TECHNICAL PAPER

Elastic waves in uniformly infinite-periodic jungles of single-walledcarbon nanotubes under action of longitudinal magnetic fields

Keivan Kiani1

Received: 14 April 2018 / Accepted: 30 August 2019� The Brazilian Society of Mechanical Sciences and Engineering 2019

AbstractExploring applicable ways to control characteristics of transverse waves in periodic jungles of single-walled carbon

nanotubes (SWCNTs) has been of interest to nanotechnologists and applied mechanics community. Herein, the theoretical-

mechanical aspects of the influence of the longitudinal magnetic field on such highly conductive nanosystems are going to

be examined. Using nonlocal Rayleigh and Timoshenko beam models, the discrete and continuous versions of equations of

motion of magnetically affected nanosystems are derived. Commonly, the discrete models suffer from both labor costs and

computational efforts for highly populated nanosystems. To conquer these special deficiencies of discrete nanosystems,

appropriate continuous models have been established and their efficiency in capturing frequencies of discrete models is

proved. The roles of wavenumber, radius of SWCNTs, magnetic field strength, nonlocality, and intertube distance in

flexural and shear frequencies as well as their corresponding phase and group velocities are displayed and discussed. The

obtained results confirm this fact that the longitudinal magnetic field could be employed as an efficient way to control

characteristics of both flexural and shear waves in periodic jungles of SWCNTs.

Keywords Periodic jungles of SWCNTs � Longitudinal magnetic field � Control of transverse waves � Nonlocal elasticitytheory � Assumed mode method

1 Introduction

Due to the astonishing mechanical strength, electrical, and

thermal properties of carbon nanotubes (CNTs), exploiting

them as the reinforcing phase of polymer matrix has been

substantially increased in recent years. It has been revealed

experimentally that mechanical behavior as well as the elec-

trical conductivity of the polymer is enhanced by introducing

CNTs into the matrix [1–5]. On the other hand, there exist

evidences that the polymer CNT-based composites with

aligned nanotubes exhibit superior mechanical and electrical

properties [6–8]. Application of the magnetic field to a group

of CNTs leads to improvement of their arrangement such that

they are moderately aligned along the direction of the

magnetic field [9–11]. Further, the mechanical behavior of the

resulted nanocomposite is enhanced by applying the magnetic

field [12, 13]. As a result, understanding vibrations of mag-

netically affected single-walled carbon annotates (SWCNTs)

is a crucial step in better realization of mechanical behavior of

magnetically affected nanocomposites. In this view, this work

is devoted to explore physical characteristics of elastic waves

in vertically aligned ensembles of SWCNTs with three-di-

mensional configuration.

Since characteristics of transverse waves within periodic

jungles of SWCNTs are mostly of interest, each nanotube is

modeled on the basis of the Rayleigh or Timoshenko beam

theory in this study. To incorporate the nonlocality into the

proposed models, the nonlocal elasticity theory of Eringen

[14–16] is adopted. The main feature of the nonlocal con-

tinuum theory (NCT) with respect to the classical continuum

theory (CCT) is the appearance of the small-scale parameter

in the nonlocal formulations of the problem. This parameter

affects the natural frequencies and mechanical response of the

nanosystem, and its value is commonly evaluated by com-

paring the obtained dispersion curves by the proposed non-

local model with those of an atomic methodology. Up to now,

Technical Editor: Wallace Moreira Bessa, D.Sc.

& Keivan Kiani

k_kiani@kntu.ac.ir; keivankiani@yahoo.com

1 Department of Civil Engineering, K. N. Toosi University of

Technology, P.O. Box 15875-4416, Valiasr Ave., Tehran,

Iran

123

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418 https://doi.org/10.1007/s40430-019-1897-2(0123456789().,-volV)(0123456789().,-volV)

free and forced vibrations of individual CNTs [17–28] as well

as elastic transverse wave characteristics in them [29–39]

have been widely scrutinized using nonlocal beams. Never-

theless, the undertaken works on vibrations of a group of

CNTs are restricted to several works and the need for further

explorations is highly needed. With regard to the improve-

ment effect of the magnetic field on the mechanical behavior

of aligned CNTs, herein, the author tries to develop several

appropriate nonlocal models to display transverse sound

waves characteristics within such magnetically affected tiny

elements.

Concerning mechanical modeling of membranes and jun-

gles of vertically aligned SWCNTs, Kiani [40] studied forced

vibrations of two- and three-dimensional configurations of

ensembles of SWCNTs. The potential lateral dynamic insta-

bility of these nanostructures under harmonically transverse

distributed loads was explained via nonlocal Rayleigh beam

theory. The axial buckling of slender groups of aligned

SWCNTs was researched by Kiani [41] via nonlocal continu-

ous and discrete models on the basis of the Rayleigh beam. In

another work, the in-plane and out-of-plane vibrations of

membranes made from vertically aligned SWCNTs were

addressed theoretically using appropriate nonlocal Rayleigh,

Timoshenko, andhigher-order beam theories [42].Basedon the

nonlocal continuousmodels, the fundamental in-plane and out-

of-plane frequencieswere analytically displayed and the role of

influential factors on them is discussed. Kiani [43] explored

nonlocal transverse vibrations of vertically aligned jungles of

SWCNTs by exploiting nonlocal Rayleigh, Timoshenko, and

higher-order beam models. To this end, the developed discrete

models were generalized. Then, newly continuous nonlocal

models were introduced and their capabilities in capturing the

results of nonlocal discrete models were explained in some

detail. Further, Kiani [44] examined nonlocal column buckling

ofmembranes fromperiodic-alignedSWCNTsvia discrete and

continuous models. For this purpose, nonlocal Rayleigh,

Timoshenko, and higher-order beams were employed and the

explicit expressionsof critical buckling loadsof thenanosystem

were evaluated. Recently, free transverse vibrations of in-

plane-aligned membranes of SWCNTs immersed in longitu-

dinal magnetic fields have been cultivated by Kiani [45] using

nonlocal Rayleigh, Timoshenko, and higher-order beam theo-

ries. The role of nonlocality and shear deformation as well as

other crucial factors on the fundamental frequency of the

nanosystem was addressed. As it is seen from the existing lit-

erature, vibrations of—and characteristics of elasticwaves in—

vertically aligned periodic jungles of SWCNTs in the presence

of longitudinal magnetic field have not been displayed yet.

In this work, studying characteristics of transverse waves

within three-dimensional periodic jungles of SWCNTs acted

upon by a longitudinal magnetic field is of great interest. For

this purpose, the vdW forces due to transverse motions of each

pair of nanotubes with infinite length are appropriately

calculated via a linear model. In fact, such interactional forces

betweenadjacent tubes are idealizedvia appropriate continuous

linear-virtual springs. By modeling of each nanotube via

appropriate beammodels, we confront a complex beam-spring

system. In the context of the nonlocal continuum theory of

Eringen [14, 15], the nonlocal equations of motion pertinent to

transverse vibration of the nanosystem are obtained using

Rayleigh and Timoshenko beams in the light of Hamilton’s

principle. These are called nonlocal discrete models (NDMs)

since the governing equations for each nanotube of the

nanosystem should be explicitly provided. To reduce the

computational efforts of such models, nonlocal continuous

models (NCMs) are presented. Flexural and shear frequencies

as well as corresponding phase and group velocities of elastic

waves withinmagnetically affected nanosystem are calculated,

and the roles of influential factors on thesewaves characteristics

are explained. The present paper could be regarded as a basic

work for better realizing of mechanical behavior of magneti-

cally affected nanotubes clusters as a vial part of advanced

micro-/nanoelectromechanical systems (MEMS/NEMS)

which are one of the promising applications of vertically

aligned periodic jungles of SWCNTs.

2 The details of the under-investigationproblem

Consider a periodic array of SWCNTs of infinite length

whose intertube distance along both y axis and z axis and its

radius in order are denoted by d and rm as shown in Fig. 1a.

The array consists of Ny and Nz tubes along the y and z axes,

respectively, and it is under action of a longitudinal magnetic

field of strength Hx. To model each nanotube via the nonlocal

elasticity theory of Eringen, an equivalent continuum struc-

ture (ECS) with wall’s thickness tb ¼ 0:34 nm is employed

[46]. The main geometrical data of such an infinite-length

ECS are the mean radius, cross-sectional area, and second

moment inertia; these factors are represented by rm, Ab, and

Ib, respectively. Furthermore, the most important mechanical

properties of ECS used in calculations are their density,

Poisson’s ratio, Young’s modulus, shear elastic modulus

which are, respectively, denoted by qb, mb, Eb, and Gb.

3 Modeling vdW forces between twoadjacent infinite-length SWCNTs

The SWCNTs of the infinite-periodic jungle interact tightly

with each other through the intertube vdW forces. The inter-

atomic 6-12 potential of Lennard-Jones for a pair of two neutral

atoms is expressed by: UijðkÞ ¼ 4�rk

� �12� r

k

� �6� �, where

418 Page 2 of 25 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418

123

k denotes the distance between ith atom and jth one, r ¼ raffiffiffi26

p

represents the distance at which the potential is zero, ra is the

distance atwhich the potential reaches itsminimum, and � is the

depth of the potential well. The vdW force between atom i and

atom j is given by: f ij ¼ � dUdk

ek, in which ek is the unit vector

associated with the position vector k. For example, let

ðx1; rm cos/1; rm sin/1Þ be the coordinate of the ith atomof an

arbitrary tube, and ðx2; rm cos/2; d þ rm sin/2Þ be that of thejth atom of its nearest tube. The position vector accounting for

transverse vibrations of these neighboring tubes is provided by

[40, 43]:

k~¼ x2 � x1ð Þ ex þ rm cosu2 � cosu1ð Þ � DVð Þ eyþ rm sinu2 � sinu1ð Þ þ d � DWð Þ ez;

ð1Þ

where DWðx; tÞ ¼ W1ðx; tÞ �W2ðx; tÞ, DVðx; tÞ ¼ V1ðx; tÞ�V2ðx; tÞ are the relative transverse displacements along

the y and z axes, respectively, W1ðx; tÞ=V1ðx; tÞ and

W2ðx; tÞ=V2ðx; tÞ are the transverse displacements of these

tubes along the z / y axis, ex, ey, and ez are the unit base

vectors pertinent to the rectangular coordinate system. By

introducing the position vector to the above-mentioned

definition of the vdW force, and by taking the integral of

such a force over the surfaces of the tubes for an arbitrary

length L, the components of the total vdW force in

Cartesian coordinate system are evaluated by:

Hx

d

x

d

y

z d

d

(a)

V(m+1)(n+1)

(x,t)

C v||

& C v⊥

( m+1)th row

C d||

& C d⊥

nth column

C v||

& C

v⊥

y

W(m+1)(n+1)

(x,t)

mth row

( m−1)th row

( n+1)th column ( n−1)th column z

(b)

Fig. 1 a Schematic

representation of a three-

dimensional periodic array of

SWCNTs immersed in a

longitudinal magnetic field; b a

representation of the

continuum-based discrete model

of the magnetically affected

nanosystem

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418 Page 3 of 25 418

123

Fx ¼24�r2CNTr

2m

r2L

Z L

0

Z 1

�1

Z 2p

0

Z 2p

0

2rk

� �14� r

k

� �8� �

x2 � x1ð Þ du1 du2 dx1 dx2;

Fy ¼24�r2CNTr

2m

r2L

Z L

0

Z 1

�1

Z 2p

0

Z 2p

0

2rk

� �14� r

k

� �8� �

rm cosu2 � cosu1ð Þ � DVð Þ du1 du2 dx1 dx2;

Fz ¼24�r2CNTr

2m

r2L

Z L

0

Z 1

�1

Z 2p

0

Z 2p

0

2rk

� �14� r

k

� �8� �

rm sinu2 � sinu1ð Þþd � DW

� �du1 du2 dx1 dx2;

ð2Þ

where rCNT ¼ 4ffiffiffi3

p

9a2is the carbon atoms’ surface density,

and a represents the bond length of carbon–carbon. By

linear modeling of variation of vdW force due to the rel-

ative transverse displacements of two adjacent tubes, the

components of such a crucially interactional force are

derived as follows:

MFy ¼ Cv? MV;

MFz ¼ Cvk MW ;ð3Þ

where

Cv?ðrm; dÞ ¼ � 256 � r2m9a4

�Z 2p

0

Z 2p

0

r12 !1K�13 � 14!2K

�15 rm cosu2 � cosu1ð Þð Þ2h i

�

r6

2!3K

�7 � 8!4K�9 rm cosu2 � cosu1ð Þð Þ2

h i

8><>:

9>=>;

du1 du2;

ð4aÞ

Cvkðrm; dÞ ¼ � 256 � r2m9a4

�Z 2p

0

Z 2p

0

r12 !1K�13 � 14!2K

�15 d þ rm sinu2 � sinu1ð Þð Þ2h i

�

r6

2!3K

�7 � 8!4K�9 d þ rm sinu2 � sinu1ð Þð Þ2

h i

8><>:

9>=>;

du1 du2;

ð4bÞ

and

!1 ¼231p1024

;!2 ¼429p2048

;!3 ¼5p16

;!4 ¼35p128

; ð5aÞ

Kðu1;u2; rm;dÞ

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2r2m 1� cosðu2�u1Þð Þþ d2þ 2rmd sinu2� sinu1ð Þ

q:

ð5bÞ

The constants pertinent to the vdW forces between diag-

onal tubes with the intertube distance dffiffiffi2

pare represented

by Cd? and Cdk (see Fig. 1b).

4 Establishment of NDMs using NRBMand NTBM

In discrete modeling of the problem, the governing equa-

tions of each SWCNT are displayed individually by con-

sidering its transverse dynamic interactions with the

neighboring tubes. In this section, based on the NRBM and

NTBM, the equations of motion of transverse vibrations of

the three-dimensional periodic array of SWCNTs in the

presence of a longitudinal magnetic field are obtained via

NDMs. Subsequently, by considering a harmonic version

of the dynamic deformations, the frequencies and their

corresponding phase and group velocities in terms of the

wavenumber are evaluated.

4.1 Wave motion in magnetically affectedperiodic jungles of SWCNTs using discreteNRBM

According to the nonlocal Rayleigh beam model and in

view of Refs. [40, 43, 45], the kinetic energy, TRðtÞ, thetotal elastic strain energy of the magnetically affected

periodic array of vertically aligned SWCNTs, URðtÞ, andthe work done by the applied longitudinal magnetic field on

the nanosystem, WRðtÞ, are provided by:

TRðtÞ ¼ 1

2

XNy

m¼1

XNz

n¼1

Z 1

�1qb

Ab

oVRmn

ot

� �2

þ oWRmn

ot

� �2 !

þ Ibo2VR

mn

otox

� �2

þ o2WRmn

otox

� �2 !!

dx;

ð6aÞ

418 Page 4 of 25 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418

123

URðtÞ¼1

2

XNy

m¼1

XNz

n¼1

Z 1

�1

�o2VRmn

ox2Mnl

bzmn

� �R�o2WR

mn

ox2Mnl

bymn

� �Rþ

Cvk VRmn�VR

ðmþ1Þn

� �21�dmNy

þ VR

mn�VRðm�1Þn

� �21�d1mð Þ

� �þ

Cv? VRmn�VR

mðn�1Þ

� �21�d1nð Þþ VR

mn�VRmðnþ1Þ

� �21�dnNz

� �þ

Cdk XRmn�XR

ðm�1Þðn�1Þ

� �21�d1mð Þ 1�d1nð Þþ

Cdk XRmn�XR

ðmþ1Þðnþ1Þ

� �21�dmNy

1�dnNz

þ

Cdk YRmn�YR

ðm�1Þðnþ1Þ

� �21�d1mð Þ 1�dnNz

þ

Cdk YRmn�YR

ðmþ1Þðn�1Þ

� �21�dmNy

1�d1nð Þþ

Cd? XRmn�XR

ðm�1Þðnþ1Þ

� �21�d1mð Þ 1�dnNz

þ

Cd? XRmn�XR

ðmþ1Þðn�1Þ

� �21�dmNy

1�d1nð Þþ

Cd? YRmn�YR

ðm�1Þðn�1Þ

� �21�d1mð Þ 1�d1nð Þþ

Cd? YRmn�YR

ðmþ1Þðnþ1Þ

� �21�dmNy

1�dnNz

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

dx;

ð6bÞ

WR ¼XNy

m¼1

XNz

n¼1

Z 1

�1gAbH

2x

o2WRmn

ox2WR

mn þo2VR

mn

ox2VRmn

� �dx;

ð6cÞ

where dmn is the Kronecker delta tensor, VRmn and WR

mn are

transverse deflections of the (m, n)th tube along the y and

z directions, respectively, Mnlbymn

� �Rand Mnl

bzmn

� �Rdenote the

nonlocal bending moments of the (m, n)th SWCNT about the

y and z axis, g is the magnetic permeability of the nanotube,

XRmn ¼

ffiffiffi2

p

2WR

mn þ VRmn

, and YR

mn ¼ffiffiffi2

p

2�WR

mn þ VRmn

. By

employing NRBM, the above-mentioned bending moments

of the SWCNT based on the NRBT are related to their local

ones as follows:

N Mnlbymn

� �R� �¼ �EbIb

o2WRmn

ox2; ð7aÞ

N Mnlbzmn

� �R� �¼ �EbIb

o2VRmn

ox2; ð7bÞ

where N½:� is the nonlocal operator defined by:

N½:� ¼ ½:� � ðe0aÞ2 o2½:�=ox2, and e0a represents the small-

scale parameter. The value of this parameter is commonly

determined by comparing the predicted dispersion curves

by the nonlocal model and those of an appropriate atomic

method [47].

By implementing the Hamilton’s principle (i.e.,Z t2

t1

dTR � dUR þWR

dt ¼ 0 where t1 and t2 are the ini-

tial and final times, and the symbol d behind the energy

terms represents the variation sign), and introducing

Eqs. (7a) and (7b) to (6a)–(6c), the nonlocal governing

equations that display transverse vibrations of the mag-

netically affected periodic jungles of SWCNTs are derived

as:

EbIbo4VR

mn

ox4þ N qb Ab

o2VRmn

ot2� Ib

o4VRmn

ot2ox2

� ��

� gAbH2x

o2VRmn

ox2

Cvk VRmn � VR

ðmþ1Þn

� �1� dmNy

h

þ VRmn � VR

ðm�1Þn

� �1� d1mð Þ

i

þ Cv? VRmn � VR

mðn�1Þ

� �1� d1nð Þ

h

þ VRmn � VR

mðnþ1Þ

� �1� dnNz

i

þ 0:5Cdk WRmn þ VR

mn �WRðm�1Þðn�1Þ � VR

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5Cdk WRmn þ VR

mn �WRðmþ1Þðnþ1Þ � VR

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5Cdk VRmn �WR

mn � VRðmþ1Þðn�1Þ þWR

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

þ 0:5Cdk VRmn �WR

mn � VRðm�1Þðnþ1Þ þWR

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5Cd? VRmn �WR

mn þWRðm�1Þðn�1Þ � VR

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5Cd? VRmn �WR

mn þWRðmþ1Þðnþ1Þ � VR

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5Cd? VRmn þWR

mn � VRðm�1Þðnþ1Þ �WR

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5Cd? VRmn þWR

mn � VRðmþ1Þðn�1Þ �WR

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

¼ 0;

ð8aÞ

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418 Page 5 of 25 418

123

EbIbo4WR

mn

ox4þ N qb Ab

o2WRmn

ot2� Ib

o4WRmn

ot2ox2

� ��

þ gAbH2x

o2WRmn

ox2Cvk WR

mn �WRmðnþ1Þ

� �1� dnNz

h

þ WRmn �WR

mðn�1Þ

� �1� d1nð Þ

iþ Cv? WR

mn �WRðm�1Þn

� �1� d1mð Þ

h

þ WRmn �WR

ðmþ1Þn

� �1� dmNy

i

þ 0:5Cdk WRmn þ VR

mn �WRðm�1Þðn�1Þ � VR

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5Cdk WRmn þ VR

mn �WRðmþ1Þðnþ1Þ � VR

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5Cdk WRmn � VR

mn �WRðm�1Þðnþ1Þ þ VR

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5Cdk WRmn � VR

mn �WRðmþ1Þðn�1Þ þ VR

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

þ 0:5Cd? WRmn � VR

mn �WRðm�1Þðn�1Þ þ VR

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5Cd? WRmn � VR

mn �WRðmþ1Þðnþ1Þ þ VR

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5Cd? WRmn þ VR

mn �WRðm�1Þðnþ1Þ � VR

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5Cd? WRmn þ VR

mn �WRðmþ1Þðn�1Þ � VR

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

¼ 0:

ð8bÞ

Equations (8a) and (8b) display free transverse vibrations of

the constitutive SWCNTs of the magnetically affected peri-

odic array of SWCNTs on the basis of the NRBM. These

relations construct a set of 2NyNz partial differential equations

(PDEs) that should be appropriately solved. To this end, the

following dimensionless quantities are taken into account:

n ¼ x

tb;V

R

mn ¼VRmn

tb;W

R

mn ¼WR

mn

tb; c ¼ z

lz;

s ¼ 1

t2b

ffiffiffiffiffiffiffiffiffiffiEbIb

qbAb

st; l ¼ e0a

tb;

CR

v½:� ¼Cv½:�t

4b

EbIb;C

R

d½:� ¼Cd½:�t

4b

EbIb; k ¼ tbffiffiffiffiffiffiffiffiffiffiffi

Ib=Ab

p ;

d ¼ d

lz;H

R

x ¼ Hx

ffiffiffiffiffiffiffiffiffiffiffigAbt

2b

EbIb

s; ½:� ¼k or ? :

ð9Þ

By introducing Eqs. (9) to (8a) and (8b), the dimensionless

discrete-nonlocal equations of motion of the magnetically

affected ensemble of tubes with three-dimensional config-

uration are obtained as:

o4VR

mn

on4þ N

(o2V

R

mn

os2� k�2 o

4VR

mn

os2on2:

� HR

x

� �2 o2VR

mn

on2

þ CR

vk VR

mn � VR

ðmþ1Þn

� �1� dmNy

h

þ VR

mn � VR

ðm�1Þn

� �1� d1mð Þ

i

þ CR

v? VR

mn � VR

mðn�1Þ

� �1� d1nð Þ

h

þ VR

mn � VR

mðnþ1Þ

� �1� dnNz

i

þ 0:5CR

dk WR

mn þ VR

mn �WR

ðm�1Þðn�1Þ � VR

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5CR

dk WR

mn þ VR

mn �WR

ðmþ1Þðnþ1Þ � VR

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5CR

dk VR

mn �WR

mn � VR

ðmþ1Þðn�1Þ þWR

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

þ 0:5CR

dk VR

mn �WR

mn � VR

ðm�1Þðnþ1Þ þWR

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5CR

d? VR

mn �WR

mn þWR

ðm�1Þðn�1Þ � VR

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5CR

d? VR

mn �WR

mn þWR

ðmþ1Þðnþ1Þ � VR

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5CR

d? VR

mn þWR

mn � VR

ðm�1Þðnþ1Þ �WR

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5CR

d? VR

mn þWR

mn � VR

ðmþ1Þðn�1Þ �WR

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

)¼ 0;

ð10aÞ

418 Page 6 of 25 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418

123

o4WR

mn

on4þ N

(o2W

R

mn

os2� k�2 o

4WR

mn

os2on2:

� HR

x

� �2 o2WR

mn

on2

þ CR

vk WR

mn �WR

mðnþ1Þ

� �1� dnNz

h

þ WR

mn �WR

mðn�1Þ

� �1� d1nð Þ

i

þ CR

v? WR

mn �WR

ðm�1Þn

� �1� d1mð Þ

h

þ WR

mn �WR

ðmþ1Þn

� �1� dmNy

i

þ 0:5CR

dk WR

mn þ VR

mn �WR

ðm�1Þðn�1Þ � VR

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5CR

dk WR

mn þ VR

mn �WR

ðmþ1Þðnþ1Þ � VR

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5CR

dk WR

mn � VR

mn �WR

ðm�1Þðnþ1Þ þ VR

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5CR

dk WR

mn � VR

mn �WR

ðmþ1Þðn�1Þ þ VR

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

þ 0:5CR

d? WR

mn � VR

mn �WR

ðm�1Þðn�1Þ þ VR

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5CR

d? WR

mn � VR

mn �WR

ðmþ1Þðnþ1Þ þ VR

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5CR

d? WR

mn þ VR

mn �WR

ðm�1Þðnþ1Þ � VR

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5CR

d? WR

mn þ VR

mn �WR

ðmþ1Þðn�1Þ � VR

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

)

¼ 0; ð10bÞ

in which N½:� ¼ ½:� � l2 ½:�;nn.The elastic transverse waves in the (m, n)th SWCNT are

now considered as follows:

\VR

mn;WR

mn [ ¼ \VR

mn0;WR

mn0 [ ei -Rs�kx nð Þ; ð11Þ

where VR

mn0 and WR

mn0 are the dimensionless amplitudes,

-R denotes the dimensionless frequency, and kx represents

the dimensionless wave number. By substituting Eqs. (11)

into (10a) and (10b), one can arrive at:

� -R 2

MR þK

Rh i

xR0 ¼ 0 where MR

and KR

can be

evaluated readily. The if and only if condition for existence

of a nontrivial solution to the resulting equations is

det � -R 2

MR þK

Rh i

¼ 0. By solving this relation for

-R, frequencies of the propagated wave with the dimen-

sionless wavenumber kx are calculated by:

xR ¼ -R

t2b

ffiffiffiffiffiffiffiffiffiffiEbIb

qbAb

s: ð12Þ

4.2 Wave motion in magnetically affectedperiodic jungles of SWCNTs using discreteNTBM

By adopting the Timoshenko beam model, the kinetic energy,

TT , the elastic strain energy, UT , of the magnetically affected

nanosystem as well as the work done by the applied magnetic

field on SWCNTs in the framework of the nonlocal contin-

uum field theory of Eringen are stated by:

TTðtÞ ¼ 1

2

XNy

m¼1

XNz

n¼1

Z 1

�1qb Ib

oHTymn

ot

!20@

0@

þoHT

zmn

ot

!21Aþ Ab

oVTmn

ot

� �2

þ oWTmn

ot

� �2 !1

Adx;

ð13aÞ

UTðtÞ¼1

2

XNy

m¼1

XNz

n¼1

Z 1

�1

�oHT

zmn

oxMnl

bzmn

� �Tþ oVT

mn

ox�HT

zmn

� �Qnl

bymn

� �Tþ

�oHT

ymn

oxMnl

bymn

� �Tþ oWT

mn

ox�HT

ymn

� �Qnl

bzmn

� �Tþ

Cvk VTmn�VT

ðmþ1Þn

� �21�dmNz

þ VT

mn�VTðm�1Þn

� �21�d1mð Þ

� �þ

Cv? VTmn�VT

mðn�1Þ

� �21�d1nð Þþ VT

mn�VTmðnþ1Þ

� �21�dnNz

� �þ

Cdk XTmn�XT

ðm�1Þðn�1Þ

� �21�d1mð Þ 1�d1nð Þþ

Cdk XTmn�XT

ðmþ1Þðnþ1Þ

� �21�dmNy

1�dnNz

þ

Cdk YTmn�YT

ðm�1Þðnþ1Þ

� �21�d1mð Þ 1�dnNz

þ

Cdk YTmn�YT

ðmþ1Þðn�1Þ

� �21�dmNy

1�d1nð Þþ

Cd? XTmn�XT

ðm�1Þðnþ1Þ

� �21�d1mð Þ 1�dnNz

þ

Cd? XTmn�XT

ðmþ1Þðn�1Þ

� �21�dmNy

1�d1nð Þþ

Cd? YTmn�YT

ðm�1Þðn�1Þ

� �21�d1mð Þ 1�d1nð Þþ

Cd? YTmn�YT

ðmþ1Þðnþ1Þ

� �21�dmNy

1�dnNz

0BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA

dx;

ð13bÞ

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418 Page 7 of 25 418

123

WTðtÞ ¼XNy

m¼1

XNz

n¼1

Z 1

�1gAbH

2x

o2WTmn

ox2WT

mn þo2VT

mn

ox2VTmn

� �dx;

ð13cÞ

where VTmn and WT

mn denote the transverse displacements of

the (m, n)th SWCNT along the y and z directions, respec-

tively, HTymn

and HTzmn

represent angles of deformation about

the y and z directions, Qnlbymn

� �Tand Qnl

bzmn

� �Tin order are

the nonlocal shear forces along the y and z axes, Mnlbymn

� �T

and Mnlbzmn

� �Tare the nonlocal bending moments about the

y and z axes, respectively, XTmn ¼

ffiffiffi2

p

2WT

mn þ VTmn

, and

YTmn ¼

ffiffiffi2

p

2�WT

mn þ VTmn

. The nonlocal shear forces and

nonlocal bending moments of the (m, n)th SWCNT mod-

eled based on the NTBM are related to their corresponding

local ones as:

N Qnlbymn

� �T� �¼ ksGbAb

oVTmn

ox�HT

zmn

� �; ð14aÞ

N Qnlbzmn

� �T� �¼ ksGbAb

oWTmn

ox�HT

ymn

� �; ð14bÞ

N Mnlbymn

� �T� �¼ �EbIb

oHTymn

ox; ð14cÞ

N Mnlbzmn

� �T� �¼ �EbIb

oHTzmn

ox: ð14dÞ

By exploiting the Hamilton’s principle (i.e.,Z t2

t1

dTT � dUT þ dWT

dt ¼ 0), in view of Eqs. (14a)–

(14d), the nonlocal discrete equations of motion of the

nanosystem acted upon by a longitudinal magnetic field in

terms of deformation fields of the NTBM are obtained as:

N qbIbo2HT

zmn

ot2

( )� ksGbAb

oVTmn

ox�HT

zmn

� �

� EbIbo2HT

zmn

ox2¼ 0;

ð15aÞ

� ksGbAb

o2VTmn

ox2�oHT

zmn

ox

!

þ N qbAb

o2VTmn

ot2� gAbH

2x

o2VTmn

ox2

�

þ Cvk VTmn � VT

ðmþ1Þn

� �1� dmNz

h

þ VTmn � VT

ðm�1Þn

� �1� d1mð Þ

i

þ Cv? VTmn � VT

mðn�1Þ

� �1� d1nð Þ

h

þ VTmn � VT

mðnþ1Þ

� �1� dnNz

i

þ 0:5Cdk WTmn þ VT

mn �WTðm�1Þðn�1Þ � VT

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5Cdk WTmn þ VT

mn �WTðmþ1Þðnþ1Þ � VT

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5Cdk VTmn �WT

mn � VTðmþ1Þðn�1Þ þWT

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

þ 0:5Cdk VTmn �WT

mn � VTðm�1Þðnþ1Þ þWT

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5Cd? VTmn �WT

mn þWTðm�1Þðn�1Þ � VT

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5Cd? VTmn �WT

mn þWTðmþ1Þðnþ1Þ � VT

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5Cd? VTmn þWT

mn � VTðm�1Þðnþ1Þ �WT

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5Cd? VTmn þWT

mn � VTðmþ1Þðn�1Þ �WT

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

¼ 0;

ð15bÞ

N qbIbo2HT

ymn

ot2

( )� ksGbAb

oWTmn

ox�HT

ymn

� �

� EbIbo2HT

ymn

ox2¼ 0;

ð15cÞ

418 Page 8 of 25 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418

123

� ksGbAb

o2WTmn

ox2�oHT

ymn

ox

!

þ N qbAb

o2WTmn

ot2� gAbH

2x

o2WTmn

ox2

�

þ Cvk WTmn �WT

mðnþ1Þ

� �1� dnNz

h

þ WTmn �WT

mðn�1Þ

� �1� d1nð Þ

i

þ Cv? WTmn �WT

ðm�1Þn

� �1� d1mð Þ

h

þ WTmn �WT

ðmþ1Þn

� �1� dmNy

i

þ 0:5Cdk WTmn þ VT

mn �WTðm�1Þðn�1Þ � VT

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5Cdk WTmn þ VT

mn �WTðmþ1Þðnþ1Þ � VT

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5Cdk WTmn � VT

mn �WTðm�1Þðnþ1Þ þ VT

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5Cdk WTmn � VT

mn �WTðmþ1Þðn�1Þ þ VT

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

þ 0:5Cd? WTmn � VT

mn �WTðm�1Þðn�1Þ þ VT

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5Cd? WTmn � VT

mn �WTðmþ1Þðnþ1Þ þ VT

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5Cd? WTmn þ VT

mn �WTðm�1Þðnþ1Þ � VT

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5Cd? WTmn þ VT

mn �WTðmþ1Þðn�1Þ � VT

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

¼ 0:

ð15dÞ

To investigate vibrations of the magnetically affected

nanosystem in a more general context, we employ the

following dimensionless quantities:

VT

mn ¼VTmn

tb;W

T

mn ¼WT

mn

tb;H

T

ymn¼ HT

ymn;H

T

zmn¼ HT

zmn;

s ¼ 1

tb

ffiffiffiffiffiffiffiffiffiffiksGb

qb

st; v ¼ EbIb

ksGbAbt2b

;CT

v½:� ¼Cv½:�t

2b

ksGbAb

;

CT

d½:� ¼Cd½:�t

2b

ksGbAb

HT

x ¼ HTx

ffiffiffiffiffiffiffiffiffiffigAb

ksGb

r; ½:� ¼ kor?: ð16Þ

By introducing Eqs. (16) to (15a)–(15d), the dimensionless

discrete equations of motion of the three-dimensional

nanosystem in the presence of a longitudinal magnetic field

according to the NTBM are obtained:

N k�2o2H

T

zmn

os2

( )� oV

T

mn

on�H

T

zmn

!� v

o2HT

zmn

on2¼ 0; ð17aÞ

� o2VT

mn

on2�oH

T

zmn

on

!þ N

o2VT

mn

os2� H

T

x

� �2 o2VT

mn

on2

(

þ CT

vk VT

mn � VT

ðmþ1Þn

� �1� dmNz

h

þ VT

mn � VT

ðm�1Þn

� �1� d1mð Þ

i

þ CT

v? VT

mn � VT

mðn�1Þ

� �1� d1nð Þ

h

þ VT

mn � VT

mðnþ1Þ

� �1� dnNz

i

þ 0:5CT

dk WT

mn þ VT

mn �WT

ðm�1Þðn�1Þ � VT

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5CT

dk WT

mn þ VT

mn �WT

ðmþ1Þðnþ1Þ � VT

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5CT

dk VT

mn �WT

mn � VT

ðmþ1Þðn�1Þ þWT

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

þ 0:5CT

dk VT

mn �WT

mn � VT

ðm�1Þðnþ1Þ þWT

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5CT

d? VT

mn �WT

mn þWT

ðm�1Þðn�1Þ � VT

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5CT

d? VT

mn �WT

mn þWT

ðmþ1Þðnþ1Þ � VT

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5CT

d? VT

mn þWT

mn � VT

ðm�1Þðnþ1Þ �WT

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5CT

d? VT

mn þWT

mn � VT

ðmþ1Þðn�1Þ �WT

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

¼ 0;

ð17bÞ

N k�2o2H

T

ymn

os2

( )� oW

T

mn

on�H

T

ymn

!

� vo2H

T

ymn

on2¼ 0;

ð17cÞ

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418 Page 9 of 25 418

123

� o2WT

mn

on2�oH

T

ymn

on

!þ N

o2WT

mn

os2

(

� HT

x

� �2 o2WT

mn

on2

þ CT

vk WT

mn �WT

mðnþ1Þ

� �1� dnNz

h

þ WT

mn �WT

mðn�1Þ

� �1� d1nð Þ

i

þ CT

v? WT

mn �WT

ðm�1Þn

� �1� d1mð Þ

h

þ WT

mn �WT

ðmþ1Þn

� �1� dmNy

i

þ 0:5CT

dk WT

mn þ VT

mn �WT

ðm�1Þðn�1Þ � VT

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5CT

dk WT

mn þ VT

mn �WT

ðmþ1Þðnþ1Þ � VT

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5CT

dk WT

mn � VT

mn �WT

ðm�1Þðnþ1Þ þ VT

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5CT

dk WT

mn � VT

mn �WT

ðmþ1Þðn�1Þ þ VT

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

þ 0:5CT

d? WT

mn � VT

mn �WT

ðm�1Þðn�1Þ þ VT

ðm�1Þðn�1Þ

� �

1� d1nð Þ 1� d1mð Þ

þ 0:5CT

d? WT

mn � VT

mn �WT

ðmþ1Þðnþ1Þ þ VT

ðmþ1Þðnþ1Þ

� �

1� dnNz

1� dmNy

þ 0:5CT

d? WT

mn þ VT

mn �WT

ðm�1Þðnþ1Þ � VT

ðm�1Þðnþ1Þ

� �

1� dnNz

1� d1mð Þ

þ 0:5CT

d? WT

mn þ VT

mn �WT

ðmþ1Þðn�1Þ � VT

ðmþ1Þðn�1Þ

� �

1� d1nð Þ 1� dmNy

¼ 0: ð17dÞ

Equations (17a)–(17d) represent a set of 4NyNz coupled

PDEs. In order to evaluate the dispersion curves, elastic

waves within the (m, n)th tube are considered to be harmonic:

\VT

mn;HT

zmn;W

T

mn;HT

ymn

[ ¼ \VR

mn0;HT

zmn0;W

R

mn0;HT

ymn0[ ei -T s�kx nð Þ;

ð18Þ

where VT

mn0, HT

zmn0, W

T

mn0, and HT

ymn0are the dimensionless

amplitudes of the transverse waves, and -T is the dimen-

sionless frequency of the elastic transverse wave within the

nanotube modeled on the basis of the NTBM. By intro-

ducing Eqs. (18) to (17a)–(17d), it is obtainable:

� -T 2

MT þK

Th i

xT0 ¼ 0 where MT

and KT

could be

readily calculated. A nontrivial solution to the recently

obtained equations exists if and only if:

det � -T 2

MT þK

Th i

¼ 0. By solving this relation for

-T , frequencies of the transverse waves for each value of

the dimensionless wavenumber kx are given by:

xT ¼ -T

tb

ffiffiffiffiffiffiffiffiffiffiksGb

qb

s: ð19Þ

5 Establishment of NCMs using NRBMand NTBM

5.1 Wave motion in magnetically affectedperiodic jungles of SWCNTs using continuousNRBM

Based on the discrete governing equations of the periodic

array of SWCNTs subjected to a longitudinal magnetic field

modeled via NRBM, Eqs. (8a) and (8b), the nonlocal equa-

tions of motion of the magnetically affected (m, n)th SWCNT

of the nanosystem are rewritten more concisely as:

EbIbo4VR

mn

ox4þ N qb Ab

o2VRmn

ot2� Ib

o4VRmn

ot2ox2

� ��

� gAbH2x

o2VRmn

ox2þ Cvk 2VR

mn � VRðmþ1Þn � VR

ðm�1Þn

� �

þ Cv? 2VRmn � VR

mðn�1Þ � VRmðnþ1Þ

� �

þ 1

2Cdk � Cd?

WRðm�1Þðnþ1Þ þWR

ðmþ1Þðn�1Þ

�

�WRðmþ1Þðnþ1Þ �WR

ðm�1Þðn�1Þ

�

þ 1

2Cdk þ Cd?

4VRmn � VR

ðm�1Þðnþ1Þ

�

�VRðmþ1Þðnþ1Þ � VR

ðmþ1Þðn�1Þ � VRðm�1Þðn�1Þ

��¼ 0;

ð20aÞ

EbIbo4WR

mn

ox4þ N qb Ab

o2WRmn

ot2� Ib

o4WRmn

ot2ox2

� ��

� gAbH2x

o2WRmn

ox2

þ Cvk 2WRmn �WR

mðnþ1Þ �WRmðn�1Þ

� �

þ Cv? 2WRmn �WR

ðm�1Þn �WRðmþ1Þn

� �

þ 1

2Cdk � Cd?

VRðm�1Þðnþ1Þ þ VR

ðmþ1Þðn�1Þ

�

�VRðmþ1Þðnþ1Þ � VR

ðm�1Þðn�1Þ

�

þ 1

2Cdk þ Cd?

4WRmn �WR

ðm�1Þðnþ1Þ

�

�WRðmþ1Þðnþ1Þ �WR

ðmþ1Þðn�1Þ �WRðm�1Þðn�1Þ

��¼ 0:

ð20bÞ

418 Page 10 of 25 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418

123

To fabric a continuous model based on the discrete rela-

tions in Eqs. (20a) and (20b), two continuous displace-

ments of the form v ¼ vðx; y; z; tÞ and w ¼ wðx; y; z; tÞ are

considered such that:

½��mnðx; tÞ � ½:�ðx; ymn; zmn; tÞ;½��ðm�1Þðn�1Þðx; tÞ � ½:�ðx; ymn � d; zmn � d; tÞ;½��ðm�1Þðnþ1Þðx; tÞ � ½:�ðx; ymn þ d; zmn � d; tÞ;½��ðmþ1Þðn�1Þðx; tÞ � ½:�ðx; ymn � d; zmn þ d; tÞ;½��ðmþ1Þðnþ1Þðx; tÞ � ½:�ðx; ymn þ d; zmn þ d; tÞ;

ð21Þ

where ½��ð½:�Þ ¼ V ½�� v½��

orW ½�� w½�� and ½�� ¼ RorT .

Using Taylor series expansion up to the sixth order, the

transverse displacements of neighboring tubes of the

(m, n)th SWCNT are approximated by:

½:�ðx; ymn d; zmn d; tÞ

¼X6i¼1

Xij¼0

i

i� j

� �oi½:�ðx; ymn; zmn; tÞ

ozjoyi�j

dð Þj dð Þi�j;

ð22Þ

where

o0½:�ðx; y; z; tÞoy0oz0

¼ ½:�ðx; y; z; tÞ; ½:� ¼ v½��orw½��

. By introducing Eqs. (22) to (20a) and (20b), the nonlocal

continuous equations of motion of the magnetically affec-

ted nanosystem made from vertically aligned SWCNTs

according to the NRBM take the following form:

EbIbo4vR

ox4þ N qb Ab

o2vR

ot2� Ib

o4vR

ot2ox2

� �� gAbH

2x

o2vR

ox2

�

� Cvkd2 o2vR

oy2þ d2

12

o4vR

oy4þ d4

360

o6vR

oy6þ d6

20160

o8vR

oy8

� �

� Cv?d2 o2vR

oz2þ d2

12

o4vR

oz4þ d4

360

o6vR

oz6þ d6

20160

o8vR

oz8

� �

� Cdk þ Cd?

d2

o2vR

oy2þ o2vR

oz2þ d2

12

o4vR

oz4þ 6

o4vR

oz2oy2þ o4vR

oy4

� �

þ d4

360

o6vR

oz6þ 15

o6vR

oz4oy2þ 15

o6vR

oz2oy4þ o6vR

oy6

� �

26664

37775

� Cdk � Cd?

d2 2o2wR

oyozþ d2

3

o4wR

oyoz3þ o4wR

oy3oz

� ��

þ d4

1803o6wR

oy5ozþ 10

o6wR

oy3oz3þ 3

o6wR

oyoz5

� ���¼ 0;

ð23aÞ

EbIbo4wR

ox4þ N qb Ab

o2wR

ot2� Ib

o4wR

ot2ox2

� �� gAbH

2x

o2wR

ox2

�

� Cvkd2 o2wR

oz2þ d2

12

o4wR

oz4þ d4

360

o6wR

oz6þ d6

20160

o8wR

oz8

� �

� Cv?d2 o2wR

oy2þ d2

12

o4wR

oy4þ d4

360

o6wR

oy6þ d6

20160

o8wR

oy8

� �

� Cdk þ Cd?

d2

o2wR

oy2þ o2wR

oz2þ d2

12

o4wR

oz4þ 6

o4wR

oz2oy2þ o4wR

oy4

� �þ

d4

360

o6wR

oz6þ 15

o6wR

oz4oy2þ 15

o6wR

oz2oy4þ o6wR

oy6

� �

26664

37775

� Cdk � Cd?

d2

2o2vR

oyozþ d2

3

o4vR

oyoz3þ o4vR

oy3oz

� ��

þ d4

1803o6vR

oy5ozþ 10

o6vR

oy3oz3þ 3

o6vR

oyoz5

� ���¼ 0:

ð23bÞ

Using Eq. (16), the recent equations are rewritten in the

dimensionless form as follows:

o4vR

on4þ N

o2vR

os2� k�2 o4vR

os2on2� H

R

x

� �2 o2vRon2

8>>>><>>>>:

� jd 2

CR

vko2vR

og2þ

jd 212

o4vR

og4

þjd 4360

o6vR

og6þ

jd 620160

o8vR

og8

!

� d2C

R

v?o2vR

oc2þ d

2

12

o4vR

oc4þ d

4

360

o6vR

oc6þ d

6

20160

o8vR

oc8

!

� CR

dk þ CR

d?

� �

j2o2vR

og2þ o2vR

oc2þ d

2

12

o4vR

oc4þ 6j2

o4vR

oc2og2þ j4

o4vR

og4

� �

þ d4

360

o6vR

oc6þ 15j2

o6vR

oc4og2þ 15j4

o6vR

oc2og4þ j6

o6vR

og6

� �

266664

377775

� j CR

dk � CR

d?

� �

2o2wR

ogocþ d

2

3

o4wR

ogoc3þ j2

o4wR

og3oc

� �

þ d4

1803j4

o6wR

og5ocþ 10j2

o6wR

og3oc3þ 3

o6wR

ogoc5

� �

266664

377775

9>>>>=>>>>;

¼ 0;

ð24aÞ

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418 Page 11 of 25 418

123

o4wR

on4þ N

o2wR

os2� k�2 o4wR

os2on2� H

R

x

� �2 o2wR

on2

8>>>><>>>>:

� d2CR

vko2wR

oc2þ d

2

12

o4wR

oc4

þ d4

360

o6wR

oc6þ d

6

20160

o8wR

oc8

!

� jd 2

CR

v?o2wR

og2

�

þjd 212

o4wR

og4þ

jd 4360

o6wR

og6þ

jd 620160

o8wR

og8

!

� CR

dk þ CR

d?

� �

j2o2wR

og2þ o2wR

oc2þ d

2

12

o4wR

oc4þ 6j2

o4wR

oc2og2þ j4

o4wR

og4

� �

þ d4

360

o6wR

oc6þ 15j2

o6wR

oc4og2þ 15j4

o6wR

oc2og4þ j6

o6wR

og6

� �

266664

377775

�j CR

dk � CR

d?

� �

2o2vR

ogocþ d

2

3

o4vR

ogoc3þ j2

o4vR

og3oc

� �

þ d4

1803j4

o6vR

og5ocþ 10j2

o6vR

og3oc3þ 3

o6vR

ogoc5

� �

266664

377775

9>>>>=>>>>;

¼ 0;

ð24bÞ

where

g ¼ y

ly; j ¼ lz

ly;C

R

d½:� ¼Cd½:�d

2t4bEbIbl2z

;

vR ¼ vR

tb;wR ¼ wR

tb;

vR ¼ vRðn; g; c; sÞ;wR ¼ wRðn; g; c; sÞ; ½:� ¼k or ? :

ð25Þ

The dimensionless displacement functions of the suggested

continuousmodel canbe stated in the followingharmonic form:

\vR;wR [ ¼ \vR0 ;wR0 [ ei -Rs�k:rð Þ; ð26Þ

where vR0 and wR0 are the dimensionless amplitudes of the

transverse waves, -R is the dimensionless frequency, k ¼kxex þ kyey þ kzez is the dimensionless vector of the

wavenumber whose components are: kx ¼ tb kx, ky ¼ tb ky,

kz ¼ tb kz, and r ¼ nex þ gey þ cez is the dimensionless

position vector. By substituting Eqs. (26) into (24a) and (24b),

we obtain:

� -R 2 1 0

0 1

� �þ

C1 C2

C2 C3

� �� �vR0

wR0

( )¼

0

0

� �; ð27Þ

where

C1 ¼k4

x þ kxHx

21þ lkx

2� �

1þ k�1kx 2� �

1þ lkx 2� �þ C

R

vk jkyd

1þ k�1kx 2

1�jkyd 2

12þ

jkyd 4360

�jkyd 620160

" #

þCR

v? kzd 2

1þ k�1kx 2 1�

kzd 212

þkzd 4360

�kzd 620160

" #

þCR

dk þ CR

d?

1þ k�1kx 2 jky

2þk2

z

h

� d2

12k4

z þ 6j2k2

yk2

z þ j4k4

y

� �

þ d4

360k6

z þ 15j2k4

z k2

y þ 15j4k2

z k4

y þ j6k6

y

� �#;

ð28aÞ

C2 ¼jkykz C

R

dk � CR

d?

� �

1þ k�1kx 2 2� d

2

3k2

z þ jky 2� �"

þ d4

1803 jky 4þ10 jky

2k2

z þ 3k4

z

� �#;

ð28bÞ

C3 ¼k4

x þ kxHx

21þ lkx

2� �

1þ k�1kx 2� �

1þ lkx 2� �þ C

R

v? jkyd

1þ k�1kx 2

1�jkyd 2

12þ

jkyd 4360

�jkyd 620160

" #

þCR

vk kzd 2

1þ k�1kx 2 1�

kzd 212

þkzd 4360

�kzd 620160

" #

þCR

dk þ CR

d?

1þ k�1kx 2 jky

2þk2

z

h

� d2

12k4

z þ 6j2k2

yk2

z þ j4k4

y

� �

þ d4

360k6

z þ 15j2k4

z k2

y þ 15j4k2

z k4

y þ j6k6

y

� �#:

ð28cÞ

A nontrivial solution to the set of equations in Eq. (27) is

obtained if the determinant of the coefficient matrix of the

418 Page 12 of 25 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418

123

amplitude vector is set equal to zero. Hence, the dispersion

relation of the magnetically affected periodic array of

SWCNTs using the NRBM would be:

-R 4� -R

2C1 þ C3ð Þ þ C1C3 � C2

2 ¼ 0: ð29Þ

By solving Eq. (29), two dimensionless flexural frequen-

cies for transverse waves within the vertically aligned

periodic array of SWCNTs immersed in longitudinal

magnetic fields are obtained:

-R1 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC1 þ C3

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC1 � C3

2

� �2

þC22

svuut;

-R2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC1 þ C3

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC1 � C3

2

� �2

þC22

svuut:

ð30Þ

The phase velocity of the transverse waves is defined by

vRp ¼ xR

kxwhere xR and kx are the frequency and the

wavenumber of the transverse waves in the nanosystem

modeled according to the NRBM. SincexR

kx¼ -R

kkx

ffiffiffiffiffiEb

qb

s, in

view of Eq. (30), the phase velocities associated with -R1

and -R2 are provided by:

vRp1 ¼1

kxk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEb

qb

C1 þ C3

2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC1 � C3

2

� �2

þC22

s0@

1A

vuuut ;

vRp2 ¼1

kxk

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEb

qb

C1 þ C3

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC1 � C3

2

� �2

þC22

s0@

1A

vuuut :

ð31Þ

Furthermore, the group velocity is defined by vRg ¼ oxR

okx. In

view of the relation:oxR

okx¼ 1

k

ffiffiffiffiffiEb

qb

so-R

okxas well as

Eqs. (28) and (30), the group velocities of the transverse

waves in periodic array of SWCNTs acted upon by a

longitudinal magnetic field are evaluated by:

vRg1 ¼1

4k-R1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Eb

qb

oC1

okxþ oC4

okx� C1 � C3

2

� �2

þC22

!�12

�

1

2

oC1

okx� oC3

okx

� �C1 � C3ð Þ þ 2C2

oC2

okx

� �

0BBBB@

1CCCCA

vuuuuuuut;

vRg2 ¼1

4k-R1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Eb

qb

oC1

okxþ oC4

okxþ C1 � C3

2

� �2

þC22

!�12

�

1

2

oC1

okx� oC3

okx

� �C1 � C3ð Þ þ 2C2

oC2

okx

� �

0BBBB@

1CCCCA

vuuuuuuut;

ð32Þ

where

5.2 Wave motion in magnetically affectedperiodic jungles of SWCNTs using continuousNTBM

Consider the (m, n)th nanotube of the nanosystem, except

the edges, in the presence of a longitudinal magnetic field

(i.e., 1\m\Nz and 1\n\Ny). By employing Eqs. (15a)–

(15d), the nonlocal governing equations that display

transverse waves within such a tube are given by:

oC1

okx¼

4k3

x 1þ lkx 2� �

� 2l2k5

x � 2kx k�2C1 � H2

x

� �1þ lkx

2� �2þ kxH

H

x

� �2

1þ k�1kx 2� �

1þ lkx 2� � ;

oC2

okx¼ � 2kxk

�2C2

1þ k�1kx 2 ;

oC3

okx¼

4k3

x 1þ lkx 2� �

� 2l2k5

x � 2kx k�2C3 � H2

x

� �1þ lkx

2� �2

1þ k�1kx 2� �

1þ lkx 2� � :

ð33Þ

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418 Page 13 of 25 418

123

N qbIbo2HT

zmn

ot2

( )� ksGbAb

oVTmn

ox�HT

zmn

� �

� EbIbo2HT

zmn

ox2¼ 0;

ð34aÞ

� ksGbAb

o2VTmn

ox2�oHT

zmn

ox

!þ N qbAb

o2VTmn

ot2

�

� gAbH2x

o2VTmn

ox2

þ Cvk 2VTmn � VT

ðmþ1Þn � VTðm�1Þn

� �

þ Cv? 2VTmn � VT

mðn�1Þ � VTmðnþ1Þ

� �

þ 1

2Cdk � Cd?

WTðm�1Þðnþ1Þ þWT

ðmþ1Þðn�1Þ

�

�WTðmþ1Þðnþ1Þ �WT

ðm�1Þðn�1Þ

�

þ 1

2Cdk þ Cd?

4VTmn � VT

ðm�1Þðnþ1Þ

�

�VTðmþ1Þðnþ1Þ � VT

ðmþ1Þðn�1Þ � VTðm�1Þðn�1Þ

��¼ 0;

ð34bÞ

N qbIbo2HT

ymn

ot2

( )� ksGbAb

oWTmn

ox�HT

ymn

� �

� EbIbo2HT

ymn

ox2¼ 0;

ð34cÞ

� ksGbAb

o2WTmn

ox2�oHT

ymn

ox

!

þ N qbAb

o2WTmn

ot2� gAbH

2x

o2WTmn

ox2

�

þ Cvk 2WTmn �WT

mðnþ1Þ �WTmðn�1Þ

� �

þ Cv? 2WTmn �WT

ðm�1Þn �WTðmþ1Þn

� �

þ 1

2Cdk � Cd?

VTðm�1Þðnþ1Þ þ VT

ðmþ1Þðn�1Þ

�

�VTðmþ1Þðnþ1Þ � VT

ðm�1Þðn�1Þ

�

þ 1

2Cdk þ Cd?

4WTmn �WT

ðm�1Þðnþ1Þ

�

�WTðmþ1Þðnþ1Þ �WT

ðmþ1Þðn�1Þ �WTðm�1Þðn�1Þ

�o¼ 0:

ð34dÞ

Define the following continuous functions for the angle of

deflections:

hTy ðx; ymn; zmn; tÞ � HTymn

ðx; tÞ;hTz ðx; ymn; zmn; tÞ � HT

zmnðx; tÞ:

ð35Þ

By exploiting Eqs. (21), (22), (35) and (34a)–(34d), the

continuous form of the governing equations of the mag-

netically affected nanosystem on the basis of the NTBM is

provided by:

N qbIbo2hTzot2

( )� ksGbAb

ovT

ox� hTz

� �� EbIb

o2hTzox2

¼ 0;

ð36aÞ

� ksGbAb

o2vT

ox2�ohTzox

!þ N qbAb

o2vT

ot2� gAbH

2x

o2vT

ox2

�

� Cvkd2 o2vT

oy2þ d2

12

o4vT

oy4þ d4

360

o6vT

oy6þ d6

20160

o8vT

oy8

� �

� Cv?d2 o2vT

oz2þ d2

12

o4vT

oz4þ d4

360

o6vT

oz6þ d6

20160

o8vT

oz8

� �

� Cdk þ Cd?

d2

o2vT

oy2þ o2vT

oz2þ d2

12

o4vT

oz4þ 6

o4vT

oz2oy2þ o4vT

oy4

� �þ

d4

360

o6vT

oz6þ 15

o6vT

oz4oy2þ 15

o6vT

oz2oy4þ o6vT

oy6

� �

266664

377775

� Cdk � Cd?

d2

2o2wT

oyozþ d2

3

o4wT

oyoz3þ o4wT

oy3oz

� ��

þ d4

1803o6wT

oy5ozþ 10

o6wT

oy3oz3þ 3

o6wT

oyoz5

� ���¼ 0;

ð36bÞ

N qbIbo2hTyot2

( )� ksGbAb

owT

ox� hTy

� �� EbIb

o2hTyox2

¼ 0;

ð36cÞ

418 Page 14 of 25 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418

123

� ksGbAb

o2wT

ox2�ohTyox

!þ N qbAb

o2wT

ot2� gAbH

2x

o2wT

ox2

�

� Cvkd2 o2wT

oz2þ d2

12

o4wT

oz4þ d4

360

o6wT

oz6þ d6

20160

o8wT

oz8

� �

� Cv?d2 o2wT

oy2þ d2

12

o4wT

oy4þ d4

360

o6wT

oy6þ d6

20160

o8wT

oy8

� �

� Cdk þ Cd?

d2

o2wT

oy2þ o2wT

oz2þ d2

12

o4wT

oz4þ 6

o4wT

oz2oy2þ o4wT

oy4

� �þ

d4

360

o6wT

oz6þ 15

o6wT

oz4oy2þ 15

o6wT

oz2oy4þ o6wT

oy6

� �

26664

37775

� Cdk � Cd?

d2 2o2vT

oyozþ d2

3

o4vT

oyoz3þ o4vT

oy3oz

� ��

þ d4

1803o6vT

oy5ozþ 10

o6vT

oy3oz3þ 3

o6vT

oyoz5

� ���¼ 0:

ð36dÞ

or in a dimensionless manner as follows:

N k�2 o2hT

z

os2

( )� ovT

on� h

T

z

� �� v

o2hT

z

on2¼ 0; ð37aÞ

� o2vT

on2�oh

T

z

on

!þ N

o2vT

os2� H

T

x

� �2 o2vTon2

�

� jd 2

CT

vk

o2vT

og2þ

jd 212

o4vT

og4þ

jd 4360

o6vT

og6þ

jd 620160

o8vT

og8

!

� d2CT

v?o2vT

oc2þ d

2

12

o4vT

oc4þ d

4

360

o6vT

oc6þ d

6

20160

o8vT

oc8

!

� CT

dk þ CT

d?

� �

j2o2vT

og2þ o2vT

oc2þ d

2

12

o4vT

oc4þ 6j2

o4vT

oc2og2þ j4

o4vT

og4

� �þ

d4

360

o6vT

oc6þ 15j2

o6vT

oc4og2þ 15j4

o6vT

oc2og4þ j6

o6vT

og6

� �

266664

377775

� j CT

dk � CT

d?

� �

2o2wT

ogocþ d

2

3

o4wT

ogoc3þ j2

o4wT

og3oc

� �þ

d4

1803j4

o6wT

og5ocþ 10j2

o6wT

og3oc3þ 3

o6wT

ogoc5

� �

266664

377775

9>>>>=>>>>;

¼ 0;

ð37bÞ

N k�2o2h

T

y

os2

( )� owT

on� h

T

y

� �� v

o2hT

y

on2¼ 0; ð37cÞ

� o2wT

on2�oh

T

y

on

!þ N

o2wT

os2� H

T

x

� �2 o2wT

on2

�

� d2CT

vk

o2wT

oc2þ d

2

12

o4wT

oc4þ d

4

360

o6wT

oc6þ d

6

20160

o8wT

oc8

!

� jd 2

CT

v?

o2wT

og2þ

jd 212

o4wT

og4þ

jd 4360

o6wT

og6þ

jd 620160

o8wT

og8

!

� CT

dk þ CT

d?

� �

j2o2wT

og2þ o2wT

oc2þ d

2

12

o4wT

oc4þ 6j2

o4wT

oc2og2þ j4

o4wT

og4

� �þ

d4

360

o6wT

oc6þ 15j2

o6wT

oc4og2þ 15j4

o6wT

oc2og4þ j6

o6wT

og6

� �

266664

377775

� j CT

dk � CT

d?

� �

2o2vT

ogocþ d

2

3

o4vT

ogoc3þ j2

o4vT

og3oc

� �þ

d4

1803j4

o6vT

og5ocþ 10j2

o6vT

og3oc3þ 3

o6vT

ogoc5

� �

266664

377775

9>>>>=>>>>;

¼ 0;

ð37dÞ

where

CT

d½:� ¼Cd½:�d

2t2bksGbAbl2z

; vT ¼ vT

tb;

wT ¼ wR

tb; h

T

y ¼ hTy ; hT

z ¼ hTz ; ½:� ¼k or ?;

hT

z ¼ hT

z ðn; g; c; sÞ; vT ¼ vTðn; g; c; sÞ;

hT

y ¼ hT

y ðn; g; c; sÞ;wT ¼ wTðn; g; c; sÞ:

ð38Þ

The transverse waves in vertically aligned periodic jungles

of SWCNTs subjected to a longitudinal magnetic field

using the continuous NTBM are considered as:

\vT ; hT

z ;wT ; h

T

y [ ¼ \vT0 ; hT

z0;wT0 ; h

T

y0

[ ei -T s�k:rð Þ:ð39Þ

By substituting this harmonic form of the deformation field

into Eqs. (37a)–(37d), we derive:

� -T 2

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

26664

37775þ

m1 m2 m3 0

m4 m5 0 0

m6 0 m7 m80 0 m9 m10

26664

37775

0BBB@

1CCCA

vT0

hT

z0

wT0

hT

y0

8>>>>><>>>>>:

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

¼

0

0

0

0

8>>><>>>:

9>>>=>>>;;

ð40Þ

where

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418 Page 15 of 25 418

123

m1 ¼k2

x

1þ lkx 2 þ kxH

T

x

� �2þC

T

vk jkyd

1�jkyd 2

12þ

jkyd 4360

�jkyd 620160

" #

þ CT

v? kzd 2

1�kzd 212

þkzd 4360

�kzd 620160

" #

þ CT

dk þ CT

d?

� �jky 2þk

2

z

h� d

2

12k4

z þ 6j2k2

yk2

z þ j4k4

y

� �

þ d4

360k6

z þ 15j2k4

z k2

y þ 15j4k2

z k4

y þ j6k6

y

� �#;

m3 ¼ jkykz CT

dk � CT

d?

� �2� d

2

3k2

z þ jky 2� �"

þ d4

1803 jky 4þ10 jky

2k2

z þ 3k4

z

� �#;

m2 ¼ �ikx

1þ lkx 2 ; m4 ¼ i

kxk2

1þ lkx 2 ; m5 ¼

k2 1þ vk2

x

� �

1þ lkx 2 ;

m6 ¼ jkykz CT

dk � CT

d?

� �2� d

2

3k2

z þ jky 2� �"

þ d4

1803 jky 4þ10 jky

2k2

z þ 3k4

z

� �#;

m7 ¼k2

x

1þ lkx 2 þ kxH

T

x

� �2þC

T

v? jkyd

1�jkyd 2

12þ

jkyd 4360

�jkyd 620160

" #

þ CT

vk kzd 2

1�kzd 212

þkzd 4360

�kzd 620160

" #

þ CT

dk þ CT

d?

� �jky 2þk

2

z

h

� d2

12k4

z þ 6j2k2

yk2

z þ j4k4

y

� �

þ d4

360k6

z þ 15j2k4

z k2

y þ 15j4k2

z k4

y þ j6k6

y

� �#;

m8 ¼ �ikx

1þ lkx 2 ; m9 ¼ i

kxk2

1þ lkx 2 ; m10 ¼

k2 1þ vk2

x

� �

1þ lkx 2 :

ð41Þ

A nontrivial solution to Eq. (40) would exist if and only if

the determinant of the coefficient matrix pertinent to the

dimensionless amplitude vector would be zero. By doing

so, one can arrive at the dispersion relation of the mag-

netically affected periodic array of SWCNTs modeled on

the basis of the NTBM:

PT8 -T 8þPT

6 -T 6þPT

4 -T 4

þ PT2 -T 2þPT

0 ¼ 0;ð42Þ

where

PT8 ¼ 1;PT

6 ¼ � m1 þ m5 þ m7 þ m10ð Þ;PT4 ¼ �m2m4 þ m1m5 � m3m6 þ m1m7 þ m5m7

� m8m9 þ m1m10 þ m5m10 þ m7m10;

PT2 ¼ m3m5m6 þ m2m4m7 � m1m5m7 þ m1m8m9 þ m5m8m9

þ m2m4m10 � m1m5m10 þ m3m6m10� m1m7m10 � m5m7m10;

PT0 ¼ m2m4m8m9 � m1m5m8m9 � m3m5m6m10

� m2m4m7m10 þ m1m5m7m10:

ð43Þ

Generally, Eq. (42) has four positive roots and four nega-

tive roots. The positive roots, -Ti ; i ¼ 1; 2; 3; 4, are the

frequencies associated with the transverse waves in the

magnetically affected periodic array of SWCNTs accord-

ing to the NTBM. Additionally, the corresponding phase

and group velocities in terms of wavenumbers and fre-

quencies are evaluated in the following form:

vTpi ¼xT

i

kx¼

ffiffiffiffiffiffiffiffiffiffiksGb

qb

s-T

i

kx;

vTgi ¼oxT

i

ok¼

ffiffiffiffiffiffiffiffiffiffiksGb

qb

so-T

i

okx;

ð44Þ

where

o-Ti

okx

¼ �oPT

8

okx-T

i

8þ oPT6

okx-T

i

6þ oPT4

okx-T

i

4þ oPT2

okx-T

i

2þ oPT0

okx

8PT8 -T

ið Þ7þ6PT6 -T

ið Þ5þ4PT4 -T

ið Þ3þ2PT2 -

Ti

:

ð45Þ

6 Results and discussion

Consider a magnetically affected periodic array of

SWCNTs with the following data: Eb ¼ 1 TPa,

qb ¼ 2500 kg/m3, mb ¼ 0:2, rm ¼ 1 nm, and d ¼ 2rm þ tbwhere tb ¼ 0:34 nm. In the following parts, the capabilities

of the suggested NCMs in reproducing the results predicted

by the NDMs are explained and discussed in some detail.

After ensuring the efficiency of such continuous models,

the roles of the wavenumber, magnetic field strength,

intertube distance, number and radius of constitutive

SWCNTs, and nonlocality on the characteristics of both

418 Page 16 of 25 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418

123

flexural and shear waves are examined using continuous-

based NRBM and NTBM.

6.1 NCMs versus NDMs

In Table 1, the lowest frequencies corresponding to kx ¼ky ¼ kz ¼ p have been provided for various radii of

SWCNTs and different numbers of constitutive SWCNTs

of the nanosystem. The predicted results based on both

NDMs and NCMs using the NRBM and NTBM are pre-

sented. A brief survey of the obtained results shows that the

NCMs could capture the results of the NDMs with a fairly

good accuracy for all considered levels of the nanotube’s

radius and various populations of the nanosystem. For a

nanosystem with a specific number of SWCNTs, by

increasing the radius of SWCNTs, the relative discrepan-

cies between the results of the NRBM and those of the

NTBM would generally reduce. By growing the population

of the nanosystem used for transferring of high-frequency

waves with kx ¼ ky ¼ kz ¼ p, these discrepancies would

also lessen slightly. As it is seen in Table 1, both NRBM

and NTBM predict that the lowest frequency of the wave

(i.e., flexural frequency) would commonly reduce by

increasing the radius of the constitutive SWCNTs of the

nanosystem. More detailed influences of radius and number

of SWCNTs on the characteristics of the transverse waves

are displayed in the upcoming parts.

Table 2 displays the lowest frequencies of transverse

waves which are going to be propagated within the

nanosystem based on the NDMs/NCMs by employing the

NRBM and NTBM. The results are given for magnetically

affected nanosystems with e0a ¼ 2 nm, rm ¼ 1 nm, four

levels of the population (i.e., Ny ¼ Nz ¼ 5, 7, 9, and 11),

and three values of the magnetic field strength (i.e.,

HR

x ¼ 0, 0.1, and 0.2). A brief comparison of the results of

the proposed NCMs and those of the NDMs reveals that the

NCMs are very capable in capturing the results of the

NDMs for ensembles of various populations which are

acted upon by various levels of the magnetic field strength.

As a general trend, the frequencies of the transverse waves

would grow by an increase in the magnetic field strength.

Nevertheless, by growing the population of the nanosys-

tem, the lowest frequency of the waves would decrease.

6.2 Effect of the wavenumberon the frequencies, phase velocities,and group velocities

Figure 2 presents the variation of flexural and shear

frequencies of transverse waves as well as their corre-

sponding phase and group velocities in terms of the

dimensionless wavenumber. The obtained results are

associated with a nanosystem with Ny ¼ Nz ¼ 100 sub-

jected to a longitudinal magnetic field of strength

HR

x ¼ 0:05. The results are provided for three levels of the

small-scale parameter (i.e., e0a ¼ 0; 1, and 2 nm) in the

case of ky ¼ kz ¼ p20. For the considered range of the

wavenumber, both NRBM and NTBM predict that the

flexural frequencies (i..e, x1 and x2) would magnify as the

Table 1 A verification of the

predicted fundamental

frequencies by the suggested

continuous models and those of

the proposed discrete models for

different levels of population

and radius of SWCNTs

(kx ¼ ky ¼ kz ¼ p, e0a ¼ 2 nm,

HR

x ¼ HR

x0

ffiffiffiffiffiffiffiffiffiffiffiAbIb0

Ab0Ib

r,

rm0 ¼ 0:7 nm, HR

x0 ¼ 0:1)

rm Ny ¼ Nz ¼ 5 Ny ¼ Nz ¼ 7 Ny ¼ Nz ¼ 9 Ny ¼ Nz ¼ 11

Discrete models

NRBM 1.0 10.704619 10.703652 10.703300 10.703134

1.5 10.320215 10.319830 10.319690 10.319624

2.0 10.176923 10.176728 10.176657 10.176624

3.0 10.071554 10.071481 10.071455 10.071442

NTBM 1.0 10.010368 10.010367 10.010367 10.010366

1.5 9.996409 9.996408 9.996408 9.996408

2.0 9.991571 9.991571 9.991571 9.991571

3.0 9.988132 9.988132 9.988132 9.988132

Continuous models

NRBM 1.0 10.704619 10.703652 10.703300 10.703134

1.5 15.265008 10.319830 10.319690 10.319624

2.0 13.501602 10.176728 10.176657 10.176624

3.0 15.053047 10.071481 10.071455 10.071442

NTBM 1.0 10.010368 10.010367 10.010367 10.010366

1.5 9.996409 9.996408 9.996408 9.996408

2.0 9.991571 9.991571 9.991571 9.991571

3.0 9.988132 9.988132 9.988132 9.988132

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418 Page 17 of 25 418

123

wavenumber increases. Irrespective of the value of the

small-scale parameter, the relative discrepancies between

the results of the NRBM and those of the NTBM would

grow in terms of the wavenumber. Additionally, variation

of the small-scale parameter has a greater impact on such

discrepancies for higher wavenumbers. For a given

wavenumber and a small-scale parameter, the predicted

flexural frequencies and their pertinent phase and group

velocities by the NRBM are generally greater than those of

the NTBM. This is mainly related to the consideration of

shear deformation by the NTBM. In other words, the

transverse stiffness of the nanosystem modeled based on

Table 2 A verification of the

predicted fundamental

frequencies by the suggested

continuous models and those of

the proposed discrete models for

different levels of population

and magnetic field strength

(kx ¼ ky ¼ kz ¼ p, e0a ¼ 2 nm)

HR

xNy ¼ Nz ¼ 5 Ny ¼ Nz ¼ 7 Ny ¼ Nz ¼ 9 Ny ¼ Nz ¼ 11

Discrete models

NRBM 0 2.336702 2.148021 2.075057 2.039878

0.1 2.338196 2.149646 2.076739 2.041589

0.2 2.342672 2.154515 2.081778 2.046714

NTBM 0 2.116685 1.896214 1.809233 1.766884

0.1 2.118409 1.898139 1.811251 1.768951

0.2 2.123572 1.903903 1.817292 1.775136

Continuous models

NRBM 0 2.336702 2.148021 2.075057 2.039878

0.1 2.338196 2.149647 2.076739 2.041589

0.2 2.342673 2.154515 2.081778 2.046714

NTBM 0 2.116686 1.896214 1.809233 1.766884

0.1 2.118410 1.898139 1.811251 1.768951

0.2 2.123573 1.903903 1.817292 1.775136

0 0.05 0.10

10

20

ω1 (

TH

z)

0 0.05 0.10

10

20

vp 1 (

km/s

)

0 0.05 0.10

10

20

vg 1 (

km/s

)0 0.05 0.1

0

10

20

ω2 (

TH

z)

0 0.05 0.10

10

20

vp 2 (

km/s

)

0 0.05 0.10

10

20

vg 2 (

km/s

)

0 0.05 0.110

20

30

ω3 (

TH

z)

0 0.05 0.10

1000

2000

vp 3 (

km/s

)

0 0.05 0.10

5

10 v

g 3 (km

/s)

0 0.05 0.110

20

30

kx

ω4 (

TH

z)

0 0.05 0.10

1000

2000

kx

vp 4 (

km/s

)

0 0.05 0.10

5

10

kx

vg 4 (

km/s

)

Fig. 2 Plots of crucial

characteristics of elastic waves

in terms of longitudinal

wavenumber for three levels of

the small-scale parameter: ((. . .)NRBM, (��) NTBM; (�)e0a ¼ 0:2, (h) e0a ¼ 0:5, (M)

e0a ¼ 1 nm; ky ¼ kz ¼p20

;

HR

x ¼ 0:05; Ny ¼ Nz ¼ 100)

418 Page 18 of 25 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418

123

the NTBM is lesser than that obtained via the NRBM. Such

an issue is more apparent for the nanosystems exploited for

transferring waves with higher wavenumbers since these

are actually high-frequency waves and their wavelength is

so tiny. For example, the wavelength of the longitudinal

component of a wave with kx ¼ 0:1 is calculated by

k ¼ 2ptbkx

� 21:36 nm. For such a wave, the effect of shear

deformation in explaining the characteristics of the waves

becomes so important. In this case, the NRBM overesti-

mates the flexural, phase velocity, and group velocity of

flexural waves based on the NTBM with relative error of

about 43, 43, and 88%, respectively. Concerning the shear

waves, the shear frequencies and the corresponding phase

velocities would reduce as the small-scale parameter

grows. Such a fact is more obvious for higher values of

wavenumbers. Further, the shear frequencies would

increase by an increase in the wavenumber; however, the

phase velocity would decrease as the wavenumber increa-

ses. The plots of the nonlocal group velocities take their

peaks at particular levels of the wavenumber. Actually, for

wavenumbers lower than these particular values, the group

velocities would increase with the wavenumber; never-

theless, for wavenumbers greater than these special values,

the group velocities would decrease mildly as the

wavenumber grows.

6.3 Effect of the magnetic field strengthon the frequencies, phase velocities,and group velocities

We are also interested in finding out that how variation of

the magnetic field strength could influence the character-

istics of the transverse waves in magnetically affected

nanosystem. To this end, the plots of frequencies, phase

velocities, and group velocities as a function of dimen-

sionless magnetic field are demonstrated in Fig. 3. The

plotted results are for fairly high populated ensembles with

Ny ¼ Nz ¼ 100 which is aimed to be used for transferring

transverse waves with kx ¼ ky ¼ kz ¼p10

. According to the

obtained results, the flexural frequencies as well as their

corresponding phase velocities would magnify by an

increase in the magnetic field strength. The continuous

models based on the NRBM and NTBM confirm this fact.

However, the frequencies and phase velocities of shear

transverse waves would vary slightly with magnetic field

strength up to a specific level. One of the influential factors

on such a special level is the small-scale parameter.

Commonly, this special level would lessen by increasing

the small-scale parameter. For those values of the magnetic

field strength greater than this special level, the afore-

mentioned characteristics of shear waves would almost

increase linearly in terms of the magnetic field strength.

0 0.25 0.50

10

20

ω1 (

TH

z)

0 0.25 0.50

20

40

vp 1 (

km/s

)

0 0.25 0.50

20

40

vg 1 (

km/s

)

0 0.25 0.50

10

20

ω2 (

TH

z)

0 0.25 0.50

20

40

vp 2 (

km/s

)

0 0.25 0.50

20

40

vg 2 (

km/s

)

0 0.25 0.510

20

30

ω3 (

TH

z)

0 0.25 0.510

20

30

vp 3 (

km/s

)

0 0.25 0.50

10

20

vg 3 (

km/s

)

0 0.25 0.510

20

30

Hx R

ω4 (

TH

z)

0 0.25 0.510

20

30

Hx R

vp 4 (

km/s

)

0 0.25 0.50

10

20

Hx R

vg 4 (

km/s

)

Fig. 3 Plots of crucial

characteristics of elastic waves

in terms of strength of magnetic

field for three values of the

small-scale parameter:

((. . .) NRBM, (��) NTBM; (�)e0a ¼ 0, (h) e0a ¼ 1, (M)

e0a ¼ 2 nm; kx ¼ ky ¼ kz ¼p10;

Ny ¼ Nz ¼ 100)

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418 Page 19 of 25 418

123

Additionally, the rate of variation of these characteristics in

terms of the magnetic field strength is more obvious for

higher levels of the small-scale parameter. Generally, the

predicted flexural frequencies as well as their correspond-

ing phase and group velocities by the NRBM are greater

than those of the NTBM. A close scrutiny of the plotted

results indicates that for a given small-scale parameter, the

relative discrepancies between the flexural frequencies and

their phase velocities by the NRBM and those of the

NTBM would reduce by growing of the magnetic field

strength up to the above-mentioned special value. For

magnetic field strength greater than this special value, these

discrepancies would grow as the magnetic field strength

increases. Such an important issue is only interpreted by

the nonlocal models, while the CET predicts that the

flexural frequencies and phase velocities of the Rayleigh

beam model become closer to those of the Timoshenko

beam model by increasing the magnetic field strength. Such

an interesting issue is attributed to the incorporation of the

nonlocality to the transverse stiffness of the nanosystem.

6.4 Effect of the intertube distanceon the frequencies, phase velocities,and group velocities

A fascinating study has been conducted to explain the role

of the intertube distance on the characteristics of waves

within magnetically affected nanosystem of vertical

SWCNTs using the suggested continuous-based models. In

Fig. 4, the plots of the frequencies, phase velocities, and

group velocities of the nanosystem in terms of the intertube

distance have been presented in the case of ky ¼ kz ¼p2,

kx ¼p50

, Ny ¼ Nz ¼ 5, and e0a ¼ 1 nm. The plotted results

have been given for three levels of the magnetic field

strength (i.e., HR

x ¼ 0:05; 0:1, and 0.15). Since the longi-

tudinal wavelength of the wave is high enough (i.e.,

k ¼ 34 nm), the predicted flexural frequencies as well as

their phase and group velocities by the NRBM and those of

the NTBM are close to each other for all considered levels

of the magnetic field strength. As it is obvious in Fig. 4, the

flexural frequencies and their phase velocities would sub-

stantially decrease with the intertube distance up to a

particular level. For intertube distances greater than this

special level, these waves’ characteristics would grow

mildly by increasing the intertube distance. Such a strange

scenario is mainly related to the mechanism of the in-plane

vdW interactional force between two tubes. Actually, a

close survey of the plot Cvjj-d shows that the above-men-

tioned explanations also hold true for this graph. (For the

sake of conciseness, this graph has not been presented.)

Regarding the shear effect, for all considered levels of the

magnetic field strength, the discrepancies between the

predicted results by the NRBM and those of the NTBM

2.34 2.52 2.70

1

2

ω1 (

TH

z)

2.34 2.52 2.70

5

10

vp 1 (

km/s

)

2.34 2.52 2.70

5

10

vg 1 (

km/s

)

2.34 2.52 2.70

1

2

ω2 (

TH

z)

2.34 2.52 2.70

5

10

vp 2 (

km/s

)

2.34 2.52 2.70

5

10

vg 2 (

km/s

)

2.34 2.52 2.713.674

13.676

13.678

ω3 (

TH

z)

2.34 2.52 2.774

74.01

74.02

vp 3 (

km/s

)

2.34 2.52 2.71

1.5

2

vg 3 (

km/s

)

2.34 2.52 2.713.674

13.676

13.678

d (nm)

ω4 (

TH

z)

2.34 2.52 2.774

74.01

74.02

d (nm)

vp 4 (

km/s

)

2.34 2.52 2.71

1.5

2

d (nm)

vg 4 (

km/s

)

Fig. 4 Plots of crucial

characteristics of elastic waves

in terms of intertube distance for

three levels of the strength of

magnetic field: ((. . .) NRBM,

(��) NTBM; (�) HR

x ¼ 0:05,

(h) HR

x ¼ 0:1, (M) HR

x ¼ 0:15;

kx ¼p50, ky ¼ kz ¼

p2;

Ny ¼ Nz ¼ 5; e0a ¼ 1 nm)

418 Page 20 of 25 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418

123

would increase by growing the intertube distance up to the

above-mentioned particular levels. By passing these special

levels, the shear effect would diminish by increasing the

intertube distance. Further, for a given intertube distance,

the shear effect would lessen by an increase in the magnetic

field strength. The plotted results clearly demonstrate that

the group velocities pertinent to the flexural frequencies

increase in terms of the intertube distance up to a specific

level, and thereafter, their values would slightly decrease as

a function of the intertube distance until taking their locally

minimum points, and finally, such group velocities would

slightly grow by increasing the intertube distance. The

above-mentioned scenarios are also valid for shear fre-

quencies and their corresponding phase velocities; how-

ever, their variations are lower sensitive to the variation of

the intertube distance with respect to those of the flexural

waves. Additionally, no detectable variation is observed for

the plots of group velocities of shear waves in terms of the

intertube distance.

6.5 Effect of the number of SWCNTson the frequencies, phase velocities,and group velocities

The designers of the magnetically affected periodic jungles

of SWCNTs for the considered jobs should also reply to

this crucial question: how population of the nanosystem

would influence its vibrational behavior? To answer this,

the plots of the frequencies, phase velocities, and group

velocities as a function of the number of rows of SWCNTs

in the y direction are provided in Fig. 5. The results have

been plotted for a nanosystem with e0a ¼ 2 nm which is

acted upon by a longitudinal magnetic field with

HR

x ¼ 0:05. Such a magnetically influenced nanosystem is

going to be exploited for transferring of elastic waves with

ky ¼ kz ¼p10

for three levels of the wavenumber of the

longitudinal component (i.e., kx ¼p100

,2p25

, and p). Con-

cerning flexural waves, frequencies as well as their corre-

sponding phase velocities would commonly reduce by an

increase in the nanosystem’s population. Generally, the

rate of reduction is more obvious for lower levels of the

population as well as wavenumber. For lower values of the

wavenumber (i.e., higher wavelengths), the longitudinal

wave tends to propagate in a flexural manner within the

nanosystem. In such a circumference, the share of shear

strain energy to the flexural one would reduce, and thereby,

the predicted results by the NRBM become close to those

of the NTBM. It is worth mentioning that for low levels of

the wavenumber and the population, the predicted flexural

frequencies and corresponding phase velocities by the

NTBM are underestimated by the NRBM. By increasing

the population of the magnetically affected nanosystem,

the relative discrepancies between the above-mentioned

results of the NRBM and those of the NTBM would usually

increase. Furthermore, the group velocities associated with

the flexural frequencies commonly increase by growing of

4 9 140

1

2

ω1 (

TH

z)

4 9 140

5

10

vp 1 (

km/s

)

4 9 140

5

10 v

g 1 (km

/s)

4 9 140

1

2

ω2 (

TH

z)

4 9 140

5

10

vp 2 (

km/s

)

4 9 140

5

10

vg 2 (

km/s

)

4 9 1412

13

14

ω3 (

TH

z)

4 9 140

100

200

vp 3 (

km/s

)

4 9 140.4

0.6

0.8

vg 3 (

km/s

)

4 9 1412

13

14

Ny

ω4 (

TH

z)

4 9 140

100

200

Ny

vp 4 (

km/s

)

4 9 140.4

0.6

0.8

Ny

vg 4 (

km/s

)Fig. 5 Plots of crucial

characteristics of elastic waves

in terms of population of the

nanosystem for three values of

the wavenumber: ((. . .) NRBM,

(��) NTBM; (�) kx ¼p100

, (h)

kx ¼2p25

, (M) kx ¼ p;

ky ¼ kz ¼p10; e0a ¼ 2 nm;

HR

x ¼ 0:054)

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418 Page 21 of 25 418

123

the nanosystem’s population and the rate of growth is more

apparent for lower levels of the population. Regarding

shear waves, the variation of the population of the

nanosystem has a slight influence on the variation of the

shear frequencies as well as their pertinent phase and group

velocities.

6.6 Effect of the radius of constitutive SWCNTson the frequencies, phase velocities,and group velocities

One of the major geometrical factors that could influence

the characteristics of transverse waves is the radius of the

constitutive SWCNTs. In Fig. 6, the plots of both flexural

and shear frequencies as well as their corresponding phase

and group velocities in terms of the radius of SWCNTs

have been demonstrated. The results of both continuous-

based NRBM and NTBM are provided for three levels of

the magnetic field strength (i.e., HR

x ¼ 0; 0:25, and 0.5)

when the magnetically affected nanosystem with 200�200 SWCNTs has been exploited for transferring of waves

with kx ¼ ky ¼ kz ¼ p10. In the absence of the longitudinal

magnetic field, the flexural frequencies as well as their

pertinent phase velocities would magnify by an increase in

the radius of SWCNTs; however, their corresponding

group velocities would somewhat reduce by increasing the

radius of SWCNTs according to the NRBM. In general, the

rate of decrease/increase of the obtained results by the

NRBM is more apparent with respect to that of the NTBM.

By applying the longitudinal magnetic field with strength

HR

x0 ¼ 0:25 or 0.5, the flexural frequencies and the corre-

sponding phase velocities decrease in terms of the radius of

SWCNTs. In the case of no magnetic field, the relative

discrepancies between the predicted flexural frequencies as

well as corresponding phase and group velocities by the

NRBM and those of the NTBM would generally increase

as the radius of SWCNTs grows. For example, for a set of

radii of SWCNTs (1, 2, 3) nm, the NRBM could reproduce

the flexural frequencies and their phase velocities of the

NTBM with relative error about (46.7, 82.4, 97.3)%,

respectively. In the case of HR

x0 ¼ 0:25, the NRBM could

reproduce the flexural frequencies and the phase velocities

of the NTBM with relative error lower than 1.5% for all

considered levels of the radius of SWCNTs. Concerning

the case of HR

x0 ¼ 0:5, the predicted flexural frequencies

and the phase velocities by the NRBM would approach to

those of the NTBM as the radius of SWCNTs increases.

For instance, for magnetically affected periodic jungles of

SWCNTs with rm ¼ 1, 2, and 3 nm, the relative discrep-

ancies between the flexural frequencies and the phase

velocities by the NRBM and those of the NTBM in order

are approximately equal to 58.2, 47.2, and 30.9%. The

main reason behind this fact is that the exertion of a

1 2 30

10

20

ω1 (

TH

z)

1 2 30

10

20

vp 1 (

km/s

)

1 2 30

10

20

vg 1 (

km/s

)

1 2 30

10

20

ω2 (

TH

z)

1 2 30

10

20

vp 2 (

km/s

)

1 2 30

10

20

vg 2 (

km/s

)

1 2 30

20

40

ω3 (

TH

z)

1 2 30

20

40

vp 3 (

km/s

)

1 2 30

5

10

vg 3 (

km/s

)

1 2 30

20

40

rm

(nm)

ω4 (

TH

z)

1 2 30

20

40

rm

(nm)

vp 4 (

km/s

)

1 2 30

5

10

rm

(nm)

vg 4 (

km/s

)

Fig. 6 Plots of crucial

characteristics of elastic waves

in terms of radius of SWCNTs

for three levels of the strength of

magnetic field: ((. . .) NRBM,

(��) NTBM; (�) HR

x0 ¼ 0, (h)

HR

x0 ¼ 0:25, (M) HR

x0 ¼ 0:5;

HR

x ¼ HR

x0

ffiffiffiffiffiffiffiffiffiffiffiAbIb0

Ab0Ib

r, rm0 ¼ 1 nm,

kx ¼ ky ¼ kz ¼p10;

Ny ¼ Nz ¼ 100; e0a ¼ 2 nm)

418 Page 22 of 25 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418

123

longitudinal magnetic field on periodic arrays of SWCNTs

leads to an increase in the transverse stiffness of each tube

of the nanosystem. The role of magnetic fields on the lat-

eral stiffness of highly conductive beam-like nanostruc-

tures is somehow similar to that of pre-tensioning forces.

By increasing the strength of the magnetic field, the ratio of

the flexural or shear lateral stiffness of the nanosystem to

that caused by the exertion of the longitudinal magnetic

field would reduce. In fact, for high levels of the magnetic

field strength, the major portion of the transverse stiffness

is provided by the magnetic field. Thereby, it is anticipated

that the results of the NRBM become close to the results of

the NTBM with an increase in the strength of magnetic

field. Regarding shear waves, the predicted shear fre-

quencies and the corresponding phase velocities by the

NTBM would slightly reduce by increasing the radius of

SWCNTs. Such a lessening is less apparent for higher

levels of the magnetic field strength. Furthermore, the

group velocities of shear waves grow with the radius of

SWCNTs and the growth’s rate is more apparent for lower

levels of the nanotube’s radius.

6.7 Effect of the nonlocality on the frequencies,phase velocities, and group velocities

The nonlocality plays an important role in the mechanical

response of nanostructures; however, its effect on wave

analysis of magnetically affected periodic arrays of

SWCNTs has not been thoroughly understood. To explain

this effect, the plots of both flexural and shear frequencies

as well as corresponding phase and group velocities as a

function of the small-scale parameter are presented in

Fig. 7. The results are given for a nanosystem with 100�100 tubes subjected to a longitudinal magnetic field of

strength HR

x =0.05. The wavenumbers in the y--z plane have

been kept fixed (i.e., ky ¼ kz ¼p20

), while three levels for

the wavenumber in the longitudinal direction have been

considered (i.e., kx ¼p20

,p10

, andp5). Commonly, the pre-

dicted flexural frequencies, phase and group velocities

would reduce by increasing the small-scale parameter. The

rate of reduction of the results of the NRBM is more

apparent with respect to that of the NTBM’s results.

Additionally, such a fact becomes more obvious for lower

levels of the small-scale parameter. A detailed scrutiny of

the plotted results indicates that the discrepancies between

the predicted results by the NRBM and those of the NTBM

would generally increase by growing of the longitudinal

wavenumber. Such a fact is mainly attributed to this fact

that the influence of the shear deformation on the vibration

of the nanosystem becomes more crucial as the wavelength

of the longitudinal wave would decrease. In general, the

discrepancies between the flexural frequencies as well as

corresponding phase and group velocities by the NTBM

and those of the NRBM would lessen as the small-scale

0 1 20

20

40

ω1 (

TH

z)

0 1 20

10

20

vp 1 (

km/s

)

0 1 20

20

40

vg 1 (

km/s

)

0 1 20

20

40

ω2 (

TH

z)

0 1 20

10

20

vp 2 (

km/s

)

0 1 20

20

40

vg 2 (

km/s

)

0 1 20

20

40

ω3 (

TH

z)

0 1 20

20

40

vp 3 (

km/s

)

0 1 20

5

10

vg 3 (

km/s

)

0 1 20

20

40

e0 a (nm)

ω4 (

TH

z)

0 1 20

20

40

e0 a (nm)

vp 4 (

km/s

)

0 1 20

5

10

e0 a (nm)

vg 4 (

km/s

)

Fig. 7 Plots of crucial

characteristics of elastic waves

in terms of small-scale

parameter for three values of the

longitudinal wavenumber: ((. . .)NRBM, (��) NTBM; (�)kx ¼ p

20, (h) kx ¼ p

10, (M) kx ¼ p

5;

ky ¼ kz ¼ p20; Ny ¼ Nz ¼ 100;

HR

x ¼ 0:05)

Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418 Page 23 of 25 418

123

parameter increases. Such a lessening is more obvious for

higher longitudinal wavenumbers. According to the

demonstrated results in Fig. 7, the shear frequencies and

their corresponding phase and group velocities would

reduce by an increase in the small-scale parameter.

7 Concluding remarks

Mechanical aspects of transverse waves in infinite-periodic

jungles of SWCNTs subjected to longitudinal magnetic

fields were studied using nonlocal continuum field theory

of Eringen. The interactional transverse vdW forces

between each pair of infinite tubes were evaluated. Such

forces are modeled by linearly continuous springs. Using

NRBM and NTBM, the governing equations of the

nanosystem were derived by exploiting discrete modeling

as well as continuous modeling of the problem. According

to the suggested NDMs, vibrations of each tube are

investigated accounting for the vdW interactional forces

between the tube and its neighboring ones. From the con-

tinuous modeling point of view, transverse waves within all

constitutive nanotubes of the nanosystem are displayed by

establishing two and four governing equations according to

the NRBM and NTBM, respectively. It implies that the

needed time and computational efforts of the developed

NCMs are generally lower than those of the NDMs. Such a

fact becomes more obvious for highly populated nanosys-

tem in magnetic fields. The capabilities of the proposed

NCMs in capturing the frequencies of waves were inves-

tigated, and their efficiency was also explained. By

employing these efficient models, the influences of the

wavenumber, strength of magnetic field, small-scale

parameter, radius of SWCNT, intertube distance, and

population of the nanosystem on the main characteristics of

transverse waves were scrutinized. It was proved that the

longitudinal magnetic field could be exploited as an effi-

cient way to control transverse waves in periodic jungles of

vertically aligned SWCNTs.

Compliance with ethical standards

Conflict of interest The author declares that he has no conflict of

interest.

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