- Email: [email protected]
Nonlocal beam model for nonlinear analysis of carbon nanotubes on elastomeric substrates
Computational Materials Science 50 (2011) 1022–1029
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Nonlocal beam model for nonlinear analysis of carbon nanotubes on elastomeric substrates Hui-Shen Shen a,b,⇑, Chen-Li Zhang a a b
Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 7 October 2010 Received in revised form 28 October 2010 Accepted 29 October 2010 Available online 20 November 2010 Keywords: Carbon nanotube Nonlocal beam model Postbuckling Nonlinear bending Nonlinear vibration Elastic foundation
a b s t r a c t Postbuckling, nonlinear bending and nonlinear vibration analyses are presented for single-wall carbon nanotubes (SWCNTs) resting on a two-parameter elastic foundation in thermal environments. The SWCNT is modeled as a nonlocal nanobeam which contains small scale effects. The elastomeric substrate with finite depth is modeled as a two-parameter elastic foundation. The thermal effects are included and the material properties of both SWCNTs and the substrate are assumed to be temperature-dependent. The governing equation that includes beam–foundation interaction is solved by a two-step perturbation technique. The numerical results reveal that the small scale parameter e0a reduces the postbuckling equilibrium paths, the static large deflections and natural frequencies of SWCNTs resting on an elastic foundation. The results also reveal that the effect of the small scale parameter is significant for compressive buckling, but less pronounced for static bending and marginal for free vibration of SWCNTs resting on an elastic foundation. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Miniaturized beams are the core structures widely used in sensors, actuators, and Micro- and Nano-Electro-Mechanical Systems (MEMS/NEMS) [1,2]. Recently, many investigations focused on the modern nanotechnology involving carbon nanotubes (CNTs) embedded in an elastic matrix [3–5] or resting on an elastomeric substrate [6], in which the CNTs are modeled as a beam pinned at both ends and resting on an elastic foundation. One of the major differences between these two problems of CNTs lies in that the former assumes that the elastic foundation has an infinite depth, whereas in the latter the elastic foundation has only a finite depth. It has been reported that the small scale effect must play an important role in the nanoscale structures, e.g. carbon nanotubes and single layer graphene sheets. An important class of modified continuum models is that based on the concept of nonlocal elasticity [7]. These models allow the integration of small scale effects into classical continuum models. Unlike classical continuum models, the nonlocal elasticity theory assumes that the stress at a reference point in a body depends not only on the strains at that point, but also on strains at all other points of the body. The use of ⇑ Corresponding author at: Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China. Tel./fax: +86 21 34206197. E-mail address: [email protected] (H.-S. Shen). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.10.042
nonlocal elasticity to study size-effects in micro and nanoscale structures was first performed by Pedisson et al. [8]. They studied the bending of micro- and nanoscale beams with the concept of nonlocal elasticity and concluded that size-effects could be significant for nanoscale structures and that the magnitude of the sizeeffects strongly depends on the value of the small scale parameter e0. Since then, considerable attention has been directed to the development and application of nonlocal beam models for the buckling [9–11], vibration [12,13] and wave propagation [14,15] analyses of CNTs embedded in an elastic matrix. It is worthy to note that in most investigations [10–14] the Young’s modulus of CNTs was assumed to be about 1 TPa. Such a low value of Young’s modulus is due to the fact that the effective thickness of CNT is assumed to be 0.34 nm or more. However, as reported recently the effective thickness of SWCNTs should be smaller than 0.142 nm [16]. Therefore, the wide used value of 0.34 nm for tube wall thickness is thoroughly inappropriate to SWCNTs. On the other hand, the buckling configuration of individual single-wall carbon nanotubes (SWCNTs) resting on an elastomeric substrate is likely nonlinear under axial compression [17]. However, it remains unclear if the small scale parameter has the same effect on the postbuckling behavior of SWCNTs resting on an elastomeric substrate and this motivates the current investigation. The present work focuses attention on the postbuckling, nonlinear bending and nonlinear vibration of SWCNTs resting on an elastomeric substrate in thermal environments. The SWCNT is modeled as a nonlocal nanobeam which contains small scale
1023
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029
effects. The elastomeric substrate with finite depth is modeled as a two-parameter elastic foundation. The thermal effects are included and the material properties of both SWCNTs and the substrate are assumed to be temperature-dependent. The governing equation that includes beam–foundation interaction is solved by a two-step perturbation technique. The numerical illustrations show the nonlinear behavior of SWCNTs resting on elastic foundations under different sets of environmental conditions.
where DT = T T0 is temperature rise from some reference temperature T0 at which there are no thermal strains. In order to incorporate the small scale effect, continuum beam models need to be refined. This may be accomplished by using the nonlocal continuum theory. In the theory of nonlocal elasticity, the constitutive relations of nonlocal elasticity for 3D problems are expressed as [7]
ð1 s2 L2 r2 Þrij ¼ C ijkl ekl
ð3Þ
2. Nonlocal beam model for nonlinear analysis of SWCNTs
ð1aÞ
where s = e0a/L, rij and eij are the stress and strain tensors, and Cijkl is the elastic modulus tensor of classical isotropic elasticity, e0 is a material constant, and a and L are the internal and external characteristic lengths, respectively. The distinct difference between the classical and nonlocal elasticity theories lies in the presence of small scale parameter e0a in the nonlocal theory. For CNTs the characteristic length a may be taken as the length of the CAC bond, i.e. a = 0.142 nm, or taken as other material properties. It is a better way to use e0a as a single scale coefficient that captures the small scale effect on the response of structures in nanosize [10–15]. Shen [19] proposed a nonlinear beam theory for Euler–Bernoulli beams resting on elastic foundations. Applying Eq. (2) to this theory, the nonlinear motion equations for the nanobeam, including beam–foundation interaction, small scale effects and thermal effects, have readily been derived and can be expressed by
ð1bÞ
8 2 2 !2 31 !2 32 <@ 4 W 3 2 @W @ W @ W @W @W 41 5 þ 41 5 EI : @X 4 @X @X @X 3 @X 2 @X
Consider an SWCNT modeled as a nanobeam with length L, mean radius R and constant thickness h that rests on an elastic foundation. The beam is exposed to elevated temperature and is subjected to axial compressive loads P only in the X-direction or combined with transverse static or dynamic load Q. Let U be the displacement in the longitudinal direction, and W be the deflection of the beam (Fig. 1). As is customary [10–15], the foundation is assumed to be a compliant foundation, which means that no part of the beam lifts off the foundation in the large deflection region. The load–displacement relationship of the foundation is assumed to be 2 2 p ¼ K 1 W K 2 ðd W=dX Þ, where p is the force per unit area, K 1 is the Winkler foundation stiffness and K 2 is the shearing layer stiffness of the foundation. Since the elastomeric substrate is relatively soft, the foundation stiffness K 1 may be expressed by [18]
K1 ¼
E0 5 2c21 þ 6c1 þ 5 expð2c1 Þ 4L 1 m20 ð2 c1 Þ2
in which
c1 ¼ ðc1 þ 2Þ expðc1 Þ
and K 2 may be determined later in the numerical calculations. In Eq. (1), c1 ¼ Hs =L; E0 ¼ Es = 1 m2s , m0 ¼ ms =ð1 ms Þ; ES is Young’s modulus of the foundation, ms the Poisson’s ratio of the foundation, and Hs the depth of the foundation. It is assumed that the Young’s modulus of the foundation ES is temperature-dependent, whereas Poisson’s ratio ms depends weakly on temperature change and is assumed to be a constant. It is also assumed that the effective Young’s modulus E and thermal expansion coefficient a of SWCNTs are temperaturedependent. The thermal forces N T and moments M T caused by elevated temperature are defined by
ðNT ; MT Þ ¼
Z
ð1; ZÞEðTÞaðTÞTðX; ZÞdA
ð2aÞ
A
The temperature field is assumed to be uniform, and in such a case T
T
ðN ; M Þ ¼ EaDT
Z
ð1; ZÞdA
ð2bÞ
A
!2 32 !2 33 9 !3 2 = 41 þ 3 @W 541 @W 5 ; @X @X @X 2
@2W
þ
1 s2 L 2
¼
2
P
@2W 4 @X 2
@2 @X
!( Q K1W K2
2
@W 1 @X
@X
!2 33=2 5
EaDT
! qA
2
@2 @X 2
Z A
@2W @t2
9 = ZdA ;
ð4Þ
Note that Eq. (4) is valid for the case of nanobeam with movable end conditions. In contrast, for the case of nanobeam with immovable end conditions, the motion equation becomes
8 2 2 !2 31 !2 32 <@ 4 W 3 2 @W @ W @ W @W @W 41 5 þ 41 5 EI : @X 4 @X @X @X 3 @X 2 @X
þ
!2 32 !2 33 9 !3 2 = 41 þ 3 @W 541 @W 5 ; @X @X @X 2
@2W
1 s2 L 2
¼
@2 @X
!( Q K1W K2
2
2
EAaDT
Fig. 1. An SWCNT with pinned ends resting on a two-parameter elastic foundation.
@2W
@2W @X 2
@2W 4 @X 2
EaDT
@W 1 @X @2
@X 2
Z
!2 33=2 5
@2W @X
2
! qA
@2W @t 2
2 !2 3 Z EA 4 L @W þ dX 5 2L 0 @X
) ZdA
ð5Þ
A
where I = pR3h is the second moment of area, A = 2pRh is the area of the cross section, q is the mass density.
1024
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029
The solutions of Eq. (9) are now determined by a two-step perturbation technique, where the small perturbation parameter has no physical meaning at the first step, and is then replaced by a dimensionless deflection at the second step. The essence of this procedure, in the present case, is to assume that
Introducing following dimensionless quantities
x¼
g¼
X ; L L2 A
p2 E
W¼
W ; L
kP ¼
;
ðK 1 ; K 2 Þ ¼
PL2
p2 EI
;
kq ¼
!
K 1 L4 ; p4 EI
K 2 L2 ; p2 EI
sffiffiffiffi pt E t¼ ; L q
QL3
ð6Þ
p4 EI
enables nonlinear Eqs. (4) and (5) to be written in dimensionless form as
"
# 4 @ W 2 @W 4 @W 1 þ p þ p þ @x4 @x @x " # 2 4 3 2 @ W @ W @W 2 @W 4 @W þ 4p2 1 þ 2 p þ 3 p þ @x3 @x2 @x @x @x !3 " # 2 4 2 2 @ W 2 @W 4 @W 1 þ 6p þ 15p þ þp @x2 @x @x !( ! @2 @2W @2W kq K 1 W K 2 g 2 ¼ 1 s2 p2 2 2 @x @x @t " #) 2 4 2 @ W 3 2 @W 15 4 @W 1 þ p þ p þ kP @x2 2 @x 8 @x 4
2
ðfor movable end conditionsÞ "
2
ðfor immovable end conditionsÞ
ð8Þ
"
# 2 4 dW 4 dW þ p þ 4 dx dx dx " # 2 4 3 2 2 d W d W dW 2 dW 4 dW 1 þ 2p þ 3p þ þ 4p 3 2 dx dx dx dx dx 1 þ p2
#
2 4 dW 4 dW þ 15 p þ 2 dx dx dx !( " # 2 4 2 @2 d W 3 2 dW 15 4 dW kP 1 þ p þ p þ þ 1 s 2 p2 2 2 2 dx 8 dx @x dx !) 2 d W ¼0 ð9Þ þ K 1W K2 2 dx 2
þ p2
d W
ej kj
ð10Þ
j¼0
where e is a small perturbation parameter. Substituting Eq. (10) into Eq. (9), collecting the terms of the same order of e, a set of perturbation equations is obtained and can be solved step by step. The solution methodology is described in Shen [19]. As a result, the asymptotic solutions can be obtained as
3 ð1Þ ð1Þ WðxÞ ¼ A10 e sin mx þ A10 e ½A330 sin 3mx
5 ð1Þ þ A10 e ½A530 sin 3mx þ A550 sin 5mx þ Oðe6 Þ
ð11Þ
2
4 ð1Þ ð1Þ kP ¼ k0 þ k2 A10 e þ k4 A10 e þ Oðe6 Þ
ð12Þ
ð1Þ In Eqs. (11) and (12), A10 e is taken as the second perturbation parameter relating to the dimensionless maximum deflection Wm. From Eq. (11), taking (x = p/2m) yields
3 2 p a330 W 2m þ 8
ð13Þ
2 4 ð2Þ ð4Þ kP ¼ kð0Þ p þ kp W m þ kp W m þ
In such a case kq = 0, W is only a function of x. The end of the beam is movable, and Eq. (7) can be written in the simple form as
!3 "
X
Substituting Eq. (13) into Eq. (12), the postbuckling equilibrium path can be written as
3. Postbuckling problem
4
ej wj ðxÞ; kP ¼
j¼1
ð1Þ
Eqs. (7) and (8) are for movable and immovable end conditions respectively, and are adopted in the following nonlinear analysis.
d W
X
A10 e ¼ W m
#
4 @ W 2 @W 4 @W 1 þ p þ p þ @x4 @x @x " # 2 4 3 2 2 @ W @ W @W 2 @W 4 @W þ 4p 1 þ 2p þ 3p þ @x3 @x2 @x @x @x !3 " # 2 4 2 2 @ W 2 @W 4 @W 1 þ 6p þ 15p þ þp @x2 @x @x !( ! @2 @2W @2W g kq K 1 W K 2 ¼ 1 s2 p2 2 @x2 @x @t 2 "Z # 2 pg p @W @2W þ dx @x @x2 2 0 " #) 2 4 @2W 3 2 @W 15 4 @W gaDT 1þ p þ p þ @x2 2 @x 8 @x 4
ð7Þ
Wðx; eÞ ¼
ð14Þ
In Eq. (14) , kpðiÞ ði ¼ 0; 2; 4; . . .Þ are described in detail in Appendix A. For all cases discussed below, three kinds of SWCNTs are selected as nanobeams. It has been reported that the material properties of CNTs are chirality- and size-dependent [20] and temperature-dependent [21]. From MD simulations the effective material properties for armchair (12, 12) and (10, 10)-tubes and zigzag (17, 0)-tube are obtained, and the details of MD simulations may be found in [21]. Typical results are listed in Table 1. It is noted that the effective wall thickness obtained is h = 0.0671 nm, 0.0674 nm and 0.0871 nm for (12, 12), (10, 10) and (17, 0) tubes, respectively. Poly (dimethylsiloxane), referred to as PDMS, is selected for the substrate, and the material properties of which are assumed to be mS = 0.48 and ES = (3.22 0.0034T) GPa, in which T = T0 + DT and T0 = 300 K (room temperature). In such a way, ES = 2.2 GPa at T = 300 K. Since ES is temperature-dependent, the foundation stiffness is also temperature dependent. Taking HS = 1 mm and L/R = 100, from Eq. (1) K 1 (in GPa/nm) and its alternative dimensionless form K1 can be obtained. K2 is taken to be 1–10th of the value of K1 for the same tube in thermal environmental conditions. The results obtained are listed in Table 2, and are adopted in the following calculations. Table 3 presents the buckling loads (nN) and the corresponding buckling mode (m), which determines the number of half-waves in
Table 1 Temperature-dependent material properties for (12, 12), (10, 10) and (17, 0)-tubes. T (K)
1 þ 6 p2
300 500 700 a b c
(12, 12)-tubea
(10, 10)-tubeb 6
(17, 0)-tubec
E (TPa)
a(10 /K)
E (TPa)
a(10 /K)
E (TPa)
a(106/K)
5.5272 5.3783 5.3351
3.8164 4.7113 4.5959
5.6466 5.5308 5.4744
3.4584 4.5361 4.6677
2.5176 2.5066 2.4890
3.9676 4.5904 4.0577
R = 0.816 nm, h = 0.0671 nm. R = 0.681 nm, h = 0.0674 nm. R = 0.669 nm, h = 0.0871 nm.
6
1025
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029 Table 2 The foundation stiffnesses for (12, 12), (10, 10) and (17, 0)-tubes resting on elastomeric substrate in thermal environments (HS = 1 mm and L/R = 100). T (K)
(12, 12)-tube
300 500 700
(10, 10)-tube
K 1 (GPa/nm)
(K1, K2)
K 1 (GPa/nm)
(K1, K2)
K 1 (GPa/nm)
(K1, K2)
0.07 0.05 0.03
(53.21, 5.32) (37.78, 3.78) (21.05, 2.10)
0.09 0.06 0.03
(51.85, 5.18) (36.57, 3.66) (20.42, 2.04)
0.09 0.06 0.03
(89.99, 9.00) (62.45, 6.24) (34.75, 3.47)
Table 3 Buckling loads Pcr (nN) for CNTs resting on an elastic foundation under axial compression in thermal environments (HS = 1 mm and L/R = 100).
a
(17, 0)-tube
T (K)
e0a (nm)
(12, 12)-tube
(10, 10)-tube
(17, 0)-tube
300
0.0 1.0 2.0 4.0
18.986 18.874 18.558 17.500
(3)a (3) (3) (3)
16.029 15.893 15.514 14.332
(3) (3) (3) (3)
12.735 12.656 12.434 11.303
500
0.0 1.0 2.0 4.0
15.500 15.393 15.084 14.055
(3) (3) (3) (3)
13.162 13.029 12.685 11.500
(3) (3) (3) (3)
10.046 (3) 9.967 (3) 9.747 (3) 9.064 (3)
700
0.0 1.0 2.0 4.0
10.295 (2) 10.274 (2) 10.211 (2) 9.981 (2)
8.685 8.658 8.582 8.311
(3) (3) (3) (3)
7.269 7.253 7.049 6.371
(3) (3) (3) (4)
(2) (2) (2) (3)
The number in brackets indicates the buckling mode (m) in X-direction.
the X-direction, for SWCNTs with movable end conditions under axial compression and resting on an elastic foundation in thermal environments. Three thermal environmental conditions, i.e. T = 300, 500 and 700 K, are considered. The key issue for successful application of the nonlocal continuum mechanics models to nanoscale structures is to determine the magnitude of the small scale parameter e0a. However, there are no experiments conducted to determine the value of e0a for CNTs. In the present example, the small scale parameter e0a is taken to be 0, 1.0, 2.0 and 4.0 nm. Note that here e0a = 0 represents the local beam model. It can be seen that the buckling loads are sensitive to the small scale parameter e0a. The results show that the (12, 12)-tube has the highest buckling load, whereas (17, 0)-tube has the lowest buckling load among the three tubes. The results confirm that the buckling loads are decreased with increase in temperature. Usually, the beam without any foundation will be buckled with m = 1. In contrast, in the present example, the buckling mode m P 2 when it rests on an elastic foundation. It can also be seen that the buckling mode m may be increased with increase in e0a, which means the wavelength kx (=2 L/m) becomes small when e0a is sufficiently larger.
Fig. 2 shows the effect of the small scale parameter e0a (=0.0, 2.0 and 4.0 nm) on the postbuckling behavior of a (10, 10)-tube subjected to axial compression and resting on an elastic foundation at T = 300 K. It can be seen that the postbuckling equilibrium path first goes down smoothly and then rises sharply when the deflection of the tube is sufficiently larger. The results confirm that the postbuckling equilibrium paths are virtually stable for nanobeams resting on elastic foundations. The small scale parameter e0a reduces the buckling loads, but increases the postbuckling strength when the nanobeam deflection is greater than 6 nm. Fig. 3 presents the postbuckling responses of a (12, 12)-tube under three different sets of thermal environmental conditions T = 300, 500 and 700 K, when the SWCNT is supported by a soft elastic foundation. The small scale parameter e0a is taken to be 2.0 nm. It can be seen that the buckling loads as well as postbuckling strength are reduced with increase in temperature. It is found that the effect of temperature changes on the postbuckling behavior is more pronounced when the temperature-dependent material properties are taken into account. In other words, the difference of the buckling loads (as well as postbuckling equilibrium paths) for the SWCNT at two thermal environmental conditions becomes large when the material properties are temperature-dependent. Fig. 4 compares the postbuckling behavior of three kinds of SWCNTs resting on elastic foundations at T = 300 K. The small scale parameter e0a is taken to be 2.0 nm. It can be seen that the (17, 0)tube has the lowest buckling load among the three tubes, while it has a higher postbuckling strength than (12, 12)-tube when the deflection is greater than 12 nm. 4. Nonlinear bending problem The static transverse load is assumed to be uniform, and Q(X) = q0. Two cases are considered in this section. Case 1, one end of the beam is movable and initial axial compressive loads are applied. In such a case, kP will be replaced by P/Pcr and Eq. (7) can be written in the simple form as
40
40
(10, 10)-tube
(12, 12)-tube
L/R =100, R = 0.681 nm, h = 0.0674 nm
L/R =100, R = 0.816 nm h = 0.0671 nm, e0a =2.0 nm HS=1 mm
HS=1 mm, T= 300 K
30
Px (nN)
Px (nN)
30 20
e0a =0.0 nm e0a =2.0 nm e0a =4.0 nm
10
20
T= 300 K T= 500 K T= 700 K
10
0 0
5
10
15
Wm (nm) Fig. 2. Comparisons of postbuckling behavior of (10, 10)-tube with different values of small scale parameter e0a subjected to axial compression and resting on an elastic foundation.
0 0
5
10
15
20
25
Wm (nm) Fig. 3. The effect of temperature rise on the postbuckling behavior of (12, 12)-tube resting on an elastic foundation in thermal environments.
1026
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029
From Eqs. (18) and (19), the load–central deflection relationship can be written as
40 L/R =100, e0a =2.0 nm HS=1 mm, T= 300 K
Px (nN)
30
q0 L3 ð1Þ W m ¼ AW p4 EI L
! ð3Þ
þ AW
Wm L
!3 þ
ð20Þ
20 ðiÞ
0
(12, 12)-tube, R = 0.816 nm, h = 0.0671 nm (10, 10)-tube, R = 0.681 nm, h = 0.0674 nm (17, 0)-tube, R = 0.669 nm, h = 0.0871 nm
0
5
10
15
Wm (nm) Fig. 4. Comparisons of postbuckling behavior of three kinds of SWCNTs resting on elastic foundations.
# 4 dW 1þp þp þ 4 dx dx " # 2 4 3 2 d W d W dW 2 dW 4 dW þ 4p2 1 þ 2 p þ 3 p þ 3 2 dx dx dx dx dx # !3 " 2 4 2 2 d W 2 dW 4 dW 1 þ 6p þ 15p þ þp 2 dx dx dx !( ! 2 @2 d W kq K 1 W K 2 ¼ 1 s2 p2 2 2 @x dx " #) 2 4 2 P d W 3 2 dW 15 4 dW 1 þ p þ p þ ð15Þ Pcr dx2 2 dx 8 dx 4
d W
"
2
dW dx
2
4
Case 2, two ends of the beam are immovable and no axial compressive loads are applied. In such a case, Eq. (8) can be written in the simple form as
" # 2 4 @4W 2 @W 4 @W 1þp þp þ @x4 @x @x " # 2 4 3 2 2 @ W @ W @W 2 @W 4 @W 1 þ 2p þ 3p þ þ 4p @x3 @x2 @x @x @x # !3 " 2 4 @2W 2 @W 4 @W 1 þ 6 p þ 15 p þ þ p2 @x2 @x @x ( ! 2 2 @ @ W ¼ ð1 s2 p2 2 Þ kq K 1 W K 2 @x2 @x "Z # pg p @W 2 @ 2 W dx þ @x @x2 2 0 " #) 2 4 @2W 3 2 @W 15 4 @W gaDT 1þ p þ p þ ð16Þ @x2 2 @x 8 @x
0.6 (10, 10)-tube L/R =100, R = 0.681 nm, h = 0.0674 nm HS=1 mm, T= 300 K
0.4
Wm/L
10
In Eq. (20), AW ði ¼ 1; 3; . . .Þ are described in detail in Appendix B. Fig. 5 shows the effect of the small scale parameter e0a (=0.0, 4.0 and 8.0 nm) on the nonlinear bending behavior of the (10, 0)-tube subjected to a transverse uniform load resting on an elastic foundation at T = 300 K. The temperature-dependent material properties of SWCNTs and substrate are adopted, as given in Section 3. The immovable end conditions are considered in this example. It can be seen that for a given load, the deflection decreases with increase in e0a. It is found that the effect of the small scale parameter on the nonlinear bending response is weaker when compared to the effect of the small scale parameter on the postbuckling response. Fig. 6 shows the effect of temperature rise on the nonlinear bending behavior of the (12, 12)-tube subjected to a uniform pressure and resting on an elastic foundation. Three sets of thermal environmental conditions, i.e. T = 300, 500 and 700 K, are considered. The end condition is assumed to be immovable. The small scale parameter e0a is taken to be 4.0 nm. The results show that the deflections are increased with increase in temperature.
e 0a =0.0 nm e 0a =4.0 nm e0a =8.0 nm
0.0
j¼1
e wj ðxÞ; kq ¼
X
j
e kj
ð17Þ
j¼1
The solution methodology is similar to that used in Section 3, and the asymptotic solutions are obtained as
3 ð1Þ ð1Þ WðxÞ ¼ A10 e sin mx þ A10 e ½A330 sin 3mx
5 ð1Þ þ A10 e ½A530 sin 3mx þ A550 sin 5mx þ Oðe6 Þ
ð18Þ
3 ð1Þ ð1Þ kq ¼ k1 A10 e þ k3 A10 e þ
ð19Þ
100
200
300
400
500
3
Fig. 5. Comparisons of nonlinear bending behavior of (10, 10)-tube with different values of small scale parameter e0a subjected to a uniform pressure and resting on an elastic foundation.
0.6 (12, 12)-tube L/R =100, R = 0.816 nm h = 0.0671 nm, e0a =4.0 nm HS=1 mm
0.4
Wm/L
Wðx; eÞ ¼
j
0
q0L /EI
The solutions of Eqs. (15) and (16) can also be determined by a twostep perturbation technique. In the present case, we assume that
X
immovable end condition
0.2
immovable end condition
0.2
0.0
T= 300 K T= 500 K T= 700 K
0
100
300
200
400
500
3
q0L /EI Fig. 6. The effect of temperature rise on the nonlinear bending behavior of (12, 12)tube resting on an elastic foundation in thermal environments.
1027
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029
The solutions of Eqs. (7) and (8) can also be determined by a twostep perturbation technique, and the asymptotic solutions are obtained as a similar form of Eq. (18)
8 (17, 0)-tube L/R =100, R = 0.669 nm h = 0.0871 nm, HS=1 mm T= 500 K, e0a =4.0 nm
Wm/L
6
3 ð1Þ ð1Þ Wðx; tÞ ¼ A10 e sin mx þ A10 e ½A330 sin 3mx
5 ð1Þ þ A10 e ½A530 sin 3mx þ A550 sin 5mx þ Oðe6 Þ
4
ð22Þ
ð1Þ
2
0
Note that now A10 is a function of t. For nonlinear free vibration, since Q ðX; tÞ ¼ 0, we also have
immovable end condition movable end condition, P/Pcr=0.0 movable end condition, P/Pcr=0.4
ð1Þ g 1 A10
0
100
200
300
400
e sin mx þ
ð1Þ g 3 A10
3
e sin mx þ g 2
ð1Þ @ 2 A10 e
500
@t 2
sin mx þ ¼ 0 ð23Þ
3
q0L /EI
Let
Fig. 7. Comparisons of nonlinear bending behavior of (17, 0)-tube with two different end conditions and resting on an elastic foundation.
2
Z p
4g 1
0
Fig. 7 shows the effect of end conditions on the nonlinear bending behavior of the (17, 0)-tube subjected to a uniform pressure and resting on elastic foundations at T = 500 K. The small scale parameter e0a is taken to be 4.0 nm. To this end, the load-deflection curves of nanobeams under movable and immovable end conditions are displayed. For the case of movable end condition, the initial axial compressive loads are taken to be P/Pcr = 0.0 and 0.4. The results confirm that the initial stress due to axial compression has a significant effect on the nonlinear bending behavior of the nanobeam. The results show that the nanobeam with immovable ends will undergo much less deflections.
5. Nonlinear vibration problem In this section Eqs. (7) and (8) are directly used for two kinds of end conditions. The initial conditions are assumed to be
Wjt¼0 ¼
@W ¼0 @t t¼0
ð21Þ
ð1Þ A10
e sin mx þ g 3
ð1Þ A10
3
e sin mx þ g 2
ð1Þ @ 2 A10 e @t2
3 sin mx þ 5
ðsin mxÞdx ¼ 0
ð24Þ
From which one has 2
d g2
eAð1Þ 10
dt
2
þ g1
ð1Þ eAð1Þ þ g 3 eA10 10
3
¼0
ð25Þ
In Eq. (25), g1, g2 and g3 are described in detail in Appendix C, and the solution of which may be written as
2
xNL
3 g3 W m ¼ xL 4 1 þ 4 g1 L
!2 31=2 5
ð26Þ
where xL = [g1/g2]1/2 is the dimensionless linear frequency. According to Eq. (6), the corresponding linear frequency can be expressed as XL = xL(p/L)(E/q)1/2. It is worth noting that the linear frequencies are the same for these two kinds of end conditions when no axial load is applied. Table 4 shows the effect of the small scale parameter e0a (=0.0, 5.0 and 10.0 nm) as well as temperature change on the nonlinear to linear frequency ratios xNL/xL of the same two (10, 10) and (17, 0)
Table 4 Comparisons of nonlinear to linear frequency ratios for CNTs with immovable end conditions and resting on an elastic foundation in thermal environments (HS = 1 mm and L/ R = 100). T (K)
(10, 10)-tube 300
e0a (nm)
xL (GHz)
xNL/xL W m =L ¼ 0
W m =L ¼ 0:1
W m =L ¼ 0:15
W m =L ¼ 0:2
W m =L ¼ 0:25
W m =L ¼ 0:3
0.0 5.0 10.0
495.8115 495.5956 495.0613
1.0 1.0 1.0
1.2833 1.2835 1.2840
1.5669 1.5673 1.5683
1.8940 1.8945 1.8959
2.2455 2.2463 2.2481
2.6117 2.6126 2.6149
500
0.0 5.0 10.0
404.2776 404.0184 403.3762
1.0 1.0 1.0
1.3970 1.7974 1.3985
1.7723 1.7730 1.7749
2.1923 2.1934 2.1961
2.6358 2.6372 2.6406
3.0926 3.0943 3.0986
700
0.0 5.0 10.0
284.2779 283.9128 283.0076
1.0 1.0 1.0
1.7036 1.7050 1.7085
2.2978 2.3001 2.3060
2.9340 2.9373 2.9454
3.5900 3.5942 3.6046
4.2567 4.2617 4.2744
0.0 5.0 10.0
442.3508 442.2352 441.9510
1.0 1.0 1.0
1.1728 1.1729 1.1730
1.3582 1.3583 1.3587
1.5817 1.5819 1.5824
1.8293 1.8296 1.8303
2.0925 2.0929 2.0938
500
0.0 5.0 10.0
363.5445 363.4045 363.0600
1.0 1.0 1.0
1.2460 1.2462 1.2466
1.4978 1.4981 1.4988
1.7918 1.7922 1.7933
2.1104 2.1110 2.1124
2.4441 2.4448 2.4466
700
0.0 5.0 10.0
263.6971 263.5053 263.0334
1.0 1.0 1.0
1.4289 1.4294 1.4307
1.8287 1.8296 1.8318
2.2732 2.2745 2.2777
2.7407 2.7424 2.7465
3.2213 3.2233 3.2284
(17, 0)-tube 300
1028
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029
Table 5 Comparisons of nonlinear to linear frequency ratios for (12, 12)-tube with movable and immovable end conditions and resting on an elastic foundation in thermal environments (HS = 1 mm and L/R = 100). T (K)
e0a (nm)
Immovable end conditions 300 0.0 5.0 10.0
xL (GHz)
xNL/xL W m =L ¼ 0
W m =L ¼ 0:1
W m =L ¼ 0:15
W m =L ¼ 0:2
W m =L ¼ 0:25
W m =L ¼ 0:3
414.6166 414.4922 414.1669
1.0 1.0 1.0
1.2769 1.2771 1.2774
1.5552 1.5555 1.5562
1.8768 1.8772 1.8781
2.2228 2.2233 2.2247
2.5836 2.5842 2.5859
500
0.0 5.0 10.0
337.9666 337.8180 337.4295
1.0 1.0 1.0
1.3864 1.3867 1.3874
1.7535 1.7540 1.7553
2.1653 2.1660 2.1679
2.6007 2.6016 2.6041
3.0495 3.0506 3.0537
700
0.0 5.0 10.0
238.6169 238.4081 237.8617
1.0 1.0 1.0
1.6830 1.6840 1.6864
2.2635 2.2650 2.2691
2.8862 2.8884 2.8941
3.5290 3.5318 3.5391
4.1825 4.1859 4.1948
414.6166 414.4922 414.1669
1.0 1.0 1.0
1.0003 1.0003 1.0003
1.0007 1.0007 1.0006
1.0012 1.0012 1.0011
1.0019 1.0019 1.0017
1.0028 1.0027 1.0024
Movable end conditions 300 0.0 5.0 10.0 500
0.0 5.0 10.0
345.8134 345.6682 345.2285
1.0 1.0 1.0
1.0004 1.0004 1.0004
1.0010 1.0009 1.0009
1.0017 1.0017 1.0015
1.0027 1.0026 1.0024
1.0039 1.0038 1.0034
700
0.0 5.0 10.0
259.4649 259.2729 258.7705
1.0 1.0 1.0
1.0008 1.0007 1.0007
1.0017 1.0017 1.0015
1.0031 1.0030 1.0027
1.0048 1.0046 1.0042
1.0069 1.0066 1.0060
tubes resting on an elastic foundation in thermal environments. In the present example, the mass density of CNTs is taken to be 1400 kg/m3 [22]. It is found that the natural frequencies are decreased with increase in e0a, but among the three, i.e. postbuckling, nonlinear bending and nonlinear vibration, the effect of the small scale parameter on the nonlinear vibration behavior is weakest. It can be seen that the temperature rise decreases the natural frequencies but increases the nonlinear to linear frequency ratios. The results confirm that the nonlinear to linear frequency ratios are increased with increase in the non-dimensional vibration amplitude of the SWCNTs. Table 5 shows the effect of the beam end conditions on the nonlinear to linear frequency ratios xNL/xL of the same (12, 12)-tube with movable and immovable end conditions and resting on an elastic foundation in thermal environments. It can be seen that although the linear frequencies are the same for these two kinds of end conditions (at T = 300 K), the nonlinear to linear frequency ratios for movable end conditions are much less than those for immovable end conditions. It can also be seen that the nonlinear to linear frequency ratios are increased slightly for immovable end conditions, whereas are decreased slightly for movable end conditions when the small scale parameter e0a is increased. 6. Concluding remarks A nonlocal beam model is proposed for nonlinear analysis of SWCNTs resting on a two-parameter elastic foundation in thermal environments. The nonlocal stress incorporating the small scale parameter e0a is introduced into the governing equations. The postbuckling equilibrium paths of an axially loaded nanobeam, the static large deflections of a bending nanobeam with or without initial stresses subjected to a uniform transverse pressure, and the nonlinear frequencies of a nanobeam have been presented. Perturbation solutions are obtained in an explicit form that can be easy to program and produces full nonlinear responses. The numerical results reveal that the small scale parameter e0a reduces the postbuckling equilibrium paths, the static large deflections and natural frequencies of SWCNTs resting on an elastic foundation. In contrast, it increases and reduces the nonlinear to linear frequency ratios slightly for the nanobeam with immovable and mo-
vable end conditions, respectively. The results also reveal that the effect of the small scale parameter is significant for compressive buckling, but less pronounced for static bending and marginal for free vibration of SWCNTs resting on an elastic foundation. Appendix A In Eq. (14)
m2 1 þ ðK 1 þ K 2 m2 Þ; R10 m2
kð0Þ p ¼
kð4Þ p ¼
kð2Þ p ¼
p2 8
m2
m2 3 2 ðK 1 þ K 2 m2 Þ ; R10 m
3 4 2 m2 m2 33 p m 2 ð4m2 81a330 Þ þ a330 25 þ 2 ðK 1 þ K 2 m2 Þ 64 R10 R10 m 4
m 3ðK 1 þ K 2 m2 Þ R10
in which
2
a330
3 ð0Þ 2 4m R k 30 p h i5; ¼ 4m4 81m4 þ R30 ðK 1 þ 9K 2 m2 Þ 9m2 kpð0Þ
R10 ¼ 1 þ s2 p2 m2 ;
R30 ¼ 1 þ 9s2 p2 m2
ðA:2Þ
Appendix B In Eq. (20), for case 1 ð1Þ
AW ¼ ð3Þ
AW ¼
P 2 m m m4 þ R10 ðK 1 þ K 2 m2 Þ Pcr 4
p
2
p 4 2 3 P R10 m m m 4 Pcr 4 2 3 2 P 2 4 m p a330 m þ R10 ðK 1 þ K 2 m2 Þ 8 Pcr
p
a330 ¼ m4
4m2 R30 P=Pcr 4 81m þ R30 ½ðK 1 þ 9K 2 m2 Þ 9m2 P=Pcr
ðB:1Þ
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029
and for case 2 ð1Þ
AW ¼
ð3Þ
AW ¼
p 4
1029
References
m m4 þ R10 ðK 1 þ K 2 m2 Þ m2 gaDT
2 p 4 2 1 3 a DT m m m þ R10 g 2 4 4 2 4 3 2 p a330 m þ R10 ðK 1 þ K 2 m2 Þ m2 gaDT 8
p
ðB:2Þ
Appendix C In Eq. (25)
g 2 ¼ R10 g
ðC:1Þ
and for case 1
P 2 ; g 1 ¼ m4 þ R10 ðK 1 þ K 2 m2 Þ m Pcr
p2 4 2 3 P R10 m m g3 ¼ 4 Pcr 2 and for case 2
g 1 ¼ m4 þ R10 ðK 1 þ K 2 m2 Þ m2 gaDT ;
p2 4 2 1 3 g3 ¼ aDT m m þ R10 g 2 4 2
ðC:3Þ
[1] E.M. Abdel-Rahman, M.I. Younis, A.H. Nayfeh, J. Micromech. Microeng. 12 (2002) 759. [2] X. Li, B. Bhushan, K. Takashima, C.-W. Baek, Y.-K. Kim, Ultramicroscopy 97 (2003) 481. [3] O. Lourie, D.M. Cox, H.D. Wagner, Phys. Rev. Lett. 81 (1998) 1638. [4] J. Yoon, C.Q. Ru, A. Mioduchowski, Compos. Sci. Technol. 65 (2005) 1326. [5] Y.M. Fu, J.W. Hong, X.Q. Wang, J. Sound Vib. 296 (2006) 746. [6] J. Xiao, H. Jiang, D.-Y. Khang, J. Wu, Y. Huang, J.A. Rogers, J. Appl. Phys. 104 (2008) 033543. [7] A.C. Eringen, J. Appl. Phys. 54 (1983) 4703. [8] J. Peddieson, G.R. Buchanan, R.P. McNitt, Int. J. Eng. Sci. 41 (2003) 305. [9] S. Adali, Phys. Lett. A 372 (2008) 5701. [10] L.J. Sudak, J. Appl. Phys. 94 (2003) 7281. [11] T. Murmu, S.C. Pradhan, Physica E 41 (2009) 1232. [12] S.C. Pradhan, T. Murmu, J. Appl. Phys. 105 (2009) 124306. [13] H.-L. Lee, W.-J. Chang, Physica E 41 (2009) 529. [14] X.-F. Li, B.-L. Wang, Y.-W. Mai, J. Appl. Phys. 103 (2008) 074309. [15] S. Narendar, S. Gopalakrishnan, Physica E 42 (2010) 1706. [16] C.Y. Wang, L.C. Zhang, Nanotechnology 19 (2008) 075705. [17] D.-Y. Khang, J. Xiao, C. Kocabas, S. MacLaren, T. Banks, H. Jiang, Y.Y. Huang, J.A. Rogers, Nano Lett. 8 (2008) 124. [18] J. Dong, L. Peng, J. Wuhan Univ. Hydraulic Electric Eng. 24 (1991) 427. [19] H.-S. Shen, Int. J. Struct. Stability Dyn. 11 (2011) (in press). [20] T. Chang, J. Geng, X. Guo, Appl. Phys. Lett. 87 (2005) 251929. [21] C.-L. Zhang, H.-S. Shen, Appl. Phys. Lett. 89 (2006) 081904. [22] J.D. Fidelus, E. Wiesel, F.H. Gojny, K. Schulte, H.D. Wagner, Compos. A 36 (2005) 1555.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Nonlocal beam model for nonlinear analysis of carbon nanotubes on elastomeric substrates Hui-Shen Shen a,b,⇑, Chen-Li Zhang a a b
Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China
a r t i c l e
i n f o
Article history: Received 7 October 2010 Received in revised form 28 October 2010 Accepted 29 October 2010 Available online 20 November 2010 Keywords: Carbon nanotube Nonlocal beam model Postbuckling Nonlinear bending Nonlinear vibration Elastic foundation
a b s t r a c t Postbuckling, nonlinear bending and nonlinear vibration analyses are presented for single-wall carbon nanotubes (SWCNTs) resting on a two-parameter elastic foundation in thermal environments. The SWCNT is modeled as a nonlocal nanobeam which contains small scale effects. The elastomeric substrate with finite depth is modeled as a two-parameter elastic foundation. The thermal effects are included and the material properties of both SWCNTs and the substrate are assumed to be temperature-dependent. The governing equation that includes beam–foundation interaction is solved by a two-step perturbation technique. The numerical results reveal that the small scale parameter e0a reduces the postbuckling equilibrium paths, the static large deflections and natural frequencies of SWCNTs resting on an elastic foundation. The results also reveal that the effect of the small scale parameter is significant for compressive buckling, but less pronounced for static bending and marginal for free vibration of SWCNTs resting on an elastic foundation. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction Miniaturized beams are the core structures widely used in sensors, actuators, and Micro- and Nano-Electro-Mechanical Systems (MEMS/NEMS) [1,2]. Recently, many investigations focused on the modern nanotechnology involving carbon nanotubes (CNTs) embedded in an elastic matrix [3–5] or resting on an elastomeric substrate [6], in which the CNTs are modeled as a beam pinned at both ends and resting on an elastic foundation. One of the major differences between these two problems of CNTs lies in that the former assumes that the elastic foundation has an infinite depth, whereas in the latter the elastic foundation has only a finite depth. It has been reported that the small scale effect must play an important role in the nanoscale structures, e.g. carbon nanotubes and single layer graphene sheets. An important class of modified continuum models is that based on the concept of nonlocal elasticity [7]. These models allow the integration of small scale effects into classical continuum models. Unlike classical continuum models, the nonlocal elasticity theory assumes that the stress at a reference point in a body depends not only on the strains at that point, but also on strains at all other points of the body. The use of ⇑ Corresponding author at: Department of Engineering Mechanics, Shanghai Jiao Tong University, Shanghai 200030, People’s Republic of China. Tel./fax: +86 21 34206197. E-mail address: [email protected] (H.-S. Shen). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.10.042
nonlocal elasticity to study size-effects in micro and nanoscale structures was first performed by Pedisson et al. [8]. They studied the bending of micro- and nanoscale beams with the concept of nonlocal elasticity and concluded that size-effects could be significant for nanoscale structures and that the magnitude of the sizeeffects strongly depends on the value of the small scale parameter e0. Since then, considerable attention has been directed to the development and application of nonlocal beam models for the buckling [9–11], vibration [12,13] and wave propagation [14,15] analyses of CNTs embedded in an elastic matrix. It is worthy to note that in most investigations [10–14] the Young’s modulus of CNTs was assumed to be about 1 TPa. Such a low value of Young’s modulus is due to the fact that the effective thickness of CNT is assumed to be 0.34 nm or more. However, as reported recently the effective thickness of SWCNTs should be smaller than 0.142 nm [16]. Therefore, the wide used value of 0.34 nm for tube wall thickness is thoroughly inappropriate to SWCNTs. On the other hand, the buckling configuration of individual single-wall carbon nanotubes (SWCNTs) resting on an elastomeric substrate is likely nonlinear under axial compression [17]. However, it remains unclear if the small scale parameter has the same effect on the postbuckling behavior of SWCNTs resting on an elastomeric substrate and this motivates the current investigation. The present work focuses attention on the postbuckling, nonlinear bending and nonlinear vibration of SWCNTs resting on an elastomeric substrate in thermal environments. The SWCNT is modeled as a nonlocal nanobeam which contains small scale
1023
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029
effects. The elastomeric substrate with finite depth is modeled as a two-parameter elastic foundation. The thermal effects are included and the material properties of both SWCNTs and the substrate are assumed to be temperature-dependent. The governing equation that includes beam–foundation interaction is solved by a two-step perturbation technique. The numerical illustrations show the nonlinear behavior of SWCNTs resting on elastic foundations under different sets of environmental conditions.
where DT = T T0 is temperature rise from some reference temperature T0 at which there are no thermal strains. In order to incorporate the small scale effect, continuum beam models need to be refined. This may be accomplished by using the nonlocal continuum theory. In the theory of nonlocal elasticity, the constitutive relations of nonlocal elasticity for 3D problems are expressed as [7]
ð1 s2 L2 r2 Þrij ¼ C ijkl ekl
ð3Þ
2. Nonlocal beam model for nonlinear analysis of SWCNTs
ð1aÞ
where s = e0a/L, rij and eij are the stress and strain tensors, and Cijkl is the elastic modulus tensor of classical isotropic elasticity, e0 is a material constant, and a and L are the internal and external characteristic lengths, respectively. The distinct difference between the classical and nonlocal elasticity theories lies in the presence of small scale parameter e0a in the nonlocal theory. For CNTs the characteristic length a may be taken as the length of the CAC bond, i.e. a = 0.142 nm, or taken as other material properties. It is a better way to use e0a as a single scale coefficient that captures the small scale effect on the response of structures in nanosize [10–15]. Shen [19] proposed a nonlinear beam theory for Euler–Bernoulli beams resting on elastic foundations. Applying Eq. (2) to this theory, the nonlinear motion equations for the nanobeam, including beam–foundation interaction, small scale effects and thermal effects, have readily been derived and can be expressed by
ð1bÞ
8 2 2 !2 31 !2 32 <@ 4 W 3 2 @W @ W @ W @W @W 41 5 þ 41 5 EI : @X 4 @X @X @X 3 @X 2 @X
Consider an SWCNT modeled as a nanobeam with length L, mean radius R and constant thickness h that rests on an elastic foundation. The beam is exposed to elevated temperature and is subjected to axial compressive loads P only in the X-direction or combined with transverse static or dynamic load Q. Let U be the displacement in the longitudinal direction, and W be the deflection of the beam (Fig. 1). As is customary [10–15], the foundation is assumed to be a compliant foundation, which means that no part of the beam lifts off the foundation in the large deflection region. The load–displacement relationship of the foundation is assumed to be 2 2 p ¼ K 1 W K 2 ðd W=dX Þ, where p is the force per unit area, K 1 is the Winkler foundation stiffness and K 2 is the shearing layer stiffness of the foundation. Since the elastomeric substrate is relatively soft, the foundation stiffness K 1 may be expressed by [18]
K1 ¼
E0 5 2c21 þ 6c1 þ 5 expð2c1 Þ 4L 1 m20 ð2 c1 Þ2
in which
c1 ¼ ðc1 þ 2Þ expðc1 Þ
and K 2 may be determined later in the numerical calculations. In Eq. (1), c1 ¼ Hs =L; E0 ¼ Es = 1 m2s , m0 ¼ ms =ð1 ms Þ; ES is Young’s modulus of the foundation, ms the Poisson’s ratio of the foundation, and Hs the depth of the foundation. It is assumed that the Young’s modulus of the foundation ES is temperature-dependent, whereas Poisson’s ratio ms depends weakly on temperature change and is assumed to be a constant. It is also assumed that the effective Young’s modulus E and thermal expansion coefficient a of SWCNTs are temperaturedependent. The thermal forces N T and moments M T caused by elevated temperature are defined by
ðNT ; MT Þ ¼
Z
ð1; ZÞEðTÞaðTÞTðX; ZÞdA
ð2aÞ
A
The temperature field is assumed to be uniform, and in such a case T
T
ðN ; M Þ ¼ EaDT
Z
ð1; ZÞdA
ð2bÞ
A
!2 32 !2 33 9 !3 2 = 41 þ 3 @W 541 @W 5 ; @X @X @X 2
@2W
þ
1 s2 L 2
¼
2
P
@2W 4 @X 2
@2 @X
!( Q K1W K2
2
@W 1 @X
@X
!2 33=2 5
EaDT
! qA
2
@2 @X 2
Z A
@2W @t2
9 = ZdA ;
ð4Þ
Note that Eq. (4) is valid for the case of nanobeam with movable end conditions. In contrast, for the case of nanobeam with immovable end conditions, the motion equation becomes
8 2 2 !2 31 !2 32 <@ 4 W 3 2 @W @ W @ W @W @W 41 5 þ 41 5 EI : @X 4 @X @X @X 3 @X 2 @X
þ
!2 32 !2 33 9 !3 2 = 41 þ 3 @W 541 @W 5 ; @X @X @X 2
@2W
1 s2 L 2
¼
@2 @X
!( Q K1W K2
2
2
EAaDT
Fig. 1. An SWCNT with pinned ends resting on a two-parameter elastic foundation.
@2W
@2W @X 2
@2W 4 @X 2
EaDT
@W 1 @X @2
@X 2
Z
!2 33=2 5
@2W @X
2
! qA
@2W @t 2
2 !2 3 Z EA 4 L @W þ dX 5 2L 0 @X
) ZdA
ð5Þ
A
where I = pR3h is the second moment of area, A = 2pRh is the area of the cross section, q is the mass density.
1024
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029
The solutions of Eq. (9) are now determined by a two-step perturbation technique, where the small perturbation parameter has no physical meaning at the first step, and is then replaced by a dimensionless deflection at the second step. The essence of this procedure, in the present case, is to assume that
Introducing following dimensionless quantities
x¼
g¼
X ; L L2 A
p2 E
W¼
W ; L
kP ¼
;
ðK 1 ; K 2 Þ ¼
PL2
p2 EI
;
kq ¼
!
K 1 L4 ; p4 EI
K 2 L2 ; p2 EI
sffiffiffiffi pt E t¼ ; L q
QL3
ð6Þ
p4 EI
enables nonlinear Eqs. (4) and (5) to be written in dimensionless form as
"
# 4 @ W 2 @W 4 @W 1 þ p þ p þ @x4 @x @x " # 2 4 3 2 @ W @ W @W 2 @W 4 @W þ 4p2 1 þ 2 p þ 3 p þ @x3 @x2 @x @x @x !3 " # 2 4 2 2 @ W 2 @W 4 @W 1 þ 6p þ 15p þ þp @x2 @x @x !( ! @2 @2W @2W kq K 1 W K 2 g 2 ¼ 1 s2 p2 2 2 @x @x @t " #) 2 4 2 @ W 3 2 @W 15 4 @W 1 þ p þ p þ kP @x2 2 @x 8 @x 4
2
ðfor movable end conditionsÞ "
2
ðfor immovable end conditionsÞ
ð8Þ
"
# 2 4 dW 4 dW þ p þ 4 dx dx dx " # 2 4 3 2 2 d W d W dW 2 dW 4 dW 1 þ 2p þ 3p þ þ 4p 3 2 dx dx dx dx dx 1 þ p2
#
2 4 dW 4 dW þ 15 p þ 2 dx dx dx !( " # 2 4 2 @2 d W 3 2 dW 15 4 dW kP 1 þ p þ p þ þ 1 s 2 p2 2 2 2 dx 8 dx @x dx !) 2 d W ¼0 ð9Þ þ K 1W K2 2 dx 2
þ p2
d W
ej kj
ð10Þ
j¼0
where e is a small perturbation parameter. Substituting Eq. (10) into Eq. (9), collecting the terms of the same order of e, a set of perturbation equations is obtained and can be solved step by step. The solution methodology is described in Shen [19]. As a result, the asymptotic solutions can be obtained as
3 ð1Þ ð1Þ WðxÞ ¼ A10 e sin mx þ A10 e ½A330 sin 3mx
5 ð1Þ þ A10 e ½A530 sin 3mx þ A550 sin 5mx þ Oðe6 Þ
ð11Þ
2
4 ð1Þ ð1Þ kP ¼ k0 þ k2 A10 e þ k4 A10 e þ Oðe6 Þ
ð12Þ
ð1Þ In Eqs. (11) and (12), A10 e is taken as the second perturbation parameter relating to the dimensionless maximum deflection Wm. From Eq. (11), taking (x = p/2m) yields
3 2 p a330 W 2m þ 8
ð13Þ
2 4 ð2Þ ð4Þ kP ¼ kð0Þ p þ kp W m þ kp W m þ
In such a case kq = 0, W is only a function of x. The end of the beam is movable, and Eq. (7) can be written in the simple form as
!3 "
X
Substituting Eq. (13) into Eq. (12), the postbuckling equilibrium path can be written as
3. Postbuckling problem
4
ej wj ðxÞ; kP ¼
j¼1
ð1Þ
Eqs. (7) and (8) are for movable and immovable end conditions respectively, and are adopted in the following nonlinear analysis.
d W
X
A10 e ¼ W m
#
4 @ W 2 @W 4 @W 1 þ p þ p þ @x4 @x @x " # 2 4 3 2 2 @ W @ W @W 2 @W 4 @W þ 4p 1 þ 2p þ 3p þ @x3 @x2 @x @x @x !3 " # 2 4 2 2 @ W 2 @W 4 @W 1 þ 6p þ 15p þ þp @x2 @x @x !( ! @2 @2W @2W g kq K 1 W K 2 ¼ 1 s2 p2 2 @x2 @x @t 2 "Z # 2 pg p @W @2W þ dx @x @x2 2 0 " #) 2 4 @2W 3 2 @W 15 4 @W gaDT 1þ p þ p þ @x2 2 @x 8 @x 4
ð7Þ
Wðx; eÞ ¼
ð14Þ
In Eq. (14) , kpðiÞ ði ¼ 0; 2; 4; . . .Þ are described in detail in Appendix A. For all cases discussed below, three kinds of SWCNTs are selected as nanobeams. It has been reported that the material properties of CNTs are chirality- and size-dependent [20] and temperature-dependent [21]. From MD simulations the effective material properties for armchair (12, 12) and (10, 10)-tubes and zigzag (17, 0)-tube are obtained, and the details of MD simulations may be found in [21]. Typical results are listed in Table 1. It is noted that the effective wall thickness obtained is h = 0.0671 nm, 0.0674 nm and 0.0871 nm for (12, 12), (10, 10) and (17, 0) tubes, respectively. Poly (dimethylsiloxane), referred to as PDMS, is selected for the substrate, and the material properties of which are assumed to be mS = 0.48 and ES = (3.22 0.0034T) GPa, in which T = T0 + DT and T0 = 300 K (room temperature). In such a way, ES = 2.2 GPa at T = 300 K. Since ES is temperature-dependent, the foundation stiffness is also temperature dependent. Taking HS = 1 mm and L/R = 100, from Eq. (1) K 1 (in GPa/nm) and its alternative dimensionless form K1 can be obtained. K2 is taken to be 1–10th of the value of K1 for the same tube in thermal environmental conditions. The results obtained are listed in Table 2, and are adopted in the following calculations. Table 3 presents the buckling loads (nN) and the corresponding buckling mode (m), which determines the number of half-waves in
Table 1 Temperature-dependent material properties for (12, 12), (10, 10) and (17, 0)-tubes. T (K)
1 þ 6 p2
300 500 700 a b c
(12, 12)-tubea
(10, 10)-tubeb 6
(17, 0)-tubec
E (TPa)
a(10 /K)
E (TPa)
a(10 /K)
E (TPa)
a(106/K)
5.5272 5.3783 5.3351
3.8164 4.7113 4.5959
5.6466 5.5308 5.4744
3.4584 4.5361 4.6677
2.5176 2.5066 2.4890
3.9676 4.5904 4.0577
R = 0.816 nm, h = 0.0671 nm. R = 0.681 nm, h = 0.0674 nm. R = 0.669 nm, h = 0.0871 nm.
6
1025
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029 Table 2 The foundation stiffnesses for (12, 12), (10, 10) and (17, 0)-tubes resting on elastomeric substrate in thermal environments (HS = 1 mm and L/R = 100). T (K)
(12, 12)-tube
300 500 700
(10, 10)-tube
K 1 (GPa/nm)
(K1, K2)
K 1 (GPa/nm)
(K1, K2)
K 1 (GPa/nm)
(K1, K2)
0.07 0.05 0.03
(53.21, 5.32) (37.78, 3.78) (21.05, 2.10)
0.09 0.06 0.03
(51.85, 5.18) (36.57, 3.66) (20.42, 2.04)
0.09 0.06 0.03
(89.99, 9.00) (62.45, 6.24) (34.75, 3.47)
Table 3 Buckling loads Pcr (nN) for CNTs resting on an elastic foundation under axial compression in thermal environments (HS = 1 mm and L/R = 100).
a
(17, 0)-tube
T (K)
e0a (nm)
(12, 12)-tube
(10, 10)-tube
(17, 0)-tube
300
0.0 1.0 2.0 4.0
18.986 18.874 18.558 17.500
(3)a (3) (3) (3)
16.029 15.893 15.514 14.332
(3) (3) (3) (3)
12.735 12.656 12.434 11.303
500
0.0 1.0 2.0 4.0
15.500 15.393 15.084 14.055
(3) (3) (3) (3)
13.162 13.029 12.685 11.500
(3) (3) (3) (3)
10.046 (3) 9.967 (3) 9.747 (3) 9.064 (3)
700
0.0 1.0 2.0 4.0
10.295 (2) 10.274 (2) 10.211 (2) 9.981 (2)
8.685 8.658 8.582 8.311
(3) (3) (3) (3)
7.269 7.253 7.049 6.371
(3) (3) (3) (4)
(2) (2) (2) (3)
The number in brackets indicates the buckling mode (m) in X-direction.
the X-direction, for SWCNTs with movable end conditions under axial compression and resting on an elastic foundation in thermal environments. Three thermal environmental conditions, i.e. T = 300, 500 and 700 K, are considered. The key issue for successful application of the nonlocal continuum mechanics models to nanoscale structures is to determine the magnitude of the small scale parameter e0a. However, there are no experiments conducted to determine the value of e0a for CNTs. In the present example, the small scale parameter e0a is taken to be 0, 1.0, 2.0 and 4.0 nm. Note that here e0a = 0 represents the local beam model. It can be seen that the buckling loads are sensitive to the small scale parameter e0a. The results show that the (12, 12)-tube has the highest buckling load, whereas (17, 0)-tube has the lowest buckling load among the three tubes. The results confirm that the buckling loads are decreased with increase in temperature. Usually, the beam without any foundation will be buckled with m = 1. In contrast, in the present example, the buckling mode m P 2 when it rests on an elastic foundation. It can also be seen that the buckling mode m may be increased with increase in e0a, which means the wavelength kx (=2 L/m) becomes small when e0a is sufficiently larger.
Fig. 2 shows the effect of the small scale parameter e0a (=0.0, 2.0 and 4.0 nm) on the postbuckling behavior of a (10, 10)-tube subjected to axial compression and resting on an elastic foundation at T = 300 K. It can be seen that the postbuckling equilibrium path first goes down smoothly and then rises sharply when the deflection of the tube is sufficiently larger. The results confirm that the postbuckling equilibrium paths are virtually stable for nanobeams resting on elastic foundations. The small scale parameter e0a reduces the buckling loads, but increases the postbuckling strength when the nanobeam deflection is greater than 6 nm. Fig. 3 presents the postbuckling responses of a (12, 12)-tube under three different sets of thermal environmental conditions T = 300, 500 and 700 K, when the SWCNT is supported by a soft elastic foundation. The small scale parameter e0a is taken to be 2.0 nm. It can be seen that the buckling loads as well as postbuckling strength are reduced with increase in temperature. It is found that the effect of temperature changes on the postbuckling behavior is more pronounced when the temperature-dependent material properties are taken into account. In other words, the difference of the buckling loads (as well as postbuckling equilibrium paths) for the SWCNT at two thermal environmental conditions becomes large when the material properties are temperature-dependent. Fig. 4 compares the postbuckling behavior of three kinds of SWCNTs resting on elastic foundations at T = 300 K. The small scale parameter e0a is taken to be 2.0 nm. It can be seen that the (17, 0)tube has the lowest buckling load among the three tubes, while it has a higher postbuckling strength than (12, 12)-tube when the deflection is greater than 12 nm. 4. Nonlinear bending problem The static transverse load is assumed to be uniform, and Q(X) = q0. Two cases are considered in this section. Case 1, one end of the beam is movable and initial axial compressive loads are applied. In such a case, kP will be replaced by P/Pcr and Eq. (7) can be written in the simple form as
40
40
(10, 10)-tube
(12, 12)-tube
L/R =100, R = 0.681 nm, h = 0.0674 nm
L/R =100, R = 0.816 nm h = 0.0671 nm, e0a =2.0 nm HS=1 mm
HS=1 mm, T= 300 K
30
Px (nN)
Px (nN)
30 20
e0a =0.0 nm e0a =2.0 nm e0a =4.0 nm
10
20
T= 300 K T= 500 K T= 700 K
10
0 0
5
10
15
Wm (nm) Fig. 2. Comparisons of postbuckling behavior of (10, 10)-tube with different values of small scale parameter e0a subjected to axial compression and resting on an elastic foundation.
0 0
5
10
15
20
25
Wm (nm) Fig. 3. The effect of temperature rise on the postbuckling behavior of (12, 12)-tube resting on an elastic foundation in thermal environments.
1026
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029
From Eqs. (18) and (19), the load–central deflection relationship can be written as
40 L/R =100, e0a =2.0 nm HS=1 mm, T= 300 K
Px (nN)
30
q0 L3 ð1Þ W m ¼ AW p4 EI L
! ð3Þ
þ AW
Wm L
!3 þ
ð20Þ
20 ðiÞ
0
(12, 12)-tube, R = 0.816 nm, h = 0.0671 nm (10, 10)-tube, R = 0.681 nm, h = 0.0674 nm (17, 0)-tube, R = 0.669 nm, h = 0.0871 nm
0
5
10
15
Wm (nm) Fig. 4. Comparisons of postbuckling behavior of three kinds of SWCNTs resting on elastic foundations.
# 4 dW 1þp þp þ 4 dx dx " # 2 4 3 2 d W d W dW 2 dW 4 dW þ 4p2 1 þ 2 p þ 3 p þ 3 2 dx dx dx dx dx # !3 " 2 4 2 2 d W 2 dW 4 dW 1 þ 6p þ 15p þ þp 2 dx dx dx !( ! 2 @2 d W kq K 1 W K 2 ¼ 1 s2 p2 2 2 @x dx " #) 2 4 2 P d W 3 2 dW 15 4 dW 1 þ p þ p þ ð15Þ Pcr dx2 2 dx 8 dx 4
d W
"
2
dW dx
2
4
Case 2, two ends of the beam are immovable and no axial compressive loads are applied. In such a case, Eq. (8) can be written in the simple form as
" # 2 4 @4W 2 @W 4 @W 1þp þp þ @x4 @x @x " # 2 4 3 2 2 @ W @ W @W 2 @W 4 @W 1 þ 2p þ 3p þ þ 4p @x3 @x2 @x @x @x # !3 " 2 4 @2W 2 @W 4 @W 1 þ 6 p þ 15 p þ þ p2 @x2 @x @x ( ! 2 2 @ @ W ¼ ð1 s2 p2 2 Þ kq K 1 W K 2 @x2 @x "Z # pg p @W 2 @ 2 W dx þ @x @x2 2 0 " #) 2 4 @2W 3 2 @W 15 4 @W gaDT 1þ p þ p þ ð16Þ @x2 2 @x 8 @x
0.6 (10, 10)-tube L/R =100, R = 0.681 nm, h = 0.0674 nm HS=1 mm, T= 300 K
0.4
Wm/L
10
In Eq. (20), AW ði ¼ 1; 3; . . .Þ are described in detail in Appendix B. Fig. 5 shows the effect of the small scale parameter e0a (=0.0, 4.0 and 8.0 nm) on the nonlinear bending behavior of the (10, 0)-tube subjected to a transverse uniform load resting on an elastic foundation at T = 300 K. The temperature-dependent material properties of SWCNTs and substrate are adopted, as given in Section 3. The immovable end conditions are considered in this example. It can be seen that for a given load, the deflection decreases with increase in e0a. It is found that the effect of the small scale parameter on the nonlinear bending response is weaker when compared to the effect of the small scale parameter on the postbuckling response. Fig. 6 shows the effect of temperature rise on the nonlinear bending behavior of the (12, 12)-tube subjected to a uniform pressure and resting on an elastic foundation. Three sets of thermal environmental conditions, i.e. T = 300, 500 and 700 K, are considered. The end condition is assumed to be immovable. The small scale parameter e0a is taken to be 4.0 nm. The results show that the deflections are increased with increase in temperature.
e 0a =0.0 nm e 0a =4.0 nm e0a =8.0 nm
0.0
j¼1
e wj ðxÞ; kq ¼
X
j
e kj
ð17Þ
j¼1
The solution methodology is similar to that used in Section 3, and the asymptotic solutions are obtained as
3 ð1Þ ð1Þ WðxÞ ¼ A10 e sin mx þ A10 e ½A330 sin 3mx
5 ð1Þ þ A10 e ½A530 sin 3mx þ A550 sin 5mx þ Oðe6 Þ
ð18Þ
3 ð1Þ ð1Þ kq ¼ k1 A10 e þ k3 A10 e þ
ð19Þ
100
200
300
400
500
3
Fig. 5. Comparisons of nonlinear bending behavior of (10, 10)-tube with different values of small scale parameter e0a subjected to a uniform pressure and resting on an elastic foundation.
0.6 (12, 12)-tube L/R =100, R = 0.816 nm h = 0.0671 nm, e0a =4.0 nm HS=1 mm
0.4
Wm/L
Wðx; eÞ ¼
j
0
q0L /EI
The solutions of Eqs. (15) and (16) can also be determined by a twostep perturbation technique. In the present case, we assume that
X
immovable end condition
0.2
immovable end condition
0.2
0.0
T= 300 K T= 500 K T= 700 K
0
100
300
200
400
500
3
q0L /EI Fig. 6. The effect of temperature rise on the nonlinear bending behavior of (12, 12)tube resting on an elastic foundation in thermal environments.
1027
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029
The solutions of Eqs. (7) and (8) can also be determined by a twostep perturbation technique, and the asymptotic solutions are obtained as a similar form of Eq. (18)
8 (17, 0)-tube L/R =100, R = 0.669 nm h = 0.0871 nm, HS=1 mm T= 500 K, e0a =4.0 nm
Wm/L
6
3 ð1Þ ð1Þ Wðx; tÞ ¼ A10 e sin mx þ A10 e ½A330 sin 3mx
5 ð1Þ þ A10 e ½A530 sin 3mx þ A550 sin 5mx þ Oðe6 Þ
4
ð22Þ
ð1Þ
2
0
Note that now A10 is a function of t. For nonlinear free vibration, since Q ðX; tÞ ¼ 0, we also have
immovable end condition movable end condition, P/Pcr=0.0 movable end condition, P/Pcr=0.4
ð1Þ g 1 A10
0
100
200
300
400
e sin mx þ
ð1Þ g 3 A10
3
e sin mx þ g 2
ð1Þ @ 2 A10 e
500
@t 2
sin mx þ ¼ 0 ð23Þ
3
q0L /EI
Let
Fig. 7. Comparisons of nonlinear bending behavior of (17, 0)-tube with two different end conditions and resting on an elastic foundation.
2
Z p
4g 1
0
Fig. 7 shows the effect of end conditions on the nonlinear bending behavior of the (17, 0)-tube subjected to a uniform pressure and resting on elastic foundations at T = 500 K. The small scale parameter e0a is taken to be 4.0 nm. To this end, the load-deflection curves of nanobeams under movable and immovable end conditions are displayed. For the case of movable end condition, the initial axial compressive loads are taken to be P/Pcr = 0.0 and 0.4. The results confirm that the initial stress due to axial compression has a significant effect on the nonlinear bending behavior of the nanobeam. The results show that the nanobeam with immovable ends will undergo much less deflections.
5. Nonlinear vibration problem In this section Eqs. (7) and (8) are directly used for two kinds of end conditions. The initial conditions are assumed to be
Wjt¼0 ¼
@W ¼0 @t t¼0
ð21Þ
ð1Þ A10
e sin mx þ g 3
ð1Þ A10
3
e sin mx þ g 2
ð1Þ @ 2 A10 e @t2
3 sin mx þ 5
ðsin mxÞdx ¼ 0
ð24Þ
From which one has 2
d g2
eAð1Þ 10
dt
2
þ g1
ð1Þ eAð1Þ þ g 3 eA10 10
3
¼0
ð25Þ
In Eq. (25), g1, g2 and g3 are described in detail in Appendix C, and the solution of which may be written as
2
xNL
3 g3 W m ¼ xL 4 1 þ 4 g1 L
!2 31=2 5
ð26Þ
where xL = [g1/g2]1/2 is the dimensionless linear frequency. According to Eq. (6), the corresponding linear frequency can be expressed as XL = xL(p/L)(E/q)1/2. It is worth noting that the linear frequencies are the same for these two kinds of end conditions when no axial load is applied. Table 4 shows the effect of the small scale parameter e0a (=0.0, 5.0 and 10.0 nm) as well as temperature change on the nonlinear to linear frequency ratios xNL/xL of the same two (10, 10) and (17, 0)
Table 4 Comparisons of nonlinear to linear frequency ratios for CNTs with immovable end conditions and resting on an elastic foundation in thermal environments (HS = 1 mm and L/ R = 100). T (K)
(10, 10)-tube 300
e0a (nm)
xL (GHz)
xNL/xL W m =L ¼ 0
W m =L ¼ 0:1
W m =L ¼ 0:15
W m =L ¼ 0:2
W m =L ¼ 0:25
W m =L ¼ 0:3
0.0 5.0 10.0
495.8115 495.5956 495.0613
1.0 1.0 1.0
1.2833 1.2835 1.2840
1.5669 1.5673 1.5683
1.8940 1.8945 1.8959
2.2455 2.2463 2.2481
2.6117 2.6126 2.6149
500
0.0 5.0 10.0
404.2776 404.0184 403.3762
1.0 1.0 1.0
1.3970 1.7974 1.3985
1.7723 1.7730 1.7749
2.1923 2.1934 2.1961
2.6358 2.6372 2.6406
3.0926 3.0943 3.0986
700
0.0 5.0 10.0
284.2779 283.9128 283.0076
1.0 1.0 1.0
1.7036 1.7050 1.7085
2.2978 2.3001 2.3060
2.9340 2.9373 2.9454
3.5900 3.5942 3.6046
4.2567 4.2617 4.2744
0.0 5.0 10.0
442.3508 442.2352 441.9510
1.0 1.0 1.0
1.1728 1.1729 1.1730
1.3582 1.3583 1.3587
1.5817 1.5819 1.5824
1.8293 1.8296 1.8303
2.0925 2.0929 2.0938
500
0.0 5.0 10.0
363.5445 363.4045 363.0600
1.0 1.0 1.0
1.2460 1.2462 1.2466
1.4978 1.4981 1.4988
1.7918 1.7922 1.7933
2.1104 2.1110 2.1124
2.4441 2.4448 2.4466
700
0.0 5.0 10.0
263.6971 263.5053 263.0334
1.0 1.0 1.0
1.4289 1.4294 1.4307
1.8287 1.8296 1.8318
2.2732 2.2745 2.2777
2.7407 2.7424 2.7465
3.2213 3.2233 3.2284
(17, 0)-tube 300
1028
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029
Table 5 Comparisons of nonlinear to linear frequency ratios for (12, 12)-tube with movable and immovable end conditions and resting on an elastic foundation in thermal environments (HS = 1 mm and L/R = 100). T (K)
e0a (nm)
Immovable end conditions 300 0.0 5.0 10.0
xL (GHz)
xNL/xL W m =L ¼ 0
W m =L ¼ 0:1
W m =L ¼ 0:15
W m =L ¼ 0:2
W m =L ¼ 0:25
W m =L ¼ 0:3
414.6166 414.4922 414.1669
1.0 1.0 1.0
1.2769 1.2771 1.2774
1.5552 1.5555 1.5562
1.8768 1.8772 1.8781
2.2228 2.2233 2.2247
2.5836 2.5842 2.5859
500
0.0 5.0 10.0
337.9666 337.8180 337.4295
1.0 1.0 1.0
1.3864 1.3867 1.3874
1.7535 1.7540 1.7553
2.1653 2.1660 2.1679
2.6007 2.6016 2.6041
3.0495 3.0506 3.0537
700
0.0 5.0 10.0
238.6169 238.4081 237.8617
1.0 1.0 1.0
1.6830 1.6840 1.6864
2.2635 2.2650 2.2691
2.8862 2.8884 2.8941
3.5290 3.5318 3.5391
4.1825 4.1859 4.1948
414.6166 414.4922 414.1669
1.0 1.0 1.0
1.0003 1.0003 1.0003
1.0007 1.0007 1.0006
1.0012 1.0012 1.0011
1.0019 1.0019 1.0017
1.0028 1.0027 1.0024
Movable end conditions 300 0.0 5.0 10.0 500
0.0 5.0 10.0
345.8134 345.6682 345.2285
1.0 1.0 1.0
1.0004 1.0004 1.0004
1.0010 1.0009 1.0009
1.0017 1.0017 1.0015
1.0027 1.0026 1.0024
1.0039 1.0038 1.0034
700
0.0 5.0 10.0
259.4649 259.2729 258.7705
1.0 1.0 1.0
1.0008 1.0007 1.0007
1.0017 1.0017 1.0015
1.0031 1.0030 1.0027
1.0048 1.0046 1.0042
1.0069 1.0066 1.0060
tubes resting on an elastic foundation in thermal environments. In the present example, the mass density of CNTs is taken to be 1400 kg/m3 [22]. It is found that the natural frequencies are decreased with increase in e0a, but among the three, i.e. postbuckling, nonlinear bending and nonlinear vibration, the effect of the small scale parameter on the nonlinear vibration behavior is weakest. It can be seen that the temperature rise decreases the natural frequencies but increases the nonlinear to linear frequency ratios. The results confirm that the nonlinear to linear frequency ratios are increased with increase in the non-dimensional vibration amplitude of the SWCNTs. Table 5 shows the effect of the beam end conditions on the nonlinear to linear frequency ratios xNL/xL of the same (12, 12)-tube with movable and immovable end conditions and resting on an elastic foundation in thermal environments. It can be seen that although the linear frequencies are the same for these two kinds of end conditions (at T = 300 K), the nonlinear to linear frequency ratios for movable end conditions are much less than those for immovable end conditions. It can also be seen that the nonlinear to linear frequency ratios are increased slightly for immovable end conditions, whereas are decreased slightly for movable end conditions when the small scale parameter e0a is increased. 6. Concluding remarks A nonlocal beam model is proposed for nonlinear analysis of SWCNTs resting on a two-parameter elastic foundation in thermal environments. The nonlocal stress incorporating the small scale parameter e0a is introduced into the governing equations. The postbuckling equilibrium paths of an axially loaded nanobeam, the static large deflections of a bending nanobeam with or without initial stresses subjected to a uniform transverse pressure, and the nonlinear frequencies of a nanobeam have been presented. Perturbation solutions are obtained in an explicit form that can be easy to program and produces full nonlinear responses. The numerical results reveal that the small scale parameter e0a reduces the postbuckling equilibrium paths, the static large deflections and natural frequencies of SWCNTs resting on an elastic foundation. In contrast, it increases and reduces the nonlinear to linear frequency ratios slightly for the nanobeam with immovable and mo-
vable end conditions, respectively. The results also reveal that the effect of the small scale parameter is significant for compressive buckling, but less pronounced for static bending and marginal for free vibration of SWCNTs resting on an elastic foundation. Appendix A In Eq. (14)
m2 1 þ ðK 1 þ K 2 m2 Þ; R10 m2
kð0Þ p ¼
kð4Þ p ¼
kð2Þ p ¼
p2 8
m2
m2 3 2 ðK 1 þ K 2 m2 Þ ; R10 m
3 4 2 m2 m2 33 p m 2 ð4m2 81a330 Þ þ a330 25 þ 2 ðK 1 þ K 2 m2 Þ 64 R10 R10 m 4
m 3ðK 1 þ K 2 m2 Þ R10
in which
2
a330
3 ð0Þ 2 4m R k 30 p h i5; ¼ 4m4 81m4 þ R30 ðK 1 þ 9K 2 m2 Þ 9m2 kpð0Þ
R10 ¼ 1 þ s2 p2 m2 ;
R30 ¼ 1 þ 9s2 p2 m2
ðA:2Þ
Appendix B In Eq. (20), for case 1 ð1Þ
AW ¼ ð3Þ
AW ¼
P 2 m m m4 þ R10 ðK 1 þ K 2 m2 Þ Pcr 4
p
2
p 4 2 3 P R10 m m m 4 Pcr 4 2 3 2 P 2 4 m p a330 m þ R10 ðK 1 þ K 2 m2 Þ 8 Pcr
p
a330 ¼ m4
4m2 R30 P=Pcr 4 81m þ R30 ½ðK 1 þ 9K 2 m2 Þ 9m2 P=Pcr
ðB:1Þ
H.-S. Shen, C.-L. Zhang / Computational Materials Science 50 (2011) 1022–1029
and for case 2 ð1Þ
AW ¼
ð3Þ
AW ¼
p 4
1029
References
m m4 þ R10 ðK 1 þ K 2 m2 Þ m2 gaDT
2 p 4 2 1 3 a DT m m m þ R10 g 2 4 4 2 4 3 2 p a330 m þ R10 ðK 1 þ K 2 m2 Þ m2 gaDT 8
p
ðB:2Þ
Appendix C In Eq. (25)
g 2 ¼ R10 g
ðC:1Þ
and for case 1
P 2 ; g 1 ¼ m4 þ R10 ðK 1 þ K 2 m2 Þ m Pcr
p2 4 2 3 P R10 m m g3 ¼ 4 Pcr 2 and for case 2
g 1 ¼ m4 þ R10 ðK 1 þ K 2 m2 Þ m2 gaDT ;
p2 4 2 1 3 g3 ¼ aDT m m þ R10 g 2 4 2
ðC:3Þ
[1] E.M. Abdel-Rahman, M.I. Younis, A.H. Nayfeh, J. Micromech. Microeng. 12 (2002) 759. [2] X. Li, B. Bhushan, K. Takashima, C.-W. Baek, Y.-K. Kim, Ultramicroscopy 97 (2003) 481. [3] O. Lourie, D.M. Cox, H.D. Wagner, Phys. Rev. Lett. 81 (1998) 1638. [4] J. Yoon, C.Q. Ru, A. Mioduchowski, Compos. Sci. Technol. 65 (2005) 1326. [5] Y.M. Fu, J.W. Hong, X.Q. Wang, J. Sound Vib. 296 (2006) 746. [6] J. Xiao, H. Jiang, D.-Y. Khang, J. Wu, Y. Huang, J.A. Rogers, J. Appl. Phys. 104 (2008) 033543. [7] A.C. Eringen, J. Appl. Phys. 54 (1983) 4703. [8] J. Peddieson, G.R. Buchanan, R.P. McNitt, Int. J. Eng. Sci. 41 (2003) 305. [9] S. Adali, Phys. Lett. A 372 (2008) 5701. [10] L.J. Sudak, J. Appl. Phys. 94 (2003) 7281. [11] T. Murmu, S.C. Pradhan, Physica E 41 (2009) 1232. [12] S.C. Pradhan, T. Murmu, J. Appl. Phys. 105 (2009) 124306. [13] H.-L. Lee, W.-J. Chang, Physica E 41 (2009) 529. [14] X.-F. Li, B.-L. Wang, Y.-W. Mai, J. Appl. Phys. 103 (2008) 074309. [15] S. Narendar, S. Gopalakrishnan, Physica E 42 (2010) 1706. [16] C.Y. Wang, L.C. Zhang, Nanotechnology 19 (2008) 075705. [17] D.-Y. Khang, J. Xiao, C. Kocabas, S. MacLaren, T. Banks, H. Jiang, Y.Y. Huang, J.A. Rogers, Nano Lett. 8 (2008) 124. [18] J. Dong, L. Peng, J. Wuhan Univ. Hydraulic Electric Eng. 24 (1991) 427. [19] H.-S. Shen, Int. J. Struct. Stability Dyn. 11 (2011) (in press). [20] T. Chang, J. Geng, X. Guo, Appl. Phys. Lett. 87 (2005) 251929. [21] C.-L. Zhang, H.-S. Shen, Appl. Phys. Lett. 89 (2006) 081904. [22] J.D. Fidelus, E. Wiesel, F.H. Gojny, K. Schulte, H.D. Wagner, Compos. A 36 (2005) 1555.