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
[email protected]; [email protected]
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.
References
1. Coleman JN, Khan U, Blau WJ, Gun’ko YK (2006) Small but
strong: a review of the mechanical properties of carbon nanotube-
polymer composites. Carbon 44:1624–1652
2. Fritzsche J, Lorenz H, Klueppel M (2009) CNT based elastomer-
hybrid-nanocomposites with promising mechanical and electrical
properties. Macromol Mater Eng 294:551–560
3. Sahoo NG, Rana S, Cho JW, Li L, Chan SH (2010) Polymer
nanocomposites based on functionalized carbon nanotubes. Prog
Polym Sci 35:837–867
4. Qian H, Greenhalgh ES, Shaffer MSP, Bismarck A (2010) Car-
bon nanotube-based hierarchical composites: a review. J Mater
Chem 20:4751–4762
5. Thostenson ET, Chou TW (2002) Aligned multi-walled carbon
nanotube-reinforced composites: processing and mechanical
characterization. J Phys D Appl Phys 35:L77
6. Haggenmueller R, Gommans HH, Rinzler AG, Fischer JE, Winey
KI (2000) Aligned single-wall carbon nanotubes in composites by
melt processing methods. Chem Phys Lett 330:219–225
7. Xie XL, Mai YW, Zhou XP (2005) Dispersion and alignment of
carbon nanotubes in polymer matrix: a review. Mater Sci Eng R
49:89–112
8. Wang D, Song P, Liu C, Wu W, Fan S (2008) Highly oriented
carbon nanotube papers made of aligned carbon nanotubes.
Nanotechnology 19:075609
9. Kimura T, Ago H, Tobita M, Ohshima S, Kyotani M, Yumura M
(2002) Polymer composites of carbon nanotubes aligned by a
magnetic field. Adv Mater 14:1380–1383
10. Garmestani H, Al-Haik MS, Dahmen K, Tannenbaum R, Li D,
Sablin SS, Hussaini MY (2003) Polymer-mediated alignment of
carbon nanotubes under high magnetic fields. Adv Mater
15:1918–1921
11. Correa-Duarte MA, Grzelczak M, Salgueirino-Maceira V, Gier-
sig M, Liz-Marzan LM, Farle M, Sierazdki K, Diaz R (2005)
Alignment of carbon nanotubes under low magnetic fields
through attachment of magnetic nanoparticles. J Phys Chem B
109:19060–19063
12. Choi ES, Brooks JS, Eaton DL, Al-Haik MS, Hussaini MY,
Garmestani H, Li D, Dahmen K (2003) Enhancement of thermal
and electrical properties of carbon nanotube polymer composites
by magnetic field processing. J Appl Phys 94:6034–6039
13. Camponeschi E, Vance R, Al-Haik M, Garmestani H, Tannen-
baum R (2007) Properties of carbon nanotube-polymer compos-
ites aligned in a magnetic field. Carbon 45:2037–2046
14. Eringen AC (1972) Linear theory of nonlocal elasticity and dis-
persion of plane waves. Int J Eng Sci 10:425–435
15. Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci
10:1–16
16. Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng
Sci 10:233–248
17. Wang Q, Varadan VK (2006) Vibration of carbon nanotubes
studied using nonlocal continuum mechanics. Smart Mater Struct
15:659
18. Wang Q, Wang CM (2007) The constitutive relation and small
scale parameter of nonlocal continuum mechanics for modelling
carbon nanotubes. Nanotechnology 18:075702
19. Reddy JN, Pang SD (2008) Nonlocal continuum theories of
beams for the analysis of carbon nanotubes. J Appl Phys
103:023511
20. Aydogdu M (2009) A general nonlocal beam theory: its appli-
cation to nanobeam bending, buckling and vibration. Physica E
41:1651–1655
21. Murmu T, Pradhan SC (2009) Thermo-mechanical vibration of a
single-walled carbon nanotube embedded in an elastic medium
based on nonlocal elasticity theory. Comput Mater Sci
46:854–859
22. Yang J, Ke LL, Kitipornchai S (2010) Nonlinear free vibration of
single-walled carbon nanotubes using nonlocal Timoshenko
beam theory. Phys E 42:1727–1735
418 Page 24 of 25 Journal of the Brazilian Society of Mechanical Sciences and Engineering (2019) 41:418
123
23. Wang CM, Zhang Z, Challamel N, Duan WH (2013) Calibration
of Eringen’s small length scale coefficient for initially stressed
vibrating nonlocal Euler beams based on microstructured beam
model. J Phys D Appl Phys 46:345501
24. Ke LL, Wang YS (2014) Free vibration of size-dependent mag-
neto-electro-elastic nanobeams based on the nonlocal theory.
Phys E 63:52–61
25. Mehdipour I, Erfani-Moghadam A, Mehdipour C (2013) Appli-
cation of an electrostatically actuated cantilevered carbon nan-
otube with an attached mass as a bio-mass sensor. Curr Appl Phys
13:1463–1469
26. Fang B, Zhen YX, Zhang CP, Tang Y (2013) Nonlinear vibration
analysis of double-walled carbon nanotubes based on nonlocal
elasticity theory. Appl Math Model 37:1096–1107
27. Karlicic D, Kozic P, Pavlovic R (2016) Nonlocal vibration and
stability of a multiple-nanobeam system coupled by the Winkler
elastic medium. Appl Math Model 40:1599–1614
28. Stamenkovic M, Karlicic D, Goran J, Kozic P (2016) Nonlocal
forced vibration of a double single-walled carbon nanotube sys-
tem under the influence of an axial magnetic field. J Mech Mater
Struct 11:279–307
29. Wang Q (2005) Wave propagation in carbon nanotubes via
nonlocal continuum mechanics. J Appl Phys 98:124301
30. Heireche H, Tounsi A, Benzair A, Maachou M, Adda Bedia EA
(2008) Sound wave propagation in single-walled carbon nan-
otubes using nonlocal elasticity. Phys E 40:2791–2799
31. Li XF, Wang BL, Mai YW (2008) Effects of a surrounding elastic
medium on flexural waves propagating in carbon nanotubes via
nonlocal elasticity. J Appl Phys 103:074309
32. Hu YG, Liew KM, Wang Q (2009) Nonlocal elastic beam models
for flexural wave propagation in double-walled carbon nanotubes.
J Appl Phys 106:044301
33. Kiani K (2012) Transverse wave propagation in elastically con-
fined single-walled carbon nanotubes subjected to longitudinal
magnetic fields using nonlocal elasticity models. Phys E
45:86–96
34. Yang Y, Zhang L, Lim CW (2011) Wave propagation in double-
walled carbon nanotubes on a novel analytically nonlocal
Timoshenko-beam model. J Sound Vib 330:1704–1717
35. Besseghier A, Tounsi A, Houari MSA, Benzair A, Boumia L,
Heireche H (2011) Thermal effect on wave propagation in
double-walled carbon nanotubes embedded in a polymer matrix
using nonlocal elasticity. Phys E 43:1379–1386
36. Wang YZ, Li FM, Kishimoto K (2012) Effects of axial load and
elastic matrix on flexural wave propagation in nanotube with
nonlocal Timoshenko beam model. J Vib Acoust 134:031011
37. Wang B, Deng Z, Ouyang H, Zhang K (2013) Wave character-
istics of single-walled fluid-conveying carbon nanotubes sub-
jected to multi-physical fields. Phys E 52:97–105
38. Wang B, Deng Z, Ouyang H, Xu X (2015) Free vibration of wavy
single-walled fluid-conveying carbon nanotubes in multi-physics
fields. Appl Math Model 39:6780–6792
39. Arani AG, Roudbari MA, Amir S (2016) Longitudinal magnetic
field effect on wave propagation of fluid-conveyed SWCNT using
Knudsen number and surface considerations. Appl Math Model
40:2025–2038
40. Kiani K (2014) Nonlocal continuous models for forced vibration
analysis of two- and three-dimensional ensembles of single-
walled carbon nanotubes. Phys E 60:229–245
41. Kiani K (2014) Axial buckling analysis of vertically aligned
ensembles of single-walled carbon nanotubes using nonlocal
discrete and continuous models. Acta Mech 225:3569–3589
42. Kiani K (2014) In- and out-of-plane dynamic flexural behaviors
of two-dimensional ensembles of vertically aligned single-walled
carbon nanotubes. Phys B 449:164–180
43. Kiani K (2014) Nonlocal discrete and continuous modeling of
free vibration of stocky ensembles of single-walled carbon nan-
otubes. Curr Appl Phys 14:1116–1139
44. Kiani K (2015) Nonlocal and shear effects on column buckling of
single-layered membranes from stocky single-walled carbon
nanotubes. Compos Part B Eng 79:535–552
45. Kiani K (2016) Free vibration of in-plane-aligned membranes of
single-walled carbon nanotubes in the presence of in-plane-uni-
directional magnetic fields. J Vib Control 22:3736–3766
46. Gupta SS, Batra RC (2008) Continuum structures equivalent in
normal mode vibrations to single-walled carbon nanotubes.
Comput Mater Sci 43:715–723
47. Eringen AC (2002) Nonlocal continuum field theories. Springer,
New York
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