International Journal of Mechanical Scienceswp.kntu.ac.ir/k_kiani/papers-2018/Nonlocal vibrations...

International Journal of Mechanical Sciences 144 (2018) 576–599

Contents lists available at ScienceDirect

International Journal of Mechanical Sciences

journal homepage: www.elsevier.com/locate/ijmecsci

Nonlocal vibrations and potential instability of monolayers from

double-walled carbon nanotubes subjected to temperature gradients

Keivan Kiani ∗ , Hossein Pakdaman

Department of Civil Engineering, K.N. Toosi University of Technology, Valiasr Ave., Tehran P.O. Box 15875-4416, Iran

a r t i c l e i n f o

Keywords:

Thermo-elastic transverse vibrations

Discrete and continuous modeling

Nonlocal elasticity theory

Shear deformation effect

Critical temperature gradient

Galerkin method

a b s t r a c t

Using nonlocal elasticity theory of Eringen, free transverse thermo-elastic vibrations of vertically aligned double-

walled carbon nanotubes (DWCNTs) with a membrane configuration is going to be explored methodically. Ac-

counting for nonlocal heat conduction along the side walls of DWCNTs with allowance of heat dissipation from

the outer surfaces, the nonlocal temperature fields within the constitutive tubes are assessed for a steady-state

regime. The van der Waals interactional forces between atoms of each pair of DWCNTs as well as the innermost

and outermost tubes are displayed by laterally continuous springs whose constants are appropriately evaluated.

By establishing suitable discrete and continuous models on the basis of the Rayleigh and higher-order beam the-

ories, nonlocal in-plane and out-of-plane vibrations of thermally affected nanosystems of monolayers of DWCNTs

are examined and discussed. The critical values of the temperature change are stated explicitly and the role of

influential factors on this crucial factor as well as the free vibration behavior are investigated in some detail.

Through conducting a fairly comprehensive numerical study, the influences of the slenderness ratio, temperature

change in the low and high temperatures, number of nanotubes, small-scale parameter, intertube distance, and

stiffness of the surrounding environment on the nonlocal-fundamental frequency are explained. The obtained

results from this work could provide crucial guidelines for the design of membranes or even jungles of vertically

aligned DWCNTs as thermal interface nanostructures.

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. Introduction

A monolayer of vertically aligned double-walled carbon nanotubes

DWCNTs) is a nanosystem consists of parallel DWCNTs located adja-

ently in a flat plane. Actually it represents a single-layered membrane

f tubes that could have many applications in nanotechnology including

train nanosensors [1,2] , nano-electro-mechanical systems (NEMs) [3] ,

nd nano-scaled heat-sinks [4,5] . Several research groups have pro-

osed ensembles of carbon nanotubes (CNTs) as the next generation of

hermal interface materials [4–7] . Further composites from carbon nan-

tubes (CNTs) have been fabricated for engineering-based management

nd improvement of thermal conductivity [8,9] . With regard to these

uperior thermal properties, CNTs as well as their composites have been

f focus of attention of interdisciplinary scholars, however, thermo-

echanical behavior of forests or even membranes made from CNTs has

ot been methodically displayed yet. To fill some part of this scientific

ap, herein, we are interested in thermo-mechanical analysis of mem-

ranes of vertically aligned DWCNTs. The main objective of this work is

o theoretically estimate their dominant frequencies for such nanosys-

ems when they are subjected to longitudinally thermal gradients.

∗ Corresponding author.

E-mail address: [email protected] (K. Kiani).

m

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ttps://doi.org/10.1016/j.ijmecsci.2018.06.018

eceived 18 March 2018; Received in revised form 23 May 2018; Accepted 7 June 2

vailable online 15 June 2018

020-7403/© 2018 Elsevier Ltd. All rights reserved.

In a monolayer of vertically aligned DWCNTs, not only the atoms of

he innermost and the outermost tubes of each DWCNT would interact

ynamically with each other due to the existing van der Waals (vdW)

orces, but also the atoms of two nearby DWCNTs would interact.

ore calculations reveal that the variation of vdW forces due to the

n-plane deflections of DWCNTs is not the same as that of the out-of-

lane deflections. This leads to dissimilar vibrational behavior of such

anosystems for the in-plane and out-of-plane cases. Herein a linear

ontinuum-based vdW model is established for two adjacent DWCNTs

nd it is appropriately extended to thermo-mechanical modelling of

ertically aligned DWCNTs embedded in an elastic matrix. It should

e noticed that only the transverse motions of DWCNTs are taken

nto account in evaluating the variations of vdW forces. Further, the

ariations of such forces for both in-plane and out-of-plane vibrations

re linearly linked to their relative in-plane and out-of-plane transverse

isplacements, respectively. In other words, the nonlinear terms in

ach direction as well as the coupled terms are excluded since linear

nalysis is of interest. The role of nonlinear intertube vdW forces on

he local and nonlocal transverse vibrations of individual DWCNTs has

een studied by Xu et al. [10] and Khosrozadeh and Hajabasi [11] . For

ore accurate prediction of nonlinear chaotic vibrations of thermally

ffected monolayers of vertically aligned DWCNTs as well as highly

onlinear forced vibration cases, consideration of the aforementioned

018

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K. Kiani, H. Pakdaman International Journal of Mechanical Sciences 144 (2018) 576–599

Fig. 1. Schematic representation of an elastically embedded membranes of vertically aligned DWCNTs subjected to a longitudinal temperature gradient.

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ffects would result in a more rational estimation of the model. This

ssue could be regarded as a hot topic for future works.

Through contacting the ends of the constitutive DWCNTs of a mono-

ayer to heat sources, a nonlocal longitudinal heat flux could be gener-

ted within each tube due to the gradient of temperature at the ends.

n fact, this is a nonlocal flux since the continuum theory of Eringen

12] displays that the thermal field at each point of the nanostructure

ould be affected by the temperatures of its nearby points. Due to the

xistence of heat exchange between the atoms of the outermost tubes of

he monolayer with the surrounding environment, some part of the heat

nergy would be dissipated through the outer surface. Accounting for

oth nonlocality and energy dissipation across the length, we present

steady-state model to explain temperature profile along the consti-

utive tubes more accurately. Thereafter we proceed in calculating the

esulted mechanical forces in the constitutive tubes due to the thermal

oading. From theory of elastic beams’ points of view, major resulted

orces within tubes would be axial thermal force and thermal bending

oment. The axially thermal forces within tubes are incorporated into

he transverse equations of motion when the nanosystem has been

omehow restrained longitudinally at the nanotubes’ ends while the

hermal bending moments have a tendency to bend nanotubes progres-

ively. In the context of linear analysis, the axial thermal forces could

nfluence on the natural frequencies of elastically restrained nanosys-

ems, however, the thermal bending moments are highly incorporated

nto the pseudo-dynamic response of the nanosystem, and, generally, it

annot affect the frequencies of the beam-like nanosystem. The thermal

ending moments could also influence on the interactional vdW forces

etween atoms of each pair of tubes. In the present study, temperature

ariation across the transverse direction of constitutive tubes does not

xist, and thereby, no bending moment is generated within nanotubes.

Application of atomistic-based approaches to vibrations of mono-

ayers of vertically aligned DWCNTs is surely compromised with a

uge amount of labor costs and computational efforts. To conquer this

ilestone of atomic models, we employ the nonlocal continuum theory

f Eringen [12–14] for modelling of both in-plane and out-of-plane

ynamics of nanosystems. Until now this theory has been widely imple-

ented for continuum-based modeling of single-, double-, and multi-

alled carbon nanotubes. Concerning mechanical modeling of DWC-

Ts, free vibration [15–22] , elastic wave analysis [23–29] , nonlinear

ibration [30] , buckling and postbuckling [31–33] , and vibrations in

he presence of magnetic fields [34–36] have been addressed suitably.

egarding thermally affected individual DWCNTs, free transverse

ibrations [37] , instability of fluid-conveying [38] , propagation of

lastic waves [39] , and buckling analysis [40] have been of concern of
D
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anomechanics community during the past decade. Most of these works

ave been conducted in the context of the nonlocal continuum theory

f Eringen by employing beam or shell models. Furthermore, transverse

ibrations of membranes and forests of single-walled carbon nanotubes

41,42] , their free dynamic analysis in the presence of longitudinal

agnetic field [43,44] , and their axial buckling [45,46] have been

nvestigated using nonlocal Rayleigh, Timoshenko, and higher-order

eam theories. This brief survey shows that the mechanical behavior

f thermally affected membranes of vertically aligned DWCNTs has

ot been scrutinized yet. With regard to the vast applications of such

anosystems as mentioned in above, we are interested in examining

heir nonlocal vibrations using well-known beam theories.

In the present work, free nonlocal-transverse vibrations of elastically

mbedded single-layered membranes of vertically aligned DWCNTs

cted upon by a longitudinal heat flux are reported. The continuum-

ased vdW forces for both statics and linear dynamics are displayed

nd the resulted elastic energy of such crucial forces is appropriately

ncorporated into the total elastic energy of the nanosystem. Using

onlocal elasticity theory of Eringen in conjunction with the Rayleigh

nd the higher-order beam theories, the equations of motion associated

ith both in-plane and out-of-plane vibrations of the thermally affected

anosystem are obtained. By implementing assumed mode method

AMM), the resulted equations are discretized and solved for natural

requencies on the basis of the nonlocal-discrete models. Subsequently,

he appropriate nonlocal-continuous models are established and ana-

yzed. The main feature of the newly developed models is their high

apability in evaluating free dynamic response of thermally affected

anosystems with high numbers of DWCNTs. The efficacy of these

ewly developed models in capturing both in-plane and out-of-plane

undamental frequencies are displayed and proved through various

umerical studies. A fairly comprehensive parametric study is then

iven and the roles of influential factors on the free dynamic response

f the nanosystem are investigated and explained in some detail.

y employing the newly sophisticated continuous-based models, the

ritical values of the temperature change for the nanosystem at hand

re also derived explicitly, and the key roles of the slenderness ratio,

mall-scale parameter, intertube distance, population, and radius of

WCNTs on this crucial factor are revealed.

. Defining the nanomechanical problem

Consider a membrane with N vertically aligned DWCNTs whose

engths are l b and the distance between the major axes of each pair of

WCNTs is denoted by d (see Fig. 1 ). The bottom and the top ends of


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he constitutive tubes are subjected to the temperature change of T 1 and

2 , respectively, while the environmental temperature is represented

y T en . Each DWCNTs consists of doubly coaxial nanotubes of the same

ength such that the radii of the innermost and the outermost tubes

n order are 𝑟 𝑚 1 and 𝑟 𝑚 2 . The equivalent continuum structure (ECS)

ssociated with each tube is a circular cylindrical shell of mean radius

qual to the radius of the tube and the wall’s thickness, t b , is set equal

o 0.34 nm. The innermost and the outermost tubes interact statically

nd dynamically due to existing vdW forces between their constitutive

toms. Additionally, each tube interacts with that of the nearby DWC-

Ts due to the vdW forces. A detailed scrutiny on such statics forces as

ell as purely dynamical part of vdW forces due to the lateral motion

f neighboring tubes has been given in supplementary material, part

. Since flexural and shear transverse vibrations of thermally affected

embranes from DWCNTs are of particular interest in this study, we

odel each tube via beam-like equivalent structures by considering the

onlocal effect. The deflections of the i th tube based on the nonlocal

ayleigh beam model (NRBM) in the y and z directions, 𝑉 𝑅 𝑖 ( 𝑥, 𝑡 ) and

𝑅 𝑖 ( 𝑥, 𝑡 ) , while the deflections and angles of deflections of such a tube

n the basis of the nonlocal higher-order beam model (NHOBM) are

epresented by 𝑉 𝐻

𝑖 ( 𝑥, 𝑡 ) / 𝑊

𝐻

𝑖 ( 𝑥, 𝑡 ) and Ψ𝐻

𝑖 𝑦 ( 𝑥, 𝑡 ) / Ψ𝐻

𝑖 𝑧 ( 𝑥, 𝑡 ) , respectively.

urther, 𝐸 𝑏 𝑗 , 𝐺 𝑏 𝑗

, 𝐼 𝑏 𝑗 , 𝐴 𝑏 𝑗

, and 𝜌𝑏 𝑗 in order represent the Young’s

odulus, shear elastic modulus, moment inertia of cross-section of the

CS, cross-sectional area, and density of the j th tube from the DWCNTs

j = 1,2). In general, the membrane is rested on an elastic foundation

nd it could be interacted with its surrounding medium due to ex-

sting bonds. By exploiting Pasternak spring model, the constants of

ontinuous transverse and rotational springs are denoted by K t and K r ,

espectively.

In this paper, we examine the dominant in-plane and out-of-

lane vibrational behavior of the vertically aligned DWCNTs acted

pon by temperature gradients between tubes’ ends. By employ-

ng Hamilton’s principle, nonlocal-discrete-based models using

ayleigh and higher-order beam theories are developed to pro-

ide us with the nonlocal-continuum-based equations of motion of

he nanosystem. Subsequently, suitable nonlocal-continuous-based

odels are established and their capabilities are explained in some

etails.

. Nonlocal temperature field analysis of nanotubes

According to the Fourier’s law for solid structures, the longitudinal

eat conduction within a continuum-based DWCNTs accounting for

oth nonlocality and heat conduction through its innermost and

utermost walls is displayed by:

𝑏 𝐶 𝑏

𝜕𝑇

𝜕𝑡 = −

𝜕 𝑞 𝑛𝑙 𝑇

𝜕𝑥 − 𝑄 𝑑𝑖𝑠𝑠 ,

𝑑𝑖𝑠𝑠 =

ℎ 𝑏 𝑃 𝑏

𝐴 𝑏

(𝑇 − 𝑇 𝑒𝑛

), (1)

here 𝜕 is the partial sign, t is the time factor, T denotes the tempera-

ure field within the ECS associated with the nanotube, T en represents

he temperature field of the environment, C b is the specific heat

apacity, h b is the heat transfer coefficient of the nanotube with its

urrounding environment, P b is the sum of innermost and outermost

yramids of the ECS’s cross-section (i.e., 𝑃 𝑏 = 2 𝜋( 𝑟 𝑚 1 + 𝑟 𝑚 2 ) ), and A b is

he whole cross-sectional area of both tubes (i.e., 𝐴 𝑏 = 2 𝜋𝑡 𝑏 ( 𝑟 𝑚 1 + 𝑟 𝑚 2 ) ).

n presenting Eq. (1) , we have assumed that the temperature fields

ithin both innermost and outermost tubes are the same and the lateral

eat exchange between the constitutive tubes of the DWCNTs has been

gnored.

In the context of the nonlocal continuum filed theory of Eringen

14] , the nonlocal heat flux ( 𝑞 𝑛𝑙 𝑇

) is related to the local one ( 𝑞 𝑙 𝑇

) by a

578

imple relation of the following form:

𝑛𝑙 𝑇 − ( 𝑒 0 𝑎 ) 2

𝜕 2 𝑞 𝑛𝑙 𝑇

𝜕𝑥 2 = 𝑞 𝑙

𝑇 , (2)

here e 0 a represents the small-scale factor, 𝑞 𝑙 𝑇 = − 𝜅𝑏

𝜕𝑇

𝜕𝑥 is the longitu-

inal local heat flux, and k b is the thermal conductivity of the nanotube.

y combining Eqs. (1) and (2) , the nonlocal governing equation of heat

ransfer within the nanotube takes the following form:

𝑏 𝐶 𝑏

(

𝜕𝑇

𝜕𝑡 − ( 𝑒 0 𝑎 ) 2

𝜕 3 𝑇

𝜕 𝑡𝜕 𝑥 2

)

= 𝑘 𝑏 𝜕 2 𝑇

𝜕𝑥 2 −

ℎ 𝑏 𝑃 𝑏

𝐴 𝑏

( (𝑇 − 𝑇 𝑒𝑛

)− ( 𝑒 0 𝑎 ) 2

𝜕 2 𝑇

𝜕𝑥 2

)

.

(3)

or steady-state heat conduction, Eq. (3) is reduced to:

d 2 𝑇

d 𝜉2 − ℎ 𝑇

(

𝑇 − 𝜇2 d 2 𝑇

d 𝜉2

)

= 0 , (4)

here

( 𝜉) = 𝑇 ( 𝜉) − 𝑇 𝑒𝑛 , 𝜇 =

𝑒 0 𝑎

𝑙 𝑏 , ℎ 𝑇 =

ℎ 𝑏 𝑃 𝑏 𝑙 2 𝑏

𝑘 𝑏 𝐴 𝑏

. (5)

et’s assume the following boundary conditions for the ends of

anotubes:

(0) = 𝑇 1 + 𝑇 𝑒𝑛 , 𝑇 ( 𝑙 𝑏 ) = 𝑇 2 + 𝑇 𝑒𝑛 . (6)

y solving Eq. (4) for the given essential conditions in Eq. (6) in view

f Eq. (5) , the steady-state temperature field within all tubes of the

anosystem is evaluated by:

( 𝜉) =

⎧ ⎪ ⎨ ⎪ ⎩ (

𝑇 2 − 𝑇 1 cosh ( 𝛽) sinh ( 𝛽)

)

sinh ( 𝛽𝜉) + 𝑇 1 cosh ( 𝛽𝜉); ℎ 𝑇 ≠ 0 , (𝑇 2 − 𝑇 1

)𝜉 + 𝑇 1 ; ℎ 𝑇 = 0 ,

(7)

here 𝛽 =

√

ℎ 𝑇

1+ 𝜇2 ℎ 𝑇 . In Eq. (7) , the second case (i.e., h T = 0) displays

he circumference that the lateral surfaces of both the innermost

nd outermost tubes have been fully insulated (i.e., no heat flux is

llowed to be transferred through the lateral surface of the nanotube)

hile the first case is the more general one (i.e., it could be searched

athematically that the second case would be concluded from the first

ase using L’Hopital’s rule).

If both ends of each tube have been fixed along the longitudinal

irection, the resulted axial thermal force within the j th tube could be

eadily calculated by:

𝑇 𝑗 = − 𝐸 𝑏 𝐴 𝑏 𝑗 ∫

1

0 𝛼𝑇 ( 𝑇 ) 𝑇 ( 𝜉) d 𝜉, (8)

here 𝛼T is the coefficient of longitudinal thermal expansion of

NTs which is generally temperature dependent. By assuming

𝑇 = 𝛼𝑇 ( 𝑇 1 , 𝑇 2 ) , and by substituting Eq. (7) into Eq. (8) , the axial

hermal forces within the constitutive nanotubes are explicitly provided

y:

𝑇 𝑗 = − 𝐸 𝑏 𝑗

𝐴 𝑏 𝑗 𝛼𝑇

(𝑇 1 + 𝑇 2

)⎧ ⎪ ⎨ ⎪ ⎩ coth ( 𝛽) − csch ( 𝛽)

𝛽; ℎ 𝑇 ≠ 0 ,

1 2 ; ℎ 𝑇 = 0 ,

. (9)


4 DWCNTs membranes

4

modeled based on the NRBM, the kinetic energy ( T R ) as well as the strain

e the interactional effect with the surrounding elastic medium are written as:

𝑇

2 𝑊

𝑅 𝑖

𝜕 𝑡𝜕 𝑥

) 2 ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠ d 𝑥, (10a)

𝑈

𝛿(2 𝑁)( 𝑖 ) ) − 𝑉 𝑅

𝑖 −2 )2 (1 − 𝛿2 𝑖 )

𝜕𝑉 𝑅 𝑖

𝜕𝑥

) 2 + 𝑁 𝑇 𝑖

(

𝜕𝑉 𝑅 𝑖

𝜕𝑥

) 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ d 𝑥

𝛿(2 𝑁−1)( 𝑖 ) )

𝑅 𝑖

− 𝑉 𝑅 𝑖 −2 )2 (1 − 𝛿1 𝑖 )

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ d 𝑥

1 − 𝛿(2 𝑁)( 𝑖 ) )

− 𝑊

𝑅 𝑖 −3 )2 (1 − 𝛿2 𝑖 )

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ d 𝑥

1 − 𝛿(2 𝑁−1)( 𝑖 ) )

(𝑊

𝑅 𝑖

− 𝑊

𝑅 𝑖 −2 )2

𝑖

(

𝜕𝑊

𝑅 𝑖

𝜕𝑥

) 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ d 𝑥, (10b)

w bout the y and z axes according to the Rayleigh beam theory, respectively,

a

4] , the nonlocal bending moments within the i th tube are simply linked to

t

((11a)

((11b)

view of Eqs. (10 a) and ( 10 b), the equations of motion of thermally affected

m M could be expressed by:

𝑅

)⎫ ⎪ ⎬ ⎪ ⎭ + 𝐸 𝑏 𝑗 𝐼 𝑏 𝑗

𝜕 4 𝑉 𝑅 𝑗

𝜕𝑥 4 = 0; 𝑗 = 1 , 2 , (12a)

𝑊

𝑅 3 )⎫ ⎪ ⎬ ⎪ ⎭ + 𝐸 𝑏 𝑗

𝐼 𝑏 𝑗

𝜕 4 𝑊

𝑅 𝑗

𝜕𝑥 4 = 0 , (12b)

. Implementation of discrete NRBM model for thermally affected

.1. Establishment of nonlocal equations of motion

For the thermally affected membrane of vertically aligned DWCNTs

nergy ( U

R ) by considering the nonlocality, the intertube vdW forces, and

𝑅 ( 𝑡 ) =

1 2

2 𝑁 ∑𝑖 =1

∫𝑙 𝑏

0 𝜌𝑏 𝑖

⎛ ⎜ ⎜ ⎝ 𝐴 𝑏 𝑖

⎛ ⎜ ⎜ ⎝ (

𝜕𝑉 𝑅 𝑖

𝜕𝑡

) 2

+

(

𝜕𝑊

𝑅 𝑖

𝜕𝑡

) 2 ⎞ ⎟ ⎟ ⎠ + 𝐼 𝑏 𝑖

⎛ ⎜ ⎜ ⎝ (

𝜕 2 𝑉 𝑅 𝑖

𝜕 𝑡𝜕 𝑥

) 2

+

(

𝜕

𝑅 ( 𝑡 ) =

1 2

2 𝑁 ∑𝑖 =1

∫𝑙 𝑏

0

[ −

𝜕 2 𝑉 𝑅 𝑖

𝜕𝑥 2

(𝑀

𝑛𝑙 𝑏𝑧 𝑖

)𝑅 −

𝜕 2 𝑊

𝑅 𝑖

𝜕𝑥 2

(𝑀

𝑛𝑙 𝑏𝑦 𝑖

)𝑅 ] d 𝑥

+

1 2

2 𝑁 ∑𝑖 =2 , 4 , …

∫𝑙 𝑏

0

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝐶 𝑣𝑦 ( 𝑖,𝑖 −1)

(𝑉 𝑅 𝑖

− 𝑉 𝑅 𝑖 −1 )2 + 𝐶 𝑣𝑦 ( 𝑖,𝑖 +2)

(𝑉 𝑅 𝑖

− 𝑉 𝑅 𝑖 +2 )2 (1 −

+ 𝐶 𝑣𝑦 ( 𝑖,𝑖 +1)

(𝑉 𝑅 𝑖

− 𝑉 𝑅 𝑖 +1 )2 (1 − 𝛿(2 𝑁)( 𝑖 ) ) + 𝐶 𝑣𝑦 ( 𝑖,𝑖 −2)

(𝑉 𝑅𝑖

+ 𝐶 𝑣𝑦 ( 𝑖,𝑖 −3)

(𝑉 𝑅 𝑖

− 𝑉 𝑅 𝑖 −3 )2 (1 − 𝛿2 𝑖 ) + 𝐾 𝑡

(𝑉 𝑅 𝑖

)2 + 𝐾 𝑟

(

+

1 2

2 𝑁−1 ∑𝑖 =3 , 5…

∫𝑙 𝑏

0

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝐶 𝑣𝑦 ( 𝑖,𝑖 +1)

(𝑉 𝑅 𝑖

− 𝑉 𝑅 𝑖 +1 )2 + 𝐶 𝑣𝑦 ( 𝑖,𝑖 +2)

(𝑉 𝑅 𝑖

− 𝑉 𝑅 𝑖 +2 )2 (1 −

+ 𝐶 𝑣𝑦 ( 𝑖,𝑖 +3)

(𝑉 𝑅 𝑖

− 𝑉 𝑅 𝑖 +3 )2 (1 − 𝛿(2 𝑁−1)( 𝑖 ) ) + 𝐶 𝑣𝑦 ( 𝑖,𝑖 −2)

(𝑉

+ 𝐶 𝑣𝑦 ( 𝑖,𝑖 −1)

(𝑉 𝑅 𝑖

− 𝑉 𝑅 𝑖 −1 )2 (1 − 𝛿1 𝑖 ) + 𝑁 𝑇 𝑖

(

𝜕𝑉 𝑅 𝑖

𝜕𝑥

) 2

+

1 2

2 𝑁 ∑𝑖 =2 , 4 , …

∫𝑙 𝑏

0

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𝐶 𝑣𝑧 ( 𝑖,𝑖 −1)

(𝑊

𝑅 𝑖

− 𝑊

𝑅 𝑖 −1 )2 + 𝐶 𝑣𝑧 ( 𝑖,𝑖 +2)

(𝑊

𝑅 𝑖

− 𝑊

𝑅 𝑖 +2 )2 (

+ 𝐶 𝑣𝑧 ( 𝑖,𝑖 +1)

(𝑊

𝑅 𝑖

− 𝑊

𝑅 𝑖 +1 )2 (1 − 𝛿(2 𝑁)( 𝑖 ) )

+ 𝐶 𝑣𝑧 ( 𝑖,𝑖 −2)

(𝑊

𝑅 𝑖

− 𝑊

𝑅 𝑖 −2 )2 (1 − 𝛿2 𝑖 ) + 𝐶 𝑣𝑧 ( 𝑖,𝑖 −3)

(𝑊

𝑅 𝑖

+ 𝐾 𝑡

(𝑊

𝑅 𝑖

)2 + 𝐾 𝑟

(

𝜕𝑊

𝑅 𝑖

𝜕𝑥

) 2 + 𝑁 𝑇 𝑖

(

𝜕𝑊

𝑅 𝑖

𝜕𝑥

) 2

+

1 2

2 𝑁−1 ∑𝑖 =3 , 5 , …

∫𝑙 𝑏

0

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝐶 𝑣𝑧 ( 𝑖,𝑖 +1)

(𝑊

𝑅 𝑖

− 𝑊

𝑅 𝑖 +1 )2 + 𝐶 𝑣𝑧 ( 𝑖,𝑖 +2)

(𝑊

𝑅 𝑖

− 𝑊

𝑅 𝑖 +2 )2 (

+ 𝐶 𝑣𝑧 ( 𝑖,𝑖 +3)

(𝑊

𝑅 𝑖

− 𝑊

𝑅 𝑖 +3 )2 (1 − 𝛿(2 𝑁−1)( 𝑖 ) ) + 𝐶 𝑣𝑧 ( 𝑖,𝑖 −2)

× (1 − 𝛿1 𝑖 ) + 𝐶 𝑣𝑧 ( 𝑖,𝑖 −1)

(𝑊

𝑅 𝑖

− 𝑊

𝑅 𝑖 −1 )2 (1 − 𝛿1 𝑖 ) + 𝑁 𝑇

here (𝑀


)𝑅 and

(𝑀


)𝑅 represent the nonlocal bending moments a

nd 𝛿ij denotes the Kronecker delta tensor.

In the framework of the nonlocal elasticity theory of Eringen [12–1

heir corresponding local values by the following relations [47–49] :

𝑀


)𝑅 − ( 𝑒 0 𝑎 ) 2 𝜕 2 (𝑀


)𝑅 𝜕𝑥 2

= − 𝐸 𝑏 𝑖 𝐼 𝑏 𝑖

𝜕 2 𝑊

𝑅 𝑖

𝜕𝑥 2 ,

𝑀


)𝑅 − ( 𝑒 0 𝑎 ) 2 𝜕 2 (𝑀


)𝑅 𝜕𝑥 2

= − 𝐸 𝑏 𝑖 𝐼 𝑏 𝑖

𝜕 2 𝑉 𝑅 𝑖

𝜕𝑥 2 .

By employing the Hamilton’s principle, ∫ 𝑡

0 ( 𝛿𝑇 𝑅 ( 𝑡 ) − 𝛿𝑈

𝑅 ( 𝑡 )) d 𝑡 = 0, in

embrane made of vertically aligned DWCNTs on the basis of the NRB

• The first thermally affected DWCNTs:

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝜌𝑏 𝑗

(

𝐴 𝑏 𝑗

𝜕 2 𝑉 𝑅 𝑗

𝜕𝑡 2 − 𝐼 𝑏 𝑗

𝜕 4 𝑉 𝑅 𝑗

𝜕 𝑡 2 𝜕 𝑥 2

)

+ 𝐶 𝑣𝑦 ( 𝑗 , 3− 𝑗 )

(𝑉 𝑅 𝑗

− 𝑉 𝑅 3− 𝑗 )+ 𝐶 𝑣𝑦 ( 𝑗, 3)

(𝑉 𝑅 𝑗

− 𝑉 3

+ 𝐶 𝑣𝑦 ( 𝑗, 4)

(𝑉 𝑅 𝑗

− 𝑉 𝑅 4 )+

(

𝐾 𝑡 𝑉 𝑅 𝑗

− 𝐾 𝑟

𝜕 2 𝑉 𝑅 𝑗

𝜕𝑥 2

)

(1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

𝜕 2 𝑉 𝑅 𝑗

𝜕𝑥 2

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝜌𝑏 𝑗

(

𝐴 𝑏 𝑗

𝜕 2 𝑊

𝑅 𝑗

𝜕𝑡 2 − 𝐼 𝑏 𝑗

𝜕 4 𝑊

𝑅 𝑗

𝜕 𝑡 2 𝜕 𝑥 2

)

+ 𝐶 𝑣𝑧 ( 𝑗 , 3− 𝑗 )

(𝑊

𝑅 𝑗

− 𝑊

𝑅 3− 𝑗

)+ 𝐶 𝑣𝑧 ( 𝑗, 3)

(𝑊

𝑅 𝑗

−

+ 𝐶 𝑣𝑧 ( 𝑗, 4)

(𝑊

𝑅 𝑗

− 𝑊

𝑅 4 )+

(

𝐾 𝑡 𝑊

𝑅 𝑗

− 𝐾 𝑟

𝜕 2 𝑊

𝑅 𝑗

𝜕𝑥 2

)

(1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

𝜕 2 𝑊

𝑅 𝑗

𝜕𝑥 2

579


− 𝑗 ) + 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 −2) ( 𝑉 𝑅 2 𝑛 −2+ 𝑗 − 𝑉 𝑅 2 𝑛 −2 ) + 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 −3)

( 𝑉 𝑅 2 𝑛 −2+ 𝑗 − 𝑉 𝑅 2 𝑛 −3 ) ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(13a)

(13a)

𝑛 +1− 𝑗 ) + 𝐶 𝑣𝑧 (2 𝑛 −2+ 𝑗, 2 𝑛 −2) ( 𝑊

𝑅 2 𝑛 −2+ 𝑗 − 𝑊

𝑅 2 𝑛 −2 ) + 𝐶 𝑣𝑧 (2 𝑛 −2+ 𝑗, 2 𝑛 −3)

( 𝑊

𝑅 2 𝑛 −2+ 𝑗 − 𝑊

𝑅 2 𝑛 −3 )

2 )

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ (13b)

+1− 𝑗 ) + 𝐶 𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 −2) ( 𝑉 𝑅 2 𝑁−2+ 𝑗 − 𝑉 𝑅 2 𝑁−2) ) + 𝐶 𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 −3)

( 𝑉 𝑅 2 𝑁−2+ 𝑗 − 𝑉 𝑅 2 𝑁−3) ) ⎫ ⎪ ⎬ ⎪ ⎭

(14a)

𝑊

𝑅 2 𝑁+1− 𝑗 )

𝑊

𝑅 2 𝑁−3 )

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + 𝐸 𝑏 𝑗

𝐼 𝑏 𝑗

𝜕 4 𝑊

𝑅 2 𝑁−2+ 𝑗

𝜕𝑥 4 = 0 , (14b)

vestigate free vibration of the thermally affected nanosystem irrespective of

onless parameters:

2 1 =

𝜌𝑏 2 𝐴 𝑏 2


,

𝑚,𝑛 ) =

𝐶 𝑣𝑦 ( 𝑚,𝑛 ) 𝑙 4 𝑏

𝐸 𝑏 1 𝐼 𝑏 1

,

, (15)

Eq. (15) , the dimensionless NRBM-based equations of motion of thermally

ith plain membrane configuration take the following form:

𝑅

𝑣𝑦 ( 𝑗, 3)

− 𝑁

𝑅

𝑇 𝑗

𝜕 2 𝑉 𝑅

𝑗

𝜕𝜉2

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + ( 𝜚 2 3 𝛿𝑗2 + 𝛿𝑗1 )

𝜕 4 𝑉 𝑅 𝑗

𝜕𝜉4 = 0 , (16a)

𝑅

𝑣𝑧 ( 𝑗, 3)

− 𝑁

𝑅

𝑇 𝑗

𝜕 2 𝑊

𝑅

𝑗

𝜕𝜉2

⎫ ⎪ ⎬ ⎪ ⎭ + ( 𝜚 2 3 𝛿𝑗2 + 𝛿𝑗1 ) 𝜕 4 ��

𝑅 𝑗

𝜕𝜉4 = 0 , (16b)

+ 𝑗, 2 𝑛 −3)

2 𝑛 +2)

𝑅

𝑇 𝑗

𝜕 2 𝑉 𝑅

2 𝑛 −2+ 𝑗 𝜕𝜉2

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ + ( 𝜚 2 3 𝛿𝑗2 + 𝛿𝑗1 )

𝜕 4 𝑉 𝑅

2 𝑛 −2+ 𝑗

𝜕𝜉4 = 0; 𝑛 = 2 , 3 , … , 𝑁 − 1 , (17a)

• The intermediate thermally affected DWCNTs:

Ξ

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝜌𝑏 𝑗

(

𝐴 𝑏 𝑗

𝜕 2 𝑉 𝑅 2 𝑛 −2+ 𝑗 𝜕𝑡 2

− 𝐼 𝑏 𝑗

𝜕 4 𝑉 𝑅 2 𝑛 −2+ 𝑗 𝜕 𝑡 2 𝜕 𝑥 2

)

+ 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗 , 2 𝑛 +1− 𝑗 ) ( 𝑉 𝑅 2 𝑛 −2+ 𝑗 − 𝑉 𝑅 2 𝑛 +1

+ 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 +1) ( 𝑉 𝑅 2 𝑛 −2+ 𝑗 − 𝑉 𝑅 2 𝑛 +1 ) + 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 +2)

( 𝑉 𝑅 2 𝑛 −2+ 𝑗 − 𝑉 𝑅 2 𝑛 +2 )

+

(

𝐾 𝑡 𝑉 𝑅 2 𝑛 −2+ 𝑗 − 𝐾 𝑟

𝜕 2 𝑉 𝑅 2 𝑛 −2+ 𝑗 𝜕𝑥 2

)

(1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

𝜕 2 𝑉 𝑅 2 𝑛 −2+ 𝑗 𝜕𝑥 2

+ 𝐸 𝑏 𝑗 𝐼 𝑏 𝑗

𝜕 4 𝑉 𝑅 2 𝑛 −2+ 𝑗

𝜕𝑥 4 = 0; 𝑛 = 2 , 3 , … , 𝑁 − 1 ,

Ξ

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝜌𝑏 𝑗

(

𝐴 𝑏 𝑗

𝜕 2 𝑊

𝑅 2 𝑛 −2+ 𝑗 𝜕𝑡 2

− 𝐼 𝑏 𝑗

𝜕 4 𝑊

𝑅 2 𝑛 −2+ 𝑗

𝜕 𝑡 2 𝜕 𝑥 2

)

+ 𝐶 𝑣𝑧 (2 𝑛 −2+ 𝑗 , 2 𝑛 +1− 𝑗 ) ( 𝑊

𝑅 2 𝑛 −2+ 𝑗 − 𝑊

𝑅2

+ 𝐶 𝑣𝑧 (2 𝑛 −2+ 𝑗, 2 𝑛 +1) ( 𝑊

𝑅 2 𝑛 −2+ 𝑗 − 𝑊

𝑅 2 𝑛 +1 ) + 𝐶 𝑣𝑧 (2 𝑛 −2+ 𝑗, 2 𝑛 +2)

( 𝑊

𝑅 2 𝑛 −2+ 𝑗 − 𝑊

𝑅 2 𝑛 +

+

(

𝐾 𝑡 𝑊

𝑅 2 𝑛 −2+ 𝑗 − 𝐾 𝑟

𝜕 2 𝑊

𝑅 2 𝑛 −2+ 𝑗 𝜕𝑥 2

)

(1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

𝜕 2 𝑊

𝑅 2 𝑛 −2+ 𝑗 𝜕𝑥 2


𝜕 4 𝑊

𝑅 2 𝑛 −2+ 𝑗

𝜕𝑥 4 = 0 ,

• The N th thermally affected DWCNTs:

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝜌𝑏 𝑗

(

𝐴 𝑏 𝑗

𝜕 2 𝑉 𝑅 2 𝑁−2+ 𝑗 𝜕𝑡 2

− 𝐼 𝑏 𝑗

𝜕 4 𝑉 𝑅 2 𝑁−2+ 𝑗 𝜕 𝑡 2 𝜕 𝑥 2

)

+ 𝐶 𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 +1− 𝑗) ( 𝑉 𝑅 2 𝑁−2+ 𝑗 − 𝑉 𝑅 2 𝑁

+

(

𝐾 𝑡 𝑉 𝑅 2 𝑁−2+ 𝑗 − 𝐾 𝑟

𝜕 2 𝑉 𝑅 2 𝑁−2+ 𝑗 𝜕𝑥 2

)

(1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

𝜕 2 𝑉 𝑅 2 𝑁−2+ 𝑗 𝜕𝑥 2


𝜕 4 𝑉 𝑅 2 𝑁−2+ 𝑗

𝜕𝑥 4 = 0 ,

Ξ

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝜌𝑏 𝑗

(

𝐴 𝑏 𝑗

𝜕 2 𝑊

𝑅 2 𝑁−2+ 𝑗 𝜕𝑡 2

− 𝐼 𝑏 𝑗

𝜕 4 𝑊

𝑅 2 𝑁−2+ 𝑗

𝜕 𝑡 2 𝜕 𝑥 2

)

+ 𝐶 𝑣𝑧 (2 𝑁 −2+ 𝑗, 2 𝑁 +1− 𝑗) ( 𝑊

𝑅 2 𝑁−2+ 𝑗 −

+ 𝐶 𝑣𝑧 (2 𝑁 −2+ 𝑗, 2 𝑁 −2) ( 𝑊

𝑅 2 𝑁−2+ 𝑗 − 𝑊

𝑅 2 𝑁−2 ) + 𝐶 𝑣𝑧 (2 𝑁 −2+ 𝑗, 2 𝑁 −3)

( 𝑊

𝑅 2 𝑁−2+ 𝑗 −

+

(

𝐾 𝑡 𝑊

𝑅 2 𝑁−2+ 𝑗 − 𝐾 𝑟

𝜕 2 𝑊

𝑅 2 𝑁−2+ 𝑗 𝜕𝑥 2

)

(1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

𝜕 2 𝑊

𝑅 2 𝑁−2+ 𝑗 𝜕𝑥 2

where Ξ[ ⋅] = [ ⋅] − ( 𝑒 0 𝑎 ) 2 𝜕 2

𝜕𝑥 2 [ ⋅] represents the nonlocal operator. To in

the time and space dimensions, we introduce the following dimensi

𝜉 =

𝑥

𝑙 𝑏 , 𝛾 =

𝑧

𝑙 𝑧 , 𝑉

𝑅

𝑖 =

𝑉 𝑅 𝑖

𝑙 𝑏 , 𝑊

𝑅

𝑖 =

𝑊

𝑅 𝑖

𝑙 𝑏 , 𝜏 =

𝑡

𝑙 2 𝑏

√

𝐸 𝑏 1 𝐼 𝑏 1


, 𝜚

𝜚 2 2 =

𝜌𝑏 2 𝐼 𝑏 2

𝜌𝑏 1 𝐼 𝑏 1

, 𝜚 2 3 =

𝐸 𝑏 2 𝐼 𝑏 2

𝐸 𝑏 1 𝐼 𝑏 1

, 𝐾

𝑅

𝑡 =

𝐾 𝑡 𝑙 4 𝑏

𝐸 𝑏 1 𝐼 𝑏 1

, 𝐾

𝑅

𝑟 =

𝐾 𝑟 𝑙 2 𝑏

𝐸 𝑏 1 𝐼 𝑏 1

, 𝐶

𝑅

𝑣𝑦 (

𝐶

𝑅

𝑣𝑧 ( 𝑚,𝑛 ) =

𝐶 𝑣𝑧 ( 𝑚,𝑛 ) 𝑙 4 𝑏

𝐸 𝑏 1 𝐼 𝑏 1

, 𝑙 𝑧 = ( 𝑁 − 1) 𝑑, 𝜆1 =

𝑙 𝑏 √

𝐼 𝑏 1 𝐴 𝑏 1

, 𝑁

𝑅

𝑇 𝑗 =

𝑁 𝑇 𝑗 𝑙 2 𝑏

𝐸 𝑏 1 𝐼 𝑏 1

by mixing Eqs. (12 a) , ( 12 b), ( 13 a), ( 13 b), ( 14 a), and ( 14 b) with

affected-vertically aligned DWCNTs embedded in an elastic matrix w• The first thermally affected DWCNTs:

Ξ

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ( 𝜚 2 1 𝛿𝑗2 + 𝛿𝑗1 )

𝜕 2 𝑉 𝑅

𝑗

𝜕𝜏2 −

(

𝜚 2 2 𝛿𝑗2 + 𝛿𝑗1 𝜆2 1

)

𝜕 4 𝑉 𝑅

𝑗

𝜕 𝜏2 𝜕 𝜉2 + 𝐶

𝑅

𝑣𝑦 ( 𝑗 , 3− 𝑗 ) ( 𝑉

𝑅

𝑗 − 𝑉

𝑅

3− 𝑗 ) + 𝐶

× ( 𝑉 𝑅

𝑗 − 𝑉

𝑅

3 ) + 𝐶

𝑅

𝑣𝑦 ( 𝑗, 4) ( 𝑉

𝑅

𝑗 − 𝑉

𝑅

4 ) +

(

𝐾

𝑅

𝑡 𝑉 𝑅

𝑗 − 𝐾

𝑅

𝑟

𝜕 2 𝑉 𝑅

𝑗

𝜕𝜉2

)

(1 − 𝛿𝑗1 )

Ξ⎧ ⎪ ⎨ ⎪ ⎩ ( 𝜚 2 1 𝛿𝑗2 + 𝛿𝑗1 )

𝜕 2 𝑊

𝑅

𝑗

𝜕𝜏2 −

(

𝜚 2 2 𝛿𝑗2 + 𝛿𝑗1 𝜆2 1

)

𝜕 4 𝑊

𝑅

𝑗

𝜕 𝜏2 𝜕 𝜉2 + 𝐶

𝑅

𝑣𝑧 ( 𝑗 , 3− 𝑗 ) ( 𝑊

𝑅

𝑗 − 𝑊

𝑅

3− 𝑗 ) + 𝐶

× ( 𝑊

𝑅

𝑗 − 𝑊

𝑅

3 ) + 𝐶

𝑅

𝑣𝑧 ( 𝑗, 4) ( 𝑊

𝑅

𝑗 − 𝑊

𝑅

4 ) + ( 𝐾

𝑅

𝑡 𝑊

𝑅

𝑗 − 𝐾

𝑅

𝑟

𝜕 2 𝑊

𝑅

𝑗

𝜕𝜉2 )(1 − 𝛿𝑗1 )

• The intermediate thermally affected DWCNTs:

Ξ

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪

( 𝜚 2 1 𝛿𝑗2 + 𝛿𝑗1 ) 𝜕 2 𝑉

𝑅

2 𝑛 −2+ 𝑗 𝜕𝜏2

−

(

𝜚 2 2 𝛿𝑗2 + 𝛿𝑗1 𝜆2 1

)

𝜕 4 𝑉 𝑅

2 𝑛 −2+ 𝑗 𝜕 𝜏2 𝜕 𝜉2

+ 𝐶

𝑅

𝑣𝑦 (2 𝑛 −2+ 𝑗 , 2 𝑛 +1− 𝑗 )

× ( 𝑉 𝑅

2 𝑛 −2+ 𝑗 − 𝑉 𝑅

2 𝑛 +1− 𝑗 ) + 𝐶

𝑅

𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 −2) ( 𝑉

𝑅

2 𝑛 −2+ 𝑗 − 𝑉 𝑅

2 𝑛 −2 ) + 𝐶

𝑅

𝑣𝑦 (2 𝑛 −2

× ( 𝑉 𝑅

2 𝑛 −2+ 𝑗 − 𝑉 𝑅

2 𝑛 −3 ) + 𝐶

𝑅

𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 +1) ( 𝑉

𝑅

2 𝑛 −2+ 𝑗 − 𝑉 𝑅

2 𝑛 +1 ) + 𝐶

𝑅

𝑣𝑦 (2 𝑛 −2+ 𝑗,

× ( 𝑉 𝑅

2 𝑛 −2+ 𝑗 − 𝑉 𝑅

2 𝑛 −3 ) +

(

𝐾

𝑅

𝑡 𝑉 𝑅

2 𝑛 −2+ 𝑗 − 𝐾

𝑅

𝑟

𝜕 2 𝑉 𝑅

2 𝑛 −2+ 𝑗 𝜕𝜉2

)

(1 − 𝛿𝑗1 ) − 𝑁
⎩ ⎭ 580


2+ 𝑗, 2 𝑛 −3)

𝑗, 2 𝑛 +2)

𝑅

𝑇 𝑗

𝜕 2 𝑊

𝑅

2 𝑛 −2+ 𝑗 𝜕𝜉2

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ + ( 𝜚 2 3 𝛿𝑗2 + 𝛿𝑗1 )

𝜕 4 𝑊

𝑅

2 𝑛 −2+ 𝑗

𝜕𝜉4 = 0 , (17b)

𝑁 −2+ 𝑗, 2 𝑁 −3)

𝑗

𝜕 2 𝑉 𝑅

2 𝑁−2+ 𝑗 𝜕𝜉2

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + ( 𝜚 2 3 𝛿𝑗2 + 𝛿𝑗1 )

𝜕 4 𝑉 𝑅

2 𝑁−2+ 𝑗

𝜕𝜉4 = 0 , (18a)

𝑅

𝑣𝑧 (2 𝑁 −2+ 𝑗, 2 𝑁 −3)

− 𝑁

𝑅

𝑇 𝑗

𝜕 2 𝑊

𝑅

2 𝑁−2+ 𝑗 𝜕𝜉2

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ + ( 𝜚 2 3 𝛿𝑗2 + 𝛿𝑗1 )

𝜕 4 𝑊

𝑅

2 𝑁−2+ 𝑗

𝜕𝜉4 = 0 , (18b)

d ( 18 b) present 4 N fourth-order partial differential equations. For vertically

in an elastic matrix, these coupled equations can be exploited for examining

sing the number of constitutive DWCNTs, the number of governing equations

tion is highly required. To conquer this deficiency of the proposed discrete

ing section.

4

be simple. Additionally, the outermost carbon nanotubes of the first and the

N 𝜏) = 𝑊

𝑅

2 𝑁

( 𝜉, 𝜏) = 0 , 𝑉 𝑅

2 ( 𝜉, 𝜏) = 𝑊

𝑅

2 ( 𝜉, 𝜏) = 0 . We employ AMM for frequency

a elastic matrix. Therefore, the deflections fields of the constitutive tubes of

t

𝑉 (19a)

𝑊 (19b)

w the m th mode of vibration of the i th nanotube, N my and N mz in order denote

t nosystem along the y and z directions. By introducing Eqs. (19 a) and ( 19 b)

t f second-order ordinary differential equations is obtained:

(20)

w

𝐱 𝑜𝑟 𝑊 , (21)

a . To compute natural frequencies, we consider 𝐱 𝑅 ( 𝜏) = 𝐱 𝑅 0 exp (i 𝜛

𝑅 𝜏) where

i ϖR denotes dimensionless natural frequency. By substituting this form into

E ving the resulted set of equations.

5 d DWCNTs membranes

5

unction with the nonlocal elasticity theory of Eringen [12–14] , the kinetic

e edded membrane of vertically aligned DWCNTs subjected to longitudinal

Ξ

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

( 𝜚 2 1 𝛿𝑗2 + 𝛿𝑗1 ) 𝜕 2 𝑊

𝑅

(2 𝑛 −2+ 𝑗) 𝜕𝜏2

−

(

𝜚 2 2 𝛿𝑗2 + 𝛿𝑗1 𝜆2 1

)

𝜕 4 𝑊

𝑅

2 𝑛 −2+ 𝑗 𝜕 𝜏2 𝜕 𝜉2

+ 𝐶

𝑅

𝑣𝑧 (2 𝑛 −2+ 𝑗 , 2 𝑛 +1− 𝑗 )

× ( 𝑊

𝑅

2 𝑛 −2+ 𝑗 − 𝑊

𝑅

2 𝑛 +1− 𝑗 ) + 𝐶

𝑅

𝑣𝑧 (2 𝑛 −2+ 𝑗, 2 𝑛 −2) ( 𝑊

𝑅

2 𝑛 −2+ 𝑗 − 𝑊

𝑅

2 𝑛 −2 ) + 𝐶

𝑅

𝑣𝑧 (2 𝑛 −

× ( 𝑊

𝑅

2 𝑛 −2+ 𝑗 − 𝑊

𝑅

2 𝑛 −3 ) + 𝐶

𝑅

𝑣𝑧 (2 𝑛 −2+ 𝑗, 2 𝑛 +1) ( 𝑊

𝑅

2 𝑛 −2+ 𝑗 − 𝑊

𝑅

2 𝑛 +1 ) + 𝐶

𝑅

𝑣𝑧 (2 𝑛 −2+

× ( 𝑊

𝑅

2 𝑛 −2+ 𝑗 − 𝑊

𝑅

2 𝑛 +2 ) +

(

𝐾

𝑅

𝑡 𝑊

𝑅

2 𝑛 −2+ 𝑗 − 𝐾

𝑅

𝑟

𝜕 2 𝑊

𝑅

2 𝑛 −2+ 𝑗 𝜕𝜉2

)

(1 − 𝛿𝑗1 ) − 𝑁

• The N th thermally affected DWCNTs:

Ξ

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ( 𝜚 2 1 𝛿𝑗2 + 𝛿𝑗1 )

𝜕 2 𝑉 𝑅

2 𝑁−2+ 𝑗 𝜕𝜏2

−

(

𝜚 2 2 𝛿𝑗2 + 𝛿𝑗1 𝜆2 1

)

𝜕 4 𝑉 𝑅

2 𝑁−2+ 𝑗 𝜕 𝜏2 𝜕 𝜉2

+ 𝐶

𝑅

𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 +1− 𝑗)

× ( 𝑉 𝑅

2 𝑁−2+ 𝑗 − 𝑉 𝑅

2 𝑁+1− 𝑗 ) + 𝐶

𝑅

𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 −2) ( 𝑉

𝑅

2 𝑁−2+ 𝑗 − 𝑉 𝑅

2 𝑁−2 ) + 𝐶

𝑅

𝑣𝑦 (2

× ( 𝑉 𝑅

2 𝑁−2+ 𝑗 − 𝑉 𝑅

2 𝑁−3 ) + ( 𝐾

𝑅

𝑡 𝑉 𝑅

2 𝑁−2+ 𝑗 − 𝐾

𝑅

𝑟

𝜕 2 𝑉 𝑅

2 𝑁−2+ 𝑗 𝜕𝜉2

)(1 − 𝛿𝑗1 ) − 𝑁

𝑅

𝑇

Ξ

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

( 𝜚 2 1 𝛿𝑗2 + 𝛿𝑗1 ) 𝜕 2 𝑊

𝑅

2 𝑁−2+ 𝑗 𝜕𝜏2

−

(

𝜚 2 2 𝛿𝑗2 + 𝛿𝑗1 𝜆2 1

)

𝜕 4 𝑊

𝑅

2 𝑁−2+ 𝑗 𝜕 𝜏2 𝜕 𝜉2

+ 𝐶

𝑅

𝑣𝑧 (2 𝑁 −2+ 𝑗, 2 𝑁 +1− 𝑗)

× ( 𝑊

𝑅

2 𝑁−2+ 𝑗 − 𝑊

𝑅

2 𝑁+1− 𝑗 ) + 𝐶

𝑅

𝑣𝑧 (2 𝑁 −2+ 𝑗, 2 𝑁 −2) ( 𝑊

𝑅

2 𝑁−2+ 𝑗 − 𝑊

𝑅

2 𝑁−2 ) + 𝐶

× ( 𝑊

𝑅

2 𝑁−2+ 𝑗 − 𝑊

𝑅

2 𝑁−3 ) +

(

𝐾

𝑅

𝑡 𝑊

𝑅

2 𝑁−2+ 𝑗 − 𝐾

𝑅

𝑟

𝜕 2 𝑊

𝑅

2 𝑁−2+ 𝑗 𝜕𝜉2

)

(1 − 𝛿𝑗1 )

where Ξ[ ⋅] = [ ⋅] − 𝜇2 𝜕 2 𝜕𝜉2

[ ⋅] . Eqs. (16 a) , ( 16 b), ( 17 a), ( 17 b), ( 18 a), an

aligned membranes made from a low number of DWCNTs embedded

transverse vibrations of the thermally affected nanosystem. By increa

would also drastically increase and the development of another solu

model, we will develop appropriate continuous models in an upcom

.2. Evaluation of natural frequencies

Assume that the ends of all tubes of the vertically aligned DWCNTs

th DWCNTs have been prevented from any lateral motion, i.e., 𝑉 𝑅

2 𝑁

( 𝜉,nalysis of vertically aligned membranes of DWCNTs embedded in an

he membrane are discretized by:

𝑅

𝑖 ( 𝜉, 𝜏) =

𝑁 𝑚𝑦 ∑𝑖 =1

𝑉 𝑅

𝑖𝑚 ( 𝜏) sin ( 𝑚𝜋𝜉) ,

𝑅

𝑖 ( 𝜉, 𝜏) =

𝑁 𝑚𝑧 ∑𝑖 =1

𝑊

𝑅


here 𝑉 𝑅

𝑖𝑚 ( 𝜏) and 𝑊

𝑅

𝑖𝑚 ( 𝜏) are the time-dependent quantities pertinent to

he number of modes taken into account for lateral vibrations of the na

o Eqs. (16 a) , ( 16 b), ( 17 a), ( 17 b), ( 18 a), and ( 18 b), the following set o

d 2 𝐱 𝑅

d 𝜏2 + 𝚪

𝑅 𝐱 𝑅 = 0 ,

here 𝐱 𝑅 is defined as follows:

𝑅 =

{

[ . ] 𝑅

1 𝑚 , [ . ] 𝑅

3 𝑚 , [ . ] 𝑅

4 𝑚 , [ . ] 𝑅

5 𝑚 , …[ . ] 𝑅

(2 𝑁−2) 𝑚 , [ . ] 𝑅

(2 𝑁−1) 𝑚

} 𝑇

; [ . ] = 𝑉

nd the elements of 𝚪𝑅

are provided in supplementary material, part B

=

√−1 , 𝐱 𝑅 0 is the dimensionless vector of unknown parameters, and

q. (20) , the natural frequencies could be readily evaluated through sol

. Implementation of discrete NHOBM model for thermally affecte

.1. Establishment of nonlocal equations of motion

Using higher-order beam theory of Reddy–Bickford [50,51] in conj

nergy ( T H ) and the elastic strain energy ( U

H ) of the elastically emb

581


t

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ d 𝑥, (22a)

𝑈

)

⎤ ⎥ ⎥ ⎥ ⎦ d 𝑥 𝛿(2 𝑁)( 𝑖 ) )

1 − 𝛿2 𝑖 )

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ d 𝑥

𝛿(2 𝑁−1)( 𝑖 ) )

1 − 𝛿1 𝑖 ) + 𝑁 𝑇 𝑖

(

𝜕𝑉 𝐻 𝑖

𝜕𝑥

) 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ d 𝑥

(1 − 𝛿(2 𝑁)( 𝑖 ) )

2 + 𝑁 𝑇 𝑖

(

𝜕𝑊

𝐻 𝑖

𝜕𝑥

) 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ d 𝑥

(1 − 𝛿(2 𝑁−1)( 𝑖 ) )

)

(𝑊

𝐻

𝑖 − 𝑊

𝐻

𝑖 −2 )2

𝑖

(

𝜕𝑊

𝐻 𝑖

𝜕𝑥

) 2

⎤ ⎥ ⎥ ⎥ ⎥ ⎦ d 𝑥, (22b)

w nd z axes, respectively, Ψ𝐻

𝑦 𝑖 and Ψ𝐻

𝑧 𝑖 in order denote the angle of deflections

a represent the whole shear force within the i th tube along the y and z axes,

r of the i th tube about the y -axis and the z -axis, respectively. By implementing

t rnal forces are related to their corresponding local ones via the following

r(𝑖 𝐽 4 𝑖

𝜕 2 Ψ𝐻

𝑧 𝑖

𝜕𝑥 2 − 𝛼2

𝑖 𝐽 6 𝑖

(

𝜕 2 Ψ𝐻

𝑧 𝑖

𝜕𝑥 2 +

𝜕 3 𝑉 𝐻

𝑖

𝜕𝑥 3

)

, (23a)

(𝛼𝑖 𝐽 4 𝑖

𝜕 2 Ψ𝐻

𝑦 𝑖

𝜕𝑥 2 − 𝛼2

𝑖 𝐽 6 𝑖

(

𝜕 2 Ψ𝐻

𝑦 𝑖

𝜕𝑥 2 +

𝜕 3 𝑊

𝐻

𝑖

𝜕𝑥 3

)

, (23b)

𝑀 (23c)

𝑀 (23d)

w

𝐼 (24)

N hrough using integration by part technique in conjunction with Eqs. (23

a tions of the thermally affected-embedded membranes of DWCNTs subjected

t :

emperature gradient are written as follows:

𝑇 𝐻 ( 𝑡 ) =

1 2

2 𝑁 ∑𝑖 =1

∫𝑙 𝑏

0

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

𝐼 0 𝑖

( (

𝜕 2 𝑉 𝐻 𝑖

𝜕 𝑡𝜕 𝑥

) 2 +

(

𝜕 2 𝑊

𝐻 𝑖

𝜕 𝑡𝜕 𝑥

) 2 )

+ 𝐼 2 𝑖

(

𝜕Ψ𝐻 𝑦 𝑖

𝜕𝑡

) 2

+ 𝛼2 𝑖 𝐼 6 𝑖

(


𝜕𝑡 +

𝜕 2 𝑊

𝐻 𝑖

𝜕 𝑡𝜕 𝑥

)

− 2 𝛼𝑖 𝐼 4 𝑖 𝜕Ψ𝐻

𝑦 𝑖

𝜕𝑡

(


𝜕𝑡 +

𝜕 2 𝑊

𝐻 𝑖

𝜕 𝑡𝜕 𝑥

)+ 𝐼 2 𝑖

(

𝜕Ψ𝐻 𝑧 𝑖

𝜕𝑡

) 2 + 𝛼2

𝑖 𝐼 6 𝑖

(


𝜕𝑡 +

𝜕 2 𝑉 𝐻 𝑖

𝜕 𝑡𝜕 𝑥

)

− 2 𝛼𝑖 𝐼 4 𝑖 𝜕Ψ𝐻

𝑧 𝑖

𝜕𝑡

(


𝜕𝑡 +

𝜕 2 𝑉 𝐻 𝑖

𝜕 𝑡𝜕 𝑥

)

𝐻 ( 𝑡 ) =

1 2

2 𝑁 ∑𝑖 =1

∫𝑙 𝑏

0

⎡ ⎢ ⎢ ⎢ ⎣ 𝜕Ψ𝐻

𝑦 𝑖

𝜕𝑥

(𝑀


)𝐻 +

(Ψ𝐻

𝑦 𝑖 +

𝜕𝑊

𝐻 𝑖

𝜕𝑥

)(

𝛼𝑖

𝜕( 𝑃 𝑛𝑙 𝑏𝑧 𝑖

) 𝐻

𝜕𝑥 + ( 𝑄


) 𝐻

)

+


𝜕𝑥

(𝑀


)𝐻 +

(

Ψ𝐻

𝑧 𝑖 +

𝜕𝑉 𝐻 𝑖

𝜕𝑥

) (

𝛼𝑖

𝜕( 𝑃 𝑛𝑙 𝑏𝑦 𝑖

) 𝐻

𝜕𝑥 + ( 𝑄


) 𝐻

+

1 2

2 𝑁 ∑𝑖 =2 , 4 , …

∫𝑙 𝑏

0

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𝐶 𝑣𝑦 ( 𝑖,𝑖 −1)

(𝑉 𝐻

𝑖 − 𝑉 𝐻

𝑖 −1 )2 + 𝐶 𝑣𝑦 ( 𝑖,𝑖 +2)

(𝑉 𝐻

𝑖 − 𝑉 𝐻

𝑖 +2 )2 (1 −

+ 𝐶 𝑣𝑦 ( 𝑖,𝑖 +1)

(𝑉 𝐻

𝑖 − 𝑉 𝐻

𝑖 +1 )2 (1 − 𝛿(2 𝑁)( 𝑖 ) ) + 𝐶 𝑣𝑦 ( 𝑖,𝑖 −2)

×(𝑉 𝐻

𝑖 − 𝑉 𝐻

𝑖 −2 )2 (1 − 𝛿2 𝑖 ) + 𝐶 𝑣𝑦 ( 𝑖,𝑖 −3)

(𝑉 𝐻

𝑖 − 𝑉 𝐻

𝑖 −3 )2 (

+ 𝐾 𝑡

(𝑉 𝐻

𝑖

)2 + 𝐾 𝑟

(Ψ𝐻

𝑧 𝑖

)2 + 𝑁 𝑇 𝑖

(

𝜕𝑉 𝐻 𝑖

𝜕𝑥

) 2

+

1 2

2 𝑁−1 ∑𝑖 =3 , 5…

∫𝑙 𝑏

0

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝐶 𝑣𝑦 ( 𝑖,𝑖 +1)

(𝑉 𝐻

𝑖 − 𝑉 𝐻

𝑖 +1 )2 + 𝐶 𝑣𝑦 ( 𝑖,𝑖 +2)

(𝑉 𝐻

𝑖 − 𝑉 𝐻

𝑖 +2 )2 (1 −

+ 𝐶 𝑣𝑦 ( 𝑖,𝑖 +3)

(𝑉 𝐻

𝑖 − 𝑉 𝐻

𝑖 +3 )2 (1 − 𝛿(2 𝑁−1)( 𝑖 ) ) + 𝐶 𝑣𝑦 ( 𝑖,𝑖 −2)

×(𝑉 𝐻

𝑖 − 𝑉 𝐻

𝑖 −2 )2 (1 − 𝛿1 𝑖 ) + 𝐶 𝑣𝑦 ( 𝑖,𝑖 −1)

(𝑉 𝐻

𝑖 − 𝑉 𝐻

𝑖 −1 )2 (

+

1 2

2 𝑁 ∑𝑖 =2 , 4 , …

∫𝑙 𝑏

0

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

𝐶 𝑣𝑧 ( 𝑖,𝑖 −1)

(𝑊

𝐻

𝑖 − 𝑊

𝐻

𝑖 −1 )2 + 𝐶 𝑣𝑧 ( 𝑖,𝑖 +2)

(𝑊

𝐻

𝑖 − 𝑊

𝐻

𝑖 +2 )2

+ 𝐶 𝑣𝑧 ( 𝑖,𝑖 +1)

(𝑊

𝐻

𝑖 − 𝑊

𝐻

𝑖 +1 )2 (1 − 𝛿(2 𝑁)( 𝑖 ) )

+ 𝐶 𝑣𝑧 ( 𝑖,𝑖 −2)

(𝑊

𝐻

𝑖 − 𝑊

𝐻

𝑖 −2 )2 (1 − 𝛿2 𝑖 ) + 𝐶 𝑣𝑧 ( 𝑖,𝑖 −3)

×(𝑊

𝐻

𝑖 − 𝑊

𝐻

𝑖 −3 )2 (1 − 𝛿2 𝑖 ) + 𝐾 𝑡

(𝑊

𝐻

𝑖

)2 + 𝐾 𝑟 (Ψ𝐻

𝑦 𝑖 )

+

1 2

2 𝑁−1 ∑𝑖 =3 , 5 , …

∫𝑙 𝑏

0

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝐶 𝑣𝑧 ( 𝑖,𝑖 +1)

(𝑊

𝐻

𝑖 − 𝑊

𝐻

𝑖 +1 )2 + 𝐶 𝑣𝑧 ( 𝑖,𝑖 +2)

(𝑊

𝐻

𝑖 − 𝑊

𝐻

𝑖 +2 )2

+ 𝐶 𝑣𝑧 ( 𝑖,𝑖 +3)

(𝑊

𝐻

𝑖 − 𝑊

𝐻

𝑖 +3 )2 (1 − 𝛿(2 𝑁−1)( 𝑖 ) ) + 𝐶 𝑣𝑧 ( 𝑖,𝑖 −2

× (1 − 𝛿1 𝑖 ) + 𝐶 𝑣𝑧 ( 𝑖,𝑖 −1) ( 𝑊

𝐻

𝑖 − 𝑊

𝐻

𝑖 −1 ) 2 (1 − 𝛿1 𝑖 ) + 𝑁 𝑇

here 𝑉 𝐻

𝑖 and 𝑊

𝐻

𝑖 are the deflections of the i th nanotube along the y a

bout the y -axis and z -axis, ( 𝑄


) 𝐻 + 𝛼𝑖

𝜕( 𝑃 𝑛𝑙 𝑏𝑦 𝑖

) 𝐻

𝜕𝑥 and ( 𝑄


) 𝐻 + 𝛼𝑖

𝜕( 𝑃 𝑛𝑙 𝑏𝑧 𝑖

) 𝐻

𝜕𝑥

espectively, ( 𝑀


) 𝐻 and ( 𝑀


) 𝐻 denote the nonlocal bending moment

he nonlocal elasticity theory of Eringen [12–14] , these nonlocal inte

elations [20,35,36,52] :

𝑄

𝐻

𝑏𝑦 𝑖 + 𝛼𝑖

𝜕𝑃 𝐻

𝑏𝑦 𝑖

𝜕𝑥

)

− ( 𝑒 0 𝑎 ) 2 𝜕 2

𝜕𝑥 2

(

𝑄

𝐻

𝑏𝑦 𝑖 + 𝛼𝑖

𝜕𝑃 𝐻

𝑏𝑦 𝑖

𝜕𝑥

)

= 𝜅𝑖

(

Ψ𝐻

𝑏𝑧 𝑖 +

𝜕𝑉 𝐻

𝑖

𝜕𝑥

)

+ 𝛼

𝑄

𝐻

𝑏𝑧 𝑖 + 𝛼𝑖

𝜕𝑃 𝐻

𝑏𝑧 𝑖

𝜕𝑥

)

− ( 𝑒 0 𝑎 ) 2 𝜕 2

𝜕𝑥 2

(

𝑄

𝐻

𝑏𝑧 𝑖 + 𝛼𝑖

𝜕𝑃 𝐻

𝑏𝑧 𝑖

𝜕𝑥

)

= 𝜅𝑖

(

Ψ𝐻

𝑏𝑦 𝑖 +

𝜕𝑊

𝐻

𝑖

𝜕𝑥

)

+

𝐻

𝑏𝑦 𝑖 − ( 𝑒 0 𝑎 ) 2

𝜕 2 𝑀

𝐻

𝑏𝑦 𝑖

𝜕𝑥 2 = 𝐽 2 𝑖

𝜕Ψ𝐻

𝑦 𝑖

𝜕𝑥 − 𝛼𝑖 𝐽 4 𝑖

(

𝜕Ψ𝐻

𝑦 𝑖

𝜕𝑥 +

𝜕 2 𝑊

𝐻

𝑖

𝜕𝑥 2

)

,

𝐻

𝑏𝑧 𝑖 − ( 𝑒 0 𝑎 ) 2

𝜕 2 𝑀

𝐻

𝑏𝑧 𝑖

𝜕𝑥 2 = 𝐽 2 𝑖

𝜕Ψ𝐻

𝑧 𝑖

𝜕𝑥 − 𝛼𝑖 𝐽 4 𝑖

(

𝜕Ψ𝐻

𝑧 𝑖

𝜕𝑥 +

𝜕 2 𝑉 𝐻

𝑖

𝜕𝑥 2

)

,

here 𝛼i , 𝜅 i , 𝐼 𝑛 𝑖 , and 𝐽 𝑛 𝑖 are defined by:

𝛼𝑖 =

1 3 𝑟 2

𝑜 𝑖

, 𝜅𝑖 = ∫𝐴 𝑏 𝑖 𝐺 𝑏 𝑖

(1 − 3 𝛼𝑖 𝑧 2 ) d 𝐴 ,

𝑛 𝑖 = ∫𝐴 𝑏 𝑖

𝜌𝑏 𝑖 𝑧 𝑛 d 𝐴 , 𝐽 𝑛 𝑖

= ∫𝐴 𝑏 𝑖 𝐸 𝑏 𝑖

𝑧 𝑛 d 𝐴 ; 𝑛 = 0 , 2 , 4 , 6 .

ow by employing Hamilton’s principle, ∫ 𝑡

0 (𝛿𝑇 𝐻 ( 𝑡 ) − 𝛿𝑈

𝐻 ( 𝑡 ) )d 𝑡 = 0, t

) –( 23 d), the nonlocal equations of motion that display transverse vibra

o a longitudinal heat flux on the basis of the NHOBM are displayed by

582


𝛿𝑗1 ) ⎫ ⎪ ⎬ ⎪ ⎭

6 𝑗 ) 𝜕 3 𝑉 𝐻

𝑗

𝜕𝑥 3 = 0 , (25a)

𝐻

− 𝑗 )

𝜕 2 𝑉 𝐻 𝑗

𝜕𝑥 2

⎫ ⎪ ⎬ ⎪ ⎭ (25b)

𝛿𝑗1 ) ⎫ ⎪ ⎬ ⎪ ⎭

𝐽 6 𝑗 ) 𝜕 3 𝑊

𝐻

𝑗

𝜕𝑥 3 = 0 , (25c)

𝐻

3− 𝑗 )

𝑁 𝑇 𝑗

𝜕 2 𝑊

𝐻 𝑗

𝜕𝑥 2

⎫ ⎪ ⎬ ⎪ ⎭ . (25d)

𝑗 𝐽 4 𝑗 − 𝛼2 𝑗 𝐽 6 𝑗

) 𝜕 3 𝑉 𝐻

2 𝑛 −2+ 𝑗

𝜕𝑥 3 = 0 , (26a)

−2 )

2+ 𝑗

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ +

𝜕 4 𝑉 𝐻

2 𝑛 −2+ 𝑗

𝜕𝑥 4

⎞ ⎟ ⎟ ⎠ = 0 , (26b)

𝑗 𝐽 4 𝑗 − 𝛼2 𝑗 𝐽 6 𝑗

)𝜕 3 𝑊

𝐻

2 𝑛 −2+ 𝑗

𝜕𝑥 3 = 0 , (26c)

𝐻

2 𝑛 −2 )

1 )

𝐻 2 𝑛 −2+ 𝑗 𝑥 2

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ +

𝜕 4 𝑊

𝐻

2 𝑛 −2+ 𝑗

𝜕𝑥 4

⎞ ⎟ ⎟ = 0 . (26d)

• The first thermally affected DWCNTs

Ξ⎧ ⎪ ⎨ ⎪ ⎩ (𝐼 2 𝑗 − 2 𝛼𝑗 𝐼 4 𝑗 + 𝛼2

𝑗 𝐼 6 𝑗 ) 𝜕 2 Ψ𝐻

𝑧 𝑗

𝜕𝑡 2 +

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 𝑉 𝐻

𝑗

𝜕 𝑡 2 𝜕 𝑥 + 𝐾 𝑟 Ψ𝐻

𝑧 𝑗 (1 −

+ 𝜅𝑗

(

Ψ𝐻

𝑧 𝑗 +

𝜕𝑉 𝐻

𝑗

𝜕𝑥

)

−

(𝐽 2 𝑗 − 2 𝛼𝑗 𝐽 4 𝑗 + 𝛼2

𝑗 𝐽 6 𝑗

) 𝜕 2 Ψ𝐻

𝑧 𝑗

𝜕𝑥 2 +

(𝛼𝑗 𝐽 4 𝑗 − 𝛼2

𝑗 𝐽

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝐼 0 𝑗

𝜕 2 𝑉 𝐻 𝑗

𝜕𝑡 2 −

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 Ψ𝐻 𝑧 𝑗

𝜕 𝑡 2 𝜕 𝑥 − 𝛼2

𝑗 𝐼 6 𝑗

𝜕 4 𝑉 𝐻 𝑗

𝜕 𝑡 2 𝜕 𝑥 2 + 𝐶 𝑣𝑦 ( 𝑗 , 3− 𝑗 )

(𝑉 𝐻

𝑗 − 𝑉 3

+ 𝐶 𝑣𝑦 ( 𝑗, 3)

(𝑉 𝐻

𝑗 − 𝑉 𝐻

3 )+ 𝐶 𝑣𝑦 ( 𝑗, 4)

(𝑉 𝐻

𝑗 − 𝑉 𝐻

4 )+ 𝐾 𝑡 𝑉

𝐻

𝑗 (1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

− 𝜅𝑗

⎛ ⎜ ⎜ ⎝ 𝜕Ψ𝐻

𝑧 𝑗

𝜕𝑥 +

𝜕 2 𝑉 𝐻

𝑗

𝜕𝑥 2

⎞ ⎟ ⎟ ⎠ − 𝛼𝑗 𝐽 4 𝑗

𝜕 3 Ψ𝐻

𝑧 𝑗

𝜕𝑥 3 + 𝛼2

𝑗 𝐽 6 𝑗

⎛ ⎜ ⎜ ⎝ 𝜕 3 Ψ𝐻

𝑧 𝑗

𝜕𝑥 3 +

𝜕 4 𝑉 𝐻

𝑗

𝜕𝑥 4

⎞ ⎟ ⎟ ⎠ = 0 ,

Ξ⎧ ⎪ ⎨ ⎪ ⎩ (𝐼 2 𝑗 − 2 𝛼𝑗 𝐼 4 𝑗 + 𝛼2

𝑗 𝐼 6 𝑗 ) 𝜕 2 Ψ𝐻

𝑦 𝑗

𝜕𝑡 2 +

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 𝑊

𝐻

𝑗

𝜕 𝑡 2 𝜕 𝑥 + 𝐾 𝑟 Ψ𝐻

𝑦 𝑗 (1 −

+ 𝜅𝑗

(

Ψ𝐻

𝑦 𝑗 +

𝜕𝑊

𝐻

𝑗

𝜕𝑥

)

−

(𝐽 2 𝑗 − 2 𝛼𝑗 𝐽 4 𝑗 + 𝛼2

𝑗 𝐽 6 𝑗

) 𝜕 2 Ψ𝐻

𝑦 𝑗

𝜕𝑥 2 +

(𝛼𝑗 𝐽 4 𝑗 − 𝛼2

𝑗

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝐼 0 𝑗

𝜕 2 𝑊

𝐻 𝑗

𝜕𝑡 2 −

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 Ψ𝐻 𝑦 𝑗

𝜕 𝑡 2 𝜕 𝑥 − 𝛼2

𝑗 𝐼 6 𝑗

𝜕 4 𝑊

𝐻 𝑗

𝜕 𝑡 2 𝜕 𝑥 2 + 𝐶 𝑣𝑧 ( 𝑗 , 3− 𝑗 )

(𝑊

𝐻

𝑗 − 𝑊

+ 𝐶 𝑣𝑧 ( 𝑗, 3)

(𝑊

𝐻

𝑗 − 𝑊

𝐻

3 )+ 𝐶 𝑣𝑧 ( 𝑗, 4)

(𝑊

𝐻

𝑗 − 𝑊

𝐻

4 )+ 𝐾 𝑡 𝑊

𝐻

𝑗 (1 − 𝛿𝑗1 ) −

− 𝜅𝑗

⎛ ⎜ ⎜ ⎝ 𝜕Ψ𝐻

𝑦 𝑗

𝜕𝑥 +

𝜕 2 𝑊

𝐻

𝑗

𝜕𝑥 2

⎞ ⎟ ⎟ ⎠ − 𝛼𝑗 𝐽 4 𝑗

𝜕 3 Ψ𝐻

𝑦 𝑗

𝜕𝑥 3 + 𝛼2

𝑗 𝐽 6 𝑗

⎛ ⎜ ⎜ ⎝ 𝜕 3 Ψ𝐻

𝑦 𝑗

𝜕𝑥 3 +

𝜕 4 𝑊

𝐻

𝑗

𝜕𝑥 4

⎞ ⎟ ⎟ ⎠ = 0

• The intermediate thermally affected DWCNTs

Ξ⎧ ⎪ ⎨ ⎪ ⎩ (𝐼 2 𝑗 − 2 𝛼𝑗 𝐼 4 𝑗 + 𝛼2

𝑗 𝐼 6 𝑗 ) 𝜕 2 Ψ𝐻

𝑧 2 𝑛 −2+ 𝑗 𝜕𝑡 2

+

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 𝑉 𝐻 (2 𝑛 −2+ 𝑗) 𝜕 𝑡 2 𝜕 𝑥

+ 𝐾 𝑟 Ψ𝐻

𝑧 2 𝑛 −2+ 𝑗 (1 − 𝛿𝑗1 )

⎫ ⎪ ⎬ ⎪ ⎭ + 𝜅𝑗

×

(

Ψ𝐻

𝑧 (2 𝑛 −2+ 𝑗) +

𝜕𝑉 𝐻

2 𝑛 −2+ 𝑗

𝜕𝑥

)

−

(𝐽 2 𝑗 − 2 𝛼𝑗 𝐽 4 𝑗 + 𝛼2

𝑗 𝐽 6 𝑗

) 𝜕 2 Ψ𝐻

𝑧 2 𝑛 −2+ 𝑗

𝜕𝑥 2 +

(𝛼

Ξ

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝐼 0 𝑗

𝜕 2 𝑉 𝐻 (2 𝑛 −2+ 𝑗) 𝜕𝑡 2

−

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 Ψ𝐻 𝑧 2 𝑛 −2+ 𝑗 𝜕 𝑡 2 𝜕 𝑥

− 𝛼2 𝑗 𝐼 6 𝑗

𝜕 4 𝑉 𝐻 2 𝑛 −2+ 𝑗 𝜕 𝑡 2 𝜕 𝑥 2

+ 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗 , 2 𝑛 +1− 𝑗 )

(𝑉 𝐻

2 𝑛 −2+ 𝑗 − 𝑉 𝐻

2 𝑛 +1− 𝑗 )+ 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 −2)

(𝑉 𝐻

2 𝑛 −2+ 𝑗 − 𝑉 𝐻2 𝑛+ 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 −3)

(𝑉 𝐻

2 𝑛 −2+ 𝑗 − 𝑉 𝐻

2 𝑛 −3 )+ 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 +1)

(𝑉 𝐻

2 𝑛 −2+ 𝑗 − 𝑉 𝐻

2 𝑛 +1 )

+ 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 +2)

(𝑉 𝐻

2 𝑛 −2+ 𝑗 − 𝑉 𝐻

2 𝑛 +2 )+ 𝐾 𝑡 𝑉

𝐻

2 𝑛 −2+ 𝑗 (1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

𝜕 2 𝑉 𝐻 2 𝑛 −𝜕𝑥 2

+ 𝜅𝑗

(

𝜕Ψ𝐻 𝑧 2 𝑛 −2+ 𝑗 𝜕𝑥

+

𝜕 2 𝑉 𝐻 2 𝑛 −2+ 𝑗 𝜕𝑥 2

)

− 𝛼𝑗 𝐽 4 𝑗

𝜕 3 Ψ𝐻

𝑧 2 𝑛 −2+ 𝑗

𝜕𝑥 3 + 𝛼2

𝑗 𝐽 6 𝑗

⎛ ⎜ ⎜ ⎝ 𝜕 3 Ψ𝐻

𝑧 2 𝑛 −2+ 𝑗

𝜕𝑥 3

Ξ⎧ ⎪ ⎨ ⎪ ⎩ ( 𝐼 2 𝑗 − 2 𝛼𝑗 𝐼 4 𝑗 + 𝛼2

𝑗 𝐼 6 𝑗 )

𝜕 2 Ψ𝐻 𝑦 2 𝑛 −2+ 𝑗 𝜕𝑡 2

+ ( 𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗 )

𝜕 3 𝑊

𝐻 2 𝑛 −2+ 𝑗

𝜕 𝑡 2 𝜕 𝑥 + 𝐾 𝑟 Ψ𝐻

𝑦 2 𝑛 −2+ 𝑗 (1 − 𝛿𝑗1 )

⎫ ⎪ ⎬ ⎪ ⎭ + 𝜅𝑗

×

(

Ψ𝐻

𝑦 (2 𝑛 −2+ 𝑗) +

𝜕𝑊

𝐻

2 𝑛 −2+ 𝑗

𝜕𝑥

)

− ( 𝐽 2 𝑗 − 2 𝛼𝑗 𝐽 4 𝑗 + 𝛼2 𝑗 𝐽 6 𝑗 )

𝜕 2 Ψ𝐻

𝑦 2 𝑛 −2+ 𝑗

𝜕𝑥 2 +

(𝛼

Ξ

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝐼 0 𝑗

𝜕 2 𝑊

𝐻 2 𝑛 −2+ 𝑗 𝜕𝑡 2

− ( 𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗 )

𝜕 3 Ψ𝐻 𝑦 2 𝑛 −2+ 𝑗 𝜕 𝑡 2 𝜕 𝑥

− 𝛼2 𝑗 𝐼 6 𝑗

𝜕 4 𝑊

𝐻 2 𝑛 −2+ 𝑗

𝜕 𝑡 2 𝜕 𝑥 2

+ 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗 , 2 𝑛 +1− 𝑗 ) ( 𝑊

𝐻

2 𝑛 −2+ 𝑗 − 𝑊

𝐻

2 𝑛 +1− 𝑗 ) + 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑁−2) ( 𝑊

𝐻

2 𝑛 −2+ 𝑗 − 𝑊

+ 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 −3) ( 𝑊

𝐻

2 𝑛 −2+ 𝑗 − 𝑊

𝐻

2 𝑛 −3 ) + 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 +1) ( 𝑊

𝐻

2 𝑛 −2+ 𝑗 − 𝑊

𝐻

2 𝑛 +

+ 𝐶 𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 +2) ( 𝑊

𝐻

2 𝑛 −2+ 𝑗 − 𝑊

𝐻

2 𝑛 +2 ) + 𝐾 𝑡 𝑊

𝐻

2 𝑛 −2+ 𝑗 (1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

𝜕 2 𝑊

𝜕

− 𝜅𝑗

(

𝜕Ψ𝐻 𝑦 2 𝑛 −2+ 𝑗 𝜕𝑥

+

𝜕 2 𝑊

𝐻 2 𝑛 −2+ 𝑗 𝜕𝑥 2

)


𝜕 3 Ψ𝐻

𝑦 2 𝑛 −2+ 𝑗

𝜕𝑥 3 + 𝛼2

𝑗 𝐽 6 𝑗

⎛ ⎜ ⎜ 𝜕 3 Ψ𝐻

𝑦 2 𝑛 −2+ 𝑗

𝜕𝑥 3
⎝ ⎠ 583


𝑗 𝐽 4 𝑗 − 𝛼2 𝑗 𝐽 6 𝑗

) 𝜕 3 𝑉 𝐻

2 𝑁−2+ 𝑗

𝜕𝑥 3 = 0 , (27a)

− 𝑉 𝐻

2 𝑁−2 )

𝑉 𝐻 2 𝑁−2+ 𝑗 𝜕𝑥 2

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + 𝑗 +

𝜕 4 𝑉 𝐻

2 𝑁−2+ 𝑗

𝜕𝑥 4

⎞ ⎟ ⎟ ⎠ = 0 , (27b)

𝑗

(𝛼𝑗 𝐽 4 𝑗 − 𝛼2

𝑗 𝐽 6 𝑗

) 𝜕 3 𝑊

𝐻

(2 𝑁−2+ 𝑗)

𝜕𝑥 3 = 0 , (27c)

+ 𝑗 − 𝑊

𝐻

2 𝑁−2 )

𝜕 2 𝑊

𝐻 2 𝑁−2+ 𝑗 𝜕𝑥 2

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ + 𝑗 +

𝜕 4 𝑊

𝐻

2 𝑁−2+ 𝑗

𝜕𝑥 4

⎞ ⎟ ⎟ ⎠ = 0 . (27d)

𝑖 , (28)

ith the thermally affected membranes of vertically aligned DWCNTs in the

𝜕 𝑉 𝐻

𝑗

𝜕𝜉

⎞ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 8 𝛾2 8

𝜕 2 Ψ𝐻

𝑧 𝑗

𝜕𝜉2 + 𝜗

2 𝑗−2 5 𝛾2 9

𝜕 3 𝑉 𝐻

𝑗

𝜕𝜉3 = 0 , (29a)

)

𝐻

𝑇 𝑗

𝜕 2 𝑉 𝐻

𝑗

𝜕𝜉2

⎫ ⎪ ⎬ ⎪ ⎭ (29b)

(29b)

𝜕 𝑊

𝐻

𝑗

𝜕𝜉

⎞ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 8 𝛾2 8

𝜕 2 Ψ𝐻

𝑦 𝑗

𝜕𝜉2 + 𝜗

2 𝑗−2 5 𝛾2 9

𝜕 3 𝑊

𝐻

𝑗

𝜕𝜉3 = 0 , (29c)

− 𝑗 ) 𝑁

𝐻

𝑇 𝑗

𝜕 2 𝑊

𝐻

𝑗

𝜕𝜉2

⎫ ⎪ ⎬ ⎪ ⎭ (29d)

• The N th thermally affected DWCNTs

Ξ⎧ ⎪ ⎨ ⎪ ⎩ (𝐼 2 𝑗 − 2 𝛼𝑗 𝐼 4 𝑗 + 𝛼2

𝑗 𝐼 6 𝑗 ) 𝜕 2 Ψ𝐻

𝑧 2 𝑁−2+ 𝑗 𝜕𝑡 2

+

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 𝑉 𝐻 2 𝑁−2+ 𝑗 𝜕 𝑡 2 𝜕 𝑥

+ 𝐾 𝑟 Ψ𝐻

𝑧 2 𝑁−2+ 𝑗 (1 − 𝛿𝑗1 )

⎫ ⎪ ⎬ ⎪ ⎭ + 𝜅𝑗

×

(

Ψ𝐻

𝑧 2 𝑁−2+ 𝑗 +

𝜕𝑉 𝐻

2 𝑁−2+ 𝑗

𝜕𝑥

)

−

(𝐽 2 𝑗 − 2 𝛼𝑗 𝐽 4 𝑗 + 𝛼2

𝑗 𝐽 6 𝑗

) 𝜕 2 Ψ𝐻

𝑧 2 𝑁−2+ 𝑗

𝜕𝑥 2 +

(𝛼

Ξ

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝐼 0 𝑗

𝜕 2 𝑉 𝐻 2 𝑁−2+ 𝑗 𝜕𝑡 2

−

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 Ψ𝐻 𝑧 2 𝑁−2+ 𝑗 𝜕 𝑡 2 𝜕 𝑥

− 𝛼2 𝑗 𝐼 6 𝑗

𝜕 4 𝑉 𝐻 2 𝑁−2+ 𝑗 𝜕 𝑡 2 𝜕 𝑥 2

+ 𝐶 𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 +1− 𝑗)

(𝑉 𝐻

2 𝑁−2+ 𝑗 − 𝑉 𝐻

2 𝑁+1− 𝑗 )+ 𝐶 𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 −2)

(𝑉 𝐻

2 𝑁−2+ 𝑗

+ 𝐶 𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 −3)

(𝑉 𝐻

2 𝑁−2+ 𝑗 − 𝑉 𝐻

2 𝑁−3 )+ 𝐾 𝑡 𝑉

𝐻

2 𝑁−2+ 𝑗 (1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

𝜕 2

− 𝜅𝑗

(

𝜕Ψ𝐻 𝑧 2 𝑁−2+ 𝑗 𝜕𝑥

+

𝜕 2 𝑉 𝐻 2 𝑁−2+ 𝑗 𝜕𝑥 2

)


𝜕 3 Ψ𝐻

𝑧 2 𝑁−2+ 𝑗

𝜕𝑥 3 + 𝛼2

𝑗 𝐽 6 𝑗

⎛ ⎜ ⎜ ⎝ 𝜕 3 Ψ𝐻

𝑧 2 𝑁−2

𝜕𝑥 3

Ξ⎧ ⎪ ⎨ ⎪ ⎩ (𝐼 2 𝑗 − 2 𝛼𝑗 𝐼 4 𝑗 + 𝛼2

𝑗 𝐼 6 𝑗 ) 𝜕 2 Ψ𝐻

𝑦 2 𝑁−2+ 𝑗 𝜕𝑡 2

+

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 𝑊

𝐻 2 𝑁−2+ 𝑗 𝜕 𝑡 2 𝜕 𝑥

+ 𝐾 𝑟 Ψ𝐻

𝑦 2 𝑁−2+ 𝑗 (1 − 𝛿𝑗1 )

⎫ ⎪ ⎬ ⎪ ⎭ + 𝜅

×

(

Ψ𝐻

𝑦 2 𝑁−2+ 𝑗 +

𝜕𝑊

𝐻

2 𝑁−2+ 𝑗

𝜕𝑥

)

−

(𝐽 2 𝑗 − 2 𝛼𝑗 𝐽 4 𝑗 + 𝛼2

𝑗 𝐽 6 𝑗

) 𝜕 2 Ψ𝐻

𝑦 2 𝑁−2+ 𝑗

𝜕𝑥 2 +

Ξ

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝐼 0 𝑗

𝜕 2 𝑊

𝐻 2 𝑁−2+ 𝑗 𝜕𝑡 2

−

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 Ψ𝐻 𝑦 2 𝑁−2+ 𝑗 𝜕 𝑡 2 𝜕 𝑥

− 𝛼2 𝑗 𝐼 6 𝑗

𝜕 4 𝑊

𝐻 2 𝑁−2+ 𝑗

𝜕 𝑡 2 𝜕 𝑥 2

+ 𝐶 𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 +1− 𝑗)

(𝑊

𝐻

2 𝑁−2+ 𝑗 − 𝑊

𝐻

2 𝑁+1− 𝑗 )+ 𝐶 𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 −2)

(𝑊

𝐻

2 𝑁−2

+ 𝐶 𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 −3)

(𝑊

𝐻

2 𝑁−2+ 𝑗 − 𝑊

𝐻

2 𝑁−3 )+ 𝐾 𝑡 𝑊

𝐻

2 𝑁−2+ 𝑗 (1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

− 𝜅𝑗

(

𝜕Ψ𝐻 𝑦 2 𝑁−2+ 𝑗 𝜕𝑥

+

𝜕 2 𝑊

𝐻 2 𝑁−2+ 𝑗 𝜕𝑥 2

)


𝜕 3 Ψ𝐻

𝑦 2 𝑁−2+ 𝑗

𝜕𝑥 3 + 𝛼2

𝑗 𝐽 6 𝑗

⎛ ⎜ ⎜ ⎝ 𝜕 3 Ψ𝐻

𝑦 2 𝑁−2

𝜕𝑥 3

Now by considering the following dimensionless quantities:

𝑉 𝐻

𝑖 =

𝑉 𝐻

𝑖

𝑙 𝑏 , 𝑊

𝐻

𝑖 =

𝑊

𝐻

𝑖

𝑙 𝑏 , 𝜏 =

𝛼1 𝑡

𝑙 2 𝑏

√

𝐽 6 1 𝐼 0 1

, Ψ𝐻

𝑦 𝑖 = Ψ𝐻

𝑦 𝑖 , Ψ

𝐻

𝑧 𝑖 = Ψ𝐻

𝑧

the dimensionless NHOBM-based governing equations associated w

presence of longitudinal heat flux would be derived as: • The first thermally affected DWCNTs

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝜗

2 𝑗−2 7

𝜕 2 Ψ𝐻

𝑧 𝑗

𝜕𝜏2 − 𝜗

2 𝑗−2 2 𝛾2 6

𝜕 3 𝑉 𝐻

𝑗

𝜕 𝜏2 𝜕 𝜉+ 𝐾

𝐻

𝑟 Ψ𝐻

𝑧 𝑗 (1 − 𝛿𝑗1 )

⎫ ⎪ ⎬ ⎪ ⎭ + 𝜗 2 𝑗−2 4 𝛾2 7

⎛ ⎜ ⎜ ⎝ Ψ𝐻

𝑧 𝑗 +

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝜗 2 𝑗−2 1

𝜕 2 𝑉 𝐻

𝑗

𝜕𝜏2 + 𝜗

2 𝑗−2 2 𝛾2 1

𝜕 3 Ψ𝐻

𝑧 𝑗

𝜕 𝜏2 𝜕 𝜉− 𝜗

2 𝑗−2 3 𝛾2 2

𝜕 4 𝑉 𝐻

𝑗

𝜕 𝜏2 𝜕 𝜉2 + 𝐶

𝐻

𝑣𝑦 ( 𝑗 , 3− 𝑗 )

(𝑉 𝐻

𝑗 − 𝑉

𝐻

3− 𝑗

+ 𝐶

𝐻

𝑣𝑦 ( 𝑗, 3)

(𝑉 𝐻

𝑗 − 𝑉

𝐻

3 )+ 𝐶

𝐻

𝑣𝑦 ( 𝑗, 4)

(𝑉 𝐻

𝑗 − 𝑉

𝐻

4 )+ 𝐾

𝐻

𝑡 𝑉 𝐻

𝑗 (1 − 𝛿𝑗1 ) − 𝑁

− 𝜗 2 𝑗−2 4 𝛾2 3

⎛ ⎜ ⎜ ⎜ ⎝ 𝜕 Ψ

𝐻

𝑧 𝑗

𝜕𝜉+

𝜕 2 𝑉 𝐻

𝑗

𝜕𝜉2

⎞ ⎟ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 5 𝛾2 4

𝜕 3 Ψ𝐻

𝑧 𝑗

𝜕𝜉3 + 𝜗

2 𝑗−2 6

𝜕 4 𝑉 𝐻

𝑗

𝜕𝜉4 = 0 ,

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝜗

2 𝑗−2 7

𝜕 2 Ψ𝐻

𝑦 𝑗

𝜕𝜏2 − 𝜗

2 𝑗−2 2 𝛾2 6

𝜕 3 𝑊

𝐻

𝑗

𝜕 𝜏2 𝜕 𝜉+ 𝐾

𝐻

𝑟 Ψ𝐻

𝑦 𝑗 (1 − 𝛿𝑗1 )

⎫ ⎪ ⎬ ⎪ ⎭ + 𝜗 2 𝑗−2 4 𝛾2 7

⎛ ⎜ ⎜ ⎝ Ψ𝐻

𝑦 𝑗 +

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝜗 2 𝑗−2 1

𝜕 2 𝑊

𝐻

𝑗

𝜕𝜏2 + 𝜗

2 𝑗−2 2 𝛾2 1

𝜕 3 Ψ𝐻

𝑦 𝑗

𝜕 𝜏2 𝜕 𝜉− 𝜗

2 𝑗−2 3 𝛾2 2

𝜕 4 𝑊

𝐻

𝑗

𝜕 𝜏2 𝜕 𝜉2 + 𝐶

𝐻

𝑣𝑧 ( 𝑗 , 3− 𝑗 )

(𝑊

𝐻

𝑗 − 𝑊

𝐻

3

+ 𝐶

𝐻

𝑣𝑧 ( 𝑗, 3)

(𝑊

𝐻

𝑗 − 𝑊

𝐻

3 )+ 𝐶

𝐻

𝑣𝑧 ( 𝑗, 4)

(𝑊

𝐻

𝑗 − 𝑊

𝐻

4 )+ 𝐾

𝐻

𝑡 𝑊

𝐻

𝑗 (1 − 𝛿𝑗1 ) −

− 𝜗 2 𝑗−2 4 𝛾2 3

⎛ ⎜ ⎜ ⎜ 𝜕 Ψ

𝐻

𝑦 𝑗

𝜕𝜉+

𝜕 2 𝑊

𝐻

𝑗

𝜕𝜉2

⎞ ⎟ ⎟ ⎟ − 𝜗 2 𝑗−2 5 𝛾2 4

𝜕 3 Ψ𝐻

𝑦 𝑗

𝜕𝜉3 + 𝜗

2 𝑗−2 6

𝜕 4 𝑊

𝐻

𝑗

𝜕𝜉4 = 0 .
⎝ ⎠
584


𝑗−2

𝛾2 7

0 , (30a)

𝐻

2 𝑛 −2 )

1 )

𝐻

2 𝑛 −2+ 𝑗 𝜕𝜉2

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

𝐻

2 𝑛 −2+ 𝑗

𝜕𝜉4 = 0 , (30b)

2 𝛾2 7

= 0 , (30c)

𝑊

𝐻

2 𝑛 −2 )

𝐻

2 𝑛 +1 )

𝜕 2 𝑊

𝐻

2 𝑛 −2+ 𝑗 𝜕𝜉2

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ 𝑊

𝐻

2 𝑛 −2+ 𝑗

𝜕𝜉4 = 0 . (30d)

𝜗 2 𝑗−2 4 𝛾2 7

+ 𝑗 = 0 , (31a)

𝑗 − 𝑉

𝐻

2 𝑁−2 )

𝑗

𝜕 2 𝑉 𝐻

2 𝑁−2+ 𝑗 𝜕𝜉2

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

4 𝑉 𝐻

2 𝑁−2+ 𝑗

𝜕𝜉4 = 0 , (31b)

2 𝑗−2 4 𝛾2 7

2+ 𝑗 = 0 , (31c)

2+ 𝑗, 2 𝑁 +1− 𝑗)

𝐻

𝑇 𝑗

𝜕 2 𝑊

𝐻

2 𝑁−2+ 𝑗 𝜕𝜉2

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

• The intermediate thermally affected DWCNTs

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝜗

2 𝑗−2 7

𝜕 2 Ψ𝐻

𝑧 2 𝑛 −2+ 𝑗

𝜕𝜏2 − 𝜗

2 𝑗−2 2 𝛾2 6

𝜕 3 𝑉 𝐻

2 𝑛 −2+ 𝑗

𝜕 𝜏2 𝜕 𝜉+ 𝐾

𝐻

𝑟 Ψ𝐻

𝑧 2 𝑛 −2+ 𝑗 (1 − 𝛿𝑗1 )

⎫ ⎪ ⎬ ⎪ ⎭ + 𝜗 24

×(

Ψ𝐻

𝑧 2 𝑛 −2+ 𝑗 +

𝜕 𝑉 𝐻

2 𝑛 −2+ 𝑗 𝜕𝜉

)

− 𝜗 2 𝑗−2 8 𝛾2 8

𝜕 2 Ψ𝐻

𝑧 2 𝑛 −2+ 𝑗

𝜕𝜉2 + 𝜗

2 𝑗−2 5 𝛾2 9

𝜕 3 𝑉 𝐻

2 𝑛 −2+ 𝑗

𝜕𝜉3 =

Ξ

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝜗 2 𝑗−2 1

𝜕 2 𝑉 𝐻

2 𝑛 −2+ 𝑗 𝜕𝜏2

+ 𝜗 2 𝑗−2 2 𝛾2 1

𝜕 3 Ψ𝐻

𝑧 2 𝑛 −2+ 𝑗 𝜕 𝜏2 𝜕 𝜉

− 𝜗 2 𝑗−2 3 𝛾2 2

𝜕 4 𝑉 𝐻

2 𝑛 −2+ 𝑗 𝜕 𝜏2 𝜕 𝜉2

+ 𝐶

𝐻

𝑣𝑦 (2 𝑛 −2+ 𝑗 , 2 𝑛 +1− 𝑗 )

(𝑉 𝐻

2 𝑛 −2+ 𝑗 − 𝑉 𝐻

2 𝑛 +1− 𝑗 )+ 𝐶

𝐻

𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 −2)

(𝑉 𝐻

2 𝑛 −2+ 𝑗 − 𝑉

+ 𝐶

𝐻

𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 −3)

(𝑉 𝐻

2 𝑛 −2+ 𝑗 − 𝑉 𝐻

2 𝑛 −3 )+ 𝐶

𝐻

𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 +1)

(𝑉 𝐻

2 𝑛 −2+ 𝑗 − 𝑉 𝐻

2 𝑛 +

+ 𝐶

𝐻

𝑣𝑦 (2 𝑛 −2+ 𝑗, 2 𝑛 +2)

(𝑉 𝐻

2 𝑛 −2+ 𝑗 − 𝑉 𝐻

2 𝑛 +2 )+ 𝐾

𝐻

𝑡 𝑉 𝐻

2 𝑛 −2+ 𝑗 (1 − 𝛿𝑗1 ) − 𝑁

𝐻

𝑇 𝑗

𝜕 2 𝑉

− 𝜗 2 𝑗−2 4 𝛾2 3

⎛ ⎜ ⎜ ⎜ ⎝ 𝜕 Ψ

𝐻

𝑧 2 𝑛 −2+ 𝑗

𝜕𝜉+

𝜕 2 𝑉 𝐻

2 𝑛 −2+ 𝑗

𝜕𝜉2

⎞ ⎟ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 5 𝛾2 4

𝜕 3 Ψ𝐻

𝑧 2 𝑛 −2+ 𝑗

𝜕𝜉3 + 𝜗

2 𝑗−2 6

𝜕 4 𝑉

Ξ{

𝜗 2 𝑗−2 7

𝜕 2 Ψ𝐻

𝑦 2 𝑛 −2+ 𝑗 𝜕𝜏2

− 𝜗 2 𝑗−2 2 𝛾2 6

𝜕 3 𝑊

𝐻

2 𝑛 −2+ 𝑗 𝜕 𝜏2 𝜕 𝜉

+ 𝐾

𝐻

𝑟 Ψ𝐻

𝑦 2 𝑛 −2+ 𝑗 (1 − 𝛿𝑗1 )

}

+ 𝜗 2 𝑗−4

×⎛ ⎜ ⎜ ⎝ Ψ

𝐻

𝑦 2 𝑛 −2+ 𝑗 +

𝜕 𝑊

𝐻

2 𝑛 −2+ 𝑗

𝜕𝜉

⎞ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 8 𝛾2 8

𝜕 2 Ψ𝐻

𝑦 2 𝑛 −2+ 𝑗

𝜕𝜉2 + 𝜗

2 𝑗−2 5 𝛾2 9

𝜕 3 𝑊

𝐻

2 𝑛 −2+ 𝑗

𝜕𝜉3

Ξ

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝜗 2 𝑗−2 1

𝜕 2 𝑊

𝐻

2 𝑛 −2+ 𝑗 𝜕𝜏2

+ 𝜗 2 𝑗−2 2 𝛾2 1

𝜕 3 Ψ𝐻

𝑦 2 𝑛 −2+ 𝑗 𝜕 𝜏2 𝜕 𝜉

− 𝜗 2 𝑗−2 3 𝛾2 2

𝜕 4 𝑊

𝐻

2 𝑛 −2+ 𝑗 𝜕 𝜏2 𝜕 𝜉2

+ 𝐶

𝐻

𝑣𝑧 (2 𝑛 −2+ 𝑗 , 2 𝑛 +1− 𝑗 )

(𝑊

𝐻

2 𝑛 −2+ 𝑗 − 𝑊

𝐻

2 𝑛 +1− 𝑗 )+ 𝐶

𝐻

𝑣𝑧 (2 𝑛 −2+ 𝑗, 2 𝑛 −2)

(𝑊

𝐻

2 𝑛 −2+ 𝑗 −

+ 𝐶

𝐻

𝑣𝑧 (2 𝑛 −2+ 𝑗, 2 𝑛 −3)

(𝑊

𝐻

2 𝑛 −2+ 𝑗 − 𝑊

𝐻

2 𝑛 −3 )+ 𝐶

𝐻

𝑣𝑧 (2 𝑛 −2+ 𝑗, 2 𝑛 +1)

(𝑊

𝐻

2 𝑛 −2+ 𝑗 − 𝑊

+ 𝐶

𝐻

𝑣𝑧 (2 𝑛 −2+ 𝑗, 2 𝑛 +2)

(𝑊

𝐻

2 𝑛 −2+ 𝑗 − 𝑊

𝐻

2 𝑛 +2 )+ 𝐾

𝐻

𝑡 𝑊

𝐻

2 𝑛 −2+ 𝑗 (1 − 𝛿𝑗1 ) − 𝑁

𝐻

𝑇 𝑗

− 𝜗 2 𝑗−2 4 𝛾2 3

⎛ ⎜ ⎜ ⎜ ⎝ 𝜕 Ψ

𝐻

𝑦 2 𝑛 −2+ 𝑗

𝜕𝜉+

𝜕 2 𝑊

𝐻

2 𝑛 −2+ 𝑗

𝜕𝜉2

⎞ ⎟ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 5 𝛾2 4

𝜕 3 Ψ𝐻

𝑦 2 𝑛 −2+ 𝑗

𝜕𝜉3 + 𝜗

2 𝑗−2 6

𝜕 4

• The N th thermally affected DWCNTs

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝜗

2 𝑗−2 7

𝜕 2 Ψ𝐻

𝑧 2 𝑁−2+ 𝑗

𝜕𝜏2 − 𝜗

2 𝑗−2 2 𝛾2 6

𝜕 3 𝑉 𝐻

2 𝑁−2+ 𝑗

𝜕 𝜏2 𝜕 𝜉+ 𝐾

𝐻

𝑟 Ψ𝐻

𝑧 2 𝑁−2+ 𝑗 (1 − 𝛿𝑗1 )

⎫ ⎪ ⎬ ⎪ ⎭ +

×⎛ ⎜ ⎜ ⎝ Ψ

𝐻

𝑧 2 𝑁−2+ 𝑗 +

𝜕 𝑉 𝐻

2 𝑁−2+ 𝑗

𝜕𝜉

⎞ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 8 𝛾2 8

𝜕 2 Ψ𝐻

𝑧 (2 𝑁−2+ 𝑗)

𝜕𝜉2 + 𝜗

2 𝑗−2 5 𝛾2 9

𝜕 3 𝑉 𝐻

2 𝑁−2

𝜕𝜉3

Ξ

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝜗 2 𝑗−2 1

𝜕 2 𝑉 𝐻

2 𝑁−2+ 𝑗 𝜕𝜏2

+ 𝜗 2 𝑗−2 2 𝛾2 1

𝜕 3 Ψ𝐻

𝑧 2 𝑁−2+ 𝑗 𝜕 𝜏2 𝜕 𝜉

− 𝜗 2 𝑗−2 3 𝛾2 2

𝜕 4 𝑉 𝐻

2 𝑁−2+ 𝑗 𝜕 𝜏2 𝜕 𝜉2

+ 𝐶

𝐻

𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 +1− 𝑗)

(𝑉 𝐻

2 𝑁−2+ 𝑗 − 𝑉 𝐻

2 𝑁+1− 𝑗 )+ 𝐶

𝐻

𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 −2)

(𝑉 𝐻

2 𝑁−2+

+ 𝐶

𝐻

𝑣𝑦 (2 𝑁 −2+ 𝑗, 2 𝑁 −3)

(𝑉 𝐻

2 𝑁−2+ 𝑗 − 𝑉 𝐻

2 𝑁−3 )+ 𝐾

𝐻

𝑡 𝑉 𝐻

2 𝑁−2+ 𝑗 (1 − 𝛿𝑗1 ) − 𝑁

𝐻

𝑇

− 𝜗 2 𝑗−2 4 𝛾2 3

⎛ ⎜ ⎜ ⎜ ⎝ 𝜕 Ψ

𝐻

𝑧 2 𝑁−2+ 𝑗

𝜕𝜉+

𝜕 2 𝑉 𝐻

2 𝑁−2+ 𝑗

𝜕𝜉2

⎞ ⎟ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 5 𝛾2 4

𝜕 3 Ψ𝐻

𝑧 2 𝑁−2+ 𝑗

𝜕𝜉3 + 𝜗

2 𝑗−2 6

𝜕

Ξ{

𝜗 2 𝑗−2 7

𝜕 2 Ψ𝐻

𝑦 2 𝑁−2+ 𝑗 𝜕𝜏2

− 𝜗 2 𝑗−2 2 𝛾2 6

𝜕 3 𝑊

𝐻

2 𝑁−2+ 𝑗 𝜕 𝜏2 𝜕 𝜉

+ 𝐾

𝐻

𝑟 Ψ𝐻

𝑦 2 𝑁−2+ 𝑗 (1 − 𝛿𝑗1 )

}

+ 𝜗

×⎛ ⎜ ⎜ ⎝ Ψ

𝐻

𝑦 2 𝑁−2+ 𝑗 +

𝜕 𝑊

𝐻

2 𝑁−2+ 𝑗

𝜕𝜉

⎞ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 8 𝛾2 8

𝜕 2 Ψ𝐻

𝑦 2 𝑁−2+ 𝑗

𝜕𝜉2 + 𝜗

2 𝑗−2 5 𝛾2 9

𝜕 3 𝑊

𝐻

2 𝑁−

𝜕𝜉3

Ξ

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 𝜗 2 𝑗−2 1

𝜕 2 𝑊

𝐻

2 𝑁−2+ 𝑗 𝜕𝜏2

+ 𝜗 2 𝑗−2 2 𝛾2 1

𝜕 3 Ψ𝐻

𝑦 2 𝑁−2+ 𝑗 𝜕 𝜏2 𝜕 𝜉

− 𝜗 2 𝑗−2 3 𝛾2 2

𝜕 4 𝑊

𝐻

2 𝑁−2+ 𝑗 𝜕 𝜏2 𝜕 𝜉2

+ 𝐶

𝐻

𝑣𝑧 (2 𝑁 −

×(𝑊

𝐻

2 𝑁−2+ 𝑗 − 𝑊

𝐻

2 𝑁+1− 𝑗 )+ 𝐶

𝐻

𝑣𝑧 (2 𝑁 −2+ 𝑗, 2 𝑁 −2)

(𝑊

𝐻

2 𝑁−2+ 𝑗 − 𝑊

𝐻

2 𝑁−2 )

+ 𝐶

𝐻

𝑣𝑧 (2 𝑁 −2+ 𝑗, 2 𝑁 −3)

(𝑊

𝐻

2 𝑁−2+ 𝑗 − 𝑊

𝐻

2 𝑁−3 )+ 𝐾

𝐻

𝑡 𝑊

𝐻

2 𝑁−2+ 𝑗 (1 − 𝛿𝑗1 ) − 𝑁

585


𝐻

2 𝑁−2+ 𝑗

𝜕𝜉4 = 0 , (31d)

𝑙 2 𝑏

6 1

,

𝐶 𝑣𝑧 ( 𝑚,𝑛 ) 𝑙 4 𝑏

𝛼2 1 𝐽 6 1

,

𝐻

𝑇 𝑗 =

𝑁 𝑇 𝑗 𝑙 2 𝑏

𝛼2 1 𝐽 6 1

,

6 2

6 1

,

(32)

5

OBM, it is assumed that all constitutive tubes have simple ends and the first

a 𝑉 𝐻

2 ( 𝜉, 𝜏) = 𝑊

𝐻

2 ( 𝜉, 𝜏)= 0, 𝑉 𝐻

2 𝑁

( 𝜉, 𝜏) = 𝑊

𝐻

2 𝑁

( 𝜉, 𝜏)= 0, Ψ𝐻

𝑧 2 ( 𝜉, 𝜏) = Ψ

𝐻

𝑦 2 ( 𝜉, 𝜏)= 0,

Ψ following admissible form for deformation fields of nanotubes:

𝑉 (33a)

Ψ (33b)

𝑊 (33c)

Ψ (33d)

w parameters associated with deflections and angles of deflections. Now by

s e can arrive at the following set of linear second-order ordinary differential

e

𝐌 (34)

w

𝐱 −1) 𝑚

} T

, (35)

i

are provided in supplementary material, part C. Finally, by following the

g affected membranes of vertically aligned DWCNTs are easily determined.

6 NTs membranes

s and stiffness matrices for discrete models based on the NRBM or NHOBM

w f the nanosystem. It implies that the evaluation of natural frequencies for

h ational efforts. In order to reduce these expenditures, herein, we develop

s nd, let us to define continuous deformation fields of the nanosystem such

t he deformation fields of the discrete models for the tube of concern (i.e.,

− 𝜗 2 𝑗−2 4 𝛾2 3

⎛ ⎜ ⎜ ⎜ ⎝ 𝜕 Ψ

𝐻

𝑦 2 𝑁−2+ 𝑗

𝜕𝜉+

𝜕 2 𝑊

𝐻

2 𝑁−2+ 𝑗

𝜕𝜉2

⎞ ⎟ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 5 𝛾2 4

𝜕 3 Ψ𝐻

𝑦 2 𝑁−2+ 𝑗

𝜕𝜉3 + 𝜗

2 𝑗−2 6

𝜕 4 𝑊

in which

𝛾2 7 =

𝜅1 𝐼 0 1 𝑙 4 𝑏 (

𝐼 2 1 − 2 𝛼1 𝐼 4 1 + 𝛼2 1 𝐼 6 1 )𝛼2 1 𝐽 6 1

, 𝛾2 8 =

(𝐽 2 1 − 2 𝛼1 𝐽 4 1 + 𝛼2 1 𝐽 6 1

)𝐼 0 1 (

𝐼 2 1 − 2 𝛼1 𝐼 4 1 + 𝛼2 1 𝐼 6 1 )𝛼2 1 𝐽

𝛾2 9 =

(𝛼1 𝐽 4 1 − 𝛼2 1 𝐽 6 1

)𝐼 0 1 𝑙

2 𝑏 (

𝐼 2 1 − 2 𝛼1 𝐼 4 1 + 𝛼2 1 𝐼 6 1 )𝛼2 1 𝐽 6 1

, 𝐶

𝐻

𝑣𝑦 ( 𝑚,𝑛 ) =

𝐶 𝑣𝑦 ( 𝑚,𝑛 ) 𝑙 4 𝑏

𝛼2 1 𝐽 6 1

, 𝐶

𝐻

𝑣𝑧 ( 𝑚,𝑛 ) =

𝐾

𝐻

𝑡 =

𝐾 𝑡 𝑙 4 𝑏

𝛼2 1 𝐽 6 1

, 𝐾

𝐻

𝑟 =

𝐾 𝑟 𝐼 0 1 𝑙 4 𝑏

𝛼2 1 𝐽 6 1 (𝐼 2 1 − 2 𝛼1 𝐼 4 1 + 𝛼2 1 𝐼 6 1

) , 𝜗 2 1 =

𝐼 0 2 𝐼 0 1

, 𝑁

𝜗 2 2 =

𝛼2 𝐼 4 2 − 𝛼2 2 𝐼 6 2

𝛼1 𝐼 4 1 − 𝛼2 1 𝐼 6 1

, 𝜗 2 3 =

𝛼2 2 𝐼 6 2

𝛼2 1 𝐼 6 1

, 𝜗 2 4 =

𝜅2 𝜅1

, 𝜗 2 5 =

𝛼2 𝐽 4 2 − 𝛼2 2 𝐽 6 2

𝛼1 𝐽 4 1 − 𝛼2 1 𝐽 6 1

,

𝜗 2 6 =

𝛼2 2 𝐽 6 2

𝛼2 1 𝐽 6 1

, 𝜗 2 7 =

𝐼 2 2 − 2 𝛼2 𝐼 4 2 + 𝛼2 2 𝐼 6 2

𝐼 2 1 − 2 𝛼1 𝐼 4 1 + 𝛼2 1 𝐼 6 1

, 𝜗 2 8 =

𝐽 2 2 − 2 𝛼2 𝐽 4 2 + 𝛼2 2 𝐽

𝐽 2 1 − 2 𝛼1 𝐽 4 1 + 𝛼2 1 𝐽

𝛾2 1 =

𝛼1 𝐽 4 1 − 𝛼2 1 𝐽 6 1

𝐼 0 1 𝑙 2 𝑏

, 𝛾2 2 =

𝛼2 1 𝐼 6 1

𝐼 0 1 𝑙 2 𝑏

, 𝛾2 3 =

𝜅1 𝑙 2 𝑏

𝛼2 1 𝐽 6 1

,

𝛾2 4 =

𝛼1 𝐽 4 1 − 𝛼2 1 𝐽 6 1

𝛼2 1 𝐽 6 1

, 𝛾2 6 =

𝛼1 𝐼 4 1 − 𝛼2 1 𝐼 6 1

𝐼 2 1 − 2 𝛼1 𝐼 4 1 + 𝛼2 1 𝐼 6 1

.

.2. Evaluation of natural frequencies

For the thermally affected nanosystem modeled according to the NH

nd the last DWCNTs are prevented from any transverse motion (i.e.,𝐻

𝑧 2 𝑁 ( 𝜉, 𝜏) = Ψ

𝐻

𝑦 2 𝑁 ( 𝜉, 𝜏)= 0). By implementing the AMM, we consider the

𝐻

𝑖 ( 𝜉, 𝜏) =

𝑁𝑀 ∑𝑚 =1

𝑉 𝐻


𝐻

𝑧 𝑖 ( 𝜉, 𝜏) =

𝑁𝑀 ∑𝑚 =1

Ψ𝐻

𝑧 𝑖𝑚 ( 𝜏) cos ( 𝑚𝜋𝜉) ,

𝐻

𝑖 ( 𝜉, 𝜏) =

𝑁𝑀 ∑𝑚 =1

𝑊

𝐻


𝐻

𝑦 𝑖 ( 𝜉, 𝜏) =

𝑁𝑀 ∑𝑚 =1

Ψ𝐻

𝑦 𝑖𝑚 ( 𝜏) cos ( 𝑚𝜋𝜉) ,

here 𝑉 𝐻

𝑖𝑚 ( 𝜏) , 𝑊

𝐻

𝑖𝑚 ( 𝜏) , Ψ

𝐻

𝑦 𝑖𝑚 ( 𝜏) , and Ψ

𝐻

𝑧 𝑖𝑚 ( 𝜏) denote the time-dependent

ubstituting Eqs. (33 a) , ( 33 b), ( 33 c), and ( 33 d) into Eqs. (29) –(31) , on

quations:

𝐻 d 2 𝐱 𝐻

d 𝜏2 + 𝐊

𝐻

𝐱 𝐻 = 𝟎 ,

here 𝐱 𝐻

is given by:

𝐻 =

{

[∙] 𝐻

1 𝑚 , [ ⋅] 𝐻

1 𝑚 , [∙] 𝐻

3 𝑚 , [ ⋅] 𝐻

3 𝑚 , [∙] 𝐻

4 𝑚 , [ ⋅] 𝐻

4 𝑚 , … , [∙] 𝐻

(2 𝑁−1) 𝑚 , [ ⋅] 𝐻

(2 𝑁

n which [∙] = 𝑉 or W , [ ⋅] =Ψz or Ψy , and the elements of 𝐌

𝐻

and 𝐊

𝐻

iven procedure in Section 4.2 , the natural frequencies of the thermally

. Development of continuous models for thermally affected DWC

As it was explained in the earlier parts, the dimensions of both mas

ould substantially increase by increasing the number of DWCNTs o

ighly populated nanosystems would compromise with great comput

everal continuous models based on the NRBM and NHOBM. To this e

hat their values at the neutral axis of each tube would be equal to t

586


𝑉 𝑡 ) ; j = 1,2). For constructing the aforementioned continuous functions, it is

n

,

(36)

(36)

a bes are approximated by:

[ (37)

w

eveloped on the basis of the nonlocal-discrete-based models established in

t

6

6

the nonlocal-continuous equations of motion of thermally affected DWCNTs

m g to the NRBM take the following form:

Ξ 𝑣 𝑅 𝑗

2

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ + 𝐸 𝑏 𝑗

𝐼 𝑏 𝑗

𝜕 4 𝑣 𝑅 𝑗

𝜕𝑥 4 = 0 , (38a)

Ξ

𝜕 2 𝑤 𝑅 𝑗

𝜕𝑥 2

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ + 𝐸 𝑏 𝑗

𝐼 𝑏 𝑗

𝜕 4 𝑤

𝑅 𝑗

𝜕𝑥 4 = 0 , (38b)

a a) and ( 38 b), the dimensionless-nonlocal-continuous governing equations

t based on the NRBM are derived as:

Ξ

𝜕 2 𝑣 𝑅 𝑗 𝜕𝜉2

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ + 𝜚

2 𝑗−2 3

𝜕 4 𝑣 𝑅 𝑗

𝜕𝜉4 = 0 , (39a)

Ξ

𝑅

𝑇 𝑗

𝜕 2 𝑤 𝑅 𝑗 𝜕𝜉2

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ + 𝜚

2 𝑗−2 3

𝜕 4 𝑤

𝑅

𝑗

𝜕𝜉4 = 0 . (39b)

6

ith simple ends at 𝑥 = 0 and 𝑥 = 𝑙 𝑏 whose edges have been fixed at 𝑧 = 0 and

𝑧 ections and nonlocal bending moments of nanotubes modeled according to

t

0 ,

[ ⋅] 𝑗

= 𝑣 [ ⋅] 𝑗 ( 𝑥, 𝑧, 𝑡 ) , 𝑊

[ ⋅] 𝑗

= 𝑤

[ ⋅] 𝑗 ( 𝑥, 𝑧, 𝑡 ) , Ψ𝐻

𝑦 𝑗 = 𝜓 𝐻

𝑦 𝑗 ( 𝑥, 𝑧, 𝑡 ) , Ψ𝐻

𝑧 𝑗 = 𝜓 𝐻

𝑧 𝑗 ( 𝑥, 𝑧,

oticed that they should satisfy the following conditions:

𝑣 [ ⋅] 𝑗 ( 𝑥, 𝑧 2 𝑛 −2+ 𝑗 , 𝑡 ) ≈ 𝑉

[ ⋅] 2 𝑛 −2+ 𝑗 ( 𝑥, 𝑡 ) , 𝑣

[ ⋅] 𝑗 ( 𝑥, 𝑧 2 𝑛 −4+ 𝑗 − 𝑑, 𝑡 ) ≈ 𝑉

[ ⋅] 2 𝑛 −4+ 𝑗 ( 𝑥, 𝑡 ) ,

𝑣 [ ⋅] 𝑗 ( 𝑥, 𝑧 2 𝑛 + 𝑗 + 𝑑, 𝑡 ) ≈ 𝑉

[ ⋅] 2 𝑛 + 𝑗 ( 𝑥, 𝑡 ); [ ⋅] = 𝑅, 𝐻,

𝑤

[ ⋅] 𝑗 ( 𝑥, 𝑧 2 𝑛 −2+ 𝑗 , 𝑡 ) ≈ 𝑊

[ ⋅] 2 𝑛 −2+ 𝑗 ( 𝑥, 𝑡 ) , 𝑤

[ ⋅] 𝑗 ( 𝑥, 𝑧 2 𝑛 −4+ 𝑗 − 𝑑, 𝑡 ) ≈ 𝑊

[ ⋅] (2 𝑛 −4+ 𝑗) ( 𝑥, 𝑡 )

𝑤

[ ⋅] 𝑗 ( 𝑥, 𝑧 2 𝑛 + 𝑗 + 𝑑, 𝑡 ) ≈ 𝑊

[ ⋅] 2 𝑛 + 𝑗 ( 𝑥, 𝑡 ) ,

𝜓 𝐻

𝑦 𝑗 ( 𝑥, 𝑧 2 𝑛 −2+ 𝑗 , 𝑡 ) ≈ Ψ𝐻

𝑦 2 𝑛 −2+ 𝑗 ( 𝑥, 𝑡 ) , 𝜓 𝐻

𝑧 𝑗 ( 𝑥, 𝑧 2 𝑛 −2+ 𝑗 , 𝑡 ) ≈ Ψ𝐻

𝑧 2 𝑛 −2+ 𝑗 ( 𝑥, 𝑡 ) ,

nd the deformation fields at the neutral axis of the neighboring nanotu

⋆ ] 𝑗 ( 𝑥, 𝑧 𝑚 ± 𝑑, 𝑡 ) =

6 ∑𝑖 =0

(± 𝑑) 𝑖

𝑖 ! 𝜕 𝑖 [ ⋆ ] 𝑗 𝜕𝑧 𝑖

( 𝑥, 𝑧 𝑚 , 𝑡 ) ,

here [ ⋆ ] 𝑗 = 𝑣 [ ⋅] or 𝑤

[ ⋅] and 𝑚 = 2 𝑛 − 4 + 𝑗, 2 𝑛 + 𝑗.

In the following parts, the nonlocal-continuous-based models are d

he previous sections.

.1. Continuous NRBM model for thermally affected DWCNTs membranes

.1.1. Development of nonlocal-continuous equations of motion

By introducing Eq. (37) to Eq. (13 a) and ( 13 b) in view of Eq. (36) ,

embranes acted upon by a longitudinal temperature gradient accordin

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝜌𝑏 𝑗

(

𝐴 𝑏 𝑗

𝜕 2 𝑣 𝑅 𝑗

𝜕𝑡 2 − 𝐼 𝑏 𝑗

𝜕 4 𝑣 𝑅 𝑗

𝜕 𝑡 2 𝜕 𝑥 2

)

+ 𝐶 𝑣𝑦 ( 𝑗 +2 , 5− 𝑗 )

(𝑣 𝑅 𝑗 − 𝑣 𝑅 3− 𝑗

)+ 𝐶 𝑣𝑦 ( 𝑗+2 , 2)

×(

2 (𝑣 𝑅 𝑗 − 𝑣 𝑅 3− 𝑗

)− 𝑑 2

𝜕 2 𝑣 𝑅 3− 𝑗 𝜕𝑧 2

−

𝑑 4

12

𝜕 4 𝑣 𝑅 3− 𝑗 𝜕𝑧 4

−

𝑑 6

360

𝜕 6 𝑣 𝑅 3− 𝑗 𝜕𝑧 6

)

− 𝐶 𝑣𝑦 ( 𝑗+2 , 1)

×(

𝑑 2 𝜕 2 𝑣 𝑅

𝑗

𝜕𝑧 2 +

𝑑 4

12 𝜕 4 𝑣 𝑅

𝑗

𝜕𝑧 4 +

𝑑 6

360 𝜕 6 𝑣 𝑅

𝑗

𝜕𝑧 6

)

+

(

𝐾 𝑡 𝑣 𝑅 𝑗 − 𝐾 𝑟

𝜕 2 𝑣 𝑅 𝑗

𝜕𝑥 2

)

(1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

𝜕 2

𝜕𝑥

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝜌𝑏 𝑗

(

𝐴 𝑏 𝑗

𝜕 2 𝑤 𝑅 𝑗

𝜕𝑡 2 − 𝐼 𝑏 𝑗

𝜕 4 𝑤 𝑅 𝑗

𝜕 𝑡 2 𝜕 𝑥 2

)

+ 𝐶 𝑣𝑧 ( 𝑗 +2 , 5− 𝑗 )

(𝑤

𝑅 𝑗 − 𝑤

𝑅 3− 𝑗

)+ 𝐶 𝑣𝑧 ( 𝑗+2 , 2)

×(

2 (𝑤

𝑅 𝑗 − 𝑤

𝑅 3− 𝑗

)− 𝑑 2

𝜕 2 𝑤 𝑅 3− 𝑗 𝜕𝑧 2

−

𝑑 4

12

𝜕 4 𝑤 𝑅 3− 𝑗 𝜕𝑧 4

−

𝑑 6

360

𝜕 6 𝑤 𝑅 3− 𝑗 𝜕𝑧 6

)

− 𝐶 𝑣𝑧 ( 𝑗+2 , 1)

×(

𝑑 2 𝜕 2 𝑤 𝑅

𝑗

𝜕𝑧 2 +

𝑑 4

12 𝜕 4 𝑤 𝑅

𝑗

𝜕𝑧 4 +

𝑑 6

360 𝜕 6 𝑤 𝑅

𝑗

𝜕𝑧 6

)

+

(

𝐾 𝑡 𝑤

𝑅 𝑗 − 𝐾 𝑟

𝜕 2 𝑤 𝑅 𝑗

𝜕𝑥 2

)

(1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

nd by introducing the dimensionless quantities in Eq. (15) to Eq. (38

hat display transverse vibrations of the thermally affected nanosystem

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝜚 2 𝑗−2 1

𝜕 2 𝑣 𝑅 𝑗 𝜕𝜏2

−

𝜚 2 𝑗−2 2 𝜆2 1

𝜕 4 𝑣 𝑅 𝑗 𝜕 𝜏2 𝜕 𝜉2

+ 𝐶

𝑅

𝑣𝑦 ( 𝑗 +2 , 5− 𝑗 )

(𝑣 𝑅

𝑗 − �� 𝑅 2

)+ 𝐶

𝑅

𝑣𝑦 ( 𝑗+2 , 2)

×(

2 (𝑣 𝑅

𝑗 − 𝑣

𝑅

3− 𝑗 )− 𝑑

2 𝜕 2 𝑣 𝑅 3− 𝑗 𝜕𝛾2

−

𝑑 4

12 𝜕 4 𝑣 𝑅 3− 𝑗 𝜕𝛾4

−

𝑑 6

360 𝜕 6 𝑣 𝑅 3− 𝑗 𝜕𝛾6

)

− 𝐶

𝑅

𝑣𝑦 ( 𝑗+2 , 1)

×(

𝑑 2 𝜕 2 𝑣 𝑅 𝑗

𝜕𝛾2 +

𝑑 4

12 𝜕 4 𝑣 𝑅 𝑗 𝜕𝛾4

+

𝑑 6

360 𝜕 6 𝑣 𝑅 𝑗 𝜕𝛾6

)

+

(

𝐾

𝑅

𝑡 𝑣 𝑅

𝑗 − 𝐾

𝑅

𝑟

𝜕 2 𝑣 𝑅 𝑗 𝜕𝜉2

)

(1 − 𝛿𝑗1 ) − 𝑁

𝑅

𝑇 𝑗

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝜚 2 𝑗−2 1

𝜕 2 𝑤 𝑅 𝑗 𝜕𝜏2

−

𝜚 2 𝑗−2 2 𝜆2 1

𝜕 4 𝑤 𝑅 𝑗 𝜕 𝜏2 𝜕 𝜉2

+ 𝐶

𝑅

𝑣𝑧 ( 𝑗 +2 , 5− 𝑗 )

(𝑤

𝑅

𝑗 − ��

𝑅 2 )+ 𝐶

𝑅

𝑣𝑧 ( 𝑗+2 , 2)

×(

2 (𝑤

𝑅

𝑗 − 𝑤

𝑅

3− 𝑗 )− 𝑑

2 𝜕 2 𝑤 𝑅 3− 𝑗 𝜕𝛾2

−

𝑑 4

12 𝜕 4 𝑤 𝑅 3− 𝑗 𝜕𝛾4

−

𝑑 6

360 𝜕 6 𝑤 𝑅 3− 𝑗 𝜕𝛾6

)

− 𝐶

𝑅

𝑣𝑧 ( 𝑗+2 , 1)

×(

𝑑 2 𝜕 2 𝑤 𝑅 𝑗

𝜕𝛾2 +

𝑑 4

12 𝜕 4 𝑤 𝑅 𝑗 𝜕𝛾4

+

𝑑 6

360 𝜕 6 𝑤 𝑅 𝑗 𝜕𝛾6

)

+

(

𝐾

𝑅

𝑡 𝑤

𝑅

𝑗 − 𝐾

𝑅

𝑟

𝜕 2 𝑤 𝑅 𝑗 𝜕𝜉2

)

(1 − 𝛿𝑗1 ) − 𝑁

.1.2. Evaluation of natural frequencies

For elastically embedded membranes under longitudinal heat flux w

= ( 𝑁 𝑧 − 1) 𝑑, see Fig. 2 , the following conditions should be met by defl

he NRBM:

𝑣 𝑅

𝑗 ( 𝜉 = 0 , 𝛾, 𝜏) = �� 𝑅

𝑗 ( 𝜉 = 1 , 𝛾, 𝜏) = 0 , 𝑤

𝑅

𝑗 ( 𝜉 = 0 , 𝛾, 𝜏) = ��

𝑅 𝑗 ( 𝜉 = 1 , 𝛾, 𝜏) = (

𝑀

𝑅

𝑏𝑦 𝑗

)𝑛𝑙 ( 𝜉 = 0 , 𝛾, 𝜏) =

(𝑀

𝑅

𝑏𝑦 𝑗

)𝑛𝑙 ( 𝜉 = 1 , 𝛾, 𝜏) = 0 ,

587


Fig. 2. An illustration of the spatial domain of the nonlocal-continuous-based models employed for vibration analysis of thermally affected membranes consist of N

vertically aligned DWCNTs.

0 . (40)

T ven boundary conditions in Eq. (40) , the continuous deflection fields of the

t

𝑣 (41a)

𝑤 (41b)

w )th vibration mode, ��

𝑅 𝑦

and ��

𝑅 𝑦

denote the dimensionless natural frequencies

p ively. By substituting Eqs. (41 a) and ( 41 b) into Eqs. (39 a) and ( 39 b), the

c ould be readily evaluated (see supplementary material, part D).

6 s

6

–( 26 d), the nonlocal-continuous equations that display transverse vibrations

o ix are obtained as follows:

) ⎫ ⎪ ⎬ ⎪ ⎭

3 𝑣 𝐻

𝑗

𝜕𝑥 3 = 0 , (42a)

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (42b)

) ⎫ ⎪ ⎬ ⎪ ⎭ 𝜕 3 𝑤

𝐻

𝑗

𝜕𝑥 3 = 0 , (42c)

(𝑀

𝑅

𝑏𝑧 𝑗

)𝑛𝑙 ( 𝜉 = 0 , 𝛾, 𝜏) =

(𝑀

𝑅

𝑏𝑧 𝑗

)𝑛𝑙 ( 𝜉 = 1 , 𝛾, 𝜏) = 0 ,

𝑤

𝑅

𝑗 ( 𝜉, 𝛾 = 0 , 𝜏) = ��

𝑅 𝑗 ( 𝜉, 𝛾 = 1 , 𝜏) = 0 , 𝑣

𝑅

𝑗 ( 𝜉, 𝛾 = 0 , 𝜏) = �� 𝑅

𝑗 ( 𝜉, 𝛾 = 1 , 𝜏) =

o solve Eq. (39 a) and ( 39 b) for natural frequencies in view of the gi

hermally affected nanosystem are stated by:

𝑅

𝑗 ( 𝜉, 𝛾, 𝜏) =

∞∑𝑚 =1

∞∑𝑛 =1

𝑣 𝑅

( 𝑚𝑛 0) 𝑗 sin ( 𝑚𝜋𝜉) sin ( 𝑛𝜋𝛾) 𝑒 i 𝜔

𝑅 𝑦 𝜏 ,

𝑅

𝑗 ( 𝜉, 𝛾, 𝜏) =

∞∑𝑚 =1

∞∑𝑛 =1

𝑤

𝑅

( 𝑚𝑛 0) 𝑗 sin ( 𝑚𝜋𝜉) sin ( 𝑛𝜋𝛾) 𝑒 i 𝜔 𝑅 𝑧 𝜏 ,

here i =

√−1 , 𝑣 𝑅 ( 𝑚𝑛 0) 𝑗 and 𝑤

𝑅

( 𝑚𝑛 0) 𝑗 represent the amplitudes of the ( m, n

ertinent to the out-of-plane and the in-plane vibration modes, respect

haracteristic equation and the natural frequencies of the nanosystem c

.2. Continuous NHOBM model for thermally affected DWCNTs membrane

.2.1. Development of nonlocal-continuous equations of motion

By virtue of Eq. (36) and through introducing Eq. (37) to Eq. (26 a)

f thermally affected DWCNTs membranes embedded in an elastic matr

Ξ⎧ ⎪ ⎨ ⎪ ⎩ (𝐼 2 𝑗 − 2 𝛼𝑗 𝐼 4 𝑗 + 𝛼2

𝑗 𝐼 6 𝑗 ) 𝜕 2 𝜓 𝐻

𝑧 𝑗

𝜕𝑡 2 +

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 𝑣 𝐻

𝑗

𝜕 𝑡 2 𝜕 𝑥 + 𝐾 𝑟 𝜓

𝐻

𝑧 𝑗 (1 − 𝛿𝑗1

+ 𝜅𝑗

(

𝜓 𝐻

𝑧 𝑗 +

𝜕𝑣 𝐻

𝑗

𝜕𝑥

)

−

(𝐽 2 𝑗 − 2 𝛼𝑗 𝐽 4 𝑗 + 𝛼2

𝑗 𝐽 6 𝑗

) 𝜕 2 𝜓 𝐻

𝑧 𝑗

𝜕𝑥 2 +

(𝛼𝑗 𝐽 4 𝑗 − 𝛼2

𝑗 𝐽 6 𝑗

) 𝜕

Ξ

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝐼 0 𝑗 𝜕 2 𝑣 𝐻

𝑗

𝜕𝑡 2 −

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 𝜓 𝐻 𝑧 𝑗

𝜕 𝑡 2 𝜕 𝑥 − 𝛼2

𝑗 𝐼 6 𝑗

𝜕 4 𝑣 𝐻 𝑗

𝜕 𝑡 2 𝜕 𝑥 2 + 𝐶 𝑣𝑦 ( 𝑗 +2 , 5− 𝑗 )

(𝑣 𝐻

𝑗 − 𝑣 𝐻

3− 𝑗 )

+ 𝐶 𝑣𝑦 ( 𝑗+2 , 2)

(

2 (𝑣 𝐻

𝑗 − 𝑣 𝐻

3− 𝑗 )− 𝑑 2

𝜕 2 𝑣 𝐻 3− 𝑗 𝜕𝑧 2

−

𝑑 4

12

𝜕 4 𝑣 𝐻 3− 𝑗 𝜕𝑧 4

−

𝑑 6

360

𝜕 6 𝑣 𝐻 3− 𝑗 𝜕𝑧 6

)

− 𝐶 𝑣𝑦 ( 𝑗+2 , 1)

(

𝑑 2 𝜕 2 𝑣 𝐻

𝑗

𝜕𝑧 2 +

𝑑 4

12 𝜕 4 𝑣 𝐻

𝑗

𝜕𝑧 4 +

𝑑 6

360 𝜕 6 𝑣 𝐻

𝑗

𝜕𝑧 6

)

+ 𝐾 𝑡 𝑣 𝐻

𝑗 (1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

𝜕 2 𝑣 𝐻𝑗

𝜕𝑥 2

− 𝜅𝑗

⎛ ⎜ ⎜ ⎝ 𝜕𝜓 𝐻

𝑧 𝑗

𝜕𝑥 +

𝜕 2 𝑣 𝐻

𝑗

𝜕𝑥 2

⎞ ⎟ ⎟ ⎠ − 𝛼𝑗 𝐽 4 𝑗

𝜕 3 𝜓 𝐻

𝑧 𝑗

𝜕𝑥 3 + 𝛼2

𝑗 𝐽 6 𝑗

⎛ ⎜ ⎜ ⎝ 𝜕 3 𝜓 𝐻

𝑧 𝑗

𝜕𝑥 3 +

𝜕 4 𝑣 𝐻

𝑗

𝜕𝑥 4

⎞ ⎟ ⎟ ⎠ = 0 ,

Ξ⎧ ⎪ ⎨ ⎪ ⎩ (𝐼 2 𝑗 − 2 𝛼𝑗 𝐼 4 𝑗 + 𝛼2

𝑗 𝐼 6 𝑗 ) 𝜕 2 𝜓 𝐻

𝑦 𝑗

𝜕𝑡 2 +

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 𝑤

𝐻

𝑗

𝜕 𝑡 2 𝜕 𝑥 + 𝐾 𝑟 𝜓

𝐻

𝑦 𝑗 (1 − 𝛿𝑗1

+ 𝜅𝑗

(

𝜓 𝐻

𝑦 𝑗 +

𝜕𝑤

𝐻

𝑗

𝜕𝑥

)

−

(𝐽 2 𝑗 − 2 𝛼𝑗 𝐽 4 𝑗 + 𝛼2

𝑗 𝐽 6 𝑗

) 𝜕 2 𝜓 𝐻

𝑦 𝑗

𝜕𝑥 2 +

(𝛼𝑗 𝐽 4 𝑗 − 𝛼2

𝑗 𝐽 6 𝑗

)
588


𝑗

) 𝑤 𝐻 𝑗

𝑥 2

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ (42d)

a

⎞ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 8 𝛾2 8

𝜕 2 𝜓 𝐻

𝑧 𝑗

𝜕𝜉2 + 𝜗

2 𝑗−2 5 𝛾2 9

𝜕 3 𝑣 𝐻

𝑗

𝜕𝜉3 = 0 , (43a)

Ξ

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ − 𝜗

2 𝑗−2 4 𝛾2 3

⎛ ⎜ ⎜ ⎝ 𝜕 𝜓

𝐻

𝑧 𝑗

𝜕𝜉+

𝜕 2 𝑣 𝐻

𝑗

𝜕𝜉2

⎞ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 5 𝛾2 4

𝜕 3 𝜓 𝐻

𝑧 𝑗

𝜕𝜉3 + 𝜗

2 𝑗−2 6

𝜕 4 𝑣 𝐻

𝑗

𝜕𝜉4 = 0 , (43b)

⎞ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 8 𝛾2 8

𝜕 2 𝜓 𝐻

𝑦 𝑗

𝜕𝜉2 + 𝜗

2 𝑗−2 5 𝛾2 9

𝜕 3 𝑤

𝐻

𝑗

𝜕𝜉3 = 0 , (43c)

Ξ 𝑤 𝐻 𝑗

𝜉2

⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭ − 𝜗

2 𝑗−2 4 𝛾2 3

⎛ ⎜ ⎜ ⎝ 𝜕 𝜓

𝐻

𝑦 𝑗

𝜕𝜉+

𝜕 2 𝑤

𝐻

𝑗

𝜕𝜉2

⎞ ⎟ ⎟ ⎠ − 𝜗 2 𝑗−2 5 𝛾2 4

𝜕 3 𝜓 𝐻

𝑦 𝑗

𝜕𝜉3 + 𝜗

2 𝑗−2 6

𝜕 4 𝑤

𝐻

𝑗

𝜕𝜉4 = 0 . (43d)

tions of motion of vertically aligned DWCNTs membranes subjected to a

l coupled equations and an efficient methodology should be implemented for

a

6

ports at the ends of the constitutive nanotubes and the outermost tubes of

t om any transverse motion; therefore,

0 ,

0 . (44)

I system according to Eq. (43 a) –( 43 d) with the given conditions in Eq. (44) ,

w e NHOBM:

𝑣 (45a)

𝜓 (45b)

𝑤 (45c)

𝜓 (45d)

w ertinent to the ( m, n )th vibration mode, ��

𝐻

𝑦 and ��

𝐻

𝑧 are the dimensionless

f by substituting Eq. (45 a) –( 45 d) into the linearly coupled Eq. (43 a) –( 43 d),

t rane embedded in an elastic matrix is derived and the natural frequencies

w given in supplementary material, part E.

Ξ

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝐼 0 𝑗 𝜕 2 𝑤 𝐻

𝑗

𝜕𝑡 2 −

(𝛼2 𝑗 𝐼 6 𝑗 − 𝛼𝑗 𝐼 4 𝑗

) 𝜕 3 𝜓 𝐻 𝑦 𝑗

𝜕 𝑡 2 𝜕 𝑥 − 𝛼2

𝑗 𝐼 6 𝑗

𝜕 4 𝑤 𝐻 𝑗

𝜕 𝑡 2 𝜕 𝑥 2 + 𝐶 𝑣𝑧 ( 𝑗 +2 , 5− 𝑗 )

(𝑤

𝐻

𝑗 − 𝑤

𝐻

3−

+ 𝐶 𝑣𝑧 ( 𝑗+2 , 2)

(

2 (𝑤

𝐻

𝑗 − 𝑤

𝐻

3− 𝑗 )− 𝑑 2

𝜕 2 𝑤 𝐻 3− 𝑗 𝜕𝑧 2

−

𝑑 4

12

𝜕 4 𝑤 𝐻 3− 𝑗 𝜕𝑧 4

−

𝑑 6

360

𝜕 6 𝑤 𝐻 3− 𝑗 𝜕𝑧 6

)

− 𝐶 𝑣𝑧 ( 𝑗+2 , 1)

(

𝑑 2 𝜕 2 𝑤 𝐻

𝑗

𝜕𝑧 2 +

𝑑 4

12 𝜕 4 𝑤 𝐻

𝑗

𝜕𝑧 4 +

𝑑 6

360 𝜕 6 𝑤 𝐻

𝑗

𝜕𝑧 6

)

+ 𝐾 𝑡 𝑤

𝐻

𝑗 (1 − 𝛿𝑗1 ) − 𝑁 𝑇 𝑗

𝜕 2

𝜕

− 𝜅𝑗

⎛ ⎜ ⎜ ⎝ 𝜕𝜓 𝐻

𝑦 𝑗

𝜕𝑥 +

𝜕 2 𝑤

𝐻

𝑗

𝜕𝑥 2

⎞ ⎟ ⎟ ⎠ − 𝛼𝑗 𝐽 4 𝑗

𝜕 3 𝜓 𝐻

𝑦 𝑗

𝜕𝑥 3 + 𝛼2

𝑗 𝐽 6 𝑗

⎛ ⎜ ⎜ ⎝ 𝜕 3 𝜓 𝐻

𝑦 𝑗

𝜕𝑥 3 +

𝜕 4 𝑤

𝐻

𝑗

𝜕𝑥 4

⎞ ⎟ ⎟ ⎠ = 0 ,

nd by introducing Eq. (32) to Eq. (42 a) –( 42 d), it is obtainable:

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝜗

2 𝑗−2 7

𝜕 2 𝜓 𝐻

𝑧 𝑗

𝜕𝜏2 − 𝜗

2 𝑗−2 2 𝛾2 6

𝜕 3 𝑣 𝐻

𝑗

𝜕 𝜏2 𝜕 𝜉+ 𝐾

𝐻

𝑟 𝜓 𝐻

𝑧 𝑗 (1 − 𝛿𝑗1 )

⎫ ⎪ ⎬ ⎪ ⎭ + 𝜗 2 𝑗−2 4 𝛾2 7

⎛ ⎜ ⎜ ⎝ 𝜓 𝐻

𝑧 𝑗 +

𝜕 𝑣 𝐻

𝑗

𝜕𝜉

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝜗 2 𝑗−2 1

𝜕 2 𝑣 𝐻 𝑗

𝜕𝜏2 + 𝜗

2 𝑗−2 2 𝛾2 1

𝜕 3 Ψ𝐻

𝑧 𝑗

𝜕 𝜏2 𝜕 𝜉− 𝜗

2 𝑗−2 3 𝛾2 2

𝜕 4 𝑣 𝐻 𝑗

𝜕 𝜏2 𝜕 𝜉2 + 𝐶

𝐻

𝑣𝑦 ( 𝑗 +2 , 5− 𝑗 )

(𝑣 𝐻

𝑗 − 𝑣

𝐻

3− 𝑗 )

+ 𝐶

𝐻

𝑣𝑦 ( 𝑗+2 , 2)

(

2 (𝑣 𝐻

𝑗 − 𝑣

𝐻

3− 𝑗 )− 𝑑

2 𝜕 2 𝑣 𝐻 3− 𝑗 𝜕𝛾2

−

𝑑 4

12 𝜕 4 𝑣 𝐻 3− 𝑗 𝜕𝛾4

−

𝑑 6

360 𝜕 6 𝑣 𝐻 3− 𝑗 𝜕𝛾6

)

− 𝐶

𝐻

𝑣𝑦 ( 𝑗+2 , 1)

(

𝑑 2 𝜕 2 𝑣 𝐻 𝑗

𝜕𝛾2 +

𝑑 4

12 𝜕 4 𝑣 𝐻 𝑗

𝜕𝛾4 +

𝑑 6

360 𝜕 6 𝑣 𝐻 𝑗

𝜕𝛾6

)

+ 𝐾

𝐻

𝑡 𝑣 𝐻

𝑗 (1 − 𝛿𝑗1 ) − 𝑁

𝐻

𝑇 𝑗

𝜕 2 𝑣 𝐻𝑗 𝜕𝜉2

Ξ⎧ ⎪ ⎨ ⎪ ⎩ 𝜗

2 𝑗−2 7

𝜕 2 𝜓 𝐻

𝑦 𝑗

𝜕𝜏2 − 𝜗

2 𝑗−2 2 𝛾2 6

𝜕 3 𝑤

𝐻

𝑗

𝜕 𝜏2 𝜕 𝜉+ 𝐾

𝐻

𝑟 𝜓 𝐻

𝑦 𝑗 (1 − 𝛿𝑗1 )

⎫ ⎪ ⎬ ⎪ ⎭ + 𝜗 2 𝑗−2 4 𝛾2 7

⎛ ⎜ ⎜ ⎝ 𝜓 𝐻

𝑦 𝑗 +

𝜕 𝑤

𝐻

𝑗

𝜕𝜉

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

𝜗 2 𝑗−2 1

𝜕 2 𝑤 𝐻 𝑗

𝜕𝜏2 + 𝜗

2 𝑗−2 2 𝛾2 1

𝜕 3 𝜓 𝐻 𝑦 𝑗

𝜕 𝜏2 𝜕 𝜉− 𝜗

2 𝑗−2 3 𝛾2 2

𝜕 4 𝑣 𝐻 𝑗

𝜕 𝜏2 𝜕 𝜉2 + 𝐶

𝐻

𝑣𝑧 ( 𝑗 +2 , 5− 𝑗 )

(𝑤

𝐻

𝑗 − 𝑤

𝐻

3− 𝑗 )

+ 𝐶

𝐻

𝑣𝑧 ( 𝑗+2 , 2)

(

2 (𝑤

𝐻

𝑗 − 𝑤

𝐻

3− 𝑗 )− 𝑑

2 𝜕 2 𝑤 𝐻 3− 𝑗 𝜕𝛾2

−

𝑑 4

12 𝜕 4 𝑤 𝐻 3− 𝑗 𝜕𝛾4

−

𝑑 6

360 𝜕 6 𝑤 𝐻 3− 𝑗 𝜕𝛾6

)

− 𝐶

𝐻

𝑣𝑧 ( 𝑗+2 , 1)

(

𝑑 2 𝜕 2 𝑤 𝐻 𝑗

𝜕𝛾2 +

𝑑 4

12 𝜕 4 𝑤 𝐻 𝑗

𝜕𝛾4 +

𝑑 6

360 𝜕 6 𝑤 𝐻 𝑗

𝜕𝛾6

)

+ 𝐾

𝐻

𝑡 𝑤

𝐻

𝑗 (1 − 𝛿𝑗1 ) − 𝑁

𝐻

𝑇 𝑗

𝜕 2

𝜕

Eq. (43 a) –( 43 d) furnish us regarding dimensionless-nonlocal equa

ongitudinal temperature gradient according to the NHOBM. These are

ssessing the natural frequencies of the nanosystem.

.2.2. Evaluation of natural frequencies

It is assumed that the thermally affected nanosystem has simple sup

he first and the last DWCNTs of the nanosystem have been provoked fr

𝑣 𝐻

𝑗 ( 𝜉 = 0 , 𝛾, 𝜏) = �� 𝐻

𝑗 ( 𝜉 = 1 , 𝛾, 𝜏) = 0 , 𝑤

𝐻

𝑗 ( 𝜉 = 0 , 𝛾, 𝜏) = ��

𝐻

𝑗 ( 𝜉 = 1 , 𝛾, 𝜏) =(

𝑀

𝐻

𝑏𝑦 𝑗

)𝑛𝑙 ( 𝜉 = 0 , 𝛾, 𝜏) =

(𝑀

𝐻

𝑏𝑦 𝑗

)𝑛𝑙 ( 𝜉 = 1 , 𝛾, 𝜏) = 0 , (𝑀

𝐻

𝑏𝑧 𝑗

)𝑛𝑙 ( 𝜉 = 0 , 𝛾, 𝜏) =

(𝑀

𝐻

𝑏𝑧 𝑗

)𝑛𝑙 ( 𝜉 = 1 , 𝛾, 𝜏) = 0 ,

𝑣 𝐻

𝑗 ( 𝜉, 𝛾 = 0 , 𝜏) = �� 𝐻

𝑗 ( 𝜉, 𝛾 = 1 , 𝜏) = 0 , 𝑤

𝐻

𝑗 ( 𝜉, 𝛾 = 0 , 𝜏) = ��

𝐻

𝑗 ( 𝜉, 𝛾 = 1 , 𝜏) =

n order to examine natural frequencies of the thermally affected nano

e take into account the following form for the deformation fields of th

𝐻

𝑗 ( 𝜉, 𝛾, 𝜏) =

∞∑𝑚 =1

∞∑𝑛 =1

𝑣 𝐻

( 𝑚𝑛 0) 𝑗 sin ( 𝑚𝜋𝜉) sin ( 𝑛𝜋𝛾) 𝑒 i 𝜔

𝐻 𝑦 𝜏 ,

𝐻

𝑧 𝑗 ( 𝜉, 𝛾, 𝜏) =

∞∑𝑚 =1

∞∑𝑛 =1

𝜓 𝐻

𝑧 ( 𝑚𝑛 0) 𝑗 cos ( 𝑚𝜋𝜉) sin ( 𝑛𝜋𝛾) 𝑒 i 𝜔

𝐻 𝑦 𝜏 ,

𝐻

𝑗 ( 𝜉, 𝛾, 𝜏) =

∞∑𝑚 =1

∞∑𝑛 =1

𝑤

𝐻

( 𝑚𝑛 0) 𝑗 sin ( 𝑚𝜋𝜉) sin ( 𝑛𝜋𝛾) 𝑒 i 𝜔 𝐻 𝑧 𝜏 ,

𝐻

𝑦 𝑗 ( 𝜉, 𝛾, 𝜏) =

∞∑𝑚 =1

∞∑𝑛 =1

𝜓 𝐻

𝑦 ( 𝑚𝑛 0) 𝑗 cos ( 𝑚𝜋𝜉) sin ( 𝑛𝜋𝛾) 𝑒 i 𝜔 𝐻 𝑧 𝜏 ,

here 𝑣 𝐻

( 𝑚𝑛 0) 𝑗 , 𝑤

𝐻

( 𝑚𝑛 0) 𝑗 , 𝜓

𝐻

𝑧 ( 𝑚𝑛 0) 𝑗 , and 𝜓

𝐻

𝑦 ( 𝑚𝑛 0) 𝑗 represent the amplitudes p

requencies associated with the vibrations in the y and z directions. Now

he characteristic equation of the thermally influenced DWCNTs memb

ould be readily evaluated. The details of such calculations have been

589


Table 1

Comparison of the out-of-plane fundamental frequencies of the discrete-based models and those of the continuous-based models in the case of

uniform temperature change (low temperature case; 𝑇 1 = 𝑇 2 = Δ𝑇 ).

Discrete ΔT = 30 °C ΔT = 300 °C

models 𝜆1 𝑁 = 5 𝑁 = 10 𝑁 = 30 𝑁 = 50 𝑁 = 5 𝑁 = 10 𝑁 = 30 𝑁 = 50

NRBM 20 1105.311 1095.239 1092.868 1092.702 1113.043 1103.042 1100.687 1100.522

30 560.609 539.760 534.762 534.411 567.519 546.934 542.001 541.655

100 171.544 90.916 57.547 54.454 173.610 94.758 63.443 60.651

1000 140.837 63.914 19.950 11.840 140.863 63.969 20.129 12.138

5000 313.007 142.032 44.282 26.216 313.007 142.0331 44.285 26.222

NHOBM 20 1041.031 1030.147 1027.592 1027.414 1049.312 1038.514 1035.979 1035.802

30 545.847 524.342 519.182 518.819 552.956 531.739 526.651 526.293

100 171.501 90.832 57.414 54.313 173.568 94.677 63.322 60.525

1000 140.837 63.914 19.950 11.840 140.863 63.969 20.129 12.138

5000 313.007 142.032 44.282 26.216 313.007 142.033 44.285 26.222

Continuous

models

NRBM 20 1105.308 1095.236 1092.865 1092.699 1113.018 1103.017 1100.662 1100.498

30 560.609 539.760 534.761 534.410 567.513 546.928 541.996 541.649

100 171.546 90.9165 57.547 54.454 173.612 94.758 63.443 60.651

1000 140.839 63.914 19.950 11.840 140.865 63.970 20.129 12.138

5000 313.011 142.032 44.282 26.216 313.011 142.033 44.285 26.222

NHOBM 20 1040.945 1030.138 1027.592 1027.414 1049.226 1038.505 1035.979 1035.802

30 545.824 524.340 519.182 518.819 552.933 531.736 526.651 526.293

100 171.502 90.832 57.414 54.313 173.569 94.677 63.322 60.525

1000 140.839 63.914 19.950 11.840 140.865 63.970 20.129 12.138

5000 313.011 142.032 44.282 26.216 313.011 142.033 44.285 26.222

7

ous models based on the Rayleigh and high-order shear deformable beam

t tem embedded in an elastic environment are going to be discussed here. The

p hermally affected membranes of DWCNTs are as follows: 𝑟 𝑚 1 = 0.7 nm, 𝜈b = 0.2,

e b = 0.34 nm. According to the previous studies [53] , the thermal expansion

c w temperatures. This coefficient is considered to be −1 . 6 × 10 −6 ◦ C

−1 for

l rformed numerical studies, the nanosystem is isolated from the surrounding

e g thermal conductivity and heat transfer coefficients of CNTs, Hsu et al.

[ f SWCNTs bundles in air. It is reported that this factor would be in the range

o m 9.89 to 12.54 nm. Using micro-Raman spectroscopy technique, Wang et al.

[ the coefficient of heat transfer of suspended individual SWCNTs would grow

f Che et al. [56] showed that the thermal conductivity of CNTs with diameter

g rk, Han and Fina [57] proposed that the coefficient of thermal conductivity

o vironment temperature being in the range of 100–400 o C. Based on this brief

s ctivity of CNTs to be 8.9 ×10 4 W/m

2 C and 3500 W/mC, respectively.

7 sed ones

s of thermally affected membranes of DWCNTs according to the established

c . The results are provided for slenderness ratios of 20, 30, 100, 1000, and

5 bles 1 and 2 are given for the case of the nanosystem subjected to uniform

t 3 are provided for nanosystem acted upon by temperature gradient (i.e.,

𝑇 the continuous models’ results and those of the discrete models shows that

t damental frequencies of discrete models’ results with acceptable accuracy.

F does not affect the computational time of continuous models, so such models

c ly populated nanosystems made from DWCNTs under temperature gradients.

temperature variation are lower than those of the uniform temperature case,

w can be concluded that the results of the two proposed models are closer to

e es show that increasing the temperature of the nanosystem from 30 to 300°C

y ems exposed to low temperatures, the resulted thermal tensile-forces within

n sing the geometrical stiffness, the total transverse stiffness of the thermally

a from 5 to 50, the fundamental frequency decreases, which is more apparent

i fluence of population growth on the fundamental frequencies is higher in

n riation of frequency decreases with increasing the number of nanotubes in

h d out-of-plane vibrations show that the increase of the slenderness ratio up

t cies, and such a fact is more obvious in nanosystems with more population.

F fundamental frequency commonly increases with the slenderness ratio.

those of the NRBM. This fact is mainly attributed to the incorporation of the

s deformation in the total transverse stiffness of the constitutive nanotubes.

. Results and discussion

Based on the given solutions for the nonlocal discrete and continu

heories, the free thermo-elastic vibration characteristics of the nanosys

arameters used to calculate the in-plane and out-of-plane vibrations of t

0 a = 2 nm, 𝐸 𝑏 1 = 𝐸 𝑏 2

= 1 TPa, 𝑑 = 2 𝑟 𝑚 2 + 𝑡 𝑏 , 𝜌𝑏 1 = 𝜌𝑏 2

= 2300 kg/m

3 , t

oefficient of CNTs for high temperatures is different from that for lo

ow temperatures and 1 . 1 × 10 −6 ◦ C

−1 for high temperatures. In all pe

nvironment (i.e., 𝐾 𝑟 = 𝐾 𝑡 = 0) unless explicitly referred to. Concernin

54] proposed two-laser approach to measure heat transfer coefficient o

f 1 . 5 × 10 3 − 7 . 9 × 10 4 W/m

2 C for nanotubes whose diameter ranges fro

55] reported that by increasing the diameter from 0.97 nm to 1.47 nm,

rom 7.5 ×10 4 to 8.9 ×10 4 W/m

2 C. By employing molecular dynamic,

reater than 10 nm would be about 2980 W/mC. In another review wo

f CNT(11,11) would be in the range of 2000–7500 W/mC when the en

urvey, we consider the coefficients of heat transfer and thermal condu

.1. A comparison between the continuous-based results and the discrete-ba

In Tables 1–3 , the fundamental in-plane and out-of-plane frequencie

ontinuous and discrete models are presented for different populations

000, at two temperature levels of 30 and 300 °C. The given data in Ta

emperature change (i.e., 𝑇 1 = 𝑇 2 = Δ𝑇 ) while those data in Table

2 = 𝑇 1 + Δ𝑇 ) at low environmental temperatures. The comparison of

he continuous model can estimate both in-plane and out-of-plane fun

urther calculations reveal that the increase in the number of nanotubes

an be meritoriously used for exploring vibrational behavior of extreme

As shown in Tables 1 and 3 , the predicted results in the case of linear

hich is especially more apparent for the NRBM’s results. Generally, it

ach other for the case of the linear temperature variation. Further studi

ields increasing of the fundamental frequencies. Actually, for nanosyst

anotubes would magnify with the temperature change and by increa

ffected membrane would reduce. By increasing the number of DWCNTs

n the case of in-plane vibration. It should be also noticed that the in

anosystems with higher slenderness ratios. In addition, the rate of va

igh-population nanosystems. The obtained results for both in-plane an

o a certain level would result in a decrease of the fundamental frequen

or slenderness ratios greater than above-mentioned certain value, the

In all cases, the obtained results based on the NHOBM are less than

mall-scale parameter into the shear stresses as well as the role of shear

590


Table 2

Comparison of the in-plane fundamental frequencies of the discrete-based models and those of the continuous-based models in the case of uniform

temperature change (low temperature case; 𝑇 1 = 𝑇 2 = Δ𝑇 ).



NRBM 20 1457.886 1177.587 1101.168 1095.617 1463.758 1184.848 1108.929 1103.417

30 1114.558 695.369 551.977 540.511 1118.049 700.952 558.993 547.675

100 1157.499 528.889 172.441 110.579 1157.807 529.563 174.497 113.759

1000 3925.708 1787.115 557.595 330.12 3925.709 1787.116 557.601 330.139

5000 8778.505 3996.270 1246.869 738.216 8778.505 3996.270 1246.869 738.216

NHOBM 20 1413.057 1118.162 1036.507 1030.546 1419.168 1125.875 1044.822 1038.910

30 1108.136 683.780 536.935 525.114 1111.655 689.468 544.160 532.500

100 1157.504 528.879 172.398 110.510 1157.813 529.553 174.454 113.692

1000 3925.707 1787.115 557.595 330.128 3925.707 1787.117 557.601 330.139

5000 8778.504 3996.270 1246.869 738.215 8778.504 3996.270 1246.869 738.216

Continuous

models

NRBM 20 1457.918 1177.588 1101.166 1095.614 1463.735 1184.818 1108.903 1103.392

30 1114.606 695.376 551.976 540.511 1118.073 700.947 558.987 547.669

100 1157.543 528.897 172.442 110.579 1157.849 529.570 174.497 113.759

1000 3925.744 1787.118 557.595 330.128 3925.745 1787.120 557.601 330.139

5000 8778.586 3996.277 1246.869 738.216 8778.586 3996.277 1246.869 738.216

NHOBM 20 1412.981 1118.157 1036.507 1030.546 1419.092 1125.870 1044.822 1038.910

30 1108.139 683.785 536.935 525.114 1111.658 689.473 544.160 532.500

100 1157.543 528.887 172.399 110.511 1157.851 529.561 174.455 113.692

1000 3925.742 1787.118 557.595 330.128 3925.743 1787.120 557.601 330.139

5000 8778.584 3996.277 1246.869 738.216 8778.584 3996.277 1246.869 738.216

Table 3

Comparison of the out-of-plane fundamental frequencies of the discrete-based models and those of the continuous-based models in the case of

existence of longitudinal temperature gradient (low temperature case; T 1 = 0, 𝑇 2 = Δ𝑇 ).



NRBM 20 1104.879 1094.804 1092.432 1092.265 1108.754 1098.714 1096.350 1096.184

30 560.222 539.359 534.357 534.005 563.690 542.960 537.992 537.642

100 171.428 90.698 57.202 54.089 172.465 92.643 60.239 57.291

1000 140.836 63.910 19.940 11.824 140.849 63.938 20.030 11.974

5000 313.007 142.032 44.282 26.216 313.007 142.032 44.283 26.219

NHOBM 20 1040.569 1029.680 1027.124 1026.946 1044.719 1033.874 1031.328 1031.150

30 545.449 523.928 518.764 518.401 549.018 527.642 522.514 522.154

100 171.385 90.614 57.068 53.947 172.423 92.561 60.112 57.157

1000 140.836 63.910 19.940 11.824 140.849 63.938 20.030 11.974

5000 313.007 142.032 44.282 26.216 313.007 142.032 44.283 26.219

Continuous

models

NRBM 20 1104.879 1094.803 1092.430 1092.264 1108.741 1098.701 1096.337 1096.183

30 560.223 539.359 534.357 534.005 563.688 542.957 537.989 537.640

100 171.430 90.698 57.202 53.947 172.467 92.643 60.239 57.157

1000 140.838 63.911 19.941 11.824 140.850 63.939 20.030 11.974

5000 313.011 142.032 44.282 26.216 313.011 142.033 44.283 26.219

NHOBM 20 1040.484 1029.672 1027.124 1026.946 1044.634 1033.865 1031.328 1031.150

30 545.426 523.926 518.764 518.401 548.995 527.640 522.514 522.154

100 171.387 90.614 57.068 53.947 172.424 92.561 60.112 57.157

1000 140.838 63.911 19.941 11.824 140.850 63.939 20.030 11.974

5000 313.011 142.032 44.282 26.216 313.011 142.033 44.283 26.219

7

ed models with those of another work is of concern. Since thermo-elastic

v n addressed yet, we restrict our verification study to a special case. Tounsi

e al DWCNTs in the presence of a uniform temperature field using nonlocal

E the small-scale parameter as well as the temperature change on the charac-

t udy, the following properties are considered: e 0 a = 2 nm, 𝜌b = 2300 kg/m

3 ,

E ne the ratio: 𝜁1 =

𝜔 𝑛𝐼

( 𝜔 𝑛𝐼 ) 0 where 𝜔 nI (( 𝜔 nI )

0 ) denotes the NEBM-based lowest

f uniform temperature change. In Fig. 3 , the predicted results by the proposed

n models based on the NRBM and the NHOBM for three levels of the uniform

t nt with a low temperature. As it is seen, in the case of n = 2, there exists a

r the NRBM-based discrete model of this work and those of the NEBM-based

m HOBM are slightly lower than those of the NEBM. Such discrepancies in the

r

.2. Another comparison study in a particular case

In this section, comparing of the results obtained from the propos

ibrations of a nanosystem consists of multiple DWCNTs have not bee

t al. [39] studied properties of elastic transverse waves in an individu

uler–Bernoulli’s beam theory (NEBM). They proved the importance of

eristics of waves in such nanostructures. For the sake of verification st

b = 1 TPa, t b = 0.35 nm, d 2 = 1.4 nm, and d 1 = 0.7 nm. Let us to defi

requency of the n th vibration mode with(without) consideration of the

onlocal model of Tounsi et al. [39] and those of our suggested discrete

emperature change have been demonstrated in the case of environme

easonably good agreement between the predicted frequency ratios by

odel of Tounsi et al. [39] . Furthermore, the predicted results by the N

esults of these models are related to the effect of shear deformation.

591


Fig. 3. Comparison of the predicted frequency ratio of the second vibration mode by the proposed NRBM-based and NHOBM-based discrete models and those of

Tounsi et al. [39] for different values of the uniform temperature change; (( □) ΔT = 20, ( ○) ΔT = 40, ( △) ΔT = 60 °C; (...) Tounsi et al. [39] , ( − − − ) present study

based on the NRBM, ( —) present study based on the NHOBM; e 0 a = 2 nm).

Fig. 4. Influence of the temperature change in high temperatures on the out-of-plane fundamental frequency for different slenderness ratios: (a) 𝜆1 = 20, (b) 𝜆1 = 50,

(c) 𝜆1 = 100; (( − − − ) NRBM, ( —) NHOBM; ( ○) e 0 a = 0, ( □) e 0 a = 1, ( Δ) e 0 a = 2 nm; N = 100, 𝑇 1 = 𝑇 2 = Δ𝑇 ).

7 ty of the nanosystem

rse vibrations of the nanosystem is evaluated. For this purpose, in Fig. 4 (a)–

4 frequencies is plotted for thermally influenced nanosystems in the presence

o ntinuous models for three slenderness ratios of 20, 50 and 100, as well as

t ous theories show that the fundamental frequency of the nanosystem would

r n of the nanosystem’s transverse stiffness due to the increase in temperature.

F undamental frequency of slenderer nanosystems is more obvious. As shown

i ly the fundamental frequency lessens, but also the predicted results by the

N of 𝜆1 = 20 and e 0 a = 2 nm, the NRBM predicts the NHOBM results with a

r al change, while in the case of 𝜆1 = 50 and e 0 a = 2 nm, the NRBM predicts

t ll proposed models, the fundamental frequencies reduce by increasing the

s of the transverse stiffness to the mass. The effect of the small-scale parameter

i ifferent models approach each other, which is clearly seen in Fig. 4 (c). The

p an those of the NRBM. As shown in Fig. 4 (c), for a nanosystem with 𝜆1 = 100

s out-of-plane fundamental frequency is zero, and thereby, the nanosystem

w The lowest value of the temperature change with zero frequency is called

c an the critical value, the nonlinear deformation would be generated and the

.3. Role of the temperature gradient and a discussion on potential instabili

In this part, the effect of the temperature change on the free transve

(c), the effect of temperature variation on the fundamental out-of-plate

f high temperatures. The plotted results are based on the nonlocal co

hree values of the small-scale parameter 0, 1 and 2. The results of vari

educe by increasing the thermal change. This is because of the reductio

urther, the influence of the temperature changes on the variation of f

n Fig. 4 (b) and 4 (c), with the increase of the slenderness ratio, not on

RBM become closer to those of the NHOBM. For instance, in the case

elative error lower than 7 percent for the considered range of the therm

he NHOBM’s results with a relative error lesser than 2 percent. In a

mall-scale parameter. This behavior is due to the reduction of the ratio

s lowered by increasing the slenderness ratio such that the results of d

lotted results confirm that the NHOBM’s graphs are commonly lower th

ubjected to a temperature change greater than a specific value, the

ould be no longer stable according to the linear nonlocal elasticity.

ritical temperature change . Actually, for temperature changes greater th

592


Fig. 5. Influence of the temperature change in high temperatures on the in-plane fundamental frequency for different slenderness ratios: (a) 𝜆1 = 20, (b) 𝜆1 = 50,

(c) 𝜆1 = 100; (( − − − ) NRBM, ( —) NHOBM; ( ○) e 0 a = 0, ( □) e 0 a = 1, ( Δ) e 0 a = 2 nm; N = 100, 𝑇 1 = 𝑇 2 = Δ𝑇 ).

n strain relations. The large deformation/postbuckling behavior of thermally

a ould be pursued by the nanomechanics community in the near future.

re change of a nanosystem with slenderness ratio of 100 and the small-scale

p he critical temperature changes on the basis of the NHOBM are 1526, 1521,

a e predicted critical temperature change by the NRBM are greater than those

o change would grow with the small-scale parameter. This brief discussion

e Using the proposed nonlocal-continuous models by setting the fundamental

f re of the nanosystem based on the NRBM and NHOBM are derived as:

Δ

2 𝜚 𝑅 6 ( 𝜚 𝑅 8 )

2 𝜚 𝑅 2

)

1 2

−

𝜚 𝑅 5 𝜚 𝑅 9 − 𝜚 𝑅 6 𝜚

𝑅 8 − 𝜚 𝑅 8 𝜚

𝑅 2 + 𝜚 𝑅 9 𝜚

𝑅 4

2 𝜚 𝑅 7 𝜚 𝑅 8 𝜚

𝑅 9

, (46a)

Δ

𝜚 𝐻

11 𝜚 𝐻

10 𝜚 𝐻

12 𝜚 𝐻

6 𝜚 𝐻

7

𝐻

4 𝜚 𝐻

11 𝜚 𝐻

10 𝜚 𝐻

12 𝜚 𝐻

6 𝜚 𝐻

7

𝐻

11 𝜚 𝐻

10 𝜚 𝐻

12 ( 𝜚 𝐻

6 ) 2 𝜚 𝐻

2

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

1 2

, (46b)

w tary material, part F. Eq. (46 a) and ( 46 b) display that the critical values of

t ical data of the membrane as well as mechanical properties of both membrane

a the nanosystem would result in a higher critical temperature change.

res on the in-plane fundamental frequency has been demonstrated. A close

v plane frequencies are generally higher than those of the out-of-plane ones,

e

f-plane fundamental frequencies as a function of temperature change have

b se of temperature changes at high temperature situation, in this case, both

m increase as the temperature change grows. By comparing the frequencies

a at the rate of change of the out-of-plane frequency in terms of temperature

c redicted in-plane fundamental frequencies are commonly larger than those

o derness ratio, in addition to the reduction of the fundamental frequencies,

t the cases, the NHOBM’s results are lower than the NRBM’s results which is

m in energy of the nanosystem modeled by the NHOBM.

7

e of ensembles of CNTs is the slenderness ratio. The graphs of the out-of-plane

f own in Fig. 7 (a)–7 (c) for three values of the temperature change and three

onlinear strain terms should be incorporated into the nonlocal stress-

ffected membranes made from CNTs is also an interesting topic that sh

According to the NRBM, the critical values of the uniform temperatu

arameters 0, 1, and 2 nm in order are 1535, 1529, and 1511°C, while t

nd 1504°C, respectively. The comparison of these results shows that th

btained based on the NHOBM. Additionally, the critical temperature

ncourages us to evaluate the critical temperature changes analytically.

requency equal to zero, the explicit expression of the critical temperatu

𝑇 𝑅 𝑐𝑟

=

(

( 𝜚 𝑅 5 𝜚 𝑅 9 )

2 + 2 𝜚 𝑅 5 𝜚 𝑅 6 𝜚

𝑅 8 𝜚

𝑅 9 + 2 𝜚 𝑅 5 𝜚

𝑅 8 𝜚

𝑅 2 𝜚

𝑅 9 + 2 𝜚 𝑅 5 ( 𝜚

𝑅 9 )

2 𝜚 𝑅 4 + ( 𝜚 𝑅 6 𝜚 𝑅 8 )

2 ++ 2 𝜚 𝑅 6 𝜚

𝑅 8 𝜚

𝑅 9 𝜚

𝑅 4 + ( 𝜚 𝑅 2 𝜚

𝑅 8 )

2 + 2 𝜚 𝑅 8 𝜚 𝑅 2 𝜚

𝑅 9 𝜚

𝑅 4 + 4 𝜚 𝑅 1 𝜚

𝑅 3 𝜚

𝑅 8 𝜚

𝑅 9 + ( 𝜚 𝑅 9 𝜚

𝑅 4 )

2

2 𝜚 𝑅 7 𝜚 𝑅 8 𝜚

𝑅 9

𝑇 𝐻

𝑐𝑟 = −

⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ( 𝜚 𝐻

1 ) 2 ( 𝜚 𝐻

4 ) 2 ( 𝜚 𝐻

10 ) 2 ( 𝜚 𝐻

7 ) 2 + 2 𝜚 𝐻

1 𝜚 𝐻

9 𝜚 𝐻

8 𝜚 𝐻

4 ( 𝜚 𝐻

10 ) 2 ( 𝜚 𝐻

7 ) 2 − 2 𝜚 𝐻

1 ( 𝜚 𝐻

4 ) 2

− 2 𝜚 𝐻

1 ( 𝜚 𝐻

4 ) 2 ( 𝜚 𝐻

10 ) 2 𝜚 𝐻

6 𝜚 𝐻

2 𝜚 𝐻

7 + ( 𝜚 𝐻

9 ) 2 ( 𝜚 𝐻

8 ) 2 ( 𝜚 𝐻

10 ) 2 ( 𝜚 𝐻

7 ) 2 − 2 𝜚 𝐻

9 𝜚 𝐻

8 𝜚

− 2 𝜚 𝐻

9 𝜚 𝐻

8 𝜚 𝐻

4 ( 𝜚 𝐻

10 ) 2 𝜚 𝐻

6 𝜚 𝐻

2 𝜚 𝐻

7 + ( 𝜚 𝐻

4 ) 2 ( 𝜚 𝐻

11 ) 2 ( 𝜚 𝐻

10 ) 2 ( 𝜚 𝐻

6 ) 2 + 2( 𝜚 𝐻

4 ) 2 𝜚

+ ( 𝜚 𝐻

4 ) 2 ( 𝜚 𝐻

10 ) 2 ( 𝜚 𝐻

6 ) 2 ( 𝜚 𝐻

2 ) 2 + 4( 𝜚 𝐻

3 ) 2 ( 𝜚 𝐻

4 ) 2 ( 𝜚 𝐻

10 ) 2 𝜚 𝐻

6 𝜚 𝐻

7

(2 𝜚 𝐻

4 𝜚 𝐻

10 𝜚 𝐻

6 𝜚 𝐻

7 𝜚 𝐻

5 )

+

𝜚 𝐻

4 𝜚 𝐻

11 𝜚 𝐻

12 𝜚 𝐻

6

(2 𝜚 𝐻

4 𝜚 𝐻

10 𝜚 𝐻

6 𝜚 𝐻

7 𝜚 𝐻

5 ) +

𝜚 𝐻

4 𝜚 𝐻

10 𝜚 𝐻

6 𝜚 𝐻

2 + 𝜚 𝐻

1 𝜚 𝐻

4 𝜚 𝐻

10 𝜚 𝐻

7 + 𝜚 𝐻

9 𝜚 𝐻

8 𝜚 𝐻

10 𝜚 𝐻7

(2 𝜚 𝐻

4 𝜚 𝐻

10 𝜚 𝐻

6 𝜚 𝐻

7 𝜚 𝐻

5 )

here 𝜚 𝑅 𝑖 ; 𝑖 = 1 , 2 , … , 9 and 𝜚 𝐻

𝑗 ; 𝑗 = 1 , 2 , … , 12 are given in supplemen

emperature change for membranes made from DWCNTs rely on geometr

nd elastic matrix. In fact, each factor that assists transverse stiffness of

In Fig. 5 (a)–(c), the effect of temperature change at high temperatu

erification of the plotted results in Figs. 4 and 5 indicate that the in-

specially in slenderer nanosystems.

In Fig. 6 , the diagrams of the variations of both in-plane and out-o

een demonstrated for nanosystems at low temperatures. Unlike the ca

odels predict that both in-plane and out-of-plane frequencies would

ssociated with the in-plane and out-of-plane vibrations, it is realized th

hange is greater than that of the in-plane frequency. In addition, the p

f the out-of-plane ones. The results show that by increasing the slen

he results of the two proposed models approach each other. In most of

ainly related to the incorporation of shear stresses into the elastic stra

.4. Role of the slenderness ratio

One of the most important parameters affecting free dynamic respons

undamental frequency as a function of slenderness ratio have been sh

593


Fig. 6. Influence of the temperature change in low temperatures on the in-plane and out-of-plane fundamental frequencies for different slenderness ratios; (( ○)

𝜆1 = 40, ( □) 𝜆1 = 50, ( Δ) 𝜆1 = 60; N = 80, 𝑇 1 = 𝑇 2 = Δ𝑇 ).

p

e

w

v

i

l

s

l

c

o

v

p

t

I

t

o

l

i

s

s

i

p

s

t

s

r

n

I

N

f

b

p

g

m

r

c

i

f

t

F

i

c

n

t

o

p

t

b

w

d

o

N

e

7

o

F

a

t

f

o

b

o

w

t

N

2

N

Δ

p

o

d

r

r

g

i

t

f

t

t

t

opulations. The obtained results based on the proposed nonlocal mod-

ls indicate that the fundamental frequency of the nanosystem decreases

ith the increase of slenderness ratio. The rate of reduction is more ob-

ious in nanosystems with more DWCNTs. On the other hand, with the

ncrease in the number of DWCNTs, the fundamental frequency would

essen, particularly in nanosystems with higher slenderness ratio. As

hown in Fig. 7 (a), the results predicted by the NRBM, especially in

ower slenderness ratios, are higher than the NHOBM’s results. By in-

reasing the slenderness ratio, the results of these models approach each

ther. This reflects the great effect of the shear deformation on the free

ibration of nanosystems with low slenderness ratio. By comparing the

rovided graphs in Fig. 7 (a)–7 (c), it is understood that by increasing the

emperature of the nanosystem, the fundamental frequency decreases.

n addition, the relative difference between the results of the NRBM and

hose of the NHOBM increases at high temperatures.

In Fig. 7 (b), in the case of ΔT = 2100 and N = 20, the frequency

f the nanosystem becomes zero in the slenderness ratio of 105. The

owest slenderness ratio in which the nanosystem frequency vanishes

s called the critical slenderness ratio . Actually, the nanosystem with

lenderness ratio great than the critical one would be dynamically un-

table. Therefore, identifying the critical slenderness ratio is important

n optimal design of the membranes from DWCNTs affected by the tem-

erature gradients. As it is seen in the demonstrated graphs, the critical

lenderness ratio would reduce as the temperature change increases for

hose nanosystems in environments with high temperatures. The critical

lenderness ratio in the cases of N = 30 and N = 100 reaches 90 and 85,

espectively. This means that the instability of the thermally affected

anosystems with higher population occurs at smaller slenderness ratio.

n the case of ΔT = 700, the relative difference between the results of the

RBM and NHOBM would reduced by increasing the slenderness ratio;

or example, in the cases of 𝜆1 = 10 and 110, the relative differences

etween the results of these models are, respectively, about 20 and 0.5

ercent. In the case of ΔT = 700 and 2100, as the slenderness ratio

rows the relative discrepancies between the results of the proposed

odels decrease up to a certain slenderness ratio. For the slenderness

atios greater than this particular level, the aforementioned discrepan-

ies increase with growing of the slenderness ratio such that reaches to

ts maximum value at the zero frequency. This issue is more apparent

or the thermally affected nanosystems with higher populations.

With regard to the crucial role of the slenderness ratio on the critical

emperature change of the nanosystem, this fact has been illustrated in

ig. 8 . The results are given for a membrane consists of 4000 DWCNTs

n the cases of e a = 0, 1 and 2 nm. The results of both nonlocal-
0 o
594

ontinuous-based theories indicate that the critical temperature of the

anosystem decreases with increasing the slenderness ratio. In addition,

he critical temperature of the nanosystem decreases with the increase

f the small-scale parameter. By comparing the results of the two

roposed models, the results of the NHOBM are generally lower than

hose of the NRBM. By increasing the slenderness ratio, the difference

etween the predicted critical temperatures by suggested models

ould reduce for all considered levels of the small-scale parameter. As

iscussed in previous parts, this issue is mainly related to the reduction

f the shear effect. The predicted results show that for 𝜆1 > 30, the

RBM could reproduce the results of the NHOBM model with relative

rror lesser than 7 percent, irrespective of the small-scale parameter.

.5. Role of the radius of constitutive tubes

In this part, the aim is to investigate the effect of the radius variation

f constitutive DWCNTs on the out-of-plane fundamental frequency.

or this purpose, three nanosystems with nanotube’s lengths of 20, 30

nd 50 nm are considered in a high temperature medium. According

o Fig. 9 , the results of proposed models indicate that the fundamental

requency of the nanosystem would increase by growing of the radius

f DWCNTs. This fact is more obvious at high temperatures; such vi-

rational behavior of the nanosystem can be attributed to the increase

f vdW forces due to the growth of radius. In all plots in Fig. 9 (a)–(c),

ith increasing radius of DWCNTs, the relative difference between

he fundamental frequencies predicted by the NRBM and those of the

HOBM generally increases. For example, for nanosystems of length

0 nm subjected to ΔT = 700 °C, NRBM can estimate the results of the

HOBM with relative error in the range of 1–5 percent. However, at

T = 2100 °C, the relative error of these models ranges from 2 to 5.5

ercent, which reveals an increase in discrepancies between the results

f the proposed models due to an increasing of the temperature. At

ifferent temperatures, it is also clearly observed that the variations of

elative discrepancies of two models’ results in terms of the radius would

educe by increasing the length of nanotubes. In all proposed models, by

rowing the temperature change from 700 to 2100°C, and also through

ncreasing the length of the nanotubes from 20 to 50 nm, the fundamen-

al frequency has dropped. Further studies show that the variation of

undamental frequency is higher in slenderer nanosystems. According

o the demonstrated results in Fig. 9 (b) and 9 (c), for a specific value of

he radius of the nanotubes, the fundamental frequency becomes zero in

he case of l b = 50 nm. The radius in which the fundamental frequency

f the nanosystem would be zero is called the critical radius. Based on


Fig. 7. Influence of the slenderness ratio in high temperatures on the out-of-plane fundamental frequency for different levels of the temperature change and

number of DWCNTs: (a) 𝑇 1 = 𝑇 2 = Δ𝑇 = 700, (b) 𝑇 1 = 𝑇 2 = Δ𝑇 = 1400, (c) 𝑇 1 = 𝑇 2 = Δ𝑇 = 2100 °C; (( − − − ) NRBM, ( —) NHOBM; ( ○) N = 20, ( □) N = 30,

( Δ) N = 100; e 0 a = 2 nm).

t

l

s

t

7

t

p

s

v

1

a

i

t

i

o

t

o

b

e

N

r

a

g

o

h

d

b

t

o

r

he suggested nonlocal theories, the critical radii for a nanosystem of

ength 50 nm at ΔT = 1400 and 2100°C are about 0.66 and 0.86 nm, re-

pectively. In other words, the critical radius of the nanosystem at high

emperatures would grow by an increase of the temperature change.

.6. Role of the adjacent elastic environment

Using nonlocal-continuous models based on the NRBM and NHOBM,

he effect of transverse stiffness of the elastic environment on the in-

lane and out-of-plane fundamental frequencies of the nanosystem is

hown in Fig. 10 . The results are provided for a nanosystem with 300

ertically aligned DWCNTs under temperature changes of 0, 800, and

600°C. For both proposed models, the fundamental frequencies associ-

ted with the in-plane and out-of-plane vibrations increase with increas-

ng the transverse stiffness. In addition, the fundamental frequencies of

he elastically embedded nanosystem at high temperatures reduce by

ncreasing the temperature. The influence of the temperature increase

595

n the estimated results by both models is approximately the same. Also,

he fundamental frequencies of the in-plane vibration and those of the

ut-of-plane vibration are nearly equal in most of the cases, which can

e attributed to the high level of the transverse stiffness of surrounding

nvironment in both y and z directions. The presented plots show that

HOBM’s results are always lower than the NRBM’s results. Also, the

elative discrepancies between the plotted results based on the NRBM

nd those of the NHOBM would decrease as the transverse stiffness

rows.

The role of the rotational stiffness of the neighboring environment

n the fundamental frequencies of the thermally affected nanosystem

as been shown in Fig. 11 . The results of the proposed models in-

icate that the fundamental frequencies of the nanosystem increase

y growing the rotational stiffness. In addition, by an increase of the

emperature change from zero to 1600°C, the fundamental frequencies

f the nanosystem would decrease. The plotted diagrams show that the

elative discrepancies between the results of the NRBM and those of the


Fig. 8. Variation of the critical temperature change as a function of the slenderness ratio for different values of the small-scale parameter; (( − − − ) NRBM, ( —)

NHOBM; ( ○) e 0 a = 0, ( □) e 0 a = 1, ( Δ) e 0 a = 2 nm; N = 4000, 𝑇 1 = 𝑇 2 = Δ𝑇 ).

Fig. 9. Influence of the mean radius of the innermost tube in high temperatures on the out-of-plane fundamental frequency for different values of nanotube’s length

and temperature change: (a) ΔT = 700, (b) ΔT = 1400, (c) ΔT = 2100 °C; (( − − − ) NRBM, ( —) NHOBM; ( ○) l b = 20, ( □) l b = 30, ( Δ) l b = 50 nm; e 0 a = 2 nm,

𝑇 1 = 𝑇 2 = Δ𝑇 ).

N

e

r

r

i

t

7

a

p

b

t

B

a

a

p

t

a

b

t

i

s

c

p

t

n

i

v

i

o

r

o

a

f

HOBM grow by an increase of the rotational stiffness of the adjacent

nvironment. For instance, in the case of ΔT = 800 and 𝐾

𝑅 𝑟

= 0, the

elative difference between the two models is about 3 percent, which

eaches 20 percent at 𝐾

𝑅 𝑟

= 100. The main reason of this fact is the

ncorporation of the rotational stiffness into the shear strain energy of

he thermally affected nanosystem modeled by the NHOBM.

.7. Role of the intertube distance

Herein the influence of the intertube distance on the thermally

ffected nanosystem’s vibration behavior is considered. For this pur-

ose, the effect of gap between DWCNTs at high temperatures on

oth in-plane and out-of-plane fundamental frequencies is studied for

hree nanosystems of populations N = 10, 20 and 80 (see Fig. 12 ).

y increasing the temperature change, the fundamental frequencies

ssociated with the in-plane and out-of-plane vibrations would reduce

ccording to the proposed nonlocal-continuous models. In a low-

opulation nanosystem, the in-plane frequency decreases with growing

596

he intertube distance such that it reaches to its locally minimum value

t d = 2.49 nm. As the distance increases, in contrast to the previous

ranch, the frequency increases, and for intertube distances greater

han 2.65 nm, the variation of the frequency as a function of the

ntertube distance becomes almost negligible. Further investigations

how that this trend is very similar to the variation of vdW force

oefficient in terms of the intertube distance. As the nanosystem’s

opulation increases, the rate of change of in-plane frequencies in

erms of intertube distance decreases such that in largely populated

anosystems, the fundamental frequency almost does not alter with the

ntertube distance. In contrast to the results obtained for the in-plane

ibration, the out-of-plane frequency has fairly no sensitive to the

ntertube distance. Generally, this behavior is due to the low variation

f the vdW forces in terms of the intertube distance. Additionally, the

elative discrepancy between the predicted in-plane frequencies based

n the NRBM and those of the NHOBM reaches to its maximum value

t the locally minimum points of the plots. This issue is more apparent

or thermally affected nanosystems with low number of nanotubes.


Fig. 10. Influence of the lateral stiffness of the elastic matrix in high temperatures on the in-plane and out-of-plane fundamental frequencies for different levels of

the temperature change; (( ○) 𝑇 1 = 𝑇 2 = Δ𝑇 = 0, ( □) 𝑇 1 = 𝑇 2 = Δ𝑇 = 800, ( Δ) 𝑇 1 = 𝑇 2 = Δ𝑇 = 1600 °C; ( − − − ) NRBM, ( —) NHOBM; N = 300, e 0 a = 2 nm).

Fig. 11. Influence of the rotational stiffness of the elastic matrix on the in-plane and out-of-plane fundamental frequencies of the nanosystem in high temperatures

for different temperature changes; (( ○) 𝑇 1 = 𝑇 2 = Δ𝑇 = 0, ( □) 𝑇 1 = 𝑇 2 = Δ𝑇 = 800, ( Δ) 𝑇 1 = 𝑇 2 = Δ𝑇 = 1600 °C; ( − − − ) NRBM, ( —) NHOBM; N = 300, e 0 a = 2 nm).

8

b

t

t

i

t

v

B

fi

t

h

i

n

s

T

r

o

o

o

c

fl

i

f

o

a

m

o

n

j

a

f

m

g

. Concluding remarks

Transverse vibrations of elastically confined monolayers mem-

ranes from vertically aligned DWCNTs in the presence of longitudinal

emperature field are examined using nonlocal elasticity theory. To

his end, a continuum-based model is developed for evaluating vdW

nteractional forces within the nanosystem. This model explains simply

hat how variation of such forces could be linearly linked to the

ariations of the in-plane and out-of-plane deflections of nanotubes.

y deploying a nonlocal Fourier law model, the nonlocal temperature

elds and the resulted thermal forces within nanotubes because of

he longitudinal temperature gradients are calculated accounting for

eat dissipation through the surface of constitutive DWCNTs. Then, by

mplementing NRBM and NHOBM and the Hamilton’s principle, both

onlocal-discrete and nonlocal-continuous models are developed and

olved for fundamental frequencies using assumed mode approach.

he recently novel continuous models enable us to study free dynamic

esponse of thermally affected membranes of DWCNTs irrespective

597

f their populations. The explicit expressions of the in-plane and the

ut-of-plane frequencies based on the continuous-based models are

btained on the basis of the NRBM and NHOBM, and thereafter, the

ritical values of temperature change are evaluated analytically. The in-

uences of temperature change, slenderness ratio, radius of nanotubes,

ntertube distance, transverse and rotational stiffness of the elastic

oundation on both in-plane and out-of-plane fundamental frequencies

f the nanosystem acted upon by uniform temperature changes as well

s longitudinal temperature gradients are examined comprehensively.

The proposed nonlocal continuum-based models and the suggested

ethodologies in this work could be extended to vibrational analysis

f thermally influenced membranes made from multi-walled carbon

anotubes. Further, vibrational examination of three-dimensional

ungles of vertically aligned DWCNTs in the presence of longitudinal

nd transverse temperature gradients could be considered as a hot topic

or the future explorations. Through advancements in continuum-based

odeling of doubly orthogonal DWCNTs subjected to temperature

radients, we could also explore free vibrations of thermally affected


Fig. 12. Influence of the intertube distance on the in-plane and out-of-plane fundamental frequencies of the thermally affected nanosystem in high temperatures

for different temperature changes and numbers of DWCNTs: (a) N = 10, (b) N = 20, (c) N = 80; (( ○) 𝑇 1 = 𝑇 2 = Δ𝑇 = 0, ( □) 𝑇 1 = 𝑇 2 = Δ𝑇 = 500, ( Δ) 𝑇 1 = 𝑇 2 = Δ𝑇 = 1000 °C; ( − − − ) NRBM, ( —) NHOBM; e 0 a = 2 nm).

d

f

p

f

A

(

k

S

i

R

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[

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ouble-layered or even multi-layered orthogonal-membranes made

rom DWCNTs by some efforts. These important issues could be

aid attention to by the applied mechanics community in the near

uture.

cknowledgments

The financial supports of the Iran National Science Foundation

INSF) as well as K.N. Toosi University of Technology is gratefully ac-

nowledged.

upplementary material

Supplementary material associated with this article can be found,

n the online version, at 10.1016/j.ijmecsci.2018.06.018 .

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