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Nonlinear buckling and free vibration of curved CNTs by doublet mechanics

  • Eltaher, Mohamed A. (Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University) ;
  • Mohamed, Nazira (Department of Engineering Mathematics, Faculty of Engineering, Zagazig University) ;
  • Mohamed, Salwa A. (Department of Engineering Mathematics, Faculty of Engineering, Zagazig University)
  • Received : 2019.11.17
  • Accepted : 2020.03.28
  • Published : 2020.08.25
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Abstract

In this manuscript, static and dynamic behaviors of geometrically imperfect carbon nanotubes (CNTs) subject to different types of end conditions are investigated. The Doublet Mechanics (DM) theory, which is length scale dependent theory, is used in the analysis. The Euler-Bernoulli kinematic and nonlinear mid-plane stretching effect are considered through analysis. The governing equation of imperfect CNTs is a sixth order nonlinear integro-partial-differential equation. The buckling problem is discretized via the differential-integral-quadrature method (DIQM) and then it is solved using Newton's method. The equation of linear vibration problem is discretized using DIQM and then solved as a linear eigenvalue problem to get natural frequencies and corresponding mode shapes. The DIQM results are compared with analytical ones available in the literature and excellent agreement is obtained. The numerical results are depicted to illustrate the influence of length scale parameter, imperfection amplitude and shear foundation constant on critical buckling load, post-buckling configuration and linear vibration behavior. The current model is effective in designing of NEMS, nano-sensor and nano-actuator manufactured by CNTs.

Keywords

Acknowledgement

This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (G-051-135-1441). The authors, therefore, gratefully acknowledge the DSR technical and financial support.

References

  1. Agwa, M.A. and Eltaher, M.A. (2016), "Vibration of a carbyne nanomechanical mass sensor with surface effect", Appl. Phys. A, 122(4), 335. https://doi.org/10.1007/s00339-016-9934-9
  2. Akgoz, B. and Civalek, O. (2011), "Application of strain gradient elasticity theory for buckling analysis of protein microtubules", Current Appl. Phys., 11(5), 1133-1138. https://doi.org/10.1016/j.cap.2011.02.006
  3. Akgoz, B. and Civalek, O. (2018), "Vibrational characteristics of embedded microbeams lying on a two-parameter elastic foundation in thermal environment", Compos. Part B: Eng., 150, 68-77. https://doi.org/10.1016/j.compositesb.2018.05.049
  4. Ameri, A., Ajori, S. and Ansari, R. (2020), "On the buckling behavior of functionalized single-and double-walled carbon nanotubes with azobenzene in the aqueous environment: a molecular dynamics study", Struct. Chem., 31(1), 371-384. https://doi.org/10.1007/s11224-019-01418-6
  5. Amir, S., Arshid, E. and Arani, M.R.G. (2019), "Size-dependent magneto-electro-elastic vibration analysis of FG saturated porous annular/circular micro sandwich plates embedded with nanocomposite face sheets subjected to multi-physical pre loads", Smart Struct. Syst., Int. J., 23(5), 429-447. https://doi.org/10.12989/sss.2019.23.5.429
  6. Arani, A.G., Pourjamshidian, M., Arefi, M. and Arani, M.R. (2019), "Application of nonlocal elasticity theory on the wave propagation of flexoelectric functionally graded (FG) timoshenko nano-beams considering surface effects and residual surface stress", Smart Struct. Syst., Int. J., 23(2), 141-153. https://doi.org/10.12989/sss.2019.23.2.141
  7. Arda, M. and Aydogdu, M. (2020), "Vibration analysis of carbon nanotube mass sensors considering both inertia and stiffness of the detected mass", Mech. Based Des. Struct. Mach., 1-17. https://doi.org/10.1080/15397734.2020.1728548
  8. Atkins, P.W. and Friedman, R.S. (2011), Molecular Quantum Mechanics, Oxford University Press.
  9. Aydogdu, M. and Gul, U. (2018), "Buckling analysis of double nanofibers embeded in an elastic medium using doublet mechanics theory", Compos. Struct., 202, 355-363. https://doi.org/10.1016/j.compstruct.2018.02.015
  10. Aydogdu, M. and Gul, U. (2020), "Longitudinal vibration of double nanorod systems using doublet mechanics theory", Struct. Eng. Mech., Int. J., 73(1), 37-52. https://doi.org/10.12989/sem.2020.73.1.037
  11. Boussoula, A., Boucham, B., Bourada, M., Bourada, F., Tounsi, A., Bousahla, A.A. and Tounsi, A. (2020), "A simple nth-order shear deformation theory for thermomechanical bending analysis of different configurations of FG sandwich plates", Smart Struct. Syst., Int. J., 25(2), 197-218. https://doi.org/10.12989/sss.2020.25.2.197
  12. Civalek, O., Uzun, B., Yayli, M.O. and Akgoz, B. (2020), "Sizedependent 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
  13. Eberhardt, O. and Wallmersperger, T. (2014), "Mechanical properties and deformation behavior of carbon nanotubes calculated by a molecular mechanics approach", Smart Struct. Syst., Int. J., 13(4), 685-709. https://doi.org/10.12989/sss.2014.13.4.685
  14. Ellali, M., Amara, K., Bouazza, M. and Bourada, F. (2018), "The buckling of piezoelectric plates on pasternak elastic foundation using higher-order shear deformation plate theories", Smart Struct. Syst., Int. J., 21(1), 113-122. https://doi.org/10.12989/sss.2018.21.1.113
  15. Eltaher, M.A. and Agwa, M.A. (2016), "Analysis of sizedependent mechanical properties of CNTs mass sensor using energy equivalent model", Sensors Actuat. A: Phys., 246, 9-17. https://doi.org/10.1016/j.sna.2016.05.009
  16. Eltaher, M.A. and Mohamed, N (2020a), "Vibration of nonlocal perforated nanobeams with general boundary conditions", Smart Struct. Syst., Int. J., 25(4), 501-514. https://doi.org/10.12989/sss.2020.25.4.501
  17. Eltaher, M.A. and Mohamed, N. (2020b), "Nonlinear stability and vibration of imperfect CNTs by Doublet mechanics", Appl. Mathe. Computat., 382, 125311. https://doi.org/10.1016/j.amc.2020.125311
  18. Eltaher, M.A., Mahmoud, F.F., Assie, A.E. and Meletis, E.I. (2013), "Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams", Appl. Mathe. Computat., 224, 760-774. https://doi.org/10.1016/j.amc.2013.09.002
  19. Eltaher, M.A., Khairy, A., Sadoun, A.M. and Omar, F.A. (2014a), "Static and buckling analysis of functionally graded Timoshenko nanobeams", Appl. Mathe. Computat., 229, 283-295. http://dx.doi.org/10.1016/j.amc.2013.12.072
  20. Eltaher, M.A., Hamed, M.A., Sadoun, A.M. and Mansour, A. (2014b), "Mechanical analysis of higher order gradient nanobeams", Appl. Mathe. Computat., 229, 260-272. http://dx.doi.org/10.1016/j.amc.2013.12.076
  21. Eltaher, M.A., El-Borgi, S. and Reddy, J.N. (2016a), "Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs", Compos. Struct., 153, 902-913. http://dx.doi.org/10.1016/j.compstruct.2016.07.013
  22. Eltaher, M.A., Agwa, M.A. and Mahmoud, F.F. (2016b), "Nanobeam sensor for measuring a zeptogram mass", Int. J. Mech. Mater. Des., 12(2), 211-221. https://doi.org/10.1007/s10999-015-9302-5
  23. Eltaher, M.A., Agwa, M. and Kabeel, A. (2018), "Vibration analysis of material size-dependent CNTs using energy equivalent model", J. Appl. Computat. Mech., 4(2), 75-86. https://doi.org/10.22055/JACM.2017.22579.1136
  24. Eltaher, M.A., Almalki, T.A., Ahmed, K.I. and Almitani, K.H. (2019a), "Characterization and behaviors of single walled carbon nanotube by equivalent-continuum mechanics approach", Adv.in Nano Res., Int. J., 7(1), 39-49. https://doi.org/10.12989/anr.2019.7.1.039
  25. Eltaher, M.A., Almalki, T.A., Almitani, K.H., Ahmed, K.I.E. and Abdraboh, A.M. (2019b), "Modal participation of fixed-fixed single-walled carbon nanotube with vacancies", Int. J. Adv. Struct. Eng., 11(2), 151-163. https://doi.org/10.1007/s40091-019-0222-8
  26. Eltaher, M.A., Almalki, T.A., Almitani, K.H. and Ahmed, K.I.E. (2019c), "Participation factor and vibration of carbon nanotube with vacancies", J. Nano Res., 57, 158-174. https://doi.org/10.4028/www.scientific.net/JNanoR.57.158
  27. Eltaher, M.A., Mohamed, N., Mohamed, S. and Seddek, L.F. (2019d), "Postbuckling of curved carbon nanotubes using energy equivalent model", J. Nano Res., 57, 136-157. https://doi.org/10.12989/sem.2019.70.6.737
  28. Eltaher, M.A., Mohamed, N., Mohamed, S.A. and Seddek, L.F. (2019e), "Periodic and nonperiodic modes of postbuckling and nonlinear vibration of beams attached to nonlinear foundations", Appl. Mathe. Model., 75, 414-445. https://doi.org/10.1016/j.apm.2019.05.026
  29. Eltaher, M.A., Omar, F.A., Abdraboh, A.M., Abdalla, W.S. and Alshorbagy, A.E. (2020), "Mechanical behaviors of piezoelectric nonlocal nanobeam with cutouts", Smart Struct. Syst., Int. J., 25(2), 219-228. https://doi.org/10.12989/sss.2020.25.2.219
  30. Emam, S.A., Eltaher, M.A., Khater, M.E. and Abdalla, W.S. (2018), "Postbuckling and free vibration of multilayer imperfect nanobeams under a pre-stress load", Appl. Sci., 8(11), 2238. https://doi.org/10.3390/app8112238
  31. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  32. Fatahi-Vajari, A. and Imam, A. (2016a), "Axial vibration of singlewalled carbon nanotubes using doublet mechanics", Indian J. Phys., 90(4), 447-455. https://doi.org/10.1007/s12648-015-0775-8
  33. Fatahi-Vajari, A. and Imam, A. (2016b), "Torsional vibration of single-walled carbon nanotubes using doublet mechanics", Zeitschrift fur angewandte Mathematik und Physik, 67(4), 81. https://doi.org/10.1007/s00033-016-0675-6
  34. Ferrari, M., Granik, V.T., Granik, V.T., Imam, A. and Nadeau, J.C. (Eds.) (1997), Advances in Doublet Mechanics, (Vol. 45), Springer Science & Business Media.
  35. Gao, G., Cagin, T. and Goddard III, W.A. (1998), "Energetics, structure, mechanical and vibrational properties of single-walled carbon nanotubes", Nanotechnology, 9(3), 184. https://doi.org/10.1088/0957-4484/9/3/007
  36. 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
  37. Gul, U. and Aydogdu, M. (2017), "Wave propagation in double walled carbon nanotubes by using doublet mechanics theory", Physica E: Low-dimens. Syst. Nanostruct., 93, 345-357. https://doi.org/10.1016/j.physe.2017.07.003
  38. Gul, U. and Aydogdu, M. (2018a), "Structural modelling of nanorods and nanobeams using doublet mechanics theory", Int. J. Mech. Mater. Des., 14(2), 195-212. https://doi.org/10.1007/s10999-017-9371-8
  39. Gul, U. and Aydogdu, M. (2018b), "Noncoaxial vibration and buckling analysis of embedded double-walled carbon nanotubes by using doublet mechanics", Compos. Part B: Eng., 137, 60-67. https://doi.org/10.1016/j.compositesb.2017.11.005
  40. Gul, U. and Aydogdu, M. (2019), "Vibration analysis of Love nanorods using doublet mechanics theory", J. Brazil. Soc. Mech. Sci. Eng., 41(8), 351. https://doi.org/10.1007/s40430-019-1849-x
  41. 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
  42. Gul, U., Aydogdu, M. and Gaygusuzoglu, G.J.J.o.E.M. (2018), "Vibration and buckling analysis of nanotubes (nanofibers) embedded in an elastic medium using Doublet Mechanics", J. Eng. Mathe., 109(1), 85-111. https://doi.org/10.1007/s10665-017-9908-8
  43. Gurtin, M.E. and Murdoch, A.I. (1975), "A continuum theory of elastic material surfaces", Arch. Rational Mech. Anal., 57(4), 291-323. https://doi.org/10.1007/BF00261375
  44. Hehre, W.J. (1976), "Ab initio molecular orbital theory", Accounts Chem. Res., 9(11), 399-406. https://doi.org/10.1021/ar50107a003
  45. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354(6348), 56. https://doi.org/10.1038/354056a0
  46. 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. Methods Appl. Mech. Eng., 200(13-16), 1446-1454. https://doi.org/10.1016/j.cma.2011.01.001
  47. Kresse, G. and Hafner, J. (1993), "Ab initio molecular dynamics for liquid metals", Phys. Rev. B, 47(1), 558. https://doi.org/10.1103/PhysRevB.47.558
  48. Li, C. and Chou, T.W. (2003), "A structural mechanics approach for the analysis of carbon nanotubes", Int. J. Solids Struct., 40(10), 2487-2499. https://doi.org/10.1016/S0020-7683(03)00056-8
  49. Li, C. and Chou, T.W. (2004), "Modeling of elastic buckling of carbon nanotubes by molecular structural mechanics approach", Mech. Mater., 36(11), 1047-1055. https://doi.org/10.1016/j.mechmat.2003.08.009
  50. Mehralian, F., Beni, Y.T. and Kiani, Y. (2017), "Molecular dynamics study on the thermal buckling of carbon nanotubes in the presence of pre-load", Mater. Res. Express, 4(1), 015011. https://doi.org/10.1088/2053-1591/aa576a
  51. Mindlin, R.D. (1963), "Influence of couple-stresses on stress concentrations", Experim. Mech., 3(1), 1-7. https://doi.org/10.1007/BF02327219
  52. Mohamed, N., Eltaher, M.A., Mohamed, S.A. and Seddek, L.F. (2018), "Numerical analysis of nonlinear free and forced vibrations of buckled curved beams resting on nonlinear elastic foundations", Int. J. Non-Linear Mech., 101, 157-173. https://doi.org/10.1016/j.ijnonlinmec.2018.02.014
  53. Mohamed, N., Eltaher, M.A., Mohamed, S.A. and Seddek, L.F. (2019), "Energy equivalent model in analysis of postbuckling of imperfect carbon nanotubes resting on nonlinear elastic foundation", Struct. Eng. Mech., Int. J., 70(6), 737-750. https://doi.org/10.12989/sem.2019.70.6.737
  54. Mohamed, N., Mohamed, S.A. and Eltaher, M.A. (2020), "Buckling and post-buckling behaviors of higher order carbon nanotubes using energy-equivalent model", Eng. Comput. https://doi.org/10.1007/s00366-020-00976-2
  55. Peng, S. and Cho, K. (2003), "Ab initio study of doped carbon nanotube sensors", Nano Lett., 3(4), 513-517. https://doi.org/10.1021/nl034064u
  56. Rapaport, D.C. and Rapaport, D.C.R. (2004), The Art of Molecular Dynamics Simulation, Cambridge University Press.
  57. Shokravi, M. (2018), "Dynamic buckling of smart sandwich beam subjected to electric field based on hyperbolic piezoelasticity theory", Smart Struct. Syst., Int. J., 22(3), 327-334. https://doi.org/10.12989/sss.2018.22.3.327
  58. Shokravi, M. and Jalili, N. (2017), "Dynamic buckling response of temperature-dependent functionally graded-carbon nanotubesreinforced sandwich microplates considering structural damping", Smart Struct. Syst., Int. J., 20(5), 583-593. https://doi.org/10.12989/sss.2017.20.5.583
  59. Tabbakh, M. and Nasihatgozar, M. (2018), "Buckling analysis of nanocomposite plates coated by magnetostrictive layer", Smart Struct. Syst., Int. J., 22(6), 743-751. https://doi.org/10.12989/sss.2018.22.6.743
  60. Tounsi, A., Benguediab, S., Adda, B., Semmah, A. and Zidour, M. (2013), "Nonlocal effects on thermal buckling properties of double-walled carbon nanotubes", Adv. Nano Res., Int. J., 1(1), 1-11. https://doi.org/10.12989/anr.2013.1.1.001
  61. Toupin, R.A. (1962), "Elastic materials with couple-stresses", Arch. Rational Mech. Anal., 11(1), 385-414. https://doi.org/10.1007/BF00253945
  62. Xiao, J.R., Gama, B.A. and Gillespie Jr, J.W. (2005), "An analytical molecular structural mechanics model for the mechanical properties of carbon nanotubes", Int. J. Solids Struct., 42(11-12), 3075-3092. https://doi.org/10.1016/j.ijsolstr.2004.10.031
  63. Yang, F.A.C.M., 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
  64. Yayli, M.O. and Asa, E. (2019), "Longitudinal vibration of carbon nanotubes with elastically restrained ends using doublet mechanics", Microsyst. Technol., 1-10. https://doi.org/10.1007/s00542-019-04512-1
  65. Youcef, D.O., Kaci, A., Benzair, A., Bousahla, A.A. and Tounsi, A. (2018), "Dynamic analysis of nanoscale beams including surface stress effects", Smart Struct. Syst., Int. J., 21(1), 65-74. https://doi.org/10.12989/sss.2018.21.1.065
  66. Zhou, L.G. and Shi, S.Q. (2002), "Molecular dynamic simulations on tensile mechanical properties of single-walled carbon nanotubes with and without hydrogen storage", Computat. Mater. Sci., 23(1-4), 166-174. https://doi.org/10.1016/S0927-0256(01)00233-6

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