Advances in nano research

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Finite element analysis for longitudinal vibration of nanorods based on doublet mechanics

  • Ufuk Gul (Department of Mechanical Engineering, Trakya University) ;
  • Metin Aydogdu (Department of Mechanical Engineering, Trakya University)
  • Received : 2021.10.17
  • Accepted : 2023.08.28
  • Published : 2023.11.25
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Abstract

In the present study, the axial vibration of the nanorods is investigated in the framework of the doublet mechanics theory. The equations of motion and boundary conditions of nanorods are derived by applying the Hamilton principle. A finite element method is developed to obtain the vibration frequencies of nanorods for different boundary conditions. A two-noded higher order rod finite element is used to solve the vibration problem. The natural frequencies of nanorods obtained with the present finite element analysis are validated by comparing the results of classical doublet mechanics and nonlocal strain gradient theories. The effects of rod length, mode number and boundary conditions on the axial vibration frequencies of nanorods are examined in detail. Mode shapes of the nanorods are presented for the different boundary conditions. It is shown that the doublet mechanics model can be used for the dynamic analysis of nanotubes, and the presented finite element formulation can be used for mechanical problems of rods with unavailable analytical solutions. These new results can also be used as references for the future studies.

Keywords

References

  1. Adhikari, S., Murmu, T. and McCarthy, M.A. (2013), "Dynamic finite element analysis of axially vibrating nonlocal rods", Finite Elem. Anal. Des., 63, 42-50. https://doi.org/10.1016/j.finel.2012.08.001.
  2. Adhikari, S., Murmu, T. and McCarthy, M.A. (2014), "Frequency domain analysis of nonlocal rods embedded in an elastic medium", Physica E, 59, 33-40. https://doi.org/10.1016/j.physe.2013.11.001.
  3. Aifantis, E.C. (1999), "Gradient deformation models at nano, micro, and macro scales", ASME J. Eng. Mater. Technol., 121(2), 189-202. https://doi.org/10.1115/1.2812366.
  4. Akgoz, B. and Civalek, O . (2014), "Longitudinal vibration analysis for microbars based on strain gradient elasticity theory", J. Vib. Control, 20(4), 606-616. https://doi.org/10.1177/1077546312463752.
  5. Anderson, W. and Lakes, R. (1994), "Size effects due to Cosserat elasticity and surface damage in closed-cell polymethacrylimide foam", J. Mater. Sci., 29, 6413-6419. https://doi.org/10.1007/BF00353997.
  6. Aydogdu, M. (2009), "Axial vibration of the nanorods with the nonlocal continuum rod model", Physica E, 41(5), 861-864. https://doi.org/10.1016/j.physe.2009.01.007.
  7. Aydogdu, M. (2012), "Axial vibration analysis of nanorods (carbon nanotubes) embedded in an elastic medium using nonlocal elasticity", Mech. Res. Commun., 43, 34-40. https://doi.org/10.1016/j.mechrescom.2012.02.001.
  8. Aydogdu, M. and Gul, U. (2020), "Longitudinal vibration of double nanorod systems using doublet mechanics theory", Struct. Eng. Mech., 73(1), 37-52. http://doi.org/10.12989/sem.2020.73.1.037.
  9. Challamel, N. (2013), "Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams", Compos. Struct., 105, 351-368. https://doi.org/10.1016/j.compstruct.2013.05.026.
  10. Demir, C . and Civalek, O . (2013), "Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models", Appl. Math. Model., 37(22), 9355-9367. https://doi.org/10.1016/j.apm.2013.04.050.
  11. Eghbali, M., Hosseini, S.A. and Rahmani, O. (2021), "Free vibration of axially functionally graded nanobeam with an attached mass based on nonlocal strain gradient theory via new ADM numerical method", Amirkabir J. Mech. Eng., 53(2), 275-276. https://doi.org/10.22060/mej.2020.17013.6495.
  12. Eghbali, M., Hosseini, S.A. and Pourseifi, M. (2022a), "Free transverse vibrations analysis of size-dependent cracked piezoelectric nano-beam based on the strain gradient theory under mechanic-electro forces", Eng. Anal. Bound. Elem., 143, 606-612. http://doi.org/10.1016/j.enganabound.2022.07.006.
  13. Eghbali, M., Hosseini, S.A. and Hamidi, B.A. (2022b), "Mantari's higher-order shear deformation theory of sandwich beam with CNTRC face layers with porous core under thermal loading", Int. J. Struct. Stab. Dyn., 22(16), 2250181. https://doi.org/10.1142/S0219455422501814.
  14. Eringen, A.C. (1972a), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5.
  15. Eringen, A.C. (1972b), "Linear theory of nonlocal elasticity and dispersion of plane waves", Int. J. Eng. Sci., 10(5), 425-435. https://doi.org/10.1016/0020-7225(72)90050-X.
  16. Fatahi-Vajari, A. and Azimzadeh, Z. (2020), "Axial vibration of single-walled carbon nanotubes with fractional damping using doublet mechanics", Indian J. Phys., 94, 975-986. https://doi.org/10.1007/s12648-019-01547-y.
  17. Fleck, N., Muller, G., Ashby, M. and Hutchinson, J. (1994), "Strain gradient plasticity: theory and experiment", Acta Metall. Mater., 42(2), 475-487. https://doi.org/10.1016/0956-7151(94)90502-9.
  18. Granik, V. (1978), "Microstructural mechanics of granular media", Technique Report IM/MGU Institute of Mechanics of Moscow State University,78-241.
  19. Granik, V.T. and Ferrari, M. (1993), "Microstructural mechanics of granular media", Mech. Mater., 15(4), 301-322. https://doi.org/10.1016/0167-6636(93)90005-C.
  20. Gul, U., Aydogdu, M. and Gaygusuzoglu, G. (2017), "Axial dynamics of a nanorod embedded in an elastic medium using doublet mechanics", Compos. Struct., 160, 1268-1278. https://doi.org/10.1016/j.compstruct.2016.11.023.
  21. Gul, U. and Aydogdu, M. (2018), "Structural modelling of nanorods and nanobeams using doublet mechanics theory", Int. J. Mech. Mater. Des., 14, 195-212. https://doi.org/10.1007/s10999-017-9371-8.
  22. Gul, U. and Aydogdu, M. (2019), "Vibration analysis of Love nanorods using doublet mechanics theory", J. Braz. Soc. Mech. Sci. Eng., 41, 351. https://doi.org/10.1007/s40430-019-1849-x.
  23. Guven, U. (2014), "Love-Bishop rod solution based on strain gradient elasticity theory", Comptes Rendus Mecanique, 342(1), 8-16. https://doi.org/10.1016/j.crme.2013.10.011.
  24. Hutchinson, J.W. (2000), "Plasticity at the micron scale", Int. J. Solid. Struct., 37(1-2), 225-238. https://doi.org/10.1016/S0020-7683(99)00090-6.
  25. Kahrobaiyan, M.H., Asghari, M. and Ahmadian, M.T. (2013), "Longitudinal behavior of strain gradient bars", Int. J. Eng. Sci., 66-67, 44-59. https://doi.org/10.1016/j.ijengsci.2013.02.005.
  26. Karamanli, A. (2021), "Structural behaviours of zigzag and armchair nanobeams using finite element doublet mechanics", Europ. J. Mech. A Solid., 89, 104287. https://doi.org/10.1016/j.euromechsol.2021.104287.
  27. Khorshidi, M.A. (2018), "The material length scale parameter used in couple stress theories is not a material constant", Int. J. Eng. Sci., 133, 15-25. https://doi.org/10.1016/j.ijengsci.2018.08.005.
  28. Kojic, M., Vlastelica, I., Decuzzi, P., Granik, V.T. and Ferrari, M. (2011), "A finite element formulation for the doublet mechanics modeling of microstructural materials", Comput. Meth. Appl. Mech. Eng., 200(13-16), 1446-1454. https://doi.org/10.1016/j.cma.2011.01.001.
  29. Lam, D., Yang, F., Chong, A., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solid., 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X.
  30. Li, L., Hu, Y. and Li, X. (2016), "Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory", Int. J. Mech. Sci., 115-116, 135-144. https://doi.org/10.1016/j.ijmecsci.2016.06.011.
  31. Lim, C.W., Zhang, G. and Reddy, J.N. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its application in wave propagation", J. Mech. Phys. Solid., 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001.
  32. Mindlin, R. and Tiersten, H. (1962), "Effects of couple-stresses in linear elasticity", Columbia University, New York., U.S.A.
  33. Murmu, T., Adhikari, S. and Wang, C.Y. (2011), "Torsional vibration of carbon nanotube-buckyball systems based on nonlocal elasticity theory", Physica E, 43(6), 1276-1280. https://doi.org/10.1016/j.physe.2011.02.017.
  34. Mustapha, K.B. and Ruan, D. (2015), "Size-dependent axial dynamics of magnetically-sensitive strain gradient microbars with end attachments", Int. J. Mech. Sci., 94-95, 96-110. https://doi.org/10.1016/j.ijmecsci.2015.02.010.
  35. Numanoglu, H.M. and Civalek, O . (2019), "On the torsional vibration of nanorods surrounded by elastic matrix via nonlocal FEM", Int. J. Mech. Sci., 161-162, 105076. https://doi.org/10.1016/j.ijmecsci.2019.105076.
  36. Reddy J.N. (2002), "Energy Principles and Variational Methods in Applied Mechanics", John Wiley & Sons Inc., Hoboken.
  37. Sharma, R., Jadon, V.K. and Singh, B. (2015), "A review on the finite element methods for heat conduction in functionally graded materials", J. Inst. Eng. India Ser. C, 96, 73-81. https://doi.org/10.1007/s40032-014-0125-1.
  38. Sears, A. and Batra, R.C. (2004), "Macroscopic properties of carbon nanotubes from molecular-mechanics simulations", Phys. Rev. B., 69, 235406. https://doi.org/10.1103/PhysRevB.69.235406.
  39. Vajari, A.F. and Imam, A. (2016), "Axial vibration of single-walled carbon nanotubes using doublet mechanics", Indian J. Phys., 90, 447-455. https://doi.org/10.1007/s12648-015-0775-8.
  40. Zhang, Y.Y., Wang, C.M. and Challamel, N. (2009), "Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model", J. Eng. Mech., 136(5), 562-574. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000107.