- Email: [email protected]
Vibration analysis of fluid-conveying nanotubes embedded in an elastic medium considering surface effects
THEORETICAL & APPLIED MECHANICS LETTERS 2, 031011 (2012)
Vibration analysis of fluid-conveying nanotubes embedded in an elastic medium considering surface effects Yuhang Li,1 Bo Fang,1 Jiazhong Zhang,1 and Jizhou Song2, a) 1) 2)
School of Astronautics, Harbin Institute of Technology, Harbin 150001, China Department of Mechanical and Aerospace Engineering, University of Miami, Coral Gables, FL 33146, USA
(Received 2 February 2012; accepted 19 March 2012; published online 10 May 2012) Abstract An analytical model is developed to study the surface effects on the vibration behavior including the natural frequency and the critical flow velocity of fluid-conveying nanotubes embedded in an elastic medium. The effects of surface elasticity and residual surface stress are accounted through the surface elasticity model and the Young-Laplace equation. A Winkler-type foundation is employed to model the interaction of nanotubes and the surrounding medium. The results show that the surface effects have more prominent influences on the nature frequency with smaller nanotube thickness, larger aspect ratio and larger elastic medium constants. Both surface layers and the elastic medium enhance the stability of nanotubes. This study might be helpful for designing the c 2012 The Chinese Society of fluid-conveying nanotube devices in NEMS and MEMS systems. Theoretical and Applied Mechanics. [doi:10.1063/2.1203111] Keywords surface effect, fluid-conveying nanotube, vibration, stability Novel fabrication techniques have opened up new possibilities for the development of small-scale devices. One example is the fluid-conveying nanotubes with potential applications in molecular separation and detection, biocatalysis, and drug delivery.1–7 For fluidconveying nanotubes, one of the important issues is to accurately predict the vibration behavior such as natural frequency and critical flow velocity. Many researchers have studied this issue. Reddy et al.8 obtained the free vibration of carbon nanotubes using atomistic simulations and the continuum beam model. Yoon et al.9,10 studied the influence of the internal moving fluid on the free vibration frequency and the structural instability of carbon nanotubes based on a continuum elastic model. Chang and Lee11 developed the Timoshenko beam model for the vibration of the fluid-conveying carbon nanotubes. Recently, the nonlocal beam model has been adopted to study the vibration behavior of fluidconveying nanotubes. Lee and Chang12 studied the effects of fluid velocity on the vibration frequency and mode shape of the fluid-conveying carbon nanotubes. Wang13 developed an analytical model for the doublewalled carbon nanotubes with conveying fluid using the theory of nonlocal elasticity. Unlike the macroscopic counterparts, fluidconveying nanotubes would show different behavior due to the large ratio of surface area to volume at nanoscale. It is believed that surface plays a key role to accurately predict the vibration behavior of fluidconveying nanotubes. Surface effects on the behavior of nanostructures (e.g., nanobeam and nanofilm) have been studied extensively.14–23 However, the investigation of surface effects on fluid-conveying nanotubes is very limited. Wang24 studied the surface effects on vibrations of free-standing fluid-conveying nanotubes a) Corresponding
author. Email: [email protected].
based on the non-local elasticity theory and found the surface effects with positive elastic constant and positive residual tension tend to increase the natural frequency of and critical flow velocity, which is very useful to design nanofluidic devices. Most applications will either require fluid-conveying tubes to be on or embedded in a medium. As far as we know, there has been no investigation on the fluid-conveying nanotubes considering both surface effects and substrate effects. In this paper, the vibration of fluid-conveying nanotubes embedded in an elastic medium considering surface effects including surface elasticity and surface residual stress is analyzed using a surface elasticity model and the Young-Laplace equation. We consider an L-long circular fluid-conveying nanotubes with the outer diameter do and inner diameter di embedded in an elastic medium as shown in Fig. 1. The nanotube is modeled as a straight and slender beam and the medium is modeled as a Winkler-type elastic foundation with the Winkler constant k.25 The fluid in the tube has a constant velocity U moving from one end to the other. The surface layer model is employed to study the fluid-conveying nanotubes with both inner and outer surface layers with negligible thickness (Fig. 1(b)). It is well known that the effects of surface elasticity and surface residual stress can be accounted by the surface elastic model and Young-Laplace equation, respectively.16,26,27 The surface elasticity increases the bending rigidity and the additional bending rigidity can be expressed as24 α=
1 πE s (d3i + d3o ), 8
(1)
where E s is the surface elastic modulus. The surface residual stress induces a distributed transverse load through the Young-Laplace equations as24,26,27 p(x) = P
∂2w , ∂x2
(2)
031011-2
Theor. Appl. Mech. Lett. 2, 031011 (2012)
Y. H. Li, B. Fang, J. Z. Zhang, and J. Z. Song
(kˆ − ω ˆ 2 )ϕˆ = 0,
(7)
∂ ϕ(0) ˆ ∂ ϕ(1) ˆ = ϕ(0) ˆ = = ϕ(1) ˆ = 0. ∂x ˆ ∂x ˆ
(8)
The solution of Eq. (7) can be expressed as ϕ(ˆ ˆ x) = Ceiγ xˆ ,
(9)
where C is a constant. Substitution of Eq. (9) into Eq. (7) gives (1 + α ˆ )γ 4 − (ˆ u2 − Pˆ π 2 )γ 2 − 2γ ω ˆu ˆβ 1/2 + (kˆ − ω ˆ 2 ) = 0.
(10)
The solution of Eq. (7) can be rewritten as Fig. 1. Schematic diagram of (a) a fluid-conveying nanotube embedded in an elastic medium; (b) the circular cross section with surface layers.
where P = 2τ 0 (di + do ), w denotes the deflection of the nanotube, x is the axial coordinate and τ 0 is the surface residual stress. The dynamic equation of the embedded fluid-conveying nanotube considering surface effects such as surface elasticity and surface residual stress can be obtained as24 ∂2w ∂2w ∂4w 2 + (M U − P ) + 2M U + ∂x4 ∂x2 ∂x∂τ 2 ∂ w (m + M ) 2 + kw = 0, (3) ∂τ
(EI + α)
where EI denotes the bending rigidity of the nanotube without surface effects, M and m are the mass of the fluid and the nanotube per unit length, respectively, τ is the time. Introducing w ˆ = w/L, x ˆ = x/L, τˆ = [(EI)/(M + m)]1/2 τ /L2 , u ˆ = [M/(EI)]1/2 U L, α ˆ = α/(EI), β = M/(M + m), Pˆ = P L2 /[π 2 (EI)], kˆ = kL4 /(EI) and ω ˆ = [(M + m)/(EI)]1/2 L2 ω normalizes Eq. (3) as 2
4
2
∂ w ˆ ∂ w ˆ ∂ w ˆ + u2 − Pˆ π 2 ) 2 + 2ˆ uβ 1/2 (1 + α ˆ ) 4 + (ˆ ∂x ˆ ∂x ˆ ∂x ˆ∂ τˆ 2 ∂ w ˆ ˆ + kw ˆ = 0, (4) 2 ∂ τˆ where ω is the natural frequency of the nanotube. The doubly clamped fluid-conveying nanotube is considered in this paper to demonstrate our approach. The boundaries at the ends (ˆ x = 0, 1) satisfy ∂ w(0, ˆ τˆ) ∂ w(1, ˆ τˆ) = w(0, ˆ τˆ) = = w(1, ˆ τˆ) = 0. ∂x ˆ ∂x ˆ
ϕ(ˆ ˆ x) =
w(ˆ ˆ x, τˆ) = ϕ(ˆ ˆ x)e
.
Then Eqs. (4) and (5) become d4 ϕˆ d2 ϕˆ dϕˆ (1 + α ˆ ) 4 + (ˆ u2 − Pˆ π 2 ) 2 + 2iˆ ωu ˆβ 1/2 + dˆ x dˆ x dˆ x
Cn eiγn xˆ ,
(11)
n=1
where γn (n = 1, 2, 3, 4) are determined by Eq. (10) and the constants Cn (n = 1, 2, 3, 4) by the boundary conditions Eq. (8). Substituting Eq. (11) into Eq. (8) yields 1 1 1 1 C1 γ2 γ3 γ4 C2 γ1 = 0. (12) eiγ1 eiγ2 eiγ3 eiγ4 C3 iγ1 iγ2 iγ3 iγ4 C4 γ1 e γ2 e γ3 e γ4 e
The existence of the non-trivial solution of Eq. (12), i.e., the determinant of coefficient matrix is equal to zero, gives the natural frequency ω ˆ. To obtain the critical flow velocity, the terms involving time are set to be zero and we have (1 + α ˆ)
2 ˆ ˆ ∂4w ˆ 2 ˆ π2 ) ∂ w + (ˆ u − P + kw ˆ = 0, ∂x ˆ4 ∂x ˆ2
(13)
with the boundaries ∂ w(0) ˆ ∂ w(1) ˆ = w(0) ˆ = = w(1) ˆ = 0. ∂x ˆ ∂x ˆ
(14)
The solution of Eq. (13) takes the form w ˆ = A cos(λ1 x ˆ) + B sin(λ1 x ˆ) + C cos(λ2 x ˆ) + D sin(λ2 x ˆ),
(15)
where
(5) λ1 =
The solution of Eq. (4) takes the form of iˆ ω τˆ
4 X
(6) λ2 =
v v u !2 u 2 ˆ 2 u 2 u u ˆ π2 uu ˆ − P π ˆ − P 4kˆ t u + − u 1+α ˆ 1+α ˆ 1+α ˆ t 2 v v u !2 u 2 ˆ 2 u 2 u u ˆ π2 uu ˆ − P π ˆ − P 4kˆ t u − − u 1+α ˆ 1+α ˆ 1+α ˆ t 2
,
. (16)
031011-3
Vibration analysis of fluid-conveying nanotubes embedded
Theor. Appl. Mech. Lett. 2, 031011 (2012)
Fig. 2. The normalized natural frequency of the fluidconveying nanotube versus the normalized flow velocity.
Fig. 3. The normalized critical flow velocity versus the thickness of the fluid-conveying nanotube.
Substituting Eq. (15) into the boundary condition Eq. (14), we have cos(λ1 ) − cos(λ2 ) sin(λ1 ) − λλ13 sin(λ2 ) · λ2 sin(λ2 ) − λ1 sin(λ1 ) λ1 (cos(λ1 ) − cos(λ2 )) A = 0. (17) B
length of nanotube is chosen as 20 times as do . It is found that surface layers and substrate both enhance the stability of fluid-conveying nanotubes. For example, for t = 4 nm, the normalized critical velocity is 6.28 without considering the surface effects, and it increases to 7.66 after considering surface effects but not substrate effect and 9.73 after considering both surface and substrate (kˆ = 500) effects. It is also observed that as the thickness increases, the critical flow velocity decreases. For example, the critical velocity increases 62% of the velocity without surface effects for thickness 1 nm and kˆ = 0, while it only increases 13% for thickness 8 nm and kˆ = 0. These results clearly show the underlying physics of the surface and substrate effects on the dynamic behavior of fluid-conveying nanotubes. Surface effects increase the bending rigidity of tube, which makes the tube more stable and increases the natural frequency and the critical flow velocity. The substrate has a similar effect because it constrains the deformation of the tube and therefore it makes the tube more stable than a free one. In summary, an analytical model is developed to investigate the surface effects on the vibration behaviors (e.g. natural frequency and stability) of fluid-conveying nanotubes embedded in an elastic medium. The results show that the elastic coefficient of medium, thickness of nanotubes and aspect ratios have significant effects on the dynamic behaviors of the nanotubes after considering the surface effects. This article might be helpful for designing the nanotubes for fluid-conveying applications embedded in elastic medium in NEMS and MEMS.
The critical flow velocity of the fluid-conveying nanotubes can then be obtained by letting the determinant of the coefficient matrix equal to zero. Alumina with crystallographic of [1 1 1] is taken as an example to show the surface effects. We follow 3 Wang24 to choose E = 70 GPa, ρt = 2 700 kg/m s 0 (mass density of the tube), E = 5.188 2 N/m, τ = 3 0.910 8 N/m, ρw = 1 000 kg/m (mass density of the internal water), and di = 20 nm. It should be noted that the predicted trend of surface effects depends on the surface properties. The trend may be different for other materials. Figure 2 shows the normalized natural frequency of the fluid-conveying nanotube from Eq. (12) versus the normalized flow velocity for different thicknesses t = (do − di )/2, aspect ratios L/do , and different Winˆ The normalized natural frequency dekler constants k. creases as the flow velocity increases. The natural frequency becomes zero (i.e., the nanotube buckles) when the flow velocity increases to a critical velocity. It can be observed that surface effects generally increase the normalized natural frequency comparing to the counterpart without surface effects. The thickness has a negative influence on the natural frequency, i.e., the smaller the thickness, the larger the natural frequency. On the contrary, the aspect ratio and Winkler constant have opposite influences, i.e., the larger the aspect ratio and the stiffer the elastic medium, the larger the natural frequency. Figure 3 shows the normalized critical flow velocity of fluid-conveying nanotubes from Eq. (17) as the function of thickness for different elastic medium. The
This work was supported by YL acknowledges China Scholarship Council (CSC), the Provost Award (University of Miami), the Ralph E. Powe Junior Faculty Enhancement Award (ORAU) and NSF (OISE1043161). 1. K. B. Jirage, J. C. Hulteen, and C. R. Martin, Science 278, 655 (1997).
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Y. H. Li, B. Fang, J. Z. Zhang, and J. Z. Song
2. D. T. Mitchell, S. B. Lee, and L. N. Li, et al., J. Am. Chem. Soc. 124, 11864 (2002). 3. D. W. Deamer, and M. Akeson, Trends Biotechnol. 18, 147 (2000). 4. P. M. Ajayan, and S. Iijima, Nature 361, 333 (1993). 5. P. M. Ajayan, O. Stephan, and P. Redlich, et al., Nature 375, 564 (1995). 6. A. I. Skoulidas, D. M. Ackerman, and J. K. Johnson, et al., Phys. Rev. Lett. 89, 185901 (2002). 7. J. K. Holt, H. G. Park, and Y. Wang, et al., Science 312, 1034 (2006). 8. C. D. Reddy, C. Lu, and S. Rajendran, et al., Appl. Phys. Lett. 90, 133122 (2007). 9. J. Yoon, C. Q. Ru, and A. Mioduchowski, Int. J. Solids. Struct. 43, 3337 (2006). 10. J. Yoon, C. Q. Ru, and A. Mioduchowski, Compos. Sci. Technol. 65, 1326 (2005). 11. W. J. Chang, and H. L. Lee, Phys. Lett. A 373, 982 (2009). 12. H. L. Lee, and W. J. Chang, J. Appl. Phys. 103, 024302 (2008). 13. L. Wang, Comput. Mater. Sci. 45, 584 (2009). 14. M. E. Gurtin, X. Markenscoff, and R. N. Thurston, Appl. Phys. Lett. 29, 529 (1976). 15. M. E. Gurtin, J. Weissmuller, and F. Larche, Philos. Mag. A 78, 1093 (1998).
Theor. Appl. Mech. Lett. 2, 031011 (2012)
16. G. F. Wang, and X. Q. Feng, Appl. Phys. Lett. 90, 231904 (2007). 17. G. F. Wang, and X. Q. Feng, J. Phys. D: Appl. Phys. 42, 155411 (2009). 18. C. F. Lu, W. Q. Chen, and C. W. Lim, Compos. Sci. Technol. 69 1124 (2009). 19. C. F. Lu, C. W. Lim, and W. Q. Chen, Int. J. Solids. Struct. 46, 1176 (2009). 20. Z. Yan, and L. Y. Jiang, J. Phys. D: Appl. Phys. 44, 075404 (2011). 21. G. F. Wang, and X. Q. Feng, EPL 91, 56007 (2010). 22. Z. Yan, and L. Y. Jiang, Nanotechnology 22, 245703 (2011). 23. B. Farshi, A. Assadi, and A. Alinia-ziazi, Appl. Phys. Lett. 96, 093105 (2010). 24. L. Wang, Physica E 43, 437 (2010). 25. S. P. Timoshenko, and J. M. Geer, Theory of Elastic Stability (McGraw-Hill, New York, 1961) 26. Y. Li, J. Song, and B. Fang, et al., J. Phys. D: Appl. Phys. 44, 425304 (2011). 27. Y. Li, B. Fang, and J. Zhang, et al., Thin Solid Films 520, 2077 (2012). 28. T. Chen, M. S. Chiu, and C. N. Weng, J. Appl. Phys. 100, 074308 (2006).
Vibration analysis of fluid-conveying nanotubes embedded in an elastic medium considering surface effects Yuhang Li,1 Bo Fang,1 Jiazhong Zhang,1 and Jizhou Song2, a) 1) 2)
School of Astronautics, Harbin Institute of Technology, Harbin 150001, China Department of Mechanical and Aerospace Engineering, University of Miami, Coral Gables, FL 33146, USA
(Received 2 February 2012; accepted 19 March 2012; published online 10 May 2012) Abstract An analytical model is developed to study the surface effects on the vibration behavior including the natural frequency and the critical flow velocity of fluid-conveying nanotubes embedded in an elastic medium. The effects of surface elasticity and residual surface stress are accounted through the surface elasticity model and the Young-Laplace equation. A Winkler-type foundation is employed to model the interaction of nanotubes and the surrounding medium. The results show that the surface effects have more prominent influences on the nature frequency with smaller nanotube thickness, larger aspect ratio and larger elastic medium constants. Both surface layers and the elastic medium enhance the stability of nanotubes. This study might be helpful for designing the c 2012 The Chinese Society of fluid-conveying nanotube devices in NEMS and MEMS systems. Theoretical and Applied Mechanics. [doi:10.1063/2.1203111] Keywords surface effect, fluid-conveying nanotube, vibration, stability Novel fabrication techniques have opened up new possibilities for the development of small-scale devices. One example is the fluid-conveying nanotubes with potential applications in molecular separation and detection, biocatalysis, and drug delivery.1–7 For fluidconveying nanotubes, one of the important issues is to accurately predict the vibration behavior such as natural frequency and critical flow velocity. Many researchers have studied this issue. Reddy et al.8 obtained the free vibration of carbon nanotubes using atomistic simulations and the continuum beam model. Yoon et al.9,10 studied the influence of the internal moving fluid on the free vibration frequency and the structural instability of carbon nanotubes based on a continuum elastic model. Chang and Lee11 developed the Timoshenko beam model for the vibration of the fluid-conveying carbon nanotubes. Recently, the nonlocal beam model has been adopted to study the vibration behavior of fluidconveying nanotubes. Lee and Chang12 studied the effects of fluid velocity on the vibration frequency and mode shape of the fluid-conveying carbon nanotubes. Wang13 developed an analytical model for the doublewalled carbon nanotubes with conveying fluid using the theory of nonlocal elasticity. Unlike the macroscopic counterparts, fluidconveying nanotubes would show different behavior due to the large ratio of surface area to volume at nanoscale. It is believed that surface plays a key role to accurately predict the vibration behavior of fluidconveying nanotubes. Surface effects on the behavior of nanostructures (e.g., nanobeam and nanofilm) have been studied extensively.14–23 However, the investigation of surface effects on fluid-conveying nanotubes is very limited. Wang24 studied the surface effects on vibrations of free-standing fluid-conveying nanotubes a) Corresponding
author. Email: [email protected].
based on the non-local elasticity theory and found the surface effects with positive elastic constant and positive residual tension tend to increase the natural frequency of and critical flow velocity, which is very useful to design nanofluidic devices. Most applications will either require fluid-conveying tubes to be on or embedded in a medium. As far as we know, there has been no investigation on the fluid-conveying nanotubes considering both surface effects and substrate effects. In this paper, the vibration of fluid-conveying nanotubes embedded in an elastic medium considering surface effects including surface elasticity and surface residual stress is analyzed using a surface elasticity model and the Young-Laplace equation. We consider an L-long circular fluid-conveying nanotubes with the outer diameter do and inner diameter di embedded in an elastic medium as shown in Fig. 1. The nanotube is modeled as a straight and slender beam and the medium is modeled as a Winkler-type elastic foundation with the Winkler constant k.25 The fluid in the tube has a constant velocity U moving from one end to the other. The surface layer model is employed to study the fluid-conveying nanotubes with both inner and outer surface layers with negligible thickness (Fig. 1(b)). It is well known that the effects of surface elasticity and surface residual stress can be accounted by the surface elastic model and Young-Laplace equation, respectively.16,26,27 The surface elasticity increases the bending rigidity and the additional bending rigidity can be expressed as24 α=
1 πE s (d3i + d3o ), 8
(1)
where E s is the surface elastic modulus. The surface residual stress induces a distributed transverse load through the Young-Laplace equations as24,26,27 p(x) = P
∂2w , ∂x2
(2)
031011-2
Theor. Appl. Mech. Lett. 2, 031011 (2012)
Y. H. Li, B. Fang, J. Z. Zhang, and J. Z. Song
(kˆ − ω ˆ 2 )ϕˆ = 0,
(7)
∂ ϕ(0) ˆ ∂ ϕ(1) ˆ = ϕ(0) ˆ = = ϕ(1) ˆ = 0. ∂x ˆ ∂x ˆ
(8)
The solution of Eq. (7) can be expressed as ϕ(ˆ ˆ x) = Ceiγ xˆ ,
(9)
where C is a constant. Substitution of Eq. (9) into Eq. (7) gives (1 + α ˆ )γ 4 − (ˆ u2 − Pˆ π 2 )γ 2 − 2γ ω ˆu ˆβ 1/2 + (kˆ − ω ˆ 2 ) = 0.
(10)
The solution of Eq. (7) can be rewritten as Fig. 1. Schematic diagram of (a) a fluid-conveying nanotube embedded in an elastic medium; (b) the circular cross section with surface layers.
where P = 2τ 0 (di + do ), w denotes the deflection of the nanotube, x is the axial coordinate and τ 0 is the surface residual stress. The dynamic equation of the embedded fluid-conveying nanotube considering surface effects such as surface elasticity and surface residual stress can be obtained as24 ∂2w ∂2w ∂4w 2 + (M U − P ) + 2M U + ∂x4 ∂x2 ∂x∂τ 2 ∂ w (m + M ) 2 + kw = 0, (3) ∂τ
(EI + α)
where EI denotes the bending rigidity of the nanotube without surface effects, M and m are the mass of the fluid and the nanotube per unit length, respectively, τ is the time. Introducing w ˆ = w/L, x ˆ = x/L, τˆ = [(EI)/(M + m)]1/2 τ /L2 , u ˆ = [M/(EI)]1/2 U L, α ˆ = α/(EI), β = M/(M + m), Pˆ = P L2 /[π 2 (EI)], kˆ = kL4 /(EI) and ω ˆ = [(M + m)/(EI)]1/2 L2 ω normalizes Eq. (3) as 2
4
2
∂ w ˆ ∂ w ˆ ∂ w ˆ + u2 − Pˆ π 2 ) 2 + 2ˆ uβ 1/2 (1 + α ˆ ) 4 + (ˆ ∂x ˆ ∂x ˆ ∂x ˆ∂ τˆ 2 ∂ w ˆ ˆ + kw ˆ = 0, (4) 2 ∂ τˆ where ω is the natural frequency of the nanotube. The doubly clamped fluid-conveying nanotube is considered in this paper to demonstrate our approach. The boundaries at the ends (ˆ x = 0, 1) satisfy ∂ w(0, ˆ τˆ) ∂ w(1, ˆ τˆ) = w(0, ˆ τˆ) = = w(1, ˆ τˆ) = 0. ∂x ˆ ∂x ˆ
ϕ(ˆ ˆ x) =
w(ˆ ˆ x, τˆ) = ϕ(ˆ ˆ x)e
.
Then Eqs. (4) and (5) become d4 ϕˆ d2 ϕˆ dϕˆ (1 + α ˆ ) 4 + (ˆ u2 − Pˆ π 2 ) 2 + 2iˆ ωu ˆβ 1/2 + dˆ x dˆ x dˆ x
Cn eiγn xˆ ,
(11)
n=1
where γn (n = 1, 2, 3, 4) are determined by Eq. (10) and the constants Cn (n = 1, 2, 3, 4) by the boundary conditions Eq. (8). Substituting Eq. (11) into Eq. (8) yields 1 1 1 1 C1 γ2 γ3 γ4 C2 γ1 = 0. (12) eiγ1 eiγ2 eiγ3 eiγ4 C3 iγ1 iγ2 iγ3 iγ4 C4 γ1 e γ2 e γ3 e γ4 e
The existence of the non-trivial solution of Eq. (12), i.e., the determinant of coefficient matrix is equal to zero, gives the natural frequency ω ˆ. To obtain the critical flow velocity, the terms involving time are set to be zero and we have (1 + α ˆ)
2 ˆ ˆ ∂4w ˆ 2 ˆ π2 ) ∂ w + (ˆ u − P + kw ˆ = 0, ∂x ˆ4 ∂x ˆ2
(13)
with the boundaries ∂ w(0) ˆ ∂ w(1) ˆ = w(0) ˆ = = w(1) ˆ = 0. ∂x ˆ ∂x ˆ
(14)
The solution of Eq. (13) takes the form w ˆ = A cos(λ1 x ˆ) + B sin(λ1 x ˆ) + C cos(λ2 x ˆ) + D sin(λ2 x ˆ),
(15)
where
(5) λ1 =
The solution of Eq. (4) takes the form of iˆ ω τˆ
4 X
(6) λ2 =
v v u !2 u 2 ˆ 2 u 2 u u ˆ π2 uu ˆ − P π ˆ − P 4kˆ t u + − u 1+α ˆ 1+α ˆ 1+α ˆ t 2 v v u !2 u 2 ˆ 2 u 2 u u ˆ π2 uu ˆ − P π ˆ − P 4kˆ t u − − u 1+α ˆ 1+α ˆ 1+α ˆ t 2
,
. (16)
031011-3
Vibration analysis of fluid-conveying nanotubes embedded
Theor. Appl. Mech. Lett. 2, 031011 (2012)
Fig. 2. The normalized natural frequency of the fluidconveying nanotube versus the normalized flow velocity.
Fig. 3. The normalized critical flow velocity versus the thickness of the fluid-conveying nanotube.
Substituting Eq. (15) into the boundary condition Eq. (14), we have cos(λ1 ) − cos(λ2 ) sin(λ1 ) − λλ13 sin(λ2 ) · λ2 sin(λ2 ) − λ1 sin(λ1 ) λ1 (cos(λ1 ) − cos(λ2 )) A = 0. (17) B
length of nanotube is chosen as 20 times as do . It is found that surface layers and substrate both enhance the stability of fluid-conveying nanotubes. For example, for t = 4 nm, the normalized critical velocity is 6.28 without considering the surface effects, and it increases to 7.66 after considering surface effects but not substrate effect and 9.73 after considering both surface and substrate (kˆ = 500) effects. It is also observed that as the thickness increases, the critical flow velocity decreases. For example, the critical velocity increases 62% of the velocity without surface effects for thickness 1 nm and kˆ = 0, while it only increases 13% for thickness 8 nm and kˆ = 0. These results clearly show the underlying physics of the surface and substrate effects on the dynamic behavior of fluid-conveying nanotubes. Surface effects increase the bending rigidity of tube, which makes the tube more stable and increases the natural frequency and the critical flow velocity. The substrate has a similar effect because it constrains the deformation of the tube and therefore it makes the tube more stable than a free one. In summary, an analytical model is developed to investigate the surface effects on the vibration behaviors (e.g. natural frequency and stability) of fluid-conveying nanotubes embedded in an elastic medium. The results show that the elastic coefficient of medium, thickness of nanotubes and aspect ratios have significant effects on the dynamic behaviors of the nanotubes after considering the surface effects. This article might be helpful for designing the nanotubes for fluid-conveying applications embedded in elastic medium in NEMS and MEMS.
The critical flow velocity of the fluid-conveying nanotubes can then be obtained by letting the determinant of the coefficient matrix equal to zero. Alumina with crystallographic of [1 1 1] is taken as an example to show the surface effects. We follow 3 Wang24 to choose E = 70 GPa, ρt = 2 700 kg/m s 0 (mass density of the tube), E = 5.188 2 N/m, τ = 3 0.910 8 N/m, ρw = 1 000 kg/m (mass density of the internal water), and di = 20 nm. It should be noted that the predicted trend of surface effects depends on the surface properties. The trend may be different for other materials. Figure 2 shows the normalized natural frequency of the fluid-conveying nanotube from Eq. (12) versus the normalized flow velocity for different thicknesses t = (do − di )/2, aspect ratios L/do , and different Winˆ The normalized natural frequency dekler constants k. creases as the flow velocity increases. The natural frequency becomes zero (i.e., the nanotube buckles) when the flow velocity increases to a critical velocity. It can be observed that surface effects generally increase the normalized natural frequency comparing to the counterpart without surface effects. The thickness has a negative influence on the natural frequency, i.e., the smaller the thickness, the larger the natural frequency. On the contrary, the aspect ratio and Winkler constant have opposite influences, i.e., the larger the aspect ratio and the stiffer the elastic medium, the larger the natural frequency. Figure 3 shows the normalized critical flow velocity of fluid-conveying nanotubes from Eq. (17) as the function of thickness for different elastic medium. The
This work was supported by YL acknowledges China Scholarship Council (CSC), the Provost Award (University of Miami), the Ralph E. Powe Junior Faculty Enhancement Award (ORAU) and NSF (OISE1043161). 1. K. B. Jirage, J. C. Hulteen, and C. R. Martin, Science 278, 655 (1997).
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Y. H. Li, B. Fang, J. Z. Zhang, and J. Z. Song
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