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Nonlocal wave propagation in rotating nanotube
Results in Physics 1 (2011) 17–25
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Nonlocal wave propagation in rotating nanotube S. Narendar a,⇑, S. Gopalakrishnan b a b
Defence Research and Development Laboratory, Kanchanbagh, Hyderabad 500 058, India Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560 012, India
a r t i c l e
i n f o
Article history: Received 28 April 2011 Accepted 16 June 2011 Available online 13 July 2011 Keywords: Nanobeam Wavenumber Rotational speed Frequency Centrifugal force Nonlocal elasticity
a b s t r a c t The present work deals with the wave dispersion behavior of a rotating nanotube using the nonlocal elasticity theory. The rotating nanobeam is modeled as an Euler–Bernoulli theory. The governing partial differential equation for a uniform rotating beam is derived incorporating the nonlocal scale effects. The spatial variation in centrifugal force is modeled in an average sense. Even though this averaging seems to be a crude approximation, one can use this as a powerful model in analyzing the wave dispersion characteristics of the rotating nanobeam. Spectrum and dispersion curves are obtained as a function of rotating speed and nonlocal scaling parameter. It has been shown that the dispersive flexural wave tends to behave non-dispersively at very high rotation speeds. Understanding the dynamic behavior of rotating nanostructures is important for practical development of nanomachines. At the nanoscale, the nonlocal effects often become more prominent. The numerical results are simulated for a rotating nanobeam as a waveguide. The results can provide useful guidance for the study and design of the next generation of nanodevices such as blades of a nanoturbine, nanogears, nanoscale molecular bearings etc, that make use of the wave propagation properties of rotating single-walled carbon nanotubes. Ó 2011 Published by Elsevier B.V.
1. Introduction Carbon nanotubes were first observed and identified as such by Iijima using transmission electron microscopy in 1991 [1]. As we see, nanotubes are cylindrical molecules which may consist of several concentric sheets of graphene (Multi Walled Carbon Nanotubes – MWCNTs). Since the discovery of Iijima, research on carbon nanotubes has become a paramount activity. Their geometry was studied and it was observed that their diameter can range from a few to hundreds of nanometers and their length from a few to tens of micrometers [2]. The nanotubes are promising materials in a number of new areas. These include molecular electronics where the nanotubes can be used both for wiring and as electronic devices after modification of the nanotube itself. They can also be used as parts of mechanical devices at the nanoscale such as the shaft of nanomotors [3,4]. Moreover, their high strength and flexibility are exploited to reinforce materials such as cement. Many production methods were developed such as the laser vaporization method combined with transition metal catalyst [5] or the carbon arc method [6]. Nowadays, dozens of companies are specialized in the production of single- and multi-walled carbon nanotubes. They can produce up to hundreds of grams of SWCNTs per day and
⇑ Corresponding author. Tel.: +91 88976 25977, +91 99493 91823; fax: +91 40 24342309. E-mail addresses: [email protected], [email protected] (S. Narendar), [email protected] (S. Gopalakrishnan). 2211-3797/$ - see front matter Ó 2011 Published by Elsevier B.V. doi:10.1016/j.rinp.2011.06.002
hundreds of kilograms of MWCNTs. Forecasts predict that global demand for nanotubes will expand rapidly [7]. Flat panel displays for both computers and televisions will be the first widely commercialized application. It was observed that there is a wide variety of carbon nanotubes [2]. Their diameters vary between 0.7 and 10 nm, though most of the observed tubes have diameters smaller than 2 nm. The length of single-walled nanotubes goes from a few nanometers up to 20 cm [8]. Moreover nanotubes can be chiral or achiral. These varieties of SWCNT’s come from the fact that the nanotubes are constructed by the folding of a graphite plane on itself, and have many ways of folding arise. Recently, Kral and Sadeghpour [9] and Krim et al. [10] showed the possibility to spin nanotubes with circularly polarized light, further motivating the usage of nanotubes as possible axis of rotation at the nanoscale. However, no systematic analytical and numerical study of rotation has yet been done, except the work of Zhang et al. [11], where nonequilibrium molecular dynamics simulations were carried out to calculate the energy dissipation rate during the rotational motion. In their work, the usage of the thermostat would however add dissipation which is not intrinsic to the system. In their work, simulations were performed by Hamiltonian dynamics in the microcanonical ensemble, and in that process they avoided additional sources of dissipation. Because of future promising exploration of nanotechnology, focus is being put in the miniaturization of mechanical and electromechanical devices. Attention is sought toward the development
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S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25
Fig. 1. Carbon nanotube gears: molecular diagrams [17].
of nanodevices and nanomachines [12]. Nanomachines are systems in the nanometer realm with moving parts [13].Nanostructures undergoing rotation include nanoturbines, nanoscale molecular bearings, shaft and gear, and multiple gear systems. These nanostructure machines are expected to receive considerable attention in the near future.Researchers have thus reported the feasibility of nanoscale rotating structures. Examples include study of molecular gears [12], fullerene gears [14], and carbon nanotubes gears [15,16]. Srivastava [17] has reported the rotational dynamics of carbon nanotubes and carbon nanotubes gears under a single applied laser field. A typical carbon nanotubes gear is shown in Fig. 1. Lohrasebi and Tabar [18] carried out computational modeling of rotating nanomotor. Dynamics of the rotary nanomotor was simulated using stochastic molecular dynamics (MD) method. Zhang et al. [11] carried out atomistic simulations of doublewalled carbon nanotubes as rotational bearings. Recently Fennimore et al. [3] reported the feasibility of rotating nanostructures. They showed the construction and successful operation of a nanoscale electromechanical actuator. The rotating nanostructural system consists of a rotatable metal plate, with a multi-walled carbon nanotube serving as the key motion-enabling element. For efficient design of these rotating nanomachines,proper understanding of its mechanical behavior such as bending, vibration, and buckling is required. The development of simplified models for the dynamics of complex nano-technological systems is thus necessary. This is because for many cases, fully atomistic simulations would be computationally expensive and prohibitive. The length scales associated with nanostructures like CNTs are such that to apply any classical continuum techniques, we need to consider the small length scales such as lattice spacing between individual atoms, surface properties, grain size etc. This makes a physically consistent classical continuum model formulation very challenging. The Eringen’s nonlocal elasticity theory [19–21] is useful tool in treating phenomena whose origins lie in the regimes smaller than the classical continuum models. In this theory, the internal size or scale could be represented in the constitutive equations simply as material parameters. Such a nonlocal continuum mechanics has been widely accepted and has been applied to many problems including wave propagation, dislocation, crack problems, etc [22]. Recently, there has been great interest in the application of nonlocal continuum mechanics for modeling and analysis of CNTs. Nonlocal elasticity theory gives a good prediction for modeling the behavior of nanostructures. Literature shows that the theory of nonlocal elasticity is receiving increasing attention for efficient analysis of nanostructures which include nanobeams, nanoplates, nanorings, carbon nanotubes, graphenes and microtubules. Wave propagation analyses of CNTs are relevant due to their various applications [23] which include sensing superconductivity, transport and optical phenomena. The actual value of the fre-
quency (THz in this case) does not justify the treatment of high frequency analysis. It is the number of modes (i.e., the size of wavelength) what matters, not the frequency value itself. In nanoscale structures one does not expect small wavelength vibration with frequencies going up to tens or hundreds of THz. Therefore, the scientific need for spectral methods in the nano-scale is much less compared to usual macro-scale structures. Lattice dynamics for direct observation of phonons [24,25] and spectral finite element type method are efficient and consistent to analyze macroscale situation [26,27]. With these theories and method of analysis, the present study brings out several interesting features of wave propagation in rotating nanotubes. Recently, there has been great work done on vibration analysis of CNTS under rotation. Pradhan and Murmu [28], developed a single nonlocal beam model and applied to investigate the flap wise bending-vibration characteristics of a rotating nanocantilever. Differential quadrature method (DQM) was utilized and nondimensional nonlocal frequencies were obtained. They also discussed the effects of the nonlocal small-scale, angular velocity and hub radius on vibration characteristics of the nanocantilever. Murmu and Adhikari [29], investigated the nonlocal effects in bending-vibration of an initially prestressed single-walled carbon nanotube via nonlocal elasticity. The carbon nanotube was assumed to be attached to a molecular hub and was undergoing rotation. They also utilized the Differential quadrature method and the nonlocal bending frequencies of the rotating system were determined. The effects of the initial preload on vibration characteristics of rotating carbon nanotube were examined. Further, influence of nonlocal effects, angular velocities, hub radii and higher mode frequencies were also studied. There are some works done by the author [30–33] on the wave propagation in nanotubes without rotation. So it is worth looking into the wave dispersion characteristics of the nanotubes under rotation, which are found as blades of a nanoturbine and also important for practical development of nanomachines. Regarding longitudinal vibration of nanostructures, Wang et al. [34] explored the group velocities of longitudinal and flexural waves in single- and multi-walled carbon nanotubes (CNTs) within the framework of continuum mechanics. The effect of microsize of the structure was incorporated into the formulation of the problem using the second-order gradient of strain and a parameter of microstructure. It was revealed that the microstructure’s parameter plays an important role in the dispersion of both longitudinal and flexural waves. Furthermore, the proposed nonlocal models predicted that the cut-off wave number of the single- and multi walled carbon nanotubes would be in the order of 2 1010 m1 for both longitudinal and flexural waves. Aydogdu [35] investigated free axial vibration of uniform nanorods using nonlocal continuum theory of Eringen. The results showed that the nonlocal rod model overestimates the natural frequencies of the nanorod with respect to the classical
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S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25
one. Narendar and Gopalakrishnan [32] also studied longitudinal vibration of nanorods using a nonlocal bar model. The obtained results indicated that small scale parameter of the nonlocal model causes a certain band gap region in the longitudinal wave mode such that no wave propagation would occur. This issue was shown in the illustrated spectrum curves as the regions where the wave number tends to an infinite value. Recently, Filiz and Aydogdu [36] explored axial vibration of hetero junction CNTs in the framework of nonlocal continuum theory of Eringen. Effect of nonlocality and lengths of both CNTs and their segments were also examined in some detail. It was indicated that such nanotube structures could be designed for the desired frequencies by changing the lengths and the cross sections of the selected segments. Narendar and Gopalakrishnan [33], studied the ultrasonic wave dispersion characteristics of a nanorods based on Nonlocal strain gradient models (both second and fourth order). They derived the explicit expressions for wave numbers and the wave speeds of the nanorod. Their analysis shown that the fourth order strain gradient model gives approximate results over the second order strain gradient model for dynamic analysis. The second order strain gradient model gives a critical wave number at certain wave frequency, where the wave speeds are zero.They also explained the dynamic response behavior of nanorods based on both the strain gradient models. More recently, Narendar [37], studied the terahertz wave dispersion characteristics of nanorods based on the nonlocal elasticity incorporating the lateral-inertia under the umbrella of continuum mechanics theory. In that he shown that, the unstable second order strain gradient model can be replaced by considering the inertia gradient terms in the formulations. The effects of both the nonlocal scale and the diameter of the nanorod on spectrum curves are highlighted in that work. Murmu and Adhikari [38], studied the longitudinal vibration of a double-nanorod-system (DNRS) based on nonlocal elasticity theory. Their study highlights that the nonlocal effect considerably influences the axial vibration of DNRS. To date, the nonlocal continuum theory of Eringen has been exploited to study diverse problems of the nanotube structures, including free flexural vibration and wave propagation of nanotubes and graphene sheets [39–43], nanotubes conveying fluid [44–48], nanotube structures under a moving nanoparticle [49], and buckling analysis of nanotubes [50–52]. In the present work, the wave dispersion characteristics of a rotating nanotube modeled as SWCNT are studied using the spectral analysis. The rotating SWCNT is modeled as a Euler–Bernoulli beam. The governing partial differential equation for a uniform rotating beam is derived incorporating the nonlocal scale effects and the variable coefficient for the centrifugal term is replaced by the maximum centrifugal force. The rotating beam problem is now transformed to a case of beam subjected to an axial force. Even though this averaging seems to be a crude approximation, one can use this as a powerful model in analyzing the wave dispersion characteristics of the rotating CNT. The present paper is organized as follows. In Section 2, Eringen’s nonlocal elasticity theory is explained and the governing partial differential equation is derived for the rotating nanobeam. In Section 3, the wave dispersion analysis in rotating nanobeams is performed though spectral analysis. The expressions for the wave numbers are also derived. Section 4 gives a discussion on the numerical results obtained from both the local and nonlocality cases. The paper ends with some important observations and conclusions.
at x but also on the strain states at all other points x0 of the body. The most general form of the constitutive relation in the nonlocal elasticity type representation involves an integral over the entire region of interest. The integral contains a nonlocal kernel function, which describes the relative influences of the strains at various locations on the stress at a given location. The constitutive equations of linear, homogeneous, isotropic, non-local elastic solid with body forces are given by [19,20]
rij ;i þ qðfj u€j Þ ¼ 0 rij ðxÞ ¼
Z
V
ð1Þ
aðjx x0 j; nÞrcij ðx0 Þ dVðx0 Þ
rcij ¼ C ijkl ekl eij ðx0 Þ ¼
1 oui ðx0 Þ ouj ðx0 Þ þ 2 ox0j ox0i
!
ð2Þ ð3Þ ð4Þ
Eq. (1) is the equilibrium equation, where rij, q, fj and uj are the stress tensor, mass density, body force density and displacement vector at a reference point x in the body, respectively, at time t. Eq. (3) is the classical constitutive relation where rcij ðx0 Þ is the classical stress tensor at any point x0 in the body, which is related to the linear strain tensor ekl(x0 ) at the same point. Eq. (4) is the classical strain–displacement relationship. The only difference between Eqs. (1)–(4) and the corresponding equations of classical elasticity is the introduction of Eq. (2), which relates the global (or nonlocal) stress tensor rij to the classical stress tensor rcij ðx0 Þ using the modulus of nonlocalness. The modulus of nonlocalness or the nonlocal modulus a(jx xj,n) is the kernel of the integral Eq. (2) and contains parameters which correspond to the nonlocalness [20]. A dimensional analysis of Eq. (2) clearly shows that the nonlocal modulus has dimensions of (length)3 and so it depends on a characteristic length ratio a/‘ where a is an internal characteristic length (lattice parameter, size of grain, granular distance, etc.) and ‘ is an external characteristic length of the system (wavelength, crack length, size or dimensions of sample, etc.). Therefore the nonlocal modulus can be written in the following form:
a ¼ aðjx x0 j; nÞ; n ¼
e0 a ‘
ð5Þ
where e0 is a constant appropriate to the material and has to be determined for each material independently [20]. Making certain assumptions [20], the integro-partial differential equations of nonlocal elasticity can be simplified to partial differential equations. For example, Eq. (2) takes the following simple form:
ð1 n2 ‘2 r2 Þrij ðxÞ ¼ rcij ðxÞ ¼ C ijkl ekl ðxÞ
ð6Þ
2. Mathematical formulations
where Cijkl is the elastic modulus tensor of classical isotropic elasticity and eij is the strain tensor, where r2 denotes the second order spatial gradient applied on the stress tensor rij and n = e0a/‘. A method of identifying the small scaling parameter e0 in the nonlocal theory is not known yet. As defined by Eringen [20], e0 is a constant appropriate to each material. Eringen proposed e0 = 0.39 by the matching of the dispersion curves via nonlocal theory for plane wave and Born–Karman model of lattice dynamics at the end of the Brillouin zone (ka = p), where a is the distance between atoms and k is the wave number in the phonon analysis [20]. On the other hand, Eringen proposed e0 = 0.31 in his study [21] for Rayleigh surface wave via nonlocal continuum mechanics and lattice dynamics.
2.1. A review on nonlocal elasticity theory of solids
2.2. Nonlocal governing equations of rotating nanotube
This theory assumes that the stress state at a reference point x in the body is regarded to be dependent not only on the strain state
Nanotubes are central to new rotating devices such as miniature motor. A rotating CNT can be represented as a cantilever beam
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S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25
Fig. 2. (a) A schematic rotating carbon nanotube system with mathematical idealization to rotating nanobeam and (b) coordinate system and degree of freedom.
(see Fig. 2(a)) having displacements perpendicular to the plane of rotation. The detailed coordinate system on this rotating beam are shown in Fig. 2(b). Considering the elementary Euler–Bernoulli theory of beams, the axial and transverse displacement fields of a rotating beam can be represented
ow uðx; y; z; tÞ ¼ u0 z ox wðx; y; z; tÞ ¼ wðx; tÞ
ð7Þ ð8Þ
where w is transverse displacements of the point (x, 0) on the middle plane (i.e., z = 0) of the beam. The only nonzero strain of the Euler–Bernoulli beam theory, accounting for the von Kármán strain is
exx ¼
o2 Q ou ¼ EA ox2 ox 2 2o M M ðe0 aÞ ¼ EIje ox2
Q ðe0 aÞ2
ð16Þ
R where I ¼ A z2 dA is the moment of inertia of the beam cross section 2 and je ¼ ooxw2 is the curvature of the beam. With the help of the nonlocal constitutive relations and the equations of motion presented, the moment can be expressed in terms of the generalized displacements as, by substituting Eq. (16) into Eq. (11), we get
M ¼ EI
ou ou0 o2 w ¼ z 2 ox ox ox
ð15Þ
ð9Þ
" # o2 w o2 w o ow 2 TðxÞ þ ðe aÞ q A 0 ox2 ox ot2 ox
ð17Þ
This is also called as bending strain (or curvature). The equations of motion of the Euler–Bernoulli beam theory are given by
Substituting M from Eq. (17) into Eq. (11), we obtain the equation of motion of rotating nonlocal Euler beams as
oQ o2 u 0 ¼ qA 2 ox ot o2 M o ow o2 w þ TðxÞ ¼ qA 2 2 ox ox ox ot
" # 2 o4 w o2 w o ow 2 o TðxÞ EI 4 þ ðe0 aÞ qA 2 ox ox ox ox2 ot 2 o ow o w ¼ qA 2 þ TðxÞ ox ox ot
ð10Þ ð11Þ
where
Q¼
Z
rxx dA; M ¼
A
Z
zrxx dA
ð18Þ
ð12Þ
A
TðxÞ ¼
Z
L
qAX2 x dx
ð13Þ
x
where q is the mass density, A is the beam cross section area and X is the rotation speed. Using Eq. (6), we can express stress resultants of Euler–Bernoulli beam theory in terms of the strains in that theory. As opposed to the linear algebraic equations between the stress resultants and strains in a local theory, the nonlocal constitutive relations lead to differential relations involving the stress resultants and the strains. In the following, we present these relations for homogeneous isotropic beams. The nonlocal constitutive relations in Eq. (6) take the following special form for beams:
o2 rxx rxx ðe0 aÞ ¼ Eexx ox2 2
ð14Þ
where E is the Young’s modulus of the beam. Using Eqs. (12) and (14), we have
Nondimensional wavenumber (k*h)
and rxx is the axial stress on the yz-section in the direction of x, Q is the axial force, M is the bending moment and T(x) is the axial force due to centrifugal stiffening and is given as 4 3.5 3 2.5 2 1.5 1 0.5 0 50 40
20 30
15 20
Rotating Speed (Ω/ωstr)
10 10
5 0
0
Frequency (ω/ωstr)
Fig. 3. Wavenumber dispersion in rotating nanotube for s = e0a/L = 0 for different values of X/xstr.
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S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25 10
10 0
9
τ = e0a/L = 0.1
8
τ = e a/L = 0.2
7
τ = e a/L = 0.3
0 0
τ = e0a/L = 0.5
6 5 4 3 2 1 0
τ = e0a/L = 0
9
Nondimensional wavenumber (k × h)
Nondimensional wavenumber (k × h)
τ = e a/L = 0
τ = e a/L = 0.1 0
8
τ = e a/L = 0.2
7
τ = e0a/L = 0.3
0
τ = e a/L = 0.5 0
6 5 4 3 2 1
0
5
10
15
0
20
0
5
Frequency (ω/ω )
(a)
(b)
str
20
15
20
10 τ = e a/L = 0 0
9
τ = e0a/L = 0.1
8
τ = e0a/L = 0.2
7
τ = e0a/L = 0.3
Nondimensional wavenumber (k × h)
Nondimensional wavenumber (k × h)
15 str
10
τ = e a/L = 0.5 0
6 5 4 3 2 1 0
10
Frequency (ω/ω )
9
τ = e a/L = 0
8
τ = e0a/L = 0.1
0
τ = e a/L = 0.2 0
7
τ = e a/L = 0.3 0
6
τ = e0a/L = 0.5
5 4 3 2 1
0
5
10
15
0
20
0
5
10
Frequency (ω/ωstr)
Frequency (ω/ωstr)
(c)
(d)
Nondimensional wavenumber (k × h)
10 9
τ = e a/L = 0
8
τ = e a/L = 0.1
0 0
1.2
τ = e a/L = 0.2 0
7
τ = e0a/L = 0.3
6
1
τ = e0a/L = 0.5
0.8
5 0.6
4
0.4
3 2
10
12
14
16
18
20
1 0
0
5
10
15
20
Frequency (ω/ωstr)
(e) Fig. 4. Wave dispersion in rotating nanotube for various values of nonlocal scale s = 0, 0.1, 0.2, 0.3, 0.5. For the various nondimensional rotating speeds: (a) X X X X xstr ¼ 10, (c) xstr ¼ 30, (d) xstr ¼ 50 and (e) xstr ¼ 100.
We assumed a uniform rotating beam and replace T(x) by the maximum force (at the root, i.e. at x = 0)
T max ¼
Z
0
L
qAX2 x dx ¼
qAX2 L2 2
ð19Þ
X
xstr
¼ 0, (b)
This allows us to represent the governing equation as a constant coefficient nonlocal partial differential equation. Finally, the nonlocal governing differential equation for transverse displacement (w(x, t)) of a rotating cantilever beam is derived as
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S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25
o4 w o2 w o4 w o2 w EI 4 T max 2 þ T max ðe0 aÞ2 4 þ qA 2 ox ox ox ot 4 2 o w qAðe0 aÞ 2 2 ¼ 0 ot x
"
ð20Þ
For analyzing the dispersion characteristics of waves in nanotubes, we assume that a harmonic type of wave solution for the displacement field w(x, t) and it can be expressed in complex form [26,27] as N X
^ ðx; xn Þejðkxxn tÞ w
^ ðx; xn Þ is the frequency domain amplitude of the flexural where w deformation of CNTs, k is the wave number and xn is the angular frequency of the wavep motion at nth sampling point, N is the numffiffiffiffiffiffiffi ber of amples and j ¼ 1. Eliminating the time variable from Eq. (20) using the above spectral approximation of the displacement gives,
"
^ ðx; xn Þ ^ ðx; xn Þ ^ ðx; xn Þ o4 w o2 w o4 w T max þ T max ðe0 aÞ2 4 2 ox ox ox4 n¼1 # ^ ðx; xn Þ jxn t o2 w b ðx; xn Þ þ qAðe0 aÞ2 x2 ¼0 e qAx2 w ox2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 2 2 2 2 u 2 u 1 X 2 s2 x 2 1 X x X x 2 21 s þ 4 1 þ s u 2 xstr 2 xstr 2 xstr xstr xstr xstr u k ¼ u 2 t 2 1 þ s2 12 xXstr ð29Þ
The wave numbers are mainly a function of the nonlocal scaling parameter (e0a), rotational speed of the beam (X) and the wave circular frequency. speeds, Phase The
corresponding wave namely,
speed C p ¼ Real xk and Group speed C g ¼ Real ookx , are obtained from Eq. (29). A detail discussion on these parameters is presented in the next section. 4. Numerical experiments, results and discussion
EI
ð22Þ
The Eq. (22) must be satisfied for each n and hence can be written as 4 2 4 ^ ^ ^ d w d w d w b T max 2 þ T max ðe0 aÞ2 4 qAx2 w 4 dx dx dx 2 ^ d w þ qAðe0 aÞ2 x2 2 ¼ 0 dx
EI
ð23Þ
For the sake of simplicity in the analysis we express this equation in a non-dimensional form. Now define new variables as 2 3 ^ dw ~ ^ dw ^ d2 w ~ ^ x d w d w w ~ ¼ ; ~x ¼ ; w ¼ ; L 2 ¼ 2 ; L2 3 ~ ~ L dx L dx dx dx dx 3 4 ~ ^ d4 w ~ d w d w e0 a ¼ 3 ; L3 4 ¼ 4 ; s ¼ ~ dx dx d~x L
ð24Þ
The non-dimensional form of the Eq. (23) is 4 2 4 ~ ~ ~ 1 d w 1d w 2 1 d w ~ T þ T ðe aÞ qAx2 Lw max max 0 3 o~ 3 d~ 4 2 4 ~ L x o x x L L 2 ~ 1d w L d~x2
¼0
ð25Þ
Rearranging and defining a new variable
xstr ¼
sffiffiffiffiffiffiffi EI L2 qA 1
ð26Þ
Eq. (25) changes to
"
2 #
1 X 1 þ s2 2 xstr x 2~ w¼0
xstr
4 ~ 1 X 2 d2 w ~ ~ d w x 2 d2 w 2 þ s xstr d~x2 d~x4 2 xstr d~x2 ð27Þ
Since the differential equation is a constant coefficient one, it has f ejkx , on substituting this solution ~ ¼W the solution of the form w f –0), in Eq. (27) we get (for W
0
þ qAðe0 aÞ2 x2
For Numerical Experiments, a nanobeam is assumed as (5,5) SWCNT is considered and the diameter of the SWCNT is d = 0.675 nm, length L = 10d, Young’s modulus E = 5.5 TPa, and the density q = 2300 kg/m3. The spectrum curves (i.e., non-dimensional wave number (k h), h is the thickness of the nanotube vs non-dimensional wave frequency (x/xstr)) for the rotating SWCNT are shown in Fig. 3 for different values of the non-dimensional rotational speed X/xstr for s = e0a/L = 0 (i.e., local elasticity calculation). It can be seen that for non-rotating CNT (i.e., X/xstr = 0), the non-dimensional flexural wave number shows a nonlinear variation with wave frequency, i.e., the waves will change their shape as they propagate. As the rotational speed of the CNT increases, the nondimensional wave numbers tends to become non-dispersive in nature as shown in Fig. 3. It means that the waves will not change their shape as they propagate in the medium. Also, the wave number shows an inverse dependence on rotation speed. For a nonrotating CNT, the spectrum relation is dispersive in nature. But for a rotating beam, at higher speeds, the above nonlinear relation shifts to a linear nature due to the relatively negligible contribution from the x term, especially for lower values of x. That is, the
Phase Speed (Cp /C0) & Group Speed (Cg/C )
EI
where k is the wave number. This is the dispersion/characteristic equation of the rotating uniform beams. One can solve for the wave numbers as
ð21Þ
n¼1
N X
ð28Þ
xstr
3. Wave dispersion analysis in nanotubes undergoing rotation
wðx; tÞ ¼
" 2 # 2 # 1 X 1 X x 2 2 4 2 k þ k 1þs s xstr 2 xstr 2 xstr x 2 ¼0 2
0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 50 40
20 30
15 20
10 Rotating Speed (Ω/ωstr)
10 5 0
0
Frequency (ω/ω ) str
Fig. 5. Phase speed (thick lines) and group speed (thin lines) dispersion in rotating nanotube for s ¼ e0La ¼ 0 for different values of xXstr .
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S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25 0.12 Phase: τ = 0 Group: τ = 0 Phase: τ = 0.1 Group: τ = 0.1 Phase: τ = 0.2 Group: τ = 0.2 Phase: τ = 0.3 Group: τ = 0.3 Phase: τ = 0.5 Group: τ = 0.5
0.1
0.08
Phase Speed (Cp /C0) & Group Speed (Cg/C0)
0
Phase Speed (Cp /C0) & Group Speed (Cg/C )
0.12
0.06
0.04
0.02
0
0
2
4
6
8
10
12
14
16
18
Phase: τ = 0 Group: τ = 0 Phase: τ = 0.1 Group: τ = 0.1 Phase: τ = 0.2 Group: τ = 0.2 Phase: τ = 0.3 Group: τ = 0.3 Phase: τ = 0.5 Group: τ = 0.5
0.1
0.08
0.06
0.04
0.02
0
20
Frequency (ω/ω )
0
2
4
6
8 10 12 Frequency (ω/ω )
(a)
(b)
str
Phase Speed (Cp /C0) & Group Speed (Cg/C0)
0
Phase Speed (Cp /C0) & Group Speed (Cg/C )
16
18
20
14
16
18
20
0.2
0.14
0.12
0.1
0.08
0.06
Phase: τ = 0 Group: τ = 0 Phase: τ = 0.1 Group: τ = 0.1 Phase: τ = 0.2 Group: τ = 0.2 Phase: τ = 0.3 Group: τ = 0.3 Phase: τ = 0.5 Group: τ = 0.5
0.04
0.02
0
14
str
0
2
4
0.18 0.16 0.14 0.12 0.1 Phase: τ = 0 Group: τ = 0 Phase: τ = 0.1 Group: τ = 0.1 Phase: τ = 0.2 Group: τ = 0.2 Phase: τ = 0.3 Group: τ = 0.3 Phase: τ = 0.5 Group: τ = 0.5
0.08 0.06 0.04 0.02
6
8 10 12 Frequency (ω/ω )
14
16
18
0
20
0
2
4
6
8 10 12 Frequency (ω/ω )
str
str
(c)
(d)
Phase Speed (Cp /C0) & Group Speed (Cg/C0)
0.4 0.35 0.3 0.25 0.2 Phase: τ = 0 Group: τ = 0 Phase: τ = 0.1 Group: τ = 0.1 Phase: τ = 0.2 Group: τ = 0.2 Phase: τ = 0.3 Group: τ = 0.3 Phase: τ = 0.5 Group: τ = 0.5
0.15 0.1 0.05
0
0
2
4
6
8 10 12 Frequency (ω/ω )
14
16
18
20
str
(e) Fig. 6. Phase speed (thick lines) and group speed (thin lines) dispersion in rotating nanotube for for different values of X X xstr ¼ 50 and (e) xstr ¼ 100.
variation of k is dominated by the centrifugal force term at high X/ xstr.
X
xstr .
Here (a)
X
xstr
¼ 0, (b)
X
xstr
¼ 10, (c)
X
xstr
¼ 30, (d)
Fig. 4 shows the spectrum curves for rotating and nonrotating CNTs for various values of the nonlocal scaling parameter.
24
S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25
Fig. 4(a) shows that as we move from local elasticity to nonlocal elasticity solution, the spectrum curve becomes linear at higher values of the wave frequency. The wave numbers also increases as the nonlocal scaling parameter increases. The matching of local and nonlocal solution is limited only upto <0.2 THz frequency. After this frequency, the difference between the wave numbers predicted is very large. If the rotational speed of the CNT increases from X/xstr = 0 to X/xstr = 10 (Fig. 4(b)), the spectrum curve is slightly nonlinear as compared to nonrotating case. The wave numbers obtained from local and nonlocal cases is same upto 0.45 THz frequency. The wave numbers are showing an increasing tendency as the nonlocal parameter increases. If the rotational speed of the CNT increases to very high values like X/xstr = 30, 50 and 100, the local and nonlocal calculations are almost similar upto 1 THz, 1.6 THz and 2.5 THz frequencies, respectively (see Figs. 4(c)–(e)). As the rotational speed of the CNT increases to very high values, the nonlocal scaling parameter effect on the spectrum curves is negligible. It means that if the CNT rotates at very high speeds, the local elasticity and nonlocal elasticity calculations give almost similar spectrum relations. It can also be observed that as the rotational speed of the CNT increases, the wave numbers become very small and the dispersive nature changes to non-dispersive nature (see Fig. 4). The non-dimensional phase speed ffi p/C0) and non-dimensional pffiffiffiffiffiffiffiffi(C group speed (Cg/C0), where C 0 ¼ E=q, dispersions of the rotating CNT are shown in Fig. 5, obtained from the local elasticity calculations (i.e., s = 0). Thick lines represent the phase speed variation and the thin lines show the group speed variation. It can be seen that the phase speed of the rotating CNTs is higher than the group speed. Because of the nonlinear relation of the wave number with wave frequency, for non-rotating CNT, the phase and group speeds also shows a nonlinear variation with frequency. As the rotational speed of the CNT increase to higher values, both the speeds will saturate to a constant velocity, because of the linear variation of the wave number with wave frequency. The difference between the phase and group speeds of the rotating CNTs is negligible at higher rotational speeds as shown in Fig. 5, which is a characteristic of any non-dispersive system. In the limiting case, we can say that they become equal and become constant for all wave frequencies. For a nonrotating beam, both phase and group speeds are dispersive and shows that the speeds approach infinity for very high frequencies. This unreasonable limit is due to the limitation of Euler–Bernoulli beam theory. Fig. 6 shows the non-dimensional phase speed and non-dimensional group speed variation with both the wave frequency and the nonlocal scaling parameter for rotations and non-rotating CNTs. Fig. 6(a) shows that, for non-rotating CNT, the phase and group speeds will decrease as the nonlocal scaling parameter increases because the wave numbers are increasing with increase in s (see Fig. 4). Also, the difference between the phase and group speeds dips at higher values of s. As the rotational speed of the CNT increases, both phase and group speeds will also increase as shown in Figs. 6(b)–(e). For small rotational speeds, and large values of s, both speeds show a decrease in nature at smaller frequencies and they become constant at higher wave frequencies. Such difference will also vanish at higher rotational speeds as shown in Fig. 6(e). On the other hand, one more interesting feature of the nonlocality is that, the difference between both the wave speeds is considerable at smaller rotational speeds and s and also at the higher rotational speeds and s and it can be clearly seen from Fig. 6. The present work illustrated here can be extended to study the wave propagation analysis of rotating graphene sheets. This would be helpful in understanding the wave dispersion characteristics of future rotating nanomachines. One such nanomachine is the nanoturbine. For efficient designing of these nanomachines, proper
understanding of its mechanical behavior is required. The nanoturbine is characterized by a rotating blades modeled as a rotating nanotube. Rotating nanocantilever rotates at an angular velocity about the hub axis. The performances of these nanoturbines are extremely dependent on dynamic properties of their beam-like elements. In summary, this study would be useful in perspective to simple mathematical modeling and understanding the scale effects in rotating nanostructures. 5. Conclusions In the present work, the wave dispersion characteristics of a rotating nanotube (such as SWCNT) are studied using the spectral analysis and nonlocal scale theory. The rotating nanotube is modeled as a Euler–Bernoulli beam. The governing partial differential equation for a uniform rotating beam is derived incorporating the nonlocal scale effects and a powerful model is derived in analyzing the wave dispersion characteristics of the rotating nanotube. Some of the beautiful features of the wave behavior in rotating nanotubes are observed. It has been shown that the dispersive flexural wave tends to behave non-dispersively at very high rotation speeds. Understanding the dynamic behavior of rotating nanostructures is important for practical development of nanomachines. At the nanoscale, the nonlocal effects often become more prominent. Such observations are helpful in designing the nanomotors and the other CNT based rotational nano-devices where CNT acts as a basic element. Acknowledgment The authors would like to thank the reviewers for their comments and suggestions to improve the clarity of this article. References [1] Iijima S. Nature 1991;354:56. [2] Saito R, Dresselhaus G, Dresselhaus M. Physical properties of carbon nanotubes. London: Imperial College Press; 1998. [3] Fennimore A, Yuzvinsky TD, Han WQ, Fuhrer MS, Cumings J, Zettl A. Nature 2003;424:408. [4] Bourlon B, Glattli DC, Miko C, Forro L, Bachtold A. Nano Lett 2004;4:709. [5] Thess A, Lee R, Nikolaev P, Dai H, Petit P, Robert J, Xu C, Lee YH, Kim SG, Rinzler AG, Colbert DT, Scuseria GE, Tomanek D, Fischer JE, Smalley RE. Science 1996;273:483. [6] Journet C, Maser WK, Bernier P, de la chapelle ALML, Lefrant S, Deniard P, Lee R, Fischer JE. Nature 1997;388:756. [7] Freedonia H. World Nanotubes to 2009, Freedonia, Cleveland, 2006. [8] Zheng LX, O’Connell MJ, Doorn SK, Liao XZ, Zhao YH, Akhadov EA, Hobauer MA, Roop BJ, Jia QX, Dye RC, Peterson DE, Huang SM, Liu J, Zhu YT. Nat Mater 2004;3:673. [9] Kral P, Sadeghpour HR. Phys Rev B 2002;65:161401. [10] Krim J, Solina DH, Chiarello R. Phys Rev Lett 1991;66:181. [11] Zhang S, Liu WK, Ruo RS. Nano Lett 2004;4:293. [12] Drexler KE. Nanosystems: molecular machinery, manufacturing, and computation. New York: Wiley; 1992. [13] Rao PVM, Singh M, Jain V. Nanotechnology 2005;2:250. [14] Robertson DH, Dunlap BL, Brenner DW, Mintmire JW, White CT. Fullerene/ tubule based hollow carbon nano-gears, MRS symposia proceedings no. 349. Materials Research Society, Pittsburgh, 1994. p. 283. [15] Han J, Globus A, Jaffe R, Deardorff G. Nanotechnology 1997;8:95. [16] Jaffe R, Han J, Globus A. NASA Technical Report No. NASA-97-006 1997. [17] Srivastava D. Nanotechnology 1997;8:186. [18] Lohrasebi A, Raffi-Tabar H. J Mol Graph Modell 2008;27:116. [19] Eringen AC, Edelen DGB. Int J Eng Sci 1972;10:233. [20] Eringen AC. J Appl Phys 1983;54:4703. [21] Eringen AC. Int J Eng Sci 1972;10:425. [22] Eringen AC. Nonlocal continuum field theories. Berlin: Springer; 2002. [23] Wong EW, Sheehan PE, Lieber CM. Science 1997;277:1971. [24] Cleland AN. Foundation of nanomechanics, Springer, 2002. [25] Born M, Huang K. Dynamical theory of crystal lattices. Oxford: Clarendon Press; 2002. [26] Doyle JF. Wave propagation in structures. New York: Springer-Verlag Inc.; 1997. [27] Gopalakrishnan S, Chakraborty A, Roy Mahapatra D. Spectral finite element method. Springer Verlag London Ltd; 2008.
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Pradhan SC, Murmu T. Physica E 2010;42:1944–9. Murmu T, Adhikari S. J Appl Phys 2010;108:123507. Narendar S, Gopalakrishnan S. Comput Mater Sci 2009;47:526. Narendar S, Gopalakrishnan S. Physica E 2010;42:1706. Narendar S, Gopalakrishnan S. Physica E 2010;42:1601. Narendar S, Gopalakrishnan S. J Appl Phys 2010;107:084312. Wang L, Guo W, Hu H. Proc Roy Soc A 2008;464:1423. Aydogdu M. Physica E 2009;41:861–4. Filiz S, Aydogdu M. Comput Mater Sci 2010;49(3):619. Narendar S. Physica E 2011;43:1015–20. Murmu T, Adhikari S. Physica E 2010. doi:10.1016/j.physe.2010.08.023. Hu YG, Liew KM, Wang Q, He XQ, Yakobson BI. J Mech Phys Solids 2008;56(12):3475.
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Murmu T, Pradhan SC. Comput Mater Sci 2009;46(4):854. Aydogdu M. Arch Appl Mech 2008;78(9):711. Kiani K. Int J Eng Sci 2010;52(10):1343. Narendar S, Gopalakrishnan S. Comput Mater Sci 2010;49:734. Lee HL, Chang WJ. Physica E 2009;41(4):529. Wang L. Comput Mater Sci 2009;45(2):584. Wang YZ, Li FM, Kishimoto K. Comput Mater Sci 2010;48(2):413. Kuang YD, He XQ, Chen CY, Li GQ. Comput Mater Sci 2009;45(4):875. Kiani K, Mehri B. J Sound Vib 2010;329(11):2241. Kiani K. Physica E 2010;42(9):2391. Hao MJ, Guo XM, Wang Q. Eur J Mech A: Solids 2010;29(1):49. Narendar S, Gopalakrishnan S. Physica E 2011;43:1185–91. Shen HS, Zhang CL. Compos Struct 2010;92(5):1073.
25
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Results in Physics journal homepage: www.elsevier.com/locate/rinp
Nonlocal wave propagation in rotating nanotube S. Narendar a,⇑, S. Gopalakrishnan b a b
Defence Research and Development Laboratory, Kanchanbagh, Hyderabad 500 058, India Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560 012, India
a r t i c l e
i n f o
Article history: Received 28 April 2011 Accepted 16 June 2011 Available online 13 July 2011 Keywords: Nanobeam Wavenumber Rotational speed Frequency Centrifugal force Nonlocal elasticity
a b s t r a c t The present work deals with the wave dispersion behavior of a rotating nanotube using the nonlocal elasticity theory. The rotating nanobeam is modeled as an Euler–Bernoulli theory. The governing partial differential equation for a uniform rotating beam is derived incorporating the nonlocal scale effects. The spatial variation in centrifugal force is modeled in an average sense. Even though this averaging seems to be a crude approximation, one can use this as a powerful model in analyzing the wave dispersion characteristics of the rotating nanobeam. Spectrum and dispersion curves are obtained as a function of rotating speed and nonlocal scaling parameter. It has been shown that the dispersive flexural wave tends to behave non-dispersively at very high rotation speeds. Understanding the dynamic behavior of rotating nanostructures is important for practical development of nanomachines. At the nanoscale, the nonlocal effects often become more prominent. The numerical results are simulated for a rotating nanobeam as a waveguide. The results can provide useful guidance for the study and design of the next generation of nanodevices such as blades of a nanoturbine, nanogears, nanoscale molecular bearings etc, that make use of the wave propagation properties of rotating single-walled carbon nanotubes. Ó 2011 Published by Elsevier B.V.
1. Introduction Carbon nanotubes were first observed and identified as such by Iijima using transmission electron microscopy in 1991 [1]. As we see, nanotubes are cylindrical molecules which may consist of several concentric sheets of graphene (Multi Walled Carbon Nanotubes – MWCNTs). Since the discovery of Iijima, research on carbon nanotubes has become a paramount activity. Their geometry was studied and it was observed that their diameter can range from a few to hundreds of nanometers and their length from a few to tens of micrometers [2]. The nanotubes are promising materials in a number of new areas. These include molecular electronics where the nanotubes can be used both for wiring and as electronic devices after modification of the nanotube itself. They can also be used as parts of mechanical devices at the nanoscale such as the shaft of nanomotors [3,4]. Moreover, their high strength and flexibility are exploited to reinforce materials such as cement. Many production methods were developed such as the laser vaporization method combined with transition metal catalyst [5] or the carbon arc method [6]. Nowadays, dozens of companies are specialized in the production of single- and multi-walled carbon nanotubes. They can produce up to hundreds of grams of SWCNTs per day and
⇑ Corresponding author. Tel.: +91 88976 25977, +91 99493 91823; fax: +91 40 24342309. E-mail addresses: [email protected], [email protected] (S. Narendar), [email protected] (S. Gopalakrishnan). 2211-3797/$ - see front matter Ó 2011 Published by Elsevier B.V. doi:10.1016/j.rinp.2011.06.002
hundreds of kilograms of MWCNTs. Forecasts predict that global demand for nanotubes will expand rapidly [7]. Flat panel displays for both computers and televisions will be the first widely commercialized application. It was observed that there is a wide variety of carbon nanotubes [2]. Their diameters vary between 0.7 and 10 nm, though most of the observed tubes have diameters smaller than 2 nm. The length of single-walled nanotubes goes from a few nanometers up to 20 cm [8]. Moreover nanotubes can be chiral or achiral. These varieties of SWCNT’s come from the fact that the nanotubes are constructed by the folding of a graphite plane on itself, and have many ways of folding arise. Recently, Kral and Sadeghpour [9] and Krim et al. [10] showed the possibility to spin nanotubes with circularly polarized light, further motivating the usage of nanotubes as possible axis of rotation at the nanoscale. However, no systematic analytical and numerical study of rotation has yet been done, except the work of Zhang et al. [11], where nonequilibrium molecular dynamics simulations were carried out to calculate the energy dissipation rate during the rotational motion. In their work, the usage of the thermostat would however add dissipation which is not intrinsic to the system. In their work, simulations were performed by Hamiltonian dynamics in the microcanonical ensemble, and in that process they avoided additional sources of dissipation. Because of future promising exploration of nanotechnology, focus is being put in the miniaturization of mechanical and electromechanical devices. Attention is sought toward the development
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S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25
Fig. 1. Carbon nanotube gears: molecular diagrams [17].
of nanodevices and nanomachines [12]. Nanomachines are systems in the nanometer realm with moving parts [13].Nanostructures undergoing rotation include nanoturbines, nanoscale molecular bearings, shaft and gear, and multiple gear systems. These nanostructure machines are expected to receive considerable attention in the near future.Researchers have thus reported the feasibility of nanoscale rotating structures. Examples include study of molecular gears [12], fullerene gears [14], and carbon nanotubes gears [15,16]. Srivastava [17] has reported the rotational dynamics of carbon nanotubes and carbon nanotubes gears under a single applied laser field. A typical carbon nanotubes gear is shown in Fig. 1. Lohrasebi and Tabar [18] carried out computational modeling of rotating nanomotor. Dynamics of the rotary nanomotor was simulated using stochastic molecular dynamics (MD) method. Zhang et al. [11] carried out atomistic simulations of doublewalled carbon nanotubes as rotational bearings. Recently Fennimore et al. [3] reported the feasibility of rotating nanostructures. They showed the construction and successful operation of a nanoscale electromechanical actuator. The rotating nanostructural system consists of a rotatable metal plate, with a multi-walled carbon nanotube serving as the key motion-enabling element. For efficient design of these rotating nanomachines,proper understanding of its mechanical behavior such as bending, vibration, and buckling is required. The development of simplified models for the dynamics of complex nano-technological systems is thus necessary. This is because for many cases, fully atomistic simulations would be computationally expensive and prohibitive. The length scales associated with nanostructures like CNTs are such that to apply any classical continuum techniques, we need to consider the small length scales such as lattice spacing between individual atoms, surface properties, grain size etc. This makes a physically consistent classical continuum model formulation very challenging. The Eringen’s nonlocal elasticity theory [19–21] is useful tool in treating phenomena whose origins lie in the regimes smaller than the classical continuum models. In this theory, the internal size or scale could be represented in the constitutive equations simply as material parameters. Such a nonlocal continuum mechanics has been widely accepted and has been applied to many problems including wave propagation, dislocation, crack problems, etc [22]. Recently, there has been great interest in the application of nonlocal continuum mechanics for modeling and analysis of CNTs. Nonlocal elasticity theory gives a good prediction for modeling the behavior of nanostructures. Literature shows that the theory of nonlocal elasticity is receiving increasing attention for efficient analysis of nanostructures which include nanobeams, nanoplates, nanorings, carbon nanotubes, graphenes and microtubules. Wave propagation analyses of CNTs are relevant due to their various applications [23] which include sensing superconductivity, transport and optical phenomena. The actual value of the fre-
quency (THz in this case) does not justify the treatment of high frequency analysis. It is the number of modes (i.e., the size of wavelength) what matters, not the frequency value itself. In nanoscale structures one does not expect small wavelength vibration with frequencies going up to tens or hundreds of THz. Therefore, the scientific need for spectral methods in the nano-scale is much less compared to usual macro-scale structures. Lattice dynamics for direct observation of phonons [24,25] and spectral finite element type method are efficient and consistent to analyze macroscale situation [26,27]. With these theories and method of analysis, the present study brings out several interesting features of wave propagation in rotating nanotubes. Recently, there has been great work done on vibration analysis of CNTS under rotation. Pradhan and Murmu [28], developed a single nonlocal beam model and applied to investigate the flap wise bending-vibration characteristics of a rotating nanocantilever. Differential quadrature method (DQM) was utilized and nondimensional nonlocal frequencies were obtained. They also discussed the effects of the nonlocal small-scale, angular velocity and hub radius on vibration characteristics of the nanocantilever. Murmu and Adhikari [29], investigated the nonlocal effects in bending-vibration of an initially prestressed single-walled carbon nanotube via nonlocal elasticity. The carbon nanotube was assumed to be attached to a molecular hub and was undergoing rotation. They also utilized the Differential quadrature method and the nonlocal bending frequencies of the rotating system were determined. The effects of the initial preload on vibration characteristics of rotating carbon nanotube were examined. Further, influence of nonlocal effects, angular velocities, hub radii and higher mode frequencies were also studied. There are some works done by the author [30–33] on the wave propagation in nanotubes without rotation. So it is worth looking into the wave dispersion characteristics of the nanotubes under rotation, which are found as blades of a nanoturbine and also important for practical development of nanomachines. Regarding longitudinal vibration of nanostructures, Wang et al. [34] explored the group velocities of longitudinal and flexural waves in single- and multi-walled carbon nanotubes (CNTs) within the framework of continuum mechanics. The effect of microsize of the structure was incorporated into the formulation of the problem using the second-order gradient of strain and a parameter of microstructure. It was revealed that the microstructure’s parameter plays an important role in the dispersion of both longitudinal and flexural waves. Furthermore, the proposed nonlocal models predicted that the cut-off wave number of the single- and multi walled carbon nanotubes would be in the order of 2 1010 m1 for both longitudinal and flexural waves. Aydogdu [35] investigated free axial vibration of uniform nanorods using nonlocal continuum theory of Eringen. The results showed that the nonlocal rod model overestimates the natural frequencies of the nanorod with respect to the classical
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S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25
one. Narendar and Gopalakrishnan [32] also studied longitudinal vibration of nanorods using a nonlocal bar model. The obtained results indicated that small scale parameter of the nonlocal model causes a certain band gap region in the longitudinal wave mode such that no wave propagation would occur. This issue was shown in the illustrated spectrum curves as the regions where the wave number tends to an infinite value. Recently, Filiz and Aydogdu [36] explored axial vibration of hetero junction CNTs in the framework of nonlocal continuum theory of Eringen. Effect of nonlocality and lengths of both CNTs and their segments were also examined in some detail. It was indicated that such nanotube structures could be designed for the desired frequencies by changing the lengths and the cross sections of the selected segments. Narendar and Gopalakrishnan [33], studied the ultrasonic wave dispersion characteristics of a nanorods based on Nonlocal strain gradient models (both second and fourth order). They derived the explicit expressions for wave numbers and the wave speeds of the nanorod. Their analysis shown that the fourth order strain gradient model gives approximate results over the second order strain gradient model for dynamic analysis. The second order strain gradient model gives a critical wave number at certain wave frequency, where the wave speeds are zero.They also explained the dynamic response behavior of nanorods based on both the strain gradient models. More recently, Narendar [37], studied the terahertz wave dispersion characteristics of nanorods based on the nonlocal elasticity incorporating the lateral-inertia under the umbrella of continuum mechanics theory. In that he shown that, the unstable second order strain gradient model can be replaced by considering the inertia gradient terms in the formulations. The effects of both the nonlocal scale and the diameter of the nanorod on spectrum curves are highlighted in that work. Murmu and Adhikari [38], studied the longitudinal vibration of a double-nanorod-system (DNRS) based on nonlocal elasticity theory. Their study highlights that the nonlocal effect considerably influences the axial vibration of DNRS. To date, the nonlocal continuum theory of Eringen has been exploited to study diverse problems of the nanotube structures, including free flexural vibration and wave propagation of nanotubes and graphene sheets [39–43], nanotubes conveying fluid [44–48], nanotube structures under a moving nanoparticle [49], and buckling analysis of nanotubes [50–52]. In the present work, the wave dispersion characteristics of a rotating nanotube modeled as SWCNT are studied using the spectral analysis. The rotating SWCNT is modeled as a Euler–Bernoulli beam. The governing partial differential equation for a uniform rotating beam is derived incorporating the nonlocal scale effects and the variable coefficient for the centrifugal term is replaced by the maximum centrifugal force. The rotating beam problem is now transformed to a case of beam subjected to an axial force. Even though this averaging seems to be a crude approximation, one can use this as a powerful model in analyzing the wave dispersion characteristics of the rotating CNT. The present paper is organized as follows. In Section 2, Eringen’s nonlocal elasticity theory is explained and the governing partial differential equation is derived for the rotating nanobeam. In Section 3, the wave dispersion analysis in rotating nanobeams is performed though spectral analysis. The expressions for the wave numbers are also derived. Section 4 gives a discussion on the numerical results obtained from both the local and nonlocality cases. The paper ends with some important observations and conclusions.
at x but also on the strain states at all other points x0 of the body. The most general form of the constitutive relation in the nonlocal elasticity type representation involves an integral over the entire region of interest. The integral contains a nonlocal kernel function, which describes the relative influences of the strains at various locations on the stress at a given location. The constitutive equations of linear, homogeneous, isotropic, non-local elastic solid with body forces are given by [19,20]
rij ;i þ qðfj u€j Þ ¼ 0 rij ðxÞ ¼
Z
V
ð1Þ
aðjx x0 j; nÞrcij ðx0 Þ dVðx0 Þ
rcij ¼ C ijkl ekl eij ðx0 Þ ¼
1 oui ðx0 Þ ouj ðx0 Þ þ 2 ox0j ox0i
!
ð2Þ ð3Þ ð4Þ
Eq. (1) is the equilibrium equation, where rij, q, fj and uj are the stress tensor, mass density, body force density and displacement vector at a reference point x in the body, respectively, at time t. Eq. (3) is the classical constitutive relation where rcij ðx0 Þ is the classical stress tensor at any point x0 in the body, which is related to the linear strain tensor ekl(x0 ) at the same point. Eq. (4) is the classical strain–displacement relationship. The only difference between Eqs. (1)–(4) and the corresponding equations of classical elasticity is the introduction of Eq. (2), which relates the global (or nonlocal) stress tensor rij to the classical stress tensor rcij ðx0 Þ using the modulus of nonlocalness. The modulus of nonlocalness or the nonlocal modulus a(jx xj,n) is the kernel of the integral Eq. (2) and contains parameters which correspond to the nonlocalness [20]. A dimensional analysis of Eq. (2) clearly shows that the nonlocal modulus has dimensions of (length)3 and so it depends on a characteristic length ratio a/‘ where a is an internal characteristic length (lattice parameter, size of grain, granular distance, etc.) and ‘ is an external characteristic length of the system (wavelength, crack length, size or dimensions of sample, etc.). Therefore the nonlocal modulus can be written in the following form:
a ¼ aðjx x0 j; nÞ; n ¼
e0 a ‘
ð5Þ
where e0 is a constant appropriate to the material and has to be determined for each material independently [20]. Making certain assumptions [20], the integro-partial differential equations of nonlocal elasticity can be simplified to partial differential equations. For example, Eq. (2) takes the following simple form:
ð1 n2 ‘2 r2 Þrij ðxÞ ¼ rcij ðxÞ ¼ C ijkl ekl ðxÞ
ð6Þ
2. Mathematical formulations
where Cijkl is the elastic modulus tensor of classical isotropic elasticity and eij is the strain tensor, where r2 denotes the second order spatial gradient applied on the stress tensor rij and n = e0a/‘. A method of identifying the small scaling parameter e0 in the nonlocal theory is not known yet. As defined by Eringen [20], e0 is a constant appropriate to each material. Eringen proposed e0 = 0.39 by the matching of the dispersion curves via nonlocal theory for plane wave and Born–Karman model of lattice dynamics at the end of the Brillouin zone (ka = p), where a is the distance between atoms and k is the wave number in the phonon analysis [20]. On the other hand, Eringen proposed e0 = 0.31 in his study [21] for Rayleigh surface wave via nonlocal continuum mechanics and lattice dynamics.
2.1. A review on nonlocal elasticity theory of solids
2.2. Nonlocal governing equations of rotating nanotube
This theory assumes that the stress state at a reference point x in the body is regarded to be dependent not only on the strain state
Nanotubes are central to new rotating devices such as miniature motor. A rotating CNT can be represented as a cantilever beam
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Fig. 2. (a) A schematic rotating carbon nanotube system with mathematical idealization to rotating nanobeam and (b) coordinate system and degree of freedom.
(see Fig. 2(a)) having displacements perpendicular to the plane of rotation. The detailed coordinate system on this rotating beam are shown in Fig. 2(b). Considering the elementary Euler–Bernoulli theory of beams, the axial and transverse displacement fields of a rotating beam can be represented
ow uðx; y; z; tÞ ¼ u0 z ox wðx; y; z; tÞ ¼ wðx; tÞ
ð7Þ ð8Þ
where w is transverse displacements of the point (x, 0) on the middle plane (i.e., z = 0) of the beam. The only nonzero strain of the Euler–Bernoulli beam theory, accounting for the von Kármán strain is
exx ¼
o2 Q ou ¼ EA ox2 ox 2 2o M M ðe0 aÞ ¼ EIje ox2
Q ðe0 aÞ2
ð16Þ
R where I ¼ A z2 dA is the moment of inertia of the beam cross section 2 and je ¼ ooxw2 is the curvature of the beam. With the help of the nonlocal constitutive relations and the equations of motion presented, the moment can be expressed in terms of the generalized displacements as, by substituting Eq. (16) into Eq. (11), we get
M ¼ EI
ou ou0 o2 w ¼ z 2 ox ox ox
ð15Þ
ð9Þ
" # o2 w o2 w o ow 2 TðxÞ þ ðe aÞ q A 0 ox2 ox ot2 ox
ð17Þ
This is also called as bending strain (or curvature). The equations of motion of the Euler–Bernoulli beam theory are given by
Substituting M from Eq. (17) into Eq. (11), we obtain the equation of motion of rotating nonlocal Euler beams as
oQ o2 u 0 ¼ qA 2 ox ot o2 M o ow o2 w þ TðxÞ ¼ qA 2 2 ox ox ox ot
" # 2 o4 w o2 w o ow 2 o TðxÞ EI 4 þ ðe0 aÞ qA 2 ox ox ox ox2 ot 2 o ow o w ¼ qA 2 þ TðxÞ ox ox ot
ð10Þ ð11Þ
where
Q¼
Z
rxx dA; M ¼
A
Z
zrxx dA
ð18Þ
ð12Þ
A
TðxÞ ¼
Z
L
qAX2 x dx
ð13Þ
x
where q is the mass density, A is the beam cross section area and X is the rotation speed. Using Eq. (6), we can express stress resultants of Euler–Bernoulli beam theory in terms of the strains in that theory. As opposed to the linear algebraic equations between the stress resultants and strains in a local theory, the nonlocal constitutive relations lead to differential relations involving the stress resultants and the strains. In the following, we present these relations for homogeneous isotropic beams. The nonlocal constitutive relations in Eq. (6) take the following special form for beams:
o2 rxx rxx ðe0 aÞ ¼ Eexx ox2 2
ð14Þ
where E is the Young’s modulus of the beam. Using Eqs. (12) and (14), we have
Nondimensional wavenumber (k*h)
and rxx is the axial stress on the yz-section in the direction of x, Q is the axial force, M is the bending moment and T(x) is the axial force due to centrifugal stiffening and is given as 4 3.5 3 2.5 2 1.5 1 0.5 0 50 40
20 30
15 20
Rotating Speed (Ω/ωstr)
10 10
5 0
0
Frequency (ω/ωstr)
Fig. 3. Wavenumber dispersion in rotating nanotube for s = e0a/L = 0 for different values of X/xstr.
21
S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25 10
10 0
9
τ = e0a/L = 0.1
8
τ = e a/L = 0.2
7
τ = e a/L = 0.3
0 0
τ = e0a/L = 0.5
6 5 4 3 2 1 0
τ = e0a/L = 0
9
Nondimensional wavenumber (k × h)
Nondimensional wavenumber (k × h)
τ = e a/L = 0
τ = e a/L = 0.1 0
8
τ = e a/L = 0.2
7
τ = e0a/L = 0.3
0
τ = e a/L = 0.5 0
6 5 4 3 2 1
0
5
10
15
0
20
0
5
Frequency (ω/ω )
(a)
(b)
str
20
15
20
10 τ = e a/L = 0 0
9
τ = e0a/L = 0.1
8
τ = e0a/L = 0.2
7
τ = e0a/L = 0.3
Nondimensional wavenumber (k × h)
Nondimensional wavenumber (k × h)
15 str
10
τ = e a/L = 0.5 0
6 5 4 3 2 1 0
10
Frequency (ω/ω )
9
τ = e a/L = 0
8
τ = e0a/L = 0.1
0
τ = e a/L = 0.2 0
7
τ = e a/L = 0.3 0
6
τ = e0a/L = 0.5
5 4 3 2 1
0
5
10
15
0
20
0
5
10
Frequency (ω/ωstr)
Frequency (ω/ωstr)
(c)
(d)
Nondimensional wavenumber (k × h)
10 9
τ = e a/L = 0
8
τ = e a/L = 0.1
0 0
1.2
τ = e a/L = 0.2 0
7
τ = e0a/L = 0.3
6
1
τ = e0a/L = 0.5
0.8
5 0.6
4
0.4
3 2
10
12
14
16
18
20
1 0
0
5
10
15
20
Frequency (ω/ωstr)
(e) Fig. 4. Wave dispersion in rotating nanotube for various values of nonlocal scale s = 0, 0.1, 0.2, 0.3, 0.5. For the various nondimensional rotating speeds: (a) X X X X xstr ¼ 10, (c) xstr ¼ 30, (d) xstr ¼ 50 and (e) xstr ¼ 100.
We assumed a uniform rotating beam and replace T(x) by the maximum force (at the root, i.e. at x = 0)
T max ¼
Z
0
L
qAX2 x dx ¼
qAX2 L2 2
ð19Þ
X
xstr
¼ 0, (b)
This allows us to represent the governing equation as a constant coefficient nonlocal partial differential equation. Finally, the nonlocal governing differential equation for transverse displacement (w(x, t)) of a rotating cantilever beam is derived as
22
S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25
o4 w o2 w o4 w o2 w EI 4 T max 2 þ T max ðe0 aÞ2 4 þ qA 2 ox ox ox ot 4 2 o w qAðe0 aÞ 2 2 ¼ 0 ot x
"
ð20Þ
For analyzing the dispersion characteristics of waves in nanotubes, we assume that a harmonic type of wave solution for the displacement field w(x, t) and it can be expressed in complex form [26,27] as N X
^ ðx; xn Þejðkxxn tÞ w
^ ðx; xn Þ is the frequency domain amplitude of the flexural where w deformation of CNTs, k is the wave number and xn is the angular frequency of the wavep motion at nth sampling point, N is the numffiffiffiffiffiffiffi ber of amples and j ¼ 1. Eliminating the time variable from Eq. (20) using the above spectral approximation of the displacement gives,
"
^ ðx; xn Þ ^ ðx; xn Þ ^ ðx; xn Þ o4 w o2 w o4 w T max þ T max ðe0 aÞ2 4 2 ox ox ox4 n¼1 # ^ ðx; xn Þ jxn t o2 w b ðx; xn Þ þ qAðe0 aÞ2 x2 ¼0 e qAx2 w ox2
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u 2 2 2 2 u 2 u 1 X 2 s2 x 2 1 X x X x 2 21 s þ 4 1 þ s u 2 xstr 2 xstr 2 xstr xstr xstr xstr u k ¼ u 2 t 2 1 þ s2 12 xXstr ð29Þ
The wave numbers are mainly a function of the nonlocal scaling parameter (e0a), rotational speed of the beam (X) and the wave circular frequency. speeds, Phase The
corresponding wave namely,
speed C p ¼ Real xk and Group speed C g ¼ Real ookx , are obtained from Eq. (29). A detail discussion on these parameters is presented in the next section. 4. Numerical experiments, results and discussion
EI
ð22Þ
The Eq. (22) must be satisfied for each n and hence can be written as 4 2 4 ^ ^ ^ d w d w d w b T max 2 þ T max ðe0 aÞ2 4 qAx2 w 4 dx dx dx 2 ^ d w þ qAðe0 aÞ2 x2 2 ¼ 0 dx
EI
ð23Þ
For the sake of simplicity in the analysis we express this equation in a non-dimensional form. Now define new variables as 2 3 ^ dw ~ ^ dw ^ d2 w ~ ^ x d w d w w ~ ¼ ; ~x ¼ ; w ¼ ; L 2 ¼ 2 ; L2 3 ~ ~ L dx L dx dx dx dx 3 4 ~ ^ d4 w ~ d w d w e0 a ¼ 3 ; L3 4 ¼ 4 ; s ¼ ~ dx dx d~x L
ð24Þ
The non-dimensional form of the Eq. (23) is 4 2 4 ~ ~ ~ 1 d w 1d w 2 1 d w ~ T þ T ðe aÞ qAx2 Lw max max 0 3 o~ 3 d~ 4 2 4 ~ L x o x x L L 2 ~ 1d w L d~x2
¼0
ð25Þ
Rearranging and defining a new variable
xstr ¼
sffiffiffiffiffiffiffi EI L2 qA 1
ð26Þ
Eq. (25) changes to
"
2 #
1 X 1 þ s2 2 xstr x 2~ w¼0
xstr
4 ~ 1 X 2 d2 w ~ ~ d w x 2 d2 w 2 þ s xstr d~x2 d~x4 2 xstr d~x2 ð27Þ
Since the differential equation is a constant coefficient one, it has f ejkx , on substituting this solution ~ ¼W the solution of the form w f –0), in Eq. (27) we get (for W
0
þ qAðe0 aÞ2 x2
For Numerical Experiments, a nanobeam is assumed as (5,5) SWCNT is considered and the diameter of the SWCNT is d = 0.675 nm, length L = 10d, Young’s modulus E = 5.5 TPa, and the density q = 2300 kg/m3. The spectrum curves (i.e., non-dimensional wave number (k h), h is the thickness of the nanotube vs non-dimensional wave frequency (x/xstr)) for the rotating SWCNT are shown in Fig. 3 for different values of the non-dimensional rotational speed X/xstr for s = e0a/L = 0 (i.e., local elasticity calculation). It can be seen that for non-rotating CNT (i.e., X/xstr = 0), the non-dimensional flexural wave number shows a nonlinear variation with wave frequency, i.e., the waves will change their shape as they propagate. As the rotational speed of the CNT increases, the nondimensional wave numbers tends to become non-dispersive in nature as shown in Fig. 3. It means that the waves will not change their shape as they propagate in the medium. Also, the wave number shows an inverse dependence on rotation speed. For a nonrotating CNT, the spectrum relation is dispersive in nature. But for a rotating beam, at higher speeds, the above nonlinear relation shifts to a linear nature due to the relatively negligible contribution from the x term, especially for lower values of x. That is, the
Phase Speed (Cp /C0) & Group Speed (Cg/C )
EI
where k is the wave number. This is the dispersion/characteristic equation of the rotating uniform beams. One can solve for the wave numbers as
ð21Þ
n¼1
N X
ð28Þ
xstr
3. Wave dispersion analysis in nanotubes undergoing rotation
wðx; tÞ ¼
" 2 # 2 # 1 X 1 X x 2 2 4 2 k þ k 1þs s xstr 2 xstr 2 xstr x 2 ¼0 2
0.2 0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 50 40
20 30
15 20
10 Rotating Speed (Ω/ωstr)
10 5 0
0
Frequency (ω/ω ) str
Fig. 5. Phase speed (thick lines) and group speed (thin lines) dispersion in rotating nanotube for s ¼ e0La ¼ 0 for different values of xXstr .
23
S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25 0.12 Phase: τ = 0 Group: τ = 0 Phase: τ = 0.1 Group: τ = 0.1 Phase: τ = 0.2 Group: τ = 0.2 Phase: τ = 0.3 Group: τ = 0.3 Phase: τ = 0.5 Group: τ = 0.5
0.1
0.08
Phase Speed (Cp /C0) & Group Speed (Cg/C0)
0
Phase Speed (Cp /C0) & Group Speed (Cg/C )
0.12
0.06
0.04
0.02
0
0
2
4
6
8
10
12
14
16
18
Phase: τ = 0 Group: τ = 0 Phase: τ = 0.1 Group: τ = 0.1 Phase: τ = 0.2 Group: τ = 0.2 Phase: τ = 0.3 Group: τ = 0.3 Phase: τ = 0.5 Group: τ = 0.5
0.1
0.08
0.06
0.04
0.02
0
20
Frequency (ω/ω )
0
2
4
6
8 10 12 Frequency (ω/ω )
(a)
(b)
str
Phase Speed (Cp /C0) & Group Speed (Cg/C0)
0
Phase Speed (Cp /C0) & Group Speed (Cg/C )
16
18
20
14
16
18
20
0.2
0.14
0.12
0.1
0.08
0.06
Phase: τ = 0 Group: τ = 0 Phase: τ = 0.1 Group: τ = 0.1 Phase: τ = 0.2 Group: τ = 0.2 Phase: τ = 0.3 Group: τ = 0.3 Phase: τ = 0.5 Group: τ = 0.5
0.04
0.02
0
14
str
0
2
4
0.18 0.16 0.14 0.12 0.1 Phase: τ = 0 Group: τ = 0 Phase: τ = 0.1 Group: τ = 0.1 Phase: τ = 0.2 Group: τ = 0.2 Phase: τ = 0.3 Group: τ = 0.3 Phase: τ = 0.5 Group: τ = 0.5
0.08 0.06 0.04 0.02
6
8 10 12 Frequency (ω/ω )
14
16
18
0
20
0
2
4
6
8 10 12 Frequency (ω/ω )
str
str
(c)
(d)
Phase Speed (Cp /C0) & Group Speed (Cg/C0)
0.4 0.35 0.3 0.25 0.2 Phase: τ = 0 Group: τ = 0 Phase: τ = 0.1 Group: τ = 0.1 Phase: τ = 0.2 Group: τ = 0.2 Phase: τ = 0.3 Group: τ = 0.3 Phase: τ = 0.5 Group: τ = 0.5
0.15 0.1 0.05
0
0
2
4
6
8 10 12 Frequency (ω/ω )
14
16
18
20
str
(e) Fig. 6. Phase speed (thick lines) and group speed (thin lines) dispersion in rotating nanotube for for different values of X X xstr ¼ 50 and (e) xstr ¼ 100.
variation of k is dominated by the centrifugal force term at high X/ xstr.
X
xstr .
Here (a)
X
xstr
¼ 0, (b)
X
xstr
¼ 10, (c)
X
xstr
¼ 30, (d)
Fig. 4 shows the spectrum curves for rotating and nonrotating CNTs for various values of the nonlocal scaling parameter.
24
S. Narendar, S. Gopalakrishnan / Results in Physics 1 (2011) 17–25
Fig. 4(a) shows that as we move from local elasticity to nonlocal elasticity solution, the spectrum curve becomes linear at higher values of the wave frequency. The wave numbers also increases as the nonlocal scaling parameter increases. The matching of local and nonlocal solution is limited only upto <0.2 THz frequency. After this frequency, the difference between the wave numbers predicted is very large. If the rotational speed of the CNT increases from X/xstr = 0 to X/xstr = 10 (Fig. 4(b)), the spectrum curve is slightly nonlinear as compared to nonrotating case. The wave numbers obtained from local and nonlocal cases is same upto 0.45 THz frequency. The wave numbers are showing an increasing tendency as the nonlocal parameter increases. If the rotational speed of the CNT increases to very high values like X/xstr = 30, 50 and 100, the local and nonlocal calculations are almost similar upto 1 THz, 1.6 THz and 2.5 THz frequencies, respectively (see Figs. 4(c)–(e)). As the rotational speed of the CNT increases to very high values, the nonlocal scaling parameter effect on the spectrum curves is negligible. It means that if the CNT rotates at very high speeds, the local elasticity and nonlocal elasticity calculations give almost similar spectrum relations. It can also be observed that as the rotational speed of the CNT increases, the wave numbers become very small and the dispersive nature changes to non-dispersive nature (see Fig. 4). The non-dimensional phase speed ffi p/C0) and non-dimensional pffiffiffiffiffiffiffiffi(C group speed (Cg/C0), where C 0 ¼ E=q, dispersions of the rotating CNT are shown in Fig. 5, obtained from the local elasticity calculations (i.e., s = 0). Thick lines represent the phase speed variation and the thin lines show the group speed variation. It can be seen that the phase speed of the rotating CNTs is higher than the group speed. Because of the nonlinear relation of the wave number with wave frequency, for non-rotating CNT, the phase and group speeds also shows a nonlinear variation with frequency. As the rotational speed of the CNT increase to higher values, both the speeds will saturate to a constant velocity, because of the linear variation of the wave number with wave frequency. The difference between the phase and group speeds of the rotating CNTs is negligible at higher rotational speeds as shown in Fig. 5, which is a characteristic of any non-dispersive system. In the limiting case, we can say that they become equal and become constant for all wave frequencies. For a nonrotating beam, both phase and group speeds are dispersive and shows that the speeds approach infinity for very high frequencies. This unreasonable limit is due to the limitation of Euler–Bernoulli beam theory. Fig. 6 shows the non-dimensional phase speed and non-dimensional group speed variation with both the wave frequency and the nonlocal scaling parameter for rotations and non-rotating CNTs. Fig. 6(a) shows that, for non-rotating CNT, the phase and group speeds will decrease as the nonlocal scaling parameter increases because the wave numbers are increasing with increase in s (see Fig. 4). Also, the difference between the phase and group speeds dips at higher values of s. As the rotational speed of the CNT increases, both phase and group speeds will also increase as shown in Figs. 6(b)–(e). For small rotational speeds, and large values of s, both speeds show a decrease in nature at smaller frequencies and they become constant at higher wave frequencies. Such difference will also vanish at higher rotational speeds as shown in Fig. 6(e). On the other hand, one more interesting feature of the nonlocality is that, the difference between both the wave speeds is considerable at smaller rotational speeds and s and also at the higher rotational speeds and s and it can be clearly seen from Fig. 6. The present work illustrated here can be extended to study the wave propagation analysis of rotating graphene sheets. This would be helpful in understanding the wave dispersion characteristics of future rotating nanomachines. One such nanomachine is the nanoturbine. For efficient designing of these nanomachines, proper
understanding of its mechanical behavior is required. The nanoturbine is characterized by a rotating blades modeled as a rotating nanotube. Rotating nanocantilever rotates at an angular velocity about the hub axis. The performances of these nanoturbines are extremely dependent on dynamic properties of their beam-like elements. In summary, this study would be useful in perspective to simple mathematical modeling and understanding the scale effects in rotating nanostructures. 5. Conclusions In the present work, the wave dispersion characteristics of a rotating nanotube (such as SWCNT) are studied using the spectral analysis and nonlocal scale theory. The rotating nanotube is modeled as a Euler–Bernoulli beam. The governing partial differential equation for a uniform rotating beam is derived incorporating the nonlocal scale effects and a powerful model is derived in analyzing the wave dispersion characteristics of the rotating nanotube. Some of the beautiful features of the wave behavior in rotating nanotubes are observed. It has been shown that the dispersive flexural wave tends to behave non-dispersively at very high rotation speeds. Understanding the dynamic behavior of rotating nanostructures is important for practical development of nanomachines. At the nanoscale, the nonlocal effects often become more prominent. Such observations are helpful in designing the nanomotors and the other CNT based rotational nano-devices where CNT acts as a basic element. Acknowledgment The authors would like to thank the reviewers for their comments and suggestions to improve the clarity of this article. References [1] Iijima S. Nature 1991;354:56. [2] Saito R, Dresselhaus G, Dresselhaus M. Physical properties of carbon nanotubes. London: Imperial College Press; 1998. [3] Fennimore A, Yuzvinsky TD, Han WQ, Fuhrer MS, Cumings J, Zettl A. Nature 2003;424:408. [4] Bourlon B, Glattli DC, Miko C, Forro L, Bachtold A. Nano Lett 2004;4:709. [5] Thess A, Lee R, Nikolaev P, Dai H, Petit P, Robert J, Xu C, Lee YH, Kim SG, Rinzler AG, Colbert DT, Scuseria GE, Tomanek D, Fischer JE, Smalley RE. Science 1996;273:483. [6] Journet C, Maser WK, Bernier P, de la chapelle ALML, Lefrant S, Deniard P, Lee R, Fischer JE. Nature 1997;388:756. [7] Freedonia H. World Nanotubes to 2009, Freedonia, Cleveland, 2006. [8] Zheng LX, O’Connell MJ, Doorn SK, Liao XZ, Zhao YH, Akhadov EA, Hobauer MA, Roop BJ, Jia QX, Dye RC, Peterson DE, Huang SM, Liu J, Zhu YT. Nat Mater 2004;3:673. [9] Kral P, Sadeghpour HR. Phys Rev B 2002;65:161401. [10] Krim J, Solina DH, Chiarello R. Phys Rev Lett 1991;66:181. [11] Zhang S, Liu WK, Ruo RS. Nano Lett 2004;4:293. [12] Drexler KE. Nanosystems: molecular machinery, manufacturing, and computation. New York: Wiley; 1992. [13] Rao PVM, Singh M, Jain V. Nanotechnology 2005;2:250. [14] Robertson DH, Dunlap BL, Brenner DW, Mintmire JW, White CT. Fullerene/ tubule based hollow carbon nano-gears, MRS symposia proceedings no. 349. Materials Research Society, Pittsburgh, 1994. p. 283. [15] Han J, Globus A, Jaffe R, Deardorff G. Nanotechnology 1997;8:95. [16] Jaffe R, Han J, Globus A. NASA Technical Report No. NASA-97-006 1997. [17] Srivastava D. Nanotechnology 1997;8:186. [18] Lohrasebi A, Raffi-Tabar H. J Mol Graph Modell 2008;27:116. [19] Eringen AC, Edelen DGB. Int J Eng Sci 1972;10:233. [20] Eringen AC. J Appl Phys 1983;54:4703. [21] Eringen AC. Int J Eng Sci 1972;10:425. [22] Eringen AC. Nonlocal continuum field theories. Berlin: Springer; 2002. [23] Wong EW, Sheehan PE, Lieber CM. Science 1997;277:1971. [24] Cleland AN. Foundation of nanomechanics, Springer, 2002. [25] Born M, Huang K. Dynamical theory of crystal lattices. Oxford: Clarendon Press; 2002. [26] Doyle JF. Wave propagation in structures. New York: Springer-Verlag Inc.; 1997. [27] Gopalakrishnan S, Chakraborty A, Roy Mahapatra D. Spectral finite element method. Springer Verlag London Ltd; 2008.
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