Document Type : Original Article


Department of Mechanical Engineering, University of Guilan, Rasht, Iran


Presented herein is an investigation on the vibrational response of fractional viscoelastic carbon nanotubes (CNTs) conveying fluid and resting on a fractional viscoelastic foundation. The CNTs are modeled according to the Euler-Bernoulli beam theory, and the foundation is considered to be Winkler-type. Also, to incorporate the nanoscale effect into the model, Eringen’s nonlocal elasticity is applied. Derivation of governing equation is done by a variational principle together with the Kelvin-Voigt viscoelastic model. Two solution approaches are developed for obtaining the time response of embedded fluid-conveying CNTs. The first approach is on the basis of Galerkin’s method, while the GDQM and FDM are used in the second approach. Comprehensive numerical results are given to study the effects of elastic foundation, fractional order, damping, fluid, nonlocal parameter, geometrical properties and viscoelasticity coefficient on the time responses of CNTs subject to different boundary conditions.


[1] Iijima, S., 1991, “Helical Microtubes of Graphitic Carbon,” Nature, 354, pp. 56-58.
[2] Dillon, A. C., Jones, K. M., Bekkedahl, T. A., Klang, C. H., Bethune, D. S., and Heben, M. J., 1997, “Storage of Hydrogen in Single-Walled Carbon Nanotubes,” Nature, 386, pp. 377-379.
[3] Dalton, A. B., Collins, S., Muñoz, E., Razal, J. M., Ebron, V. H., Ferraris, J. P., Coleman, J. N., Kim, B. G., and Baughman, R. H., 2003, “Super-Tough Carbon-Nanotube Fibres,” Nature, 423, pp. 361-368.
[4] Postma, H. W. Ch., Teepen, T., Yao, Z., Grifoni, M., and Dekker, C., 2001, “Carbon Nanotube Single-Electron Transistors at Room Temperature,” Science, 293, pp. 76-79.
[5] Chen, P., Kim, H. S., Kwon, S. M., Yun, Y. S., and Jin, H. J., 2009, “Regenerated Bacterial Cellulose/Multi-Walled Carbon Nanotubes Composite Fibers Prepared by Wet-Spinning,” Curr. Appl. Phys., 9, pp. 96-99.
[6] Guldi, D. M., Rahman, G. M. A., Prato, M., Jux, N., Qin, S., and Ford, W., 2005, “Single-Wall Carbon Nanotubes as Integrative Building Blocks for Solar-Energy Conversion,” Angew. Chem., 117, pp. 2051-2054.
[7] Miaudet, P., Badaire, S., Maugey, M., Derré, A., Pichot, V., Launois, P., Poulin, P., and Zakri, C., 2005, “Hot-Drawing of Single and Multiwall Carbon Nanotube Fibers for High Toughness and Alignment,” Nano Lett., 5, pp. 2212-2215.
[8] Zhang, M., Fang, S., Zakhidov, A. A., Lee, S. B., Aliev, A. E., Williams, C. D., Atkinson, K. R., and Baughman, R. H., 2005, “Strong, Transparent, Multifunctional, Carbon Nanotube Sheets,” Science, 309, pp. 1215-1219.
[9] Nardelli, M. B., and Bernholc, J., 1999, “Mechanical Deformations and Coherent Transport in Carbon Nanotubes,” Phys. Rev. B, 60, pp. R16338-R16341.
[10] Falvo, M. R., Clary, G. J., Taylor, R. M., Chi, V., Brooks, F. P., Washburn, S., and Superfine, R., 1997, “Bending and Buckling of Carbon Nanotubes under Large Strain,” Nature, 389, pp. 582-584.
[11] Gurtin, M. E., and Murdoch, A. I., 1975, “A Continuum Theory of Elastic Material Surfaces,” Arch. Rat. Mech. Anal., 57, pp. 291-323.
[12] Gurtin, M. E., and Murdoch, A. I., 1978, “Surface Stress in Solids,” Int. J. Solids Struct., 14, pp. 431-440.
[13] Gibbs, J. W., 1906, “The Scientific Papers of J. Willard Gibbs”, Vol. 1. London, Longmans-Green.
[14] Sedighi, H. M., Keivani, M., and Abadyan, M., 2015, “Modified continuum model for stability analysis of asymmetric FGM double-sided NEMS: Corrections due to finite conductivity, surface energy and nonlocal effect,” Compos. Part B: Eng., 83, pp. 117–133.
[15] Ansari, R., Mohammadi, V., Faghih Shojaei, M., Gholami, R., and Rouhi, H., 2014, “Nonlinear Vibration Analysis of Timoshenko Nanobeams Based on Surface Stress Elasticity Theory,” Eur. J. Mech. A/Solids, 45, pp. 143-152.
[16] Rouhi, H., Ansari, R., and Darvizeh, M., 2016, “Size-dependent free vibration analysis of nanoshells based on the surface stress elasticity,” Appl. Math. Model., 40, pp. 3128–3140.
[17] Rouhi, H., Ansari, R., Darvizeh, M., 2016, “Analytical treatment of the nonlinear free vibration of cylindrical nanoshells based on a first-order shear deformable continuum model including surface influences,” Acta Mech., DOI 10.1007/s00707-016-1595-4.
[18] Sedighi, H. M., and Bozorgmehri, A., 2016, “Nonlinear vibration and adhesion instability of Casimir-induced nonlocal nanowires with the consideration of surface energy,” J. Brazilian Soc. Mech. Sci. Eng., DOI: 10.1007/s40430-016-0530-x.
[19] Rouhi, H., Ansari, R., and Darvizeh, M., 2016, “Nonlinear free vibration analysis of cylindrical nanoshells based on the Ru model accounting for surface stress effect,” Int. J. Mech. Sci., 113, pp. 1–9.
[20] Tadi Beni, Y., Koochi, A., Kazemi, A. S., and Abadyan, M., 2012, “Modeling the influence of surface effect and molecular force on pull-in voltage of rotational nano–micro mirror using 2-DOF model,” Canadian J. Phys., 90, pp. 963-974.
[21] Ansari, R., Mohammadi, V., Faghih Shojaei, M., Gholami, R., and Sahmani, S., 2014, “On the forced vibration analysis of Timoshenko nanobeams based on the surface stress elasticity theory,” Compos. Part B: Eng., 60, pp. 158–166.
[22] Eringen, A. C., 1983, “On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves,” J. Appl. Phys., 54, pp. 4703-4710.
[23] Eringen, A. C., 2002, Nonlocal Continuum Field Theories, Springer, New York.
[24] Yan, J. W., Tong, L. H., Li, C., Zhu, Y., and Wang, Z. W., 2015, “Exact solutions of bending deflections for nano-beams and nano-plates based on nonlocal elasticity theory,” Compos. Struct., 125, pp. 304–313.
[25] Zamani Nejad, M., Hadi, A., and Rastgoo, A., 2016, “Buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on nonlocal elasticity theory,” Int. J. Eng. Sci., 103, pp. 1–10.
[26] Ansari, R., Shahabodini, A., Rouhi, H., and Alipour, A., 2013, “Thermal buckling analysis of multi-walled carbon nanotubes through a nonlocal shell theory incorporating interatomic potentials,” J. Therm. Stresses, 36, pp. 56–70.
[27] Ghorbanpour Arani, A., and Kolahchi, R., 2014, “Exact solution for nonlocal axial buckling of linear carbon nanotube hetero-junctions,” Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci. 228, pp. 366-377.
[28] Rouhi, H., and Ansari, R., 2012, “Nonlocal analytical Flugge shell model for axial buckling of double-walled carbon nanotubes with different end conditions,” NANO, 7, 1250018.
[29] Ansari, R., Rouhi, H., and Mirnezhad, M., 2014, “A hybrid continuum and molecular mechanics model for the axial buckling of chiral single-walled carbon nanotubes,” Curr. Appl. Phys., 14, pp. 1360-1368.
[30] Farajpour, A., Hairi Yazdi, M. R., Rastgoo, A., Loghmani, M., and Mohammadi, M., 2016, “Nonlocal nonlinear plate model for large amplitude vibration of magneto-electro-elastic nanoplates,” Compos. Struct., 140, pp. 323–336.
[31] Ansari, R., Gholami, R., and Rouhi, H., 2015, “Size-Dependent Nonlinear Forced Vibration Analysis of Magneto-Electro-Thermo-Elastic Timoshenko Nanobeams Based upon the Nonlocal Elasticity Theory,” Compos. Struct., 126, pp. 216–226.
[32] Demir, Ç., and Civalek, Ö., 2013, “Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models,” Appl. Math. Model., 37, pp. 9355–9367.
[33] Ansari, R., Rouhi, H., and Sahmani, S., 2014, “Free Vibration analysis of single- and double-walled carbon nanotubes based on nonlocal elastic shell models,” J. Vib. Control, 20, pp. 670-678.
[34] Challamel, N., Picandet, V., Elishakoff, I., Wang, C. M., Collet, B., and Michelitsch, T., 2015, “On Nonlocal Computation of Eigenfrequencies of Beams Using Finite Difference and Finite Element Methods,” Int. J. Str. Stab. Dyn., 15, p. 1540008.
[35] Natsuki, T., Matsuyama, N., and Ni, Q., Q., 2015, “Vibration analysis of carbon nanotube-based resonator using nonlocal elasticity theory,” Appl. Phys. A, 120, pp. 1309-1313.
[36] Hummer, G., Rasaiah, J. C., and Noworyta, J. P., 2001, Water conduction through the hydrophobic channel of a carbon nanotube,” Nature, 414, pp. 188–190.
[37] Ansari, R., Mahmoudinezhad, E., Alipour, A., and Hosseinzadeh, M., 2013, “A comprehensive study on the encapsulation of methane in single-walled carbon nanotubes,” J. Comput. Theor. Nanosci., 10, pp. 2209-2215.
[38] Gao, Y., and Bando, Y., 2002, “Nanotechnology: carbon nanothermometer containing gallium,” Nature, 415, p. 599.
[39] Foldvari, M., and Bagonluri, M., 2008, “Carbon nanotubes as functional excipients for nanomedicines: II. Drug delivery and biocompat-ibility issues,” Nanomed. Nanotechnol. Biol. Med., 4, pp. 183–200.
[40] Zhang, J., and Meguid, S. A., 2016, “Effect of surface energy on the dynamic response and instability of fluid-conveying nanobeams,” Eur. J. Mech. A/Solids, 58, pp. 1–9.
[41] Ansari, R., Norouzzadeh, A., Gholami, R., Faghih Shojaei, M., and Hosseinzadeh, M., 2014, “Size-dependent nonlinear vibration and instability of embedded fluid-conveying SWBNNTs in thermal environment,” Physica E, 61, pp. 148–157.
[42] Hosseini, M., and Sadeghi-Goughari, M., 2016, “Vibration and instability analysis of nanotubes conveying fluid subjected to a longitudinal magnetic field,” Appl. Math. Model., 40, pp. 2560–2576.
[43] Ansari, R., Gholami, R., Norouzzadeh, A., and Darabi, M. A., 2015, “Surface stress effect on the vibration and instability of nanoscale pipes conveying fluid based on a size-dependent Timoshenko beam model,” Acta Mech. Sin., 31, pp. 708-719.
[44] Ansari, R., Norouzzadeh, A., Gholami, R., Faghih Shojaei, M., and Darabi, M. A., 2016, “Geometrically nonlinear free vibration and instability of fluid-conveying nanoscale pipes including surface stress effects,” Microfluidics and Nanofluidics, 20, p. 28.
[45] Chang, T. P., 2011, “Thermal-Nonlocal Vibration and Instability of Single-Walled Carbon Nanotubes Conveying Fluid,” J. Mech., 27, pp. 567-573.
[46] Ansari, R., Faraji Oskouie, M., Sadeghi, F., and Bazdid-Vahdati, M., 2015, “Free vibration of fractional viscoelastic Timoshenko nanobeams using the nonlocal elasticity theory,” Physica E, 74, pp. 318-327.
[47] Faraji Oskouie, M., Ansari, R., and Rouhi, H., 2020, “Investigating vibrations of viscoelastic fluid-conveying carbon nanotubes resting on viscoelastic foundation using a nonlocal fractional Timoshenko beam model,” Proc. IMechE Part N: J. Nanomater. Nanoeng. Nanosys., DOI: 10.1177/2397791420931701.
[48] Ansari, R., Faraji Oskouie, M., and Gholami, R., 2016, “Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory,” Physica E, 75, pp. 266-271.
[49] Lakes, R. S., 2009, Viscoelastic Materials, Cambridge University Press.
[50] Paidoussis, M. P., 1998, Fluid–Structure Interaction, vol. 1, Academic Press, San Diego.
[51] Hartley, T. T., Lorenzo, C. F., and Killory Qammer, H., 1995, “Chaos in a fractional order Chua's system,” IEEE Trans. Circuits Sys. I: Fundamental Theor. Appl., 42, pp. 485-490.
[52] Shu, C., 2000, “Differential Quadrature and Its Application in Engineering,” Springer, London.
[53] Liu, F., Meerschaert, M. M., McGough, R. J., Zhuang, P., and Liu, Q, 2013, “Numerical methods for solving the multi-term time-fractional wave-diffusion equation,” Fract. Calculus Appl. Anal., 16, pp. 9-25.
[54] Zhuang, P., and Liu, F., 2007, “Finite difference approximation for two-dimensional time fractional diffusion equation,” J. Algor. Comput. Technol., 1, pp. 1-15.
[55] Ghavanloo, E., Daneshmand, F., and Rafiei, M., 2010, “Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscoelastic Winkler foundation,” Physica E, 42, pp. 2218-2224.