- Email: [email protected]
Combined strain-inertia gradient elasticity in free vibration shell analysis of single walled carbon nanotubes using shell theory
Applied Mathematics and Computation 243 (2014) 856–869
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Combined strain-inertia gradient elasticity in free vibration shell analysis of single walled carbon nanotubes using shell theory Farhang Daneshmand ⇑ Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec H3A 2K6, Canada Department of Bioresource Engineering, McGill University, 21111 Lakeshore Road, Sainte-Anne-de-Bellevue, Quebec H9X 3V9, Canada
a r t i c l e
i n f o
Keywords: Carbon nanotubes Gradient elasticity theory Size effect Vibration Winkler/Pasternak foundation
a b s t r a c t In this paper, a gradient elasticity shell formulation is presented for free vibration analysis of single-walled carbon nanotube placed on Winkler/Pasternak foundation. The proposed formulation is based on the combined strain-inertia gradient elasticity. The combined strain-inertia gradient elasticity provides an extension to the classical equations of elasticity with additional higher-order spatial derivatives of strains and two material length scale parameters related to the inertia and strain gradients, which enable formulation to investigate the size effect on the dynamic behavior of nanotubes. The effects of the length scale parameters, aspect ratio of single-walled carbon nanotube and foundation parameters on the fundamental frequencies for different values half-axial wave number and circumferential wave number are investigated. The natural frequencies obtained from the proposed shell formulation show the effects of size-dependent properties. It can be concluded that a continuum model enriched with higher-order inertia terms has been proposed as alternative to the continuum description obtained with classical elasticity theory. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction With the advance of materials synthesis and device processing capabilities over the past decade and since the discovery of carbon nanotubes (CNTs) in the early 1990s, the importance of developing and analyzing nanoscale engineering devices has dramatically increased. Carbon nanotubes are well-known as a wealth of technological applications because of their theoretically predicted mechanical properties including high strength, high stiffness, low density and structural perfection. Carbon nanotubes are also expected to be used in nano-fluidic devices and systems, nano-sensors and drug delivery devices [18]. Among different types of CNTs, the single-walled carbon nanotubes (SWCNTs) possess outstanding physical and chemical properties when compared to other nanoscale materials. A SWCNT can be viewed as a two-dimensional graphene sheet rolled into a hollow cylindrical, shell-like macromolecules composed of carbon atoms arranged in periodic hexagonal cells [10]. A single-walled carbon nanotube is uniquely indexed by a pair of integers (n, m) representing its chirality. The two limiting cases of nanotubes are (n, 0) with chiral angle = 0 and (n, n) with chiral angle = p/6. These cases are usually known as zigzag and armchair tubes based on the geometry of carbon bonds around the circumference of the nanotube (Fig. 1). The ⇑ Address: Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec H3A 2K6, Canada. E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2014.05.094 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
857
Fig. 1. Three possible patterns along the circumference of the CNT: ‘zigzag’ (i), ‘armchair’ (ii) and ‘chiral’ (iii) Fig. 1. Three possible patterns along the circumference of the CNT: ‘zigzag’ (i), ‘armchair’ (ii) and ‘chiral’ (iii).
outer diameter of SWCNTs is 1–2 nm whereas the possible lattice symmetries are determined by the wrapping direction. Carbon nanotubes are an extremely stiff and strong material and provide superior mechanical properties such as Young’s modulus approaching 1 TPa which is five times higher than that of steel (210 GPa) strength approaching 100 GPa with a density of about 1.3 g/cm3 [19]. The ability of SWCNTs to conduct heat even better than diamond suggests their eventual use in nano-electronics. Steady progress reported in using SWCNTs in developing nanodevices and nanocircuits showing remarkable logic and amplification functions. The use of carbon nanotube electrodes and new device geometries is enhancing the performance of organic light-emitting transistors (OLETs) [22]. In their design, the source electrode was made of a porous network of single-walled carbon nanotubes. Recently fabricating electrodes made of carbon nanotubes are expected to improve electron and hole injection efficiency [12]. In general, there are two main approaches to the analysis and modeling of carbon nanotubes. The first approach is the atomic modeling. The well-known molecular dynamics (MD) method may be considered as the most common atomic modeling [14]. Molecular dynamics method requires extremely fast computing facilities and is mainly confined to systems with a limited number of molecules. This limitation is due to considering each individual molecule with its multiple mechanical or chemical interactions. Representing nanostructured materials with equivalent-continuum models can be considered as the second approach to describe their mechanical behavior. Different elastic linear and nonlinear beam and shell models based on the principles of continuum mechanics have been presented to predict the static and dynamic behavior of single-walled carbon nanotubes. Heshmati and Yas [11], Sedighi and Shirazi [23], Yan et al. [29] and Yas and Heshmati [30] investigated the flow-induced instability of double-walled carbon nanotubes using Donnell’s shell theory. A two-dimensional elastic shell model to characterize the deformation of single-walled carbon nanotubes using the in-plane rigidity, Poisson ratio, bending rigidity and off-plane torsion rigidity as independent elastic constants has been proposed by Wang and Zhang [26]. However, the validity of assumptions and limitations of these models must be carefully examined. It should be noted that the above continuumbased models may be classified as local models in which the stress at a reference point is related to the local strain field at that point. These local continuum models do not exhibit intrinsic size-dependence features of the material and do not allow inclusions and property in homogeneties that are fundamental and significant in atomic modeling. It has been reported by many investigators that size-dependent effects are significantly more prominent at the nanoscale and must be taken into account in the modeling of nanoscale structures. Some recent studies can be found in the literature addressing this issue by considering nonlocal elasticity theory as an extension of the classical equations of elasticity [1,3,4,7,8,13,20,21,24,31]. The classical constitutive equations of elasticity can also be generalized by using the Laplacian of stress or strain in the gradient elasticity theory. Different gradient elasticity formulations have been used in the literature to model the vibration analysis of carbon nanotubes with Euler–Bernoulli and Timoshenko assumptions [5,16,17]. It is worth nothing that the mathematical formulation in the nonlocal elasticity theory as proposed by Eringen [7] can also be viewed as a stress gradient elasticity theory which is quite sufficient for static problems whereas it might be inadequate in dynamic analysis of nanotubes. More recently, [28] compared two gradient elasticity formulations with stress and combined strain-inertia gradients for dynamic analysis of fluid-conveying nanotubes. The stress gradient beam models has one material length scale parameter whereas the combined strain-inertia gradient beam models include two material length scale parameters related to the inertia and strain gradients and can also be used to investigate the size effect on the dynamical behavior of nanotubes conveying fluid. More recently, the standard constitutive equations of elasticity have been generalized by using the Laplacian of stress and combined strain-inertia for isotropic materials and the first-order shear deformation shell theory [6]. It is the aim of the present article to present a shell formulation enriched with the combined strain-inertia gradient elasticity for investigating the effects of material length scales on the frequency analysis of a SWCNT. Due to this issue, the fundamental aspects of the gradient elasticity theory and combined strain-inertia gradient theory are presented in Section 2. Also, the mathematical formulation required for the modeling of the SWCNT as a thin shell structure with combined strain-inertia gradient theory and the solution method of the shell equations are given in Sections 3 and 4, respectively.
858
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
The results of the present formulation are presented in section 5. In this section, the effects of the length scale parameters, aspect ratio of the SWCNT and foundation parameters on the fundamental frequencies for different values half-axial wave number and circumferential wave number are investigated. 2. Combined strain-inertia gradient theory According to the theory of elasticity with microstructure that was represented by Mindlin [15], the both of kinematic energy density T and the deformation energy density U can be written in terms of quantities at microscopic and macroscopic scale.
T¼
1 1 qu_ i u_ i þ qL21 w_ ij wij ; 2 2
ð1Þ
U¼
1 1 1 C ijkl eij ekl þ Bijkl cij ckl þ Aijklmn jijk jlmn þ Dijklm cij jklm þ F ijklm jijk elm þ Gijkl cij ekl ; 2 2 2
ð2Þ
where q is the mass density that is assumed to be equal at macroscopic and microscopic scale. L1 is related to the size unit cell of microstructure. uij and eij are the macroscopic displacement and strain and wij , cij and jijk are the microscopic deformation, the difference between microscopic and macroscopic deformation (the relative deformation) and the gradient of microscopic deformation, respectively. The other microscopic parameters are introduced in Appendix A. Moreover, Aijklmn , Bijkl , C ijkl , Dijklm , F ijklm and Gijkl are the constitutive tensors [15]. To simplify the relations of kinematic energy and deformation energy, the microscopic deformation is assumed as the first gradient of the macroscopic strain [15]. Due to this assumption, the kinematic energy can be expressed in terms of the macroscopic displacements ui , only.
T¼
1 1 qu_ i u_ i þ qL21 u_ i;j u_ i;j : 2 2
ð3Þ
Also, the reduced form of the deformation energy can be expressed as bellow:
U¼
1 keii ejj þ leij eij þ a1 eik;i ejj;k þ a2 ejj;i ekk;i þ a3 eik;i ejk;j þ a4 ejk;i ejk;i þ a5 ejk;i eij;k ; 2
ð4Þ
k and l are the lame constant and ai is the constitutive coefficient. By using the Hamilton’s principle and the relations where of deformation energy and kinematic energy, the equation of motion can be expressed with the macroscopic displacement terms as bellow in the absence of body forces:
€ i L21 u €i;jj ; ðk þ lÞ uj;ij L22 uj;ijkk þ l ui;jj L23 ui;jjkk ¼ q u
ð5Þ
where
L22 ¼
41 þ 4a2 þ 3a3 þ 2a4 þ 3a5 ; 2ðk þ lÞ
L23 ¼
a3 þ 2a4 þ a5 : 2l
It should be noted that the additional parameters L1 , L2 and L3 have the dimension of length and they relate to principle of microstructure. The combined strain- inertia gradient elasticity theory is a special case of Mindlin theory when L2 ¼ L3 ¼ Ls [2]. It should be noted that the combination of the strain gradient and inertia gradient include two material length scale parameters related to the inertia and strain gradients. Moreover, considering the length scale parameter dependent to the inertia, allow us to investigate the dynamical behavior of structure in nano-scale accurately. According to this issue, the equation of motion with the combined strain-inertia gradient effect can be expressed as:
€i L2m u € i;mm ; C ijkl uk;jl L2s uk;jlmm ¼ q u
ð6Þ
where for isotropic linear elasticity, the constitutive tensor C ijkl is expressed as:
C ijkl ¼ kdij dkl þ ldik djl þ ldil djk :
ð7Þ
Ls and Lm ¼ L1 are the length scale parameters which are related to the strain gradients and inertia gradients, respectively. It should be noted that the two length scales are related to the Representative Volume Element (RVE) size in elastostatics and elastodynamics [9]. 3. Combined strain-inertia gradient elasticity shell models for SWCNTs An isotropic single-walled carbon nanotube (SWCNT) with length L, thickness h, and radius of the middle surface R as shown in Fig. 2 is considered in the present study. The SWCNT is viewed as a hollow cylindrical shell rested on an elastic foundation. The elastic foundation is represented by the Winkler/Pasternak model with radial elastic modulus Kw and shear modulus Gw. The problem is considered with the material properties: Young’s modulus E, Poisson’s ratio m and the mass density q. An orthogonal coordinate system (x, h, z) is assumed to be established at the middle surface of the shell where x, h and z
859
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
Fig. 2. A single-walled carbon nanotube placed on Winkler/Pasternak foundation.
represent the axial (longitudinal), circumferential (angular) coordinates and the distance from the nanotube axis in radial direction perpendicular to the nanotube axis, respectively. The axial, circumferential and radial displacement deformations are also denoted by ux , uh , uz , respectively. Using the combined strain-inertia gradient theory for an isotropic elastic material that was represented in previous section, the constitutive equations can be written as:
8 9 rxx > > > > > > < = > > > :
rhh sxh
> > > ;
2
Q 11
6 6 ¼ 6 Q 21 4 0
Q 12 Q 22 0
8 8 9 91 9 308 €exx > exx > exx > > > > > > > > > > > > > > > > < < = =C = 7B< 7B C 0 7B ehh L2s r2 ehh C þ qL2m €ehh : > > > > > > 5@> A > > > > > > > > > > : : :€ > ; ; cxh cxh Q 66 cxh ; 0
ð8Þ
For isotropic materials the reduced stiffness Q ij (i, j = 1, 2 and 6) are defined as
Q 11 ¼ Q 22 ¼
E ; 1 t2
Q 12 ¼
tE ; 1 t2
Q 66 ¼
E : 2ð1 þ tÞ
ð9Þ
Due to the Sanders–Koiter linear shell theory, the strain–displacement relationships of the middle surface and changes in the curvature for a circular cylindrical shell are given as
f e11
e22
f k11
k22
e12 g ¼
ux;x n k12 g ¼ uz;xx
ux;h þ uh;x ; 1 o 1 1 u þ 3u 4u ; x;h h;x z;xh 2 ðuh;h uz;hh Þ 2R R R
1 ðuh;h R
þ uz Þ
1 R
ð10Þ
where ðe11 ; e12 ; e22 Þ and ðk11 ; k12 ; k22 Þ are the reference surface strains and curvatures, respectively. For a vibrating SWCNT considered as a thin cylindrical shell the expression for the strain energy, S is given by
S¼
1 2
Z
L
Z 2p h
2
2
2
A11 e211 þ 2A12 e11 e22 þ A22 e222 þ A66 e212 þ D11 k11 þ 2D12 k11 k22 þ D22 k22 þ D66 k12 0 0 i €11 k11 þ k €22 k22 þ k €12 k12 Rdhdx; þ L2m I1 ð€e11 e11 þ €e22 e22 þ €e12 e12 Þ þ L2m I3 k
1 L2s r2 ð11Þ
where Aij and Dij are the membrane, coupling and flexural stiffness defined by:
Z
fAij ; Dij g ¼
h=2
Q ij f1; z2 gdz:
ð12Þ
h=2
Substituting the expressions for the surface strains and curvatures from (10) into (11), the expression for the strain energy is transformed to the following form:
Z Z u 2 1 L 2p 2A12 ux;x A22 x;h A11 u2x;x þ ðuh;h þ uz Þ þ 2 ðuh;h þ uz Þ2 þ A66 þ uh;x þ D11 u2z;xx 2 0 0 R R R 2 2D12 uz;xx D22 D66 ux;h 2 2 2 1 L þ ð u u Þ þ ð u u Þ þ þ 3u 4u r h;h z;hh h;h z;hh h;x z;xh s R R2 R4 4R2 u u
1 x;htt x;h þ uh;xtt þ uh;x þ L2m I1 ux;xtt ux;x þ 2 ðuh;htt þ uz;tt Þðuh;h þ uz Þ þ R R R u 1 1 ux;htt x;h 2 þ 3uh;xtt 4uz;xhtt þ 3uh;x 4uz;xh Rdhdx; þ Lm I3 uz;xxtt uz;xx þ 4 ðuh;htt uz;hhtt Þðuh;h uz;hh Þ þ 2 R R R 4R
S¼
ð13Þ
where t denotes time. It should be noted that I1 and I3 stand for the mass moment of inertia of SWCNT per unit area as:
fI1 ; I3 g ¼
Z
h=2
h=2
qf1; z2 gdz:
ð14Þ
860
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
The kinetic energy T for a thin-walled cylindrical shell made of single-walled carbon nanotube is given by
T¼
1 2
Z 0
L
Z 2p q u2x;t þ u2h;t þ u2h;t Rdhdx:
ð15Þ
0
Employing Hamilton’s principle to the Lagrangian energy functional and introducing the terms describing the Winkler and Pasternak foundations ðK w uz Gw r2 uz Þ in the z-direction, the dynamical equations for a functionally graded cylindrical shell can be written as
"
# A12 D11 ðD12 þ D66 Þ uh;xh þ A11 ux;xx þ ux;hh þ uz;xhh ð1 L2s r2 Þ uz;x uz;xxx R R 4R4 R3 4R3 !
" I1 I3 I3 1 4R2 ðA12 þ A66 Þ þ D66 2 2 2 3 I3 ¼ I1 ð1 Lm r Þux;tt þ Lm ux;xh uh;xhtt þ ux;hhtt þ uz;xxxtt þ 2 uz;xhhtt 4 R4 R 4R3 R R 4R3
21 D66 A22 D22 A22 D22 ð5D66 D12 Þ þ A66 þ uh;xx þ þ 4 uh;hh þ 2 uz;h 4 uz;hhh þ uz;xxh ð1 L2s r2 Þ 4 R2 R2 R2 R R R
I1 I3 I3 1 21 I1 2 uz;htt þ 3 uh;xhtt þ 2 u þ u ¼ I1 ð1 L2m r2 Þuh;tt þ L2m h;hhtt h;xxtt 4 R 4R R R2 R
I3 1 A12 D66 D11 ð2D12 þ 5D66 Þ uh;xxh ux;x þ 3 ux;xhh þ ux;xxx þ 2 5uz;xxhtt þ 2 uz;hhhtt R R R R2 R R D22 A22 D12 ð2D12 þ 4D66 Þ D22 A22 uz;xxhh 4 uz;hhhh 2 uz 1 L2s r2 þ 4 uh;hhh 2 uh;h þ 2 uz;xx D11 uz;xxxx 2 R R R R R R !
2 L I3 1 I1 I3 1 ¼ I1 1 þ m2 uz;tt þ L2m ux;xxxtt þ 2 ux;xhhtt þ 2 uh;htt 2 5uh;xxhtt þ 2 uh;hhhtt R R R R R R
1 4 þ K w uz Gw r2 uz : þ I3 4 uz;hhhhtt þ 2 uz;xxhhtt þ uz;xxxxtt ð16Þ R R ð4R2 A66 3D66 Þ
4R2 ðA12 þ A66 Þ þ 4D12 þ D66
!
Gw represents the shear modulus of the material used for the elastic foundation and K w for the Winkler foundation. The Winkler model is a special case of the Pasternak model when Gw ¼ 0. 4. Solution method for simply supported SWCNTs The displacement functions of SWCNTs with different support conditions can be obtained by solving the system of Eqs. (16). The solution to these equations satisfying boundary conditions is given in the following form,
ux ðx; h; tÞ ¼
1 X 1 X
uxmn ðtÞ cosðkm xÞ cosðnhÞ;
ð17Þ
m¼1 n¼1
uh ðx; h; tÞ ¼
1 X 1 X
uhmn ðtÞ sinðkm xÞ sinðnhÞ;
m¼1 n¼1
uz ðx; h; tÞ ¼
1 X 1 X
uzmn ðtÞ sinðkm xÞ cosðnhÞ:
m¼1 n¼1
Substituting Eqs. (17) into Eqs. (16), the following system of equations are obtained,
€ g31 þ ½K33 fqg31 ¼ f0g; ½M33 fq
ð18Þ
where fqg31 is a vector of the SWCNT generalized coordinates and ½M33 and ½K33 are mass and stiffness matrices (given in Appendix B), respectively. For complex modal analysis, fqg31 is assumed to be a harmonic function of time,
fqg31
8 9 > < Ux > = ¼ U h exp ðixtÞ; > : > ; Uz
ð19Þ
where U x , U h and U z are the displacement amplitudes in the x, h, z directions, respectively and x is the eigenvalue of the system which can be a complex number in general. Substituting Eq. (19) into Eq. (18), leads to the following equation
8 9 2 3 0 > < Ux > = 6 7 Hðn; m; xÞ33 U h ¼ 4 0 5; > : > ; 0 Uz
ð20Þ
861
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869 Table 1 Properties of some SWCNTs [25,27]. Parameters
(10, 0)
Young’s modulus E, (TPa) Shear modulus G, (TPa) Poisson ratio m Mass density per unit volume, q (g/cm3) Thickness, h (nm) Radius, R (nm)
5.25 2.27 0.17874 2.3 0.06709 0.39
Table 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dimensionless frequency X ¼ xR ð1 m2 Þq=E for SWCNT (10, 0), L=R ¼ 20, Ls ¼ Lm ¼ 0 and g w ¼ 0. kw ¼ 0
kw ¼ 0:25
kw ¼ 0:5
n
m
X1
X2
X3
X1
X2
X3
X1
X2
X3
1
1 2 3 4 5
0.0025 0.0095 0.0193 0.0306 0.0424
0.1043 0.1123 0.1236 0.1368 0.1505
0.2223 0.2240 0.2271 0.2316 0.2380
0.0531 0.0528 0.0548 0.0598 0.0672
0.1048 0.1137 0.1258 0.1396 0.1539
0.2294 0.2309 0.2335 0.2374 0.2429
0.0720 0.0706 0.0715 0.0755 0.0820
0.1055 0.1156 0.1285 0.1428 0.1576
0.2371 0.2384 0.2405 0.2438 0.2486
2
1 2 3 4 5
0.0208 0.0214 0.0229 0.0258 0.0301
0.2037 0.2067 0.2115 0.2178 0.2253
0.3519 0.3538 0.3572 0.3617 0.3676
0.0725 0.0725 0.0727 0.0734 0.0749
0.2038 0.2069 0.2119 0.2184 0.2261
0.3537 0.3556 0.3588 0.3632 0.3690
0.0998 0.0996 0.0995 0.0998 0.1007
0.2038 0.2072 0.2124 0.2192 0.2271
0.3557 0.3575 0.3606 0.3649 0.3705
where m and n are half-axial wave number and circumferential mode number, respectively. The characteristic equation of the problem can be obtained as the following polynomial function
Fðn; m; xÞ ¼ 0;
ð21Þ
where Fðn; m; xÞ is a polynomial function. The characteristic equation (21) can be used to determine the frequencies of SWCNT. Solving Eq. (21) provides three pairs of roots with different combination of m and n. 5. Results The material properties of some single-walled carbon nanotubes are given in Table 1. The thickness-to-radius ratios h=R for all SWCNTs used in the present study are less than 1=10. It allows us to consider these SWCNTs as thin shells.
Fig. 3. Variation of dimensionless frequency X1 with Winkler elastic foundation for single-walled carbon nanotube (10, 0), L=R ¼ 20, Ls ¼ Lm ¼ 0 and g w ¼ 0.
862
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
Fig. 4. Variation of dimensionless frequencies with Ls and Lm = 3Ls for single-walled carbon nanotube (10, 0), n = 1, L/R = 20, kw = 0.5 and gw = 0 (a) X1 (b) X2 (c) X3.
According to Eq. (21), for any combination of m and n, we thus have three frequencies. The lowest frequency is generally associated with the mode where the transverse component dominates while the other two frequencies are usually higher by an order of magnitude and are associated with the modes where the displacements in the tangent plane dominate. Three
863
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
Fig. 5. The effect of aspect ratio L=R on dimensionless frequency X1 for single-walled carbon nanotube (10, 0), n = 1, Ls ¼ 0:2; Lm ¼ 3Ls , kw ¼ 0:5 and g w ¼ 0Þ.
Fig. 6. The effect of aspect ratio L=R and length scale parameters Ls and Lm (Lm ¼ 3L3 ) on the dimensionless frequency X1 for single-walled carbon nanotube (10, 0), n = 1, kw ¼ 0:5 and g w ¼ 0.
Table 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dimensionless frequency X ¼ xR ð1 m2 Þq=E for SWCNT (10, 0), L=R ¼ 20, Ls ¼ Lm ¼ 0 and kw ¼ 0. gw ¼ 0
g w ¼ 0:25
g w ¼ 0:55
n
m
X1
X2
X3
X1
X2
X3
X1
X2
X3
1
1 2 3 4 5
0.0025 0.0095 0.0193 0.0306 0.0424
0.1043 0.1123 0.1236 0.1368 0.1505
0.2223 0.2240 0.2271 0.2316 0.2380
0.0537 0.0550 0.0593 0.0669 0.0771
0.1048 0.1139 0.1264 0.1408 0.1562
0.2296 0.2316 0.2350 0.2399 0.2463
0.0756 0.0758 0.0793 0.0863 0.0962
0.1058 0.1165 0.1305 0.1464 0.1635
0.2392 0.2416 0.2457 0.2514 0.2587
2
1 2 3 4 5
0.0208 0.0214 0.0229 0.0258 0.0301
0.2037 0.2067 0.2115 0.2178 0.2253
0.3519 0.3538 0.3572 0.3617 0.3676
0.1382 0.1386 0.1397 0.1418 0.1450
0.2040 0.2079 0.2140 0.2218 0.2308
0.3602 0.3620 0.3651 0.3695 0.3752
0.1933 0.1903 0.1888 0.1892 0.1915
0.2072 0.2152 0.2248 0.2357 0.2476
0.3738 0.3757 0.3788 0.3831 0.3887
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dimensionless frequencies X1 , X2 and X3 can be obtained by using the relation X ¼ xR ð1 m2 Þq=E. These dimensionless frequencies for single walled carbon nanotube (10, 0) are presented in Table 1 for different values of Winkler elastic foundation and neglecting the combined strain-inertia gradient effects (Ls ¼ Lm ¼ 0). Three dimensionless frequencies X1 , X2 and X3
864
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
generally correspond to the radial, circumferential and longitudinal mode shapes of the problem. The results given in Table 2 are obtained with assuming pinned–pinned condition as the end conditions of the SWCNT. As can be seen from this table, the dimensionless frequencies X1 , X2 and X3 are increased with increasing the non-dimensional Winkler elastic foundation constant kw , (kw ¼ K w R2 ð1 m2 Þ=Eh), when dimensionless shear foundation modulus g w ¼ 0, (g w ¼ Gw ð1 m2 Þ=Eh). Variation of dimensionless frequency X1 with Winkler elastic foundation is shown in Fig. 3 for different values of n = 1, 2 and 3 for single-walled carbon nanotube (10, 0). As it can be seen from this figure, dimensionless frequency X1 increases by increasing the elastic foundation constant. The dimensionless frequencies increase rapidly with dimensionless axial wave number k (k ¼ mpR=L) and the frequency curves seem to become parallel for higher values of kw . Moreover, the most noticeable increase in the dimensionless frequency X1 occurs for k < 0:5. The same effect can be studied for the dimensionless frequencies X2 and X3 and although it is not presented here, the constant of elastic foundation has a small effect on the frequencies X2 and X3 .
Fig. 7. Combined effects of Winkler and Pasternak foundation on the dimensionless frequency X1 for single-walled carbon nanotube (10, 0), n = 1, L=R ¼ 20 and Ls ¼ Lm ¼ 0.
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
865
Fig. 4a shows the variation of dimensionless frequency X1 with length scale parameters Ls and Lm ¼ 3Ls for a single-walled carbon nanotube (10, 0), circumferential wave number n = 1, kw ¼ 0:5 and g w ¼ 0. It should be mentioned that when Ls ¼ 0, the current formulation is reduced to the equations of motion for single-walled carbon nanotubes based on the classical shell theories. In other words, the classical shell formulation may be viewed as a special case of the gradient elasticity shell formulation presented here. As it can be seen from Fig. 4a, for dimensionless axial wave number k < 0:8, the results of the present formulation for X1 for 0 < Ls < 0:15 are very close to each other revealing that the effects of gradient elasticity can be ignored for this region. However changing the value of the length scale parameter Ls has considerable effects on the dimensionless frequency X1 predicted by the gradient elasticity model when these effects diminish when Ls < 0:05 and Ls > 0:5. In all cases, the deviation from the classical shell model (with Ls ¼ 0 depends on the value of dimensionless axial wave number k when this dependency is stronger for larger values of k. It is also interesting to note that the values of the dimensionless frequency X1 for any value of Ls P 0:8 are very close to those values for Ls ¼ 0:8. This analysis also showed similar behavior for the dimensionless frequencies X2 and X3 as shown in Fig. 4b and c. Calculation of dimensionless frequencies X1 , X2 and X3 for n = 2 and kw ¼ 0:5 when keeping g w ¼ 0 showed similar variations as presented in Fig. 4 and omitted for brevity. The effect of aspect ratio L=R on dimensionless frequency X1 for single-walled carbon nanotube (10, 0) with n = 1, Ls ¼ 0:2; Lm ¼ 3Ls , kw ¼ 0:5 and g w ¼ 0 is studied in Fig. 5. For the aspect ratio greater than 30 the graph is almost a straight line and the values of dimensionless frequency X1 are approximately independent of the half-axial wave number. Moreover, the effect of aspect ratio L=R and length scale parameters Ls and Lm on the dimensionless frequency X1 for single-walled carbon nanotube (10, 0) with n = 1, kw ¼ 0:5 and g w ¼ 0 are plotted in Fig. 6. It should be noted that for small values of L=R, the nanotube is relatively short which makes the effect of material length scale parameters more observable. As expected the SWCNT with smaller aspect ratio is more sensitive to the variation of the length scale parameters and when the aspect ratio is large enough (say L=R > 30) the length scale parameters Ls and Lm have a negligible effect on the dimensionless frequencies. Three dimensionless frequencies X1 , X2 and X3 for single walled carbon nanotube (10, 0) are presented in Table 3 for different values of foundation parameters and neglecting the combined strain-inertia gradient effects with assuming pinned– pinned condition as the end conditions of the SWCNT. As can be seen from this table, the dimensionless frequencies X1 , X2 and X3 are increased with increasing the non-dimensional shear modulus of foundation g w when dimensionless Winkler elastic foundation kw ¼ 0. It should be noted that, the rate of changes is more noticeable for dimensionless frequencies X1 in comparison with other frequencies. The combined effects of Winkler and Pasternak foundation on the dimensionless frequency X1 are studied in Fig. 7 for n = 1, L=R ¼ 20, Ls ¼ Lm ¼ 0. Elastic foundation constants are varied in each figure and their effects are analyzed. Frequencies of cylindrical shell varied with the dimensional axial wave number. This variation depends on the elastic foundation constants. It can be seen from Fig. 7(a) that when the value of g w is kept fixed, the frequency curves rise upward and appear to get parallel with one another. It is also seen from Fig. 7(a) that the effect of Winkler foundation decreases as the Pasternak
Fig. 8. Variation of dimensionless frequencies with Ls and Lm (Lm ¼ 3L3 ) for single-walled carbon nanotube (10, 0), n = 1 and 2, L=R ¼ 20, kw ¼ 0:5 and g w ¼ 0:55.
866
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
foundation constant increases. As a conclusion the influence of Winkler foundation on X1 is more pronounced for smaller values of the dimensionless axial wave number and this influence can be ignored with reasonable accuracy for dimensionless axial wave number k > 1 and g w > 0:55. Similarly, the influence of Winkler foundation on the shell frequencies is more important for smaller values of Pasternak foundation constant. Fig. 7b indicates that the influences of g w on the dimensionless frequency X1 for different values of kw are almost similar. In all cases, the dimensionless frequency X1 increases as the Pasternak foundation constant increases. However, the influence of increasing g w is more pronounced for smaller values of kw . Our investigation showed similar behavior for other values of L=R and n and omitted here for brevity. The variation of dimensionless frequencies (modes 1 and 2) with length scale parameters Ls and Lm (Lm ¼ 3L3 ) for a single-walled carbon nanotube (10, 0), circumferential wave number n = 1 and 2, kw ¼ 0:5 and g w ¼ 0:55 are shown in Fig. 8. For Ls ¼ 0, the formulation presented here is reduced to the classical shell formulation. It can be seen from Fig. 8 that for a wide range of dimensionless axial wave number, the results of the present formulation for the first two modes are very close to each other for Ls > 0:35 .revealing that the effects of gradient elasticity can be ignored for this region. However, the values of the length scale parameters Ls and Lm have considerable effects on the first two dimensionless frequencies predicted by the gradient elasticity model for Ls < 0:35 (Lm ¼ 3L3 ). It should also be noted that the deviation from the classical shell model (with Ls ¼ 0) depends on the value of dimensionless axial wave number k. 6. Conclusion It is well known that the carbon nanotubes generate both quantitative and qualitative changes in the dynamical behavior of the system, because the tubes are in the nanometer scale. The motivation of this work comes from the fact that the classical continuum theories are not capable of adequately describing heterogeneous phenomena due to lack of representation of the small length scale of material nano-structure. Moreover, the stress at a material point is assumed to be dependent only on the first-order derivative of the displacements in the classical theories and not on higher-order displacement derivatives. To overcome these types of deficiencies, the constitutive equations were enriched with the higher-order strain-inertia gradient terms and a combined strain-inertia gradient elasticity theory was introduced in the present paper for dynamic analysis of carbon nanotube. The tubes are assumed to be place on Winkler/Pasternak foundation. The proposed combination of the strain gradient and inertia gradient in this paper includes two material length scale parameters related to the inertia and strain gradients. Ls and Lm are the length scale parameters which are related to the strain gradients and inertia gradients, respectively. It is worth nothing that considering the length scale parameter dependent to the inertia, allow us to develop a more realistic investigation on the dynamical behavior of structure in nano-scale. Various influences of considering the Winkler elastic foundation with and without combined strain-inertia gradient effects were investigated in this analysis. It should be noted that the classical shell formulation may be viewed as a special case of the gradient elasticity shell formulation presented here and therefore, the results of the proposed formulation were compared with the classical shell theory (Ls ¼ 0). The effects of aspect ratio L=R and length scale parameters Ls and Lm on the dimensionless frequency X1 for single-walled carbon nanotube were also investigated to show that for small values of L=R when the nanotube is relatively short, the effect of material length scale parameters is more observable. The SWCNTs with smaller aspect ratio are therefore more sensitive to the variation of the length scale parameters. Moreover, the length scale parameters Ls and Lm have a negligible effect on the dimensionless frequencies when the aspect ratio is large enough (say L=R > 30). The combined effects of Winkler and Pasternak foundation on the dimensionless frequency X1 were also studied in this analysis and it is shown that the influence of Winkler foundation on X1 is more pronounced for smaller values of the dimensionless axial wave number and this influence can be ignored with reasonable accuracy for dimensionless axial wave number k > 1 and g w > 0:55. It was also concluded that the influence of Winkler foundation on the shell frequencies is more important for smaller values of Pasternak foundation constant. In summary, the results of this study showed that the enrichment of the continuum based models with the higher-order inertia terms can be considered as a reasonable alternative to the continuum description obtained by the classical shell theory. The achievements of this study provide a better insight to the size-dependent properties of dynamic characteristics of carbon nanotubes and more realistic understanding of the mechanical and physical properties of SWNTs in the several nano-science and nano-engineering practical applications.
Fig. A1. A typical material volume V with surface S.
867
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
Appendix A. Kinematical expressions In order to more introduce some microscopic scale parameters that were used in Section 2, we follow the procedure given by Mindlin [15]. Consider a material volume V with surface S (Fig. A1). As seen from Fig. A1, X i and xi ; ði ¼ 1; 2; 3Þ, are the components of the material position vector and the spatial position vector, respectively, from a fixed origin. The components of the macroscopic displacement of material particle ui are expressed as:
ui ¼ xi X i :
ðA1Þ 0
X 0i
x0i
For an assumed micro-volume V embedded in material particle, and are the components of the material position vector and the spatial position vector, respectively. The components of microscopic displacement u0i are defined as:
u0i ¼ x0i X 0i :
ðA2Þ
Let us assume the absolute value of the displacement-gradients are small in comparison with unity:
@ui
1;
@X i
0
@ui
1;
@X 0 i
ðA3Þ
and at time t:
@uj @uj uj;i ; @X i @xi 0 0 @uj @uj u0j;i ; @X 0i @x0i
uj ¼ uj ðxi ; tÞ; ðA4Þ u0j ¼ u0j ðxi ; x0i ; tÞ:
By expressing the microscopic displacement as a sum of product of specified function of x0i and arbitrary functions of xi andt and considering only linear terms of the series, we have:
u0j ¼ x0k wkj :
ðA5Þ
wij is the displacement-gradient in the microscopic medium and is a function of xi and t,
wij ¼ u0j;i :
ðA6Þ
The symmetric part of microscopic deformation wij is the microscopic strain,
wðijÞ ¼
1 w þ wji ; 2 ij
ðA7Þ
and the anti-symmetric part is the microscopic rotation,
w½ij ¼
1 w wji : 2 ij
ðA8Þ
The usual strain or the macroscopic strain is defined as
eij ¼
1 ui;j þ uj;i : 2
ðA9Þ
Difference between the gradient of the macroscopic deformation and the microscopic deformation is called the relative deformation cij and is expressed as:
cij ¼ ui;j wij ;
ðA10Þ
and the gradient of the microscopic deformation
jijk
jijk ¼ wi;jk :
ðA11Þ
It should be noted that, all of the tensors eij ; cij and jijk are independent of the microscopic components ponents of microscopic deformation wij and relative deformation cij are demonstrated in Fig. A2. Appendix B Elements of mass matrix, ½M33 :
n2 n2 3 M11 ¼ I1 1 þ L2m ðk2m þ 2 Þ L2m I3 ; 4 R R M13
I3 n2 ¼ km L2m k2m þ 2 ; R R
M 21 ¼ L2m
M 12 ¼ L2m I3 4R2
þ I1
I3 4R
nkm ; R
þ I1 2
nkm ; R
x0i .
Typical com-
868
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
Fig. A2. Typical components of gradient of displacement ui;j , microscopic deformation wij and relative deformation cij .
n2 I3 L2 n2 21 2 þ 2m M22 ¼ I1 1 þ L2m k2m þ 2 þ km ; 2 4 R R R M31 ¼
I3 n2 km L2m k2m þ 2 ; R R
M233 ¼ I1 1 þ
L2m R2
! þ L2m I3
M 32 ¼ L2m
I1 n R
2
þ
M23 ¼ L2m
I1 n 2
R
n2 ; 5k2m þ 2 R R
I3 n 2
n2 ; 5k2m þ 2 R R
I3 n 2
! k2m n2 4 þ 4 þ k m : R4 R2 n4
Elements of stiffness matrix, ½K33 :
"
K 11 ¼ A11 k2m þ "
! #
4R2 A66 3D66 n2 n2 2 2 1 þ L ; k þ m s 4R2 R2 R2
! #
4R2 ðA12 þ A66 Þ þ 4D12 þ D66 nkm n2 2 2 1 þ L ; k þ m s R 4R2 R2
K 12 ¼
A12 D11 2 D12 þ D66 n2 km n2 2 2 1 þ L ; K 13 ¼ þ km km k þ m s R R R R2 R2 "
! #
4R2 ðA12 þ A66 Þ þ D66 nkm n2 2 2 1 þ L ; k þ m s R 4R2 R2
K 21 ¼
K 22 ¼
K 23 ¼
A66 þ
21 D66 2 D22 n2 n2 km þ A22 þ 2 1 þ L2s k2m þ 2 ; 2 2 4 R R R R
" #
A22 n D22 n3 5D66 D12 nk2m n2 2 2 ; k þ 1 þ L m s R R R R3 R R R2
A12 D66 n2 km D11 3 n2 2 2 1 þ L ; K 31 ¼ km þ þ k k þ m s R R R2 R m R2 K 32
" #
2D12 þ 5D66 nk2m D22 n3 A22 n n2 2 2 1 þ L ; ¼ k þ m s R R R R3 R R R2
K 33 ¼
" D12 R
k2 2 m
þ
D11 k4m
þ ð2D12 þ 4D66 Þ
n2 k2m R2
#
n2 n2 1 þ L2s k2m þ 2 þ K w þ Gw k2m þ 2 : þ D22 4 þ 2 R R R R n4
A22
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
869
References [1] M. Abdollahian, A. Ghorbanpour Arani, A.A. Mosallaie Barzoki, R. Kolahchi, A. Loghman, Non-local wave propagation in embedded armchair TWBNNTs conveying viscous fluid using DQM, Phys. B: Condens. Matter 418 (2013) 1–15. [2] E. Aifantis, On the role of gradients in the localization of deformation and fracture, Int. J. Eng. Sci. 30 (1992) 1279–1299. [3] M. Ali-Asgari, H.R. Mirdamadi, M. Ghayour, Coupled effects of nano-size, stretching, and slip boundary conditions on nonlinear vibrations of nano-tube conveying fluid by the homotopy analysis method, Phys. E: Low Dimension. Syst. Nanostruct. 52 (2013) 77–85. [4] R. Ansari, S. Sahmani, Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models, Commun. Nonlinear Sci. Numer. Simul. 17 (4) (2012) 1965–1979. [5] H. Askes, E.C. Aifantis, Gradient elasticity and flexural wave dispersion in carbon nanotubes, Phys. Rev. B 80 (19) (2009) 195412. [6] F. Daneshmand, M. Rafiei, S.R. Mohebpour, M. Heshmati, Stress and strain-inertia gradient elasticity in free vibration analysis of single walled carbon nanotubes with first order shear deformation shell theory, Appl. Math. Model. 37 (16–17) (2013) 7983–8003. [7] A.C. Eringen, Nonlocal Continuum Field Theories, Springer-Verlag, New York, 2002. [8] A. Ghasemi, M. Dardel, M.H. Ghasemi, M.M. Barzegari, Analytical analysis of buckling and post-buckling of fluid conveying multi-walled carbon nanotubes, Appl. Math. Model. 37 (7) (2013) 4972–4992. [9] I.M. Gitman, H. Askes, E.C. Aifantis, The representative volume size in static and dynamic micro-macro transitions, Int. J. Fract. 135 (1–4) (2005) L3–L9. [10] V.M. Harik, T.S. Gates, M.P. Nemeth, Applicability of the continuum-shell theories to the mechanics of carbon nanotubes. NASA/CR-2002-211460. Hampton, VA, USA, NASA Langley Research Center, 2002. [11] M. Heshmati, M.H. Yas, Vibrations of non-uniform functionally graded MWCNTs-polystyrene nanocomposite beams under action of moving load, Mater. Des. 46 (2013) 206–218. [12] D. Isßik, Y. Shu, G. Tarabella, N. Coppedè, S. Iannotta, L. Lutterotti, F. Cicoira, J.E. Anthony, C. Santato, Ambipolar organic thin film transistors based on a soluble pentacene derivative, Appl. Phys. Lett. 99 (2) (2011). [13] K. Kiani, Vibration behavior of simply supported inclined single-walled carbon nanotubes conveying viscous fluids flow using nonlocal Rayleigh beam model, Appl. Math. Model. 37 (4) (2013) 1836–1850. [14] C.W. Lim, On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection, Appl. Math. Mech. 31 (1) (2010) 37–54. [15] R. Mindlin, Micro-structure in linear elasticity, Arch. Ration. Mech. Anal. 16 (1964) 52–78. [16] R.D. Mindlin, Second gradient of strain and surface-tension in linear elasticity, Int. J. Solids Struct. 1 (4) (1965) 417–438. [17] R.D. Mindlin, N.N. Eshel, On first strain-gradient theories in linear elasticity, Int. J. Solids Struct. 4 (1968) 109–124. [18] S. Movahed, D. Li, Microfluidics cell electroporation, Microfluid. Nanofluid. 10 (4) (2011) 703–734. [19] F. Pampaloni, E.-L. Florin, Microtubule architecture: inspiration for novel carbon nanotube-based biomimetic materials, Trends Biotechnol. 26 (6) (2008) 302–310. [20] J. Peddieson, G.R. Buchanan, R.P. McNitt, Application of nonlocal continuum models to nanotechnology, Int. J. Eng. Sci. 41 (3–5) (2003) 305–312. [21] M. Rafiei, S.R. Mohebpour, F. Daneshmand, Small-scale effect on the vibration of non-uniform carbon nanotubes conveying fluid and embedded in viscoelastic medium, Phys. E: Low Dimension. Syst. Nanostruct. 44 (2012) 1372–1379. [22] C. Santato, F. Cicoira, R. Martel, Organic photonics: spotlight on organic transistors, Nat. Photon 5 (7) (2011) 392–393. [23] H.M. Sedighi, K.H. Shirazi, Vibrations of micro-beams actuated by an electric field via parameter expansion method, Acta Astronaut. 85 (2013) 19–24. [24] A.R. Setoodeh, M. Khosrownejad, P. Malekzadeh, Exact nonlocal solution for postbuckling of single-walled carbon nanotubes, Phys. E: Low Dimension. Syst. Nanostruct. 43 (2011) 1730–1737. [25] C. Tienchong, A molecular based anisotropic shell model for single-walled carbon nanotubes, J. Mech. Phys. Solids 58 (9) (2010) 1422–1433. [26] C.Y. Wang, L.C. Zhang, An elastic shell model for characterizing single-walled carbon nanotubes, Nanotechnology 19 (19) (2008) 195704. [27] L. Wang, Q. Zheng, J.Z. Liu, Q. Jiang, Size dependence of the thin-shell model for carbon nanotubes, Phys. Rev. Lett. 95 (10) (2005) 105501. [28] L. Wang, Vibration analysis of nanotubes conveying fluid based on gradient elasticity theory, J. Vib. Control (2011). [29] Y. Yan, X.Q. He, L.X. Zhang, Q. Wang, Flow-induced instability of double-walled carbon nanotubes based on an elastic shell model, J. Appl. Phys. 102 (4) (2007) 044307. [30] M.H. Yas, M. Heshmati, Dynamic analysis of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotube under the action of moving load, Appl. Math. Model. 36 (4) (2012) 1371–1394. [31] Y. Zhang, G. Liu, X. Han, Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure, Phys. Lett. A 349 (5) (2006) 370–376.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Combined strain-inertia gradient elasticity in free vibration shell analysis of single walled carbon nanotubes using shell theory Farhang Daneshmand ⇑ Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec H3A 2K6, Canada Department of Bioresource Engineering, McGill University, 21111 Lakeshore Road, Sainte-Anne-de-Bellevue, Quebec H9X 3V9, Canada
a r t i c l e
i n f o
Keywords: Carbon nanotubes Gradient elasticity theory Size effect Vibration Winkler/Pasternak foundation
a b s t r a c t In this paper, a gradient elasticity shell formulation is presented for free vibration analysis of single-walled carbon nanotube placed on Winkler/Pasternak foundation. The proposed formulation is based on the combined strain-inertia gradient elasticity. The combined strain-inertia gradient elasticity provides an extension to the classical equations of elasticity with additional higher-order spatial derivatives of strains and two material length scale parameters related to the inertia and strain gradients, which enable formulation to investigate the size effect on the dynamic behavior of nanotubes. The effects of the length scale parameters, aspect ratio of single-walled carbon nanotube and foundation parameters on the fundamental frequencies for different values half-axial wave number and circumferential wave number are investigated. The natural frequencies obtained from the proposed shell formulation show the effects of size-dependent properties. It can be concluded that a continuum model enriched with higher-order inertia terms has been proposed as alternative to the continuum description obtained with classical elasticity theory. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction With the advance of materials synthesis and device processing capabilities over the past decade and since the discovery of carbon nanotubes (CNTs) in the early 1990s, the importance of developing and analyzing nanoscale engineering devices has dramatically increased. Carbon nanotubes are well-known as a wealth of technological applications because of their theoretically predicted mechanical properties including high strength, high stiffness, low density and structural perfection. Carbon nanotubes are also expected to be used in nano-fluidic devices and systems, nano-sensors and drug delivery devices [18]. Among different types of CNTs, the single-walled carbon nanotubes (SWCNTs) possess outstanding physical and chemical properties when compared to other nanoscale materials. A SWCNT can be viewed as a two-dimensional graphene sheet rolled into a hollow cylindrical, shell-like macromolecules composed of carbon atoms arranged in periodic hexagonal cells [10]. A single-walled carbon nanotube is uniquely indexed by a pair of integers (n, m) representing its chirality. The two limiting cases of nanotubes are (n, 0) with chiral angle = 0 and (n, n) with chiral angle = p/6. These cases are usually known as zigzag and armchair tubes based on the geometry of carbon bonds around the circumference of the nanotube (Fig. 1). The ⇑ Address: Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec H3A 2K6, Canada. E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2014.05.094 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
857
Fig. 1. Three possible patterns along the circumference of the CNT: ‘zigzag’ (i), ‘armchair’ (ii) and ‘chiral’ (iii) Fig. 1. Three possible patterns along the circumference of the CNT: ‘zigzag’ (i), ‘armchair’ (ii) and ‘chiral’ (iii).
outer diameter of SWCNTs is 1–2 nm whereas the possible lattice symmetries are determined by the wrapping direction. Carbon nanotubes are an extremely stiff and strong material and provide superior mechanical properties such as Young’s modulus approaching 1 TPa which is five times higher than that of steel (210 GPa) strength approaching 100 GPa with a density of about 1.3 g/cm3 [19]. The ability of SWCNTs to conduct heat even better than diamond suggests their eventual use in nano-electronics. Steady progress reported in using SWCNTs in developing nanodevices and nanocircuits showing remarkable logic and amplification functions. The use of carbon nanotube electrodes and new device geometries is enhancing the performance of organic light-emitting transistors (OLETs) [22]. In their design, the source electrode was made of a porous network of single-walled carbon nanotubes. Recently fabricating electrodes made of carbon nanotubes are expected to improve electron and hole injection efficiency [12]. In general, there are two main approaches to the analysis and modeling of carbon nanotubes. The first approach is the atomic modeling. The well-known molecular dynamics (MD) method may be considered as the most common atomic modeling [14]. Molecular dynamics method requires extremely fast computing facilities and is mainly confined to systems with a limited number of molecules. This limitation is due to considering each individual molecule with its multiple mechanical or chemical interactions. Representing nanostructured materials with equivalent-continuum models can be considered as the second approach to describe their mechanical behavior. Different elastic linear and nonlinear beam and shell models based on the principles of continuum mechanics have been presented to predict the static and dynamic behavior of single-walled carbon nanotubes. Heshmati and Yas [11], Sedighi and Shirazi [23], Yan et al. [29] and Yas and Heshmati [30] investigated the flow-induced instability of double-walled carbon nanotubes using Donnell’s shell theory. A two-dimensional elastic shell model to characterize the deformation of single-walled carbon nanotubes using the in-plane rigidity, Poisson ratio, bending rigidity and off-plane torsion rigidity as independent elastic constants has been proposed by Wang and Zhang [26]. However, the validity of assumptions and limitations of these models must be carefully examined. It should be noted that the above continuumbased models may be classified as local models in which the stress at a reference point is related to the local strain field at that point. These local continuum models do not exhibit intrinsic size-dependence features of the material and do not allow inclusions and property in homogeneties that are fundamental and significant in atomic modeling. It has been reported by many investigators that size-dependent effects are significantly more prominent at the nanoscale and must be taken into account in the modeling of nanoscale structures. Some recent studies can be found in the literature addressing this issue by considering nonlocal elasticity theory as an extension of the classical equations of elasticity [1,3,4,7,8,13,20,21,24,31]. The classical constitutive equations of elasticity can also be generalized by using the Laplacian of stress or strain in the gradient elasticity theory. Different gradient elasticity formulations have been used in the literature to model the vibration analysis of carbon nanotubes with Euler–Bernoulli and Timoshenko assumptions [5,16,17]. It is worth nothing that the mathematical formulation in the nonlocal elasticity theory as proposed by Eringen [7] can also be viewed as a stress gradient elasticity theory which is quite sufficient for static problems whereas it might be inadequate in dynamic analysis of nanotubes. More recently, [28] compared two gradient elasticity formulations with stress and combined strain-inertia gradients for dynamic analysis of fluid-conveying nanotubes. The stress gradient beam models has one material length scale parameter whereas the combined strain-inertia gradient beam models include two material length scale parameters related to the inertia and strain gradients and can also be used to investigate the size effect on the dynamical behavior of nanotubes conveying fluid. More recently, the standard constitutive equations of elasticity have been generalized by using the Laplacian of stress and combined strain-inertia for isotropic materials and the first-order shear deformation shell theory [6]. It is the aim of the present article to present a shell formulation enriched with the combined strain-inertia gradient elasticity for investigating the effects of material length scales on the frequency analysis of a SWCNT. Due to this issue, the fundamental aspects of the gradient elasticity theory and combined strain-inertia gradient theory are presented in Section 2. Also, the mathematical formulation required for the modeling of the SWCNT as a thin shell structure with combined strain-inertia gradient theory and the solution method of the shell equations are given in Sections 3 and 4, respectively.
858
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
The results of the present formulation are presented in section 5. In this section, the effects of the length scale parameters, aspect ratio of the SWCNT and foundation parameters on the fundamental frequencies for different values half-axial wave number and circumferential wave number are investigated. 2. Combined strain-inertia gradient theory According to the theory of elasticity with microstructure that was represented by Mindlin [15], the both of kinematic energy density T and the deformation energy density U can be written in terms of quantities at microscopic and macroscopic scale.
T¼
1 1 qu_ i u_ i þ qL21 w_ ij wij ; 2 2
ð1Þ
U¼
1 1 1 C ijkl eij ekl þ Bijkl cij ckl þ Aijklmn jijk jlmn þ Dijklm cij jklm þ F ijklm jijk elm þ Gijkl cij ekl ; 2 2 2
ð2Þ
where q is the mass density that is assumed to be equal at macroscopic and microscopic scale. L1 is related to the size unit cell of microstructure. uij and eij are the macroscopic displacement and strain and wij , cij and jijk are the microscopic deformation, the difference between microscopic and macroscopic deformation (the relative deformation) and the gradient of microscopic deformation, respectively. The other microscopic parameters are introduced in Appendix A. Moreover, Aijklmn , Bijkl , C ijkl , Dijklm , F ijklm and Gijkl are the constitutive tensors [15]. To simplify the relations of kinematic energy and deformation energy, the microscopic deformation is assumed as the first gradient of the macroscopic strain [15]. Due to this assumption, the kinematic energy can be expressed in terms of the macroscopic displacements ui , only.
T¼
1 1 qu_ i u_ i þ qL21 u_ i;j u_ i;j : 2 2
ð3Þ
Also, the reduced form of the deformation energy can be expressed as bellow:
U¼
1 keii ejj þ leij eij þ a1 eik;i ejj;k þ a2 ejj;i ekk;i þ a3 eik;i ejk;j þ a4 ejk;i ejk;i þ a5 ejk;i eij;k ; 2
ð4Þ
k and l are the lame constant and ai is the constitutive coefficient. By using the Hamilton’s principle and the relations where of deformation energy and kinematic energy, the equation of motion can be expressed with the macroscopic displacement terms as bellow in the absence of body forces:
€ i L21 u €i;jj ; ðk þ lÞ uj;ij L22 uj;ijkk þ l ui;jj L23 ui;jjkk ¼ q u
ð5Þ
where
L22 ¼
41 þ 4a2 þ 3a3 þ 2a4 þ 3a5 ; 2ðk þ lÞ
L23 ¼
a3 þ 2a4 þ a5 : 2l
It should be noted that the additional parameters L1 , L2 and L3 have the dimension of length and they relate to principle of microstructure. The combined strain- inertia gradient elasticity theory is a special case of Mindlin theory when L2 ¼ L3 ¼ Ls [2]. It should be noted that the combination of the strain gradient and inertia gradient include two material length scale parameters related to the inertia and strain gradients. Moreover, considering the length scale parameter dependent to the inertia, allow us to investigate the dynamical behavior of structure in nano-scale accurately. According to this issue, the equation of motion with the combined strain-inertia gradient effect can be expressed as:
€i L2m u € i;mm ; C ijkl uk;jl L2s uk;jlmm ¼ q u
ð6Þ
where for isotropic linear elasticity, the constitutive tensor C ijkl is expressed as:
C ijkl ¼ kdij dkl þ ldik djl þ ldil djk :
ð7Þ
Ls and Lm ¼ L1 are the length scale parameters which are related to the strain gradients and inertia gradients, respectively. It should be noted that the two length scales are related to the Representative Volume Element (RVE) size in elastostatics and elastodynamics [9]. 3. Combined strain-inertia gradient elasticity shell models for SWCNTs An isotropic single-walled carbon nanotube (SWCNT) with length L, thickness h, and radius of the middle surface R as shown in Fig. 2 is considered in the present study. The SWCNT is viewed as a hollow cylindrical shell rested on an elastic foundation. The elastic foundation is represented by the Winkler/Pasternak model with radial elastic modulus Kw and shear modulus Gw. The problem is considered with the material properties: Young’s modulus E, Poisson’s ratio m and the mass density q. An orthogonal coordinate system (x, h, z) is assumed to be established at the middle surface of the shell where x, h and z
859
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
Fig. 2. A single-walled carbon nanotube placed on Winkler/Pasternak foundation.
represent the axial (longitudinal), circumferential (angular) coordinates and the distance from the nanotube axis in radial direction perpendicular to the nanotube axis, respectively. The axial, circumferential and radial displacement deformations are also denoted by ux , uh , uz , respectively. Using the combined strain-inertia gradient theory for an isotropic elastic material that was represented in previous section, the constitutive equations can be written as:
8 9 rxx > > > > > > < = > > > :
rhh sxh
> > > ;
2
Q 11
6 6 ¼ 6 Q 21 4 0
Q 12 Q 22 0
8 8 9 91 9 308 €exx > exx > exx > > > > > > > > > > > > > > > > < < = =C = 7B< 7B C 0 7B ehh L2s r2 ehh C þ qL2m €ehh : > > > > > > 5@> A > > > > > > > > > > : : :€ > ; ; cxh cxh Q 66 cxh ; 0
ð8Þ
For isotropic materials the reduced stiffness Q ij (i, j = 1, 2 and 6) are defined as
Q 11 ¼ Q 22 ¼
E ; 1 t2
Q 12 ¼
tE ; 1 t2
Q 66 ¼
E : 2ð1 þ tÞ
ð9Þ
Due to the Sanders–Koiter linear shell theory, the strain–displacement relationships of the middle surface and changes in the curvature for a circular cylindrical shell are given as
f e11
e22
f k11
k22
e12 g ¼
ux;x n k12 g ¼ uz;xx
ux;h þ uh;x ; 1 o 1 1 u þ 3u 4u ; x;h h;x z;xh 2 ðuh;h uz;hh Þ 2R R R
1 ðuh;h R
þ uz Þ
1 R
ð10Þ
where ðe11 ; e12 ; e22 Þ and ðk11 ; k12 ; k22 Þ are the reference surface strains and curvatures, respectively. For a vibrating SWCNT considered as a thin cylindrical shell the expression for the strain energy, S is given by
S¼
1 2
Z
L
Z 2p h
2
2
2
A11 e211 þ 2A12 e11 e22 þ A22 e222 þ A66 e212 þ D11 k11 þ 2D12 k11 k22 þ D22 k22 þ D66 k12 0 0 i €11 k11 þ k €22 k22 þ k €12 k12 Rdhdx; þ L2m I1 ð€e11 e11 þ €e22 e22 þ €e12 e12 Þ þ L2m I3 k
1 L2s r2 ð11Þ
where Aij and Dij are the membrane, coupling and flexural stiffness defined by:
Z
fAij ; Dij g ¼
h=2
Q ij f1; z2 gdz:
ð12Þ
h=2
Substituting the expressions for the surface strains and curvatures from (10) into (11), the expression for the strain energy is transformed to the following form:
Z Z u 2 1 L 2p 2A12 ux;x A22 x;h A11 u2x;x þ ðuh;h þ uz Þ þ 2 ðuh;h þ uz Þ2 þ A66 þ uh;x þ D11 u2z;xx 2 0 0 R R R 2 2D12 uz;xx D22 D66 ux;h 2 2 2 1 L þ ð u u Þ þ ð u u Þ þ þ 3u 4u r h;h z;hh h;h z;hh h;x z;xh s R R2 R4 4R2 u u
1 x;htt x;h þ uh;xtt þ uh;x þ L2m I1 ux;xtt ux;x þ 2 ðuh;htt þ uz;tt Þðuh;h þ uz Þ þ R R R u 1 1 ux;htt x;h 2 þ 3uh;xtt 4uz;xhtt þ 3uh;x 4uz;xh Rdhdx; þ Lm I3 uz;xxtt uz;xx þ 4 ðuh;htt uz;hhtt Þðuh;h uz;hh Þ þ 2 R R R 4R
S¼
ð13Þ
where t denotes time. It should be noted that I1 and I3 stand for the mass moment of inertia of SWCNT per unit area as:
fI1 ; I3 g ¼
Z
h=2
h=2
qf1; z2 gdz:
ð14Þ
860
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
The kinetic energy T for a thin-walled cylindrical shell made of single-walled carbon nanotube is given by
T¼
1 2
Z 0
L
Z 2p q u2x;t þ u2h;t þ u2h;t Rdhdx:
ð15Þ
0
Employing Hamilton’s principle to the Lagrangian energy functional and introducing the terms describing the Winkler and Pasternak foundations ðK w uz Gw r2 uz Þ in the z-direction, the dynamical equations for a functionally graded cylindrical shell can be written as
"
# A12 D11 ðD12 þ D66 Þ uh;xh þ A11 ux;xx þ ux;hh þ uz;xhh ð1 L2s r2 Þ uz;x uz;xxx R R 4R4 R3 4R3 !
" I1 I3 I3 1 4R2 ðA12 þ A66 Þ þ D66 2 2 2 3 I3 ¼ I1 ð1 Lm r Þux;tt þ Lm ux;xh uh;xhtt þ ux;hhtt þ uz;xxxtt þ 2 uz;xhhtt 4 R4 R 4R3 R R 4R3
21 D66 A22 D22 A22 D22 ð5D66 D12 Þ þ A66 þ uh;xx þ þ 4 uh;hh þ 2 uz;h 4 uz;hhh þ uz;xxh ð1 L2s r2 Þ 4 R2 R2 R2 R R R
I1 I3 I3 1 21 I1 2 uz;htt þ 3 uh;xhtt þ 2 u þ u ¼ I1 ð1 L2m r2 Þuh;tt þ L2m h;hhtt h;xxtt 4 R 4R R R2 R
I3 1 A12 D66 D11 ð2D12 þ 5D66 Þ uh;xxh ux;x þ 3 ux;xhh þ ux;xxx þ 2 5uz;xxhtt þ 2 uz;hhhtt R R R R2 R R D22 A22 D12 ð2D12 þ 4D66 Þ D22 A22 uz;xxhh 4 uz;hhhh 2 uz 1 L2s r2 þ 4 uh;hhh 2 uh;h þ 2 uz;xx D11 uz;xxxx 2 R R R R R R !
2 L I3 1 I1 I3 1 ¼ I1 1 þ m2 uz;tt þ L2m ux;xxxtt þ 2 ux;xhhtt þ 2 uh;htt 2 5uh;xxhtt þ 2 uh;hhhtt R R R R R R
1 4 þ K w uz Gw r2 uz : þ I3 4 uz;hhhhtt þ 2 uz;xxhhtt þ uz;xxxxtt ð16Þ R R ð4R2 A66 3D66 Þ
4R2 ðA12 þ A66 Þ þ 4D12 þ D66
!
Gw represents the shear modulus of the material used for the elastic foundation and K w for the Winkler foundation. The Winkler model is a special case of the Pasternak model when Gw ¼ 0. 4. Solution method for simply supported SWCNTs The displacement functions of SWCNTs with different support conditions can be obtained by solving the system of Eqs. (16). The solution to these equations satisfying boundary conditions is given in the following form,
ux ðx; h; tÞ ¼
1 X 1 X
uxmn ðtÞ cosðkm xÞ cosðnhÞ;
ð17Þ
m¼1 n¼1
uh ðx; h; tÞ ¼
1 X 1 X
uhmn ðtÞ sinðkm xÞ sinðnhÞ;
m¼1 n¼1
uz ðx; h; tÞ ¼
1 X 1 X
uzmn ðtÞ sinðkm xÞ cosðnhÞ:
m¼1 n¼1
Substituting Eqs. (17) into Eqs. (16), the following system of equations are obtained,
€ g31 þ ½K33 fqg31 ¼ f0g; ½M33 fq
ð18Þ
where fqg31 is a vector of the SWCNT generalized coordinates and ½M33 and ½K33 are mass and stiffness matrices (given in Appendix B), respectively. For complex modal analysis, fqg31 is assumed to be a harmonic function of time,
fqg31
8 9 > < Ux > = ¼ U h exp ðixtÞ; > : > ; Uz
ð19Þ
where U x , U h and U z are the displacement amplitudes in the x, h, z directions, respectively and x is the eigenvalue of the system which can be a complex number in general. Substituting Eq. (19) into Eq. (18), leads to the following equation
8 9 2 3 0 > < Ux > = 6 7 Hðn; m; xÞ33 U h ¼ 4 0 5; > : > ; 0 Uz
ð20Þ
861
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869 Table 1 Properties of some SWCNTs [25,27]. Parameters
(10, 0)
Young’s modulus E, (TPa) Shear modulus G, (TPa) Poisson ratio m Mass density per unit volume, q (g/cm3) Thickness, h (nm) Radius, R (nm)
5.25 2.27 0.17874 2.3 0.06709 0.39
Table 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dimensionless frequency X ¼ xR ð1 m2 Þq=E for SWCNT (10, 0), L=R ¼ 20, Ls ¼ Lm ¼ 0 and g w ¼ 0. kw ¼ 0
kw ¼ 0:25
kw ¼ 0:5
n
m
X1
X2
X3
X1
X2
X3
X1
X2
X3
1
1 2 3 4 5
0.0025 0.0095 0.0193 0.0306 0.0424
0.1043 0.1123 0.1236 0.1368 0.1505
0.2223 0.2240 0.2271 0.2316 0.2380
0.0531 0.0528 0.0548 0.0598 0.0672
0.1048 0.1137 0.1258 0.1396 0.1539
0.2294 0.2309 0.2335 0.2374 0.2429
0.0720 0.0706 0.0715 0.0755 0.0820
0.1055 0.1156 0.1285 0.1428 0.1576
0.2371 0.2384 0.2405 0.2438 0.2486
2
1 2 3 4 5
0.0208 0.0214 0.0229 0.0258 0.0301
0.2037 0.2067 0.2115 0.2178 0.2253
0.3519 0.3538 0.3572 0.3617 0.3676
0.0725 0.0725 0.0727 0.0734 0.0749
0.2038 0.2069 0.2119 0.2184 0.2261
0.3537 0.3556 0.3588 0.3632 0.3690
0.0998 0.0996 0.0995 0.0998 0.1007
0.2038 0.2072 0.2124 0.2192 0.2271
0.3557 0.3575 0.3606 0.3649 0.3705
where m and n are half-axial wave number and circumferential mode number, respectively. The characteristic equation of the problem can be obtained as the following polynomial function
Fðn; m; xÞ ¼ 0;
ð21Þ
where Fðn; m; xÞ is a polynomial function. The characteristic equation (21) can be used to determine the frequencies of SWCNT. Solving Eq. (21) provides three pairs of roots with different combination of m and n. 5. Results The material properties of some single-walled carbon nanotubes are given in Table 1. The thickness-to-radius ratios h=R for all SWCNTs used in the present study are less than 1=10. It allows us to consider these SWCNTs as thin shells.
Fig. 3. Variation of dimensionless frequency X1 with Winkler elastic foundation for single-walled carbon nanotube (10, 0), L=R ¼ 20, Ls ¼ Lm ¼ 0 and g w ¼ 0.
862
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
Fig. 4. Variation of dimensionless frequencies with Ls and Lm = 3Ls for single-walled carbon nanotube (10, 0), n = 1, L/R = 20, kw = 0.5 and gw = 0 (a) X1 (b) X2 (c) X3.
According to Eq. (21), for any combination of m and n, we thus have three frequencies. The lowest frequency is generally associated with the mode where the transverse component dominates while the other two frequencies are usually higher by an order of magnitude and are associated with the modes where the displacements in the tangent plane dominate. Three
863
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
Fig. 5. The effect of aspect ratio L=R on dimensionless frequency X1 for single-walled carbon nanotube (10, 0), n = 1, Ls ¼ 0:2; Lm ¼ 3Ls , kw ¼ 0:5 and g w ¼ 0Þ.
Fig. 6. The effect of aspect ratio L=R and length scale parameters Ls and Lm (Lm ¼ 3L3 ) on the dimensionless frequency X1 for single-walled carbon nanotube (10, 0), n = 1, kw ¼ 0:5 and g w ¼ 0.
Table 3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dimensionless frequency X ¼ xR ð1 m2 Þq=E for SWCNT (10, 0), L=R ¼ 20, Ls ¼ Lm ¼ 0 and kw ¼ 0. gw ¼ 0
g w ¼ 0:25
g w ¼ 0:55
n
m
X1
X2
X3
X1
X2
X3
X1
X2
X3
1
1 2 3 4 5
0.0025 0.0095 0.0193 0.0306 0.0424
0.1043 0.1123 0.1236 0.1368 0.1505
0.2223 0.2240 0.2271 0.2316 0.2380
0.0537 0.0550 0.0593 0.0669 0.0771
0.1048 0.1139 0.1264 0.1408 0.1562
0.2296 0.2316 0.2350 0.2399 0.2463
0.0756 0.0758 0.0793 0.0863 0.0962
0.1058 0.1165 0.1305 0.1464 0.1635
0.2392 0.2416 0.2457 0.2514 0.2587
2
1 2 3 4 5
0.0208 0.0214 0.0229 0.0258 0.0301
0.2037 0.2067 0.2115 0.2178 0.2253
0.3519 0.3538 0.3572 0.3617 0.3676
0.1382 0.1386 0.1397 0.1418 0.1450
0.2040 0.2079 0.2140 0.2218 0.2308
0.3602 0.3620 0.3651 0.3695 0.3752
0.1933 0.1903 0.1888 0.1892 0.1915
0.2072 0.2152 0.2248 0.2357 0.2476
0.3738 0.3757 0.3788 0.3831 0.3887
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dimensionless frequencies X1 , X2 and X3 can be obtained by using the relation X ¼ xR ð1 m2 Þq=E. These dimensionless frequencies for single walled carbon nanotube (10, 0) are presented in Table 1 for different values of Winkler elastic foundation and neglecting the combined strain-inertia gradient effects (Ls ¼ Lm ¼ 0). Three dimensionless frequencies X1 , X2 and X3
864
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
generally correspond to the radial, circumferential and longitudinal mode shapes of the problem. The results given in Table 2 are obtained with assuming pinned–pinned condition as the end conditions of the SWCNT. As can be seen from this table, the dimensionless frequencies X1 , X2 and X3 are increased with increasing the non-dimensional Winkler elastic foundation constant kw , (kw ¼ K w R2 ð1 m2 Þ=Eh), when dimensionless shear foundation modulus g w ¼ 0, (g w ¼ Gw ð1 m2 Þ=Eh). Variation of dimensionless frequency X1 with Winkler elastic foundation is shown in Fig. 3 for different values of n = 1, 2 and 3 for single-walled carbon nanotube (10, 0). As it can be seen from this figure, dimensionless frequency X1 increases by increasing the elastic foundation constant. The dimensionless frequencies increase rapidly with dimensionless axial wave number k (k ¼ mpR=L) and the frequency curves seem to become parallel for higher values of kw . Moreover, the most noticeable increase in the dimensionless frequency X1 occurs for k < 0:5. The same effect can be studied for the dimensionless frequencies X2 and X3 and although it is not presented here, the constant of elastic foundation has a small effect on the frequencies X2 and X3 .
Fig. 7. Combined effects of Winkler and Pasternak foundation on the dimensionless frequency X1 for single-walled carbon nanotube (10, 0), n = 1, L=R ¼ 20 and Ls ¼ Lm ¼ 0.
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
865
Fig. 4a shows the variation of dimensionless frequency X1 with length scale parameters Ls and Lm ¼ 3Ls for a single-walled carbon nanotube (10, 0), circumferential wave number n = 1, kw ¼ 0:5 and g w ¼ 0. It should be mentioned that when Ls ¼ 0, the current formulation is reduced to the equations of motion for single-walled carbon nanotubes based on the classical shell theories. In other words, the classical shell formulation may be viewed as a special case of the gradient elasticity shell formulation presented here. As it can be seen from Fig. 4a, for dimensionless axial wave number k < 0:8, the results of the present formulation for X1 for 0 < Ls < 0:15 are very close to each other revealing that the effects of gradient elasticity can be ignored for this region. However changing the value of the length scale parameter Ls has considerable effects on the dimensionless frequency X1 predicted by the gradient elasticity model when these effects diminish when Ls < 0:05 and Ls > 0:5. In all cases, the deviation from the classical shell model (with Ls ¼ 0 depends on the value of dimensionless axial wave number k when this dependency is stronger for larger values of k. It is also interesting to note that the values of the dimensionless frequency X1 for any value of Ls P 0:8 are very close to those values for Ls ¼ 0:8. This analysis also showed similar behavior for the dimensionless frequencies X2 and X3 as shown in Fig. 4b and c. Calculation of dimensionless frequencies X1 , X2 and X3 for n = 2 and kw ¼ 0:5 when keeping g w ¼ 0 showed similar variations as presented in Fig. 4 and omitted for brevity. The effect of aspect ratio L=R on dimensionless frequency X1 for single-walled carbon nanotube (10, 0) with n = 1, Ls ¼ 0:2; Lm ¼ 3Ls , kw ¼ 0:5 and g w ¼ 0 is studied in Fig. 5. For the aspect ratio greater than 30 the graph is almost a straight line and the values of dimensionless frequency X1 are approximately independent of the half-axial wave number. Moreover, the effect of aspect ratio L=R and length scale parameters Ls and Lm on the dimensionless frequency X1 for single-walled carbon nanotube (10, 0) with n = 1, kw ¼ 0:5 and g w ¼ 0 are plotted in Fig. 6. It should be noted that for small values of L=R, the nanotube is relatively short which makes the effect of material length scale parameters more observable. As expected the SWCNT with smaller aspect ratio is more sensitive to the variation of the length scale parameters and when the aspect ratio is large enough (say L=R > 30) the length scale parameters Ls and Lm have a negligible effect on the dimensionless frequencies. Three dimensionless frequencies X1 , X2 and X3 for single walled carbon nanotube (10, 0) are presented in Table 3 for different values of foundation parameters and neglecting the combined strain-inertia gradient effects with assuming pinned– pinned condition as the end conditions of the SWCNT. As can be seen from this table, the dimensionless frequencies X1 , X2 and X3 are increased with increasing the non-dimensional shear modulus of foundation g w when dimensionless Winkler elastic foundation kw ¼ 0. It should be noted that, the rate of changes is more noticeable for dimensionless frequencies X1 in comparison with other frequencies. The combined effects of Winkler and Pasternak foundation on the dimensionless frequency X1 are studied in Fig. 7 for n = 1, L=R ¼ 20, Ls ¼ Lm ¼ 0. Elastic foundation constants are varied in each figure and their effects are analyzed. Frequencies of cylindrical shell varied with the dimensional axial wave number. This variation depends on the elastic foundation constants. It can be seen from Fig. 7(a) that when the value of g w is kept fixed, the frequency curves rise upward and appear to get parallel with one another. It is also seen from Fig. 7(a) that the effect of Winkler foundation decreases as the Pasternak
Fig. 8. Variation of dimensionless frequencies with Ls and Lm (Lm ¼ 3L3 ) for single-walled carbon nanotube (10, 0), n = 1 and 2, L=R ¼ 20, kw ¼ 0:5 and g w ¼ 0:55.
866
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
foundation constant increases. As a conclusion the influence of Winkler foundation on X1 is more pronounced for smaller values of the dimensionless axial wave number and this influence can be ignored with reasonable accuracy for dimensionless axial wave number k > 1 and g w > 0:55. Similarly, the influence of Winkler foundation on the shell frequencies is more important for smaller values of Pasternak foundation constant. Fig. 7b indicates that the influences of g w on the dimensionless frequency X1 for different values of kw are almost similar. In all cases, the dimensionless frequency X1 increases as the Pasternak foundation constant increases. However, the influence of increasing g w is more pronounced for smaller values of kw . Our investigation showed similar behavior for other values of L=R and n and omitted here for brevity. The variation of dimensionless frequencies (modes 1 and 2) with length scale parameters Ls and Lm (Lm ¼ 3L3 ) for a single-walled carbon nanotube (10, 0), circumferential wave number n = 1 and 2, kw ¼ 0:5 and g w ¼ 0:55 are shown in Fig. 8. For Ls ¼ 0, the formulation presented here is reduced to the classical shell formulation. It can be seen from Fig. 8 that for a wide range of dimensionless axial wave number, the results of the present formulation for the first two modes are very close to each other for Ls > 0:35 .revealing that the effects of gradient elasticity can be ignored for this region. However, the values of the length scale parameters Ls and Lm have considerable effects on the first two dimensionless frequencies predicted by the gradient elasticity model for Ls < 0:35 (Lm ¼ 3L3 ). It should also be noted that the deviation from the classical shell model (with Ls ¼ 0) depends on the value of dimensionless axial wave number k. 6. Conclusion It is well known that the carbon nanotubes generate both quantitative and qualitative changes in the dynamical behavior of the system, because the tubes are in the nanometer scale. The motivation of this work comes from the fact that the classical continuum theories are not capable of adequately describing heterogeneous phenomena due to lack of representation of the small length scale of material nano-structure. Moreover, the stress at a material point is assumed to be dependent only on the first-order derivative of the displacements in the classical theories and not on higher-order displacement derivatives. To overcome these types of deficiencies, the constitutive equations were enriched with the higher-order strain-inertia gradient terms and a combined strain-inertia gradient elasticity theory was introduced in the present paper for dynamic analysis of carbon nanotube. The tubes are assumed to be place on Winkler/Pasternak foundation. The proposed combination of the strain gradient and inertia gradient in this paper includes two material length scale parameters related to the inertia and strain gradients. Ls and Lm are the length scale parameters which are related to the strain gradients and inertia gradients, respectively. It is worth nothing that considering the length scale parameter dependent to the inertia, allow us to develop a more realistic investigation on the dynamical behavior of structure in nano-scale. Various influences of considering the Winkler elastic foundation with and without combined strain-inertia gradient effects were investigated in this analysis. It should be noted that the classical shell formulation may be viewed as a special case of the gradient elasticity shell formulation presented here and therefore, the results of the proposed formulation were compared with the classical shell theory (Ls ¼ 0). The effects of aspect ratio L=R and length scale parameters Ls and Lm on the dimensionless frequency X1 for single-walled carbon nanotube were also investigated to show that for small values of L=R when the nanotube is relatively short, the effect of material length scale parameters is more observable. The SWCNTs with smaller aspect ratio are therefore more sensitive to the variation of the length scale parameters. Moreover, the length scale parameters Ls and Lm have a negligible effect on the dimensionless frequencies when the aspect ratio is large enough (say L=R > 30). The combined effects of Winkler and Pasternak foundation on the dimensionless frequency X1 were also studied in this analysis and it is shown that the influence of Winkler foundation on X1 is more pronounced for smaller values of the dimensionless axial wave number and this influence can be ignored with reasonable accuracy for dimensionless axial wave number k > 1 and g w > 0:55. It was also concluded that the influence of Winkler foundation on the shell frequencies is more important for smaller values of Pasternak foundation constant. In summary, the results of this study showed that the enrichment of the continuum based models with the higher-order inertia terms can be considered as a reasonable alternative to the continuum description obtained by the classical shell theory. The achievements of this study provide a better insight to the size-dependent properties of dynamic characteristics of carbon nanotubes and more realistic understanding of the mechanical and physical properties of SWNTs in the several nano-science and nano-engineering practical applications.
Fig. A1. A typical material volume V with surface S.
867
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
Appendix A. Kinematical expressions In order to more introduce some microscopic scale parameters that were used in Section 2, we follow the procedure given by Mindlin [15]. Consider a material volume V with surface S (Fig. A1). As seen from Fig. A1, X i and xi ; ði ¼ 1; 2; 3Þ, are the components of the material position vector and the spatial position vector, respectively, from a fixed origin. The components of the macroscopic displacement of material particle ui are expressed as:
ui ¼ xi X i :
ðA1Þ 0
X 0i
x0i
For an assumed micro-volume V embedded in material particle, and are the components of the material position vector and the spatial position vector, respectively. The components of microscopic displacement u0i are defined as:
u0i ¼ x0i X 0i :
ðA2Þ
Let us assume the absolute value of the displacement-gradients are small in comparison with unity:
@ui
1;
@X i
0
@ui
1;
@X 0 i
ðA3Þ
and at time t:
@uj @uj uj;i ; @X i @xi 0 0 @uj @uj u0j;i ; @X 0i @x0i
uj ¼ uj ðxi ; tÞ; ðA4Þ u0j ¼ u0j ðxi ; x0i ; tÞ:
By expressing the microscopic displacement as a sum of product of specified function of x0i and arbitrary functions of xi andt and considering only linear terms of the series, we have:
u0j ¼ x0k wkj :
ðA5Þ
wij is the displacement-gradient in the microscopic medium and is a function of xi and t,
wij ¼ u0j;i :
ðA6Þ
The symmetric part of microscopic deformation wij is the microscopic strain,
wðijÞ ¼
1 w þ wji ; 2 ij
ðA7Þ
and the anti-symmetric part is the microscopic rotation,
w½ij ¼
1 w wji : 2 ij
ðA8Þ
The usual strain or the macroscopic strain is defined as
eij ¼
1 ui;j þ uj;i : 2
ðA9Þ
Difference between the gradient of the macroscopic deformation and the microscopic deformation is called the relative deformation cij and is expressed as:
cij ¼ ui;j wij ;
ðA10Þ
and the gradient of the microscopic deformation
jijk
jijk ¼ wi;jk :
ðA11Þ
It should be noted that, all of the tensors eij ; cij and jijk are independent of the microscopic components ponents of microscopic deformation wij and relative deformation cij are demonstrated in Fig. A2. Appendix B Elements of mass matrix, ½M33 :
n2 n2 3 M11 ¼ I1 1 þ L2m ðk2m þ 2 Þ L2m I3 ; 4 R R M13
I3 n2 ¼ km L2m k2m þ 2 ; R R
M 21 ¼ L2m
M 12 ¼ L2m I3 4R2
þ I1
I3 4R
nkm ; R
þ I1 2
nkm ; R
x0i .
Typical com-
868
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
Fig. A2. Typical components of gradient of displacement ui;j , microscopic deformation wij and relative deformation cij .
n2 I3 L2 n2 21 2 þ 2m M22 ¼ I1 1 þ L2m k2m þ 2 þ km ; 2 4 R R R M31 ¼
I3 n2 km L2m k2m þ 2 ; R R
M233 ¼ I1 1 þ
L2m R2
! þ L2m I3
M 32 ¼ L2m
I1 n R
2
þ
M23 ¼ L2m
I1 n 2
R
n2 ; 5k2m þ 2 R R
I3 n 2
n2 ; 5k2m þ 2 R R
I3 n 2
! k2m n2 4 þ 4 þ k m : R4 R2 n4
Elements of stiffness matrix, ½K33 :
"
K 11 ¼ A11 k2m þ "
! #
4R2 A66 3D66 n2 n2 2 2 1 þ L ; k þ m s 4R2 R2 R2
! #
4R2 ðA12 þ A66 Þ þ 4D12 þ D66 nkm n2 2 2 1 þ L ; k þ m s R 4R2 R2
K 12 ¼
A12 D11 2 D12 þ D66 n2 km n2 2 2 1 þ L ; K 13 ¼ þ km km k þ m s R R R R2 R2 "
! #
4R2 ðA12 þ A66 Þ þ D66 nkm n2 2 2 1 þ L ; k þ m s R 4R2 R2
K 21 ¼
K 22 ¼
K 23 ¼
A66 þ
21 D66 2 D22 n2 n2 km þ A22 þ 2 1 þ L2s k2m þ 2 ; 2 2 4 R R R R
" #
A22 n D22 n3 5D66 D12 nk2m n2 2 2 ; k þ 1 þ L m s R R R R3 R R R2
A12 D66 n2 km D11 3 n2 2 2 1 þ L ; K 31 ¼ km þ þ k k þ m s R R R2 R m R2 K 32
" #
2D12 þ 5D66 nk2m D22 n3 A22 n n2 2 2 1 þ L ; ¼ k þ m s R R R R3 R R R2
K 33 ¼
" D12 R
k2 2 m
þ
D11 k4m
þ ð2D12 þ 4D66 Þ
n2 k2m R2
#
n2 n2 1 þ L2s k2m þ 2 þ K w þ Gw k2m þ 2 : þ D22 4 þ 2 R R R R n4
A22
F. Daneshmand / Applied Mathematics and Computation 243 (2014) 856–869
869
References [1] M. Abdollahian, A. Ghorbanpour Arani, A.A. Mosallaie Barzoki, R. Kolahchi, A. Loghman, Non-local wave propagation in embedded armchair TWBNNTs conveying viscous fluid using DQM, Phys. B: Condens. Matter 418 (2013) 1–15. [2] E. Aifantis, On the role of gradients in the localization of deformation and fracture, Int. J. Eng. Sci. 30 (1992) 1279–1299. [3] M. Ali-Asgari, H.R. Mirdamadi, M. Ghayour, Coupled effects of nano-size, stretching, and slip boundary conditions on nonlinear vibrations of nano-tube conveying fluid by the homotopy analysis method, Phys. E: Low Dimension. Syst. Nanostruct. 52 (2013) 77–85. [4] R. Ansari, S. Sahmani, Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models, Commun. Nonlinear Sci. Numer. Simul. 17 (4) (2012) 1965–1979. [5] H. Askes, E.C. Aifantis, Gradient elasticity and flexural wave dispersion in carbon nanotubes, Phys. Rev. B 80 (19) (2009) 195412. [6] F. Daneshmand, M. Rafiei, S.R. Mohebpour, M. Heshmati, Stress and strain-inertia gradient elasticity in free vibration analysis of single walled carbon nanotubes with first order shear deformation shell theory, Appl. Math. Model. 37 (16–17) (2013) 7983–8003. [7] A.C. Eringen, Nonlocal Continuum Field Theories, Springer-Verlag, New York, 2002. [8] A. Ghasemi, M. Dardel, M.H. Ghasemi, M.M. Barzegari, Analytical analysis of buckling and post-buckling of fluid conveying multi-walled carbon nanotubes, Appl. Math. Model. 37 (7) (2013) 4972–4992. [9] I.M. Gitman, H. Askes, E.C. Aifantis, The representative volume size in static and dynamic micro-macro transitions, Int. J. Fract. 135 (1–4) (2005) L3–L9. [10] V.M. Harik, T.S. Gates, M.P. Nemeth, Applicability of the continuum-shell theories to the mechanics of carbon nanotubes. NASA/CR-2002-211460. Hampton, VA, USA, NASA Langley Research Center, 2002. [11] M. Heshmati, M.H. Yas, Vibrations of non-uniform functionally graded MWCNTs-polystyrene nanocomposite beams under action of moving load, Mater. Des. 46 (2013) 206–218. [12] D. Isßik, Y. Shu, G. Tarabella, N. Coppedè, S. Iannotta, L. Lutterotti, F. Cicoira, J.E. Anthony, C. Santato, Ambipolar organic thin film transistors based on a soluble pentacene derivative, Appl. Phys. Lett. 99 (2) (2011). [13] K. Kiani, Vibration behavior of simply supported inclined single-walled carbon nanotubes conveying viscous fluids flow using nonlocal Rayleigh beam model, Appl. Math. Model. 37 (4) (2013) 1836–1850. [14] C.W. Lim, On the truth of nanoscale for nanobeams based on nonlocal elastic stress field theory: equilibrium, governing equation and static deflection, Appl. Math. Mech. 31 (1) (2010) 37–54. [15] R. Mindlin, Micro-structure in linear elasticity, Arch. Ration. Mech. Anal. 16 (1964) 52–78. [16] R.D. Mindlin, Second gradient of strain and surface-tension in linear elasticity, Int. J. Solids Struct. 1 (4) (1965) 417–438. [17] R.D. Mindlin, N.N. Eshel, On first strain-gradient theories in linear elasticity, Int. J. Solids Struct. 4 (1968) 109–124. [18] S. Movahed, D. Li, Microfluidics cell electroporation, Microfluid. Nanofluid. 10 (4) (2011) 703–734. [19] F. Pampaloni, E.-L. Florin, Microtubule architecture: inspiration for novel carbon nanotube-based biomimetic materials, Trends Biotechnol. 26 (6) (2008) 302–310. [20] J. Peddieson, G.R. Buchanan, R.P. McNitt, Application of nonlocal continuum models to nanotechnology, Int. J. Eng. Sci. 41 (3–5) (2003) 305–312. [21] M. Rafiei, S.R. Mohebpour, F. Daneshmand, Small-scale effect on the vibration of non-uniform carbon nanotubes conveying fluid and embedded in viscoelastic medium, Phys. E: Low Dimension. Syst. Nanostruct. 44 (2012) 1372–1379. [22] C. Santato, F. Cicoira, R. Martel, Organic photonics: spotlight on organic transistors, Nat. Photon 5 (7) (2011) 392–393. [23] H.M. Sedighi, K.H. Shirazi, Vibrations of micro-beams actuated by an electric field via parameter expansion method, Acta Astronaut. 85 (2013) 19–24. [24] A.R. Setoodeh, M. Khosrownejad, P. Malekzadeh, Exact nonlocal solution for postbuckling of single-walled carbon nanotubes, Phys. E: Low Dimension. Syst. Nanostruct. 43 (2011) 1730–1737. [25] C. Tienchong, A molecular based anisotropic shell model for single-walled carbon nanotubes, J. Mech. Phys. Solids 58 (9) (2010) 1422–1433. [26] C.Y. Wang, L.C. Zhang, An elastic shell model for characterizing single-walled carbon nanotubes, Nanotechnology 19 (19) (2008) 195704. [27] L. Wang, Q. Zheng, J.Z. Liu, Q. Jiang, Size dependence of the thin-shell model for carbon nanotubes, Phys. Rev. Lett. 95 (10) (2005) 105501. [28] L. Wang, Vibration analysis of nanotubes conveying fluid based on gradient elasticity theory, J. Vib. Control (2011). [29] Y. Yan, X.Q. He, L.X. Zhang, Q. Wang, Flow-induced instability of double-walled carbon nanotubes based on an elastic shell model, J. Appl. Phys. 102 (4) (2007) 044307. [30] M.H. Yas, M. Heshmati, Dynamic analysis of functionally graded nanocomposite beams reinforced by randomly oriented carbon nanotube under the action of moving load, Appl. Math. Model. 36 (4) (2012) 1371–1394. [31] Y. Zhang, G. Liu, X. Han, Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure, Phys. Lett. A 349 (5) (2006) 370–376.