Advances in nano research

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Frequency, bending and buckling loads of nanobeams with different cross sections

  • Civalek, Omer (China Medical University) ;
  • Uzun, Busra (Bursa Uludag University, Faculty of Engineering, Department of Civil Engineering) ;
  • Yayli, M. Ozgur (Bursa Uludag University, Faculty of Engineering, Department of Civil Engineering)
  • Received : 2019.12.20
  • Accepted : 2020.07.07
  • Published : 2020.08.25
⟨ Previous Next ⟩

Abstract

The bending, stability (buckling) and vibration response of nano sized beams is presented in this study based on the Eringen's nonlocal elasticity theory in conjunction with the Euler-Bernoulli beam theory. For this purpose, the bending, buckling and vibration problem of Euler-Bernoulli nanobeams are developed and solved on the basis of nonlocal elasticity theory. The effects of various parameters such as nonlocal parameter e0a, length of beam L, mode number n, distributed load q and cross-section on the bending, buckling and vibration behaviors of carbon nanotubes idealized as Euler-Bernoulli nanobeam is investigated. The transverse deflections, maximum transverse deflections, vibrational frequency and buckling load values of carbon nanotubes are given in tables and graphs.

Keywords

References

  1. Akbas, S.D. (2019), "Axially forced vibration analysis of cracked a nanorod", J. Comput. Appl. Mech., 50(1), 63-68. https://doi.org/10.22059/jcamech.2019.281285.392.
  2. Akgoz, B. and Civalek, O. (2011), "Nonlinear vibration analysis of laminated plates resting on nonlinear two-parameters elastic foundations", Steel Compos. Struct., Int. J., 11(5), 403-421. https://doi.org/10.12989/scs.2011.11.5.403.
  3. 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.
  4. Akgoz, B. and Civalek, O. (2017a), "A size-dependent beam model for stability of axially loaded carbon nanotubes surrounded by Pasternak elastic foundation", Compos. Struct., 176, 1028-1038. https://doi.org/10.1016/j.compstruct.2017.06.039.
  5. Akgoz, B. and Civalek, O. (2017b), "Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams", Compos. Part B Eng., 129, 77-87. https://doi.org/10.1016/j.compositesb.2017.07.024.
  6. Ansari, R. Gholami, R. and Darabi, M.A. (2011), "Thermal buckling analysis of embedded single-walled carbon nanotubes with arbitrary boundary conditions using the nonlocal Timoshenko beam theory", J. Therm. Stresses, 34(12), 1271-1281. https://doi.org/10.1080/01495739.2011.616802.
  7. Arani, A.G. and Jalaei, M.H. (2016), "Transient behavior of an orthotropic graphene sheet resting on orthotropic visco-Pasternak foundation", Int. J. Eng. Sci., 103, 97-113. https://doi.org/10.1016/j.ijengsci.2016.02.006.
  8. Asemi, S.R. Farajpour, A. Asemi, H.R. and Mohammadi, M. (2014), "Influence of initial stress on the vibration of double-piezoelectric-nanoplate systems with various boundary conditions using DQM", Physica E Low Dimens. Syst. Nanostruct., 63, 169-179. https://doi.org/10.1016/j.physe.2014.05.009.
  9. Attia, A. Bousahla, A.A., Tounsi, A., Mahmoud, S.R. and Alwabli, A.S. (2018), "A refined four variable plate theory for thermoelastic analysis of FGM plates resting on variable elastic foundations", Struct. Eng. Mech., Int. J., 65(4), 453-464. https://doi.org/10.12989/sem.2018.65.4.453.
  10. Aydogdu, M. (2009), "A general nonlocal beam theory : its application to nanobeam bending, buckling and vibration", Physica E Low Dimens. Syst. Nanostruct., 41(9), 1651-1655. https://doi.org/10.1016/j.physe.2009.05.014.
  11. Aydogdu, M. and Arda, M. (2016), "Torsional vibration an alysis of double walled carbon nanotubes using nonlocal elasticity", Int. J. Mech. Mater. Des., 12(1), 71-84. https://doi.org/10.1007/s10999-014-9292-8.
  12. Barretta, R. and Marottide Sciarra, F. (2013), "A nonlocal model for carbon nanotubes under axial loads", Adv. Mater. Sci. Eng., 360935. https://doi.org/10.1155/2013/360935.
  13. Bedia, W.A., Benzair, A., Semmah, A., Tounsi, A. and Mahmoud, S.R. (2015), "On the thermal buckling characteristics of armchair single-walled carbon nanotube embedded in an elastic medium based on nonlocal continuum elasticity", Braz. J. Phys., 45(2), 225-233. https://doi.org/10.1007/s13538-015-0306-2.
  14. Besseghier, A., Heireche, H., Bousahla, A.A., Tounsi, A. and Benzair, A. (2015), "Nonlinear vibration properties of a zigzag single-walled carbon nanotube embedded in a polymer matrix", Adv. Nano Res., Int. J., 3(1), 29-37. https://doi.org/10.12989/anr.2015.3.1.029.
  15. Civalek, O. and Acar, M.H. (2007), "Discrete singular convolution method for the analysis of Mindlin plates on elastic foundations", Int. J. Press. Vessel. Pip., 84(9), 527-535. https://doi.org/10.1016/j.ijpvp.2007.07.001.
  16. Civalek, O. and Demir, C. (2011a), "Bending analysis of microtubules using nonlocal Euler-Bernoulli beam theory", Appl. Math. Model., 35(5), 2053-2067. https://doi.org/10.1016/j.apm.2010.11.004.
  17. Civalek, O. and Demir, C. (2011b), "Buckling and bending analyses of cantilever carbon nanotubes using the eulerbernoulli beam theory based on non-local continuum model", Asian J. Civ. Eng., 12(5), 651-661.
  18. Civalek, O. and Demir, C. (2016), "A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method", Appl. Math. Comput., 289, 335-352. https://doi.org/10.1016/j.amc.2016.05.034.
  19. Civalek, O. and Kiracioglu, O. (2007), "Discrete singular convolution for free vibration analysis of anisotropic rectangular plates", Math. Comput. Appl., 12(3), 151-160. https://doi.org/10.3390/mca12030151.
  20. Civalek, O., Uzun, B., Yayli, M.O. and Akgoz, B. (2020), "Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method", Eur. Phys. J. Plus, 135(4), 381. https://doi.org/10.1140/epjp/s13360-020-00385-w.
  21. Demir, C. and Civalek, O. (2013), "Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal con tinuum and nonlocal discrete models", Appl. Math. Model., 37(22), 9355-9367. https://doi.org/10.1016/j.apm.2013.04.050.
  22. Demir, C. and Civalek, O. (2017a), "A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix", Compos. Struct., 168, 872-884. https://doi.org/10.1016/j.compstruct.2017.02.091.
  23. Demir, C. and Civalek, O. (2017b), "On the analysis of microbeams", Int. J. Eng. Sci, 121, 14-33. https://doi.org/10.1016/j.ijengsci.2017.08.016.
  24. Dihaj, A., Zidour, M., Meradjah, M., Rakrak, K., Heireche, H. and Chemi, A. (2018), "Free vibration analysis of chiral double-walled carbon nanotube embedded in an elastic medium using non-local elasticity theory and Euler Bernoulli beam model", Struct. Eng. Mech., Int. J., 65(3), 335-342. https://doi.org/10.12989/sem.2018.65.3.335.
  25. Ebrahimi, F. and Nasirzadeh, P. (2015), "A nonlocal Timoshenko beam theory for vibration analysis of thick nanobeams using differential transform method", J. Theor. Appl. Mech., 53(4), 1041-1052. https://doi.org/10.15632/jtam-pl.53.4.1041.
  26. Ebrahimi, F. and Salari, E. (2015), "Thermo-mechanical vibration analysis of a single-walled carbon nanotube embedded in an elastic medium based on higher-order shear deformation beam theory", J. Mech. Sci. Technol., 29(9), 3797-3083. https://doi.org/10.1007/s12206-015-0826-2.
  27. Ece, M.C. and Aydogdu, M. (2007), "Nonlocal elasticity effect on vibration of in-plane loaded double-walled carbon nano-tubes", Acta Mechanica, 190(1-4), 185-195. https://doi.org/10.1007/s00707-006-0417-5.
  28. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-4710. https://doi.org/10.1063/1.332803.
  29. Gopalakrishnan, S. and Narendar, S. (2013), Wave Propagation in Nanostructures: Nonlocal Continuum Mechanics Formulations, Springer International Publishing, Switzerland.
  30. Gurtin, M.E. and Murdoch, A.I. (1975a), "Addend a to our paper: A continuum theory of elastic material surfaces", Arch. Ration. Mech. Anal., 59(4), 389-390. https://doi.org/10.1007/BF00250426.
  31. Gurtin, M.E. and Murdoch, A.I. (1975b), "A continuum theory of elastic material surfaces", Arch. Ration. Mech. Anal., 57(4), 291-323. https://doi.org/10.1007/BF00261375.
  32. Gurtin, M.E. and Murdoch, A.I. (1978), "Surface stress in solids", Int. J. Solids Struct., 14(6), 431-440. https://doi.org/10.1016/0020-7683(78)90008-2.
  33. Gurses, M., Akgoz, B. and Civalek, O. (2012), "Mathematical modeling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation", Appl. Math. Comput., 219(6), 3226-3240. https://doi.org/10.1016/j.amc.2012.09.062.
  34. Heydarpour, Y. and Malekzadeh, P. (2019), "Dynamic stability of cylindrical nanoshells under combined static and periodic axial loads", J. Braz. Soc. Mech. Sci. Eng., 41(4), 184. https://doi.org/10.1007/s40430-019-1675-1.
  35. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354(6348), 56. https://doi.org/10.1038/354056a0.
  36. Jalaei, M.H. and Arani, A.G. (2018), "Size-dependent static and dynamic responses of embedded double-layered graphene sheets under longitudinal magnetic field with arbitrary boundary conditions", Compos. Part B Eng., 142, 117-130. https://doi.org/10.1016/j.compositesb.2017.12.053.
  37. Jalaei, M.H., Arani, A.G. and Tourang, H. (2018), "On the dynamic stability of viscoelastic graphene sheets", Int. J. Eng. Sci., 132, 16-29. https://doi.org/10.1016/j.ijengsci.2018.07.002.
  38. Jalaei, M.H., Arani, A.G. and Nguyen-Xuan, H. (2019), "Investigation of thermal and magnetic field effects on the dynamic instability of FG Timoshenko nanobeam employing nonlocal strain gradient theory", Int. J. Mech. Sci., 161, 105043. https://doi.org/10.1016/j.ijmecsci.2019.105043.
  39. Karlicic, D., Cajic, M., Murmu, T. and Adhikari, S. (2015), "Nonlocal longitudinal vibration of viscoelastic coupled doublenanorod systems", Eur. J. Mech. A Solids, 49, 183-196. https://doi.org/10.1016/j.euromechsol.2014.07.005.
  40. Ke, L.L., Wang, Y.S., Yang, J. and Kitipornchai, S. (2014), "The size-dependent vibration of embedded magneto-electro-elastic cylindrical nanoshells", Smart Mater. Struct., 23(12), 125036. https://doi.org/10.1088/0964-1726/23/12/125036.
  41. Ke, L.L., Liu, C. and Wang, Y.S. (2015), "Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions", Physica E Low Dimens. Syst. Nanostruct., 66, 93-106. https://doi.org/10.1016/j.physe.2014.10.002.
  42. Koiter, W.T. (1964), "Couple stresses in the theory of elasticity: I and II", Proc. K. Ned. Akad. Wet. B Phys. Sci., 67, 17-44.
  43. Kong, S., Zhou, S., Nie, Z. and Wang, K. (2008), "The size-dependent natural frequency of Bernoulli-Euler micro-beams", Int. J. Eng. Sci., 46(5), 427-437. https://doi.org/10.1016/j.ijengsci.2007.10.002.
  44. Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity ", J. Mech. Phys. Solids, 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X.
  45. Mercan, K. and Civalek, O. (2017), "Buckling analysis of silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ", Compos. Part B Eng., 114, 34-45. https://doi.org/10.1016/j.compositesb.2017.01.067.
  46. Mercan, K., Numanoglu, H.M., Akgoz, B., Demir, C. and Civalek, O. (2017), "Higher-order continuum theories for buckling response of silicon carbide nanowires (SiCNWs) on elastic matrix", Arch. Appl. Mech., 87(11), 1797-1814. https://doi.org/10.1007/s00419-017-1288-z.
  47. Mindlin, R.D. (1963), "Influence of couple-stresses on stress concentrations", Exp. Mech., 3(1), 1-7. https://doi.org/10.1007/BF02327219.
  48. Mindlin, R.D. (1965), "Second gradient of strain and surface-tension in linear elasticity", Int. J. Solids Struct., 1(4), 417-438. https://doi.org/10.1016/0020-7683(65)90006-5.
  49. Mindlin, R.D. and Tiersten, H.F. (1962), "Effects of couple-stresses in lin ear elasticity", Arch. Ration. Mech. Anal., 11, 415-448. https://doi.org/10.1007/BF00253946.
  50. Mokhtar, Y., Heireche, H., Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2018), "A novel shear deformation theory for buckling analysis of single layer graphene sheet based on nonlocal elasticity theory ", Smart Struct. Syst., Int. J., 21(4), 397-405. https://doi.org/10.12989/sss.2018.21.4.397.
  51. Murmu, T. and Adhikari, S. (2010), "Scale-dependent vibration analysis of prestressed carbon nanotubes undergoing rotation", J. Appl. Phys., 108(12), 123507. https://doi.org/10.1063/1.3520404.
  52. Murmu, T. and Pradhan, S.C. (2009), "Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory", Comput. Mater. Sci., 46(4), 854-859. https://doi.org/10.1016/j.commatsci.2009.04.019.
  53. Narendar, S. and Gopalakrishnan, S. (2009), "Nonlocal scale effects on wave propagation in multi-walled carbon nanotubes", Comput. Mater. Sci., 47(2), 526-538. https://doi.org/10.1016/j.commatsci.2009.09.021.
  54. Narendar, S. (2012), "Spectral finite element and nonlocal continuum mechanics based formulation for torsional wave propagation in nanorods", Finite Elem. Anal. Des., 62, 65-75. https://doi.org/10.1016/j.finel.2012.06.012.
  55. Numanoglu, H.M. and Civalek, O. (2019), "On the dynamics of small-sized structures", Int. J. Eng. Sci., 145, 103164. https://doi.org/10.1016/j.ijengsci.2018.05.001.
  56. Numanoglu, H.M., Akgoz, B. and Civalek, O. (2018), "On dynamic analysis of nanorods", Int. J. Eng. Sci., 130, 33-50. https://doi.org/10.1016/j.ijengsci.2018.05.001.
  57. Reddy, J.N. (2002), Energy Principles and Variational Methods in Applied Mechanics, John Wiley & Sons, USA.
  58. Reddy, J.N. and Pang, S.D. (2008), "Nonlocal continuum theories of beams for the an alysis of carbon nanotubes", J. Appl. Phys., 103(2), 023511. https://doi.org/10.1063/1.2833431.
  59. Sahmani, S. and Safaei, B. (2019), "Nonlinear free vibrations of bi-directional functionally graded micro/nano-beams including nonlocal str ess and microstructural strain gradient size effects", Thin Wall. Struct., 140, 342-356. https://doi.org/10.1016/j.tws.2019.03.045.
  60. Schrlau, M.G. (2011), "Carbon nanotube-based sensors: overview", Compr. Biomater., 3, 519-528. https://doi.org/10.1016/B978-0-08-055294-1.00120-3
  61. Semmah, A., Heireche, H., Bousahla, A.A. and Tounsi, A. (2019), "Thermal buckling an alysis of SWBNNT on Winkler foundation by non local FSDT", Adv. Nano Res., Int. J., 7(2), 89-98. https://doi.org/10.12989/anr.2019.7.2.089.
  62. Tounsi, A., Benguediab, S., Semmah, A. and Zidour, M. (2013), "Nonlocal effects on th ermal buckling properties of doublewalled carbon nanotubes", Adv. Nano Res., Int. J., 1(1), 1-11. https://doi.org/10.12989/anr.2013.1.1.001.
  63. Toupin, R.A. (1962), "Elastic materials with couple-stresses", Arch. Ration. Mech. Anal., 11(1), 385-414. https://doi.org/10.1007/BF00253945.
  64. Uzun, B., Numanoglu, H. and Civalek, O. (2018), "Free vibration analysis of BNNT with different cross-Sections via nonlocal FEM", J. Comput. Appl. Mech., 49(2), 252-260. https://doi.org/10.22059/jcamech.2018.
  65. Uzun, B. and Civalek, O. (2019a), "Nonlocal FEM formulation for vibration analysis of nanowires on elastic matrix with different materials", Math. Comput. Appl., 24(2), 38. https://doi.org/10.3390/mca24020038.
  66. Uzun, B. and Civalek, O. (2019b), "Free vibration analysis silicon nanowires surrounded by elastic matrix by nonlocal finite element method", Adv. Nano Res., Int. J., 7(2), 99-108. https://doi.org/10.12989/anr.2019.7.2.099.
  67. Uzun, B. and Yayli, M.O. (2020), "Nonlocal vibration analysis of Ti-6Al-4V/$ZrO_2$ functionally graded nanobeam on elastic matrix", Arab. J. Geosci., 13(4), 1-10. https://doi.org/10.1007/s12517-020-5168-4.
  68. Uzun, B., Yayli, M.O. and Deliktas, B. (2020), "Free vibration of FG nanobeam using a finite-element method ", Micro Nano Lett., 15(1), 35-40. https://doi.org/10.1049/mnl.2019.0273.
  69. Wang, Y.Z., Li, F.M. and Kishimoto, K. (2010), "Scale effects on flexural wave propagation in nanoplate embedded in elastic matrix with initial stress", Appl. Phys. A, 99(4), 907-911. https://doi.org/10.1007/s00339-010-5666-4.
  70. Yang, F., Chong, A.C.M., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X.
  71. Yayli, M.O. (2016), "A compact analytical method for vibration analysis of single-walled carbon nanotubes with restrained boundary conditions", J. Vib. Control, 22(10), 2542-2555. https://doi.org/10.1177/1077546314549203.
  72. Yayli, M.O. (2017), "Buckling analysis of a cantilever single-walled carbon nanotube embedded in an elastic medium with an attached spring", Micro Nano Lett., 12(4), 255-259. https://doi.org/10.1049/mnl.2016.0662.
  73. Yayli, M.O. (2018), "On the torsional vibrations of restrained nanotubes embedded in an elastic medium", J. Braz. Soc. Mech. Sci. Eng., 40(9), 419. https://doi.org/10.1007/s40430-018-1346-7.
  74. Yayli, M.O. (2019), "Effects of rotational restraints on the thermal buckling of carbon nanotube", Micro Nano Lett., 14(2), 158-162. https://doi.org/10.1049/mnl.2018.5428.
  75. Yazid, M., Heireche, H., Tounsi, A., Bousahla, A.A. and Houari, M.S.A. (2018), "A novel nonlocal refined plate theory for stability response of orthotropic single-layer graphene sheet resting on elastic medium", Smart Struct. Syst., Int. J., 21(1), 15-25. https://doi.org/10.12989/sss.2018.21.1.015.
  76. Youcef, D.O., Kaci, A., Benzair, A., Bousahla, A.A. and Tounsi, A. (2018), "Dynamic analysis of nanoscale beams in cluding surface stress effects", Smart Struct. Syst., Int. J., 21(1), 65-74. https://doi.org/10.12989/sss.2018.21.1.065.

Cited by

  1. Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory vol.10, pp.3, 2020, https://doi.org/10.12989/anr.2021.10.3.281
  2. Free vibration analysis of carbon nanotube RC nanobeams with variational approaches vol.11, pp.2, 2021, https://doi.org/10.12989/anr.2021.11.2.157