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Nonlinear size-dependent longitudinal vibration of carbon nanotubes embedded in an elastic medium
Author’s Accepted Manuscript Nonlinear Size-dependent Longitudinal Vibration of Carbon Nanotubes Embedded in an Elastic Medium
R. Fernandes, S. El-Borgi, S.M. Mousavi, J.N. Reddy, A. Mechmoum www.elsevier.com/locate/physe
PII: DOI: Reference:
S1386-9477(16)31170-5 http://dx.doi.org/10.1016/j.physe.2016.11.007 PHYSE12646
To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 18 October 2016 Revised date: 2 November 2016 Accepted date: 9 November 2016 Cite this article as: R. Fernandes, S. El-Borgi, S.M. Mousavi, J.N. Reddy and A. Mechmoum, Nonlinear Size-dependent Longitudinal Vibration of Carbon Nanotubes Embedded in an Elastic Medium, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2016.11.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Nonlinear Size-dependent Longitudinal Vibration of Carbon Nanotubes Embedded in an Elastic Medium R. Fernandesa , S. El-Borgib,1 , S.M. Mousavic , J.N. Reddyd , A. Mechmoumb a
Department of Aerospace Engineering, Texas A&M University, College Station, Texas 77843-3123, USA. b Mechanical Engineering Program, Texas A&M University at Qatar, Engineering Building, P.O. Box 23874, Education City, Doha, Qatar. c Department of Engineering and Physics, Karlstad University, 65188 Karlstad, Sweden. d Department of Mechanical Engineering, Texas A&M University, College Station, Texas 77843-3123, USA.
Abstract In this paper, we study the longitudinal linear and nonlinear free vibration response of a single walled carbon nanotube (CNT) embedded in an elastic medium subjected to different boundary conditions. This formulation is based on a large deformation analysis in which the linear and nonlinear von K´arm´an strains and their gradient are included in the expression of the strain energy and the velocity and its gradient are taken into account in the expression of the kinetic energy. Therefore, static and kinetic length scales associated with both energies are introduced to model size effects. The governing motion equation along with the boundary conditions are derived using Hamilton’s principle. Closed-form solutions for the linear free vibration problem of the embedded CNT rod are first obtained. Then, the nonlinear free vibration response is investigated for various values of length scales using the method of multiple scales. Keywords: Embedded carbon nanotube rod; Nonlinear von K´arm´an strain; Strain and velocity gradient theory; Linear and nonlinear free vibration; Method of Multiple Scales.
1
Introduction
Nanomaterials are materials used at the nanoscale with outstanding mechanical, chemical, electrical, optical and electronic properties. Examples of nanomaterials include carbon nanotubes, graphene sheet, zinc oxide (ZnO) nanowires, boron nitride (BN) nanotubes and nanosheets [1, 2, 3]. Carbon nanotubes (CNTs), discovered by [4], are layers of graphite that are wrapped up into cylinders with diameters of about one nanometer and lengths up to many micrometers. CNTs have received widespread interest of researchers from many disciplines, including material science, engineering, chemistry and physics. CNTs are being used in carbon nanotube composites [5] and environment resistant coatings made of carbon nanotube reinforced materials [6, 7]. 1
Corresponding author. Tel.: +974 4423 0674, e-mail: sami.el [email protected] (Sami El-Borgi).
1
Nanomaterials are the base materials of many nanoscale structures called nanostructures. A number of 1D nanostructures have been designed including nanorods, nanowires, nanotubes, nanodots, nanowalls, nanobridges, nanobelts and nanorings. Nanorods are nanostructures ranging from 1 to 3000nm which have been used in many devices such as diodes, transistors, fuel-cell cathodes, nanosensors, electrical contacts for buffering thermal expansion and LED arrays [8]. Miniaturized nanobars are basic elements utilized in a number of applications such as micro-truss structures [9], micro-manipulators [10] and micro-milling [11]. In light of these micro-scale applications, studying the static and dynamic behavior of nanobars seems to be quite important. Molecular dynamic (MD) simulation requires intense computational labor for nanostructures having large numbers of atoms and this has motivated the intensive development of size-dependent or higher-order continuum mechanics theories. These theories include nonlocal continuum theories, strain gradient theories or a combination of both (nonlocal strain gradient theories) and modified couple stress theory. Eringen’s nonlocal elasticity differential model [12] was utilized by a number of researchers to capture size-effects in nanorods. However, these studies are relatively less than those related to nanobeams [13, 14, 15]. A nonlocal model was proposed by [16] to investigate the axial vibration response of the nanorod and the small-scale effect of the axial frequencies. This model was used to examine the axial vibration of a carbon nanotube system [11]. The nonlocal longitudinal vibration of a double nanorod system was studied by [8] and its potential applications as a nonosensor in addition to nanooptomechanical systems (NOMS) were proposed. The free axial vibration of nanorods using fractional continuum mechanics was investigated by [17]. A nonlocal model proposed by [18] to examine the axial vibration behavior of a tapered nanorod using the differential quadrature method. The free and forced damped axial vibration response of a nanorod using a frequency-dependent finite element method was studied by [19] . In another study, a similar approach was used to examine the axial vibration response of a nanorod embedded in an elastic medium [20] . The wave propagation in nonlocal nanorods was investigated by [21]. The nonlocal axial vibration problem of a CNT nanorod embedded in an elastic medium was examined by [22] . The study by [23] examined the nonlocal effects of longitudinal vibration in nanorod with internal longrange interactions and proposed closed-form solutions for the case of a uniform nonlocal kernel. The theory of nonlocal elasticity was used to investigate the small scale effect on the axial vibration of non-uniform and non-homogeneous nanorods using the differential quadrature method [24]. Few number of investigators used the strain gradient theory or the nonlocal strain gradient theory to study the behavior of nanorods. The longitudinal axial vibration of a homogenous and a functionally graded microbar based on strain gradient elasticity theory were investigated by [25] and [26], respectively. Kahrobaiyan et al. [27] studied both analytically and numerically using the finite element method the longitudinal behavior of strain gradient bars. Analytical solutions for the longitudinal free vibration response of size-dependent rods using nonlocal strain gradient theory were obtained by [28]. In addition to the nonlocal theory and strain gradient theory, the modified couple stress theory has also been extensively employed in the micro- or nano- scale structures. Ke and 2
his co-authors [29, 30, 31] made a significant contribution on the linear and nonlinear vibration of size dependent functionally graded microbeams and microplates. They found that the nonlinear frequency ratio versus amplitude curves were unsymmetrical due to the bending-extension coupling in functionally graded microbeams and microplates. They also implied that this unsymmetry was also size dependence. Farokhi et al. [32] developed a new size-dependent nonlinear model for the analysis of the pull-in behavior of carbon nanotube-based resonators based on the modified coupled stress theory. The study by Miandoab et al. [33] presented a polysilicon nano-beam model based on modified couple stress and Eringens nonlocal elasticity theories. In many practical applications, nanomaterials are embedded in an elastic medium such as polymer/CNT nanocomposites (PCNTs) to enhance the performance of engineering polymers in terms of strength and fracture toughness [34]. This is due to the excellent properties of nanomaterials. Many polymer resins have been used as a matrix for CNT-nanocomposites, such as CNT/Polypropylene [35], CNT/Nylon [36] and CNT/Polycarbonate [37]. In light of this, several investigations have been conducted to study the interaction of the nanomaterial with the elastic medium. Two types of elastic media are generally considered, namely the one-parameter Winkler type [22] and the two-parameter Pasternak type [38] elastic medium. From the reviewed literature, it can be observed that most investigators have based their work related to nanorods on linear theories. To fill this gap, the present study considers the vibration problem of a CNT nanorod embedded in a Winkler type elastic medium and proposes a large-deformation analysis based on a modified strain gradient theory that incorporates both strain and velocity gradients in addition to the von K´ arm´ an nonlinearity. The linear free vibration problem obtained by ignoring the von K´ arm´ an nonlinearity is solved analytically, while the nonlinear free and forced vibration problems are solved using the Method of Multiple Scales (MMS). This paper is outlined as follows. Following this introduction, the proposed strain gradient theory for nanorods is presented in Section 2. The governing equation of motion for CNT nanorod embedded in an elastic medium is established in Section 3. The linear vibration problem without the von K´arm´an strain is solved analytically in Section 4. The nonlinear free and forced vibration solutions obtained using MMS are given in Sections 5 and 6, respectively. Numerical results are provided in Section 7. Finally, concluding remarks are given in Section 8.
2
Strain Gradient Theory
This section summarizes the Strain Gradient Theory used in this study and additional details can be found in [39, 40]. For a body occupying a volume V and made of a linear elastic isotropic material, the strain energy U can be written as follows based on the strain gradient theory: Z 1 (1) U= σij εij + σijm εij,m dV (1) 2 V
3
Here εij and εij,m are, respectively, the classical strain and its gradient. The classical (1) stress tensor σij and the higher-order stress tensor σijm are defined as σij =Cijkl εkl (1) σijm
(2)
=ls2 Cijkl εkl,m
(3)
where Cijkl are the components of the elasticity tensor and ls is a material length scale parameter to account for the strain gradient field. The total stress field can be expressed as (1)
tij = σij − ∇σijm
(4)
in which ∇ is the Laplacian operator. For a rod-type structure, the size-dependency is only accounted for in the longitudinal direction and neglected in both width and thickness directions. The stress-strain relations (2) and (3) can then be expressed as σxx =Eεxx (1) σxxx
=ls2 Eεxx,x
(5) (6)
where E is the modulus of elasticity of the rod and εxx is the axial strain. The total axial stress txx given by (4) can be simplified to txx = 1 − ls2 ∇2 Eεxx (7) where ∇2 = ∂ 2 /∂x2
3
Equations of motion of size-dependent rods
The displacement field in a rod of volume V , length L and cross-sectional area A takes the following form: u1 = u (x, t) , u2 = 0, u3 = 0 (8) Here t denotes time, u1 , u2 and u3 denote the displacements along the x, y and zdirections, respectively. u is the axial displacement. Thus, the nonlinear nonzero strain of rods can be written as ∂u 1 ∂u 2 εxx = (9) + ∂x 2 ∂x and its variation with respect to x is δεxx =
∂δu ∂u ∂δu + ∂x ∂x ∂x
Taking the variation of (1) leads to the expression of the virtual strain energy δU Z (1) δU = σxx δεxx + σxxx δεxx,x dV V
4
(10)
(11)
Integrating the second term by parts yields Z L Z (1) (1) δU = σxx δεxx − ∇σxxx δεxx dV + σxxx δεxx dA V LZ
A
txx δεxx dAdx +
= 0
Z =
A
A
L
h
N δεxx dx + N (1) δεxx
0
0
L (1) σxxx δεxx dA
Z
Z
0
iL
(12)
0
where N and N (1) are the stress resultants of, respectively, the total stress and the higherorder stress which are given by Z Z (1) (1) N= txx dA, N = σxxx dA (13) A
A
and the stress resultant of the classical stress N (0) can be defined as Z N (0) = σxx dA
(14)
A
By using equations (7), (13) and (14), the following relationships can be established: ∂N (1) N = N (0) − " ∂x # ∂u 1 ∂u 2 (0) N = EA + ∂x 2 ∂x N (1) = ls2
∂N (0) ∂x
(15) (16) (17)
Here E is the Young’s Modulus of the nanorod and A is its cross-sectional area. It is assumed that the rod has a circular cross-section. Substituting for the variation of the axial strain δεxx from (10), the virtual strain energy given by (12) can be further expressed in terms of displacements as follows: Z L ∂u ∂δu ∂u ∂δu L (1) δU = N 1+ dx + N 1+ (18) ∂x ∂x ∂x ∂x 0 0 Considering the longitudinal motion of the rod and its velocity gradient, the kinetic energy K can be written as Z L 2 Z L 2 2 ∂u 1 ∂ u 1 2 K = ρA dx + ρAlk dx (19) 2 ∂t 2 ∂x∂t 0 0 where ρ is the density of the rod and lk is the kinetic material length parameter associated with the velocity gradient. The variation of the kinetic energy can, therefore, be written as Z L Z L 2 ∂u ∂δu ∂ u ∂ 2 δu 2 δK = ρA dx + ρAlk dx (20) ∂t ∂t 0 0 ∂x∂t ∂x∂t 5
The variation of the external work due to the surrounding medium can be written as Z L δW = Fx δudx (21) 0
where Fx is the force exerted by the medium in which the nanotube is embedded. According to Hamilton’s Principle, it can be stated that Zt2 (δK − δU + δW ) dt = 0
(22)
t1
Substitution of (18), (20) and (21) into (22) yields Zt2 ZL
∂u ∂δu ρA dxdt + ρAlk2 ∂t ∂t
t1 0
Zt2 ZL − t1 0
Zt2 ZL
∂ 2 u ∂ 2 δu dxdt ∂x∂t ∂x∂t
t1 0
Zt2 ∂u ∂δu L ∂u ∂δu dt dt − N (1) 1 + N 1+ ∂x ∂x ∂x ∂x 0 t1
Zt2 ZL +
Fx δudxdt = 0
(23)
t1 0
Assuming a Winkler type model [22], the force exerted by the embedding medium can be represented as Fx = −kEM u (24) where kEM is the stiffness of the elastic medium. Applying integration by parts with respect to t as well as x in (23), setting the initial conditions to zero and using the fundamental lemma of calculus variations, the following equation of motion can be derived: 4 ∂2u ∂ ∂u 2 ∂ u −ρA 2 + ρAlk 2 2 + N 1+ − kEM u = 0 (25) ∂t ∂x ∂t ∂x ∂x along with the associated boundary conditions specified at each of the ends x = 0 and x=L 3 ∂u 2 ∂ u ρAlk +N 1+ = 0 or u = 0 (26) ∂x∂t2 ∂x ∂u ∂u N (1) 1 + = 0 or =0 (27) ∂x ∂x Combining (15) to (17) gives the expressions of the axial normal force N and its higher order normal force N (1) in terms of displacements " " # # 2 2 3u 3u ∂ ∂u 1 ∂u 2 ∂ u ∂u ∂ N = EA + − ls2 EA + + (28) ∂x 2 ∂x ∂x3 ∂x2 ∂x ∂x3 2 ∂ u ∂u ∂ 2 u (1) 2 N = ls EA + (29) ∂x2 ∂x ∂x2 6
In the framework of nonlinear classical elasticity obtained by setting the length scale parameters ls and lk equal to zero, the purely non-classical boundary conditions (27) disappear. Further, it can be verified that the governing equation (25) and the associated classical boundary conditions (26) can be reduced to the classical equations ∂2u ∂ ∂u −ρA 2 + N 1+ − kEM u = 0 (30) ∂t ∂x ∂x subject to one of the following boundary conditions specified at each of the ends x = 0 and x = L: ∂u N 1+ = 0 or u = 0 (31) ∂x where
"
∂u 1 N = EA + ∂x 2
∂u ∂x
2 # (32)
The governing equation (25) and associated boundary conditions, (26) and (27), can be written in non-dimensional form as follows: ∂2υ ˆ 2 ∂4υ ∂ ¯ ∂υ − 2 + lk + N 1+ − kˆEM υ = 0 (33) ∂τ ∂ξ 2 ∂τ 2 ∂ξ ∂ξ subject to the boundary conditions specified at each of the ends ξ = 0 and ξ = 1 3 ˆl2 ∂ υ + N ¯ 1 + ∂υ = 0 or υ = 0 k ∂ξ∂τ 2 ∂ξ ¯ (1) 1 + ∂υ = 0 or ∂υ = 0 N ∂ξ ∂ξ
(34) (35)
¯ and its higher-order Based on (28) and (29), the non-dimensional axial normal force N (1) ¯ normal force N can be expressed as " # 2 2 2 3υ 3υ ∂υ 1 ∂υ ∂ ∂ υ ∂υ ∂ ¯ = N + − ˆls2 + + (36) ∂ξ 2 ∂ξ ∂ξ 3 ∂ξ 2 ∂ξ ∂ξ 3 2 2 ¯ (1) = ˆl2 ∂ υ + ∂υ ∂ υ N (37) s ∂ξ 2 ∂ξ ∂ξ 2 where the non-dimensional parameters introduced in (33) to (37) are given by s x u E ξ= , υ= , τ =t L L ρL2 2 ˆls = ls , ˆlk = lk , kˆEM = kEM L L L EA
7
(38)
4
Linear Free Vibration Analysis
Combining (33) and (36) and dropping the nonlinear terms yields the following linear equation of motion: −
4 2 ∂ 2 υ ˆ2 ∂ 4 υ ˆl2 ∂ υ + ∂ υ − kˆEM υ = 0 + l − s k ∂τ 2 ∂ξ 2 ∂τ 2 ∂ξ 4 ∂ξ 2
(39)
Assuming a harmonic motion for the nth linear mode shape φn (ξ), the solution can be written as υ (ξ, τ ) = φn (ξ) eiωn τ (40) √ where i = −1, ωn is the nth nondimensionl linear natural frequency. Substituting this solution into (39) yields the following differential eigenvalue problem: 4 2 ˆl2 d φn + ˆl2 ωn 2 − 1 d φn + −ωn 2 + kˆEM φn = 0 (41) s k dξ 4 dξ 2 The solution of the above linear fourth order differential equation can be expressed as φn (ξ) = C1 sin λ1 ξ + C2 cos λ1 ξ + C3 sinh λ2 ξ + C4 cosh λ2 ξ
(42)
where s λ1 =
−ˆ lk2 ωn 2 −
q ˆ ˆEM ˆ lk4 ωn 4 −2 ˆ lk2 ωn 2 +4 ˆ ls2 ωn 2 −4 k ls2 +1+1 2 ˆ 2l
(43a)
q ˆ ˆEM ˆ lk4 ωn 4 −2 ˆ lk2 ωn 2 +4 ˆ ls2 ωn 2 −4 k ls2 +1+1 2ˆ l2
(43b)
s
s λ2 =
−ˆ lk2 ωn 2 +
s
and Ci (i = 1, 2, 3, 4) are constants determined from the boundary conditions. We consider first a nanotube with both ends classically clamped v(0) = v(1) = 0 and ¯ (1) (0) = N ¯ (1) (1) = 0. This type of linear the higher linear normal force also clamped N boundary conditions is denoted CF-CF (CF: Clamped Forcing) in which the first and second boundary conditions at each end are, respectively, classical and non-classical and can be expressed as [28] φn (0) = φn (1) = 0 2 2 2 ∂ φn ˆl2 ∂ φn ˆ = l s s ∂ξ 2 ∂ξ 2 ξ=0
(44) =0
(45)
ξ=1
Employing these boundary conditions yields C2 = C3 = C4 = 0 sin λ1 = 0 or λ1 = nπ
8
(46)
Equating (46) to (43a) and solving for ωn yields s ˆl2 n4 π 4 + n2 π 2 + kˆEM s ωn = ˆl2 n2 π 2 + 1 k
(47)
and the corresponding modeshape for the CF-CF boundary conditions can be written as φn (ξ) = sin (nπξ)
(48)
¯ (1) (0) = We consider now a nanotube with the left end Clamped Forcing (CF), v(0) = N 0, and the right end Free Strained (FS). The FS boundary condition consists of a zero normal force N (1) = 0 which is classical and a zero strain ∂υ ∂ξ (1) = 0 which is non-classical. This type of linear boundary conditions is denoted CF-FS and can be expressed as [28] ∂φn ˆ2 ∂ 3 φn φn (0) = − ls =0 (49) ∂ξ ∂ξ 3 ξ=1 2 ∂φn ˆl2 ∂ φn = =0 (50) s ∂ξ 2 ∂ξ ξ=0
ξ=1
Employing these boundary conditions yields C1 = C3 = C4 = 0 cos λ1 = 0 or λ1 =
(2n − 1) π 2
Equating (51) to (43a) and solving for ωn yields s 1 (2 n − 1)4 ˆls2 π 4 + 4 (2 n − 1)2 π 2 + 16 kˆEM ωn = 2 4 + (2 n − 1)2 ˆlk2 π 2
(51)
(52)
and the corresponding modeshape for the CF-FS boundary conditions can be expressed as (2n − 1) πξ φn (ξ) = sin (53) 2 After setting lk = 0 and kEM = 0, it can be verified that the expression of the nonlinear frequencies (47) and (52) are the same as those found by [28] after assigning the nonlocal parameter ea equal to zero and applying the normalization given by (38).
5
Nonlinear Free Vibration Analysis
For the nonlinear free vibration analysis, Galerkin’s method [42] is applied to (33) to eliminate the spacial variable ξ and convert it to an ordinary differential equation. This is done by assuming the axial displacement of the nth mode as follows: υ (ξ, τ ) = φn (ξ)q(τ ) 9
(54)
where q(τ ) is the unknown time dependent function and φn (ξ) is the normalized linear modeshape corresponding to the nth mode. Substituting (54) into (33), multiplying by the modeshape and integrating from ξ = 0 to ξ = 1 results in the following ordinary differential equation with quadratic and cubic nonlinearities: q¨ + β1 q + β2 q 2 + β3 q 3 = 0
(55)
where R1 R1 4 2 φn ddξφ4n dξ − kˆEM 0 φ2n dξ + 0 φn ddξφ2n dξ β1 = (56a) R R ˆl2 1 φn d2 φ2n dξ − 1 φ2 dξ k 0 0 n dξ R1 R1 R 1 dφn d2 φn 4 2 3 n d φn −4 ˆls2 0 φn ddξφ2n ddξφ3n dξ − 2 ˆls2 0 φn dφ dξ dξ 4 dξ + 3 0 φn dξ dξ 2 dξ β2 = (56b) R R ˆl2 1 φn d2 φ2n dξ − 1 φ2 dξ n k 0 0 dξ 2 3 R R 1 dφn 2 d4 φn R 2 1 1 d φn d3 φn dφn d φn 2 ˆ dξ − 4 0 φn dξ2 dξ3 dξ dξ − 0 φn dξ dξ 2ls − 0 φn dξ2 dξ 4 + β3 = R1 R1 2 2ˆlk2 0 φn ddξφ2n dξ − 2 0 φ2n dξ R 1 n 2 d 2 φn 3 0 φn dφ dξ dξ dξ 2 (56c) R R 2 1 1 2ˆlk2 0 φn ddξφ2n dξ − 2 0 φ2n dξ −ˆls2
R1 0
Since (55) is a second order ODE, it is subject to the following two initial conditions: q (0) = qmax q˙ (0) = 0
(57)
where qmax is the normalized amplitude of vibration corresponding to the time dependent function q (τ ). It is worth mentioning that on substituting the modeshape function corresponding to the CF-CF boundary condition, the quadratic term β2 goes to zero. Substituting the modeshape function for the CF-FS boundary condition does not render β2 to be zero. For small values of qmax , the nonlinear ordinary differential equation in (55) can be solved analytically using the Method of Multiple Scales [41] and this solution procedure has been adopted in this study. For the sake of convenience, equation (55) can be further simplified by introducing the following nondimensional terms: qˆ =
q qmax
, τˆ =
p
β1 τ
(58)
The above normalization is then substituted into (55) and (57) and the following equation of motion and its associated initial conditions are obtained: d2 qˆ + qˆ + α1 qˆ2 + α2 qˆ3 = 0 dˆ τ2 qˆ(0) = 1, qˆ˙(0) = 0 10
(59a) (59b)
2
) and α2 = β3 (qβmax . where α1 = β2 qβmax 1 1 In using the Method of Multiple Scales [41], an n-order uniform expansion is assumed to model the response of the system as a function of multiple time scales
qˆ(ˆ τ ; ε) = εˆ q1 (T0 , T1 , T2 ) + ε2 qˆ2 (T0 , T1 , T2 ) + .... + εn qˆn (T0 , T1 , T2 )
(60)
where ε is a small parameter indicating the oscillation amplitude. The main oscillatory behavior occurs during the fast time scale T0 = τˆ , while the amplitude and phase modulation take place during the slow time scales Tn = εn = τˆ, n ≥ 1. In seeking the solution of the ODE (59a), a third-order uniform expansion is assumed with three time scales T0 , T1 and T2 qˆ(ˆ τ ) = εˆ q1 + ε2 qˆ2 + ε3 qˆ3 (61) Substitution of (61) into (59a) produces the following system of linear equations corresponding each to a different power of ε ε1 : D02 qˆ1 + qˆ1 = 0
(62a)
ε2 : D02 qˆ2 + qˆ2 = −2D0 D1 qˆ1 − α1 qˆ12 3
ε :
D02 qˆ3
+ qˆ3 =
−(D12
+ 2D2 D0 )ˆ q1 −
(62b) α2 qˆ13
− 2D0 D1 qˆ2 − 2α1 qˆ1 qˆ2
(62c)
The subsequent analysis using the method of multiple scales for a system with cubic and quadratic nonlinearities is thoroughly examined in [41]. Solving (62a) for qˆ1 , obtaining the solvability conditions at each order and expressing the final solvability condition in its polar form, the following expression for the final solution and the nonlinear frequency is obtained: qˆ =εˆ q1 + ε2 qˆ2 + +O ε3
=εa cos (Ωˆ τ + ψ0 ) + ε
2
α1 a2 α1 a2 cos (2Ωˆ τ + 2ψ0 ) − 6 2
+ O ε3
(63)
where Ω is the nonlinear frequency which can be written as [41] and O ε3 is the error associated with higher order terms. 5 2 2 2 3 2 2 a ε α1 + a ε α2 + O ε3 (64) 12 8 For this solution to satisfy the initial conditions in (59b), a trigonometric manipulation is introduced in which the error associated with the third order expansion is considered [43]. The resulting solution can be written as α1 a2 α1 a2 2 qˆ = εa cos (Ωˆ τ + ψ0 ) + ε cos (2Ωˆ τ + 2ψ0 ) − + b cos(ˆ τ + γ) + O ε3 (65) 6 2 Ω=1−
Applying the initial conditions (59b) to equation (65), the values of following constants are obtained: 1 α1 ψ0 = 0, a = , γ = 0, b = (66) ε 3 11
Substituting (66) into (64), the nonlinear frequency can therefore be simplified to 5 2 3 α + α2 (67) 12 1 8 √ The above expression can then be multiplied by β1 to obtain the non-dimensional form of the nonlinear frequency used thereafter in the results and discussion section p 5 2 3 ωN L = β1 1 − α1 + α2 (68) 12 8 Ω=1−
6
Results and Discussion
This section presents results of the non-dimensional linear and nonlinear frequency of the nanorod embedded in the elastic medium. Parametric studies are performed for the different normalized static and kinetic length scales and the normalized coefficient of the elastic medium. In each case, the values for the two boundary conditions (C-C and C-F) are presented. Normalized values of these parameters are assumed since the literature does not provide experimental or verified simulated values for the kinetic length scale and elastic foundation coefficient. Table 1 and Table 2 demonstrate the effect of normalized static length scale ls /L the linear and non-linear frequencies for the CF-CF boundary condition and the CF-FS boundary condition. The internal length scale range, i.e. 0 < ls /L < 1, is considered to demonstrate the effect of variation of ls /L. The accurate values of ls and lk in this gradient theory may be evaluated after fitting the results to proper experimental observations or atomistic simulations. It is observed that for both boundary conditions, the value of the linear and nonlinear frequency increase as ls /L increases. For the CF-FS boundary condition however, while the nonlinear frequency increases with the increase in the normalized static length scale, the values of the nonlinear frequency are lower than that of the linear frequency for the first mode, but generally higher for the second and third modes. This indicates that for the CF-FS boundary condition, the chosen normalized amplitude of vibration could influence different vibrational responses that may not necessarily be uniform across the various modes. Figure 2 and Figure 3 illustrate the trend observed for the normalized linear and nonlinear frequencies on increasing the value of ls /L and lk /L. It is interesting to note that when the normalized kinetic length scale is zero, the frequency increases as a function of ls /L and the trend is vastly different when it is greater than zero. What is also interesting is that when the normalized kinetic and static length scales are equal, the linear frequency remains constant for both the boundary conditions, with an imperceptible decrease in the nonlinear frequency. When lk /L is greater than zero, both the linear and nonlinear frequencies decrease as ls /L increases, but flatlines as the normalized static length scale approaches 1. Figure 4 demonstrates the effect that the stiffness of the elastic medium has on the frequency of the nanorod embedded on it. Quite clearly, increasing the stiffness increases the frequency. 12
Table 1: First three Nondimensional linear and nonlinear frequencies of a C-C nanorod (lk /L=0, kEM =0,qmax =0.1) ls /L 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Mode 1 ωn ωN L 3.1416 3.1852 3.2930 3.3400 3.7103 3.7666 4.3170 4.3863 5.0453 5.1296 5.8499 5.9504 6.7035 6.8207 7.5895 7.7239 8.4977 8.6496 9.4218 9.5913 10.3575 10.5448
Mode 2 ωn ωN L 6.2832 6.6320 7.4205 7.8713 10.0906 10.7651 13.4070 14.3449 16.9955 18.2105 20.7151 22.2132 24.5062 26.2904 28.3402 30.4122 32.2017 34.5626 36.0819 38.7325 39.9753 42.9161
Mode 3 ωn ωN L 9.4248 10.6020 12.9510 14.8224 20.1105 23.2760 28.2655 32.8423 36.7593 42.7810 45.4022 52.8825 54.1228 63.0686 62.8887 73.3040 71.6834 83.5707 80.4974 93.8583 89.3250 104.1607
Finally, a comparison was performed between the two different boundary conditions to examine the trend obtained on varying the normalized static length scale and the modeshape number. The difference between the linear and nonlinear frequencies for the CF-CF boundary condition Figure 5is much more stark than the difference between the linear and nonlinear frequencies for the CF-FS boundary condition Figure 6, suggesting that the effect of the geometric nonlinearity is more pronounced for the CF-CF boundary condition than for the CF-FS boundary condition. It is also observed that at lower modes, for the CF-FS boundary condition, the linear frequency is slightly higher than the nonlinear frequency to certain normalized static length scale value.
7
Conclusion
Strain and velocity gradient theory has been used to study the size-dependent vibration of carbon nanotubes (CNTs) embedded in an elastic medium. To deal with large deformation, von Karman assumptions have been considered, and motion equations as well as consistent boundary conditions have been derived using a variational approach. Within the gradient theory, the static and kinetic internal length scales are incorporated to capture the size effect on the carbon nanotubes. The linear and nonlinear free vibration of the CNT embedded in an elastic medium were investigated with a combination of clamped forcing and free strained boundary conditions. The results depict that the internal length scales affect considerably the vibration behavior of the CNT. It is concluded that for the dynamic analysis of structures such as CNTs, instead of a strain gradient theory, a strain as well as velocity gradient theory should be employed. The (static and dynamic) length scales stem from internal length parameters of the structure. For evaluation of these parameters, experimental and numerical experiments are to be carried out. 13
Table 2: First three Nondimensional linear and nonlinear frequencies of a C-F nanorod (lk /L=0, kEM =0,qmax =0.1) ls /L 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Mode 1 ωn ωN L 1.5708 1.5501 1.5901 1.5678 1.6465 1.6198 1.7365 1.7024 1.8551 1.8113 1.9974 1.9416 2.1585 2.0894 2.3346 2.2511 2.5227 2.4239 2.7201 2.6057 2.9250 2.7947
Mode 2 ωn ωN L 4.7124 4.7810 5.2094 5.2607 6.4755 6.4761 8.1602 8.0939 10.0552 9.9197 12.0619 11.8589 14.1328 13.8645 16.2432 15.9117 18.3797 17.9863 20.5340 20.0799 22.7011 22.1872
Mode 3 ωn ωN L 7.8540 8.4044 9.9868 10.6456 14.6249 15.4803 20.1032 21.1961 25.8939 27.2487 31.8268 33.4573 37.8352 39.7492 43.8880 46.0905 49.9691 52.4633 56.0693 58.8575 62.1830 65.2667
Acknowledgements. The first three authors acknowledge the financial support of Texas A&M University at Qatar. The last author is grateful to the Oscar S. Wyatt Endowed Chair.
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16
Fig. 1: Carbon Nanotube rod embedded in a Winkler-type elastic medium
17
12 l / L= 0 l / L
Normallized Linear and Nonlinear Frequency
k
s
l / L=1 l / L k
10
s
l / L=2 l / L k
s
8
6
4
2
0 0
0.2
0.4
0.6
0.8
1
l/L s
Fig. 2: Effect of the normalized static length scale ls /L on the CF-CF normalized linear (-) and nonlinear frequency (-x-) for different values of the normalized kinetic length scale lk /L (n=1, kˆEM =1, qmax = 0.2)
18
70 l / L=0 l / L
Normalized Linear and Nonlinear Frequency
k
s
l / L=1 l / L
60
k
s
l / L=2 l / L k
s
50
40
30
20
10
0 0
0.2
0.4
0.6
0.8
1
l/L s
Fig. 3: Effect of the normalized static length scale ls /L on the CF-FS normalized linear (-) and nonlinear frequency (-x-) for different values of the normalized kinetic length scale lk /L (n=1, kˆEM =1, qmax = 0.2)
19
5
Normalized Linear and Nonlinear Frequency
k^ k^ k^
4
=1
EM
=5
EM
= 10
EM
3
2
1
0 0
0.2
0.4
0.6
0.8
1
l/L k
Fig. 4: Effect of the stiffness of the elastic medium on the normalized linear (-) nonlinear frequency (-x-) for different values of the normalized kinetic length scale lk for the CF-CF boundary condition(n=1, ˆls =0, qmax = 0.2)
20
Normalized Linear and Nonlinear Frequency
15 Linear - n=1 Linear - n=2 Linear - n =3 Nonlinear - n=1 Nonlinear - n= 2 Nonlinear - n=3
10
5
0 0
0.2
0.4
0.6
0.8
1
l/L s
Fig. 5: Effect of the mode number n on the CF-CF normalized linear (-) and nonlinear frequency (-x-) for different values of the normalized static length scale ls /L, kˆEM =0, qmax = 0.2)
21
Normalized Linear and Nonlinear Frequency
12 Linear - n=1 Linear - n=2 Linear - n=3 Nonlinear - n=1 Nonlinear - n=2 Nonlinear - n=3
10
8
6
4
2
0 0
0.2
0.4
0.6
0.8
1
l /L s
Fig. 6: Effect of the mode number n on the CF-FS normalized linear (-) and nonlinear frequency (-x-) for different values of the normalized static length scale ls /L, kˆEM =0, qmax = 0.2)
22
R. Fernandes, S. El-Borgi, S.M. Mousavi, J.N. Reddy, A. Mechmoum www.elsevier.com/locate/physe
PII: DOI: Reference:
S1386-9477(16)31170-5 http://dx.doi.org/10.1016/j.physe.2016.11.007 PHYSE12646
To appear in: Physica E: Low-dimensional Systems and Nanostructures Received date: 18 October 2016 Revised date: 2 November 2016 Accepted date: 9 November 2016 Cite this article as: R. Fernandes, S. El-Borgi, S.M. Mousavi, J.N. Reddy and A. Mechmoum, Nonlinear Size-dependent Longitudinal Vibration of Carbon Nanotubes Embedded in an Elastic Medium, Physica E: Low-dimensional Systems and Nanostructures, http://dx.doi.org/10.1016/j.physe.2016.11.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Nonlinear Size-dependent Longitudinal Vibration of Carbon Nanotubes Embedded in an Elastic Medium R. Fernandesa , S. El-Borgib,1 , S.M. Mousavic , J.N. Reddyd , A. Mechmoumb a
Department of Aerospace Engineering, Texas A&M University, College Station, Texas 77843-3123, USA. b Mechanical Engineering Program, Texas A&M University at Qatar, Engineering Building, P.O. Box 23874, Education City, Doha, Qatar. c Department of Engineering and Physics, Karlstad University, 65188 Karlstad, Sweden. d Department of Mechanical Engineering, Texas A&M University, College Station, Texas 77843-3123, USA.
Abstract In this paper, we study the longitudinal linear and nonlinear free vibration response of a single walled carbon nanotube (CNT) embedded in an elastic medium subjected to different boundary conditions. This formulation is based on a large deformation analysis in which the linear and nonlinear von K´arm´an strains and their gradient are included in the expression of the strain energy and the velocity and its gradient are taken into account in the expression of the kinetic energy. Therefore, static and kinetic length scales associated with both energies are introduced to model size effects. The governing motion equation along with the boundary conditions are derived using Hamilton’s principle. Closed-form solutions for the linear free vibration problem of the embedded CNT rod are first obtained. Then, the nonlinear free vibration response is investigated for various values of length scales using the method of multiple scales. Keywords: Embedded carbon nanotube rod; Nonlinear von K´arm´an strain; Strain and velocity gradient theory; Linear and nonlinear free vibration; Method of Multiple Scales.
1
Introduction
Nanomaterials are materials used at the nanoscale with outstanding mechanical, chemical, electrical, optical and electronic properties. Examples of nanomaterials include carbon nanotubes, graphene sheet, zinc oxide (ZnO) nanowires, boron nitride (BN) nanotubes and nanosheets [1, 2, 3]. Carbon nanotubes (CNTs), discovered by [4], are layers of graphite that are wrapped up into cylinders with diameters of about one nanometer and lengths up to many micrometers. CNTs have received widespread interest of researchers from many disciplines, including material science, engineering, chemistry and physics. CNTs are being used in carbon nanotube composites [5] and environment resistant coatings made of carbon nanotube reinforced materials [6, 7]. 1
Corresponding author. Tel.: +974 4423 0674, e-mail: sami.el [email protected] (Sami El-Borgi).
1
Nanomaterials are the base materials of many nanoscale structures called nanostructures. A number of 1D nanostructures have been designed including nanorods, nanowires, nanotubes, nanodots, nanowalls, nanobridges, nanobelts and nanorings. Nanorods are nanostructures ranging from 1 to 3000nm which have been used in many devices such as diodes, transistors, fuel-cell cathodes, nanosensors, electrical contacts for buffering thermal expansion and LED arrays [8]. Miniaturized nanobars are basic elements utilized in a number of applications such as micro-truss structures [9], micro-manipulators [10] and micro-milling [11]. In light of these micro-scale applications, studying the static and dynamic behavior of nanobars seems to be quite important. Molecular dynamic (MD) simulation requires intense computational labor for nanostructures having large numbers of atoms and this has motivated the intensive development of size-dependent or higher-order continuum mechanics theories. These theories include nonlocal continuum theories, strain gradient theories or a combination of both (nonlocal strain gradient theories) and modified couple stress theory. Eringen’s nonlocal elasticity differential model [12] was utilized by a number of researchers to capture size-effects in nanorods. However, these studies are relatively less than those related to nanobeams [13, 14, 15]. A nonlocal model was proposed by [16] to investigate the axial vibration response of the nanorod and the small-scale effect of the axial frequencies. This model was used to examine the axial vibration of a carbon nanotube system [11]. The nonlocal longitudinal vibration of a double nanorod system was studied by [8] and its potential applications as a nonosensor in addition to nanooptomechanical systems (NOMS) were proposed. The free axial vibration of nanorods using fractional continuum mechanics was investigated by [17]. A nonlocal model proposed by [18] to examine the axial vibration behavior of a tapered nanorod using the differential quadrature method. The free and forced damped axial vibration response of a nanorod using a frequency-dependent finite element method was studied by [19] . In another study, a similar approach was used to examine the axial vibration response of a nanorod embedded in an elastic medium [20] . The wave propagation in nonlocal nanorods was investigated by [21]. The nonlocal axial vibration problem of a CNT nanorod embedded in an elastic medium was examined by [22] . The study by [23] examined the nonlocal effects of longitudinal vibration in nanorod with internal longrange interactions and proposed closed-form solutions for the case of a uniform nonlocal kernel. The theory of nonlocal elasticity was used to investigate the small scale effect on the axial vibration of non-uniform and non-homogeneous nanorods using the differential quadrature method [24]. Few number of investigators used the strain gradient theory or the nonlocal strain gradient theory to study the behavior of nanorods. The longitudinal axial vibration of a homogenous and a functionally graded microbar based on strain gradient elasticity theory were investigated by [25] and [26], respectively. Kahrobaiyan et al. [27] studied both analytically and numerically using the finite element method the longitudinal behavior of strain gradient bars. Analytical solutions for the longitudinal free vibration response of size-dependent rods using nonlocal strain gradient theory were obtained by [28]. In addition to the nonlocal theory and strain gradient theory, the modified couple stress theory has also been extensively employed in the micro- or nano- scale structures. Ke and 2
his co-authors [29, 30, 31] made a significant contribution on the linear and nonlinear vibration of size dependent functionally graded microbeams and microplates. They found that the nonlinear frequency ratio versus amplitude curves were unsymmetrical due to the bending-extension coupling in functionally graded microbeams and microplates. They also implied that this unsymmetry was also size dependence. Farokhi et al. [32] developed a new size-dependent nonlinear model for the analysis of the pull-in behavior of carbon nanotube-based resonators based on the modified coupled stress theory. The study by Miandoab et al. [33] presented a polysilicon nano-beam model based on modified couple stress and Eringens nonlocal elasticity theories. In many practical applications, nanomaterials are embedded in an elastic medium such as polymer/CNT nanocomposites (PCNTs) to enhance the performance of engineering polymers in terms of strength and fracture toughness [34]. This is due to the excellent properties of nanomaterials. Many polymer resins have been used as a matrix for CNT-nanocomposites, such as CNT/Polypropylene [35], CNT/Nylon [36] and CNT/Polycarbonate [37]. In light of this, several investigations have been conducted to study the interaction of the nanomaterial with the elastic medium. Two types of elastic media are generally considered, namely the one-parameter Winkler type [22] and the two-parameter Pasternak type [38] elastic medium. From the reviewed literature, it can be observed that most investigators have based their work related to nanorods on linear theories. To fill this gap, the present study considers the vibration problem of a CNT nanorod embedded in a Winkler type elastic medium and proposes a large-deformation analysis based on a modified strain gradient theory that incorporates both strain and velocity gradients in addition to the von K´ arm´ an nonlinearity. The linear free vibration problem obtained by ignoring the von K´ arm´ an nonlinearity is solved analytically, while the nonlinear free and forced vibration problems are solved using the Method of Multiple Scales (MMS). This paper is outlined as follows. Following this introduction, the proposed strain gradient theory for nanorods is presented in Section 2. The governing equation of motion for CNT nanorod embedded in an elastic medium is established in Section 3. The linear vibration problem without the von K´arm´an strain is solved analytically in Section 4. The nonlinear free and forced vibration solutions obtained using MMS are given in Sections 5 and 6, respectively. Numerical results are provided in Section 7. Finally, concluding remarks are given in Section 8.
2
Strain Gradient Theory
This section summarizes the Strain Gradient Theory used in this study and additional details can be found in [39, 40]. For a body occupying a volume V and made of a linear elastic isotropic material, the strain energy U can be written as follows based on the strain gradient theory: Z 1 (1) U= σij εij + σijm εij,m dV (1) 2 V
3
Here εij and εij,m are, respectively, the classical strain and its gradient. The classical (1) stress tensor σij and the higher-order stress tensor σijm are defined as σij =Cijkl εkl (1) σijm
(2)
=ls2 Cijkl εkl,m
(3)
where Cijkl are the components of the elasticity tensor and ls is a material length scale parameter to account for the strain gradient field. The total stress field can be expressed as (1)
tij = σij − ∇σijm
(4)
in which ∇ is the Laplacian operator. For a rod-type structure, the size-dependency is only accounted for in the longitudinal direction and neglected in both width and thickness directions. The stress-strain relations (2) and (3) can then be expressed as σxx =Eεxx (1) σxxx
=ls2 Eεxx,x
(5) (6)
where E is the modulus of elasticity of the rod and εxx is the axial strain. The total axial stress txx given by (4) can be simplified to txx = 1 − ls2 ∇2 Eεxx (7) where ∇2 = ∂ 2 /∂x2
3
Equations of motion of size-dependent rods
The displacement field in a rod of volume V , length L and cross-sectional area A takes the following form: u1 = u (x, t) , u2 = 0, u3 = 0 (8) Here t denotes time, u1 , u2 and u3 denote the displacements along the x, y and zdirections, respectively. u is the axial displacement. Thus, the nonlinear nonzero strain of rods can be written as ∂u 1 ∂u 2 εxx = (9) + ∂x 2 ∂x and its variation with respect to x is δεxx =
∂δu ∂u ∂δu + ∂x ∂x ∂x
Taking the variation of (1) leads to the expression of the virtual strain energy δU Z (1) δU = σxx δεxx + σxxx δεxx,x dV V
4
(10)
(11)
Integrating the second term by parts yields Z L Z (1) (1) δU = σxx δεxx − ∇σxxx δεxx dV + σxxx δεxx dA V LZ
A
txx δεxx dAdx +
= 0
Z =
A
A
L
h
N δεxx dx + N (1) δεxx
0
0
L (1) σxxx δεxx dA
Z
Z
0
iL
(12)
0
where N and N (1) are the stress resultants of, respectively, the total stress and the higherorder stress which are given by Z Z (1) (1) N= txx dA, N = σxxx dA (13) A
A
and the stress resultant of the classical stress N (0) can be defined as Z N (0) = σxx dA
(14)
A
By using equations (7), (13) and (14), the following relationships can be established: ∂N (1) N = N (0) − " ∂x # ∂u 1 ∂u 2 (0) N = EA + ∂x 2 ∂x N (1) = ls2
∂N (0) ∂x
(15) (16) (17)
Here E is the Young’s Modulus of the nanorod and A is its cross-sectional area. It is assumed that the rod has a circular cross-section. Substituting for the variation of the axial strain δεxx from (10), the virtual strain energy given by (12) can be further expressed in terms of displacements as follows: Z L ∂u ∂δu ∂u ∂δu L (1) δU = N 1+ dx + N 1+ (18) ∂x ∂x ∂x ∂x 0 0 Considering the longitudinal motion of the rod and its velocity gradient, the kinetic energy K can be written as Z L 2 Z L 2 2 ∂u 1 ∂ u 1 2 K = ρA dx + ρAlk dx (19) 2 ∂t 2 ∂x∂t 0 0 where ρ is the density of the rod and lk is the kinetic material length parameter associated with the velocity gradient. The variation of the kinetic energy can, therefore, be written as Z L Z L 2 ∂u ∂δu ∂ u ∂ 2 δu 2 δK = ρA dx + ρAlk dx (20) ∂t ∂t 0 0 ∂x∂t ∂x∂t 5
The variation of the external work due to the surrounding medium can be written as Z L δW = Fx δudx (21) 0
where Fx is the force exerted by the medium in which the nanotube is embedded. According to Hamilton’s Principle, it can be stated that Zt2 (δK − δU + δW ) dt = 0
(22)
t1
Substitution of (18), (20) and (21) into (22) yields Zt2 ZL
∂u ∂δu ρA dxdt + ρAlk2 ∂t ∂t
t1 0
Zt2 ZL − t1 0
Zt2 ZL
∂ 2 u ∂ 2 δu dxdt ∂x∂t ∂x∂t
t1 0
Zt2 ∂u ∂δu L ∂u ∂δu dt dt − N (1) 1 + N 1+ ∂x ∂x ∂x ∂x 0 t1
Zt2 ZL +
Fx δudxdt = 0
(23)
t1 0
Assuming a Winkler type model [22], the force exerted by the embedding medium can be represented as Fx = −kEM u (24) where kEM is the stiffness of the elastic medium. Applying integration by parts with respect to t as well as x in (23), setting the initial conditions to zero and using the fundamental lemma of calculus variations, the following equation of motion can be derived: 4 ∂2u ∂ ∂u 2 ∂ u −ρA 2 + ρAlk 2 2 + N 1+ − kEM u = 0 (25) ∂t ∂x ∂t ∂x ∂x along with the associated boundary conditions specified at each of the ends x = 0 and x=L 3 ∂u 2 ∂ u ρAlk +N 1+ = 0 or u = 0 (26) ∂x∂t2 ∂x ∂u ∂u N (1) 1 + = 0 or =0 (27) ∂x ∂x Combining (15) to (17) gives the expressions of the axial normal force N and its higher order normal force N (1) in terms of displacements " " # # 2 2 3u 3u ∂ ∂u 1 ∂u 2 ∂ u ∂u ∂ N = EA + − ls2 EA + + (28) ∂x 2 ∂x ∂x3 ∂x2 ∂x ∂x3 2 ∂ u ∂u ∂ 2 u (1) 2 N = ls EA + (29) ∂x2 ∂x ∂x2 6
In the framework of nonlinear classical elasticity obtained by setting the length scale parameters ls and lk equal to zero, the purely non-classical boundary conditions (27) disappear. Further, it can be verified that the governing equation (25) and the associated classical boundary conditions (26) can be reduced to the classical equations ∂2u ∂ ∂u −ρA 2 + N 1+ − kEM u = 0 (30) ∂t ∂x ∂x subject to one of the following boundary conditions specified at each of the ends x = 0 and x = L: ∂u N 1+ = 0 or u = 0 (31) ∂x where
"
∂u 1 N = EA + ∂x 2
∂u ∂x
2 # (32)
The governing equation (25) and associated boundary conditions, (26) and (27), can be written in non-dimensional form as follows: ∂2υ ˆ 2 ∂4υ ∂ ¯ ∂υ − 2 + lk + N 1+ − kˆEM υ = 0 (33) ∂τ ∂ξ 2 ∂τ 2 ∂ξ ∂ξ subject to the boundary conditions specified at each of the ends ξ = 0 and ξ = 1 3 ˆl2 ∂ υ + N ¯ 1 + ∂υ = 0 or υ = 0 k ∂ξ∂τ 2 ∂ξ ¯ (1) 1 + ∂υ = 0 or ∂υ = 0 N ∂ξ ∂ξ
(34) (35)
¯ and its higher-order Based on (28) and (29), the non-dimensional axial normal force N (1) ¯ normal force N can be expressed as " # 2 2 2 3υ 3υ ∂υ 1 ∂υ ∂ ∂ υ ∂υ ∂ ¯ = N + − ˆls2 + + (36) ∂ξ 2 ∂ξ ∂ξ 3 ∂ξ 2 ∂ξ ∂ξ 3 2 2 ¯ (1) = ˆl2 ∂ υ + ∂υ ∂ υ N (37) s ∂ξ 2 ∂ξ ∂ξ 2 where the non-dimensional parameters introduced in (33) to (37) are given by s x u E ξ= , υ= , τ =t L L ρL2 2 ˆls = ls , ˆlk = lk , kˆEM = kEM L L L EA
7
(38)
4
Linear Free Vibration Analysis
Combining (33) and (36) and dropping the nonlinear terms yields the following linear equation of motion: −
4 2 ∂ 2 υ ˆ2 ∂ 4 υ ˆl2 ∂ υ + ∂ υ − kˆEM υ = 0 + l − s k ∂τ 2 ∂ξ 2 ∂τ 2 ∂ξ 4 ∂ξ 2
(39)
Assuming a harmonic motion for the nth linear mode shape φn (ξ), the solution can be written as υ (ξ, τ ) = φn (ξ) eiωn τ (40) √ where i = −1, ωn is the nth nondimensionl linear natural frequency. Substituting this solution into (39) yields the following differential eigenvalue problem: 4 2 ˆl2 d φn + ˆl2 ωn 2 − 1 d φn + −ωn 2 + kˆEM φn = 0 (41) s k dξ 4 dξ 2 The solution of the above linear fourth order differential equation can be expressed as φn (ξ) = C1 sin λ1 ξ + C2 cos λ1 ξ + C3 sinh λ2 ξ + C4 cosh λ2 ξ
(42)
where s λ1 =
−ˆ lk2 ωn 2 −
q ˆ ˆEM ˆ lk4 ωn 4 −2 ˆ lk2 ωn 2 +4 ˆ ls2 ωn 2 −4 k ls2 +1+1 2 ˆ 2l
(43a)
q ˆ ˆEM ˆ lk4 ωn 4 −2 ˆ lk2 ωn 2 +4 ˆ ls2 ωn 2 −4 k ls2 +1+1 2ˆ l2
(43b)
s
s λ2 =
−ˆ lk2 ωn 2 +
s
and Ci (i = 1, 2, 3, 4) are constants determined from the boundary conditions. We consider first a nanotube with both ends classically clamped v(0) = v(1) = 0 and ¯ (1) (0) = N ¯ (1) (1) = 0. This type of linear the higher linear normal force also clamped N boundary conditions is denoted CF-CF (CF: Clamped Forcing) in which the first and second boundary conditions at each end are, respectively, classical and non-classical and can be expressed as [28] φn (0) = φn (1) = 0 2 2 2 ∂ φn ˆl2 ∂ φn ˆ = l s s ∂ξ 2 ∂ξ 2 ξ=0
(44) =0
(45)
ξ=1
Employing these boundary conditions yields C2 = C3 = C4 = 0 sin λ1 = 0 or λ1 = nπ
8
(46)
Equating (46) to (43a) and solving for ωn yields s ˆl2 n4 π 4 + n2 π 2 + kˆEM s ωn = ˆl2 n2 π 2 + 1 k
(47)
and the corresponding modeshape for the CF-CF boundary conditions can be written as φn (ξ) = sin (nπξ)
(48)
¯ (1) (0) = We consider now a nanotube with the left end Clamped Forcing (CF), v(0) = N 0, and the right end Free Strained (FS). The FS boundary condition consists of a zero normal force N (1) = 0 which is classical and a zero strain ∂υ ∂ξ (1) = 0 which is non-classical. This type of linear boundary conditions is denoted CF-FS and can be expressed as [28] ∂φn ˆ2 ∂ 3 φn φn (0) = − ls =0 (49) ∂ξ ∂ξ 3 ξ=1 2 ∂φn ˆl2 ∂ φn = =0 (50) s ∂ξ 2 ∂ξ ξ=0
ξ=1
Employing these boundary conditions yields C1 = C3 = C4 = 0 cos λ1 = 0 or λ1 =
(2n − 1) π 2
Equating (51) to (43a) and solving for ωn yields s 1 (2 n − 1)4 ˆls2 π 4 + 4 (2 n − 1)2 π 2 + 16 kˆEM ωn = 2 4 + (2 n − 1)2 ˆlk2 π 2
(51)
(52)
and the corresponding modeshape for the CF-FS boundary conditions can be expressed as (2n − 1) πξ φn (ξ) = sin (53) 2 After setting lk = 0 and kEM = 0, it can be verified that the expression of the nonlinear frequencies (47) and (52) are the same as those found by [28] after assigning the nonlocal parameter ea equal to zero and applying the normalization given by (38).
5
Nonlinear Free Vibration Analysis
For the nonlinear free vibration analysis, Galerkin’s method [42] is applied to (33) to eliminate the spacial variable ξ and convert it to an ordinary differential equation. This is done by assuming the axial displacement of the nth mode as follows: υ (ξ, τ ) = φn (ξ)q(τ ) 9
(54)
where q(τ ) is the unknown time dependent function and φn (ξ) is the normalized linear modeshape corresponding to the nth mode. Substituting (54) into (33), multiplying by the modeshape and integrating from ξ = 0 to ξ = 1 results in the following ordinary differential equation with quadratic and cubic nonlinearities: q¨ + β1 q + β2 q 2 + β3 q 3 = 0
(55)
where R1 R1 4 2 φn ddξφ4n dξ − kˆEM 0 φ2n dξ + 0 φn ddξφ2n dξ β1 = (56a) R R ˆl2 1 φn d2 φ2n dξ − 1 φ2 dξ k 0 0 n dξ R1 R1 R 1 dφn d2 φn 4 2 3 n d φn −4 ˆls2 0 φn ddξφ2n ddξφ3n dξ − 2 ˆls2 0 φn dφ dξ dξ 4 dξ + 3 0 φn dξ dξ 2 dξ β2 = (56b) R R ˆl2 1 φn d2 φ2n dξ − 1 φ2 dξ n k 0 0 dξ 2 3 R R 1 dφn 2 d4 φn R 2 1 1 d φn d3 φn dφn d φn 2 ˆ dξ − 4 0 φn dξ2 dξ3 dξ dξ − 0 φn dξ dξ 2ls − 0 φn dξ2 dξ 4 + β3 = R1 R1 2 2ˆlk2 0 φn ddξφ2n dξ − 2 0 φ2n dξ R 1 n 2 d 2 φn 3 0 φn dφ dξ dξ dξ 2 (56c) R R 2 1 1 2ˆlk2 0 φn ddξφ2n dξ − 2 0 φ2n dξ −ˆls2
R1 0
Since (55) is a second order ODE, it is subject to the following two initial conditions: q (0) = qmax q˙ (0) = 0
(57)
where qmax is the normalized amplitude of vibration corresponding to the time dependent function q (τ ). It is worth mentioning that on substituting the modeshape function corresponding to the CF-CF boundary condition, the quadratic term β2 goes to zero. Substituting the modeshape function for the CF-FS boundary condition does not render β2 to be zero. For small values of qmax , the nonlinear ordinary differential equation in (55) can be solved analytically using the Method of Multiple Scales [41] and this solution procedure has been adopted in this study. For the sake of convenience, equation (55) can be further simplified by introducing the following nondimensional terms: qˆ =
q qmax
, τˆ =
p
β1 τ
(58)
The above normalization is then substituted into (55) and (57) and the following equation of motion and its associated initial conditions are obtained: d2 qˆ + qˆ + α1 qˆ2 + α2 qˆ3 = 0 dˆ τ2 qˆ(0) = 1, qˆ˙(0) = 0 10
(59a) (59b)
2
) and α2 = β3 (qβmax . where α1 = β2 qβmax 1 1 In using the Method of Multiple Scales [41], an n-order uniform expansion is assumed to model the response of the system as a function of multiple time scales
qˆ(ˆ τ ; ε) = εˆ q1 (T0 , T1 , T2 ) + ε2 qˆ2 (T0 , T1 , T2 ) + .... + εn qˆn (T0 , T1 , T2 )
(60)
where ε is a small parameter indicating the oscillation amplitude. The main oscillatory behavior occurs during the fast time scale T0 = τˆ , while the amplitude and phase modulation take place during the slow time scales Tn = εn = τˆ, n ≥ 1. In seeking the solution of the ODE (59a), a third-order uniform expansion is assumed with three time scales T0 , T1 and T2 qˆ(ˆ τ ) = εˆ q1 + ε2 qˆ2 + ε3 qˆ3 (61) Substitution of (61) into (59a) produces the following system of linear equations corresponding each to a different power of ε ε1 : D02 qˆ1 + qˆ1 = 0
(62a)
ε2 : D02 qˆ2 + qˆ2 = −2D0 D1 qˆ1 − α1 qˆ12 3
ε :
D02 qˆ3
+ qˆ3 =
−(D12
+ 2D2 D0 )ˆ q1 −
(62b) α2 qˆ13
− 2D0 D1 qˆ2 − 2α1 qˆ1 qˆ2
(62c)
The subsequent analysis using the method of multiple scales for a system with cubic and quadratic nonlinearities is thoroughly examined in [41]. Solving (62a) for qˆ1 , obtaining the solvability conditions at each order and expressing the final solvability condition in its polar form, the following expression for the final solution and the nonlinear frequency is obtained: qˆ =εˆ q1 + ε2 qˆ2 + +O ε3
=εa cos (Ωˆ τ + ψ0 ) + ε
2
α1 a2 α1 a2 cos (2Ωˆ τ + 2ψ0 ) − 6 2
+ O ε3
(63)
where Ω is the nonlinear frequency which can be written as [41] and O ε3 is the error associated with higher order terms. 5 2 2 2 3 2 2 a ε α1 + a ε α2 + O ε3 (64) 12 8 For this solution to satisfy the initial conditions in (59b), a trigonometric manipulation is introduced in which the error associated with the third order expansion is considered [43]. The resulting solution can be written as α1 a2 α1 a2 2 qˆ = εa cos (Ωˆ τ + ψ0 ) + ε cos (2Ωˆ τ + 2ψ0 ) − + b cos(ˆ τ + γ) + O ε3 (65) 6 2 Ω=1−
Applying the initial conditions (59b) to equation (65), the values of following constants are obtained: 1 α1 ψ0 = 0, a = , γ = 0, b = (66) ε 3 11
Substituting (66) into (64), the nonlinear frequency can therefore be simplified to 5 2 3 α + α2 (67) 12 1 8 √ The above expression can then be multiplied by β1 to obtain the non-dimensional form of the nonlinear frequency used thereafter in the results and discussion section p 5 2 3 ωN L = β1 1 − α1 + α2 (68) 12 8 Ω=1−
6
Results and Discussion
This section presents results of the non-dimensional linear and nonlinear frequency of the nanorod embedded in the elastic medium. Parametric studies are performed for the different normalized static and kinetic length scales and the normalized coefficient of the elastic medium. In each case, the values for the two boundary conditions (C-C and C-F) are presented. Normalized values of these parameters are assumed since the literature does not provide experimental or verified simulated values for the kinetic length scale and elastic foundation coefficient. Table 1 and Table 2 demonstrate the effect of normalized static length scale ls /L the linear and non-linear frequencies for the CF-CF boundary condition and the CF-FS boundary condition. The internal length scale range, i.e. 0 < ls /L < 1, is considered to demonstrate the effect of variation of ls /L. The accurate values of ls and lk in this gradient theory may be evaluated after fitting the results to proper experimental observations or atomistic simulations. It is observed that for both boundary conditions, the value of the linear and nonlinear frequency increase as ls /L increases. For the CF-FS boundary condition however, while the nonlinear frequency increases with the increase in the normalized static length scale, the values of the nonlinear frequency are lower than that of the linear frequency for the first mode, but generally higher for the second and third modes. This indicates that for the CF-FS boundary condition, the chosen normalized amplitude of vibration could influence different vibrational responses that may not necessarily be uniform across the various modes. Figure 2 and Figure 3 illustrate the trend observed for the normalized linear and nonlinear frequencies on increasing the value of ls /L and lk /L. It is interesting to note that when the normalized kinetic length scale is zero, the frequency increases as a function of ls /L and the trend is vastly different when it is greater than zero. What is also interesting is that when the normalized kinetic and static length scales are equal, the linear frequency remains constant for both the boundary conditions, with an imperceptible decrease in the nonlinear frequency. When lk /L is greater than zero, both the linear and nonlinear frequencies decrease as ls /L increases, but flatlines as the normalized static length scale approaches 1. Figure 4 demonstrates the effect that the stiffness of the elastic medium has on the frequency of the nanorod embedded on it. Quite clearly, increasing the stiffness increases the frequency. 12
Table 1: First three Nondimensional linear and nonlinear frequencies of a C-C nanorod (lk /L=0, kEM =0,qmax =0.1) ls /L 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Mode 1 ωn ωN L 3.1416 3.1852 3.2930 3.3400 3.7103 3.7666 4.3170 4.3863 5.0453 5.1296 5.8499 5.9504 6.7035 6.8207 7.5895 7.7239 8.4977 8.6496 9.4218 9.5913 10.3575 10.5448
Mode 2 ωn ωN L 6.2832 6.6320 7.4205 7.8713 10.0906 10.7651 13.4070 14.3449 16.9955 18.2105 20.7151 22.2132 24.5062 26.2904 28.3402 30.4122 32.2017 34.5626 36.0819 38.7325 39.9753 42.9161
Mode 3 ωn ωN L 9.4248 10.6020 12.9510 14.8224 20.1105 23.2760 28.2655 32.8423 36.7593 42.7810 45.4022 52.8825 54.1228 63.0686 62.8887 73.3040 71.6834 83.5707 80.4974 93.8583 89.3250 104.1607
Finally, a comparison was performed between the two different boundary conditions to examine the trend obtained on varying the normalized static length scale and the modeshape number. The difference between the linear and nonlinear frequencies for the CF-CF boundary condition Figure 5is much more stark than the difference between the linear and nonlinear frequencies for the CF-FS boundary condition Figure 6, suggesting that the effect of the geometric nonlinearity is more pronounced for the CF-CF boundary condition than for the CF-FS boundary condition. It is also observed that at lower modes, for the CF-FS boundary condition, the linear frequency is slightly higher than the nonlinear frequency to certain normalized static length scale value.
7
Conclusion
Strain and velocity gradient theory has been used to study the size-dependent vibration of carbon nanotubes (CNTs) embedded in an elastic medium. To deal with large deformation, von Karman assumptions have been considered, and motion equations as well as consistent boundary conditions have been derived using a variational approach. Within the gradient theory, the static and kinetic internal length scales are incorporated to capture the size effect on the carbon nanotubes. The linear and nonlinear free vibration of the CNT embedded in an elastic medium were investigated with a combination of clamped forcing and free strained boundary conditions. The results depict that the internal length scales affect considerably the vibration behavior of the CNT. It is concluded that for the dynamic analysis of structures such as CNTs, instead of a strain gradient theory, a strain as well as velocity gradient theory should be employed. The (static and dynamic) length scales stem from internal length parameters of the structure. For evaluation of these parameters, experimental and numerical experiments are to be carried out. 13
Table 2: First three Nondimensional linear and nonlinear frequencies of a C-F nanorod (lk /L=0, kEM =0,qmax =0.1) ls /L 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Mode 1 ωn ωN L 1.5708 1.5501 1.5901 1.5678 1.6465 1.6198 1.7365 1.7024 1.8551 1.8113 1.9974 1.9416 2.1585 2.0894 2.3346 2.2511 2.5227 2.4239 2.7201 2.6057 2.9250 2.7947
Mode 2 ωn ωN L 4.7124 4.7810 5.2094 5.2607 6.4755 6.4761 8.1602 8.0939 10.0552 9.9197 12.0619 11.8589 14.1328 13.8645 16.2432 15.9117 18.3797 17.9863 20.5340 20.0799 22.7011 22.1872
Mode 3 ωn ωN L 7.8540 8.4044 9.9868 10.6456 14.6249 15.4803 20.1032 21.1961 25.8939 27.2487 31.8268 33.4573 37.8352 39.7492 43.8880 46.0905 49.9691 52.4633 56.0693 58.8575 62.1830 65.2667
Acknowledgements. The first three authors acknowledge the financial support of Texas A&M University at Qatar. The last author is grateful to the Oscar S. Wyatt Endowed Chair.
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16
Fig. 1: Carbon Nanotube rod embedded in a Winkler-type elastic medium
17
12 l / L= 0 l / L
Normallized Linear and Nonlinear Frequency
k
s
l / L=1 l / L k
10
s
l / L=2 l / L k
s
8
6
4
2
0 0
0.2
0.4
0.6
0.8
1
l/L s
Fig. 2: Effect of the normalized static length scale ls /L on the CF-CF normalized linear (-) and nonlinear frequency (-x-) for different values of the normalized kinetic length scale lk /L (n=1, kˆEM =1, qmax = 0.2)
18
70 l / L=0 l / L
Normalized Linear and Nonlinear Frequency
k
s
l / L=1 l / L
60
k
s
l / L=2 l / L k
s
50
40
30
20
10
0 0
0.2
0.4
0.6
0.8
1
l/L s
Fig. 3: Effect of the normalized static length scale ls /L on the CF-FS normalized linear (-) and nonlinear frequency (-x-) for different values of the normalized kinetic length scale lk /L (n=1, kˆEM =1, qmax = 0.2)
19
5
Normalized Linear and Nonlinear Frequency
k^ k^ k^
4
=1
EM
=5
EM
= 10
EM
3
2
1
0 0
0.2
0.4
0.6
0.8
1
l/L k
Fig. 4: Effect of the stiffness of the elastic medium on the normalized linear (-) nonlinear frequency (-x-) for different values of the normalized kinetic length scale lk for the CF-CF boundary condition(n=1, ˆls =0, qmax = 0.2)
20
Normalized Linear and Nonlinear Frequency
15 Linear - n=1 Linear - n=2 Linear - n =3 Nonlinear - n=1 Nonlinear - n= 2 Nonlinear - n=3
10
5
0 0
0.2
0.4
0.6
0.8
1
l/L s
Fig. 5: Effect of the mode number n on the CF-CF normalized linear (-) and nonlinear frequency (-x-) for different values of the normalized static length scale ls /L, kˆEM =0, qmax = 0.2)
21
Normalized Linear and Nonlinear Frequency
12 Linear - n=1 Linear - n=2 Linear - n=3 Nonlinear - n=1 Nonlinear - n=2 Nonlinear - n=3
10
8
6
4
2
0 0
0.2
0.4
0.6
0.8
1
l /L s
Fig. 6: Effect of the mode number n on the CF-FS normalized linear (-) and nonlinear frequency (-x-) for different values of the normalized static length scale ls /L, kˆEM =0, qmax = 0.2)
22