Electro-thermo-torsional buckling of an embedded armchair DWBNNT using nonlocal shear deformable shell model

Composites: Part B 51 (2013) 291–299

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Electro-thermo-torsional buckling of an embedded armchair DWBNNT using nonlocal shear deformable shell model A. Ghorbanpour Arani a,b,⇑, M. Abdollahian a, R. Kolahchi a, A.H. Rahmati a a b

Faculty of Mechanical Engineering, University of Kashan, Kashan, Islamic Republic of Iran Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Islamic Republic of Iran

a r t i c l e

i n f o

Article history: Received 8 May 2012 Received in revised form 7 September 2012 Accepted 10 March 2013 Available online 22 March 2013 Keywords: A. Nano-structures A. Smart materials B. Buckling B. Thermomechanical

a b s t r a c t Electro-thermo-torsional buckling response of a double-walled boron nitride nanotube (DWBNNT) has been investigated based on nonlocal elasticity and piezoelasticity theories. The effects of surrounding elastic medium such as the spring constant of the Winkler-type and the shear constant of the Pasternak-type are taken into account. The van der Waals (vdW) forces are considered between inner and outer layers of nanotube. According to the relationship between the piezoelectric coefficient of armchair boron nitride nanotubes (BNNTs) and stresses, the first order shear deformation theory (FSDT) is used. Energy method and Hamilton’s principle are employed to obtain coupled differential equations containing displacements, rotations and electric potential terms. The detailed parameter study is conducted to investigate the effects of nonlocal parameter, elastic foundation modulus, temperature change, piezoelectric and dielectric constants on the critical torsional buckling load. Results indicate that the critical buckling load decreases when piezoelectric effect is considered. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction BNNTs are an important class of nanotubes, discovered in the mid 1990s. They can be considered as a rolled up hexagonal BN layer similar to carbon nanotubes (CNTs) [1]. Apart from having high mechanical, electrical and chemical properties, BNNTs have more stable piezoelectric property and better resistance to oxidation at high temperatures. Torsional buckling and postbuckling of CNTs have been investigated by some researchers [2–6]. Sun and Liu [4] presented the results of an investigation on combined torsional buckling of multi-walled carbon nanotubes (MWCNTs) under combined torque, axial loading and radial pressure based on the continuum mechanics model. Silvestre [5] studied the suitability of shell models to assess the buckling behavior of single-walled carbon nanotubes (SWCNTs). The torsional buckling of individual MWCNTs under two different loading conditions was analyzed by Wang and Wang [6]. At nano-length scale, size effects become more important, thus, the classical continuum elasticity model, which is a scale independent theory, is not suitable as it cannot

⇑ Corresponding author at: Faculty of Mechanical Engineering, University of Kashan, Kashan, Islamic Republic of Iran. Tel.: +98 913 1626594; fax: +98 361 5912424. E-mail addresses: [email protected], [email protected] (A. Ghorbanpour Arani). 1359-8368/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesb.2013.03.017

consider scale effects. Among size-dependent continuum theories, the theory of nonlocal continuum mechanics initiated by Eringen [7], has been widely noticed and applied to a wide variety of nano-size structure problems. Shen and Zhang [8] investigated the buckling and postbuckling of double-walled carbon nanotubes (DWCNTs) subjected to torsion in thermal environments. They used nonlocal shear deformable cylindrical shell and vdW interaction forces. Khademolhosseini et al. [9] studied size-effects in the torsional buckling of CNTs based on nonlocal elasticity shell models. In many practical applications, CNTs are embedded in an elastic foundation. Winkler-type elastic foundation is used in a lot of torsional buckling of CNTs problems [10–14]. For instance, the torsional buckling of a DWCNT embedded in an elastic medium was studied by Han and Lu [10]. They considered the effects of surrounding elastic medium and vdW forces between inner and outer nanotubes. Sun and Liu [11] investigated the mechanical behavior for dynamic torsional buckling of an embedded DWCNT considering vdW interaction. The influences of the aspect ratio, the buckling modes, and the surrounding elastic medium on the torsional instability of DWCNTs were studied by Natsuki et al. [13]. Small scale effect on torsional buckling of MWCNTs was presented by Hao et al. [14]. They developed a nonlocal multiple-shell model for the MWCNTs surrounded by an elastic medium and employ it to find the small scale effect on torsional loads coupling with the change in temperature.

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Since Winkler model considers only the parameter representing the normal pressure, it cannot be taken into account as an exact approximation of the elastic medium. Considering both normal pressure and transverse shear stress is a more generalized depiction of the elastic foundation. This model is called as Winkler–Pasternak foundation type. A limited number of studies have used the Pasternak-type model for depicting the mechanical characteristics of the elastic foundation. Murmu and Pardhan [15] presented Winkler and Pasternak foundation models for buckling analysis of SWCNTs. They also implemented nonlocal elasticity theory. The effects of the surrounding elastic medium, such as the Winkler and the Pasternak models are considered by Mohammadimehr et al. [16] for torsional buckling of a DWCNT. They showed that critical buckling torque increases with increasing the Pasternak shear modulus. Ghorbanpour Arani et al. [17] studied the thermal effect on the buckling analysis of a DWCNT embedded in an elastic medium. They modeled the interaction between matrix and the outer tube as a Pasternak foundation. All the above mentioned researches have been conducted on CNTs which are not smart materials. Recently, a large amount research works have been carried out on the buckling and vibration of smart materials which possess piezoelectric properties such as PZTs [18,19]. Studies on dynamic behavior of functionally graded cylindrical shell with PZT layers under moving loads were carried out by Sheng and Wang [18] based on the FSDT. In another attempt they [19] presented the report of an investigation into thermoelastic vibration and buckling characteristics of the functionally graded piezoelectric cylindrical shell. Among smart nanostructures, BNNTs have attracted more attention due to outstanding chemical, electrical and mechanical properties. Nevertheless a couple of papers have considered the buckling of BNNTs [20,21]. However, to date, no reports have been found in the literature on the torsional buckling of an embedded DWBNNT. Since BNNTs are smart materials, they can be used in many nanoelectromechanical devices, pressure sensors, electro–nano strain gauge and smart nanorobots, therefore, buckling analysis of such nanotubes become more important in designing smart nanorobots. The problem of buckling analysis of armchair BNNTs is more useful in the arms and the legs of nanorobots where the torsion load becomes prominent. Motivated by these considerations we aim to study torsional buckling of DWBNNTs embedded in Winkler and Pasternak foundation using nonlocal elasticity and piezoelasticity deformable shell theories.

2. Nonlocal elasticity and piezoelasticity theories Two renowned families of BNNTs are the zigzag and the armchair structures. Sai and Mele [22] revealed that the armchair structures exhibit an electric dipole moment linearly coupled to torsion while the zigzag structures show a longitudinal piezoelectric response under tension or compression (uniaxial strain state). The piezoelectric constants are introduced as h11 and h14 for zigzag and armchair tubes, respectively. Since the aim of this article is to investigate the torsional buckling behavior of BNNTs, the armchair structure has been selected. On the other hand, according to nonlocal elasticity theory introduced by Eringen [7], the stress at a reference point x is considered to be function of the strain at every point in the body. Therefore the nonlocal stress tensor (rmn(m, n = x, h, z)) in terms of classical stress which contains mechanical, thermal and piezoelectric terms, with coordinates x, h and z denoting radial, circumferential and axial direction of coordinate system, respectively, can be expressed as follows [20–22]:

8 rxx 9 > > > > > > > > > r hh > > > > > > > < =

rzz

> rhz > > > > > > > > > > > rzx > > > > : ;

 l2 r2

rxh

2

8 rxx 9 > > > > > > > > > r hh > > > > > > > < =

rzz

> rhz > > > > > > > > > > > rzx > > > > : ;

rxh

3

C 11

C 12

C 13

0

0

0

6 C 12 6 6 6 C 13 ¼6 6 0 6 6 4 0

C 22

C 23

0

0

C 23

C 33

0

0

0

0

C 44

0

0

0

0

C 55

0 7 7 7 0 7 7 0 7 7 7 0 5

0 0 0 0 0 C 66 9 8 9 1 2 08 0 e a xx x > > > > > > > > > > > > C 6 0 B> > > > > > > > e a hh h > > > C 6 B> > > > > > > > C 6 B> B< ezz = < az = C 6 0 6 B B DT C  C  6h > > > 2 e 0 hz > > > > C 6 14 B> > > > > > > > C 6 B> > > > > > > > A 4 0 @> 0 2 e > > > > zx > > > : : > ; ; 0 0 2exh

0 0

3

0 07 78 E 9 7> x > 0 0 7< = 7 Eh ; 0 07 : > ; 7> 7 Ez 5 0 0

ð1Þ

0 0

where emn, Em and am (m, n = x, h, z) are respectively, strains, electric fields and thermal expansion coefficients, Cpq(p, q = 1, . . . , 6) and r2 = @ 2/@x2 + (1/R2) @ 2/@h2 are the elastic constants and the Laplace operator, DT is the temperature change and l = e0 a is the small scale parameter; e0 is the nonlocal constant appropriate to a given material and a is the internal characteristic length. Also the nonlocal electric displacement relation for armchair BNNTs can be expressed as [23,25]:

8 9 8 9 > > < Dx > < Dx > = = 2 Dh  l2 r Dh > > : > : > ; ; Dz Dz

8 9 1 08 exx 9 ax > > > > > > > > > > > C > B> > > > > > > ehh > ah > C > > 2 3B> > > > > > > > > C 0 0 0 h14 0 0 B > > > > < = C = B< e a zz z C 6 7B DT C ¼ 4 0 0 0 0 0 0 5B  > C B> 0> > > 2ehz > > > > > > > C B > > > 0 0 0 0 0 0 B> > > > > C > > > > > 2ezx > > > A @> 0> > > > > : > : ; ; > 2exh 0 2 38 9 211 0 0 > < Ex > = 6 7 þ 4 0 222 0 5 Eh ; > : > ; Ez 0 0 233

ð2Þ

in which Dm(m = x, h, z), 2pp(p = 1, 2, 3) are the electric displacements and dielectric constants, respectively. It should be noted that the process of deriving Eqs. (1) and (2) are given in detail in Appendix A. 3. Fundamental formulation 3.1. Strain displacement relationships A schematic diagram of DWBNNT embedded in a Winkle and Pasternak foundations is shown in Fig. 1 in which geometrical parameters of length, L thickness, h and radius of each layer Ri(i = 1, 2) are also indicated. Based on FSDT, the displacement components of an arbitrary point in the cylindrical shell in terms of x, h ~; v ~ and w ~ can be written as: and z coordinates, denoted by u

~ ðx; h; z; tÞ ¼ uðx; h; tÞ þ z/x ðx; h; tÞ; u

v~ ðx; h; z; tÞ ¼ v ðx; h; tÞ þ z/h ðx; h; tÞ; ~ h; z; tÞ ¼ wðx; h; tÞ; wðx;

ð3Þ

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ZZZ

1 2 i ¼ 1;2;

Ui ¼

V

ðrxxi exxi þ rhhi ehhi þ rxhi cxhi þ rxzi cxzi þ rzhi czhi Dxi Exi ÞdV; ð6Þ

where i denotes the number of layers and i = 1 and i = 2 corresponds to inner and outer DWBNNT layers. The axial electric field (Ex) can be defined as a function of electric potential, (ux) as below:

Ex ¼ 

@ ux ; @x

ð7Þ

Substituting Eqs. (4) and (7) into Eq. (6) yields the virtual electrostatic energy dUi as follow: !   Z 2p Z L " @dui @wi @dwi dwi 1 @dv i 1 @wi @dwi þ þ N hhi þ 2 dU i ¼ Nxxi þ @x @x @x Ri Ri @h Ri @h @h 0 0   @dv i 1 @dui 1 @wi @dwi 1 @wi @dwi þ þ þ  Nxhi Ri @h @x @x Ri @h Ri @x @h @d/xi M hhi @d/hi @d/hi M xhi @d/hi þ þMxhi þ þ M xxi @x Ri @h @x Ri @h      @dwi 1 @dwi dv i @dui þ Q hi  Ri dxdh: þ d/hi þ Dxi h þ Q xi d/xi þ @x Ri @x Ri @h ð8Þ The resultant forces (Nxxi, Nhhi, Nxhi), moment resultants (Mxxi, Mhhi, Mxhi) and transverse shear force resultants (Qxi, Qhi) are defined as [26]:

8 9 8 9 > < Nxxi > = Z h=2 > < rxxi > = Nhhi ¼ rhhi dz; > > > h=2 > : ; : Nxhi rxhi ;   Z h=2   rxzi Q xi ¼ dz: Q hi rzhi h=2

Fig. 1. Schematic of embedded DWBNNT.

where u, v and w are displacement components of the mid-plane in the axial, circumferential and radial directions, respectively, /x and /h are the rotations of a transverse normal about the axial and circumferential directions [24,26]. The nonlinear strain–displacement relations are given by:

exx

 2 @u 1 @w @/ þ ¼ þz x; @x 2 @x @x

ð4aÞ

ehh ¼

 2   1 @v 1 @w z @/h þ 2 wþ þ ; R R @h @h 2R @h

ð4bÞ

cxh ¼

    @ v 1 @u @w @w @/h 1 @/x þ þz þ þ ; @x R @h @h @x @x R @h

ð4cÞ

cxz ¼ /x þ

@w ; @x

ð4dÞ

  1 @w  v  z/h þ /h : czh ¼ R @h

ð4eÞ

8 9 8 9 > < M xxi > = Z h=2 > < rxxi > = M hhi ¼ rhhi zdz; > > > h=2 > : ; : M xhi rxhi ; ð9Þ

The work done by the elastic medium can be expressed as:

W Elastic

Medium

¼

Z 2p Z 0

L

ðkw w2 þ kg r2 w2 ÞR2 dxdh;

ð10Þ

0

in which w2 corresponds to the lateral displacement of the outer layer, kw and kg are the spring constant of the Winkler type and the shear constant of the Pasternak type, respectively. As shown in Fig. 1, DWBNNT can be assumed as a set of concentric cylindrical shells with vdW interaction forces between the inner and the outer tubes which are equal in magnitude and opposite in sign. Therefore, the vdW interaction forces between two adjacent layers can be expressed as [21]:

p1 ¼ p12 ¼ cðw2  w1 Þ; R1 R1 p2 ¼  p12 ¼ c ðw2  w1 Þ; R2 R2

ð11Þ

where c is vdW interaction coefficient and can be written as:

c¼

320 erg=cm2 0:16d

2

;

ðd ¼ 0:145 nmÞ:

ð12Þ

So, the work done by vdW forces can be expressed as: 3.2. Hamilton’s principle

dW v dW ¼

To derive the equations of motion of DWBNNT embedded on Winkler and Pasternak foundations, the Hamilton’s principle is utilized, which is defined as follows [26]:

Z 0

T

ðdU  dW v dW  dW Elastic

Medium Þdt

¼ 0;

0

0

L

pi wi Ri dxdh ði ¼ 1; 2Þ:

ð13Þ

Substituting Eqs. (8), (10) and (13) into Eq. (5), integrating the displacement gradients by part and setting the coefficients of dui, dvi, dwi, d/xi, d/hi and dui to zero yield the equations of motions as follow:

ð5Þ

where U is the electrostatic energy, WvdW and WElastic Medium are the works done by vdW forces and elastic medium, respectively. The electrostatic energy for each layer of the longitudinally polarized DWBNNT (Ui) can be written as follows:

Z 2p Z

dui :

@Nxxi 1 @Nxhi þ ¼ 0; Ri @h @x

ð14aÞ

dv i :

1 @Nhhi @Nxhi Q hi ¼ 0; þ þ Ri @h @x Ri

ð14bÞ

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A. Ghorbanpour Arani et al. / Composites: Part B 51 (2013) 291–299

      @ @wi Nhhi 1 @ @wi 1 @ @wi dwi : þ 2 N xxi N hhi Nxhi  þ @x Ri @h @x Ri @h @x Ri @h   Q hi;h 1 @ @wi 2 Nxhi þ pi þ kg r w3  kw w3 ¼ 0; þ þ Q xi;x þ Ri @x @h Ri

@ 2 Uh1 @ 2 Uh1 @ 2 Ux1 g þ C g g C g g g þ þ 12 66 x h h x h @X@h @X@h @h2 @X 2   @W 1  12ks C 55 gx þ Ux1 ¼ 0; @X 2 x

@ 2 Ux1

!

ð18dÞ

ð14cÞ d/xi :

@M xxi 1 @M xhi þ  Q xi ¼ 0; Ri @h @x

ð14dÞ

d/hi :

1 @M hhi @M xhi þ  Q hi ¼ 0; Ri @h @x

ð14eÞ

@Dxi dui : ¼ 0: @x

ð14fÞ

The total torque applied to the end of the DWBNNT can assumed as [10–12]:

T ¼ T1 þ T2;

 x1 @ 2 Ux1 @ 2 Uh1 @ 2 Uh1 @2/ C 12 gx gh þ C 66 gx gx þ gh þ C 22 g2h 2 2 @X@h @n @h @X x @h     @W 1 @ H1  12ks C 44 gh  V 1 þ Uh1  12H14 gx ¼ 0; @h @X H14 gh

g2x

Nxh ¼ Nxh1 ¼ Nxh2 ¼

T1 2pR21

¼

T2 2pR22

ð16Þ

:

Dimensionless parameters are then defined as:

ðui ; v i ; wi Þ x R2 ; Ux ¼ /x ; Uh ¼ /h ; X ¼ ; nh ¼ ; h L R1 h h l Nxh C ij gx ¼ ; gh ¼ ; l ¼ ; Nxh ¼ ði; j ¼ 1; . . . ; 6Þ; ; C ij ¼ L R1 h C 11 h C 11 rffiffiffiffiffiffiffiffi h14 211 ui ci h kg kw h ; Kg ¼ : ; ci ¼ ; Kw ¼ H14 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; Hi ¼ C C 11 C h C h C 11 211 11 11 11 ð17Þ Substituting Eq. (9) into Eqs. (14) and using dimensionless parameters from Eq. (17) yields the following dimensionless equations of motion in terms of dimensionless displacements and rotations as:

g

U1

@X 2

! " !# @W 1 @ 2 V 1 @2V 1 @2U1 þ C 66 gh gx ¼ 0; þ þ gh @X @X@h @X@h @h2

þ C 12 gh gx

ð18aÞ " !# ! @2 U1 @W 1 @ 2 V 1 @2V 1 @2U1 þ C g g þ g þ C 22 g2h þ 66 x x h @X@h @h @X@h @h2 @X 2     @ H1 @W 1 þ H14 gx gh ð18bÞ þ ks C 44 gh gh  V 1 þ Uh1 ¼ 0; @X @h

C 12 gx gh

  @U 1 @V 1 @2W 1 @4W 1  2 ax g4x DT  C 22 g2h W 1 þ  ax g2x DT  C 12 gx gh þl @X @h @X 2 @X 4  2 ax g2h g2x DT þl

@ Uh1 @ 2 H1 ¼ 0;  gx @X @X 2

ð18fÞ

! " !# @ 2 U 2 C 12 gh gx @W 2 @ 2 V 2 C 66 gh @2V 2 @2 U2 þ g g þ þ þ ¼ 0; x h nh2 @X @X@h nh2 @X@h @h2 @X 2 ð18gÞ

C 12 gx gh @ 2 U 2 C 22 g2h @W 2 @ 2 V 2 þ 2 þ nh2 @X@h @h nh2 @h2 " !# 2 2 @ V2 @ U2 H14 gx gh @ H2 þ þ gh þ C 66 gx gx 2 @X@h nh2 @X @X     ks C 44 gh @W 2  V 2 þ Uh2 ¼ 0; gh þ nh2 @h

@4W 1 @X 2 @h2

@4W 1

 ah g2h DT

@2 W 1 @h2

 2 ah g2x g2h DT þl

" þ ks C 44 gh gh

2

@h

2



þ c1 ðW 2  W 1 Þ  l g

@2W 2 @X 2



@2W 1 @X 2

! 2

2 h c1

l g

 2 ax g4x DT þl þ

@2W 2 @h2

@4W 2 @X

4

þ

l 2 ax g2h g2x DT @ 4 W 2

2 l 2 ah g2x g2h DT @ 4 w n2h2

@X 2 @h

n2h2 þ 2

2



2

@X @h

ah g2h DT @ 2 W 2 n2h2

@h2

l 2 ah g4h DT @ 4 W 2 2Nxh gx gh @ 2 W 2 n4h2

þ

@h4

nh2

@X@h

 2 Nxh g3x gh @ 4 W 2 2l  2 Nxh gx g3h @ 4 W 2 2l  3 nh2 n3h2 @X@h3 @X @h ! " ! # @ 2 W 2 @ Ux2 ks C 44 gh gh @ 2 W 2 @V 2 @ Uh2 þ ks C 55 gx gx þ  þ þ @X nh2 nh2 @h2 @h @h @X 2 ! c @ 2 W 2 @ 2 W 1 H14 gx gh @ 2 H2 c  2 g2x   ðW 2  W 1 Þ þ l þ nh2 @X 2 nh2 @X@h nh2 @X 2 ! l 2 g2 c @ 2 W 2 @ 2 W 1 @2W 2  2 K w g2x  KwW 2 þ l  þ 3h 2 2 @h @h nh2 @X 2 

@X 2 @h2

l 2 K w g2h @ 2 W 2 n2h2

2

@h

þ K g g2x

 2 K g g2x g2h @ 4 W 2 2l n2h2



@2W 1 @h2

! ¼ 0;

@ 2 Ux2

 @X 2 @h2

@2W 2 @X

2

þ

K g g2h @ 2 W 2 n2h2

l 2 K g g4h @ 4 W 2 n4h2

@h4

2

@h

 2 K g g4x l

@4W 2 @X 4

¼ 0;

C 12 gx gh @ 2 Uh2 C 66 gh @ 2 Uh2 gh @ 2 Ux2 g þ gx þ þ 2 nh2 @X@h nh2 @X@h nh2 @h2 @X   @W 2 þ Ux2 ¼ 0;  12ks C 55 gx @X 2 x

! # @V 1 @ Uh1 @ 2 H1 þ þ H14 gx gh @h @h @X@h

2 x c1

ð18hÞ

  C 12 gx gh @U 2 C 22 g2h @V 2 @2W 2  ax g2x DT W2 þ  2 nh2 @X @h nh2 @X 2



1 @4w

þ 2N xh gx gh

@2 W 1



þ

@2W 1 @4W  2 Nxh g3x gh 3 1  2l 4 @X@h @h @X @h ! 4 2 @ W @ W @ U 1 1 x1  2 N xh gx g3h þ ks C 55 gx gx þ  2l @X @X@h3 @X 2

 2 ah g4h DT þl

þ H14

!

ðU i ; V i ; W i Þ ¼

2

!

ð18eÞ

ð15Þ

in which T1 and T2 are the torque applied to the outer and the inner tubes, respectively. Assuming that the shear membrane forces in each layer of a DWBNNT are identical, the following relations can be written [10–12]:

2@ x

@ 2 W 1 @V 1  @X@h @X

!

ð18iÞ !

 x2 C 12 gx gh @ 2 Ux2 C 22 g2h @ 2 Uh2 @ 2 Uh2 gh @ 2 u þ C g g þ þ 2 66 x x 2 2 nh2 @X@h n @n nh2 @h @X h2 x @h     g @W 2 @ H2  12ks C 44 h  V 2 þ Uh2  12H14 gx ¼ 0; nh2 @h @X

ð18jÞ !

ð18kÞ

295

þ H14

@ Uh2 @ 2 H2 ¼ 0;  gx @X @X 2

ð18lÞ

where ks is the shear correction factor. 3.3. Solution to equations of motion The boundary conditions considered in this study are included a simply supported DWBNNT subjected to a torsional moment and zero electrical boundary condition (short-circuit [28]) at both ends of the cylinder. It should be noted that the critical buckling of the DWBNNT does not depend on the mechanical boundary condition when the length of the nanotube is long enough [11]. In order to solve Eqs. (18), the following buckling modes are assumed [16]:

U i ¼ U i cosðmpX  nhÞ;

ð19aÞ

V i ¼ V i cosðmpX  nhÞ;

ð19bÞ

W i ¼ W i sinðmpX  nhÞ;

ð19cÞ

Uxi ¼ Uxi cosðmpX  nhÞ;

ð19dÞ

Uhi ¼ Uhi cosðmpX  nhÞ;

ð19eÞ

Hi ¼ Hi sinðmpX  nhÞ;

ð19fÞ

where U i ; V i ; W i ; Uxi ; Uhi and Hi are arbitrary dimensionless coefficients, m and nare the axial half wave and circumferential wave numbers, respectively. Substituting Eqs. (19) into Eqs. (18) leads to:

½MfYg ¼ f0g;

ð20Þ

in which the matrices in Eq. (20) are expressed in detail in Appendix B. In order to obtain a non-zero solution for Eq. (20), the determinate of the coefficient matrix must be zero.

det½M ¼ 0:

ð21Þ

The solution of Eq. (21) yields the critical buckling load.

C 12

(a)

e0a = 0 (nm) e0a = 0.5 (nm)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

20

40

60

80

100

120

140

160

180

200

0.3

(b)

e0a = 0.5 (nm) e0a = 1.0 (nm) e0a = 1.5 (nm)

0.25

e0a = 2.0 (nm)

0.2

0.15

0.1

0.05

20

40

60

80

100

120

140

160

180

200

Fig. 2. Effect of small scale parameter on the dimensionless nonlocal critical buckling load for the case of high temperature and n = 1.

Et E ¼ ; C 44 ¼ C 66 ¼ C 55 ¼ ; 2ð1 þ tÞ 1  t2 ð22Þ

where the elastic modulus E = 1.8 (TPa) and the Poisson’s ratio t = 0.34.

Dimensionless Nonlocal Critical Buckling Load

0.18

In this section the numerical results are drawn in Figs. 2–10, showing the effects of Winkler and Pasternak coefficients, temperature change, small scale parameter, axial half and circumferential wave numbers on the nonlocal critical torsional buckling load of an armchair DWBNNT. Dimensions, mechanical, electrical and thermal properties used in numerical results for the embedded DWBNNTs are taken as follows [16,20,22]: The radii, thickness and length of nanotube are assumed to be R1 = 11.43 (nm) (innermost radius), R2 = 12.31 (nm) (outermost radius), h = 0.95 (nm) and L/R1 = 10. The vdW interaction coefficient c = 9.512  1019 (N/m3), the piezoelectric coefficient h14 = 0.95 (C/ m2), the dielectric constant e11 = 0.9824  108 (C/Vm), the Winkler constant kw = 8.999  1017 (N/m3) and the Pasternak constant kg = 2.07127 (N/m). The values of temperature change and thermal expansion coefficients at high temperatures are DT = 50 (K), ax = 1.2  106 (1/K) and ah = 0.6  106 (1/K). The elastic coefficient introduced in Eq. (1) can be defined as follows [20]:

C 11 ¼ C 22

0.8

The axial half wavenumber (m)

4. Numerical results and discussion

E ¼ ; 1  t2

0.9

The axial half wavenumber (m) Dimensionless Nonlocal Critical Buckling Load

H14 gh @ 2 W 2 @V 2  nh2 @X@h @X

Dimensionless Nonlocal Critical Buckling Load

A. Ghorbanpour Arani et al. / Composites: Part B 51 (2013) 291–299

!

0.16 0.14 0.12 0.1 3

0.08

k = 0 (N/m ) w

3

k = 8.9*e17 (N/m ) w

0.06

3

k = 17.9*e17 (N/m ) w

3

k = 35.9*e17 (N/m ) w

0.04 20

40

60

80

100

120

140

The axial half wavenumber (m) Fig. 3. Effect of Winkler constant on the dimensionless nonlocal critical buckling load for the case of high temperature, n = 1 and l = 1 (nm).

Fig. 2a depicts the local and nonlocal critical buckling loads under torsional load versus the axial half wave number for the circumferential wave number of 1 (n = 1). It is shown that

296

A. Ghorbanpour Arani et al. / Composites: Part B 51 (2013) 291–299

Dimensionless Nonlocal Critical Buckling Load

Dimensionless Nonlocal Critical Buckling Load

0.35 kg = 0 (N/m) kg = 2 (N/m)

0.3

kg = 4 (N/m) kg = 6 (N/m)

0.25 0.2 0.15 0.1 0.05 0

20

40

60

80

100

120

0.02 m=5 m = 10 m = 15 m = 20 m = 40 m = 60

0.015

0.01

0.005

140

5

10

The axial half wavenumber (m) Fig. 4a. Effect of Pasternak constant on the dimensionless nonlocal critical buckling load for the case of high temperature, n = 1 and l = 1 (nm).

g

0.025

0.02

80

100

120

140

160

Dimesnsionless Nonlocal Critical Buckling Load

Dimensionless Nonlocal Critical Buckling Load

k = 0 (N/m)

60

180

30

∈ = 1.0e-8 11

∈ = 1.5e-8

0.0937

11

∈ = 2.0e-8 11

0.0937 0.0937 0.0937 0.0937 0.0937 0.0937 0.0937 0.0937

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

h14

Fig. 4b. Dimensionless nonlocal critical buckling load versus the axial half wave number for the case of high temperature, l = 1 (nm) and Kg = 0.

Fig. 6. Effect of piezoelectric and dielectric constants on the dimensionless nonlocal critical buckling load for the case of high temperature, m = 20 and l = 1 (nm).

0.18

0.2 n=1 n=2 n=3 n=4

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 20

40

60

80

100

120

140

160

180

200

The axial half wavenumber (m)

Dimensionless Nonlocal Critical Buckling Load

Dimensionless Nonlocal Critical Buckling Load

25

0.0937

The axial half wavenumber (m)

0

20

Fig. 5b. Effect of axial half wave number on dimensionless nonlocal critical buckling load versus the circumferential wave number for the case of high temperature and l = 1 (nm).

0.03

0.015 40

15

The circumferential wavenumber (n)

ΔT = 0 Δ T = 100 Δ T = 200

0.16

0.14

0.12

0.1

0.08

0.06

20

40

60

80

100

120

140

160

180

200

The axial half wavenumber (m) Fig. 5a. Effect of circumferential wave number on dimensionless nonlocal critical buckling load versus the axial half wave number for the case of high temperature and l = 1 (nm).

Fig. 7. Effect temperature change on the dimensionless nonlocal critical buckling load for the case of high temperature, n = 1 and l = 1 (nm).

A. Ghorbanpour Arani et al. / Composites: Part B 51 (2013) 291–299

Dimensionless Nonlocal Critical Buckling Load

0.0386 The case of high temperature The case of low temperature

0.0384

0.0382

0.038

0.0378

0.0376

0.0374

0

20

40

60

80

100

120

140

160

180

200

Temperature change (K) Fig. 8. dimensionless nonlocal critical buckling load versus temperature change for two cases of low and high temperatures.

25 n = 1 Mohammadimehr et al. [16]

Nonlocal Critical Buckling Load (N/m)

n = 2 Mohammadimehr et al. [16] n = 3 Mohammadimehr et al. [16] n = 4 Mohammadimehr et al. [16] n = 1 Present work

20

n = 2 Present work n = 3 Present work n = 4 Present work

15

10

5

0

50

100

150

200

250

The axial half wavenumber (m) Fig. 9. Comparison present results with those obtained by Ref. [16].

5.5 nonlocal- Hao et al. [14] classical- Hao et al. [14]

5

nonlocal- Present work

4.5

classical- Present work

Ln (shear stress) (Gpa)

4

297

consideration of small scale effect, decreases critical buckling load. This suggests in nano mechanic analysis, when high precision is required, the small scale effect must be taken into account. Moreover, the difference between the local and the nonlocal critical buckling load is increased with increasing the axial half wave number. Variation of dimensionless nonlocal critical buckling load ðN xh Þ versus the axial half wave number (m) for different values of small scale parameter is illustrated in Fig. 2b. As can be seen, the N xh decreases as the small scale parameter is increased. Fig. 3 depicts the influence of Winkler constant on the N xh with respect to the axial half wave number. It is obvious from Fig. 3 that N xh increases with increasing the Winkler constant. This is perhaps because increasing Winkler constant increases the stiffness of embedded DWBNNT. In addition, it is clearly observed that the minimum N xh tends toward higher values of the axial half wave number with increasing Winkler constant. In order to show the influences of Pasternak constant on the nonlocal critical torsional load of the embedded DWBNNT, variation of N xh versus the axial half wave number is depicted in Fig. 4a. It is seen that N xh is increased with increasing the Pasternak shear constant because this leads to stiffer DWBNNT. Moreover, in lower values of axial half wave number, N xh reduces sharply down to the axial half wave number of 25 (m = 25), where a minimum is observed, before N xh increases again slightly. In order to show the minimum value N xh for Kg = 0, the N xh versus the axial half wave number is plotted in Fig. 4b. It is seen that the N xh becomes minimum at almost m = 90 when Kg = 0. Figs. 5a and 5b demonstrate the effects of axial half and circumferential wave numbers (m, n) on the N xh , respectively. It is observed that N xh decreases to its minimum value (critical buckling load) by increasing axial half wave and circumferential wave numbers, before N xh increases again slightly. Fig. 6 illustrates the effect of piezoelectric and dielectric constants on N xh for the axial half wave number of 20(m = 20). The results indicate that N xh decreases with increasing the piezoelectric constant, while it increases with increasing the dielectric constant. The influence of temperature change on the graph of dimensionless nonlocal critical buckling load N xh versus the axial half wave number is shown in Fig. 7. As can be seen, the N xh decreases slightly as the temperature change (DT) is increased, especially at higher values of axial half wave numbers. The reason is that the equivalent stiffness decreases with increasing temperature change. Fig. 8 shows the variation of dimensionless nonlocal critical buckling load in term of temperature change for two cases of low temperatures (i.e. negative thermal expansion coefficients [14]) and high temperatures (i.e. positive thermal expansion coefficients [14]). According to the results illustrated in Fig. 8, for the case of high temperature, the dimensionless nonlocal critical buckling load decreases with increasing temperature change while N xh increases as DT is increased. The results obtained are the same as those expressed in [14], indicating validation of our work.

3.5

5. Validation

3 2.5 2 1.5 1 0.5 0

5

10

15

20

25

30

35

40

45

m Fig. 10. Comparison present results with those obtained by Ref. [14].

50

To the best of the authors’ knowledge no published literature is available for comparison the torsional buckling response of the DWBNNT. However, the present results can be validated by the other published literatures in the torsional buckling of DWCNTs. In this regard, the simplified result of this paper is compared with the work of Mohammadinehr et al. [16]. The results are shown in Fig. 9 which indicates nonlocal buckling load versus axial half wave number. As can be observed results are almost the same as those in Ref. [16]. On the other hand, the trend of the critical shear stress versus axial half wave number can be qualitatively compared with Hao et al. [14]. Hence, Fig. 10 shows the comparison between present results and obtained results by Ref. [14].

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A. Ghorbanpour Arani et al. / Composites: Part B 51 (2013) 291–299

Eqs. (1) and (2) can be obtained by rewriting the expanded forms of Eqs. (A.5) and (A.6) in matrix form.

6. Conclusion Torsional buckling of an embedded armchair DWBNNT with considering small scale effects has been investigated here. The effect of surrounding elastic medium is considered by applying both Winkler-type and Pasternak-type models. The DWBNNT is assumed as a set of nested cylindrical shells in which the vdW interaction exists between all layers. Using the FSDT and the constitutive relation of a piezoelectric material, the governing equations of motion are derived. The following conclusions are made from the results obtained:  Since increasing the small scale parameter decreases nonlocal buckling load, the small scale effect on the torsional buckling load should not be neglected. Especially at high axial half wave numbers, where the difference between the nonlocal and the local buckling load increases.  The nonlocal critical torsional load is increased with increasing the Winkler and Pasternak constants.  Effect of temperature change on the critical buckling load is studied for two cases of low and high temperatures. It is found that nonlocal critical buckling load decreases with increasing temperature change for the case of high temperatures; while it increases as temperature change is increased for the case of low temperature.  It is shown that nonlocal critical buckling load decreases with increasing piezoelectric constant, while it increases with increasing dielectric constant.

Appendix B The matrices in Eq. (20) can be expressed as:

 ½M ¼

½A ½B

 ;

½A ¼ aij ; ½B ¼ bij ði ¼ 1; . . . 6Þ; ðj ¼ 1; . . . ; 12Þ ðB:1Þ

a11 ¼ g2x m2 p2  C 66 g2h n2 ;

a12 ¼ C 12 gx nmp þ C 66 gx nmp; a13

¼ C 12 gx gh mp; a21 ¼ C 12 gx gh nmp þ C 66 gx gh nmp;

a22

¼ C 22 g2h n2  C 66 g2x m2 p2  ks C 44 g2h ; a23 ¼ C 22 g2h n  ks C 44 g2h n;

a25 ¼ ks C 44 gh ; a26 ¼ H14 gx gh mp;

a31 ¼ C 12 gx gh mp ; a32 ¼ C 22 g2h n  ks C 44 g2h n;  2 ax g4x DTm4 p4 þ l  2 ax g2x g2h DTn2 m2 p2 a33 ¼ C 22 g2h þ ax g2x DTm2 p2 þ l  2 ah g2x g2h DTn2 m2 p2 þ l  2 ah g4h DTn4 þ 2N xh gx gh nmp þ ah g2h DTn2 þ l  2 N xh g3x gh nm3 p3 þ 2l  2 N xh gx g3h n3 mp  C 55 ks g2x m2 p2  ks C 44 g2h n2  c1 þ 2l  2 g2x m2 p2  c1 l  2 g2h n2 ;  c1 l

a34 ¼ C 55 ks gx mp; a35 ¼ C 44 ks gh n;

a36 ¼ H14 gx gh nmp;

Acknowledgments

 2 g2x m2 p2 þ c1 l  2 g2h n2 ; a39 ¼ c1 þ c1 l

The author thanks the reviewers for their reports to improve the clarity of this article. The authors are grateful to University of Kashan for supporting this work by Grant No. 65475/25. They would also like to thank the Iranian Nanotechnology Development Committee for their financial support.

Recently, Zhou et al. [27] extended the Eringen’s nonlocal elasticity theory to the piezoelectric nanostructures. For a homogeneous and nonlocal piezoelectric solid without body force, the following equations can be written [23,27]

Z

a44 ¼ g2x m2 p2  C 66 g2h n2  12C 55 ks gh ; ¼ C 12 gx gh nmp þ C 66 gx gh nmp a52 ¼ 12C 44 ks gh ;

aðjx0  xj  sÞ½C ijkl ðekl  ai DTÞ  ekij Ek ðx0 Þdx0 :

Di ¼

0

a53 ¼ 12C 44 ks gh n; a54

ðA:1Þ

a55 ¼ C 22 g2h n2  C 66 g2x m2 p2  12C 44 ks ; a62 ¼ H14 gh mp; a63 ¼ H14 gh nmp; b17 ¼ g2x m2 p2 

V

Z

a45

¼ C 12 gx gh nmp þ C 66 gx gh nmp

Appendix A

rij ¼

a43 ¼ 12C 55 ks gx mp;

0

0

aðjx  xj  sÞ½ekij ðekl  ai DTÞ  2ik Ek ðx Þdx :

ðA:2Þ

C 66 g2h n2 n2h2

;

b18 ¼

b19 ¼

C 12 gx gh mp nh2

b27 ¼

C 12 gx gh nmp C 66 gx gh nmp þ ; nh2 nh2

a56 ¼ 12H14 gx mp

a65 ¼ H14 mp; a66 ¼ gx m2 p2 C 12 gx nmp C 66 gx nmp þ ; nh2 nh2

V

rij;j ¼ qu€i ; Di;i ¼ 0: 1 2

ðA:3Þ ðA:4Þ

¼

In which ui is the electric potential; a(jx0  xj, s) is the nonlocal attenuation function; jx0  xj is the Euclidean distance; s = e0a/l is the scale coefficient, where l is the external characteristics of length. The integral constitutive equations can be rewritten as follows

b29 ¼ 

rij  l2 r2 rij ¼ C ijkl ðekl  ai DTÞ  ekij Ek ;

ðA:5Þ

b33 ¼

Di  l2 r2 Di ¼ eikl ðekl  ai DTÞ  2ik Ek :

ðA:6Þ

b38 ¼ 

eij ¼ ðui;j þ uj;i Þ; Ei ¼ u;i :

C 22 g2h n2 n2h2 C 22 g2h n n2h2

 C 66 g2x m2 p2 



ks C 44 g2h n ; nh2

ks C 44 g2h nh2

b211 ¼

c cl  2 g2x m2 p2 cl  2 g2h n2 þ þ ; nh2 nh2 n3h2 C 22 g2h n n2h2



ks C 44 g2h n n2h2

b28

ks C 44 gh H14 gx gh mp ; b212 ¼ nh2 nh2 b37 ¼

C 12 gx gh mp ; nh2

A. Ghorbanpour Arani et al. / Composites: Part B 51 (2013) 291–299

b39 ¼ 

C 22 g2h n2h2

þ

 2 ax g4x DTm4 p4 þ ax g2x DTm2 p2 þ l

l 2 ax g2x g2h DTn2 m2 p2

þ

n2h2 2

þ

l 2 ah g2x g2h DTn m2 p2

þ

n2h2

ah g2h DTn2 n2h2

l 2 ah g4h DTn4 n4h2

2Nxh gx gh nmp 2l g gh nm3 p3 þ nh2 nh2  2 Nxh gx g3h n3 mp c 2l ks C 44 g2h n2 þ  C 55 ks g2x m2 p2   3 nh2 nh2 n2h2 cl  2 g2x m2 p2 cl  2 g2h n2   nh2 n3h2 2

þ

 kg g2x m2 p2  

kg g2h n2 n2h2

 2 kg g2x g2h n2 m2 p2 2l n2h2

 2 g2x m2 p2   kw l b311 ¼

 2 g4x m4 p4  kg l



 2 g4h n4 kg l n4h2

 2 g2h n2 kw l n2h2

;

 kw

b310 ¼ C 55 ks gx mp;

C 44 ks gh n H14 gx gh nmp ; b312 ¼ nh2 nh2

b49 ¼ 12C 55 ks gx mp;

b411 ¼

Nxh 3x

b410 ¼ g2x m2 p2 

C 12 gx gh nmp C 66 gx gh nmp þ ; nh2 nh2

C 66 g2h n2 n2h2

 12C 55 ks gh

b58 ¼ 12C 44 ks gh ;

b59 ¼ 12C 44 ks gh n b510 ¼

C 12 gx gh nmp C 66 gx gh nmp þ ; nh2 nh2

b511 ¼ 

C 22 g2h n2 n2h2

 C 66 g2x m2 p2  12C 44 ks

b512 ¼ 12H14 gx mp;

b68 ¼

H14 gh mp H14 gh nmp ; b69 ¼ ; nh2 nh2

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