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13Nonlocal Gradient Approach on Torsional Vibration of CNTs
1 9
Arda M.1, Aydogdu M.2 iPhD, Trakya University, Edirne, Turkey 2 Professor, Trakya University, Edirne, Turkey
Abstract
The nonlocal strain gradient approach for torsional vibration of CNTs have been investigated in the present study. The effects of the stress and strain gradient small scale parameters on the non-dimensional frequencies have been obtained. Strain Gradient Theory has stiffening effect on the dynamics of CNT. Combination of both theories gives more acceptable results according to Lattice Dynamics.
Key words: torsional vibration, carbon nanotubes, nonlocal gradient theory.
Нелокальный градиентный подход для крутильных колебаний УНТ
Арда М.1, Айдогду М.2 1 К.т.н., Университет Тракья, Эдирне, Турция 2 Профессор, Университет Тракья, Эдирне, Турция
Аннотация
В данном исследовании был рассмотрен подход нелокального градиента деформации для крутильных колебаний УНТ. Было определено влияние микропараметров градиента напряжения и деформации на безразмерных частотах. Теория градиента деформации имеет эффект повышения жесткости на динамику УНТ. Сочетание обеих теорий дает более приемлемые результаты в соответствии с динамикой решетки.
Ключевые слова: Крутильные колебания, Углеродные нанотрубки, Теории нелокального градиента.
Introduction
Nowadays, scientists and engineers have great interest on CNTs, which were discovered by Iijima [1] in 1991, because of CNTs' superior physical properties. Analysis related to circumferential direction in CNTs has gained importance especially in design process of nano-motors, nano-bearings and nano-gearboxes.
Classical theories couldn't be applied in nanoscale analysis because of their size independence. Nonlocal Theory, proposed by Eringen [2, 3], is a continuum theory and includes size dependence. Stress and strain gradient approaches can be applied with the nonlocal theory. These nonlocal gradient approaches give more acceptable results when compared to the classical theories. Eringen adjust the wave dispersion curves according to Lattice Dynamics and with this assumption Eringen combined continuum and discrete theories as one theory.
In literature search, lots of study can be found about nonlocal stress gradient theory, but application of nonlocal strain gradient theory can be seen barely. Wang and Varadan developed a nonlocal continuum mechanics model and applied to the both single-walled carbon nanotubes (SWCNTs) and double-walled carbon nanotubes (DWCNTs). They investigated the small-scale effects on vibration characteristics of CNTs [4]. Duan et al. adjusted the e0 parameter for the Nonlocal Timoshenko Beam Theory in free vibration case of
E-mail: [email protected] (Arda M.), [email protected] (Aydogdu M.)
SWCNTs. They used vibration frequencies generated from Moleculer Dynamics (MD) Simulation for optimization of (e0) parameter [5]. Kumar et al. studied the buckling of CNTs using nonlocal one dimensional Euler-Bernoulli Beam Model. They used both stress and strain gradient approach and variational principle in model and boundary conditions [6]. Lim applied the Nonlocal Elasticity Field Theory to nano-mechanics and variational principle. He derived the main governing equation and boundary conditions for bending case [7]. §im§ek studied the forced vibration of simply supported SWCNT under the moving harmonic load effect. He used the Nonlocal Euler-Bernoulli Beam Theory in modeling [8]. Zhang et al. developed a hybrid Nonlocal Euler-Bernoulli Beam Model for bending, buckling and vibration analysis of nanobeams. The strain energy functional combines the local and nonlocal curvatures in the hybrid model that has two independent small-length scale parameters unlike Eringen's Nonlocal Model [9]. Ansari et al. investigated the vibrational characteristic of SWCNTs based on the gradient elasticity theory. They applied different gradient elasticity theories like stress, strain and combined one to nanotube for showing the nonlocal effect [10]. Thai proposed a Nonlocal Shear Deformation Beam Theory for bending, buckling and vibration case using Eringen's nonlocal differential constitutive relations. He didn't use shear correction factor in his model that account for both small scale effects and quadratic variation of shear strain together [11]. Narendar et al. studied the torsional vibration of nanorods using Strain Gradient Theory which is a non-classical theory and includes size effect [12]. Wang developed two beam model for vibration analysis of fluid conveying nanotubes using strain gradient elasticity combined with inertia gradients [13]. Wang and Wang investigated the vibration of nanotubes embedded in an elastic matrix by using Nonlocal Timoshenko Beam Model. They considered both stress and strain gradient approaches in formulation [14]. Arda and Aydogdu made the static and dynamic analysis of CNTs embedded in an elastic medium using Nonlocal Stress Gradient Theory [15]. Akgoz and Civelek studied the longitudinal free vibration problem of micro-bar using the Strain Gradient Elasticity Theory. They obtained the equation of motion and boundary conditions with Hamilton Principle [16]. Karlicic et al. analyzed free flexural vibration and buckling of SWCNTs under the effect of compressive axial loading. They used Reddy and Huu-Thai reformulated beam theories in equation of motions according to different gradient elasticity approaches like stress, strain and combined strain/inertia [17].
In this study, governing equations for torsional vibration will be obtained using stress, strain and combined gradient theories. Effect of the nonlocal parameters on vibration will be investigated and depicted in figures.
Analysis
1. Eringen's Nonlocal Theory
Eringen proposed the Nonlocal Elasticity Theory for including size effect and long range interactions [2, 3] in continuum. In order to account size dependence, the stress tensor in the nonlocal approach can be defined as [2]:
*ij(x) = jy z(|x - x'\,Y)Ci]kl£kldV(x'), Vx e V, (1)
where T£y is the nonlocal stress tensor, eki is the strain tensor, Cijki elastic modulus tensor, X(\x — x'\,y) is the attenuation function and |x — x'| is the Euclidean distance. ^ = e0a, where n is nonlocal parameter, a is an internal characteristic length for CNT and e0 is a constant. e0 parameter can be adjusted using the dispersion curves based on the atomic models. For a specific material, geometry or problem the nonlocal parameter can be estimated
by fitting the results of Atomic Lattice Dynamics [2, 18] and Molecular Dynamics [5, 19]. An estimation of the small scale parameter for a SWCNT was proposed in literature (eoa<2nm) [20].
One dimensional form of nonlocal stress and strain gradient relation in circumferential direction can be obtained in the light of Nonlocal Theory, as below:
(l-^B* = G(l + (2)
where y is the local shear strain and t is the local shear stress of CNT and and are the nonlocal stress and strain gradient parameter, respectively.
2. Equation of Motion
A nanotube with length (L) and diameter (d) is considered. The equation of motion in the circumferential direction can be written as [15]:
n 920 i 920
= (3)
where G is the shear modulus, p is the density, Ip is the polar moment of inertia, 0 is the angular displacement of CNT. The Ip is defined as:
1, = *^ (4)
where R1 and R2 are the inner and outer radius of CNT respectively. If Eq. (3) is rearranged according to Eq. (2), one obtains:
(-, , d2\d2e , d20/. d2\ GIp )—2= pip ^ (1 - Vr^) (5)
If Eq. (5) is rearranged, governing equation of motion for the torsional deformation according to both strain and inertia gradient theories is obtained. If the nonlocal stress gradient parameter is accepted as equal to zero = 0), the nonlocal strain gradient elasticity equation is obtained. If the nonlocal strain gradient parameter is accepted as equal to zero (uy = 0), the nonlocal stress gradient elasticity equation is obtained. If both nonlocal parameter is accepted as equal to zero = = 0), classical elasticity equation is obtained.
ri d4e d2e d4e d2e
VyGlP — + GlP — + -plPl^ = 0 (6)
With the harmonic vibration assumption in the circumferential direction, displacement for each tube can be written as (* = j)):
0i(x,t)=A(x)ie^t (7)
_ 2
where A(x) is the angular displacement function, co is the torsional frequency and j =-1. If
Eq. (7) is rearranged according to Eq. (8) and non-dimensional parameters, one obtains:
Bm+S[i-^]+w]=° (8)
where 0 is the non-dimensional frequency parameter (NDFP) and defines as:
= ^ (9)
For the Clamped-Clamped (C-C) boundary condition, the displacement function (A(x)) can be assumed as:
A(x) = Csin(nrcx) (10)
where C is the amplitude of the displacement and n is the half wave number of vibration. If Eq. (8) rearranged according to Eq. (10), one obtains:
22
[(nn)4 (£) - Cnn)2 (l - ^n2) + (ß2)] = 0 (11)
Characteristic equation in Eq. (11) must be zero and non-dimensional frequency parameter which satisfy this condition, can be found with solving this equation.
3. Numerical Results and Discussion
In this section, validation of the present model is shown firstly. After that, variation of NDFP with nonlocal parameters is investigated. Effect of the nanotube length to the nonlocal behavior is also shown in figures.
Material properties of CNTs are selected as in the Table-1. There have been many researches about physical properties of CNTs. Shear Modulus for CNTs are selected from the Ref. [21]. Density(p) of CNTs is determined using the calculation method in Ref. [22]. CNT thickness is accepted as 0.066 nm according to Ref. [23].
Table 1
Material Properties for CNT
CNT Inner Radius (nm) Density (p) (kg/m3) Shear Modulus (G) (GPa)
Armchair (10x10) 0.68 10989 0.45
The torsional wave frequency results are compared with the Lattice Dynamics for the validation. Comparison can be seen in Fig. (1). Nonlocal combined gradient theory shows much better approximation when compared to the nonlocal stress and strain gradient theories and obviously local (classical) theory. Torsional vibration wave frequencies are obtained using Eqs. (12) and (13) [24] .
(12)
= w2 (13)
Eringen determined the e0 parameter in stress gradient approach according to Lattice Dynamics results (e0x=0.39). Same parameters for the strain gradient approach (e0e=0.25) and combined gradient approach (e0x=0.20, e0e=0.21) can be determined using the Lattice Dynamics results.
MD Simulations results are obtained by Khademolhosseini's work [25]. Torsional frequencies are compared at different mode numbers (see Table-2). Present model gives very close results to MD simulation results.
Nonlocal Stress Gradient Theory Nonlocal Strain Gradient Theory1
1.5 1.6 1.7 1 3 1.9 2 2.1 2.2 2.3
Wave Number (1/m) x101D
Fig. 1. Torsional Wave Propagation Dispersion Curves for Different Theories
Table 2
Comparison of Torsional Wave Frequencies (rad/s)
Ref. [25] Present Study
Mode MD Simulation Nonlocal Stress Nonlocal Strain Nonlocal Combined
1 2.41 x 1012 2.4340 x 1012 2.4340 x 1012 2.4340 x 1012
4 9.62 x 1012 9.7286 x 1012 9.7331 x 1012 9.7320 x 1012
0 0.5 1 1.5 2 2.5 3 3.5 4
Nonlocal Stress Gradient Parameter (m5) x jg 18
Fig. 3. Variation of NDFP with Nonlocal Stress and Strain Gradient Parameters (L=5nm)
Effect of the nonlocal stress and strain gradient parameters can be seen at Figs. (2-5). Both gradient theories decrease the NDFP with increasing nonlocal parameter. But in the stress gradient approach, NDFP value approaches a limit value for higher values of
>> 4nm2) contrary to strain gradient approach. CNT became stronger for strain gradient approach rather than the stress gradient. Especially in shorter nanotubes, NDFP approaches to zero with the effect of strain gradient nonlocal parameter. Also limit value for the NDFP in strain gradient approach must be zero at longer nanotube lengths >> 4nm2).
Fig. 5. Variation of NDFP with Nonlocal Stress and Strain Gradient Parameters (L=10nm)
In Figs. 6 and 7, nanotube length effect can be seen. Both nonlocal gradient theories are effective in shorter nanotube length. The nonlocal effect vanishes with increasing nanotube length and nonlocal results approaches to the local results, especially for bigger nanotube length values from 20nm.
Fig. 6. Variation of NDFP with Nanotube Length
Nonlocal Stress Gradient Parameter (m2) Nonlocal Strain Gradient Parameter (rri)
Fig. 7. Variation of NDFP with Nonlocal Stress and Strain Gradient Parameters
Conclusion
In the present study, torsional vibration of CNTs is investigated by using nonlocal stress gradient, nonlocal strain gradient and nonlocal combined (strain/inertia) gradient theories. Nonlocal effect decreases the NDFP in all cases. CNT is become stiffer with the effect of strain gradient theory. Nonlocal effect is more pronounced for CNTs for L<20nm. Combined nonlocal gradient theory shows much more agreement with Lattice Dynamics rather than the stress and strain gradient nonlocal theories. Present results can be useful for mechanical modeling of CNTs.
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