Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field

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Authors

  • N. D. Anh Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
  • D. V. Hieu Thai Nguyen University of Technology, Thai Nguyen, Vietnam https://orcid.org/0000-0001-8789-7284

DOI:

https://doi.org/10.15625/0866-7136/15467

Keywords:

nonlinear vibration, carbon nanotube, nonlocal strain gradient, magnetic field, Galerkin method, equivalent linearization, weighted averaging

Abstract

The nonlinear free vibration of embedded nanotubes under longitudinal magnetic field is studied in this paper. The governing equation for the nanotube is formulated by employing Euler – Bernoulli beam model and the nonlocal strain gradient theory. The analytical expression of the nonlinear frequency of the nanotube is obtained by using Galerkin method and the equivalent linearization method with the weighted averaging value. The accuracy of the obtained solution has been verified by comparison with the published solutions and the exact solution. The influences of the nonlocal parameter, material length scale parameter, aspect ratio, diameter ratio, Winkler parameter and longitudinal magnetic field on the nonlinear vibration responses of the nanotubes with pinned-pinned and clamped-clamped boundary conditions are investigated and discussed.

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References

S. Iijima. Helical microtubules of graphitic carbon. Nature, 354, (6348), (1991), pp. 56–58. https://doi.org/10.1038/354056a0. https://doi.org/10.1038/354056a0.">

S. Chakraverty and S. K. Jena. Free vibration of single walled carbon nanotube resting on exponentially varying elastic foundation. Curved and Layered Structures, 5, (1), (2018), pp. 260–272. https://doi.org/10.1515/cls-2018-0019. https://doi.org/10.1515/cls-2018-0019.">

P. Poncharal, Z. L. Wang, D. Ugarte, and W. A. De Heer. Electrostatic deflections and electromechanical resonances of carbon nanotubes. Science, 283, (5407), (1999), pp. 1513–1516. https://doi.org/10.1126/science.283.5407.1513. https://doi.org/10.1126/science.283.5407.1513.">

J. A. Pelesko and D. H. Bernstein. Modeling MEMS and NEMS. FL: Chapman & Hall/CRC, Boca Raton, (2003).

K. Tsukagoshi, N. Yoneya, S. Uryu, Y. Aoyagi, A. Kanda, Y. Ootuka, and B. W. Alphenaar. Carbon nanotube devices for nanoelectronics. Physica B: Condensed Matter, 323, (1-4), (2002), pp. 107–114. https://doi.org/10.1016/s0921-4526(02)00993-6. https://doi.org/10.1016/s0921-4526(02)00993-6.">

W. B. Choi, E. Bae, D. Kang, S. Chae, B.-h. Cheong, J.-h. Ko, E. Lee, and W. Park. Aligned carbon nanotubes for nanoelectronics. Nanotechnology, 15, (10), (2004). https://doi.org/10.1088/0957-4484/15/10/003. https://doi.org/10.1088/0957-4484/15/10/003.">

M. Samadishadlou, M. Farshbaf, N. Annabi, T. Kavetskyy, R. Khalilov, S. Saghfi, A. Akbarzadeh, and S. Mousavi. Magnetic carbon nanotubes: preparation, physical properties, and applications in biomedicine. Artificial Cells, Nanomedicine, and Biotechnology, 46, (7), (2018), pp. 1314–1330. https://doi.org/10.1080/21691401.2017.1389746. https://doi.org/10.1080/21691401.2017.1389746.">

Y. Gao and Y. Bando. Carbon nanothermometer containing gallium. Nature, 415, (6872), (2002), pp. 599–599. https://doi.org/10.1038/415599a. https://doi.org/10.1038/415599a.">

G. Hummer, J. C. Rasaiah, and J. P. Noworyta. Water conduction through the hydrophobic channel of a carbon nanotube. Nature, 414, (6860), (2001), pp. 188–190. https://doi.org/10.1038/35102535. https://doi.org/10.1038/35102535.">

A. C. Eringen and D. G. B. Edelen. On nonlocal elasticity. International Journal of Engineering Science, 10, (3), (1972), pp. 233–248. https://doi.org/10.1016/0020-7225(72)90039-0. https://doi.org/10.1016/0020-7225(72)90039-0.">

A. C. Eringen. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54, (9), (1983), pp. 4703–4710. https://doi.org/10.1063/1.332803. https://doi.org/10.1063/1.332803.">

R. D. Mindlin. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16, (1), (1964), pp. 51–78. https://doi.org/10.1007/bf00248490. https://doi.org/10.1007/bf00248490.">

R. D. Mindlin. Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures, 1, (4), (1965), pp. 417–438. https://doi.org/10.1016/0020-7683(65)90006-5. https://doi.org/10.1016/0020-7683(65)90006-5.">

F. A. C. M. Yang, A. C. M. Chong, D. C. C. Lam, and P. Tong. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 39, (10), (2002), pp. 2731–2743. https://doi.org/10.1016/s0020-7683(02)00152-x. https://doi.org/10.1016/s0020-7683(02)00152-x.">

J. Yang, L. L. Ke, and S. Kitipornchai. Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. Physica E: Low-dimensional Systems and Nanostructures, 42, (5), (2010), pp. 1727–1735. https://doi.org/10.1016/j.physe.2010.01.035. https://doi.org/10.1016/j.physe.2010.01.035.">

S. Narendar, S. S. Gupta, and S. Gopalakrishnan. Wave propagation in single-walled carbon nanotube under longitudinal magnetic field using nonlocal Euler–Bernoulli beam theory. Applied Mathematical Modelling, 36, (9), (2012), pp. 4529–4538. https://doi.org/10.1016/j.apm.2011.11.073. https://doi.org/10.1016/j.apm.2011.11.073.">

Z. Zhang, Y. Liu, and B. Li. Free vibration analysis of fluid-conveying carbon nanotube via wave method. Acta Mechanica Solida Sinica, 27, (6), (2014), pp. 626–634. https://doi.org/10.1016/s0894-9166(15)60007-6. https://doi.org/10.1016/s0894-9166(15)60007-6.">

Y.-Z. Wang and F.-M. Li. Nonlinear free vibration of nanotube with small scale effects embedded in viscous matrix. Mechanics Research Communications, 60, (2014), pp. 45–51. https://doi.org/10.1016/j.mechrescom.2014.06.002. https://doi.org/10.1016/j.mechrescom.2014.06.002.">

Y.-X. Zhen and B. Fang. Nonlinear vibration of fluid-conveying single-walled carbon nanotubes under harmonic excitation. International Journal of Non-Linear Mechanics, 76, (2015), pp. 48–55. https://doi.org/10.1016/j.ijnonlinmec.2015.05.005. https://doi.org/10.1016/j.ijnonlinmec.2015.05.005.">

P. Valipour, S. E. Ghasemi, M. R. Khosravani, and D. D. Ganji. Theoretical analysis on nonlinear vibration of fluid flow in single-walled carbon nanotube. Journal of Theoretical and Applied Physics, 10, (3), (2016), pp. 211–218. https://doi.org/10.1007/s40094-016-0217-9. https://doi.org/10.1007/s40094-016-0217-9.">

M. Sadeghi-Goughari, S. Jeon, and H.-J. Kwon. Effects of magnetic-fluid flow on structural instability of a carbon nanotube conveying nanoflow under a longitudinal magnetic field. Physics Letters A, 381, (35), (2017), pp. 2898–2905. https://doi.org/10.1016/j.physleta.2017.06.054. https://doi.org/10.1016/j.physleta.2017.06.054.">

L. Wang. Size-dependent vibration characteristics of fluid-conveying microtubes. Journal of Fluids and Structures, 26, (4), (2010), pp. 675–684. https://doi.org/10.1016/j.jfluidstructs.2010.02.005. https://doi.org/10.1016/j.jfluidstructs.2010.02.005.">

L. Wang, H. T. Liu, Q. Ni, and Y. Wu. Flexural vibrations of microscale pipes conveying fluid by considering the size effects of micro-flow and micro-structure. International Journal of Engineering Science, 71, (2013), pp. 92–101. https://doi.org/10.1016/j.ijengsci.2013.06.006. https://doi.org/10.1016/j.ijengsci.2013.06.006.">

M. Tang, Q. Ni, L. Wang, Y. Luo, and Y. Wang. Nonlinear modeling and size-dependent vibration analysis of curved microtubes conveying fluid based on modified couple stress theory. International Journal of Engineering Science, 84, (2014), pp. 1–10. https://doi.org/10.1016/j.ijengsci.2014.06.007. https://doi.org/10.1016/j.ijengsci.2014.06.007.">

W. Xia and L.Wang. Microfluid-induced vibration and stability of structures modeled as microscale pipes conveying fluid based on non-classical Timoshenko beam theory. Microfluidics and Nanofluidics, 9, (4-5), (2010), pp. 955–962. https://doi.org/10.1007/s10404-010-0618-z. https://doi.org/10.1007/s10404-010-0618-z.">

M. R. Ghazavi, H. Molki, and A. A. Beigloo. Nonlinear analysis of the micro/nanotube conveying fluid based on second strain gradient theory. Applied Mathematical Modelling, 60, (2018), pp. 77–93. https://doi.org/10.1016/j.apm.2018.03.013. https://doi.org/10.1016/j.apm.2018.03.013.">

C. W. Lim, G. Zhang, and J. N. Reddy. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. Journal of the Mechanics and Physics of Solids, 78, (2015), pp. 298–313. https://doi.org/10.1016/j.jmps.2015.02.001. https://doi.org/10.1016/j.jmps.2015.02.001.">

M. Simsek. Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. International Journal of Engineering Science, 105, (2016), pp. 12–27. https://doi.org/10.1016/j.ijengsci.2016.04.013. https://doi.org/10.1016/j.ijengsci.2016.04.013.">

M. N. M. Allam and A. F. Radwan. Nonlocal strain gradient theory for bending, buckling, and vibration of viscoelastic functionally graded curved nanobeam embedded in an elastic medium. Advances in Mechanical Engineering, 11, (4), (2019), p. 1687814019837067. https://doi.org/10.1177/1687814019837067. https://doi.org/10.1177/1687814019837067.">

S. Esfahani, S. E. Khadem, and A. E. Mamaghani. Nonlinear vibration analysis of an electrostatic functionally graded nano-resonator with surface effects based on nonlocal strain gradient theory. International Journal of Mechanical Sciences, 151, (2019), pp. 508–522. https://doi.org/10.1016/j.ijmecsci.2018.11.030. https://doi.org/10.1016/j.ijmecsci.2018.11.030.">

V.-H. Dang, D.-A. Nguyen, M.-Q. Le, and T.-H. Duong. Nonlinear vibration of nanobeams under electrostatic force based on the nonlocal strain gradient theory. International Journal of Mechanics and Materials in Design, (2019), pp. 1–20. https://doi.org/10.1007/s10999-019-09468-8. https://doi.org/10.1007/s10999-019-09468-8.">

R. Bahaadini, A. R. Saidi, and M. Hosseini. Flow-induced vibration and stability analysis of carbon nanotubes based on the nonlocal strain gradient Timoshenko beam theory. Journal of Vibration and Control, 25, (1), (2019), pp. 203–218. https://doi.org/10.1177/1077546318774242. https://doi.org/10.1177/1077546318774242.">

M. Malikan, V. B. Nguyen, and F. Tornabene. Damped forced vibration analysis of single-walled carbon nanotubes resting on viscoelastic foundation in thermal environment using nonlocal strain gradient theory. Engineering Science and Technology, an International Journal, 21, (4), (2018), pp. 778–786. https://doi.org/10.1016/j.jestch.2018.06.001. https://doi.org/10.1016/j.jestch.2018.06.001.">

G.-L. She, Y.-R. Ren, F.-G. Yuan, and W.-S. Xiao. On vibrations of porous nanotubes. International Journal of Engineering Science, 125, (2018), pp. 23–35. https://doi.org/10.1016/j.ijengsci.2017.12.009. https://doi.org/10.1016/j.ijengsci.2017.12.009.">

M. Atashafrooz, R. Bahaadini, and H. R. Sheibani. Nonlocal, strain gradient and surface effects on vibration and instability of nanotubes conveying nanoflow. Mechanics of Advanced Materials and Structures, 27, (7), (2020), pp. 586–598. https://doi.org/10.1080/15376494.2018.1487611. https://doi.org/10.1080/15376494.2018.1487611.">

L. Li, Y. Hu, X. Li, and L. Ling. Size-dependent effects on critical flow velocity of fluid-conveying microtubes via nonlocal strain gradient theory. Microfluidics and Nanofluidics, 20, (5), (2016), p. 76. https://doi.org/10.1007/s10404-016-1739-9. https://doi.org/10.1007/s10404-016-1739-9.">

M. H. Ghayesh and A. Farajpour. Nonlinear mechanics of nanoscale tubes via nonlocal strain gradient theory. International Journal of Engineering Science, 129, (2018), pp. 84–95. https://doi.org/10.1016/j.ijengsci.2018.04.003. https://doi.org/10.1016/j.ijengsci.2018.04.003.">

M. H. Ghayesh and A. Farajpour. Nonlinear coupled mechanics of nanotubes incorporating both nonlocal and strain gradient effects. Mechanics of Advanced Materials and Structures, 27, (5), (2020), pp. 373–382. https://doi.org/10.1080/15376494.2018.1473537. https://doi.org/10.1080/15376494.2018.1473537.">

A. Azrar, M. Ben Said, L. Azrar, and A. A. Aljinaidi. Dynamic analysis of Carbon NanoTubes conveying fluid with uncertain parameters and random excitation. Mechanics of Advanced Materials and Structures, 26, (10), (2019), pp. 898–913. https://doi.org/10.1080/15376494.2018.1430272. https://doi.org/10.1080/15376494.2018.1430272.">

L. Li, Y. Hu, and L. Ling. Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory. Physica E: Low-dimensional Systems and Nanostructures, 75, (2016), pp. 118–124. https://doi.org/10.1016/j.physe.2015.09.028. https://doi.org/10.1016/j.physe.2015.09.028.">

L. Li and Y. Hu. Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory. Computational Materials Science, 112, (2016), pp. 282–288. https://doi.org/10.1016/j.commatsci.2015.10.044. https://doi.org/10.1016/j.commatsci.2015.10.044.">

Y. Zhen and L. Zhou. Wave propagation in fluid-conveying viscoelastic carbon nanotubes under longitudinal magnetic field with thermal and surface effect via nonlocal strain gradient theory. Modern Physics Letters B, 31, (08), (2017), https://doi.org/10.1142/s0217984917500695. https://doi.org/10.1142/s0217984917500695.">

N. D. Anh. Dual approach to averaged values of functions: A form for weighting coefficient. Vietnam Journal of Mechanics, 37, (2), (2015), pp. 145–150. https://doi.org/10.15625/0866-7136/37/2/6206. https://doi.org/10.15625/0866-7136/37/2/6206.">

N. D. Anh, N. Q. Hai, and D. V. Hieu. The equivalent linearization method with a weighted averaging for analyzing of nonlinear vibrating systems. Latin American Journal of Solids and Structures, 14, (9), (2017), pp. 1723–1740. https://doi.org/10.1590/1679-78253488. https://doi.org/10.1590/1679-78253488.">

D. V. Hieu. A new approximate solution for a generalized nonlinear oscillator. International Journal of Applied and Computational Mathematics, 5, (5), (2019), p. 126. https://doi.org/10.1007/s40819-019-0709-9. https://doi.org/10.1007/s40819-019-0709-9.">

D. V. Hieu and N. Q. Hai. Analyzing of nonlinear generalized duffing oscillators using the equivalent linearization method with a weighted averaging. Asian Research Journal of Mathematics, (2018), pp. 1–14. https://doi.org/10.9734/arjom/2018/40684. https://doi.org/10.9734/arjom/2018/40684.">

V. Hieu-Dang. An approximate solution for a nonlinear Duffing–Harmonic oscillator. Asian Research Journal of Mathematics, (2019), pp. 1–14. https://doi.org/10.9734/arjom/2019/v15i430154. https://doi.org/10.9734/arjom/2019/v15i430154.">

D. V. Hieu, N. Q. Hai, and D. T. Hung. The equivalent linearization method with a weighted averaging for solving undamped nonlinear oscillators. Journal of Applied Mathematics, 2018, (2018). https://doi.org/10.1155/2018/7487851. https://doi.org/10.1155/2018/7487851.">

D. V. Hieu and N. Q. Hai. Free vibration analysis of quintic nonlinear beams using equivalent linearization method with a weighted averaging. Journal of Applied and Computational Mechanics, 5, (1), (2019), pp. 46–57. https://doi.org/10.22055/JACM.2018.24919.1217. https://doi.org/10.22055/JACM.2018.24919.1217.">

S. S. Rao. Vibration of continuous systems. John Wiley & Sons, Inc., (2007).

M. Bayat, I. Pakar, and G. Domairry. Recent developments of some asymptotic methods and their applications for nonlinear vibration equations in engineering problems: A review. Latin American Journal of Solids and Structures, 9, (2), (2012), pp. 1–93. https://doi.org/10.1590/s1679-78252012000200003. https://doi.org/10.1590/s1679-78252012000200003.">

T.-P. Chang. Nonlinear free vibration analysis of nanobeams under magnetic field based on nonlocal elasticity theory. Journal of Vibroengineering, 18, (3), (2016), pp. 1912–1919. https://doi.org/10.21595/jve.2015.16751. https://doi.org/10.21595/jve.2015.16751.">

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Published

31-03-2021

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[1]
N. D. Anh and D. V. Hieu, Nonlinear vibration of nonlocal strain gradient nanotubes under longitudinal magnetic field, Vietnam J. Mech. 43 (2021) 55–77. DOI: https://doi.org/10.15625/0866-7136/15467.

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