Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part II: parametric study

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1 Acta ech 6, 97 6 () DOI.7/s Keivan Kiani Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part II: parametric study Received: August 9 / Revised: April / Published online: 8 July Springer-Verlag Abstract The capabilities of the proposed nonlocal beam models in the companion paper in capturing the critical velocity of a moving nanoparticle as well as the dynamic response of double-walled carbon nanotubes (DWCNTs) under a moving nanoparticle are scrutinized in some detail. The role of the small-scale effect parameter, slenderness of DWCNTs and velocity of the moving nanoparticle on dynamic deflections and nonlocal bending moments of the innermost and outermost tubes as well as their maximum values are then investigated. The results reveal that the critical velocity increases with the slenderness of DWCNTs and the magnitude of the van der Waals interaction force. Nevertheless, the critical velocity generally decreases with the small-scale effect as well as the ratio of the mean diameter to the thickness of the innermost tube. Additionally, the predicted maximum dynamic deflections and nonlocal bending moments of the innermost and outermost tubes by using the nonlocal Euler Bernoulli and Timoshenko beam theories are generally the lower and upper bounds of those obtained by the nonlocal higher-order beam theory (NHOBT). In the case of λ <, the use of the NHOBT is highly recommended for more realistic prediction of dynamic response of DWCNTs under a moving nanoparticle. Introduction The carbon nanotubes (CNTs) can be used as efficient tools for delivering drug, fuel, or in general, transport of nanoparticles because of their remarkable mechanical properties. According to hollow cylindrical shape and large aspect ratios of CNTs, they can be utilized as potential conduits for nanoscale particles. Furthermore, the smooth surface and the stiffness of CNTs could provide them as a superlative material for quick transport of nanoparticles, particularly those with higher surface energy. Therefore, vibration analysis of CNTs under such external loads had been an important topic among researchers of various disciplines during the past decade. Despite the enormous applications that could be expected for CNTs, few studies have been conducted on assessing the dynamic response of these nanostructures under moving nanoparticles. However, the dynamic analyses of CNTs under fluid flow have been appropriately addressed in the literature, but even the elastodynamic behavior of them under a moving nanoparticle has not been well understood until now. As a result, realizing the role of important parameters on vibration of CNTs under excitation of moving nanoparticles would be of great importance in optimal design of these beam-like nanostructures in the future. This article is lovingly dedicated to my father and mother, Amrullah Kiani and Kobra Ahmadi, whose love and encouragements I feel every day of my life. K. Kiani (B) Department of Civil Engineering, Sharif University of Technology, P.O. Box 65-9, Tehran, Iran k_kiani@civil.sharif.edu Tel.: Fax:

2 98 K. Kiani According to the proposed nonlocal beam models in the companion paper [], herein the effects of the ratio of the diameter to the thickness of the innermost tube, the slenderness of double-walled carbon nanotubes (DWCNTs), the small-scale effect and van der Waals (vdw) interaction force on the critical velocities of the moving nanoparticle are scrutinized in some detail. The capabilities of the nonlocal Euler Bernoulli beam theory (NEBT), nonlocal Timoshenko beam theory (NTBT) and nonlocal higher-order beam theory (NHOBT) in predicting vibration of DWCNTs are then examined through various parametric studies. Furthermore, the role of the small-scale effect, slenderness of DWCNTs and velocity of the moving nanoparticle on the maximum dynamic deflections and nonlocal bending moments of the innermost and outermost tubes are explored. Results and discussion. Comparison of the obtained results with those of other researchers Due to the lack of research on the effect of moving nanoparticles on the vibration of DWCNTs, the results of dynamic response of DWCNTs under such effects do not exist. Therefore, for the sake of comparison, we restrict our attention to study frequencies of DWCNTs according to the previously suggested double beam theories []. However, comparison of the obtained results by various nonlocal double beam models with each other would also ensure us that overall numerical procedures which are presented in the companion paper [] have been carried out with a good accuracy. This issue would be followed in some detail in the following parts. In the case of free vibration, the general form of the equations of motion of DWCNTs according to the nonlocal [ ] beam theory would be [] x [],ττ + K[ ] x [] =, () where []stands for E or T or H. By introducing x [] (τ) = x [] eiϖ []τ to Eq. (), one may arrive at [ (ϖ []) ] [] + K [] x [] =, () where ϖ [] is the nondimensional frequency of the DWCNT associated with the natural frequency of the nonlocal [ ] beam theory, ω []. By solving the set of eigenvalue equations in Eq. (), natural frequencies of the DWCNT for different nonlocal beam models are readily obtained as ωi E n = ω T j n = E b I b /(ρ b A b l 4 b )ϖe i n ; i =,, k s G b /(ρ b l b )ϖt j n ; j =,,, 4, () ω H j n = α J 6() /(I () l 4 b )ϖh j n, where ϖ [] i n = r [] i n. As one can see, the nondimensional frequencies of the DWCNT based on the nonlocal [] beam model are exactly equal to the roots of Δ [] n. If the accuracy of the evaluated roots could be checked, it assures us that the major parts of the enormous calculations of the former paper are correct. To this end, consider a DWCNT with the following data: r m =.5 nm, r m =.7nm,andh =.5 nm. Since verification of the results of the presented models with those of Aydogdu [] is of concern, only in this part, it is presumed that the vdw interaction coefficient could be estimated from the relation: C v = (r j )/.6d (erg/cm ) where r j is the inner radius of the jth tube and d =.4 nm. Furthermore, for the sake of comparability of the obtained results with those ofaydogdu [], theith dimensionless frequency parameter of the nth mode of ( ) vibration is introduced as Ω [] ρ i n = b A b lb 4 E b I b ω [] 4 i n where Ab = A b + A b, I b = I b + I b,ρ b = ρ b = ρ b, and E b = E b = E b. In the work of Aydogdu [], frequencies of simply supported multi-walled carbon nanotubes were derived analytically using higher-order beam theory. The predicted first and second dimensionless frequency parameters of various nonlocal double beam models as well as those of other researchers have been summarized in Tables and. The frequencies of DWCNTs up to the fifth vibrational mode have been provided for two levels of the aspect ratio, l b /d m = and 5. The given results in columns with the

3 Vibration of DWCNTs under a moving nanoparticle 99 Table Comparison of the predicted first five dimensionless frequencies associated with Ω n of the DWCNT by the proposed beam models with those of other researchers l b /d m n Double Euler Bernoulli beam Double Timoshenko beam Double higher-order beam AH Present work AH Present work Ref. [] Present work The parameter d m represents the average of the diameters of the innermost and outermost carbon nanotubes The word AH denotes the results obtained by Aydogdu [] according to the work of He et al. [] Table Comparison of the predicted first five dimensionless frequencies associated with Ω n of the DWCNT by the proposed beam models with those of other researchers l b /d m n Double Euler Bernoulli beam Double Timoshenko beam Double higher-order beam AH Present work AH Present work Ref. [] Present work Table Comparison of the predicted first five dimensionless frequencies associated with Ω n and Ω 4n of the DWCNT by the proposed NHOBT with those of Aydogdu [] n l b /d m = l b /d m = l b /d m = 5 Ref. [] Present work Ref. [] Present work Ref. [] Present work Ω n Ω 4n abbreviated word AH represent those generated by Aydogdu [] according to the work of He et al. []. It could be observed that the predicted frequencies by the proposed models are in a reasonable good agreement with those of Aydogdu [] and AH. As it is expected, for slender DWCNTs (i.e., l b /d m = 5), that shear strain energy has fairly no effect on the vibration frequencies. The predicted values of Ω n and Ω 4n by the NHOBT are also presented in Table for three aspect ratios of DWCNTs, l b /d m =, and 5. The obtained values of Aydogdu [] are also listed to check the accuracy of the predicted results by the NHOBT. As it could be seen, the NHOBT could reproduce successfully the obtained values of Ω n and Ω 4n by Aydogdu [] with relative error lower than 5%. oreover, a brief survey of the presented results in Tables,, indicates that the frequencies of higher modes are more sensitive to the aspect ratio with respect to those of lower modes.

4 K. Kiani. Parametric study In order to examine the capabilities of various nonlocal beam models in capturing the critical velocities of the moving nanoparticle and dynamic response of DWCNTs acted upon by a moving nanoparticle, a comprehensive parametric study is performed. To this end, consider a DWCNT with the following data: r m =.4 nm, r m =.68 nm, h =.4 nm, ρ b = ρ b = 5 (kg/m ), E b = E b = TPaandν = ν =.. The shear modulus is determined from the relation G bi = E bi /(( + ν i )); i =,. oreover, the shear correction factors of the innermost and outermost tubes in the NTBT are evaluated from k si = 6( + z i ) /(7 + 4z i + 7z 4 i ); i =, [4] inwhichz i = (r mi h)/(r mi + h). For the sake of convenience in the numerical analyses, the following normalized parameters are used: W in = 48 (E b I b + E b I b )w i /(mgl b ), b i N = 4 bi /(mgl b ) and V N = v/v E cr. Concerning the magnitude of e, its value is commonly determined such that the nonlocal continuum theory of Eringen could reproduce frequencies of atomic models with a good accuracy. Eringen [5] stated that the value of e a can be estimated by justification of the dispersion curves obtained by the nonlocal continuum theory with those of lattice dynamics. To this end, Eringen [5] used the Born Karman model of lattice dynamics and showed that if e is set equal to.9, the maximum relative error would be lower than 6% in the interesting range of the wavenumber for a one-dimensional wave propagation problem. Through this choice of e,the matching between the dispersion curves of the nonlocal model and that of the lattice dynamics would be perfect for the boundary of the Brillouin zone. Furthermore, the propagation of Rayleigh waves was studied by Eringen [5] via nonlocal continuum theory. In this two-dimensional problem, it was shown that adopting e =. would lead to a reasonable good agreement between the obtained results by the nonlocal model and those of lattice dynamics performed by other researchers for KCl crystals. In order to predict the molecular lattice behavior of KCl by an isotropic elastic solid, the Poisson s ratio in the calculations of the nonlocal model was set equal to.. Wang and Hu [6] suggested e =.88 for SWCNTs having armchair construction through comparing the predicted results by the higher-order strain gradient for elastic beams with those of molecular dynamics. In another study, Wang et al. [7] proposed e a =.7 nm for the application of the nonlocal elastic rod theory in prediction of the axial stiffness of SWCNTs. Using the proposed value of e a, the obtained results of the nonlocal continuum theory were successfully compared with those of molecular dynamics. On the other hand, the nonlocal small-scale parameter e a is usually considered lower than nm in some reference works [8,9] for the dynamic analysis of those SWCNTs, particularly when the measured frequencies of SWCNTs are larger than THz [,]. A brief survey of the literature reveals that the value of e rests on the crystal structure of the material, multi-physics nature of the problem and also geometry of the nanostructure. Due to this fact, there are some concerns about the accurate value of the small-scale effect parameter adopted in nonlocal models to study the nanostructures. Further precise research is still needed to determine the more accurate value of the small-scale effect parameter for each problem. This task could be carried out by comparison of the frequency results of the nonlocal continuum theory with those of atomisticbased models in each case study. In the present work, wherever the values of e and a have not been pointed out, e =.9 [5]anda =.4 nm [] are exploited for a vibration analysis of DWCNTs under excitation of a moving nanoparticle... Investigation of the influential parameters on the critical velocities The effects of important parameters on the predicted values of the critical velocity of the moving nanoparticle are investigated by using the proposed models for DWCNTs subjected to a moving nanoparticle. Figure a d present v cr v E as a function of the slenderness of the innermost CNT, the ratio of the mean diameter of the innermost CNT to its thickness (i.e., D m /h), the normalized small-scale effect parameter, and the vdw interaction force between two adjacent CNTs, correspondingly. As it is seen in Fig. a, the critical velocities of the moving nanoparticle associated with various nonlocal beams increase with λ regardless of the assumed value of the small-scale effect parameter. oreover, increasing the small-scale effect parameter leads to a considerable decrease in the predicted values of the critical velocities associated with different beam theories. Figure b shows that the critical velocity of the moving nanoparticle usually decreases as the magnitude of D m /h increases for all the nonlocal beam theories. The variation of v cr in terms of the normalized small-scale effect parameter of the CNTs has been depicted in Fig. c for different values of λ. As it is obvious, the critical velocities of the moving nanoparticle for DWCNTs with low slenderness (λ < ) decrease dramatically with the small-scale effect parameter. As expected, the effect of the size effect parameter on v cr lessens as the magnitude as λ grows. Figure d reveals that v cr increases with the vdw interaction force; however, the

5 Vibration of DWCNTs under a moving nanoparticle.6 v cr / v E v cr / v E λ 5 5 D /h m (d).6.4 v cr / v E v cr / v E e a C v Fig. Variation of the normalized critical velocity of the moving nanoparticle in terms of the a slenderness of the innermost tube, λ, b ratio of the mean diameter of the innermost tube to its thickness, D m /h (λ = ), c small-scale effect parameter, e a (λ = ), d vdw interaction force (λ = ) (open circle e a =., inverted triangle e a =.5nm, triangle e a = nm; plus symbol λ = 5, cross symbol λ =, asterisk λ =, dot symbol λ = 4; dotted line NEBT, dot with dashed line NTBT, solid line NHOBT) critical velocity would increase slightly with the vdw interaction force for that larger than. C v.abrief comparison of the depicted results in Fig. a d reveals that the predicted critical velocities associated with the NTBT and NHOBT are commonly close to each other such that the critical velocity of the NHOBT would be an upper bound of that of the NTBT. Furthermore, the critical velocity of the NEBT is being an upper bound of those of the NTBT and NHOBT... Time history of deflections and nonlocal bending moments for the innermost and outermost tubes In Fig. a c, the time history plots of deflection and nonlocal bending moments of the mid-point of the simply supported DWCNTs for the innermost and outermost tubes have been provided for different velocity levels of the moving nanoparticle. The understudy DWCNT is a fairly stocky continuum beam with λ = 5. The value of the parameter C v is set to be.455 Pa from Eq. (7) of the companion paper []. From depicted results in Fig., it can be observed that the predicted results of the nonlocal Euler Bernoulli beam (NEB) are generally lower than those of the nonlocal Timoshenko beam (NTB) and nonlocal higher-order beam (NHOB) since these nonlocal shear deformable beams take into account the effects of the shear and rotary inertia of DWCNTs in their formulas. For example, the NEBT underestimates the maximum dynamic deflection of the innermost tube with respect to that of the NHOBT by 7.74, 5.97 and.5% for V N =., V N =. and V N =.7, correspondingly. Comparing the dynamic response of the innermost tube with that of the outermost tube, the maximum dynamic deflection of the innermost tube is mostly higher than that of the outermost tube during the phase of excitation, particularly for low levels of the velocity of the moving nanoparticle. The difference between the dynamic deflection of the innermost tube and that of the outermost tubes decreases with the velocity of the moving nanoparticle. Besides, the outermost tube participates more effectively in bearing the nonlocal bending moment due to the applied force of the moving nanoparticle on the innermost tube. As it is expected, for low levels of the velocity of the moving nanoparticle (V N.), the maximum dynamic deflection takes place approximately at τ =.5 τ f. By increasing the velocity of the moving nanoparticle, the occurrence time of the maximum dynamic deflection of the mid-point of DWCNTs moves toward the phase of the free vibration; this issue could be inspected mathematically while a moving nanoparticle is traveling across the DWCNT with the critical velocity according to Eqs. (99), () or () of the companion paper []. As a general result from Fig. a c, the predicted results of the NTB are generally larger than those of the NEB and NHOB. In most of the cases, the obtained results of the NTB and those of the NHOB are in line and close to each other in comparison with those of the NEB.

6 K. Kiani W N /) W N /) b N /).5 b N /) τ / τ f τ /τ f 4 W N /) W N /).6 b N /).4. b N /).5..5 τ /τ f τ /τ f W N /) 4 4 W N /).5 b N /) b N /) τ /τ f τ /τ f Fig. Normalized deflections and nonlocal bending moments at the mid-span point of the DWCNT for different values of the moving nanoparticle velocity a V N =.; b V N =.; c V N =.7; (dotted line NEBT, dot with dashed line NTBT, solid line NHOBT; λ = 5)

7 Vibration of DWCNTs under a moving nanoparticle.. The effect of the slenderness of DWCNTs on the maximum dynamic deflections and nonlocal bending moments of the innermost and outermost tubes Through Figs. and 4, the effects of the slenderness of the innermost tube on the normalized maximum dynamic deflections and bending moments of both innermost and outermost tubes are illustrated for different values of the moving mass velocity and the size effect parameter. The dotted, dashed-dotted and solid lines represent the predicted results associated with the NEBT, NTBT and NHOBT, respectively. In Figs. and 4, the obtained results of various nonlocal beams for both the innermost and outermost tubes generally decrease with λ regardless of the assumed value of the size effect parameter; moreover, for a constant value of the small-scale effect parameter, the obtained results of various nonlocal beam models converge to a constant value as λ increases. Additionally, further investigation shows that for high values of λ the variation of the small-scale effect parameter has roughly no effect on the variation of the results of various nonlocal beams. As it could be seen in Fig. a c, the maximum dynamic deflection of the innermost tube is higher than that of the outermost CNT in most of the cases. By increasing the velocity of the nanoparticle, the difference between the maximum dynamic deflections of the innermost and outermost tubes decreases. Nevertheless, the outermost tube generally undergoes a bending moment more than three times of the innermost tube when the DWCNT is acted upon by a moving nanoparticle (see Fig. 4a c). oreover, the predicted results by the NEBT and the NTBT are in turn the lower and upper bounds of those of the NHOBT, in most of the cases. Regarding the capabilities of the NEBT and NTBT in predicting the maximum dynamic deflection and bending moment, their results are compared with those of the NHOBT for each level of the velocity of the moving nanoparticle. A precise exploration of Fig. a reveals that the difference of the maximum dynamic deflections of the NEB and NTB from that of the NHOB becomes lower than 5% for λ > 5 and λ > 5, correspondingly. However, the relative percentage difference in the maximum nonlocal bending moments of the NEB and NHOB, and the NTB and NHOB exceed approximately by 5% for λ < and λ <, respectively. As a result, utilizing NEBTandNTBTinturnforλ > 5 and λ > 5 leads to the trustworthy results for V N =.; moreover, it is also found from Figs. and 4 that the NEBT and NTBT could be exploited for the dynamic analysis of DWCNTs under a moving nanoparticle with λ > 4 and λ >, respectively. As a general result, the use of NHOBT is highly recommended in the case of λ < (i.e., very stocky DWCNTs) for capturing a more realistic response of DWCNTs under excitation of a moving nanoparticle, irrespective of the value of the small-scale effect parameter and the velocity of the moving nanoparticle. ( W N ( W N ( W N ( W N λ ( W N ( W N λ Fig. Normalized maximum dynamic deflections of the innermost and outermost tubes of the DWCNT in terms of the slenderness of the innermost tube for different values of the moving nanoparticle velocity and small-scale effect parameter a V N =.; b V N =.; c V N =.7 (open circle e a =, triangle e a =.nm, diamond e a =. nm; dotted line NEBT, dot with dashed line NTBT, solid line NHOBT)

8 4 K. Kiani ( b N ( b N ( b N λ ( b N ( b N ( b N λ Fig. 4 Normalized maximum dynamic nonlocal bending moments of the innermost and outermost tubes of the DWCNT in terms of the slenderness of the innermost tube for different values of the moving nanoparticle velocity and small-scale effect parameter a V N =.; b V N =.; c V N =.7 (open circle e a =, triangle e a =.nm, diamond e a =.nm; dotted line NEBT, dot with dashed line NTBT, solid line NHOBT)..4 The effect of the velocity of the moving nanoparticle on the maximum dynamic deflections and nonlocal bending moments of the innermost and outermost tubes Figures 5 and 6 present the maximum dynamic deflections and nonlocal bending moments of the innermost and outermost tubes of the DWCNT as a function of the velocity of the moving nanoparticle for different values of the small-scale effect parameter and slenderness of the innermost tube. In Figs. 5a and6a, the DWCNT is assumed to be very stocky since the magnitude of λ is taken to be. As explained earlier, using the NHOBT is strongly recommended for dynamic analyses of the nanostructure. For low values of the velocity of the moving nanoparticle (. < V N <.), the predicted results of (W N (or ( b N ) for various values of e a are obviously distinct; moreover, the difference of the predicted results of (W N and (W N decreases as the velocity of the moving nanoparticle increases, regardless of the assumed value of e a.in other words, the effect of e a on (W N,(W N,( b N and ( b N vanishes with V N.Abrief comparison of the plots of (W N with those of (W N reveals that the variation of e a has some more effect on the variation of (W N with respect to that of (W N, particularly for. < V N <.. For moderately stocky DWCNTs with λ = 5, the graphs of the dynamic response for both tubes of DWCNTs have been illustrated according to the NTBT (see Figs. 5b and 6b). Finally, for a fairly slender beam with λ = 5, the NEBT is used to study the variations of (W N and (W N in terms of V N (see Figs. 5c and 6c). A brief comparison of the plots in Figs. 5a c and 6a c shows that the variation of e a has more effect on the variation of (W N,(W N,( b N and ( b N of very stocky DWCNTs with respect to the other cases. It means that as the slenderness of the DWCNT increases, the effect of e a on the maximum dynamic deflections and nonlocal bending moments of tubes decreases. As a result, the accuracy of the results of the classical beam theory increases. Additionally, it is seen in Fig. 5a c that the difference between (W N and (W N decreases as λ increases, irrespective of V N. As expected, the magnitude of (W N is larger than that of (W N in all the cases. oreover, a comparison of the demonstrated results in Fig. 5a c with those depicted in Fig. 6a c reveals that the variation of the maximum dynamic deflections is somehow in line with the variation of the maximum nonlocal bending moments. Conclusions The capabilities of the proposed nonlocal beam theories in the former paper [], in predicting the critical velocity of the moving nanoparticle, dynamic deflections and nonlocal bending moments of the innermost and

9 Vibration of DWCNTs under a moving nanoparticle 5 (W N (W N (W N V N (W N (W N (W N V N Fig. 5 Normalized maximum dynamic deflections of the innermost and outermost tubes of the DWCNT as functions of the normalized velocity of the moving nanoparticle for different values of the slenderness of the innermost tube a λ =, analyzed based on the NHOBT; b λ = 5, analyzed based on the NTBT; c λ = 5, analyzed based on the NEBT (dotted line e a =, dot with dashed line e a =.nm,solid line e a =.nm).7 ( b N ) max ( b N ( b N ( b N ( b N.5 ( b N V V N N Fig. 6 Normalized maximum dynamic nonlocal bending moments of the innermost and outermost tubes of the DWCNT as a function of the normalized velocity of the moving nanoparticle for different values of the slenderness of the innermost tube a λ =, analyzed based on the NHOBT; b λ = 5, analyzed based on the NTBT; c λ = 5, analyzed based on the NEBT (dotted line e a =, dot with dashed line e a =.nm,solid line e a =.nm) outermost tubes are examined by various parametric studies. The results reveal that the critical velocity of the moving nanoparticle increases with the slenderness of the DWCNTs and the magnitude of the vdw interaction force, irrespective of the assumed nonlocal beam theory; however, the critical velocity of the moving nanoparticle associated with DWCNTs decreases with the ratio of the mean diameter to the thickness of the innermost tube and the small-scale effect parameter. The role of the small-scale effect parameter, slenderness of the

10 6 K. Kiani DWCNTs and velocity of the moving nanoparticle on the dynamic deflections and nonlocal bending moments as well as their maximum values for both the innermost and outermost tubes are also explored in some detail. The results indicate that the difference between the dynamic deflection of the innermost tube and that of the outermost tube decreases with the velocity of the moving nanoparticle and slenderness of the DWCNTs. It is shown that the maximum nonlocal bending moment within the outermost tube is generally larger than times of that within the innermost tube. oreover, as the slenderness of the DWCNTs decreases, the effect of the small-scale effect parameter on the maximum dynamic deflections and nonlocal bending moments intensifies, especially for low values of the velocity of the moving nanoparticle (. < V N <.). The obtained results manifest that the predicted maximum dynamic deflections and nonlocal bending moments of both tubes by the NEBT and NTBT are generally the lower and upper bounds of those obtained by the NHOBT. As a general result, the use of the NHOBT is highly recommended in the case of λ < for capturing a more realistic dynamic response of DWCNTs under excitation of a moving nanoparticle. Additionally, the predicted results of the NTBT and NEBT would be, respectively, reliable for DWCNTs with λ > 5 and λ > 5. References. Kiani, K.: Application of nonlocal beam models to double-walled carbon nanotubes under a moving nanoparticle. Part I: theoretical formulations. Acta ech. doi:.7/s (). Aydogdu,.: Vibration of multi-walled carbon nanotubes by generalized shear deformation theory. Int. J. ech. Sci. 5, (8). He, X.Q., Eisenberger,., Liew, K..: The effect of van der Waals interaction modeling on the vibration characteristics of multiwalled carbon nanotubes. J. Appl. Phys., 47 (6) 4. Zhang, H., Zhang, S.Y., Wang, T.H.: Flexural vibration analyses of piezoelectric ceramic tubes with mass loads in ultrasonic actuators. Ultrasonics 47, 8 89 (7) 5. Eringen, A.C.: On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J. Appl. Phys. 54, (98) 6. Wang, L., Hu, H.: Flexural wave propagation in single-walled carbon nanotubes. Phys. Rev. B 7, 954 (5) 7. Wang, Q., Han, Q.K., Wen, B.C.: Estimate of material property of carbon nanotubes via nonlocal elasticity. Adv. Theor. Appl. ech., (8) 8. Wang, Q., Wang, C..: The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes. Nanotechnology 8, 757 (7) 9. Aydogdu,.: A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E 4, (9). Yoon, J., Ru, C.Q., ioduchowski, A.: Timoshenko-beam effects on transverse wave propagation in carbon nanotubes. Composites B 5, 87 9 (4). Wang, Q.: Wave propagation in carbon nanotubes via nonlocal continuum mechanics. J. Appl. Phys. 98, 4 (5). Peddieson, J., Buchanan, G.R., cnitt, R.P.: Application of nonlocal continuum models to nanotechnology. Int. J. Eng. Sci. 4, 5 ()