DOI QR코드

DOI QR Code

Analytical analysis for the forced vibration of CNT surrounding elastic medium including thermal effect using nonlocal Euler-Bernoulli theory

  • Received : 2018.08.19
  • Accepted : 2019.04.24
  • Published : 2018.09.25

Abstract

This article studies the free and forced vibrations of the carbon nanotubes CNTs embedded in an elastic medium including thermal and dynamic load effects based on nonlocal Euler-Bernoulli beam. A Winkler type elastic foundation is employed to model the interaction of carbon nanotube and the surrounding elastic medium. Influence of all parameters such as nonlocal small-scale effects, high temperature change, Winkler modulus parameter, vibration mode and aspect ratio of short carbon nanotubes on the vibration frequency are analyzed and discussed. The non-local Euler-Bernoulli beam model predicts lower resonance frequencies. The research work reveals the significance of the small-scale coefficient, the vibrational mode number, the elastic medium and the temperature change on the non-dimensional natural frequency.

Keywords

References

  1. Abdelhak, Z., Hadji, L., Daouadji, T.H. and Adda, B. (2016), "Thermal buckling response of functionally graded sandwich plates with clamped boundary conditions", Smart Struct. Syst., Int. J., 18(2), 267-291. https://doi.org/10.12989/sss.2016.18.2.267
  2. Abderezak, R., Daouadji, T.H., Abbes, B., Rabia, B., Belkacem, A. and Abbes, F. (2017), "Elastic analysis of interfacial stress concentrations in CFRP-RC hybrid beams: Effect of creep and shrinkage", Adv. Mater. Res., Int. J., 6(3), 257-278.
  3. Abderezak, R., Daouadji, T.H., Rabia, B. and Belkacem, A. (2018), "Nonlinear analysis of damaged RC beams strengthened with glass fiber reinforced polymer plate under symmetric loads", Earthq. Struct., Int. J., 15(2), 113-122.
  4. Adim, B., Daouadji, T.H. and Rabahi, A. (2016a), "A simple higher order shear deformation theory for mechanical behavior of laminated composite plates", Int. J. Adv. Struct. Eng. (IJASE), 8, 103-117. https://doi.org/10.1007/s40091-016-0109-x
  5. Adim, B., Daouadji, T.H., Abbes, B. and Rabahi, A. (2016b), "Buckling and free vibration analysis of laminated composite plates using an efficient and simple higher order shear deformation theory", J. Mech. Industry, 17(5), 512. https://doi.org/10.1051/meca/2015112
  6. Aifantis, E.C. (1984), "On the micro structural origin of certain inelastic models", J. Eng. Mater. Technol., 106(4), 326-330. https://doi.org/10.1115/1.3225725
  7. Aissani, K., Bachir Bouiadjra, M., Ahouel, M. and Tounsi, A. (2015), "A new nonlocal hyperbolic shear deformation theory for nanobeams embedded in an elastic medium", Struct. Eng. Mech., Int. J., 55(4), 743-762. https://doi.org/10.12989/sem.2015.55.4.743
  8. Amara, K., Tounsi, A. and Mechab, I. (2010), "Nonlocal elasticity effect on column buckling of multi walled carbon nanotubes under temperature field", Appl. Math. Model., 34(12), 3933-3942. https://doi.org/10.1016/j.apm.2010.03.029
  9. Ansari, R. and Ramezannezhad, H. (2011), "Nonlocal Timoshenko beam model for the large-amplitude vibrations of embedded multi walled carbon nanotubes including thermal effects", Physica E: Low dimens. Syst. Nanostruct., 43(6), 1171-1178. https://doi.org/10.1016/j.physe.2011.01.024
  10. Ansari, R. and Sahmani, S. (2012), "Small scale effect on vibrational response of single-walled carbon nanotubes with different boundary conditions based on nonlocal beam models", Commun. Nonlinear Sci. Numer. Simul., 17(4), 1965-1979. https://doi.org/10.1016/j.cnsns.2011.08.043
  11. Ansari, R., Gholami, R., Hosseini, K. and Sahmani, S. (2011), "A sixth-order compact finite difference method for vibration analysis of nano-beams embedded in an elastic medium based on nonlocal beam theory", Math. Comput. Model., 54(11), 2577-2586. https://doi.org/10.1016/j.mcm.2011.06.030
  12. Baughman, R.H., Zakhidov, A.A. and De Heer, W.A. (2002), "Carbon nanotubes--the route toward applications", Science, 297(5582), 787-792. https://doi.org/10.1126/science.1060928
  13. Belkacem, A., Tahar, H.D., Abderrezak, R., Amine, B.M., Mohamed, Z. and Boussad, A. (2018), "Mechanical buckling analysis of hybrid laminated composite plates under different boundary conditions", Struct. Eng. Mech., Int. J., 66(6), 761-769.
  14. Ben Henni, M., Daouadji, T.H., Abbes, B., Li, Y.M. and Abbes, F. (2018), "Analytical and Numerical Results for Free Vibration of Laminated Composites Plates", Int. J. Chem. Molecul. Eng., 12(6), 300-304.
  15. Benferhat, R., Daouadji, T.H. and Said-Mansour, M. (2014), "A Higher Order Shear Deformation Model for Bending Analysis R. of Functionally Graded Plates", Trans Indian Inst. Metals, 68(1), 7-16. https://doi.org/10.1007/s12666-014-0428-1
  16. Benferhat, R., Daouadji, T.H. and Mansour, M.S. (2016), "Free vibration analysis of FG plates resting on the elastic foundation and based on the neutral surface concept using higher order shear deformation theory", Comptes Rendus Mecanique, 344(9), 631-641 . https://doi.org/10.1016/j.crme.2016.03.002
  17. Bensattalah, T., Daouadji, T.H., Zidour, M., Tounsi, A. and Bedia, E.A. (2016), "Investigation of thermal and chirality effects on vibration of single-walled carbon nanotubes embedded in a polymeric matrix using nonlocal elasticity theories", Mech. Compos. Mater., 52(4), 555-568. https://doi.org/10.1007/s11029-016-9606-z
  18. Bensattalah, T., Bouakkaz, K., Zidour, M. and Daouadji, T.H. (2018a), "Critical buckling loads of carbon nanotube embedded in Kerr's medium", Adv. Nano Res., Int. J., 6(4), 339-356.
  19. Bensattalah, T., Zidour, M., Meziane, M.E., Daouadji, T.H. and Tounsi, A. (2018b), "Critical buckling load of carbon nanotube with non-local Timoshenko beam using the differential transform method", Int. J. Civil Environ. Eng., 12(6), 637-644.
  20. Bouakaz, K., Daouadji, T.H., Meftah, S.A., Ameur, M., Tounsi, A. and Bedia, E.A. (2014), "A numerical analysis of steel beams strengthened with composite materials", Mech. Compos. Mater., 50(4), 685-696.
  21. Cao, G. and Chen, X.I. (2007), "The effects of chirality and boundary conditions on the mechanical properties ofsingle-walled carbon nanotubes", Int. J. Solids Struct., 44, 5447-5465. https://doi.org/10.1016/j.ijsolstr.2007.01.005
  22. Chaht, F.L., Kaci, A., Houari, M.S.A., Tounsi, A., Beg, O.A. and Mahmoud, S.R. (2015), "Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect", Steel Compos. Struct., Int. J., 18(2), 425-442. https://doi.org/10.12989/scs.2015.18.2.425
  23. Chedad, A., Daouadji, T.H., Abderezak, R., Belkacem, A., Abbes, B., Rabia, B. and Abbes, F. (2017), "A high-order closed-form solution for interfacial stresses in externally sandwich FGM plated RC beams", Adv. Mater. Res., Int. J., 6(4), 317-328.
  24. Chow, T.L. (2013), Classical Mechanics, CRC Press, Boca Raton, FL, USA.
  25. Daouadj, T.H. and Adim, B. (2017), "Mechanical behaviour of FGM sandwich plates using a quasi-3Dhigher order shear and normal deformation theory", Struct. Eng. Mech., Int. J., 61(1), 49-63. https://doi.org/10.12989/sem.2017.61.1.049
  26. Daouadji, T.H. (2017), "Analytical and numerical modeling of interfacial stresses in beams bonded with a thin plate", Adv. Computat. Des., 2(1), 57-69.
  27. Daouadji, T.H. and Adim, B. (2016), "Theoretical analysis of composite beams under uniformly distributed load", Adv. Mater. Res., Int. J., 5(1), 1-9. https://doi.org/10.18052/www.scipress.com/IJARM.5.1
  28. Daouadji, T.H. and Hadji, L. (2015), "Analytical solution of nonlinear cylindrical bending for functionally graded plates", Geomech. Eng., Int. J., 9(5), 631-644. https://doi.org/10.12989/gae.2015.9.5.631
  29. Daouadji, T.H., Rabahi, A., Abbes, B. and Adim, B. (2016a), "Theoretical and finite element studies of interfacial stresses in reinforced concrete beams strengthened by externally FRP laminates plate", J. Adhes. Sci. Technol., 30(12), 1253-1280. https://doi.org/10.1080/01694243.2016.1140703
  30. Daouadji, T.H., Benferhat, R. and Adim, B. (2016b), "Bending analysis of an imperfect advanced composite plates resting on the elastic foundations", Coupl. Syst. Mech., Int. J., 5(3), 269-285. https://doi.org/10.12989/csm.2016.5.3.269
  31. Ebrahimi, F. and Nasirzadeh, P. (2015), "Small-scale effects on transverse vibrational behavior of singlewalled carbon nanotubes with arbitrary boundary conditions", Journal of Engineering Solid Mechanics, 3(2), 131-141. https://doi.org/10.5267/j.esm.2015.1.002
  32. Eringen, A.C. (1972a), "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
  33. Eringen, A.C. (1972b), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  34. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54(9), 4703-4710. https://doi.org/10.1063/1.332803
  35. Ghorbanpourarani, A., Mohammadimehr, M., Arefmanesh, A. and Ghasemi, A. (2010), "Transverse vibration of short carbon nanotubes using cylindrical shell and beam models", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 224(3), 745-756. https://doi.org/10.1243/09544062JMES1659
  36. Gupta, S.S. and Batra, R.C. (2008), "Continuum structures equivalent in normal mode vibrations to singlewalled carbon nanotubes", Comput. Mater. Sci., 43(4), 715-723. https://doi.org/10.1016/j.commatsci.2008.01.032
  37. Hadji, L., Khelifa, Z., Daouadji, T.H. and Bedia, E.A. (2015), "Static bending and free vibration of FGM beam using an exponential shear deformation theory", Coupl. Syst. Mech., Int. J., 4(1), 99-114. https://doi.org/10.12989/csm.2015.4.1.099
  38. Hadji, L., Daouadji, T.H., Meziane, M.A.A., Tlidji, Y. and Bedia, E.A. (2016a), "Analysis of functionally graded beam using a new first-order shear deformation theory", Struct. Eng. Mech., Int. J., 57(2), 315-325. https://doi.org/10.12989/sem.2016.57.2.315
  39. Hadji, L., Daouadji, T.H. and Bedia, E.A. (2016b), "Dynamic behavior of FGM beam using a new first shear deformation theory", Eartq. Struct., Int. J., 10(2), 451-461.
  40. Hassan, I.A.H. (2002), "On solving some eigen value problems by using a differential", Appl. Math. Computat., 127(1), 1-22. https://doi.org/10.1016/S0096-3003(00)00123-5
  41. Heireche, H., Tounsi, A., Benzair, A. and Mechab, I. (2008), "Sound wave propagation in single-walled carbon nanotubes with initial axial stress", J. Appl. Phys., 104, 014301. https://doi.org/10.1063/1.2949274
  42. Heireche, H., Tounsi, A., Benzair, A., Maachou, M. and Bedia, E.A. (2009), "Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity", Physica E, 40, 2791-2799.
  43. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354(6348), 56-58. https://doi.org/10.1038/354056a0
  44. Jiang, H., Liu, B., Huang, Y. and Hwang, K.C. (2004), "Thermal expansion of single wall carbon nanotube", Journal of Engineering of Material and Technology, 26, 265-270.
  45. Ju, S.P. (2004), "Application of differential transformation to transient advective-dispersive transport equation", Appl. Math. Computat., 155(1), 25-38. https://doi.org/10.1016/S0096-3003(03)00755-0
  46. Khelifa, Z., Hadji, L., Daouadji, T.H. and Bourada, M. (2018), "Buckling response with stretching effect of carbon nanotube-reinforced composite beams resting on elastic foundation", Struct. Eng. Mech., Int. J., 67(2), 125-130.
  47. Kiani, K. (2010), "A meshless approach for free transverse vibration of embedded single-walled nanotubes with arbitrary boundary conditions accounting for nonlocal effect", Int. J. Mech. Sci., 52(10), 1343-1356. https://doi.org/10.1016/j.ijmecsci.2010.06.010
  48. Kiani, K. and Mehri, B. (2010), "Assessment of nanotube structures under a moving nano-particle using nonlocal beam theories", J. Sound Vib., 329(11), 2241-2264. https://doi.org/10.1016/j.jsv.2009.12.017
  49. Lee, H.L. and Chang, W.J. (2009a), "Vibration analysis of fluid-conveying double-walled carbon nanotubes based on nonlocal elastic theory", J. Phys.: Condens. Matter., 21, 448001. https://doi.org/10.1088/0953-8984/21/44/448001
  50. Lee, H.L. and Chang, W.J. (2009b), "Free transverse vibration of the fluid-conveying single-walled carbon nanotube using nonlocal elastic theory", J. Appl. Phys., 103, 024302.
  51. Maachou, M., Zidour, M., Baghdadi, H., Ziane, N. and Tounsi, A. (2011), "A nonlocal Levinson beam model for free vibration analysis of zigzag single-walled carbon nanotubes including thermal effects", Solid State Commun., 151, 1467-1471 https://doi.org/10.1016/j.ssc.2011.06.038
  52. Naceri, M., Zidour, M., Semmah, A., Houari, M.S.A., Benzair, A. and Tounsi, A. (2011), "Sound wave propagation in armchair single walled carbon nanotubes under thermal environment", J. Appl. Phys., 110, 124322. DOI: 10.1063/1.3671636
  53. Rabahi, A., Daouadji, T.H., Abbes, B. and Adim, B. (2016), "Analytical and numerical solution of the interfacial stress in reinforced-concrete beams reinforced with bonded prestressed composite plate", Journal of Reinforced Plastics and Composites, 35(3), 258-272. https://doi.org/10.1177/0731684415613633
  54. Rabia, B., Abderezak, R., Daouadji, T.H., Abbes, B., Belkacem, A. and Abbes, F. (2018), "Analytical analysis of the interfacial shear stress in RC beams strengthened with prestressed exponentially-varying properties plate", Adv. Mater. Res., Int. J., 7(1), 29-44.
  55. Rajoria, H. and Jalili, N. (2005), "Passive vibration damping enhacement using carbon nanotube-epoxy reinforced composites", Compos. Sci. Technol., 65, 2079-2093. https://doi.org/10.1016/j.compscitech.2005.05.015
  56. Sallai, B., Hadji, L., Daouadji, T.H. and Adda, B. (2015), "Analytical solution for bending analysis of functionally graded beam", Steel Compos. Struct., Int. J., 19(4), 829-841. https://doi.org/10.12989/scs.2015.19.4.829
  57. Silvestre, N. (2008), "Length dependence of critical measures in single-walled carbon nanotubes", Int. J. Solids Struct., 45, 4902-4920. https://doi.org/10.1016/j.ijsolstr.2008.04.029
  58. Sudak, L.J. (2003), "Column buckling of multi walled carbon nanotubes using nonlocal continuum mechanics", J. Appl. Phys., 94(11), 7281-7287. https://doi.org/10.1063/1.1625437
  59. Tlidji, Y., Daouadji, T.H., Hadji, L., Tounsi, A. and Bedia, E.A.A. (2014), "Elasticity solution for bending response of functionally graded sandwich plates under thermo mechanical loading", J. Thermal Stress., 37, 852-869. https://doi.org/10.1080/01495739.2014.912917
  60. Tounsi, A., Daouadji, T.H. and Benyoucef, S. (2008), "Interfacial stresses in FRP-plated RC beams: Effect of adherend shear deformations", Int. J. Adhes. Adhes., 29(4), 343-351. https://doi.org/10.1016/j.ijadhadh.2008.06.008
  61. Yao, X.H. and Han, Q. (2006), "Buckling analysis of multi-walled carbon nanotubes under tensional load coupling with temperature change", Journal of Engineering of Material and Technology, 128, 419-428. https://doi.org/10.1115/1.2203102
  62. Yuzhou, S. and Liew, K.M. (2008), "The buckling of single-walled carbon nanotubes upon bending: the higher order gradient continuum and mesh-free method", Comput. Methods Appl. Mech. Eng., 197, 3001-3013. https://doi.org/10.1016/j.cma.2008.02.003
  63. Zidour, M., Benrahou, K.H., Semmah, A., Naceri, M., Belhadj, H.A., Bakhti, K. and Tounsi, A. (2012), "The thermal effect on vibration of zigzag single walled carbon nanotubes using nonlocal Timoshenko beam theory", Computat. Mater. Sci., 51(1), 252-260. https://doi.org/10.1016/j.commatsci.2011.07.021
  64. Zidour, M., Daouadji, T.H., Benrahou, K.H., Tounsi, A., Bedia, E.A.A. and Hadji, L. (2014), "Buckling analysis of chiral single walled carbon nanotubes by the nonlocal Timoshenko beam theory", Mech. Compos. Mater., 50(1), 95-104. https://doi.org/10.1007/s11029-014-9396-0
  65. Zoubida, K., Daouadji, T.H., Hadji, L., Tounsi, A. and El Abbes, A.B. (2016), "A new higher order shear deformation model of functionally graded beams based on neutral surface position", Transact. Indian Inst. Metals, 69(3), 683-691. https://doi.org/10.1007/s12666-015-0540-x

Cited by

  1. Buckling behavior of a single-layered graphene sheet resting on viscoelastic medium via nonlocal four-unknown integral model vol.34, pp.5, 2018, https://doi.org/10.12989/scs.2020.34.5.643
  2. Influence of boundary conditions on the bending and free vibration behavior of FGM sandwich plates using a four-unknown refined integral plate theory vol.25, pp.3, 2020, https://doi.org/10.12989/cac.2020.25.3.225
  3. Effects of hygro-thermo-mechanical conditions on the buckling of FG sandwich plates resting on elastic foundations vol.25, pp.4, 2018, https://doi.org/10.12989/cac.2020.25.4.311
  4. Predictions of the maximum plate end stresses of imperfect FRP strengthened RC beams: study and analysis vol.9, pp.4, 2018, https://doi.org/10.12989/amr.2020.9.4.265
  5. Effect of porosity distribution rate for bending analysis of imperfect FGM plates resting on Winkler-Pasternak foundations under various boundary conditions vol.9, pp.6, 2020, https://doi.org/10.12989/csm.2020.9.6.575
  6. Influences of porosity distributions and boundary conditions on mechanical bending response of functionally graded plates resting on Pasternak foundation vol.38, pp.1, 2018, https://doi.org/10.12989/scs.2021.38.1.001
  7. Study and analysis of the free vibration for FGM microbeam containing various distribution shape of porosity vol.77, pp.2, 2018, https://doi.org/10.12989/sem.2021.77.2.217
  8. Vibration analysis of porous FGM plate resting on elastic foundations: Effect of the distribution shape of porosity vol.10, pp.1, 2018, https://doi.org/10.12989/csm.2021.10.1.061
  9. Analysis on the buckling of imperfect functionally graded sandwich plates using new modified power-law formulations vol.77, pp.6, 2018, https://doi.org/10.12989/sem.2021.77.6.797
  10. Modeling and analysis of the imperfect FGM-damaged RC hybrid beams vol.6, pp.2, 2018, https://doi.org/10.12989/acd.2021.6.2.117
  11. New solution for damaged porous RC cantilever beams strengthening by composite plate vol.10, pp.3, 2018, https://doi.org/10.12989/amr.2021.10.3.169