v6 vH ðkdÞ6 v8 vH 2 H v v 2 v jz 2 v v þ þX þ g1 2  g2 þ ðkdÞ C vk þ þ þ   12 vh4 360 vh6 20160 vh8 vt2 vt vx vh2 vx vx4 vt2 vx2 ! ! " 2 4 6 2   H 2 H 4 H 4 H 2 H v2 vH d v4 vH d v6 vH d v8 vH v2 vH d v4 vH H 2v v 2 v v 4v v þ þ þ C þ C þ þ þ 6k þ k   d C v⊥ k dk d⊥ 12 vg4 360 vg6 20160 vg8 12 vg4 vg2 vh2 vg2 vg2 vh2 vh4 " !# ! 4  v2 wH d2 v4 wH  H 6 H 6 H 6 H 4 H d v6 vH H 2 v v 4 v v 6v v 2v w 2 þ þ 15k þ 15k þ k C  C þ k  k þ dk d⊥ 360 vg6 vhvg 3 vhvg3 vg4 vh2 vg2 vh4 vh6 vh3 vg !#) 4 d v6 wH v6 wH v6 wH 3k4 5 þ 10k2 3 3 þ 3 ¼ 0; þ 180 vh vg vhvg5 vh vg H

g23

vjz v2 vH þ vx vx2

!

v g24

3 H jz 3

v4 vH

(49b) )   2 H H 3 H 3 H v jy v 2 jy vwH H 2v w 2 2 2v w  g X  g j þ þ g ¼ 0; þ g y 7 6 8 9 vx vt2 vt2 vx vx2 vx3

(49c)

( ! H 2 4 6 4 H 2 H v 3 jy 2 H v w v2 wH d v4 wH d v6 wH d v8 wH 2 2 v w þ þ þX þ g1 2  g2 þ d C vk þ þ þ vx 12 vg4 360 vg6 20160 vg8 vt2 vt vx vg2 vx2 vx3 vx4 vt2 vx2 ! ! " 2   H 2 H 4 H 4 H v2 wH ðkdÞ2 v4 wH ðkdÞ4 v6 wH ðkdÞ6 v8 wH v2 wH d v4 wH H 2 H 2v w 2 v w 4v w þ þ þ þ þ þ 6k þk  C dk þ C d⊥ k  ðkdÞ C v⊥ 12 vh4 360 vh6 20160 vh8 12 vg4 vh2 vh2 vg2 vg2 vh2 vh4 !# ! " 4  v2 vH d2 v4 vH  H 6 H 6 H 6 H d v6 wH v4 vH H 2 v w 4 v w 6v w þ þ 15k þ 15k þ k C  C þ k2 3  k þ 2 dk d⊥ 4 2 2 4 6 3 360 vg6 vhvg 3 vg vh vg vh vh vhvg vh vg !#) 4 6 H 6 H 6 H d v v v v v v ¼ 0; 3k4 5 þ 10k2 3 3 þ 3 þ 180 vh vg vhvg5 vh vg H

 g23

vjy

v2 wH

!

H

 g24

v 3 jy

v4 wH

(49d)

K. Kiani / Current Applied Physics 14 (2014) 1116e1139

1135

where

H

C d½:: ¼ H jz

¼

Cd½:: d2 l4b a2 J6 l2z

; vH ¼

H jz ðx; h; g; tÞ; vH

vH H wH H H H ;w ¼ ; jy ¼ jH y ; jz ¼ jz ; ½:: ¼k or ⊥; lb lb

H

¼v

H ðx; h; g; tÞ; jy

¼

H jy ðx; h; g; tÞ; wH

¼ w ðx; h; g; tÞ:

5.3.2. Frequency analysis of ESWCNTs using continuous NHOBT The continuous deformation fields of the ensemble of SWCNTs based on the NHOBT are discretized in terms of appropriate mode shape functions as follows: pw NX mw N nw N X X

H

jz ðx; h; g; tÞ ¼

jz H jzmnp ðtÞfmnp ðx; h; gÞ;

m¼1 n¼1 p¼1

N pv mv N nv N X X X

vH ðx; h; g; tÞ ¼

v vH mnp ðtÞfmnp ðx; h; gÞ;

m¼1 n¼1 p¼1 H jy ðx; h; g; tÞ

¼

N pv mv N nv N X X X m¼1 n¼1 p¼1

wH ðx; h; g; tÞ ¼

(51)

jy H jymnp ðtÞfmnp ðx; h; gÞ;

pw NX mw N nw N X X

(50)

H

w wH mnp ðtÞfmnp ðx; h; gÞ;

m¼1 n¼1 p¼1

where the mode shapes of the ensemble which are admissible with the geometric boundary conditions are as:

fvmnp ðx; h; gÞ ¼ fw mnp ðx; h; gÞ ¼ sinðmpxÞsinðnphÞsinðppgÞ; jy jz fmnp ðx; h; gÞ ¼ fmnp ðx; h; gÞ ¼ cosðmpxÞsinðnphÞsinðppgÞ: (52)

where the elements of the mass and stiffness matrices could be readily derived. 6. Results and discussion In this part, the efficiency of the proposed nonlocal continuous models in predicting the results of the nonlocal discrete models are firstly addressed. Subsequently, the roles of geometry, population, and small-scale data associated with ensembles of SWCNT on their flexural frequencies are investigated via the nonlocal continuous models. For this purpose, the constitutive SWCNTs of the ensemble are considered with the following properties [72]: Eb ¼ 1012 Pa, nb ¼ 0.2, rb ¼ 2500 kg/m, e0a ¼ 2 nm, and d ¼ 2rm þ tb. In all plotted results in this part, the predicted results by the NRBT, NTBT, and NHOBT are demonstrated by the dotted, dashed, and solid lines, respectively. Since the NHOBT provides a more realistic prediction of variation of shear stress across the cross sections of the constitutive SWCNTs of the ensemble, its results would be more accurate than those of the NRBT and NTBT. In the absence of experimentally observed data, the capabilities of the proposed models on the basis of the NRBT and NTBT in predicting natural frequencies of the nanostructure are checked through verifying their obtained results with those of the models based on the NHOBT.

H

By premultiplying both sides of Eqs. (49a)-(49d) in order by djz , H dv , djy , dwH , and taking the required integration by parts, H

2h

i H jz jz Mb 6 h H ivjz 6 Mb 6h i 6 MH jy jz 4 b h H iwjz Mb

2h

h H ijz v Mb h H ivv Mb h H ijy v Mb h H iwv Mb

i H jz jz Kb 6 h H ivjz 6 Kb þ 6 h ij j 6 KH y z 4 b h H iwjz Kb 8 9 0> > > < > = 0 ¼ ; 0> > > : > ; 0

i H jz jy Mb h H ivjy Mb h H ijy jy Mb h H iwjy Mb h

h H ijz v Kb h H ivv Kb h H ijy v Kb h H iwv Kb

h

z

Mb h H vw Mb 7 h H i jy w 7 Mb 5 h H iww Mb

i H jz jy Kb h H ivjy Kb h H ijy jy Kb h H iwjy Kb h

H

8 J ij w > vt > 7> >

2

2

2 H

vt2

v J > > > vt > > :v w

h

H z

2

H y

2

2

3

H

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

vt2

i H jz w Kb 8 9 H h H ivw 7> Jz > > = 7< H > Kb 7 vH h H ijy w 7 J > > Kb : Hy > ; 5> w h H iww Kb

(53)

6.1. Comparison of the obtained results with those of other works In order to check the accuracy of the proposed models, a comparison study is conducted. For this purpose, we consider doublenanobeam-systems in which their free transverse vibrations have been investigated by Murmu and Adhikari [84]. The constitutive nanobeams of the system are SWCNTs in which their geometry and mechanical properties are given in Ref. [84]. The predicted dimensionless frequencies as a function of the dimensionless small-scale parameter by Murmu and Adhikari [84] and those of the proposed discrete models are provided in Fig. 3. As it is seen in Fig. 3, there is a good accuracy between the predicted results of all models and those of Murmu and Adhikari [84]. Since the considered beam-like system has a large slenderness ratio, therefore, the results of the nonlocal shear deformable beam theories (NSDBTs) are fairly coincident with those of the NRBT. 6.2. The capabilities of the proposed nonlocal continuous models To ensure regarding the validity of the predicted results by the nonlocal continuous models, such results should be appropriately checked. To this end, the predicted fundamental frequencies of the ESWCNTs by the discrete models and those of the continuous models are provided in Table 1. These results are given for three levels of the slenderness ratio (i.e., l ¼ 10, 15, and 30) and four levels of the number of SWCNTs of the ensemble (i.e., Ny ¼ Nz ¼ 5, 7, 9, and 11). As it is obvious from Table 1, there is a reasonably good

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Fig. 7. Variation of the fundamental frequency as a function of the population of the ensemble for different levels of the slenderness ratio: ((…) NRBT, () NTBT, (d) NHOBT; (B) l ¼ 10, (,) l ¼ 15, (D) l ¼ 30).

agreement between the results of the continuous models and those of the discrete ones for all levels of the slenderness ratio and the population of the ensemble. In most of the cases, the continuous models can capture the results of the discrete ones up to five significant digits. Furthermore, by increasing the slenderness ratio, the discrepancies between the results of various models would lessen since the effect of shear deformation within the nanostructure decreases. As the number of the constitutive SWCNTs of the ensemble increases, the fundamental frequency decreases. Such a fact is more obvious for more slender ESWCNTs. A more detail explanations regarding the effects of slenderness ratio and ensemble's population on frequencies will be given in the upcoming parts. 6.3. Numerical studies 6.3.1. The role of the slenderness ratio on the fundamental flexural frequency Using proposed nonlocal continuous models, the influence of the slenderness ratio on free dynamic behavior of the nanostructure is of concern. The plots of fundamental frequency of the ensemble of SWCNTs in terms of the slenderness ratio of the

nanostructure are given in Figs. 4. The predicted results are demonstrated for a highly populated ensemble (i.e., Ny ¼ Nz ¼ 1000) with three levels of the small-scale parameter (i.e., e0a ¼ 0, 1, and 2 nm). According to Fig. 4, the fundamental flexural frequency of the ESWCNTs would lessen as the slenderness ratio increases. Furthermore, the fundamental frequency would generally decrease as the influence of the small-scale parameter becomes highlighted. In an upcoming part, the effect of the small-scale parameter on the frequency of the nanostructure will be examined in some details. In most of the cases, the predicted results by the NHOBT are between the results of the NRBT and those of the NTBT. By increasing the slenderness ratio, the discrepancies between the predicted frequencies by the proposed nonlocal continuous models would lessen. It is mainly attributed to the reduction of the ratio of shear elastic energy to the total strain energy of the nanostructure. A more close scrutiny of the obtained results shows that such discrepancies are trivially affected by the small-scale parameter. The plotted results display that the NRBT overestimates the results of the NHOBT for l > 10 with relative error lower than 10 percent. Additionally, for l > 11, the NTBT underestimates the results of the NHOBT with relative error lower than 5 percent for all levels of the small-scale parameter.

Fig. 8. Variation of the fundamental frequency as a function of the intertube distance for different levels of the slenderness ratio: (a) Ny ¼ Nz ¼ 7, (b) Ny ¼ Nz ¼ 1000; ((…) NRBT, () NTBT, (d) NHOBT; (B) l ¼ 10, (,) l ¼ 15, (D) l ¼ 20).

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6.3.2. The role of the mean radius of the constitutive SWCNTs of the ensemble on the fundamental flexural frequency Another interesting scrutiny has been carried out to determine the influence of the radius of the SWCNTs of the ensemble on its fundamental frequency. For three levels of the SWCNT's length (i.e., lb ¼ 10, 15, and 30 nm), the predicted fundamental flexural frequencies as a function of the radius of SWCNTs are demonstrated in Fig. 5. Generally, the fundamental frequency increases with the radius of SWCNTs. The main reason of this fact is that the vdW forces between two adjacent nanotubes would increase as theirs radius magnify. As a result, the flexural stiffness of the nanostructure would grow with the radius of the SWCNT. Such a fact is more obvious for stocky ESWCNTs. It implies that the fundamental frequency of stockier ESWCNTs is more influenced by the variation of the radius of SWCNTs. Concerning the capabilities of the proposed nonlocal continuous models in predicting the fundamental frequency of the nanostructure, the discrepancies between the proposed models commonly magnify as the radius of SWCNTs increases. Further, this matter is more apparent for ESWCNTs with lower lengths. Generally, the predicted results by the NTBT are closer to those of the NHOBT. For the considered nanostructure with lb ¼ 10, 15, and 30 nm, the NRBT(NTBT) can capture the results of the NHOBT with relative error lower than 6(3), 18(9.5), and 29.5(13.5) percent, respectively. 6.3.3. The role of the small-scale parameter on the fundamental flexural frequency The impact of the small-scale parameter on the free vibration behavior of the ESWCNTs is of interest. The predicted results by the proposed nonlocal continuous models are plotted in Fig. 6. Such plots display variation of the fundamental frequency in terms of the small-scale parameter for a highly populated ensemble with different levels of the slenderness ratio. As it is obvious from Fig. 6, all proposed models predict that the fundamental frequency would decrease as the small-scale parameter increases. Such a reduction behavior is also more obvious for stockier ESWCNTs. For nanostructures with higher levels of the slenderness ratios (i.e., l ¼ 30), variation of the small-scale parameter has a slight influence on the variation of the fundamental frequencies. A close examination of the obtained results reveals that the discrepancies between the results of the proposed models are trivially affected by the small-scale parameter. This fact holds true for all levels of the slenderness ratio of the nanostructure. For l ¼ 10, 15, and 30, the predicted results explain that the NRBT overestimates the results of the NHOBT with relative error lower than 10.2, 5.1, and 1.4 percent, respectively. Additionally, for such levels of the slenderness ratio, the NTBT in order underestimates the predicted frequency by the NHOBT with relative error lower than 5.85, 3.2, and 0.95 percent. 6.3.4. The role of the ensemble's population on the fundamental flexural frequency The influence of the number of constitutive SWCNTs of the ensemble on the free vibration behavior of ESWCNTs is of high interest. To this end, the predicted fundamental frequencies by the proposed nonlocal continuous models as a function of the number of SWCNTs along one side of the ensemble are demonstrated in Fig. 7 for three levels of the slenderness ratio (i.e., l ¼ 10, 15, and 30). As it is seen in Fig. 7, the predicted fundamental frequencies by the proposed models would monotonically decrease as the population of the ESWCNTs increases. However, for higher levels of the population, variation of the population would have a trivial influence on the variation of the fundamental frequency. It is also worth mentioning that the ESWCNTs with higher levels of the slenderness ratio are more influenced by the variation of the ensemble's

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population. A close scrutiny of the obtained results shows that the discrepancies between the proposed models generally magnify as the ensemble's population grows. However, such discrepancies would follow with a lower rate for highly populated ensembles. In the case of Ny ¼ 5(15), the NRBT can reproduce the results of the NHOBT with relative error lower than 6.1(9.7), 1.5(4.4), and 0.35(0.45) percent for l ¼ 10, 15, and 30, respectively. Further, the NTBT in order predicts the results of the NHOBT with relative error lower than 3.4(5.6), 0.95(2.7), and 0.03(0.3) for the abovementioned levels of the slenderness ratio as well as population of the ensemble.

6.3.5. The role of the intertube distance on the fundamental flexural frequency A fascinating scrutiny has been conducted to disclose the role of the intertube distance on the free dynamic response of ESWCNTs. In Fig. 8(a) and (b), the plots of the fundamental frequency as a function of intertube distance for three levels of the slenderness ratio (i.e., l ¼ 10, 15, and 20) have been demonstrated. Fig. 8(a) displays the plotted results for a lowly populated ensemble (i.e., Ny ¼ 7) whereas Fig. 8(b) shows the demonstrated results for a highly populated one (i.e., Ny ¼ 1000). For a lowly populated ensemble, the plots associated with various levels of the slenderness ratio consist of three obvious branches (see Fig. 8(a)). In the first branch, all proposed nonlocal continuous models predict that the fundamental frequency drastically decreases with the intertube distance. All plotted results take their relative minimum values at d z 2.37 nm. In the second branch, the fundamental frequency magnifies with the intertube distance up to d z 2.41 nm where the plots take their relative maximum points. Finally, the last branch of the plotted results displays that the fundamental frequency of the ESWCNTs would slightly decrease with the intertube distance. For highly populated ESWCNTs, variation of the intertube distance has a trivial effect on the variation of the fundamental frequency (see Fig. 8(b)). The main reason of this fact can be sought in the mechanism of the intertube vdW forces in the first mode of vibration. The mode shape functions pertinent to transverse motion of such a mode are [.] ¼ [.]0sin (px)sin (ph)sin (pg) where ½: ¼ v or w. For a highly populated ensemble in the first vibration mode, the discrepancies between the transverse displacements of two adjacent tubes are fairly negligible. As a result, the atoms of the neighboring SWCNTs would be interacted by tiny vdW forces. However, for a lowly populated ensemble, the discrepancies between the transverse displacements associated with the first vibration mode would not be negligible at all since there exists a low number of tubes along each side. Thereby the vdW interaction forces between two tubes would be highly greater than those of the highly populated ensemble in the first vibration mode. Further studies reveals that the overall trend of the plot of Cvk , the most strength component of the vdW force, in terms of the intertube distance is fairly identical to that of the plot of fundamental frequency as a function of intertube distance for lowly populated ensemble. Such evidences explain the reason behind the trend of the plots in Fig. 8(a) and (b). Regarding the lowly populated ensemble, the discrepancies between the results of various beam models generally increase with the intertube distance. For l ¼ 10, 15, and 20, the NRBT (NTBT) can capture the results of the NHOBT with relative error lower than 10(5.8), 5(3.1), and 2.8(1.8) percent, respectively. Concerning the highly populated ESWCNTs, the discrepancies between the predicted results by the proposed models trivially change with the intertube distance. For the above-mentioned levels of the slenderness ratio, the NRBT(NTBT) in order could predict the predicted results by the NHOBT with the relative error lower than 10.2(5.9), 5.1(3.2), and 3(2.0) percent.

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7. Conclusions Free dynamic analysis of ensembles of vertically aligned SWCNTs is of concern. By employing the Hamilton's principle, the discrete equations of motion corresponding to their transverse vibrations are extracted based on the nonlocal Rayleigh, Timoshenko, and higher-order beam theories. In the case of simply supported SWCNTs with immovable exterior ones, the frequency analysis of the nanostructure is performed. For a high number of SWCNTs within the ensemble, the application of the discrete models requires a large amount of computational effort. Thereby, alternative continuous models are developed and their efficiency is checked through various numerical examples. By using these novel models, the effects of the slenderness ratio, intertube distance, radius of SWCNTs, small-scale parameter, and population of the ensemble on the fundamental flexural frequency of ESWCNTs are addressed. The major obtained results are as follows:  The fundamental frequency of the nanostructure decreases with the slenderness ratio. In most of the cases, the predicted frequencies by the NRBT and NTBT are, respectively, the upper and lower bonds of those of the NHOBT. As the slenderness ratio increases, the shear strain energy vanishes. Therefore, the discrepancies between various models would decrease.  The fundamental frequency increases with the radius of the constitutive SWCNTs of the ensemble. This fact is more apparent for stockier ESWCNTs. It is mainly attributed to the increasing of the vdW forces between adjacent tubes.  The fundamental frequency reduces as the influence of the small-scale parameter becomes highlighted. Such a fact is more apparent for stockier ESWCNTs. It is mainly related to the incorporation of the small-scale parameter into the shear strain energy of the nanostructure.  The fundamental frequency decreases as the population of the ensemble increases. Additionally, the frequencies of stockier ESWCNTs are lesser affected by the ensemble's population. The discrepancies between the results of the nonlocal continuous models would generally increase with the population of the ensemble.  The influence of the intertube distance on the fundamental frequency is highly affected by the population of the ensemble. For highly populated ensembles, variation of the intertube distance has a trivial effect on the variation of the fundamental frequency of the nanostructure. However, for lowly populated ensembles, three distinct zones for the plots of frequency in terms of intertube distance can be detected. Acknowledgments The financial support of the Iranian National Science Foundation (INSF) under the Grant No. 92032128 is gratefully acknowledged. The author would also like to express his gratitude to the anonymous reviewers for their fruitful comments in which lead to the improvement of the present work. References

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