european journal of mechanics / a solidswp.kntu.ac.ir/k_kiani/papers-2018/nonlocal...
TRANSCRIPT
Contents lists available at ScienceDirect
European Journal of Mechanics / A Solids
journal homepage: www.elsevier.com/locate/ejmsol
Nonlocal magneto-thermo-vibro-elastic analysis of vertically aligned arraysof single-walled carbon nanotubes
Keivan Kiania,∗, Quan Wangb
a Department of Civil Engineering, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, IranbDepartment of Mechanics and Aerospace Engineering, Southern University of Science and Technology, Shenzhen 518055, PR China
A R T I C L E I N F O
Keywords:Magneto-thermo-elastic vibrationVertically aligned arrays of single-walledcarbon nanotubesNonlocal Rayleigh beamNonlocal Timoshenko beamNonlocal continuous modelAssumed mode method
A B S T R A C T
This study concerns with transverse vibrations of magnetically-thermally affected vertically aligned arrays ofsingle-walled carbon nanotubes (SWCNTs) using nonlocal elasticity theory of Eringen. Using nonlocal Rayleighand Timoshenko beam theories, both discrete and continuous versions of equations of motion are presented. Incontrast to the discrete models, the continuous models do not suffer from huge time and labor costs for nano-systems with high population. The capability and efficiency of the continuous models in capturing the fre-quencies of the discrete models are displayed, and a reasonably good agreement is obtained. Subsequently, theinfluences of radius of SWCNTs, slenderness ratio, population, small-scale parameter, strength of magnetic field,and variation of the temperature on the fundamental frequency are explained and discussed. Additionally, therole of shear deformation on the obtained results is explained and the limitations of the nonlocal Rayleigh beammodel are revealed.
1. Introduction
From structural mechanics standpoint, vertically aligned jungles ofcarbon nanotubes (CNTs) exhibit nanosystems consist of fairly parallelCNTs close to each other in which dynamically interacted by intertubevan der Waals (vdW) and friction forces. The vdW forces and crowdingeffects are among major factors that enforce these tiny elements to growvertically on the substrate during synthesis process. These nanosystemshave great potential applications in various branches of technologiesand science such as transistors (Xiao et al., 2003; Choi et al., 2003,2004; Qu et al., 2008), supercapacitors (Du et al., 2005; Futaba et al.,2006; Wang, 2005a,b; Zhang et al., 2008), biological sensors (Wang,2005a,b; Zhang et al., 2008), physical sensors (Tung et al., 2007; Huanget al., 2007), chemical sensors (Cottineau et al., 2013; Silva et al.,2015), fuel cells (Tian et al., 2011; Murata et al., 2014), field-emissiondevices (Fan et al., 1999; Tsai et al., 2009), blackbody absorbers(Mizuno et al., 2009), and thermal interface materials (TIMs) (Tonget al., 2007). All of these promising functionalities are indebted to ex-traordinary electrical, mechanical, physical, and chemical of CNTs. Forexample, the latter application has been of interest to engineers of high-power electronic nanosystems due to the high thermal conductivity ofCNTs. In this view, vertically aligned CNTs-polymer nanocompositeshave been proposed as effective TIMs (Ngo et al., 2004; Marconnetet al., 2011) in which a combination of high thermal conductivity, low
thermal interface resistance with the nearby microprocessor, and re-markable mechanical compliance is required. For most of the above-mentioned applications, one confronts to vertically aligned arrays ofsingle-walled carbon nanotubes (ASWCNTs) subjected to multi-physicalfields; however, their mechanical responses are still uncovered, andtheir statics, buckling, and vibration behaviors are highly needed to beexamined via appropriate models.
At the atomic level, each atom interacts with its nearby atoms due tothe inter-atomic forces such as short-range vdW forces, long-range ionicforces, metallic, covalent, and hydrogen-bond. It implies that vibrationeach atom relies on vibrations and states of its neighboring atoms. Thisfact cannot be explained by the classical continuum mechanics (CCM)at all. To overcome this shortening of the CCM, Eringen (1966, 1972,2002) proposed theory of nonlocal continuum mechanics (NCM). Onthe basis of this advanced theory, the magnitude of each physical field(for example, deformation, strain, stress, thermal, electric, and mag-netic fields) at a point of the continuum does not only depend on thatfield at that point, but also on the magnitude of that field at theneighboring points. Such a fact is commonly called nonlocality and thisplays a crucial role in mechanical analysis of nanostructures. Because ofsimplicity of the NCM of Eringen in application to the local governingequations, this theory has been frequently adopted to examine variousmechanical aspects of CNTs and nanobeams including buckling (Wanget al., 2006a,b; Adali, 2008; Khademolhosseini et al., 2010), free
https://doi.org/10.1016/j.euromechsol.2018.05.017Received 3 November 2017; Received in revised form 25 May 2018; Accepted 28 May 2018
∗ Corresponding author.E-mail addresses: [email protected] (K. Kiani), [email protected] (Q. Wang).
European Journal of Mechanics / A Solids 72 (2018) 497–515
Available online 02 June 20180997-7538/ © 2018 Elsevier Masson SAS. All rights reserved.
T
vibrations (Wang and Wang, 2007; Wang, 2005a,b; Hu et al., 2008;Wang and Varadan, 2007; Yang et al., 2010; Arash and Ansari, 2010; Keet al., 2009; Arash and Wang, 2012), and forced vibrations (Karaogluand Aydogdu, 2010; Simsek, 2011; Kiani, 2014a,b; Uymaz, 2013).Additionally, vibrational analysis of CNTs as well as beam-like nanos-tructures in thermal fields (Benzair et al., 2008; Murmu and Pradhan,2009; Ansari and Ramezannezhad, 2011; Zidour et al., 2012), magneticfields (Kiani, 2013; Narendar et al., 2012; Murmu et al., 2012), electro-magneto fields (Ke and Wang, 2014; Ke et al., 2014b), and both thermaland magnetic fields (Ke et al., 2014a; Ansari et al., 2015) have beenperformed and a sufficient knowledge on these subjects has beenemerged in the light of appropriate nonlocal models.
In a particular view, ensembles of vertically aligned SWCNTs couldbe categorized into two major groups: membranes and jungles that inorder present two- and three-dimensional patterns. Despite of manyresearches on a variety of problems relevant to mechanical response ofCNTs, there are rare research works on statics, buckling, and vibrationsof these important elements of the future technologies. Until now,buckling (Kiani, 2014c, 2015a), free vibration as well as sound wavesanalysis (Kiani, 2014d,e, 2016, 2015b), and forced vibrations (Kiani,2014f) of groups of vertically aligned SWCNTs have been investigatedusing nonlocal beam theories. However, their transverse vibrations inthe presence of multi-physical fields have not been displayed yet.
This work deals with magneto-thermo-elastic vibrations of verticallyaligned ASWCNTs in the context of the nonlocal continuum field theoryof Eringen (Eringen, 1966, 1972). Accounting for the interactional vdWforces, the discrete and continuous forms of the governing equations aredeveloped using the nonlocal Rayleigh beam model (NRBM) and non-local Timoshenko beam model (NTBM). The main feature of the con-tinuous models is fairly accurate prediction of free vibration of highly
populated nanosystems in which discrete models commonly suffer fromhuge computational efforts. The roles of influential factors on vibrationbehavior of the nanosystem under various magneto-thermal environ-ments are addressed. Special attention is also paid to the role of non-locality and shear deformation in the predicted frequencies. The pro-posed continuous models in this work would be very useful in designand analysis of thermally-magnetically affected micro-/nano-electro-mechanical systems made from vertically aligned SWCNTs arrays.
2. Nonlocal discrete models using classical and shear deformablebeam theories
Consider a vertically aligned ASWCNTs consists of Ny and Nz
nanotubes in the y and z directions, respectively. The length of all na-notubes are lb and they have been grown uniformly in both y and zdirections with intertube distance d (see Fig. 1(a)). Such a nanosystemis subjected to a steady longitudinal magnetic field of strength Hx and aconstant temperature change TΔ . Both ends of the constitutive tubes ofthe nanosystem are not allowed to move longitudinally and transver-sely. For modeling of these tiny elements, their corresponding equiva-lent continuum structures are employed whose mean radius, thickness,and length in order are rm, Nz, and lb, such that in our analysis the wall'sthickness of nanotubes is set equal to 0.34 nm.
In the following sections, appropriate nonlocal continuum-basedmodels are planning to be developed for exploring transverse vibrationsof such a magnetically-thermally affected nanosystem. For this purpose,suitable discrete and continuous models are developed based on theNRBM and NTBM.
2.1. Free transverse vibrations of magnetically-thermally affectedASWCNTs using NRBM
We are initially interested in establishing the discrete equations ofmotion of the nanosystem under both steady-longitudinal thermal andmagnetic field on the basis of the Rayleigh beam theory. For this pur-pose, we write down the expressions of the kinetic energy (TR), thestrain energy (UR), and the work done by the exerted magnetic field onthe deformed nanosystem (WR) at each time t as in the following form:
∑ ∑ ∫ ⎜
⎟
⎜ ⎟
⎜ ⎟
= ⎛⎝
⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
⎞⎠
= =
∂∂
∂∂
∂∂ ∂
∂∂ ∂
( ) ( )( ) ( )
T tN N
ρ A
I x
( )
d ,
R
m
y
n
zl
b bV
tW
t
bVt x
Wt x
12
1 10
2 2
2 2
b mnR
mnR
mnR
mnR2 2
(1a)
∑ ∑ ∫ ⎜ ⎟= ⎡
⎣⎢
⎛⎝
⎞⎠
+ ⎤
⎦⎥
= =
W tN N
f V f W x( ) ( ) d ,R
m
y
n
zl
yR
mnmnR
zR
mn mnR
1 10
b
(1c)
where VmnR =V x t( , )mn
R and WmnR =W x t( , )mn
R represent the transversedeflections of the m n( , )th nanotube along the y and z axes, respec-tively, δij is the Kronecker delta tensor, Ab is the cross-sectional area ofthe nanotube, ρb is the density, Ib is the moment inertia, f( )y
Rmn and
f( )zR
mnare the transverse applied Lorentz forces on the m n( , )th tube,
and NT is the resulted axial force within each tube due to the tem-perature change. These forces, due to the applied steady-longitudinalthermal and magnetic fields, are evaluated by Narendar et al. (2012);
∫∑ ∑=
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
− − + + +
− − + − − +
− − + − − +
− − − +
− − − +
− − − +
− − − +
− − − +
− − − +
− − − +
− − −
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
= =
∂∂
∂∂
∂∂
∂∂
+ −
⊥ − +
− −
+ +
− +
+ −
⊥ − +
⊥ + −
⊥ − −
⊥ + +
( )
U tN N
M M N
C V V δ V V δ
C V V δ V V δ
C X X δ δ
C X X δ δ
C Y Y δ δ
C Y Y δ δ
C X X δ δ
C X X δ δ
C Y Y δ δ
C Y Y δ δ
x( ) 12
( ) ( )
[( ) (1 ) ( ) (1 )]
[( ) (1 ) ( ) (1 )]
( ) (1 )(1 )
( ) (1 )(1 )
( ) (1 )(1 )( ) (1 )(1 )
( ) (1 )(1 )( ) (1 )(1 )
( ) (1 )(1 )
( ) (1 )(1 )
d ,R
m
y
n
z l
Vx bz
nl R Wx by
nl RT
Vx
Wx
v mnR
m nR
mN mnR
m nR
m
v mnR
m nR
n mnR
m nR
nN
d mnR
m nR
m n
d mnR
m nR
mN nN
d mnR
m nR
m nN
d mnR
m nR
mN n
d mnR
m nR
m nN
d mnR
m nR
mN n
d mnR
m nR
m n
d mnR
m nR
mN nN
1 10
( 1)2
( 1)2
1
( 1)2
1 ( 1)2
( 1)( 1)2
1 1
( 1)( 1)2
( 1)( 1)2
1
( 1)( 1)2
1
( 1)( 1)2
1
( 1)( 1)2
1
( 1)( 1)2
1 1
( 1)( 1)2
b
mnR
mnmnR
mnmnR
mnR
y
z
y z
z
y
z
y
y z
2
2
2
2
2
2
2
2
(1b)
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
498
Murmu et al. (2012):
=N E A α TΔ ,T b b T (2a)
= ∂∂
f ηA H( ) ,yR
mn b xVx
2 mnR2
2 (2b)
= ∂∂
f ηA H( ) ,zR
mn b xWx
2 mnR2
2 (2c)
in which Eb is the modulus of elasticity of the nanotube, αT is thecoefficient of thermal expansion which is naturally temperature-de-pendent, η is the magnetic permeability of the SWCNT, and Hx is thestrength of the longitudinal magnetic field. The coefficients of vdWforces between two adjacent tubes in the y and z directions are given byKiani (2014d):
∫ ∫ ∫ ∫
= − ×
⎧⎨⎩
− − −
− −⎫⎬⎭
⊥
− −
− −
C r d
σ χ χ r φ φ
χ χ r φ φφ φ x x
( , )
[ 14 ( (cos cos )) ]
[ 8 ( (cos cos )) ]d d d d ,
v mε r
a l
l l π π m
σm
2569
0 0 02
02
12 7 82 1
2
24 5
2 12 1 2 1 2
mb
b b
2
4
6
(3a)
∫ ∫ ∫ ∫
= − ×
⎧⎨⎩
− + − −
− + −⎫⎬⎭
− −
− −
C r d
σ χ χ d r φ φ
χ χ d r φ φφ φ x x
( , )
[ 14 ( (sin sin )) ]
[ 8 ( (sin sin )) ]d d d d ,
v mε r
a l
l l π π m
σm
2569
0 0 02
02
12 7 82 1
2
24 5
2 12 1 2 1 2
mb
b b
2
4
6
(3b)
= − + − − + + −χ x x r φ φ d r d φ φ( ) 2 (1 cos( )) 2 (sin sin ).m m2 12 2
2 12
2 1
(3c)
Additionally, the coefficients of vdW force between two neighboringtubes in the diagonal direction are defined by: =C C r d( , 2 )d v m and
=⊥ ⊥C C r d( , 2 )d v m (see Fig. 1(b)).In Eq. (1b), M( )by
nl Rmn and M( )bz
nl Rmn denote the nonlocal bending
moments of the m n( , )th tube about the y and z axis, respectively,= +X W V( )/ 2mn
RmnR
mnR , and = − +Y W V( )/ 2mn
RmnR
mnR . In the context of
the NRBM, the above-mentioned nonlocal bending moments of them n( , )th SWCNT are linked to their corresponding local ones byPeddieson et al. (2003); Sudak (2003):
− = − ∂∂
M e a M E I( ) ( ) ( ) ,bynl R
bynl
xxR
b bWx0
2,mn mn
mnR2
2 (4a)
− = − ∂∂
M e a M E I( ) ( ) ( ) ,bznl R
bznl
xxR
b bVx0
2,mn mn
mnR2
2 (4b)
where e a0 is the small-scale parameter.By applying the Hamilton's principle to Eqs. (1a)-(1c) in view of Eqs.
(2a)-(2c), (4a), and (4b), the governing equations of the magnetically-thermally affected nanosystem modeled based on the NRBM are ob-tained as:
+ ⎧⎨⎩
− − −
− − + − − +
− − + − − +
+ − − − − +
+ − − − − +
− − + − − +
− − + − − +
− + − − − +
− + − − − +
+ − − − − +
+ − − − − =
∂∂
∂∂
∂∂ ∂
∂∂
+ −
⊥ − +
− − − −
+ + + +
+ − + −
− + − +
⊥ − − − −
⊥ + + + +
⊥ − + − +
⊥ + − + −
( )E I ρ A I ηA H N
C V V δ V V δ
C V V δ V V δ
C W V W V δ δ
C W V W V δ δ
C V W V W δ δ
C V W V W δ δ
C V W W V δ δ
C V W W V δ δ
C V W V W δ δ
C V W V W δ δ
Ξ ( )
[( )(1 ) ( )(1 )]
[( )(1 ) ( )(1 )]
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )} 0
b bVx b b
Vt b
Vt x b x T
Vx
v mnR
m nR
mN mnR
m nR
m
v mnR
m nR
n mnR
m nR
nN
d mnR
mnR
m nR
m nR
n m
d mnR
mnR
m nR
m nR
nN mN
d mnR
mnR
m nR
m nR
n mN
d mnR
mnR
m nR
m nR
nN m
d mnR
mnR
m nR
m nR
n m
d mnR
mnR
m nR
m nR
nN mN
d mnR
mnR
m nR
m nR
nN m
d mnR
mnR
m nR
m nR
n mN
2
( 1) ( 1) 1
( 1) 1 ( 1)
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
mnR
mnR
mnR
mnR
y
z
z y
y
z
z y
z
y
4
4
2
2
4
2 2
2
2
(5a)
+ ⎧⎨⎩
− − − +
− − + − − +
− − + − − +
+ − − − − +
+ − − − − +
− − + − − +
− − + − − +
− − + − − +
− − + − − +
+ − − − − +
+ − − − − =
∂∂
∂∂
∂∂ ∂
∂∂
+ −
⊥ − +
− − − −
+ + + +
− + − +
+ − + −
⊥ − − − −
⊥ + + + +
⊥ − + − +
⊥ + − + −
( )E I ρ A I ηA H N
C W W δ W W δ
C W W δ W W δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
Ξ ( )
[( )(1 ) ( )(1 )]
[( )(1 ) ( )(1 )]
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )} 0
b bWx b b
Wt b
Wt x b x T
Wx
v mnR
m nR
nN mnR
m nR
n
v mnR
m nR
m mnR
m nR
mN
d mnR
mnR
m nR
m nR
n m
d mnR
mnR
m nR
m nR
nN mN
d mnR
mnR
m nR
m nR
nN m
d mnR
mnR
m nR
m nR
n mN
d mnR
mnR
m nR
m nR
n m
d mnR
mnR
m nR
m nR
nN mN
d mnR
mnR
m nR
m nR
nN m
d mnR
mnR
m nR
m nR
n mN
2
( 1) ( 1) 1
( 1) 1 ( 1)
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
mnR
mnR
mnR
mnR
z
y
z y
z
y
z y
z
y
4
4
2
2
4
2 2
2
2
(5b)
To investigate free dynamic response of the nanostructure regard-less of time and space dimensions, we make dimensionless Eqs. (5a) and(5b) by introducing the following quantities:
= = = = = = =
= = = = = = ⊥−
ξ V W γ τ t μ C
C λ d H H N
, , , , , , ,
, , , , ; [.] or .
xlb mn
R VmnR
lb mnR WmnR
lbzlz lb
EbIbρbAb
e alb v
R Cv lbEbIb
dR Cd lb
EbIb
lbIb Ab
dNz d x
Rx
ηAblbEbIb
TR NT lb
EbIb
12
0[.]
[.] 4
[.][.] 4
/ ( 1)
2 2
(6)
Fig. 1. Schematic representation of a magnetically-thermally affectedASWCNTs.
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
499
Hence, the dimensionless governing equations of the magnetically-thermally affected ASWCNTs using NRBM are derived as:
+ ⎧⎨⎩
− − − +
− − + − − +
− − + − − +
+ − − − − +
+ − − − − +
− − + − − +
− − + − − +
− + − − − +
− + − − − +
+ − − − − +
+ − − − − =
∂∂
∂∂
− ∂∂ ∂
∂∂
+ −
⊥ − +
− − − −
+ + + +
+ − + −
− + − +
⊥ − − − −
⊥ + + + +
⊥ − + − +
⊥ + − + −
λ H N
C V V δ V V δ
C V V δ V V δ
C W V W V δ δ
C W V W V δ δ
C V W V W δ δ
C V W V W δ δ
C V W W V δ δ
C V W W V δ δ
C V W V W δ δ
C V W V W δ δ
Ξ (( ) )
[( )(1 ) ( )(1 )]
[( )(1 ) ( )(1 )]
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )} 0
VmnR
ξVmnR
τVmnR
τ ξ xR
TR VmnR
ξ
vR
mnR
m nR
mNy mnR
m nR
m
vR
mnR
m nR
n mnR
m nR
nNz
dR
mnR
mnR
m nR
m nR
n m
dR
mnR
mnR
m nR
m nR
nNz mNy
dR
mnR
mnR
m nR
m nR
n mNy
dR
mnR
mnR
m nR
m nR
nNz m
dR
mnR
mnR
m nR
m nR
n m
dR
mnR
mnR
m nR
m nR
nNz mNy
dR
mnR
mnR
m nR
m nR
nNz m
dR
mnR
mnR
m nR
m nR
n mNy
44
22
2 42 2
2 22
( 1) ( 1) 1
( 1) 1 ( 1)
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1 (7a)
+ ⎧⎨⎩
− − − +
− − + − − +
− − + − − +
+ − − − − +
+ − − − − +
− − + − − +
− − + − − +
− − + − − +
− − + − − +
+ − − − − +
+ − − − − =
∂∂
∂∂
− ∂∂ ∂
∂∂
+ −
⊥ − +
− − − −
+ + + +
− + − +
+ − + −
⊥ − − − −
⊥ + + + +
⊥ − + − +
⊥ + − + −
λ H N
C W W δ W W δ
C W W δ W W δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
Ξ (( ) )
[( )(1 ) ( )(1 )]
[( )(1 ) ( )(1 )]
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )} 0
Wξ
Wτ
Wτ ξ x
RTR W
ξ
vR
mnR
m nR
nN mnR
m nR
n
vR
mnR
m nR
m mnR
m nR
mN
dR
mnR
mnR
m nR
m nR
n m
dR
mnR
mnR
m nR
m nR
nN mN
dR
mnR
mnR
m nR
m nR
nN m
dR
mnR
mnR
m nR
m nR
n mN
dR
mnR
mnR
m nR
m nR
n m
dR
mnR
mnR
m nR
m nR
nN mN
dR
mnR
mnR
m nR
m nR
nN m
dR
mnR
mnR
m nR
m nR
n mN
2 2
( 1) ( 1) 1
( 1) 1 ( 1)
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
mnR
mnR
mnR
mnR
z
y
z y
z
y
z y
z
y
4
4
2
2
4
2 2
2
2
(7b)
where = − ∂ ∂μ ξΞ[.] [.] [.]/2 2 2.The both ends of each tube are assumed to be simple, and the ex-
terior ones are prevented from any lateral motion while the interiorones can freely vibrate along the y and z directions. In order to dis-cretize the deformation filed in the spatial domain, assumed modemethod (AMM) is implemented. Let us to express the unknown de-flection fields in terms of mode shapes which are admissible with thegeometrical boundary conditions of the problem as follows:
= ∑ = ∑= =W ξ τ W τ pπξ V ξ τ V τ pπξ( , ) ( )sin( ), ( , ) ( )sin( ),mnR
pNp
mnpR
mnR
pNp
mnpR
1 1 (8)
where Np is the considered number of modes in vibration analysis of theproblem. By introducing Eq. (8) to Eqs. (7a) and (7b), one can arrive atthe following set of equations:
+ =∂∂
M K x 0,Rτ
R RxR22 (9)
where = < >W Vx ,RmnpR
mnpR T denotes the vector of time-dependent
parameters, and MR and KR are the dimensionless mass and stiffnessmatrices that could be readily evaluated and their elements have notbeen provided for the sake of conciseness. Eq. (9) is a set of
− −N N2( 2)( 2)y z second-order ordinary differential equations (ODEs)for a given mode number. By considering a harmonic form for the time-dependent vector as: =τ ex x( )R R ϖ τ
0i R where ϖ R represents the di-
mensionless natural frequency, and xR0 is the dimensionless amplitude
vector of the magnetically-thermally affected ASWCNTs modeled ac-cording to the NRBM. By substituting this form of the unknown vectorinto Eq. (9) and solving the resulted set of eigenvalue equations for ϖ R,the frequencies of the nanosystem are determined.
2.2. Free transverse vibrations of magnetically-thermally affectedASWCNTs using NTBM
The main drawback of the constructed model via NRBM is that theinfluence of shear deformation is not incorporated into the equations ofmotion. Actually, the NRBM-based model would be applicable toslender nanosystems whose any plane perpendicular to the neutral axiswould remain plane after deformation. However, for fairly slender orstocky nanosystems, this issue is violated and generally the predictedresults by the NRBM would not be trustable. To overcome this defi-ciency of the NRBM, Timoshenko suggested that to define independentrotational field in addition to the deflection field. In this part, the dis-crete shear deformable equations of motion of the nanosystem in thepresence of longitudinal thermal and magnetic field are presented.
Based on the NTBM, the kinetic energy (TT), the strain energy (UT)of the magnetically-thermally affected ASWCNTs as well as the workdone by the Lorentz's magnetic force on the constitutive tubes of thenanosystem (W T) are given by:
∑ ∑ ∫ ⎜ ⎟
⎜ ⎟
⎜ ⎟= ⎛
⎝⎜
⎛⎝
⎛⎝
⎞⎠
+ ⎛⎝
⎞⎠
⎞⎠
+ ⎛⎝
+ ⎞⎠
⎞
⎠⎟
= =
∂
∂∂
∂
∂∂
∂∂( ) ( )
T tN N
ρ I
A x
( )
d ,
T
m
y
n
zl
b b t t
bV
tW
t
12
1 10
Θ 2Θ 2
2 2
b ymnT
zmnT
mnT
mnT
(10a)
∫∑ ∑=
⎛
⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
− + − +
− + − + +
− − + − − +
− − + − − +
− − − +
− − − +
− − − +
− − − +
− − − +
− − − +
− − − +
− − −
⎞
⎠
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
= =
∂∂
∂∂
∂∂
∂
∂∂
∂∂
∂
+ −
⊥ − +
− −
+ +
− +
+ −
⊥ − +
⊥ + −
⊥ − −
⊥ + +
( )( )
U tN N
M Q N
M Q N
C V V δ V V δ
C V V δ V V δ
C X X δ δ
C X X δ δ
C Y Y δ δ
C Y Y δ δ
C X X δ δ
C X X δ δ
C Y Y δ δ
C Y Y δ δ
x( ) 12
( ) Θ ( )
( ) Θ ( )
[( ) (1 ) ( ) (1 )]
[( ) (1 ) ( ) (1 )]
( ) (1 )(1 )
( ) (1 )(1 )
( ) (1 )(1 )( ) (1 )(1 )
( ) (1 )(1 )( ) (1 )(1 )
( ) (1 )(1 )
( ) (1 )(1 )
d ,T
m
y
n
z l
x bznl T V
x zT
bynl T
TVx
x bynl T W
x yT
bznl T
TWx
v mnT
m nT
mN mnT
m nT
m
v mnT
m nT
n mnT
m nT
nN
d mnT
m nT
m n
d mnT
m nT
mN nN
d mnT
m nT
m nN
d mnT
m nT
mN n
d mnT
m nT
m nN
d mnT
m nT
mN n
d mnT
m nT
m n
d mnT
m nT
mN nN
1 10
Θ
Θ
( 1)2
( 1)2
1
( 1)2
1 ( 1)2
( 1)( 1)2
1 1
( 1)( 1)2
( 1)( 1)2
1
( 1)( 1)2
1
( 1)( 1)2
1
( 1)( 1)2
1
( 1)( 1)2
1 1
( 1)( 1)2
b
zmnT
mnmnT
mn mnmnR
ymnT
mnmnT
mn mnmnR
z
z
y z
z
y
z
y
y z
2
2
2
2
(10b)
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
500
∑ ∑ ∫ ⎜ ⎟= ⎡
⎣⎢
⎛⎝
⎞⎠
+ ⎤
⎦⎥
= =
W tN N
f V f W x( ) ( ) d ,T
m
y
n
zl
yT
mnmnT
zT
mn mnT
1 10
b
(10c)
where the components of the applied Lorentz force on the m n( , )thnanotube modeled according to the NTBM are as:
⎜ ⎟⎛⎝
⎞⎠
= =∂∂
∂∂
f ηA H f ηA H, ( ) ,yT
mnb x
Vx z
Tmn b x
Wx
2 2mnT
mnT2
2
2
2(11)
and = +X W V( )/ 2mnT
mnT
mnT , = − +Y W V( )/ 2mn
TmnT
mnT , Vmn
T , and WmnT in
order are the components of transverse displacements of the m n( , )thSWCNT along the y and z axes, Θy
Tmn and Θz
Tmn represent angles of de-
flection about the y and z axes, respectively, Q( )bynl T
mn and Q( )bznl T
mn are
the nonlocal shear forces along the y and z directions, M( )bynl T
mn , and
M( )bznl T
mn in order are the nonlocal bending moment about the y and zaxes.
By exploiting the nonlocal elasticity theory of Eringen (1966, 1972),the nonlocal resultant shear force and the nonlocal bending momentwithin the m n( , )th nanotube are linked to their classical counterpartsas follows (Kiani, 2014a,b):
− = −∂∂( )Q e a Q k G A( ) ( ) ( ) Θ ,by
nl Tbynl
xxT
s b bV
x zT
02
,mn mnmnT
mn (12a)
− = −∂∂( )Q e a Q k G A( ) ( ) ( ) Θ ,bz
nl Tbznl
xxT
s b bW
x yT
02
,mn mnmnT
mn (12b)
− = −∂
∂M e a M E I( ) ( ) ( ) ,bynl T
bynl
xxT
b b x02
,
Θmn mn
ymnT
(12c)
− = − ∂∂M e a M E I( ) ( ) ( ) .bz
nl Tbznl
xxT
b b x02
,Θ
mn mnzmnT
(12d)
In order to establish discrete governing equations of the nanosystembased on the NTBM, Hamilton's principle is employed. In view of thegiven nonlocal forces in Eqs. (12a)-(12d), the nonlocal governingequations of the magnetically-thermally affected ASWCNTs in terms ofdisplacements are derived as follows:
− − − =∂
∂∂
∂∂
∂{ } ( )ρ I k G A E IΞ Θ 0,b b t s b bV
x zT
b b x
Θ ΘzmnT
mnT
mnzmnT2
2
2
2 (13a)
− ⎛⎝
− ⎞⎠
+ ⎧⎨⎩
− − +
− − + − − +
− − + − − +
+ − − − − +
+ − − − − +
− − + − − +
− − + − − +
− + − − − +
− + − − − +
+ − − − − +
+ − − − − =
∂∂
∂∂
∂∂
∂∂
+ −
⊥ − +
− − − −
+ + + +
+ − + −
− + − +
⊥ − − − −
⊥ + + + +
⊥ − + − +
⊥ + − + −
k G A ρ A ηA H N
C V V δ V V δ
C V V δ V V δ
C W V W V δ δ
C W V W V δ δ
C V W V W δ δ
C V W V W δ δ
C V W W V δ δ
C V W W V δ δ
C V W V W δ δ
C V W V W δ δ
Ξ ( )
[( )(1 ) ( )(1 )]
[( )(1 ) ( )(1 )]
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )} 0
s b bVx x b b
Vt b x T
Vx
v mnT
m nT
mN mnT
m nT
m
v mnT
m nT
n mnT
m nT
nN
d mnT
mnT
m nT
m nT
n m
d mnT
mnT
m nT
m nT
nN mN
d mnT
mnT
m nT
m nT
n mN
d mnT
mnT
m nT
m nT
nN m
d mnT
mnT
m nT
m nT
n m
d mnT
mnT
m nT
m nT
nN mN
d mnT
mnT
m nT
m nT
nN m
d mnT
mnT
m nT
m nT
n mN
Θ 2
( 1) ( 1) 1
( 1) 1 ( 1)
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
mnT zmn
TmnT
mnT
z
z
z y
y
z
z y
z
y
2
2
2
2
2
2
(13b)
⎧⎨⎩
⎫⎬⎭
− − − =∂
∂∂
∂
∂
∂( )ρ I k G A E IΞ Θ 0,b b t s b bW
x yT
b b x
Θ ΘymnT
mnT
mnymnT2
2
2
2(13c)
⎜ ⎟− ⎛⎝
− ⎞⎠
+ ⎧⎨⎩
− − +
− − + − − +
− − + − − +
+ − − − − +
+ − − − − +
− − + − − +
− − + − − +
− − + − − +
− − + − − +
+ − − − − +
+ − − − − =
∂∂
∂
∂∂
∂∂
∂
+ −
⊥ − +
− − − −
+ + + +
− + − +
+ − + −
⊥ − − − −
⊥ + + + +
⊥ − + − +
⊥ + − + −
k G A ρ A ηA H N
C W W δ W W δ
C W W δ W W δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
Ξ ( )
[( )(1 ) ( )(1 )]
[( )(1 ) ( )(1 )]
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )} 0
s b bWx x b b
Wt b x T
Wx
v mnT
m nT
nN mnT
m nT
n
v mnT
m nT
m mnT
m nT
mN
d mnT
mnT
m nT
m nT
n m
d mnT
mnT
m nT
m nT
nN mN
d mnT
mnT
m nT
m nT
nN m
d mnT
mnT
m nT
m nT
n mN
d mnT
mnT
m nT
m nT
n m
d mnT
mnT
m nT
m nT
nN mN
d mnT
mnT
m nT
m nT
nN m
d mnT
mnT
m nT
m nT
n mN
Θ 2
( 1) ( 1) 1
( 1) 1 ( 1)
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
mnT ymn
TmnT
mnT
z
y
z y
z
y
z y
z
y
2
2
2
2
2
2
(13d)
Now we define the following dimensionless parameters:
= = = = =
= = = = =
V W τ t
χ C C H H N
, , Θ Θ , Θ Θ , ,
, , , , ,
mnT V
l mnT W
l yT
yT
zT
zT
lk G
ρ
E Ik G A l v
T C lk G A d
T C lk G A x
Tx
ηk G T
T Nk G A
1
[.] [.]
mnT
bmnT
b mn mn mn mn bs b
b
b bs b b b
v bs b b
d bs b b s b
Ts b b2
[.] 2 [.] 2
(14)
in which = ⊥[.] or . In view of the newly introduced dimensionlessparameters in Eq. (14), the dimensionless form of Eqs. (13a)-(13d) areobtained as follows:
− − − =− ∂
∂∂
∂∂
∂{ } ( )λ χΞ Θ 0,τ
Vξ z
Tξ
2 Θ ΘzmnT
mnT
mnzmnT2
2
2
2 (15a)
− ⎛⎝
− ⎞⎠
+ ⎧⎨⎩
− − +
− − + − − +
− − + − − +
+ − − − − +
+ − − − − +
− − + − − +
− − + − − +
− + − − − +
− + − − − +
+ − − − − +
+ − − − − =
∂∂
∂∂
∂∂
∂∂
+ −
⊥ − +
− − − −
+ + + +
+ − + −
− + − +
⊥ − − − −
⊥ + + + +
⊥ − + − +
⊥ + − + −
H N
C V V δ V V δ
C V V δ V V δ
C W V W V δ δ
C W V W V δ δ
C V W V W δ δ
C V W V W δ δ
C V W W V δ δ
C V W W V δ δ
C V W V W δ δ
C V W V W δ δ
Ξ (( ) )
[( )(1 ) ( )(1 )]
[( )(1 ) ( )(1 )]
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )} 0
Vξ ξ
Vτ x
TTT V
ξ
vT
mnT
m nT
mN mnT
m nT
m
vT
mnT
m nT
n mnT
m nT
nN
dT
mnT
mnT
m nT
m nT
n m
dT
mnT
mnT
m nT
m nT
nN mN
dT
mnT
mnT
m nT
m nT
n mN
dT
mnT
mnT
m nT
m nT
nN m
dT
mnT
mnT
m nT
m nT
n m
dT
mnT
mnT
m nT
m nT
nN mN
dT
mnT
mnT
m nT
m nT
nN m
dT
mnT
mnT
m nT
m nT
n mN
Θ 2
( 1) ( 1) 1
( 1) 1 ( 1)
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
mnT zmn
TmnT
mnT
z
z
z y
y
z
z y
z
y
2
2
2
2
2
2
(15b)
⎧⎨⎩
⎫⎬⎭
− − − =− ∂
∂∂
∂
∂
∂( )λ χΞ Θ 0,τ
Wξ y
Tξ
2 Θ ΘymnT
mnT
mnymnT2
2
2
2(15c)
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
501
⎜ ⎟− ⎛⎝
− ⎞⎠
+ ⎧⎨⎩
− − +
− − + − − +
− − + − − +
+ − − − − +
+ − − − − +
− − + − − +
− − + − − +
− − + − − +
− − + − − +
+ − − − − +
+ − − − − =
∂∂
∂
∂∂
∂∂
∂
+ −
⊥ − +
− − − −
+ + + +
− + − +
+ − + −
⊥ − − − −
⊥ + + + +
⊥ − + − +
⊥ + − + −
H N
C W W δ W W δ
C W W δ W W δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
C W V W V δ δ
Ξ (( ) )
[( )(1 ) ( )(1 )]
[( )(1 ) ( )(1 )]
0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )0.5 ( )(1 )(1 )
0.5 ( )(1 )(1 )} 0
Wξ ξ
Wτ x
TTT W
ξ
vT
mnT
m nT
nN mnT
m nT
n
vT
mnT
m nT
m mnT
m nT
mN
dT
mnT
mnT
m nT
m nT
n m
dT
mnT
mnT
m nT
m nT
nN mN
dT
mnT
mnT
m nT
m nT
nN m
dT
mnT
mnT
m nT
m nT
n mN
dT
mnT
mnT
m nT
m nT
n m
dT
mnT
mnT
m nT
m nT
nN mN
dT
mnT
mnT
m nT
m nT
nN m
dT
mnT
mnT
m nT
m nT
n mN
Θ 2
( 1) ( 1) 1
( 1) 1 ( 1)
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1 1
( 1)( 1) ( 1)( 1)
( 1)( 1) ( 1)( 1) 1
( 1)( 1) ( 1)( 1) 1
mnT ymn
TmnT
mnT
z
y
z y
z
y
z y
z
y
2
2
2
2
2
2
(15d)
Eqs. (15a)-(15d) furnish us regarding dimensionless-discreteequations of motion of the vertically aligned ASWCNTs acted upon byboth longitudinal thermal and magnetic fields on the basis of theNTBM. From mathematics points of view, these are N N4 y z coupledfourth-order partial differential equations (PDEs). To solve theseequations for natural frequencies, they should be appropriatelytransferred into ODEs. For this purpose, assume that the whole con-stitutive SWCNTs of the magnetically-thermally affected ASWCNTshave simple supports at their ends and the exterior nanotubes havebeen prevented from any transverse displacement. Using AMM, thedeformation fields of the constitutive nanotubes are discretized in thefollowing form:
= ∑ = ∑
= ∑
= ∑
= =
=
=
W ξ τ W τ pπξ V ξ τ V τ pπξ
ξ τ τ pπξ ξ τ
τ pπξ
( , ) ( )sin( ), ( , ) ( )sin( ),
Θ ( , ) Θ ( )cos( ), Θ ( , )
Θ ( )cos( ),
mnT
pN
mnpT
mnT
pN
mnpT
yT
pN
yT
zT
pN
zT
1 1
1
1
p p
mnp
mnp mn
pmnp
(16)
by substituting Eq. (16) into Eqs. (15a)-(15d) and recalling that theexterior SWCNTs are prohibited from any lateral movement, the fol-lowing set of ODEs is derived:
+ =∂∂
M K x 0,Tτ
T TxT22 (17)
where the vector of unknown coefficients is given by:= < >W Vx , , Θ , ΘT
mnpT
mnpT
yT
zT T
mnp mnp and the dimensionless mass andstiffness matrices (i.e., MT and KT) are easily obtained. Let us to takeinto account a harmonic form for the unknown coefficients vector:
τx ( )T = exT ϖ τ0
i T where xT0 and ϖT in order represent the dimensionless
amplitude vector and the dimensionless frequency of the magneti-cally-thermally affected ASWCNTs according to the NTBM. By in-troducing such a form of time-dependent vector to Eq. (17) and sol-ving the resulted set of eigenvalue equations for ϖT , the frequencies ofthe nanosystem are calculated.
3. Nonlocal continuous models using classical and sheardeformable beam theories
On the basis of the suggested discrete models in the previous parts,herein, suitable continuous models are developed to investigate trans-verse vibrations of magnetically-thermally affected ASWCNTs.Thereafter, appropriate solutions are presented to predict free dynamicresponse of nanosystems under various multi-physical fields.
3.1. Application of NRBM-continuous model to vibrations of magnetically-thermally affected ASWCNTs
Based on the suggested nonlocal-discrete-NRBM model to studyvibrations of the magnetically-thermally affected nanosystem, thegoverning equations pertinent to transverse vibrations of the m n( , )thtube are displayed by:
+ ⎧⎨⎩
− − −
+ − − + − −
+ − + −
−
+ + − − −
− =
∂∂
∂∂
∂∂ ∂
∂∂
+ − ⊥ − +
⊥ − + + − + +
− −
⊥ − + + + + −
− −
( )E I ρ A I ηA H N
C V V V C V V V
C C W W W
W
C C V V V V
V
Ξ ( )
(2 ) (2 )
( ) (
)
( ) (4
)} 0
b bVx b b
Vt b
Vt x b x T
Vx
v mnR
m nR
m nR
v mnR
m nR
m nR
d d m nR
m nR
m nR
m nR
d d mnR
m nR
m nR
m nR
m nR
2
( 1) ( 1) ( 1) ( 1)12 ( 1)( 1) ( 1)( 1) ( 1)( 1)
( 1)( 1)12 ( 1)( 1) ( 1)( 1) ( 1)( 1)
( 1)( 1)
mnR
mnR
mnR
mnR4
4
2
2
4
2 2
2
2
(18a)
+ ⎧⎨⎩
− − −
+ − − + − −
+ − + − −
+ + − − −
− =
∂∂
∂∂
∂∂ ∂
∂∂
+ − ⊥ − +
⊥ − + + − + + − −
⊥ − + + + + −
− −
( )E I ρ A I ηA H N
C W W W C W W W
C C V V V V
C C W W W W
W
Ξ ( )
(2 ) (2 )
( )( )
( ) 4
} 0
b bWx b b
Wt b
Wt x b x T
Wx
v mnR
m nR
m nR
v mnR
m nR
m nR
d d m nR
m nR
m nR
m nR
d d mnR
m nR
m nR
m nR
m nR
2
( 1) ( 1) ( 1) ( 1)12 ( 1)( 1) ( 1)( 1) ( 1)( 1) ( 1)( 1)
12 ( 1)( 1) ( 1)( 1) ( 1)( 1)
( 1)( 1)
mnR
mnR
mnR
mnR4
4
2
2
4
2 2
2
2
(18b)
To establish an appropriate continuous model on the basis of thesuggested discrete model, two continuous-transverse displacementfunctions =v v x y z t( , , , ) and =w w x y z t( , , , ) are introduced to Eqs.(18a) and (18b) such that the following relations are approximatedwith a reasonable accuracy:
≈≈ − −≈ + −≈ − +≈ + +
− −
− +
+ −
+ +
x t x y z tx t x y d z d tx t x y d z d tx t x y d z d tx t x y d z d t
[•] ( , ) [.]( , , , ),[•] ( , ) [.]( , , , ),[•] ( , ) [.]( , , , ),[•] ( , ) [.]( , , , ),[•] ( , ) [.]( , , , ),
mn mn mn
m n mn mn
m n mn mn
m n mn mn
m n mn mn
( 1)( 1)
( 1)( 1)
( 1)( 1)
( 1)( 1) (19)
in which y z( , )mn mn is the coordinates of the revolutionary axis of them n( , )th nanotube of the ensemble in the y-z plane, and
= ∘ ∘ ∘ ∘V v W w[•]([.]) ( ) or ( )[ ] [ ] [ ] [ ] and ∘ = R T[ ] or . Now the transversedeformations of the tubes adjacent to the m n( , )th nanotube are ap-proximated by:
∑ ∑ ⎜ ⎟± ± ≈ ⎛⎝ −
⎞⎠
± ±= =
∂∂ ∂
−−v x y d z d t i
i j d d( , , , )6 6
( ) ( ) ,mn mni j
v x y z tz y
j i j
1 0
( , , , )imn mnj i j
(20a)
∑ ∑ ⎜ ⎟± ± ≈ ⎛⎝ −
⎞⎠
± ±= =
∂∂ ∂
−−w x y d z d t i
i j d d( , , , )6 6
( ) ( ) .mn mni j
w x y z tz y
j i j
1 0
( , , , )imn mnj i j
(20b)
By virtue of Eq. (19) and through introducing Eqs. (20a) and (20b)to Eqs. (18a) and (18b), the continuous form of the equations of motionof magnetically-thermally affected ASWCNTs on the basis of the NRBMare obtained as:
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
502
+ ⎧⎨⎩
− − −
− + + +
− + + +
− +⎡
⎣
⎢⎢⎢
+ + + + +
+ + +
⎤
⎦
⎥⎥⎥
− − ⎡⎣
+ +
+ + + ⎤⎦
⎫⎬⎭
=
∂∂
∂∂
∂∂ ∂
∂∂
∂∂
∂∂
∂∂
∂∂
⊥∂∂
∂∂
∂∂
∂∂
⊥
∂∂
∂∂
∂∂
∂∂ ∂
∂∂
∂∂
∂∂ ∂
∂∂ ∂
∂∂
⊥∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
( )( )( )
( )( )
( )( )
E I ρ A I ηA H N
C d
C d
C C d
C C d
Ξ ( )
( )6
15 15
( ) 2
3 10 3 0
b bv
x b bvt b
vt x b x T
vx
vvy
d vy
d vy
d vy
vvz
d vz
d vz
d vz
d d
vy
vz
d vz
vz y
vy
d vz
vz y
vz y
vy
d dw
y zd w
y zw
y z
d wy z
wy z
wy z
2
212 360 20160
212 360 20160
212
360
23
180
R R R R
R R R R
R R R R
R R R R R
R R R R
R R R
R R R
44
22
42 2
22
22
2 44
4 66
6 88
22
2 44
4 66
6 88
22
22
2 44
42 2
44
4 66
64 2
62 4
66
2 2 43
43
4 65
63 3
65
(21a)
+ ⎧⎨⎩
− − −
− + + +
− + + +
− +⎡
⎣
⎢⎢⎢
+ + + + +
+ + +
⎤
⎦
⎥⎥⎥
− − ⎡⎣
+ +
+ + + ⎤⎦
⎫⎬⎭
=
∂∂
∂∂
∂∂ ∂
∂∂
∂∂
∂∂
∂∂
∂∂
⊥∂∂
∂∂
∂∂
∂∂
⊥
∂∂
∂∂
∂∂
∂∂ ∂
∂∂
∂∂
∂∂ ∂
∂∂ ∂
∂∂
⊥∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
( )( )( )
( )( )
( )( )
E I ρ A I ηA H N
C d
C d
C C d
C C d
Ξ ( )
( )6
15 15
( ) 2
3 10 3 0
b bwx b b
wt b
wt x b x T
wx
vwz
d wz
d wz
d wz
vwy
d wy
d wy
d wy
d d
wy
wz
d wz
wz y
wy
d wz
wz y
wz y
wy
d dv
y zd v
y zv
y z
d vy z
vy z
vy z
2
212 360 20160
212 360 20160
212
360
23
180
R R R R
R R R R
R R R R
R R R R R
R R R R
R R R
R R R
44
22
42 2
22
22
2 44
4 66
6 88
22
2 44
4 66
6 88
22
22
2 44
42 2
44
4 66
64 2
62 4
66
2 2 43
43
4 65
63 3
65
(21b)
By exploiting Eq. (6), Eqs. (21a) and (21b) are rewritten in thefollowing dimensionless manner:
+ ⎧⎨⎩
− − −
− + + +
− ⎛⎝
+ + + ⎞⎠
− +⎡
⎣
⎢⎢⎢
+ + + + +
+ + +
⎤
⎦
⎥⎥⎥
− −⎡
⎣
⎢⎢⎢
+ + +
+ +
⎤
⎦
⎥⎥⎥
⎫
⎬⎪
⎭⎪
=
∂∂
∂∂
− ∂∂ ∂
∂∂
∂∂
∂∂
∂∂
∂∂
⊥∂∂
∂∂
∂∂
∂∂
⊥
∂∂
∂∂
∂∂
∂∂ ∂
∂∂
∂∂
∂∂ ∂
∂∂ ∂
∂∂
⊥
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
( )
( )( )
( )( )
λ H N
κd C
d C
C Cκ κ κ
κ κ κ
κ C Cκ
κ κ
Ξ (( ) )
( )
( )6
15 15
( )2
3 10 30
vξ
vτ
vτ ξ x
RTR v
ξ
vR v
ηκd v
ηκd v
ηκd v
η
vR v
γd v
γd v
γd v
γ
dR
dR
vη
vγ
d vγ
vγ η
vη
d vγ
vγ η
vγ η
vη
dR
dR
wη γ
d wη γ
wη γ
d wη γ
wη γ
wη γ
2 2
2 ( )12
( )360
( )20160
212 360 20160
212
2 4
3602 4 6
32
1804 2
R R R R
R R R R
R R R R
R R R R R
R R R R
R R R
R R R
44
22
42 2
22
22
2 44
4 66
6 88
22
2 44
4 66
6 88
22
22
2 44
42 2
44
4 66
64 2
62 4
66
2 2 43
43
4 65
63 3
65
(22a)
+ ⎧⎨⎩
− − −
− + + +
− + + +
− +⎡
⎣
⎢⎢⎢
+ + + + +
+ + +
⎤
⎦
⎥⎥⎥
− −⎡
⎣
⎢⎢⎢
+ + +
+ +
⎤
⎦
⎥⎥⎥
⎫
⎬⎪
⎭⎪=
∂∂
∂∂
− ∂∂ ∂
∂∂
∂∂
∂∂
∂∂
∂∂
⊥∂∂
∂∂
∂∂
∂∂
⊥
∂∂
∂∂
∂∂
∂∂ ∂
∂∂
∂∂
∂∂ ∂
∂∂ ∂
∂∂
⊥
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
( )( )
( )( )
( )( )
λ H N
d C
κd C
C Cκ κ κ
κ κ κ
κ C Cκ
κ κ
Ξ (( ) )
( )
( )6
15 15
( )2
3 10 30
wξ
wτ
wτ ξ x
RTR w
ξ
vR w
γd w
γd w
γd w
γ
vR w
ηκd w
ηκd w
ηκd w
η
dR
dR
wη
wγ
d wγ
wγ η
wη
d wγ
wγ η
wγ η
wη
dR
dR
vη γ
d vη γ
vη γ
d vη γ
vη γ
vη γ
2 2
212 360 20160
2 ( )12
( )360
( )20160
212
2 4
3602 4 6
32
1804 2
R R R R
R R R R
R R R R
R R R R R
R R R R
R R R
R R R
44
22
42 2
22
22
2 44
4 66
6 88
22
2 44
4 66
6 88
22
22
2 44
42 2
44
4 66
64 2
62 4
66
2 2 43
43
4 65
63 3
65
(22b)
where =v v ξ η γ τ( , , , )R R and =w w ξ η γ τ( , , , )R R are dimensionlesscontinuous deflections and,
= = = = = = =η γ κ C C v w, , , , , , .yl
zl
ll v
R C d l
E I l dR C d l
E I lR v
lR w
l[.] [.]y zzy
v b
b b z
d b
b b z
R
b
R
b
[.] 2 4
2[.] 2 4
2
(23)
Eqs. (22a) and (22b) display the nonlocal-continuous equations ofmotion of the magnetically-thermally affected ASWCNTs based on theNRBM. These are coupled PDEs. For frequency analysis of the problem,AMM is employed. To this end, the dimensionless transverse displace-ments in y and z directions are discretized in terms of appropriate vi-bration modes as follows:
= ∑ ∑ ∑
= ∑ ∑ ∑
= ==
= = =
v ξ η γ τ v τ ϕ ξ η γ
w ξ η γ τ w τ ϕ ξ η γ
( , , , ) ( ) ( , , ),
( , , , ) ( ) ( , , ),
RmN
nN
p
N
mnpR
mnpv
RmN
nN
pN
mnpR
mnpw
1 11
1 1 1
mv nvpv
mw nw pw(24)
where v τ( )mnpR and w τ( )mnp
R are dimensionless time-dependent para-meters, ϕmnp
v and ϕmnpw in order are admissible mode shapes pertinent to
the deflections in the y and z directions. For magnetically-thermallyaffected ASWCNTs with simple ends and immovable exterior tubes, thefollowing modes are considered:
=
=
ϕ ξ η γ mπξ nπη pπγ
ϕ ξ η γ mπξ nπη pπγ
( , , ) sin( )sin( )sin( ),
( , , ) sin( )sin( )sin( ).mnpv
mnpw
(25)
where Nmw/Nmv, Nnw/Nnv, and Npw/Npv denote the number of vibrationmodes pertinent to the deflections in x, y, and z directions, respectively.Let us to premultiply both sides of Eqs. (23a) and (23b) by δv R and δw R,where δ is the variational sign. After taking the necessary integration byparts, it is obtainable:
⎡
⎣⎢
⎤
⎦⎥
⎧
⎨⎩
⎫
⎬⎭
+ ⎡
⎣⎢
⎤
⎦⎥
⎧⎨⎩
⎫⎬⎭
= ⎧⎨⎩
⎫⎬⎭
M MM M
K KK K
vw
[ ] [ ][ ] [ ]
[ ] [ ][ ] [ ]
00 ,b
R vvbR vw
bR wv
bR ww
τ
τ
bR vv
bR vw
bR wv
bR ww
R
R
v
w
dd
dd
R
R
22
22 (26)
where the dimensionless mass and stiffness submatrices are provided inAppendix A. By considering a harmonic form for v R and w R and fol-lowing the provided procedure in Sect. 3.1, the nonlocal natural fre-quencies of the nanosystem in the presence of both thermal and mag-netic fields are calculated.
3.2. Application of NTBM-continuous model to vibrations of magnetically-thermally affected ASWCNTs
In the light of the established discrete-based NTBM, the equations ofmotion associated with transverse vibrations of the m n( , ) interior na-notube of the vertically aligned ASWCNTs subjected to both thermaland magnetic fields are expressed by (see Eqs. (13a)-(13d)):
− − − =∂
∂∂
∂∂
∂{ } ( )ρ I k G A E IΞ Θ 0,b b t s b bV
x zT
b b x
Θ ΘzmnT
mnT
mnzmnT2
2
2
2 (27a)
− ⎛⎝
− ⎞⎠
+ ⎧⎨⎩
− − +
+ − − + − −
+ − + − −
+ + − − −
− =
∂∂
∂∂
∂∂
∂∂
+ − ⊥ − +
⊥ − + + − + + − −
⊥ − + + + + −
− −
k G A ρ A ηA H N
C V V V C V V V
C C W W W W
C C V V V V
V
Ξ ( )
(2 ) (2 )
( ) ( )
( ) (4
)} 0
s b bVx x b b
Vt b x T
Vx
v mnT
m nT
m nT
v mnT
m nT
m nT
d d m nT
m nT
m nT
m nT
d d mnT
m nT
m nT
m nT
m nT
Θ 2
( 1) ( 1) ( 1) ( 1)12 ( 1)( 1) ( 1)( 1) ( 1)( 1) ( 1)( 1)
12 ( 1)( 1) ( 1)( 1) ( 1)( 1)
( 1)( 1)
mnT zmn
TmnT
mnT2
2
2
2
2
2
(27b)
⎧⎨⎩
⎫⎬⎭
− − − =∂
∂∂
∂
∂
∂( )ρ I k G A E IΞ Θ 0,b b t s b bW
x yT
b b x
Θ ΘymnT
mnT
mnymnT2
2
2
2(27c)
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
503
⎜ ⎟− ⎛⎝
− ⎞⎠
+ ⎧⎨⎩
− − +
+ − − + − −
+ − + − −
+ + − − −
− =
∂∂
∂
∂∂
∂∂
∂
+ − ⊥ − +
⊥ − + + − + + − −
⊥ − + + + + −
− −
k G A ρ A ηA H N
C W W W C W W W
C C V V V V
C C W W W W
W
Ξ ( )
(2 ) (2 )
( )( )
( ) (4
)} 0
s b bWx x b b
Wt b x T
Wx
v mnT
m nT
m nT
v mnT
m nT
m nT
d d m nT
m nT
m nT
m nT
d d mnT
m nT
m nT
m nT
m nT
Θ 2
( 1) ( 1) ( 1) ( 1)12 ( 1)( 1) ( 1)( 1) ( 1)( 1) ( 1)( 1)
12 ( 1)( 1) ( 1)( 1) ( 1)( 1)
( 1)( 1)
mnT ymn
TmnT
mnT2
2
2
2
2
2
(27d)
Let us to define new functions θyT and θz
T such that:
≈
≈
θ x y z t x t
θ x y z t x t
( , , , ) Θ ( , ),
( , , , ) Θ ( , ),yT
mn mn yT
zT
mn mn zT
mn
mn (28)
by mixing Eqs. (19), (20a) and (20b), and (28) with Eqs. (27a)-(27d),the continuous governing equations that describe transverse vibrationsof the magnetically-thermally affected ASWCNTs take the followingform:
− − − =∂∂
∂∂
∂∂{ } ( )ρ I k G A θ E IΞ 0,b b
θt s b b
vx z
Tb b
θx
zT T z
T2
2
2
2 (29a)
− ⎛⎝
− ⎞⎠
+ ⎧⎨⎩
− − +
− + + +
− + + +
− +⎡
⎣
⎢⎢⎢
+ + + + +
+ + +
⎤
⎦
⎥⎥⎥
− − ⎡⎣
+ +
+ + + ⎤⎦
⎫⎬⎭
=
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
⊥∂∂
∂∂
∂∂
∂∂
⊥
∂∂
∂∂
∂∂
∂∂ ∂
∂∂
∂∂
∂∂ ∂
∂∂ ∂
∂∂
⊥∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
( )( )
( )( )
( )( )
k G A ρ A ηA H N
C d
C d
C C d
C C d
Ξ ( )
( )6
15 15
( ) 2
3 10 3 0
s b bvx
θx b b
vt b x T
vx
vvy
d vy
d vy
d vy
vvz
d vz
d vz
d vz
d d
vy
vz
d vz
vz y
vy
d vz
vz y
vz y
vy
d dw
y zd w
y zw
y z
d wy z
wy z
wy z
2
212 360 20160
212 360 20160
2 12
360
23
180
T zT T T
T T T T
T T T T
T T T T T
T T T T
T T T
T T T
22
22
22
22
2 44
4 66
6 88
22
2 44
4 66
6 88
22
22
2 44
42 2
44
4 66
64 2
62 4
66
2 2 43
43
4 65
63 3
65
(29b)
⎧⎨⎩
⎫⎬⎭
− − − =∂
∂∂∂
∂
∂( )ρ I k G A θ E IΞ 0,b bθ
t s b bwx y
Tb b
θ
xyT T y
T2
2
2
2 (29c)
⎜ ⎟− ⎛⎝
− ⎞⎠
+ ⎧⎨⎩
− − +
− + + +
− + + +
− +⎡
⎣
⎢⎢⎢
+ + + + +
+ + +
⎤
⎦
⎥⎥⎥
− − ⎡⎣
+ +
+ + + ⎤⎦
⎫⎬⎭
=
∂∂
− − ∂∂
+
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
⊥∂∂
∂∂
∂∂
∂∂
⊥
∂∂
∂∂
∂∂
∂∂ ∂
∂∂
∂∂
∂∂ ∂
∂∂ ∂
∂∂
⊥∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
( )( )
( )( )
( )( )
k G A ρ A ηA H N
C d
C d
C C d
C C d
ρ A wt
ηA H N wx
Ξ ( )
( )6
15 15
( ) 2
3 10 3 0
( )
s b bwx
θx b b
wt b x T
wx
vwz
d wz
d wz
d wz
vwy
d wy
d wy
d wy
d d
wy
wz
d wz
wz y
wy
d wz
wz y
wz y
wy
d dv
y zd v
y zv
y z
d vy z
vy z
vy z
b bT
b x TT
2
212 360 20160
212 360 20160
212
360
23
180
2
2
22
2
T yT T T
T T T T
T T T T
T T T T T
T T T T
T T T
T T T
22
22
22
22
2 44
4 66
6 88
22
2 44
4 66
6 88
22
22
2 44
42 2
44
4 66
64 2
62 4
66
2 2 43
43
4 65
63 3
65
(29d)
By employing Eq. (14), the dimensionless form of the recently de-veloped continuous equations is represented by:
⎧⎨⎩
⎫⎬⎭
− − − =− ∂∂
∂∂
∂∂( )λ θ χΞ 0,θ
τvξ z
T θξ
2 zT T z
T2
2
2
2(30a)
⎜ ⎟− ⎛⎝
− ⎞⎠
+ ⎧⎨⎩
− − +
− + + +
− + + +
− +⎡
⎣
⎢⎢⎢
+ + + + +
+ + +
⎤
⎦
⎥⎥⎥
− −⎡
⎣
⎢⎢⎢
+ + +
+ +
⎤
⎦
⎥⎥⎥
⎫
⎬⎪
⎭⎪=
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
⊥∂∂
∂∂
∂∂
∂∂
⊥
∂∂
∂∂
∂∂
∂∂ ∂
∂∂
∂∂
∂∂ ∂
∂∂ ∂
∂∂
⊥
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
( )( )
( )( )
( )( )
H N
κd C
d C
C Cκ κ κ
κ κ κ
κ C Cκ
κ κ
Ξ (( ) )
( )
( )6
15 15
( )2
3 10 30
vξ
θξ
vτ x
TTT v
ξ
vT v
ηκd v
ηκd v
ηκd v
η
vT v
γd v
γd v
γd v
γ
dT
dT
vη
vγ
d vγ
vγ η
vη
d vγ
vγ η
vγ η
vη
dT
dT
wη γ
d wη γ
wη γ
d wη γ
wη γ
wη γ
2
2 ( )12
( )360
( )20160
212 360 20160
212
2 4
3602 4 6
32
1804 2
T zT T T
T T T T
T T T T
T T T T T
T T T T
T T T
T T T
22
22
22
22
2 44
4 66
6 88
22
2 44
4 66
6 88
22
22
2 44
42 2
44
4 66
64 2
62 4
66
2 2 43
43
4 65
63 3
65
(30b)
⎧⎨⎩
⎫⎬⎭
− − − =− ∂
∂∂∂
∂
∂( )λ θ χΞ 0,θ
τw
ξ yT θ
ξ2 y
T T yT2
2
2
2(30c)
⎜ ⎟− ⎛⎝
− ⎞⎠
+ ⎧⎨⎩
− − +
− + + +
− + + +
− +⎡
⎣
⎢⎢⎢
+ + + + +
+ + +
⎤
⎦
⎥⎥⎥
− −⎡
⎣
⎢⎢⎢
+ + +
+ +
⎤
⎦
⎥⎥⎥
⎫
⎬⎪
⎭⎪=
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
∂∂
⊥∂∂
∂∂
∂∂
∂∂
⊥
∂∂
∂∂
∂∂
∂∂ ∂
∂∂
∂∂
∂∂ ∂
∂∂ ∂
∂∂
⊥
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
∂∂ ∂
( )( )
( )( )
( )( )
H N
d C
κd C
C Cκ κ κ
κ κ κ
κ C Cκ
κ κ
Ξ (( ) )
( )
( )6
15 15
( )2
3 10 30
wξ
θξ
wτ x
TTT w
ξ
vT w
γd w
γd w
γd w
γ
vT w
ηκd w
ηκd w
ηκd w
η
dT
dT
wη
wγ
d wγ
wγ η
wη
d wγ
wγ η
wγ η
wη
dT
dT
vη γ
d vη γ
vη γ
d vη γ
vη γ
vη γ
2
212 360 20160
2 ( )12
( )360
( )20160
212
2 4
3602 4 6
32
1804 2
T yT T T
T T T T
T T T T
T T T T T
T T T T
T T T
T T T
22
22
22
22
2 44
4 66
6 88
22
2 44
4 66
6 88
22
22
2 44
42 2
44
4 66
64 2
62 4
66
2 2 43
43
4 65
63 3
65
(30d)
where =θ θ ξ η γ τ( , , , )zT
zT =v v ξ η γ τ( , , , )T T , =θ θ ξ η γ τ( , , , )y
TyT ,
=w w ξ η γ τ( , , , )T T represent the dimensionless continuous deforma-tion fields, and
= = = = =
= = ⊥
C C v w θ θ θ
θ
, , , , ,
; [.] or ,
dT C d l
k G A l vT C d l
k G A lT v
lT w
l yT
yT
zT
zT
[.] [.]d b
s b b z
v b
s b b z
T
b
R
b
[.] 2 2
2[.] 2 2
2
(31)
Eqs. (30a)-(30d) show the dimensionless nonlocal continuousequations of motion of the vertically aligned ASWCNTs acted upon byboth externally applied magnetic and thermal fields according to theNTBM. In the following, AMM is implemented to discretize the un-known fields in the spatial domain of the problem.
Let's express the deformation fields of the continuous model basedon the NTBM as follows:
= ∑ ∑ ∑
= ∑ ∑ ∑
= ∑ ∑ ∑
= ∑ ∑ ∑
= = =
= = =
= = =
= = =
θ ξ η γ τ θ τ ϕ ξ η γ
v ξ η γ τ v τ ϕ ξ η γ
θ ξ η γ τ θ τ ϕ ξ η γ
w ξ η γ τ w τ ϕ ξ η γ
( , , , ) ( ) ( , , ),
( , , , ) ( ) ( , , ),
( , , , ) ( ) ( , , ),
( , , , ) ( ) ( , , ),
zT
mN
nN
pN
zT
mnpθ
TmN
nN
pN
mnpT
mnpv
yT
mN
nN
pN
yT
mnpθ
TmN
nN
pN
mnpT
mnpw
1 1 1
1 1 1
1 1 1
1 1 1
mw nw pwmnp
z
mv nv pv
mv nv pvmnp
y
mw nw pw(32)
where the mode shapes of the nanosystem-whose ends are simplysupported and its exterior SWCNTs could not move transversely-areconsidered in the following form:
= =
= =
ϕ ξ η γ ϕ ξ η γ mπξ nπη pπγ
ϕ ξ η γ ϕ ξ η γ mπξ nπη pπγ
( , , ) ( , , ) sin( )sin( )sin( ),
( , , ) ( , , ) cos( )sin( )sin( ).mnpv
mnpw
mnpθ
mnpθy z
(33)
In order to construct the weak form of the governing equations, we
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
504
use Galerkin method. To this end, both sides of Eqs. (30a)-(30d) arepremultiplied by δθz
T , δvT , δθyT , and δwT , respectively. After taking the
integration by parts, the following set of second-order ODEs is derived:
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
⎧
⎨
⎪⎪⎪
⎩
⎪⎪⎪
⎫
⎬
⎪⎪⎪
⎭
⎪⎪⎪
+
⎡
⎣
⎢⎢⎢⎢⎢
⎤
⎦
⎥⎥⎥⎥⎥
⎧
⎨
⎪
⎩⎪
⎫
⎬
⎪
⎭⎪
=⎧
⎨⎩
⎫
⎬⎭
M M M M
M M M M
M M M M
M M M M
K K K K
K K K K
K K K K
K K K K
ΘvΘw
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
[ ] [ ] [ ] [ ]
0000
,
bT θ θ
bT θ v
bT θ θ
bT θ w
bT vθ
bT vv
bT vθ
bT vw
bT θ θ
bT θ v
bT θ θ
bT θ w
bT wθ
bT wv
bT wθ
bT ww
τ
τ
τ
τ
bT θ θ
bT θ v
bT θ θ
bT θ w
bT vθ
bT vv
bT vθ
bT vw
bT θ θ
bT θ v
bT θ θ
bT θ w
bT wθ
bT wv
bT wθ
bT ww
zT
T
yT
T
Θ
v
Θ
w
dd
dd
d
dd
d
z z z z y z
z y
y z y y y y
z y
zT
T
yT
T
z z z z y z
z y
y z y y y y
z y
2
2
22
2
2
22
(34)
in which the dimensionless mass and stiffness submatrices are given inAppendix B. By following the procedure mentioned in the previous part,the natural frequencies of the nanosystem modeled via continuous-NTBM are readily obtained.
4. Results and discussion
Magnetically-thermally affected ASWCNTs with the followingproperties are considered: Eb =1012 Pa, νb =0.2, ρb =2300 kg/m3,rm =1nm, and d=2rm+Nz. For low and high levels of temperatures,the reference values of the coefficients of thermal expansion (CTE) ofnanotubes (αT0) are set equal to −1.5× −10 6/oC and 1.1× −10 6/oC, re-spectively (Yao and Han, 2006; Zhang and Wang, 2005). In all carriedout numerical analysis in this section, a temperature-dependent form ofthe CTE has been taken into account: = +α α T(1 0.002Δ )T T0 (Zhangand Wang, 2005). In the following parts, the efficiency of the developedcontinuous-based models constructed based on the NRBM and NTBM isproved through various numerical examples. Thereafter, the roles ofinfluential factors on free transverse vibration of the magnetically-thermally influenced nanosystem are examined and discussed in detail.
A detailed discussion on the calibration of the small-scale parameterfor modeling of carbon nanotubes has been provided by Arash andWang (2012) according to the works of the initial scholars (Ansari andRamezannezhad, 2011; Duan et al., 2007). It was emphasized that therational evaluation of the nonlocal parameter is essential for applic-ability and justification of the nonlocal continuum-based models. In thisview, in most of the performed calculations in this section, the value ofthe small-scale parameter is set equal to 2 nm. In a subsection, a dis-cussion on the role of this crucial factor in the fundamental frequency ofthe nanosystem under both longitudinal thermal and magnetic fields isalso given. Actually, such a study shows sensitivity of the suggestednonlocal continuous models to the small-scale factor.
4.1. Efficiency of the suggested continuous models
For a given magnetic field strength and temperature change, thepredicted fundamental frequencies by both discrete and continuousmodels for various values of slenderness ratio and number of SWCNTshave been presented in Table 1. As it is seen there exist a reasonablygood agreement between the predicted results by the discrete modelsand those of their corresponding models for all considered values ofslenderness ratio and number of nanotubes. Additionally, the predictedfundamental frequencies by the NRBM approach to those of the NTBMas the slenderness ratio of the constitutive SWCNTs of the magnetically-thermally affected nanosystem increases. Generally, by increasing thenumber of SWCNTs within the nanosystem, both NRBM and NTBMpredict that the fundamental frequencies would reduce. More detailedinfluences of the slenderness ratio and the number of SWCNTs of thenanosystem on its vibration behavior will be discussed in the next parts.
We are also interested in checking accuracy of the suggested con-tinuous-based NRBM and NTBM by comparing their obtained resultsand those obtained from the discrete models. In Table 2, the predictedfundamental frequencies by the NRBM and those of the NTBM havebeen provided in the case of H x
R0 =1, Ny = Nz =9, and TΔ =200 °C. A
brief review of the obtained results shows that the proposed con-tinuous-based models can efficiently capture the results of the discrete-based models with a high level of accuracy. For a given magnetic fieldstrength, the fundamental frequency of the nanosystem would increaseas the radius of SWCNTs increase. In the following subsection, moreexplanations on the role of small-scale parameter and slenderness ratioin vibrations of magnetically-thermally affected ASWCNTs will begiven.
Tables 3 and 4 present the predicted fundamental frequencies of themagnetically-thermally affected nanosystem for different levels of themagnetic field strength and change of temperatures. The results areprovided for nanosystems with Ny = Nz =7, e a0 =2nm, andrm =1nm. The predicted results by both NRBM and NTBM in the caseof low temperature change have been provided in Table 3 while thoseof high temperature change are given in Table 4. As it is seen, for allconsidered values of the magnetic field strength as well as temperaturechange, the continuous models could capture the results of the discretemodels with a reasonable good accuracy. In the case of low temperaturechange, for a given magnetic field strength, both NRBM and NTBMpredict that the fundamental frequency of the nanosystem would re-duce by an increase of the temperature change. However, for high le-vels of temperature change, the predicted fundamental frequencieswould decrease as the temperature increases. For both low and highlevels of temperature changes, the fundamental frequencies would in-crease as the magnetic field strength becomes highlighted. Given theimportance of both temperature change and magnetic field strength onvibrations of the nanosystem, their effects on fundamental frequency ofthe nanosystem will be displayed and discussed more carefully in theupcoming parts.
4.2. Parametric studies
In this subsection, it is aimed to examine the influences of magneticfield strength, temperature change, slenderness ratio, population of thenanosystem, nonlocality, and radius of SWCNTs on free vibration of themagnetically-thermally affected nanosystems. In all demonstrated re-sults, the predicted results by the NRBM and those of the NTBM in order
Table 1Predicted fundamental frequencies (THz) by both continuous and discretemodels for various levels of the slenderness ratio and number of SWCNTs;( =H H λ λ/x
RxR0 0; H x
R0=1, λ0=10, e a0 =2 nm, rm=1 nm, TΔ =200 oC,
αT0=-1.5× −10 6/oC).
λ □
5 7 9 11
Discrete models NRBM 10 2.591846 2.408772 2.338583 2.30487715 1.818904 1.531100 1.411859 1.35222720 1.563306 1.208890 1.050624 .96738940 1.399542 .970389 .755960 .630555
NTBM 10 2.384918 2.175120 2.093513 2.05405515 1.777162 1.477464 1.351960 1.28879220 1.553340 1.194301 1.033074 .94792040 1.399435 .970115 .755542 .630013
Continuous models NRBM 10 2.588019 2.406874 2.337104 2.30376115 1.813031 1.527885 1.409221 1.35017820 1.556264 1.204695 1.046970 .96443840 1.391397 .964975 .750696 .625859
NTBM 10 2.380572 2.172923 2.091785 2.05274515 1.771081 1.474093 1.349172 1.28661720 1.546225 1.190037 1.029342 .94489540 1.391288 .964698 .750273 .625312
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
505
have been demonstrated by the dotted lines and the dashed lines.
4.2.1. Effect of the magnetic field strengthIn Fig. 2, the predicted fundamental frequencies of the magneti-
cally-thermally affected ASWCNTs by the continuous-based NRBM andNTBM as a function of dimensionless magnetic field strength have beendemonstrated for three lengths of the nanotubes (i.e., lb =10, 15, and20 nm) in the case of Ny = Nz =1000, e a0 =2nm, and TΔ =200oC.According to the plotted results, the predicted fundamental frequenciesby both proposed models would increase as the magnetic field strengthincreases. The main reason of this fact is that the exertion of long-itudinal magnetic field on highly conducting nanostructures would actlike as pre-tensioning force (see Eqs. (A.1d), (A.1f), (B.1f), and (B.1j)).In fact, these equations display that the transverse stiffness of the na-nosystem would grow as the magnetic field strength increases. A briefsurvey of the plotted results reveals that the magnetic field strength ismore influential on vibrations of magnetically-affected nanosystemswith higher lengths. Generally, the predicted fundamental frequenciesby the NRBM are greater than those of the NTBM. This is mainly relatedto the incorporation of the shear deformation effect into the formula-tions of the NTBM in which leads to this fact that the transverse stiffnessof the nanosystem based on the NTBM would be overestimated by thatof the NRBM. Concerning the capability of the NRBM in capturing the
results of the NTBM, a detailed scrutiny of the obtained results showsthat the relative discrepancies between the results of the continuous-based NRBM and those of the continuous-based NTBM would decreaseby growing of the magnetic field strength. The main reason of this factis that the ratio of the flexural stiffness of the nanosystem to the shearstiffness would magnify as the magnetic field strength increases.Commonly, by growing of the length of the constitutive SWCNTs of thenanosystem, the role of the shear deformation on its vibration wouldlessen; as a result, the predicted results by the NRBM would approach tothose of the NTBM by increasing the length of nanotubes.
4.2.2. Effect of the temperature changeAn important parametric study has been conducted to explore the
role of the temperature change on free transverse vibrations of themagnetically-thermally affected ASWCNTs. For this purpose, the plotsof fundamental frequencies by the suggested nonlocal continuousmodels in terms of the temperature change have been provided inFigs. 3 and 4.
Fig. 3 displays the predicted fundamental frequencies by bothNRBM and NTBM for low levels of temperature changes. The resultshave been demonstrated for three levels of the length of SWCNTs (i.e.,lb =50, 75, and 100 nm) for a high populated nanosystem (i.e.,Ny = Nz =1000) in the case of H x
R0 =0.01. We consider the
Table 2Predicted fundamental frequencies (THz) by both continuous and discrete models for various levels of the SWCNT’s radius and magnetic field strength;( =H H λ λ/x
RxR0 0; H x
R0=1, λ0=10, Ny=Nz=9, e a0 =2nm, rm=1nm, TΔ =200 oC, αT0=-1.5× −10 6/oC).
H xR0 rm (nm)
1 1.2 1.4 1.6 2
Discrete models NRBM 0 1.400837 1.585329 1.774444 1.963441 2.3143260.1 1.402293 1.586590 1.775545 1.964410 2.3151000.2 1.406655 1.590369 1.778843 1.967313 2.3174180.4 1.423968 1.605393 1.791975 1.978884 2.326667
NTBM 0 1.310736 1.436689 1.554529 1.662092 1.8233600.1 1.312317 1.438121 1.555843 1.663315 1.8244690.2 1.317049 1.442406 1.559780 1.666979 1.8277910.4 1.335809 1.459423 1.575427 1.681556 1.841019
Continuous models NRBM 0 1.398204 1.584249 1.773959 1.963206 2.3189090.1 1.399663 1.585511 1.775060 1.964175 2.3196810.2 1.404033 1.589292 1.778359 1.967079 2.3219950.4 1.421378 1.604326 1.791495 1.978651 2.331225
NTBM 0 1.307878 1.435463 1.553949 1.661795 1.8299280.1 1.309462 1.436896 1.555264 1.663019 1.8310320.2 1.314205 1.441185 1.559202 1.666684 1.8343430.4 1.333005 1.458216 1.574855 1.681263 1.847523
Table 3Predicted fundamental frequencies (THz) by both continuous and discrete models for different values of the temperature change and magnetic field strength in thecase of low temperatures; (αT0=-1.5× −10 6/oC, λ=80, Ny = Nz =7, e a0 =2 nm, rm =1nm).
H xR TΔ (oC)
0 100 150 200 300
Discrete models NRBM 0 1.141156 1.141259 1.141323 1.141395 1.1415660.5 1.141178 1.141281 1.141345 1.141418 1.1415891 1.141245 1.141348 1.141412 1.141484 1.1416551.5 1.141356 1.141459 1.141523 1.141596 1.141767
NTBM 0 1.141156 1.141259 1.141323 1.141395 1.1415660.5 1.141178 1.141281 1.141345 1.141418 1.1415891 1.141245 1.141348 1.141412 1.141484 1.1416551.5 1.141356 1.141459 1.141523 1.141596 1.141767
Continuous models NRBM 0 1.136498 1.136601 1.136665 1.136738 1.1369100.5 1.136520 1.136623 1.136688 1.136761 1.1369321 1.136587 1.136690 1.136755 1.136828 1.1369991.5 1.136699 1.136802 1.136866 1.136939 1.137111
NTBM 0 1.136498 1.136601 1.136665 1.136738 1.1369100.5 1.136520 1.136623 1.136688 1.136761 1.1369321 1.136587 1.136690 1.136755 1.136828 1.1369991.5 1.136699 1.136802 1.136866 1.136939 1.137111
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
506
dimensionless strength of magnetic filed in the form ofH x
R = H l l( / )xR
b b0 02 to neutralize the effect of variation of length on the
magnetic field strength. As it is obvious in the plotted results in Fig. 3,the fundamental frequencies of nanosystems with different lengthswould increase as the temperature magnifies in the range of fairly lowtemperatures. This is because of this fact that the coefficient of thermalexpansion is negative for such temperature levels. Therefore, by in-creasing of the temperature, a thermo-mechanical tension force wouldbe generated within the magnetically-thermally affected fixed-fixedASWCNTs. As a result, the flexural stiffness of the nanosystem as well asits fundamental frequency would increase as the temperature increaseup to 300oC. Since the length to diameter of all considered nanosystemsis so high, the role of shear deformation on vibrations is fairly negli-gible. These interpretations explain the fairly good conformity of thepredicted results based on the NRBM and those of the NTBM. It is re-vealed from the obtained results that the discrepancies between theresults of the NRBM and those of the NTBM would generally lessen byincreasing of the temperature in the considered range. Such detailedassessments indicates that the NRBM could capture the predicted fun-damental frequencies by the NTBM with relative error lower than 0.45,0.2, and 0.1 percent for lb =50, 75, and 100 nm, respectively.
Table 4Predicted fundamental frequencies (THz) by both continuous and discrete models for different values of the temperature change and magnetic field strength in thecase of high temperatures; (αT0=1.1× −10 6/oC, λ=80, Ny = Nz =7, e a0 =2nm, rm =1nm).
H xR TΔ (oC)
600 700 800 900 1000
Discrete models NRBM 0 1.140328 1.140102 1.139851 1.139574 1.1392730.5 1.140350 1.140124 1.139873 1.139597 1.1392951 1.140417 1.140191 1.139940 1.139663 1.1393621.5 1.140528 1.140302 1.140051 1.139775 1.139474
NTBM 0 1.140328 1.140102 1.139851 1.139574 1.1392730.5 1.140350 1.140124 1.139873 1.139597 1.1392951 1.140417 1.140191 1.139940 1.139663 1.1393621.5 1.140528 1.140302 1.140051 1.139775 1.139474
Continuous models NRBM 0 1.135666 1.135439 1.135187 1.134910 1.1346070.5 1.135689 1.135462 1.135209 1.134932 1.1346291 1.135756 1.135529 1.135277 1.134999 1.1346971.5 1.135868 1.135641 1.135389 1.135111 1.134808
NTBM 0 1.135666 1.135439 1.135187 1.134910 1.1346070.5 1.135689 1.135462 1.135209 1.134932 1.1346291 1.135756 1.135529 1.135277 1.134999 1.1346971.5 1.135868 1.135641 1.135389 1.135111 1.134808
Fig. 2. Fundamental frequency in terms of the magnetic field strength for dif-ferent values of the length of nanotubes: ((…) NRBM, (− −) NTBM; (○) lb=10,(□) lb=15, (△) lb=20 nm; =H H l l/x
RxR
b b0 0 , H xR0=0.5; lb0=10 nm;
Ny=Nz=1000; TΔ =200 oC; e a0 =2 nm).
Fig. 3. Fundamental frequency in terms of the temperature change for differentvalues of the length of nanotubes: ((…) NRBM, (− −) NTBM; (○) lb=50, (□)lb=75, (△) lb=100 nm; =H H l l/x
RxR
b b0 0 , H xR0=0.01; lb0=10 nm;
Ny=Nz=1000; e a0 =2 nm; fairly low temperatures).
Fig. 4. Fundamental frequency in terms of the temperature change for differentvalues of the length of nanotubes: ((…) NRBM, (− −) NTBM; (○) lb=15, (□)lb=25, (△) lb=35 nm; =H H l l/x
RxR
b b0 0 , H xR0=0.01; lb0=10 nm; Ny=Nz=1000;
e a0 =2 nm; high temperatures).
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
507
In Fig. 3, the predicted fundamental frequencies as a function oftemperature have been provided for high levels of temperature changesfor three levels of the length of SWCNTs (i.e., lb =15, 25, and 35 nm) inthe case of H x
R0 =0.01 and e a0 =2nm. As it is seen, the predicted
fundamental frequencies of the magnetically-thermally affected nano-system would reduce by increasing of the temperature change. Sincethe coefficient of thermal expansion of the constitutive SWCNTs of thenanosystem is positive for high levels of temperatures, therefore, in-creasing of the temperature would lead to exertion of extra axiallycompressive force to the nanotubes and the flexural stiffness of thenanosystem would reduce. Consequently, the natural frequencies of thenanosystem reduce. For all considered lengths of the constitutiveSWCNTs, the relative discrepancies between the results of the NRBMand those of the NTBM would somewhat magnify as the temperatureincreases. Generally, such relative discrepancies would reduce bygrowing of the length of nanotubes. For instance, in the cases of lb =15and 25, the NRBM could capture the results of the NTBM with relativeerror lower than 5.8 and 3.2 for all considered temperatures in therange of 600–1000oC.
4.2.3. Effect of the slenderness ratioThe slenderness ratio is one of the important geometrical para-
meters of the magnetically affected nanosystem in which its role on itsvibrational behavior should be carefully addressed. To this end, theplots of the most dominant frequency of the nanosystem based on thesuggested continuous models as a function of the slenderness ratio havebeen provided in Fig. 5 for three levels of the magnetic field strength(i.e., H x
R0 =0, 1.5, and 3) in the case of TΔ =200oC, e a0 =2nm,
Ny = Nz =1000, and rm =1nm. The plotted results in this figure dis-play that the predicted fundamental frequencies by both NRBM andNTBM would decrease by growing of the slenderness ratio for all con-sidered levels of the magnetic field strength. The rate of reduction ismore apparent for higher levels of magnetic field strength. Additionally,the reduction rate of the NRBM's results is more obvious with respect tothose of the NTBM. A close scrutiny of the demonstrated results in-dicates that the relative discrepancies between the results of the NRBMand those of the NTBM would generally reduce by increasing of theslenderness ratio. In fact, as the slenderness ratio magnifies, the ratio ofthe shear strain energy to the flexural strain energy would reduce, andthereby, the role of shear deformation in free vibration behavior of themagnetically-thermally affected nanosystem would diminish. A moredetailed study shows that for slenderness ratios greater than about 9and 14.5, the NRBM could reproduce the predicted fundamental fre-quencies by the NTBM with relative error lower than 5 percent for thecases of H x
R0 =3 and 1.5, respectively. In the absence of the magnetic
field, such relative errors for λ=7, 13.5, and 20 in order are ap-proximately equal to 28.65, 10.14, and 5 percent. It implies that theNRBM could not capture the results of the NTBM with relative errorlower than 5 percent for a slenderness ratio lower than 20 when nomagnetic field is applied on the nanosystem. Generally, by increasingthe magnetic field strength, the predicted fundamental frequencies bythe NRBM would approach to those of the NTBM. Such interestingphenomena and the reasons behind this fact were discussed in section5.2.1.
4.2.4. Effect of the populationWe are also interested in determination of the role of population of
magnetically-thermally affected ASWCNTs on its free vibration beha-vior. For this purpose, the predicted fundamental frequencies by bothcontinuous-based NRBM and NTBM in terms of the number of rows oftubes in the y direction (i.e., Ny) have been plotted in Fig. 6 for threelevels of the slenderness ratio (i.e., λ=10, 15, and 30) in the case ofe a0 =2nm and TΔ =200oC. Generally, the fundamental frequencieswould reduce by growing of the nanosystem's population. The rate ofreduction is more obvious for lower levels of the population. For allconsidered populations, the predicted fundamental frequencies by the
NTBM are overestimated by those of the NRBM. Further, the dis-crepancies between the results of the NRBM and those of the NTBMwould commonly increase by an increase of the population of theASWCNTs. It is mainly because of this fact that the share of shear strainenergy of beam-like nanotubes to their total strain energy would growby increasing of the population. This fact is also more obvious for na-nosystems with lower slenderness ratio. As a result, the role of sheardeformation on free vibration of the nanosystem would magnify byincreasing the number of constitutive SWCNTs, particularly for nano-systems with shorter tubes. For instance, a careful scrutiny of theplotted results shows that the relative discrepancies between the pre-dicted fundamental frequencies by the NRBM and those of the NTBMwould be approximately equal to (9.86,14.12,15.04), (2.6,5.65,6.7),and (0.06,0.38,0.73) percent for Ny=(5,10,15) in the cases of λ=10,15, and 30, respectively.
4.2.5. Effect of the small-scale parameterIn another parametric study, it is aimed to examine the influence of
the nonlocality on the transverse vibrations of magnetically-thermallyaffected ASWCNTs. In Fig. 7, the plots of fundamental frequency as afunction of the small-scale parameter have been graphed for three le-vels of the slenderness ratio (i.e., λ=10, 15, and 20) in the case ofH x
R0 =1, TΔ =200oC, rm =1nm, and d=2rm+Nz. Generally, the
fundamental frequencies would reduce as the effect of the small-scaleparameter becomes highlighted. Such a fact is more apparent formagnetically-thermally affected nanosystems with lower levels of theslenderness ratio. By an increase of the slenderness ratio, the dis-crepancies between the results of the NRBM and those of the NTBMwould lessen since the role of shear deformation in mechanical beha-vior of the nanosystem would diminish. Additionally, by growing of thesmall-scale parameter the discrepancies between the predicted funda-mental frequencies by the NRBM and those of the NTBM wouldsomewhat reduce.
4.2.6. Effect of the radius of constitutive SWCNTsIn Fig. 8, the predicted fundamental frequencies by both the con-
tinuous-based NRBM and NTBM in terms of the radius of SWCNTs havebeen demonstrated for three values of the length of SWCNTs (i.e.,lb =10, 15, and 20 nm). The obtained results are for magnetically-thermally affected nanosystem with Ny = Nz =1000, e a0 =2nm,
TΔ =200 °C, and H xR0 =0.5 such that: =H Hx
RxR A I l
A I l0b b bb b b
0 2
0 02 where
lb0 =7nm (It is noticed that the dimensionless magnetic field strength
Fig. 5. Fundamental frequency in terms of the slenderness ratio for variousvalues of the magnetic field strength: ((…) NRBM, (− −) NTBM; (○) H x
R0=0,
(□) H xR0=1.5, (△) H x
R0=3; =H H λ λ/x
RxR0 0 , λ0=10; Ny=Nz=1000; TΔ =200
oC; e a0 =2 nm).
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
508
associated with the NRBM is defined such that the variation of thelength of the constitutive SWCNTs of the nanosystem would have noinfluence on the value of the magnetic field strength). As it is obviousfrom the plotted results, both NRBM and NTBM predict that the fun-damental frequency would increase by growing of the radius of con-stitutive SWCNTs. Such a fact is more obvious for nanosystems withlower levels of the length of SWCNTs. Actually, by increasing of theradius of nanotubes, the lateral stiffness of the nanosystem wouldmagnify substantially with respect to its mass. This is mainly related tothis fact that an increase of the nanotubes' radius not only leads to anincrease of the transverse stiffness of each tube, but also the interac-tional vdW forces between each pair tubes would magnify. Conse-quently, the flexural frequency of the magnetically-thermally affectednanosystem would grow as the radius of its constitutive SWCNTs wouldgrow. It should be also noted that the relative discrepancies betweenthe results of the NRBM and those of the NTBM would commonly in-crease as the radius of SWCNTs would increase. Such a trend is alsomore obvious for those nanosystems with lower levels of length ofnanotubes. The chief reason of this fact is the increase of the ratio of theshear strain energy to the flexural strain energy as the nanotube's radius
increases.
4.2.7. A brief discussion on the capabilities of the discrete- and continuous-based models
The major privilege of the nonlocal continuous-based models withrespect to the discrete ones is their high capabilities in capturing fun-damental frequency of highly dense vertically aligned ASWCNTs undermulti-physical fields. However, it is noticed that the continuous modelshave also several limitations which are explained briefly in the fol-lowing: (i) the discrete models could be efficiently employed for vi-brational analysis of nanosystems with complex boundary conditions.For such conditions, development of admissible mode shapes is a cri-tical milestone and this issue restricts the ease in applicability of thenonlocal continuous models on the basis of the assumed mode method;(ii) for vibrational analysis of the nanosystems with Ny or ≤N 3z , thenonlocal continuous-based models could not be exploited. The discretemodels could be easily employed for mechanical analysis of verticallyaligned nanosystems with arbitrary numbers of SWCNTs in both y and zdirections; however, their efficiency would be substantially reduced forhighly dense ASWCNTs; (iii) for the case of the nanosystem acted uponby a partial thermal or magnetic field, special treatment should beconsidered for the problems which are going to be analyzed based onthe continuous model due to the discontinuity of the applied thermal/magnetic field; however, these discontinuities could be easily taken intoaccount in the nonlocal discrete models.
5. Conclusions
Using nonlocal elasticity theory of Eringen, magneto-thermo-elasticvibrations of vertically aligned ensembles of SWCNTs were studied.Based on the NRBM and NTBM, the equations of motion of the nano-system in the presence of the longitudinal magneto-thermal fields wereobtained by employing Hamilton's principle. The vdW forces betweenany pair of deformed tubes were idealized by linear springs across theirlengths. Thereafter, the thermally-magnetically affected nanosystemwas modeled by the Rayleigh and Timoshenko beam theories withappropriate interconnecting springs. Both the discrete and continuousforms of the governing equations were developed and established. Forvarious geometries, thermal environments, and magnetic field strength,capabilities of the continuous NRBM and NTBM in predicting naturalfrequencies of their counterpart discrete models were checked and ex-plained. The effects of the radius of the constitutive SWCNTs,
Fig. 6. Fundamental frequency in terms of the population of the ensemble fordifferent levels of the slenderness ratio: ((…) NRBM, (− −) NTBM; (○) λ=10,(□) λ=15, (△) λ=30; =H H λ λ/x
RxR0 0 , λ0=10, H x
R0=0.5; TΔ =200 oC; e a0 =2
nm).
Fig. 7. Fundamental frequency in terms of the small-scale parameter for dif-ferent slenderness ratios: ((…) NRBM, (− −) NTBM; (○) λ=10, (□) λ=15, (△)λ=20; =H H λ λ/x
RxR0 0, Ny=Nz=1000; H x
R0=1; λ0=10; TΔ =200 oC).
Fig. 8. Fundamental frequency in terms of the mean radius of SWCNTs fordifferent levels of nanotube length: ((…) NRBM, (− −) NTBM; (○) lb=10, (□)
lb=15, (△) lb=20 nm; =H HxR
xR AbIb lb
Ab Iblb0
0 2
0 02 , Ny=Nz=1000; H x
R0=0.5; lb0=7
nm; TΔ =200 oC; e a0 =2 nm).
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
509
temperature gradient, magnetic field strength, slenderness ratio, non-locality, and population of the nanosystem on the free vibration beha-vior were also noticed. The important role of shear deformation indynamic analysis of stocky nanosystems under both thermal and
magnetic fields was emphasized. Further, the efficiency of the sug-gested continuous models in magneto-thermo-vibro analysis ofASWCNTs with high population was highlighted.
Appendix A. Mass and stiffness matrices of the continuous-based NRBM
The dimensionless time-dependent vectors, nonzero mass and stiffness submatrices associated with the continuous-based NRBM model areprovided in the following:
= =v wv w, ,qrsR
qrsR
qrsR
qrsR
(A.1a)
∫ ∫ ∫⎜ ⎟
⎜ ⎟
=
⎡
⎣
⎢⎢⎢⎢⎢
⎛⎝
+ ⎞⎠
+
⎛⎝
+ ⎞⎠
⎤
⎦
⎥⎥⎥⎥⎥
−
ϕ ϕ μ ϕ ϕ
λ ϕ ϕ μ ϕ ϕ
ξ η γM[ ] d d d ,bR
mnpqrsvv
mnpv
qrsv
mnp ξv
qrs ξv
mnp ξv
qrs ξv
mnp ξξv
qrs ξξv
01
01
01
2, ,
2, ,
2, ,
(A.1b)
∫ ∫ ∫⎜ ⎟
⎜ ⎟
=
⎡
⎣
⎢⎢⎢⎢⎢
⎛⎝
+ ⎞⎠
+
⎛⎝
+ ⎞⎠
⎤
⎦
⎥⎥⎥⎥⎥
−
ϕ ϕ μ ϕ ϕ
λ ϕ ϕ μ ϕ ϕ
ξ η γM[ ] d d d ,bR
mnpqrsww
mnpw
qrsw
mnp ξw
qrs ξw
mnp ξw
qrs ξw
mnp ξξw
qrs ξξw
01
01
01
2, ,
2, ,
2, ,
(A.1c)
∫ ∫ ∫ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
= ⎧⎨⎩
+ − ⎛⎝
+ ⎞⎠
+ ⎡⎣⎢
+
− ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
− ⎛⎝
+ ⎞⎠
⎤
⎦⎥
+ ⎡⎣⎢
+
− ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
− ⎛⎝
+ ⎞⎠
⎤
⎦⎥
+ + ⎧⎨⎩
+
+ ⎛⎝
+ ⎞⎠
− ⎡⎣⎢
+ +
⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
⎤
⎦⎥
+ ⎡
⎣⎢ + + ⎛
⎝+ ⎞
⎠
+ ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
⎤
⎦⎥
⎫⎬⎭
⎫⎬⎭
⊥
⊥
ϕ ϕ H N ϕ ϕ μ ϕ ϕ
κd C ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ
d C ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ
C C ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ κ ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ κ ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ ξ η γ
K[ ] (( ) )
( )
( )
6
15
15
d d d ,
bR
mnpqrsvv
mnp ξξv
qrs ξξv
xR
TR
mnp ξv
qrs ξv
mnp ξξv
qrs ξξv
vR
mnp ηv
qrs ηv
mnp ξηv
qrs ξηv
κdmnp ηηv
qrs ηηv
mnp ξηηv
qrs ξηηv κd
mnp ηηηv
qrs ηηηv
mnp ξηηηv
qrs ξηηηv
κdmnp ηηηηv
qrs ηηηηv
mnp ξηηηηv
qrs ξηηηηv
vR
mnp γv
qrs γv
mnp ξγv
qrs ξγv
dmnp γγv
qrs γγv
mnp ξγγv
qrs ξγγv d
mnp γγγv
qrs γγγv
mnp ξγγγv
qrs ξγγγv
dmnp γγγγv
qrs γγγγv
mnp ξγγγγv
qrs ξγγγγv
dR
dR
mnp γv
qrs γv
mnp ξγv
qrs ξγv
mnp ηv
qrs ηv
mnp ξηv
qrs ξηv d
mnp γγv
qrs γγv
mnp ξγγv
qrs ξγγv
mnp ηγv
qrs ηγv
mnp ξηγv
qrs ξηγv
mnp ηηv
qrs ηηv
mnp ξηηv
qrs ξηηv
dmnp γγγv
qrs γγγv
mnp ξγγγv
qrs ξγγγv
mnp ηγγv
qrs ηγγv
mnp ξηγγv
qrs ξηγγv
mnp ηηγv
qrs ηηγv
mnp ξηηγv
qrs ξηηγv
mnp ηηηv
qrs ηηηv
mnp ξηηηv
qrs ξηηηv
01
01
01
, ,2
, ,2
, ,
2, ,
2, ,
( )12 , ,
2, ,
( )360 , ,
2, ,
( )20160 , ,
2, ,
2, ,
2, ,
12 , ,2
, , 360 , ,2
, ,
20160 , ,2
, ,
, ,2
, ,
2, ,
2, , 12 , ,
2, ,
2, ,
2, ,
4, ,
2, ,
360 , ,2
, ,2
, ,2
, ,
4, ,
2, ,
6, ,
2, ,
2 4
6
2 4
6
2
4
(A.1d)
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
510
∫ ∫ ∫ ∫ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
= − ×
⎧⎨⎩
+ + ⎛⎝
+ ⎞⎠
−
⎡
⎣
⎢⎢⎢⎢⎢
∇ ⎧⎨⎩
⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
+ ∇ ⎧⎨⎩
⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
+
⎛
⎝⎜∇ ⎧
⎨⎩⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
+ ∇ ⎧⎨⎩
⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
⎞
⎠⎟
⎤
⎦
⎥⎥⎥⎥⎥
+
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎛
⎝⎜ + + ⎛
⎝+ ⎞
⎠
⎞
⎠⎟+
⎛
⎝⎜ + + ⎛
⎝+ ⎞
⎠
⎞
⎠⎟+
⎛
⎝⎜ + + ⎛
⎝+ ⎞
⎠
⎞
⎠⎟
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⊥κ C C
ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ
μ ϕ ϕ ϕ ϕ
κ ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
κ ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
ξ η γ
K[ ] ( )
. .
. .
3
10
3
} d d d ,
bR
mnpqrsvw
dR
dR
mnp ηv
qrs γw
mnp γv
qrs ηw
mnp ξηv
qrs ξγw
mnp ξγv
qrs ξηw
dηγ mnp η
vηγ qrs γ
wηγ mnp γ
vηγ qrs η
w
ηγ mnp ξηv
ηγ qrs ξγw
ηγ mnp ξγv
ηγ qrs ξηw
d
mnp γηηv
qrs ηηηw
mnp ηηηv
qrs γηηw
mnp ξηηγv
qrs ξηηηw
mnp ξηηηv
qrs ξηηγw
mnp ηηηv
qrs γγγw
mnp γγγv
qrs ηηηw
mnp ξηηηv
qrs ξγγγw
mnp ξγγγv
qrs ξηηηw
mnp ηγγv
qrs γγγw
mnp γγγv
qrs ηγγw
mnp ξηγγv
qrs ξγγγw
mnp ξγγγv
qrs ξηγγw
01
01
01
01
, , , ,2
, , , ,
6
, , , ,
2, , , ,
360
4, , , ,
2, , , ,
2, , , ,
2, , , ,
, , , ,2
, , , ,
2
4
(A.1e)
∫ ∫ ∫ ∫ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
= ⎧⎨⎩
+ − ⎛⎝
+ ⎞⎠
+ ⎡⎣⎢
+
− ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
− ⎛⎝
+ ⎞⎠
⎤
⎦⎥
+ ⎡⎣⎢
+
− ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
− ⎛⎝
+ ⎞⎠
⎤
⎦⎥
+ + ⎧⎨⎩
+
+ ⎛⎝
+ ⎞⎠
− ⎡⎣⎢
+
+ ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
⎤
⎦⎥
+ ⎡
⎣⎢ + + ⎛
⎝+ ⎞
⎠
+ ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
⎫⎬⎭
⎫⎬⎭
⎤
⎦⎥⎥
⊥
⊥
ϕ ϕ H N ϕ ϕ μ ϕ ϕ
C d ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ
C κd ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ
C C ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ κ ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ κ ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ ξ η γ
K[ ] (( ) )
( )
( )
6
15
15
d d d ,
bR
mnpqrsww
mnp ξξw
qrs ξξw
xR
TR
mnp ξw
qrs ξw
mnp ξξw
qrs ξξw
vR
mnp γw
qrs γw
mnp ξγw
qrs ξγw
dmnp γγw
qrs γγw
mnp ξγγw
qrs ξγγw d
mnp γγγw
qrs γγγw
mnp ξγγγw
qrs ξγγγw
dmnp γγγγw
qrs γγγγw
mnp ξγγγγw
qrs ξγγγγw
vR
mnp ηw
qrs ηw
mnp ξηw
qrs ξηw
κdmnp ηηw
qrs ηηw
mnp ξηηw
qrs ξηηw κd
mnp ηηηw
qrs ηηηw
mnp ξηηηw
qrs ξηηηw
κdmnp ηηηηw
qrs ηηηηw
mnp ξηηηηw
qrs ξηηηηw
dR
dR
mnp γw
qrs γw
mnp ξγw
qrs ξγw
mnp ηw
qrs ηw
mnp ξηw
qrs ξηw d
mnp γγw
qrs γγw
mnp ξγγw
qrs ξγγw
mnp ηγw
qrs ηγw
mnp ξηγw
qrs ξηγw
mnp ηηw
qrs ηηw
mnp ξηηw
qrs ξηηw
dmnp γγγw
qrs γγγw
mnp ξγγγw
qrs ξγγγw
mnp ηγγw
qrs ηγγw
mnp ξηγγw
qrs ξηγγw
mnp ηηγw
qrs ηηγw
mnp ξηηγw
qrs ξηηγw
mnp ηηηw
qrs ηηηw
mnp ξηηηw
qrs ξηηηw
01
01
01
01
, ,2
, ,2
, ,
2, ,
2, ,
12 , ,2
, , 360 , ,2
, ,
20160 , ,2
, ,
2, ,
2, ,
( )12 , ,
2, ,
( )360 , ,
2, ,
( )20160 , ,
2, ,
, ,2
, ,
2, ,
2, , 12 , ,
2, ,
2, ,
2, ,
4, ,
2, ,
360 , ,2
, ,2
, ,2
, ,
4, ,
2, ,
6, ,
2, ,
2 4
6
2 4
6
2
4
(A.1f)
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
511
∫ ∫ ∫ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
= − ×
⎧⎨⎩
+ + ⎛⎝
+ ⎞⎠
−
⎡
⎣
⎢⎢⎢⎢⎢
∇ ⎧⎨⎩
⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
+ ∇ ⎧⎨⎩
⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
+
⎛
⎝⎜∇ ⎧
⎨⎩⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
+ ∇ ⎧⎨⎩
⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
⎞
⎠⎟
⎤
⎦
⎥⎥⎥⎥⎥
+
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎛
⎝⎜ + + ⎛
⎝+ ⎞
⎠
⎞
⎠⎟+
⎛
⎝⎜ + + ⎛
⎝+ ⎞
⎠
⎞
⎠⎟+
⎛
⎝⎜ + + ⎛
⎝+ ⎞
⎠
⎞
⎠⎟
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⊥κ C C
ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ
μ ϕ ϕ ϕ ϕ
κ ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
κ ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
ξ η γ
K[ ] ( )
. .
. .
3
10
3
} d d d ,
bR
mnpqrswv
dR
dR
mnp ηw
qrs γv
mnp γw
qrs ηv
mnp ξηw
qrs ξγv
mnp ξγw
qrs ξηv
dηγ mnp η
wηγ qrs γ
vηγ mnp γ
wηγ qrs η
v
ηγ mnp ξηw
ηγ qrs ξγv
ηγ mnp ξγw
ηγ qrs ξηv
d
mnp γηηw
qrs ηηηv
mnp ηηηw
qrs γηηv
mnp ξηηγw
qrs ξηηηv
mnp ξηηηw
qrs ξηηγv
mnp ηηηw
qrs γγγv
mnp γγγw
qrs ηηηv
mnp ξηηηw
qrs ξγγγv
mnp ξγγγw
qrs ξηηηv
mnp ηγγw
qrs γγγv
mnp γγγw
qrs ηγγv
mnp ξηγγw
qrs ξγγγv
mnp ξγγγw
qrs ξηγγv
0
1
0
1
0
1
, , , ,2
, , , ,
6
, , , ,
2, , , ,
360
4, , , ,
2, , , ,
2, , , ,
2, , , ,
, , , ,2
, , , ,
2
4
(A.1g)
where the used gradient vector in these relations is given by: ∇ = + κe e[.] [.] [.]ηγ γ z η y, ,.
Appendix B. Mass and stiffness matrices of the continuous-based NTBM
The dimensionless time-dependent vectors as well as the dimensionless nonzero submatrices of mass and stiffness pertinent to the continuous-based NTBM are defined as:
= =
= =
v w
θ θ
v w
Θ Θ
, ,
, ,qrsT
qrsT
qrsT
qrsT
zT
zT
yT
yT
qrs qrs qrs qrs (B.1a)
∫ ∫ ∫ ⎜ ⎟= ⎛⎝
+ ⎞⎠
ϕ ϕ μ ϕ ϕ ξ η γM[ ] d d d ,bT
mnpqrsvv
mnpv
qrsv
mnp ξv
qrs ξv
01
01
01 2
, ,(B.1b)
∫ ∫ ∫ ⎜ ⎟= ⎛⎝
+ ⎞⎠
ϕ ϕ μ ϕ ϕ ξ η γM[ ] d d d ,bT
mnpqrsww
mnpw
qrsw
mnp ξw
qrs ξw
01
01
01 2
, ,(B.1c)
∫= +− ( )λ ϕ ϕ μ ϕ ϕ ξ η γM[ ] d d d ,bT
mnpqrsθ θ
mnpθ
qrsθ
mnp ξθ
qrs ξθ
02 2
, ,y y y y y y
(B.1d)
∫= +− ( )λ ϕ ϕ μ ϕ ϕ ξ η γM[ ] d d d ,bT
mnpqrsθ θ
mnpθ
qrsθ
mnp ξθ
qrs ξθ
01 2 2
, ,z z z z z z
(B.1e)
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
512
∫ ∫ ∫ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
= ⎧⎨⎩
+ − ⎛⎝
+ ⎞⎠
+
+ ⎡⎣⎢
+
− ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
− ⎛⎝
+ ⎞⎠
⎤
⎦⎥
+ ⎡⎣⎢
+
− ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
− ⎛⎝
+ ⎞⎠
⎤
⎦⎥
+ + ⎧⎨⎩
+
+ ⎛⎝
+ ⎞⎠
− ⎡⎣⎢
+
+ ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
⎤
⎦⎥
+ ⎡
⎣⎢ + + ⎛
⎝+ ⎞
⎠
+ ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
⎤
⎦⎥
⎫⎬⎭
⎫⎬⎭
⊥
⊥
ϕ ϕ H N ϕ ϕ μ ϕ ϕ
C d ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ
C κd ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ
C C ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ κ ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ κ ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ ξ η γ
K[ ] (( ) )
( )
( )
6
15
15
d d d ,
bT
mnpqrsww
mnp ξw
qrs ξw
xT
TT
mnp ξw
qrs ξw
mnp ξξw
qrs ξξw
vR
mnp γw
qrs γw
mnp ξγw
qrs ξγw
dmnp γγw
qrs γγw
mnp ξγγw
qrs ξγγw d
mnp γγγw
qrs γγγw
mnp ξγγγw
qrs ξγγγw
dmnp γγγγw
qrs γγγγw
mnp ξγγγγw
qrs ξγγγγw
vR
mnp ηw
qrs ηw
mnp ξηw
qrs ξηw
κdmnp ηηw
qrs ηηw
mnp ξηηw
qrs ξηηw κd
mnp ηηηw
qrs ηηηw
mnp ξηηηw
qrs ξηηηw
κdmnp ηηηηw
qrs ηηηηw
mnp ξηηηηw
qrs ξηηηηw
dR
dR
mnp γw
qrs γw
mnp ξγw
qrs ξγw
mnp ηw
qrs ηw
mnp ξηw
qrs ξηw d
mnp γγw
qrs γγw
mnp ξγγw
qrs ξγγw
mnp ηγw
qrs ηγw
mnp ξηγw
qrs ξηγw
mnp ηηw
qrs ηηw
mnp ξηηw
qrs ξηηw
dmnp γγγw
qrs γγγw
mnp ξγγγw
qrs ξγγγw
mnp ηγγw
qrs ηγγw
mnp ξηγγw
qrs ξηγγw
mnp ηηγw
qrs ηηγw
mnp ξηηγw
qrs ξηηγw
mnp ηηηw
qrs ηηηw
mnp ξηηηw
qrs ξηηηw
01
01
01
, ,2
, ,2
, ,
2, ,
2, ,
12 , ,2
, , 360 , ,2
, ,
20160 , ,2
, ,
2, ,
2, ,
( )12 , ,
2, ,
( )360 , ,
2, ,
( )20160 , ,
2, ,
, ,2
, ,
2, ,
2, , 12 , ,
2, ,
2, ,
2, ,
4, ,
2, ,
360 , ,2
, ,2
, ,2
, ,
4, ,
2, ,
6, ,
2, ,
2 4
6
2 4
6
2
4
(B.1f)
∫ ∫ ∫= − ϕ ϕ ξ η γK[ ] d d d ,bT
mnpqrswθ
mnp ξw
qrsθ
01
01
01
,y y
(B.1g)
∫ ∫ ∫= − ϕ ϕ ξ η γK[ ] d d d ,bT
mnpqrsθ w
mnpθ
qrs ξw
01
01
01
,y y
(B.1h)
∫ ∫ ∫ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
= − ×
⎧⎨⎩
+ + ⎛⎝
+ ⎞⎠
−
⎡
⎣
⎢⎢⎢⎢⎢
∇ ⎧⎨⎩
⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
+ ∇ ⎧⎨⎩
⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
+
⎛
⎝⎜∇ ⎧
⎨⎩⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
+ ∇ ⎧⎨⎩
⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
⎞
⎠⎟
⎤
⎦
⎥⎥⎥⎥⎥
+
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎛
⎝⎜ + + ⎛
⎝+ ⎞
⎠
⎞
⎠⎟+
⎛
⎝⎜ + + ⎛
⎝+ ⎞
⎠
⎞
⎠⎟+
⎛
⎝⎜ + + ⎛
⎝+ ⎞
⎠
⎞
⎠⎟
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⊥κ C C
ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ
μ ϕ ϕ ϕ ϕ
κ ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
κ ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
ξ η γ
K[ ] ( )
. .
. .
3
10
3
} d d d ,
bT
mnpqrswv
dT
dT
mnp ηw
qrs γv
mnp γw
qrs ηv
mnp ξηw
qrs ξγv
mnp ξγw
qrs ξηv
dηγ mnp η
wηγ qrs γ
vηγ mnp γ
wηγ qrs η
v
ηγ mnp ξηw
ηγ qrs ξγv
ηγ mnp ξγw
ηγ qrs ξηv
d
mnp γηηw
qrs ηηηv
mnp ηηηw
qrs γηηv
mnp ξηηγw
qrs ξηηηv
mnp ξηηηw
qrs ξηηγv
mnp ηηηw
qrs γγγv
mnp γγγw
qrs ηηηv
mnp ξηηηw
qrs ξγγγv
mnp ξγγγw
qrs ξηηηv
mnp ηγγw
qrs γγγv
mnp γγγw
qrs ηγγv
mnp ξηγγw
qrs ξγγγv
mnp ξγγγw
qrs ξηγγv
01
01
01
, , , ,2
, , , ,
6
, , , ,
2, , , ,
360
4, , , ,
2, , , ,
2, , , ,
2, , , ,
, , , ,2
, , , ,
2
4
(B.1i)
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
513
∫ ∫ ∫ ⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
= ⎧⎨⎩
+ − ⎛⎝
+ ⎞⎠
+
+ ⎡⎣⎢
+
− ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
− ⎛⎝
+ ⎞⎠
⎤
⎦⎥
+ ⎡⎣⎢
+
− ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
− ⎛⎝
+ ⎞⎠
⎤
⎦⎥
+ + ⎧⎨⎩
+
+ ⎛⎝
+ ⎞⎠
− ⎡⎣⎢
+ +
⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
⎤
⎦⎥
+ ⎡
⎣⎢ + + ⎛
⎝+ ⎞
⎠
+ ⎛⎝
+ ⎞⎠
+ ⎛⎝
+ ⎞⎠
⎤
⎦⎥
⎫⎬⎭
⎫⎬⎭
⊥
⊥
ϕ ϕ H N ϕ ϕ μ ϕ ϕ
κd C ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ
d C ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ
C C ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ κ ϕ ϕ μ ϕ ϕ
ϕ ϕ μ ϕ ϕ κ ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ
κ ϕ ϕ μ ϕ ϕ ξ η γ
K[ ] (( ) )
( )
( )
6
15
15
d d d ,
bT
mnpqrsvv
mnp ξv
qrs ξv
xT
TT
mnp ξv
qrs ξv
mnp ξξv
qrs ξξw
vT
mnp ηv
qrs ηv
mnp ξηv
qrs ξηv
κdmnp ηηv
qrs ηηv
mnp ξηηv
qrs ξηηv κd
mnp ηηηv
qrs ηηηv
mnp ξηηηv
qrs ξηηηv
κdmnp ηηηηv
qrs ηηηηv
mnp ξηηηηv
qrs ξηηηηv
vT
mnp γv
qrs γv
mnp ξγv
qrs ξγv
dmnp γγv
qrs γγv
mnp ξγγv
qrs ξγγv d
mnp γγγv
qrs γγγv
mnp ξγγγv
qrs ξγγγv
dmnp γγγγv
qrs γγγγv
mnp ξγγγγv
qrs ξγγγγv
dT
dT
mnp γv
qrs γv
mnp ξγv
qrs ξγv
mnp ηv
qrs ηv
mnp ξηv
qrs ξηv d
mnp γγv
qrs γγv
mnp ξγγv
qrs ξγγv
mnp ηγv
qrs ηγv
mnp ξηγv
qrs ξηγv
mnp ηηv
qrs ηηv
mnp ξηηv
qrs ξηηv
dmnp γγγv
qrs γγγv
mnp ξγγγv
qrs ξγγγv
mnp ηγγv
qrs ηγγv
mnp ξηγγv
qrs ξηγγv
mnp ηηγv
qrs ηηγv
mnp ξηηγv
qrs ξηηγv
mnp ηηηv
qrs ηηηv
mnp ξηηηv
qrs ξηηηv
01
01
01
, ,2
, ,2
, ,
2, ,
2, ,
( )12 , ,
2, ,
( )360 , ,
2, ,
( )20160 , ,
2, ,
2, ,
2, ,
12 , ,2
, , 360 , ,2
, ,
20160 , ,2
, ,
, ,2
, ,
2, ,
2, , 12 , ,
2, ,
2, ,
2, ,
4, ,
2, ,
360 , ,2
, ,2
, ,2
, ,
4, ,
2, ,
6, ,
2, ,
2 4
6
2 4
6
2
4
(B.1j)
∫ ∫ ∫= − ϕ ϕ ξ η γK[ ] d d d ,bT
mnpqrsvθ
mnp ξv
qrsθ
01
01
01
,z z
(B.1k)
∫ ∫ ∫= − ϕ ϕ ξ η γK[ ] d d d ,bT
mnpqrsθ v
mnpθ
qrs ξv
01
01
01
,z z
(B.1l)
∫ ∫ ∫ ⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
= − ×
⎧⎨⎩
+ + ⎛⎝
+ ⎞⎠
−
⎡
⎣
⎢⎢⎢⎢⎢
∇ ⎧⎨⎩
⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
+ ∇ ⎧⎨⎩
⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
+
⎛
⎝⎜∇ ⎧
⎨⎩⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
+ ∇ ⎧⎨⎩
⎫⎬⎭
∇ ⎧⎨⎩
⎫⎬⎭
⎞
⎠⎟
⎤
⎦
⎥⎥⎥⎥⎥
+
⎡
⎣
⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎛
⎝⎜ + + ⎛
⎝+ ⎞
⎠
⎞
⎠⎟+
⎛
⎝⎜ + + ⎛
⎝+ ⎞
⎠
⎞
⎠⎟+
⎛
⎝⎜ + + ⎛
⎝+ ⎞
⎠
⎞
⎠⎟
⎤
⎦
⎥⎥⎥⎥⎥⎥⎥⎥⎥
⊥κ C C
ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ
μ ϕ ϕ ϕ ϕ
κ ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
κ ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
ϕ ϕ ϕ ϕ μ ϕ ϕ ϕ ϕ
ξ η γ
K[ ] ( )
. .
. .
3
10
3
} d d d ,
bT
mnpqrsvw
dT
dT
mnp ηv
qrs γw
mnp γv
qrs ηw
mnp ξηv
qrs ξγw
mnp ξγv
qrs ξηw
dηγ mnp η
vηγ qrs γ
wηγ mnp γ
vηγ qrs η
w
ηγ mnp ξηv
ηγ qrs ξγw
ηγ mnp ξγv
ηγ qrs ξηw
d
mnp γηηv
qrs ηηηw
mnp ηηηv
qrs γηηw
mnp ξηηγv
qrs ξηηηw
mnp ξηηηv
qrs ξηηγw
mnp ηηηv
qrs γγγw
mnp γγγv
qrs ηηηw
mnp ξηηηv
qrs ξγγγw
mnp ξγγγv
qrs ξηηηw
mnp ηγγv
qrs γγγw
mnp γγγv
qrs ηγγw
mnp ξηγγv
qrs ξγγγw
mnp ξγγγv
qrs ξηγγw
0
1
0
1
0
1
, , , ,2
, , , ,
6
, , , ,
2, , , ,
360
4, , , ,
2, , , ,
2, , , ,
2, , , ,
, , , ,2
, , , ,
2
4
(B.1m)
References
Adali, S., 2008. Variational principles for multi-walled carbon nanotubes undergoingbuckling based on nonlocal elasticity theory. Phys. Lett. 372, 5701–5705.
Ansari, R., Gholami, R., Rouhi, H., 2015. Size-dependent nonlinear forced vibration
analysis of magneto-electro-thermo-elastic Timoshenko nanobeams based upon thenonlocal elasticity theory. Compos. Struct. 126, 216–226.
Ansari, R., Ramezannezhad, H., 2011. Nonlocal Timoshenko beam model for the large-amplitude vibrations of embedded multiwalled carbon nanotubes including thermaleffects. Physica E 43, 1171–1178.
Arash, B., Ansari, R., 2010. Evaluation of nonlocal parameter in the vibrations of single-
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
514
walled carbon nanotubes with initial strain. Physica E 42, 2058–2064.Arash, B., Wang, Q., 2012. A review on the application of nonlocal elastic models in
modeling of carbon nanotubes and graphenes. Comput. Mater. Sci. 51, 303–313.Benzair, A., Tounsi, A., Besseghier, A., Heireche, H., Moulay, N., Boumia, L., 2008. The
thermal effect on vibration of single-walled carbon nanotubes using nonlocalTimoshenko beam theory. J. Phys. D Appl. Phys. 41, 225404.
Choi, W.B., Bae, E., Kang, D., Chae, S., Cheong, B.H., Ko, J.H., Lee, E., Park, W., 2004.Aligned carbon nanotubes for nanoelectronics. Nanotechnology 15, S512.
Choi, W.B., Cheong, B.H., Kim, J.J., Chu, J., Bae, E., 2003. Selective growth of carbonnanotubes for nanoscale transistors. Adv. Funct. Mater. 13, 80–84.
Cottineau, T., Pronkin, S.N., Acosta, M., Meny, C., Spitzer, D., Keller, V., 2013. Synthesisof vertically aligned titanium dioxide nanotubes on microcantilevers for new na-nostructured micromechanical sensors for explosive detection. Sensor. Actuator. BChem. 182, 489–497.
Du, C., Yeh, J., Pan, N., 2005. High power density supercapacitors using locally alignedcarbon nanotube electrodes. Nanotechnology 16, 350.
Duan, W.H., Wang, C.M., Zhang, Y.Y., 2007. Calibration of nonlocal scaling effect para-meter for free vibration of carbon nanotubes by molecular dynamics. J. Appl. Phys.101 (2), 024305.
Eringen, A.C., 1966. Linear theory of micropolar elasticity. J. Math. Mech 15, 909–923.Eringen, A.C., 1972. Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16.Eringen, A.C., 2002. Nonlocal Continuum Field Theories. Springer-Verlag, NY.Fan, S., Chapline, M.G., Franklin, N.R., Tombler, T.W., Cassell, A.M., Dai, H., 1999. Self-
oriented regular arrays of carbon nanotubes and their field emission properties.Science 283, 512–514.
Futaba, D.N., Hata, K., Yamada, T., Hiraoka, T., Hayamizu, Y., Kakudate, Y., Tanaike, O.,Hatori, H., Yumura, M., Iijima, S., 2006. Shape-engineerable and highly denselypacked single-walled carbon nanotubes and their application as super-capacitorelectrodes. Nat. Mater. 5, 987–994.
Hu, Y.G., Liew, K.M., Wang, Q., He, X.Q., Yakobson, B.I., 2008. Nonlocal shell model forelastic wave propagation in single-and double-walled carbon nanotubes. J. Mech.Phys. Solid. 56 (12), 3475–3485.
Huang, J.R., Li, M.Q., Liu, J.H., 2007. A novel conductive humidity sensor based on fieldionization from carbon nanotubes. Sensor. Actuat. A-Phys. 133, 467–471.
Karaoglu, P., Aydogdu, M., 2010. On the forced vibration of carbon nanotubes via a non-local Euler-Bernoulli beam model. P. I. Mech. Eng. C-J Mech 224, 497–503.
Ke, L.L., Wang, Y.S., 2014. Free vibration of size-dependent magneto-electro-elastic na-nobeams based on the nonlocal theory. Physica E 63, 52–61.
Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S., 2014a. Free vibration of size-dependentmagneto-electro-elastic nanoplates based on the nonlocal theory. Acta Mech. Sin. 30,516–525.
Ke, L.L., Wang, Y.S., Yang, J., Kitipornchai, S., 2014b. The size-dependent vibration ofembedded magneto-electro-elastic cylindrical nanoshells. Smart Mater. Struct. 23,125036.
Ke, L.L., Xiang, Y., Yang, J., Kitipornchai, S., 2009. Nonlinear free vibration of embeddeddouble-walled carbon nanotubes based on nonlocal Timoshenko beam theory.Comput. Mater. Sci. 47, 409–417.
Khademolhosseini, F., Rajapakse, R.K.N.D., Nojeh, A., 2010. Torsional buckling of carbonnanotubes based on nonlocal elasticity shell models. Comput. Mater. Sci. 48,736–742.
Kiani, K., 2014a. Nonlinear vibrations of a single-walled carbon nanotube for deliveringof nanoparticles. Nonlinear Dynam. 76, 1885–1903.
Kiani, K., 2014b. Nanoparticle delivery via stocky single-walled carbon nanotubes: anonlinear-nonlocal continuum-based scrutiny. Compos. Struct. 116, 254–272.
Kiani, K., 2013. Characterization of free vibration of elastically supported double-walledcarbon nanotubes subjected to a longitudinally varying magnetic field. Acta Mech.224, 3139.
Kiani, K., 2014c. Axial buckling analysis of vertically aligned ensembles of single-walledcarbon nanotubes using nonlocal discrete and continuous models. Acta Mech. 225,3569–3589.
Kiani, K., 2015a. Nonlocal and shear effects on column buckling of single-layeredmembranes from stocky single-walled carbon nanotubes. Compos. B Eng. 79,535–552.
Kiani, K., 2014d. In- and out-of- plane dynamic flexural behaviors of two-dimensionalensembles of vertically aligned single-walled carbon nanotubes. Physica B 449,164–180.
Kiani, K., 2014e. Nonlocal discrete and continuous modeling of free vibration of stockyensembles of single-walled carbon nanotubes. Curr. Appl. Phys. 14, 1116–1139.
Kiani, K., 2016. Free vibration of in-plane-aligned membranes of single-walled carbonnanotubes in the presence of in-plane-unidirectional magnetic fields. J. Vib. Contr.22, 3736–3766.
Kiani, K., 2015b. Wave characteristics in aligned forests of single-walled carbon nano-tubes using nonlocal discrete and continuous theories. Int. J. Mech. Sci. 90, 278–309.
Kiani, K., 2014f. Nonlocal continuous models for forced vibration analysis of two- andthree-dimensional ensembles of single-walled carbon nanotubes. Physica E 60,229–245.
Marconnet, A.M., Yamamoto, N., Panzer, M.A., Wardle, B.L., Goodson, K.E., 2011.Thermal conduction in aligned carbon nanotube-polymer nanocomposites with highpacking density. ACS Nano 5, 4818–4825.
Mizuno, K., Ishii, J., Kishida, H., Hayamizu, Y., Yasuda, S., Futaba, D.N., Yumura, M.,Hata, K., 2009. A black body absorber from vertically aligned single-walled carbonnanotubes. Prog. Nat. Sci. 106, 6044–6047.
Murata, S., Imanishi, M., Hasegawa, S., Namba, R., 2014. Vertically aligned carbon na-notube electrodes for high current density operating proton exchange membrane fuelcells. J. Power Sources 253, 104–113.
Murmu, T., McCarthy, M.A., Adhikari, S., 2012. Vibration response of double-walledcarbon nanotubes subjected to an externally applied longitudinal magnetic field: anonlocal elasticity approach. J. Sound Vib. 331, 5069–5086.
Murmu, T., Pradhan, S.C., 2009. Thermo-mechanical vibration of a single-walled carbonnanotube embedded in an elastic medium based on nonlocal elasticity theory.Comput. Mater. Sci. 46, 854–859.
Narendar, S., Gupta, S.S., Gopalakrishnan, S., 2012. Wave propagation in single-walledcarbon nanotube under longitudinal magnetic field using nonlocal EulerBernoullibeam theory. Appl. Math. Model. 36, 4529–4538.
Ngo, Q., Cruden, B.A., Cassell, A.M., Sims, G., Meyyappan, M., Li, J., Yang, C.Y., 2004.Thermal interface properties of Cu-filled vertically aligned carbon nanofiber arrays.Nano Lett. 4, 2403–2407.
Peddieson, J., Buchanan, G.R., McNitt, R.P., 2003. Application of nonlocal continuummodels to nanotechnology. Int. J. Eng. Sci. 41, 305–312.
Qu, L., Du, F., Dai, L., 2008. Preferential syntheses of semiconducting vertically alignedsingle-walled carbon nanotubes for direct use in FETs. Nano Lett. 8, 2682–2687.
Silva, T.A., Zanin, H., Vicentini, F.C., Corat, E.J., Fatibello-Filho, O., 2015.Electrochemical determination of rosuvastatin calcium in pharmaceutical and humanbody fluid samples using a composite of vertically aligned carbon nanotubes andgraphene oxide as the electrode material. Sensor. Actuator. B Chem. 218, 51–59.
Simsek, M., 2011. Nonlocal effects in the forced vibration of an elastically connecteddouble-carbon nanotube system under a moving nanoparticle. Comput. Mater. Sci.50, 2112–2123.
Sudak, L.J., 2003. Column buckling of multiwalled carbon nanotubes using nonlocalcontinuum mechanics. J. Appl. Phys. 94, 7281–7287.
Tian, Z.Q., Lim, S.H., Poh, C.K., Tang, Z., Xia, Z., Luo, Z., Shen, P.K., et al., 2011. A highlyorder-structured membrane electrode assembly with vertically aligned carbon na-notubes for ultra-low Pt loading PEM fuel cells. Adv. Energy Mater 1, 1205–1214.
Tong, T., Zhao, Y., Delzeit, L., Kashani, A., Meyyappan, M., Majumdar, A., 2007. Densevertically aligned multiwalled carbon nanotube arrays as thermal interface materials.IEEE Trans. Compon. Packag. Technol. 30, 92–100.
Tsai, T.Y., Lee, C.Y., Tai, N.H., Tuan, W.H., 2009. Transfer of patterned vertically alignedcarbon nanotubes onto plastic substrates for flexible electronics and field emissiondevices. Appl. Phys. Lett. 95, 013107.
Tung, S., Rokadia, H., Li, W.J., 2007. A micro shear stress sensor based on laterallyaligned carbon nanotubes. Sensor. Actuat. A-Phys. 133, 431–438.
Uymaz, B., 2013. Forced vibration analysis of functionally graded beams using nonlocalelasticity. Compos. Struct. 105, 227–239.
Wang, J., 2005a. Carbon-nanotube based electrochemical biosensors: a review.Electroanalysis 17, 7–14.
Wang, Q., 2005b. Wave propagation in carbon nanotubes via nonlocal continuum me-chanics. J. Appl. Phys. 98, 124301.
Wang, Q., Varadan, V.K., Quek, S.T., 2006a. Small scale effect on elastic buckling ofcarbon nanotubes with nonlocal continuum models. Phys. Lett. 357, 130–135.
Wang, Q., Varadan, V.K., 2007. Application of nonlocal elastic shell theory in wavepropagation analysis of carbon nanotubes. Smart Mater. Struct. 16 (1), 178.
Wang, Q., Wang, C.M., 2007. The constitutive relation and small scale parameter ofnonlocal continuum mechanics for modelling carbon nanotubes. Nanotechnology 18,075702.
Wang, C.M., Zhang, Y.Y., Ramesh, S.S., Kitipornchai, S., 2006b. Buckling analysis ofmicro-and nano-rods/tubes based on nonlocal Timoshenko beam theory. J. Phys. DAppl. Phys. 39, 3904.
Xiao, K., Liu, Y., Hu, P., Yu, G., Wang, X., Zhu, D., 2003. High-mobility thin-film tran-sistors based on aligned carbon nanotubes. Appl. Phys. Lett. 83, 150–152.
Yang, J., Ke, L.L., Kitipornchai, S., 2010. Nonlinear free vibration of single-walled carbonnanotubes using nonlocal Timoshenko beam theory. Physica E 42, 1727–1735.
Yao, X., Han, Q., 2006. Buckling analysis of multiwalled carbon nanotubes under tor-sional load coupling with temperature change. J. Eng. Mater. Technol. 128, 419–427.
Zhang, H., Cao, G., Wang, Z., Yang, Y., Shi, Z., Gu, Z., 2008. Growth of manganese oxidenanoflowers on vertically-aligned carbon nanotube arrays for high-rate electro-chemical capacitive energy storage. Nano Lett. 8, 2664–2668.
Zhang, Y.C., Wang, X., 2005. Thermal effects on interfacial stress transfer characteristicsof carbon nanotubes/polymer composites. Int. J. Solid Struct. 42, 5399–5412.
Zidour, M., Benrahou, K.H., Semmah, A., Naceri, M., Belhadj, H.A., Bakhti, K., Tounsi, A.,2012. The thermal effect on vibration of zigzag single walled carbon nanotubes usingnonlocal Timoshenko beam theory. Comput. Mater. Sci. 51, 252–260.
K. Kiani, Q. Wang European Journal of Mechanics / A Solids 72 (2018) 497–515
515