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Wave Characteristics of Nanotubes Conveying Fluid Based on the Non-classical Timoshenko Beam Model Incorporating Surface Energies

  • Research Article - Mechanical Engineering
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Abstract

The aim of this paper was to investigate the wave propagation of nanotubes conveying fluid by considering the surface stress effect. To this end, the nanotube is modeled as a Timoshenko nanobeam. According to the Gurtin–Murdoch continuum elasticity, the surface stress effect is incorporated into the governing equations of motion obtained from the Hamilton principle. The governing differential equations are solved by generalized differential quadrature method. Then, the effects of the thickness, material and surface stress modulus, residual surface stress, surface density and flow velocity on spectrum curves of nanotubes predicted by both classical and non-classical theories are studied. The first three fundamental modes including flexural, axial, and shear waves of nanotubes are considered.

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Abbreviations

\({L}\) :

Length (\({{m})}\)

\({h}\) :

Thickness (\({{m})}\)

\({d_{\rm i} }\) :

Inner diameter (\({{m})}\)

\({d_{\rm o} }\) :

Outer diameter (\({{m})}\)

\({E}\) :

Young’s modulus (\({{Pa})}\)

\({\nu }\) :

Poisson’s ratio

\({\lambda ,\mu }\) :

Lame’s constants (\({{Pa})}\)

\({\rho }\) :

Mass density (\({{kg}/{m}^{3})}\)

\({A}\) :

Cross-sectional area (\({{m}^{2})}\)

\({I}\) :

Second moment of inertia (\({{m}^{4})}\)

\({\rho _{\rm f} }\) :

Mass density (\({{kg}/{m}^{3})}\)

\({V}\) :

Velocity (\({{m}/{s})}\)

\({A_{\rm f} }\) :

Cross-sectional area (\({{m}^{2})}\)

\({I_{\rm f} }\) :

Second moment of inertia (\({{m}^{4})}\)

\({\left( {U,W,{\varPsi}} \right)}\) :

Amplitude of displacement field

\({k}\) :

Wave number

\({\omega }\) :

Frequency

\({{\bf M}}\) :

Inertia matrix

\({{\bf C}}\) :

Damping matrix

\({{\bf K}}\) :

Stiffness matrix

\({{\bf I}}\) :

Identity matrix

\({{\bf S}}\) :

State-space matrix

\({E_{\rm s} }\) :

Elasticity modulus (\({{Pa})}\)

\({\nu _{\rm s} }\) :

Poisson’s ratio

\({\rho _{\rm s} }\) :

Mass density (\({{kg}/{m}^{2})}\)

\({\tau _{\rm s} }\) :

Residual tension (\({{N}/{m})}\)

\({\lambda _{\rm s} ,\mu _{\rm s} }\) :

Lame’s constants (\({{N}/{m})}\)

\({\left( {x,y,z} \right)}\) :

Cartesian coordinate system

\({\left( {u,w,\psi } \right)}\) :

Displacement field \({\left( {m,m,-} \right)}\)

\({\varepsilon _{xx} }\) :

Strain

\({\sigma _{ij} ,\sigma _{ij}^s }\) :

Stresses (\({{Pa},{N}/{m})}\)

\({N_{xx} ,\bar{N}_{xx} }\) :

Resultant normal forces (\({{N})}\)

\({M_{xx} ,\bar{M}_{xx} }\) :

Resultant bending moments (\({{Nm})}\)

\({Q_x ,\bar{Q}_x}\) :

Resultant shear forces (\({{N})}\)

\({\Pi_{\rm s} }\) :

Strain energy (\({{J})}\)

\({\Pi_T }\) :

Kinetic energy (\({{J})}\)

\({\Pi_{T_{f}} }\) :

Fluid kinetic energy (\({{J})}\)

\({A_{11} ,A_{33} , A_{55} ; D_{11} ,E_{11} }\) :

Stiffness components (in Eq. 15) (\({{N}}\); \({{Nm}^{2})}\)

\({I_0 ; I_2 ,G}\) :

Inertia components (in Eq. 15) (kg/m; kg m)

\({u,w,x, \eta ,\tau , I_0^\ast ,I_2^\ast ,g, a_{11}}\) :

Non-dimensional parameters

\({a_{13} , a_{33} , d_{11} ,e_{11} , K_b ,v}\) :

(in Eq. 17)

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

    R. Ansari, A. Norouzzadeh & M. A. Darabi

  2. Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran

    R. Gholami

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  1. R. Ansari
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Correspondence to R. Gholami.

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Ansari, R., Gholami, R., Norouzzadeh, A. et al. Wave Characteristics of Nanotubes Conveying Fluid Based on the Non-classical Timoshenko Beam Model Incorporating Surface Energies. Arab J Sci Eng 41, 4359–4369 (2016). https://doi.org/10.1007/s13369-016-2132-4

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