Advances in nano research

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Free vibration analysis of chiral double-walled carbon nanotube using non-local elasticity theory

  • Rakrak, Kaddour (Laboratoire de Modelisation et Simulation Multi-echelle, Universite de Sidi Bel Abbes) ;
  • Zidour, Mohamed (Laboratoire des Materiaux et Hydrologie, Universite de Sidi Bel Abbes) ;
  • Heireche, Houari (Laboratoire de Modelisation et Simulation Multi-echelle, Universite de Sidi Bel Abbes) ;
  • Bousahla, Abdelmoumen Anis (Laboratoire de Modelisation et Simulation Multi-echelle, Universite de Sidi Bel Abbes) ;
  • Chemi, Awda (Laboratoire de Modelisation et Simulation Multi-echelle, Universite de Sidi Bel Abbes)
  • Received : 2015.12.24
  • Accepted : 2016.03.02
  • Published : 2016.03.25
⟨ Previous Next ⟩

Abstract

This article is concerned with the free vibration problem for chiral double-walled carbon nanotube (DWCNTs) modelled using the non-local elasticity theory and Euler Bernoulli beam model. According to the governing equations of non-local Euler Bernoulli beam theory and the boundary conditions, the analytical solution is derived and two branches of transverse wave propagating are obtained. The numerical results obtained provide better representations of the vibration behaviour of double-walled carbon nanotube, where the aspect ratio of the (DWCNTs), the vibrational mode number, the small-scale coefficient and chirality of double-walled carbon nanotube on the frequency ratio (${\chi}^N$) of the (DWCNTs) are significant. In this work, the numerical results obtained can be used to predict and prevent the phenomenon of resonance for the forced vibration analyses of double -walled carbon nanotubes.

Keywords

References

  1. 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., 55(4), 743-762. https://doi.org/10.12989/sem.2015.55.4.743
  2. Ait Amar Meziane, M., Abdelaziz, H.H. and Tounsi, A. (2014), "An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions", J. Sandw. Struct. Mater., 16(3), 293-318. https://doi.org/10.1177/1099636214526852
  3. Ait Yahia, S., Ait Atmane, H., Houari, M.S.A. and Tounsi, A. (2015), "Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories", Struct. Eng. Mech., 53(6), 1143-1165. https://doi.org/10.12989/sem.2015.53.6.1143
  4. Al-Basyouni, K.S., Tounsi, A. and Mahmoud, S.R. (2015), "Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position", Compos. Struct., 125, 621-630. https://doi.org/10.1016/j.compstruct.2014.12.070
  5. Amara, K., Tounsi, A., Mechab, I. and Adda Bedia, E. (2010), "Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field", Appl. Math. Model., 34, 3933-3942. https://doi.org/10.1016/j.apm.2010.03.029
  6. Baghdadi, H., Tounsi, A., Zidour, M. and Benzair, A. (2015), "Thermal effect on vibration characteristics of armchair and zigzag single-walled carbon nanotubes using nonlocal parabolic beam theory", Full. Nanotub. Carbon Nanostr., 23, 266-272. https://doi.org/10.1080/1536383X.2013.787605
  7. Bao, W.X., Zhu, Ch.Ch. and Cui, W.Zh. (2004), "Simulation of Young's modulus of single-walled carbon nanotubes by molecular dynamics", Physica B, 352, 156-163. https://doi.org/10.1016/j.physb.2004.07.005
  8. Benguediab, S., Tounsi, A., Zidour, M. and Semmah, A. (2014), "Chirality and scale effects on mechanical buckling properties of zigzag double-walled carbon nanotubes", Compos. Part B, 57, 21-24. https://doi.org/10.1016/j.compositesb.2013.08.020
  9. 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., 3(1), 29-37. https://doi.org/10.12989/anr.2015.3.1.029
  10. Belabed, Z., Houari, M.S.A., Tounsi, A., Mahmoud, S.R. and Anwar Beg, O. (2014), "An efficient and simple higher order shear and normal deformation theory for functionally graded material (FGM) plates", Compos. Part B, 60, 274-283. https://doi.org/10.1016/j.compositesb.2013.12.057
  11. Bellifa, H., Benrahou, K.H., Hadji, L., Houari, M.S.A. and Tounsi, A. (2016), "Bending and free vibration analysis of functionally graded plates using a simple shear deformation theory and the concept the neutral surface position", J. Braz. Soc. Mech. Sci. Eng., 38, 265-275. https://doi.org/10.1007/s40430-015-0354-0
  12. Belkorissat, I., Houari, M.S.A., Tounsi, A., Adda Bedia, E.A. and Mahmoud, S.R. (2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model", Steel Compos. Struct., 18(4), 1063-1081. https://doi.org/10.12989/scs.2015.18.4.1063
  13. Bennoun, M., Houari, M.S.A. and Tounsi, A. (2016), "A novel five variable refined plate theory for vibration analysis of functionally graded sandwich plates", Mech. Adv. Mater. Struct., 23(4), 423-431. https://doi.org/10.1080/15376494.2014.984088
  14. Benzair, A., Tounsi, A., Besseghier, A., Heireche, H., Moulay, N. and Boumia, L. (2008), "The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory", J. Phys. D., 41, 225404. https://doi.org/10.1088/0022-3727/41/22/225404
  15. Berrabah, H.M., Tounsi, A., Semmah, A. and El Adda, B. (2013), "Comparison of various refined nonlocal beam theories for bending, vibration and buckling analysis of nanobeams", Struct. Eng. Mech., 48(3), 351-365. https://doi.org/10.12989/sem.2013.48.3.351
  16. Bouazza, M., Amara, K., Zidour, M., Tounsi, A. and El Adda, B. (2014), "Thermal effect on buckling of multiwalled carbon nanotubes using different gradient elasticity theories", Nanosci. Nanotech., 4(2) 27-33.
  17. Bouderba, B., Houari, M.S.A. and Tounsi, A. (2013), "Thermomechanical bending response of FGM thick plates resting on Winkler-Pasternak elastic foundations", Steel Compos. Struct., 14(1), 85-104. https://doi.org/10.12989/scs.2013.14.1.085
  18. Boumia, L., Zidour, M., Benzair, A. and Tounsi, A. (2014), "Timoshenko beam model for vibration analysis of chiral single-walled carbon nanotubes", Physica E, 59, 186-191. https://doi.org/10.1016/j.physe.2014.01.020
  19. Bounouara, F., Benrahou, K.H., Belkorissat, I. and Tounsi, A. (2016), "A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation", Steel Compos. Struct., 20(2), 227-249. https://doi.org/10.12989/scs.2016.20.2.227
  20. Bourada, M., Kaci, A., Houari, M.S.A. and Tounsi, A. (2015), "A new simple shear and normal deformations theory for functionally graded beams", Steel Compos. Struct., 18(2), 409-423. https://doi.org/10.12989/scs.2015.18.2.409
  21. Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Adda Bedia, E.A. (2014), "A novel higher order shear and normal deformation theory based on neutral surface position for bending analysis of advanced composite plates", Int. J. Comput. Meth., 11(6), 1350082. https://doi.org/10.1142/S0219876213500825
  22. Chemi, A., Heireche, H., Zidour, M., Rakrak, K. and Bousahla, A.A. (2015), "Critical buckling load of chiral double-walled carbon nanotube using non-local theory elasticity", Adv. Nano Res., 3(4), 193-206. https://doi.org/10.12989/anr.2015.3.4.193
  23. Cornwell, C.F. and Wille, L.T. (1997), "Elastic properties of single-walled carbon nanotubes in compression", Solid. State. Commun., 101, 555. https://doi.org/10.1016/S0038-1098(96)00742-9
  24. Doyle, J.F. (1997), "Wave Propagation in Structures", 2nd edition, Springer, New York,.
  25. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 101-106.
  26. 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
  27. Fu, Y.M., Hong, J.W. and Wang, X.Q. (2006), "Analysis of nonlinear vibration for embedded carbon nanotubes", J. Sound Vib., 296, 746-756. https://doi.org/10.1016/j.jsv.2006.02.024
  28. Gafour,Y., Zidour, M., Tounsi, A., Heireche, H. and Semmah, A. (2013), "Sound wave propagation in zigzag double-walled carbon nanotubes embedded in an elastic medium using nonlocal elasticity theory", Physica E, 48, 118-123. https://doi.org/10.1016/j.physe.2012.11.006
  29. Hamidi, A., Houari, M.S.A., Mahmoud, S.R. and Tounsi, A. (2015), "A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates", Steel Compos. Struct., 18(1), 235-253. https://doi.org/10.12989/scs.2015.18.1.235
  30. Hebali, H., Tounsi, A., Houari, M.S.A., Bessaim, A. and Adda Bedia, E.A. (2014), "A new quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech., ASCE, 140, 374-383. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000665
  31. Heireche, H., Tounsi, A., Benzair, A., Maachou, M. and Adda Bedia, E.A. (2008), "Sound wave propagation in single-walled carbon nanotubes using nonlocal elasticity", Physica E., 40, 2791. https://doi.org/10.1016/j.physe.2007.12.021
  32. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354, 56-8. https://doi.org/10.1038/354056a0
  33. Iijima, S. and Ichihashi, T. (1993), "Single-shell carbon nanotubes of 1 nm diameter", Nature, 363, 603. https://doi.org/10.1038/363603a0
  34. Jin, Y. and Yuan, F.G. (2003), "Simulation of elastic properties of single-walled carbon nanotubes", Compos. Sci. Technol., 63, 1507. https://doi.org/10.1016/S0266-3538(03)00074-5
  35. Larbi Chaht, F., Kaci, A., Houari, M.S.A., Tounsi, A., Anwar Beg, O. 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., 18(2), 425-442. https://doi.org/10.12989/scs.2015.18.2.425
  36. Lau, K.T. and Hui, D. (2002), "The revolutionary creation of new advanced materials-carbon nanotube composites", Compos. Part B: Eng., 33(4), 263-277. https://doi.org/10.1016/S1359-8368(02)00012-4
  37. Liu, J.Z., Zheng, Q.S. and Jiang, Q. (2001), "Effect of a rippling mode on resonances of carbon nanotubes", Phys. Rev. Lett., 86, 4843. https://doi.org/10.1103/PhysRevLett.86.4843
  38. Lu, P., Lee, H. P., Lu C. and Zhang, P. Q. (2006), "Dynamic properties of flexural beams using a nonlocal elasticity model", J. Appl. Phys., 99, 073510. https://doi.org/10.1063/1.2189213
  39. 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
  40. Murmu, T. and Adhikari, S. (2010), "Thermal effects on the stability of embedded carbon nanotubes", Physica E, 43, 415-422. https://doi.org/10.1016/j.physe.2010.08.023
  41. Naceri, M., Zidour, M., Semmah, A., Houari, 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. https://doi.org/10.1063/1.3671636
  42. Natsuki, T., Hayashi, T. and M. Endo, (2005), "Wave propagation of carbon nanotubes embedded in an elastic medium", J. Appl. Phys., 97, 044307. https://doi.org/10.1063/1.1849823
  43. Ould Youcef, D., Kaci, A., Houari, M.S.A., Tounsi, A., Benzair, A. and Heireche, H. (2015), "On the bending and stability of nanowire using various HSDTs", Adv. Nano Res., 3(4), 177-191. https://doi.org/10.12989/anr.2015.3.4.177
  44. Qian, D., Wagner, G.J., Liu, W.K., Yu, M.F. and Ruoff, R.S. (2002), "Mechanics of carbon nanotubes ", Appl. Mech. Rev., 55(6), 495-533. https://doi.org/10.1115/1.1490129
  45. Ranjbartoreh, A.R., Ghorbanpour, A. and Soltani, B. (2007), "Double-walled carbon nanotube with surrounding elastic medium under axial pressure", Physica E, 39, 230-239. https://doi.org/10.1016/j.physe.2007.04.010
  46. Tagrara et al. (2015), "On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams", Steel Compos. Struct., 19(5), 1259-1277. https://doi.org/10.12989/scs.2015.19.5.1259
  47. Tokio, Y. (1995), "Recent development of carbon nanotube", Synth. Met., 70, 1511-8. https://doi.org/10.1016/0379-6779(94)02939-V
  48. Tombler, T.W., Zhou, C.W., Alexseyev, L. et al. (2000), "Reversible nanotube electro-mechanical characteristics under local probe manipulation", Nature, 405, 769. https://doi.org/10.1038/35015519
  49. Tounsi, A., Benguediab, S., Bedia, El A., Semmah, A. and Zidour, M. (2013), "Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes", Adv. Nano Res., 1(1), 1-11. https://doi.org/10.12989/anr.2013.1.1.001
  50. Tu, Z.C. and Ou-Yang, Z.C. (2002), "Single-walled and multiwalled carbon nanotubes viewed as elastic tubes with the effective Young's modulus dependent on layer number", Phys. Rev. B, 65, 233407. https://doi.org/10.1103/PhysRevB.65.233407
  51. Wong, E.W., Sheehan, P.E. and Lieber, C.M. (1997), "Nanobeam mechanics: elasticity, strength and toughness of nanorods and nanotubes", Science, 2771971.
  52. Yoon, J., Ru, C.Q. and Mioduchowski, A. (2003), "Vibration of an embedded multiwall carbon nanotube", Compos. Sci. Tech., 63, 1533-1545. https://doi.org/10.1016/S0266-3538(03)00058-7
  53. Zemri, A., Houari, M.S.A., Bousahla, A.A. and Tounsi, A. (2015), "A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory", Struct. Eng. Mech., 54(4), 693-710. https://doi.org/10.12989/sem.2015.54.4.693
  54. Zhang, Y.Q., Liu, G.R. and Xie, X.Y. (2005), "Free transverse vibrations of double-walled carbon nanotubes using a theory of nonlocal elasticity", Phys. Rev., 71, 195404. https://doi.org/10.1103/PhysRevB.71.195404
  55. 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", Comput. Mater. Sci., 51, 252-260. https://doi.org/10.1016/j.commatsci.2011.07.021
  56. Zidour, M., Daouadji, T.H., Benrahou, K.H., Tounsi, A., Adda Bedia, El A. and Hadji, L. (2014), "Buckling analysis of chiral single-walled carbon nanotubes by using the nonlocal timoshenko beam theory", Mech. Compos. Mater., 50(1), 95-104. https://doi.org/10.1007/s11029-014-9396-0
  57. Zidour, M., Hadji, L., Bouazza, M., Tounsi, A. and Adda Bedia, El A. (2015), "The mechanical properties of Zigzag carbon nanotube using the energy-equivalent model", J. Chem. Mater. Res., 3, 9-14.

Cited by

  1. Vibration Analysis of Nano Beam Using Differential Transform Method Including Thermal Effect vol.54, pp.1661-9897, 2018, https://doi.org/10.4028/www.scientific.net/JNanoR.54.1
  2. A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams vol.19, pp.2, 2017, https://doi.org/10.12989/sss.2017.19.2.115
  3. Buckling temperature of a single-walled boron nitride nanotubes using a novel nonlocal beam model vol.5, pp.1, 2017, https://doi.org/10.12989/anr.2017.5.1.001
  4. Effect of high energy ball milling on the structure of iron - multiwall carbon nanotubes (MWCNT) composite vol.6, pp.3, 2017, https://doi.org/10.12989/amr.2017.6.3.245
  5. Load transfer and energy absorption in transversely compressed multi-walled carbon nanotubes vol.6, pp.3, 2016, https://doi.org/10.12989/csm.2017.6.3.273
  6. A new nonlocal HSDT for analysis of stability of single layer graphene sheet vol.6, pp.2, 2016, https://doi.org/10.12989/anr.2018.6.2.147
  7. The critical buckling load of reinforced nanocomposite porous plates vol.67, pp.2, 2016, https://doi.org/10.12989/sem.2018.67.2.115
  8. Analysis of boundary conditions effects on vibration of nanobeam in a polymeric matrix vol.67, pp.5, 2016, https://doi.org/10.12989/sem.2018.67.5.517
  9. Small-scale effect on the forced vibration of a nano beam embedded an elastic medium using nonlocal elasticity theory vol.6, pp.1, 2019, https://doi.org/10.12989/aas.2019.6.1.001
  10. Vibration analysis of different material distributions of functionally graded microbeam vol.69, pp.6, 2016, https://doi.org/10.12989/sem.2019.69.6.637
  11. Static and Dynamic Behavior of Nanotubes-Reinforced Sandwich Plates Using (FSDT) vol.57, pp.None, 2016, https://doi.org/10.4028/www.scientific.net/jnanor.57.117
  12. A Non-Linear Spring Model for Predicting Modal Behavior of Oscillators Built from Double Walled Carbon Nanotubes vol.60, pp.None, 2016, https://doi.org/10.4028/www.scientific.net/jnanor.60.21
  13. A Non-Linear Spring Model for Predicting Modal Behavior of Oscillators Built from Double Walled Carbon Nanotubes vol.60, pp.None, 2016, https://doi.org/10.4028/www.scientific.net/jnanor.60.21
  14. Vibrational analysis of an irregular single-walled carbon nanotube incorporating initial stress effects vol.9, pp.1, 2020, https://doi.org/10.1515/ntrev-2020-0114
  15. Porosity-dependent free vibration analysis of FG nanobeam using non-local shear deformation and energy principle vol.8, pp.1, 2020, https://doi.org/10.12989/anr.2020.8.1.037
  16. Buckling of carbon nanotube reinforced composite plates supported by Kerr foundation using Hamilton's energy principle vol.73, pp.2, 2016, https://doi.org/10.12989/sem.2020.73.2.209
  17. Prediction and assessment of nonlocal natural frequencies of DWCNTs: Vibration analysis vol.25, pp.2, 2016, https://doi.org/10.12989/cac.2020.25.2.133
  18. Critical Buckling Load of Triple-Walled Carbon Nanotube Based on Nonlocal Elasticity Theory vol.62, pp.None, 2020, https://doi.org/10.4028/www.scientific.net/jnanor.62.108
  19. Critical Buckling Load of Triple-Walled Carbon Nanotube Based on Nonlocal Elasticity Theory vol.62, pp.None, 2020, https://doi.org/10.4028/www.scientific.net/jnanor.62.108
  20. Bending analysis of magneto-electro piezoelectric nanobeams system under hygro-thermal loading vol.8, pp.3, 2016, https://doi.org/10.12989/anr.2020.8.3.203
  21. Eringen's nonlocal model sandwich with Kelvin's theory for vibration of DWCNT vol.25, pp.4, 2016, https://doi.org/10.12989/cac.2020.25.4.343
  22. Theoretical impact of Kelvin's theory for vibration of double walled carbon nanotubes vol.8, pp.4, 2016, https://doi.org/10.12989/anr.2020.8.4.307
  23. Deformation in a nonlocal magneto-thermoelastic solid with hall current due to normal force vol.22, pp.2, 2016, https://doi.org/10.12989/gae.2020.22.2.109
  24. Wave propagation analysis for a second strain gradient rod theory vol.33, pp.10, 2016, https://doi.org/10.1016/j.cja.2019.10.006
  25. Torsional vibration of irregular single-walled carbon nanotube incorporating compressive initial stress effects vol.37, pp.None, 2021, https://doi.org/10.1093/jom/ufab002
  26. Influence of the Catalyst Supporting Material on the Growth of Carbon Nanotubes vol.1163, pp.None, 2016, https://doi.org/10.4028/www.scientific.net/amr.1163.117
  27. Influence of the Catalyst Supporting Material on the Growth of Carbon Nanotubes vol.1163, pp.None, 2016, https://doi.org/10.4028/www.scientific.net/amr.1163.117