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Application of Kelvin's approach for material structure of CNT: Polynomial volume fraction law

  • Hussain, Muzamal (Department of Mathematics, Govt. College University Faisalabad)
  • 투고 : 2020.03.19
  • 심사 : 2020.05.27
  • 발행 : 2020.10.10

초록

In this piece of work, carbon nanotubes motion equations are framed by Kelvin's method. Employment of the Kelvin's method procedure gives birth to the tube frequency equation. It is also exhibited that the effect of frequencies is investigated by varying the different index of polynomial function. By using volume fraction for power law index, the fundamental natural frequency spectra for two forms of single-walled carbon nanotubes are calculated. The influence of frequencies against length-to-diameter ratios with varying power law index are investigated in detail for these tubes. Throughout the computation, it is observed that the frequency behavior for the boundary conditions follow as; clamped-clamped, simply supported-simply supported and these frequency curves are higher than that of clamped-free curves. Computer software MATLAB is utilized for the frequencies of single-walled carbon nanotubes.

키워드

참고문헌

  1. Ahmed, R. A., Fenjan, R. M., and Faleh, N. M. (2019), "Analyzing post-buckling behavior of continuously graded FG nanobeams with geometrical imperfections", Geomech. Eng., 17(2), 175-180.-https://doi.org/10.12989/gae.2019.17.2.175.
  2. Ansari, R., and Ajori, S. (2014), "Molecular dynamics study of the torsional vibration characteristics of boron-nitride nanotubes", Physics Lett. A, 378(38-39), 2876-2880. https://doi.org/10.1016/j.physleta.2014.08.006
  3. Ansari, R., and Rouhi, H. (2013), "Nonlocal analytical Flugge shell model for the vibrations of double-walled carbon nanotubes with different end conditions", Int. J. Appl. Mech., 80(2), 021006. https://doi.org/10.1142/S179329201250018X.
  4. Arani, A. J., and Kolahchi, R. (2016), "Buckling analysis of embedded concrete columns armed with carbon nanotubes", Comput. Concrete, 17(5), 567-578. https://doi.org/10.12989/cac.2016.17.5.567
  5. Arani, Jafarian A., and Kolahchi R. (2016), "Buckling analysis of embedded concrete columns armed with carbon nanotubes", Comput Concr., 17(5), 567-578. https://doi.org/10.12989/cac.2016.17.5.567.
  6. Batou, B., Nebab, M., Bennai, R., Atmane, H. A., Tounsi, A. and Bouremana, M. (2019), "Wave dispersion properties in imperfect sigmoid plates using various HSDTs", Steel Compos. Struct., 33(5), 699. https://doi.org/10.12989/scs.2019.33.5.699
  7. Batou, B., Nebab, M., Bennai, R., Atmane, H.A., Tounsi, A. and Bouremana, M. (2019), "Wave dispersion properties in imperfect sigmoid plates using various HSDTs", Steel Compos. Struct., 33(5), 699-716. https://doi.org/10.12989/scs.2019.33.5.699
  8. Behera, S. and Kumari, P. (2018), "Free vibration of Levy-type rectangular laminated plates using efficient zig-zag theory", Adv. Comput. Des., 3(3), 213-232. https://doi.org/10.12989/acd.2018.3.3.213
  9. Benguediab, S., Tounsi, A., Ziadour, and Semmah, A. (2014), "Chirality and scale effects on mechanical and buckling properties of zigzag double-walled carbon nanotubes", Composites Part B, 57, 21-24. https://doi.org/10.1016/j.compositesb.2013.08.020.
  10. Bilouei, B. S., Kolahchi, R., and Bidgoli, M. R. (2016), "Buckling of concrete columns retrofitted with "Nano-Fiber Reinforced Polymer (NFRP)", Comput. Concrete, 18(5), 1053-1063. https://doi.org/10.12989/cac.2016.18.5.1053
  11. Bilouei, Safari B, Kolahchi, R., and Bidgoli, M. R. (2016), " Buckling of concrete columns retrofitted with Nano-Fiber Reinforced Polymer (NFRP)", Comput. Concrete, 18(5), 1053- 1063. https://doi.org/10.12989/cac.2016.18.5.1053.
  12. Bisen, H.B., Hirwani, C.K., Satankar, R.K., Panda, S.K., Mehar, K. and Patel, B. (2018), "Numerical study of frequency and deflection responses of natural fiber (Luffa) reinforced polymer composite and experimental validation", J. Nat. Fib., 1-15. https://doi.org/10.1080/15440478.2018.1503129
  13. Brischotto, S. (2015), "A continuum shell model including van der Waals interaction for free vibrations of double-walled carbon nanotubes", CMES, 104, 305-327.
  14. Chemi, A., Zidour, M., Heireche, H., Rakrak, K. and Bousahla, A.A. (2018), "Critical buckling load of chiral double-walled carbon nanotubes embedded in an elastic medium", Mech. Compos. Mater., 53(6), 827-836. https://doi.org/10.1007/s11029-018-9708-x
  15. Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer Science and Business Media, Germany.
  16. Faleh, N. M., Fenjan, R. M., and Ahmed, R. A. (2020), "Forced Vibrations of Multi-phase Crystalline Porous Shells Based on Strain Gradient Elasticity and Pulse Load Effects", J. Vib. Eng. Technol., 1-9.- 10.1007/s42417-020-00203-8.
  17. Fatahi-Vajari. A., Azimzadeh, Z., Hussain. M., (2019), "Nonlinear coupled axial-torsional vibration of single-walled carbon nanotubes using Galerkin and Homotopy perturbation method", Micro Nano Lett., https://doi.org/10.1049/mnl.2019.0203.
  18. Fenjan, R. M., Ahmed, R. A., Alasadi, A. A., and Faleh, N. M. (2019c), "Nonlocal strain gradient thermal vibration analysis of double-coupled metal foam plate system with uniform and non-uniform porosities", Coupled. Syst. Mech., 8(3), 247-257.
  19. Fenjan, R. M., Ahmed, R. A., Alasadi, A. A., and Faleh, N. M. (2019b), "Nonlocal strain gradient thermal vibration analysis of double-coupled metal foam plate system with uniform and non-uniform porosities. Coupled. Syst. Mech., 8(3), 247-257.- https://doi.org/10.12989/csm.2019.8.3.247.
  20. Fenjan, R. M., Ahmed, R. A., and \Faleh, N. M. (2019a), "Investigating dynamic stability of metal foam nanoplates under periodic in-plane loads via a three-unknown plate theory", Adv. Aircraft Spacecraft Sci., 6(4), 297-314.- https://doi.org/10.12989/aas.2019.6.4.297.
  21. Flugge, S., (1973), Stresses in Shells, Springer 2nd Edition, Germany.
  22. Gao, Y., and An, L. (2010), "A nonlocal elastic anisotropic shell model for microtubule buckling behaviors in cytoplasm", Physica E, 42(9), 2406-2415. https://doi.org/10.1016/j.physe.2010.05.022.
  23. Georgantzinos, S. K., Giannopoulos, G. I. and Anifantis, N. K. (2009), "An efficient numerical model for vibration analysis of single-walled carbon nanotubes", Comput/ Mech., 43(6), 731- 741. https://doi.org/10.1007/s00466-008-0341-8
  24. Ghadiri, M., Ebrahimi, F., Salari, E., Hosseini, S. A. H., and Shaghaghi, G. R. (2015), "Electro-thermo-mechanical vibration analysis of embedded single-walled boron nitride nanotubes based on nonlocal third-order beam theory", J. Multiscale Comput. Eng., 13(5),
  25. Ghavanloo, E., Daneshmand, F., and Rafiei, M. (2010), "Vibration and instability analysis of carbon nanotubes conveying fluid and resting on a linear viscous elastic Winkler foundation", Physica E, 42, 2218-2224. https://doi.org/10.1016/j.physe.2010.04.024.
  26. Gibson, R.F., Ayorinde, E.O. and Wen, Y.F. (2007), "Vibrations of carbon nanotubes and their composites: A review", Compos. Sci. Technol., 67(1),1-28. https://doi.org/10.1016/j.compscitech.2006.03.031.
  27. Goncalves, P.B., DA silva, F.M.A. and Prado, Z.J.G.N. (2006), "Transient stability of empty and fluid-filled cylindrical shells", J. Braz. Soc. Mech. Sci. Eng, 28(3), 331-333. http://dx.doi.org/10.1590/S1678-58782006000300011.
  28. Gupta, S.S., Bosco, F.G., and Batra, R.C. (2010), "Wall thickness and elastic moduli of single-walled carbon nanotubes from frequencies of axial, torsional and inextensional modes of vibration", Comput. Mater. Sci., 47(4), 1049-1059. https://doi.org/10.1016/j.commatsci.2009.12.007.
  29. Hayati, H., Hosseini, S. A., and Rahmani, O. (2017), "Coupled twist-bending static and dynamic behavior of a curved single-walled carbon nanotube based on nonlocal theory", Microsyst. Technol., 23(7), 2393-2401. https://doi.org/10.1007/s00542-016-2933-0
  30. He, X.Q., Kitipornchai, S., and Liew, K.M. (2005), "Buckling analysis of multi-walled carbon nanotubes: a continuum model accounting for van der Waals interaction", J. Mech. Phys. Solids, 53, 303-326. https://doi.org/10.1016/j.jmps.2004.08.003.
  31. Heydarpour, Y., Aghdam, M.M., and Malekzadeh, P. (2014), "Free vibration analysis of rotating functionally graded carbon nanotube-reinforced composite truncated conical shells", Compos. Struct., 117, 187-200. https://doi.org/10.1016/j.compstruct.2014.06.023.
  32. Hsu, J. C., Chang, R. P., and Chang, W. J. (2008), "Resonance frequency of chiral single-walled carbon nanotubes using Timoshenko beam theory", Physics Letters A, 372(16), 2757- 2759. https://doi.org/10.1016/j.physleta.2008.01.007
  33. Hu, Y.G., Liew, K.M., Wang, Q., He, X.Q., and Yakobson, B.I. (2008), "Nonlocal shell model for elastic wave propagation in single- and double-walled carbon nanotubes", J. Mech. Phys. Solids, 56, 3475-3485. https://doi.org/10.1016/j.jmps.2008.08.010.
  34. Hussain M., Naeem M. N. (2020a), "Mass density effect on vibration of zigzag and chiral SWCNTs", J. Sandwich Struct. Mater. https://doi.org/10.1177/1099636220906257
  35. Hussain, M., and Naeem, M., (2018), "Vibration of single-walled carbon nanotubes based on Donnell shell theory using wave propagation approach", Novel Nanomaterials - Synthesis and Applications, Intechopen, United Kingdom.
  36. Hussain, M., and Naeem, M., (2019a), "Vibration characteristics of single-walled carbon nanotubes based on non-local elasticity theory using wave propagation approach (WPA) including chirality" IntechOpen. United Kingdom.
  37. Hussain, M., and Naeem, M.N. (2019b), "Effects of ring supports on vibration of armchair and zigzag FGM rotating carbon nanotubes using Galerkin's method", Compos. Part B, 163, 548-561. https://doi.org/10.1016/j.compositesb.2018.12.144
  38. Hussain, M., Naeem., M.N. 2017, "Vibration analysis of single-walled carbon nanotubes using wave propagation approach", Mech. Sci., 8(1), 155-164. https://doi.org/10.5194/ms-8-155-2017
  39. Hussain, M., Naeem., M.N., Shahzad, A., and He, M. (2017), "Vibrational behavior of single-walled carbon nanotubes based on cylindrical shell model using wave propagation approach", AIP Advances, 7(4). https://doi.org/10.1063/1.4979112.
  40. Iijima, S. (1991), "Helical microtubules of graphitic carbon", Nature, 354(7), 56-58. https://doi.org/10.1038/354056a0
  41. Kar, V. R., Panda, S. K., and Pandey, H. K. (2018), "Numerical study of temperature dependent eigenfrequency responses of tilted functionally graded shallow shell structures", Struct. Eng. Mech., 68(5), 527-536. https://doi.org/10.12989/sem.2018.68.5.527
  42. Karami, H., and Farid, M. (2015), "A new formulation to study in-plane vibration of curved carbon nanotubes conveying viscous fluid", J. Vib. Control, 21(12), 2360-2371. https://doi.org/10.1177/1077546313511137
  43. Kiani, K. (2010), "Longitudinal and transverse vibration of a single-walled carbon nanotube subjected to a moving nanoparticle accounting for both nonlocal and inertial effects", Physica E, 42(9), 2391-2401. https://doi.org/10.1016/j.physe.2010.05.021
  44. Kocal, T., and Akbarov, S. D. (2019), "The influence of the rheological parameters on the dispersion of the flexural waves in a viscoelastic bi-layered hollow cylinder", Struct. Eng. Mech., 71(5), 577-601. https://doi.org/10.12989/sem.2019.71.5.577
  45. Kolahchi, R. (2017), "A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods", Aerosp. Sci. Technol., 66, 235-248. https://doi.org/10.1016/j.ast.2017.03.016
  46. Kolahchi, R., and Bidgoli, A. M. (2016), "Size-dependent sinusoidal beam model for dynamic instability of single-walled carbon nanotubes", Appl. Math. Mech., 37(2), 265-274. https://doi.org/10.1007/s10483-016-2030-8
  47. Kolahchi, R., and Cheraghbak, A. (2017), "Agglomeration effects on the dynamic buckling of viscoelastic microplates reinforced with SWCNTs using Bolotin method", Nonlinear Dynam., 90(1), 479-492. https://doi.org/10.1007/s11071-017-3676-x
  48. Kolahchi, R., Hosseini, H., and Esmailpour, M. (2016a), "Differential cubature and quadrature-Bolotin methods for dynamic stability of embedded piezoelectric nanoplates based on visco-nonlocal-piezoelasticity theories", Compos. Struct., 157, 174-186. https://doi.org/10.1016/j.compstruct.2016.08.032
  49. Kolahchi, R., Hosseini, H., Fakhar, M. H., Taherifar, R., and Mahmoudi, M. (2019), "A numerical method for magneto-hygro-thermal postbuckling analysis of defective quadrilateral graphene sheets using higher order nonlocal strain gradient theory with different movable boundary conditions", Comput. Math. Appl., 78(6), 2018-2034. https://doi.org/10.1016/j.camwa.2019.03.042
  50. Kolahchi, R., Keshtegar, B., and Fakhar, M. H. (2020), "Optimization of dynamic buckling for sandwich nanocomposite plates with sensor and actuator layer based on sinusoidal-visco-piezoelasticity theories using Grey Wolf algorithm", J. Sandwich Struct. Mater., 22(1), 3-27. https://doi.org/10.1177/1099636217731071
  51. Kolahchi, R., Safari, M., and Esmailpour, M. (2016b), "Dynamic stability analysis of temperature-dependent functionally graded CNT-reinforced visco-plates resting on orthotropic elastomeric medium", Compos. Struct., 150, 255-265. https://doi.org/10.1016/j.compstruct.2016.05.023
  52. Kolahchi, R., Zarei, M. S., Hajmohammad, M. H., and Nouri, A. (2017), "Wave propagation of embedded viscoelastic FG-CNT-reinforced sandwich plates intebgrated with sensor and actuator based on refined zigzag theory", J. Mech. Sci., 130, 534-545. https://doi.org/10.1016/j.ijmecsci.2017.06.039
  53. Kolahchi, R., Zarei, M. S., Hajmohammad, M. H., and Oskouei, A. N. (2017), "Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods", Thin-Walled Struct., 113, 162-169. https://doi.org/10.1016/j.tws.2017.01.016
  54. Kroner, E. (1967), "Elasticity theory of materials with long range cohesive forces", J. Solid. Struct., 3(5), 731-742. https://doi.org/10.1016/0020-7683(67)90049-2.
  55. Lee, H. L., and Chang, W. J. (2009), "Vibration analysis of a viscous-fluid-conveying single-walled carbon nanotube embedded in an elastic medium", Physica E, 41(4), 529-532. https://doi.org/10.1016/j.physe.2008.10.002
  56. Loy, C.T., Lam, K.L., Shu, C. (1997), "Analysis of cylindrical shells using generalized differential quadrature", Shock Vib., 4(3), 193-198. https://doi.org/10.1155/1997/538754
  57. Loy, C.T., Lam, K.Y., and Reddy, J.N. (1999), "Vibration of functionally graded cylindrical shells", Int J Mech Sci, 1, 309- 324. https://doi.org/10.1016/S0020-7403(98)00054-X.
  58. Madani, H., Hosseini, H., and Shokravi, M. (2016), "Differential cubature method for vibration analysis of embedded FG-CNT-reinforced piezoelectric cylindrical shells subjected to uniform and non-uniform temperature distributions", Steel Compos. Struct., 22(4), 889-913. https://doi.org/10.12989/scs.2016.22.4.889
  59. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2017a), "Thermoelastic nonlinear frequency analysis of CNT reinforced functionally graded sandwich structure", Eur. J. Mech. A/Solid., 65, 384-396. https://doi.org/10.1016/j.euromechsol.2017.05.005.
  60. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2017b), "Theoretical and experimental investigation of vibration characteristic of carbon nanotube reinforced polymer composite structure", Int. J. Mech. Sci., 133, 319-329. https://doi.org/10.1016/j.ijmecsci.2017.08.057
  61. Mohammadimehr, M., and Alimirzaei, S. (2017), "Buckling and free vibration analysis of tapered FG-CNTRC micro Reddy beam under longitudinal magnetic field using FEM", Smart Struct. Syst., 19(3), 309-322. https://doi.org/10.12989/sss.2017.19.3.309
  62. Mohsen, M., and Eyvazian A. (2020), "Post-buckling analysis of Mindlin Cut out-plate reinforced by FG-CNTs." Steel Compos. Struct. 34, no. 2 (2020): 289. https://doi.org/10.12989/scs.2020.34.2.289
  63. Motezaker M and Eyvazian A. (2020), "Buckling load optimization of beam reinforced by nanoparticles", Struct. Eng. Mech., 73(5), 481-486 https://doi.org/10.12989/sem.2020.73.5.481
  64. Motezaker, M., and Kolahchi, R. (2017a), "Seismic response of concrete columns with nanofiber reinforced polymer layer" Computers and Concrete, 20(3), 361-368. https://doi.org/10.12989/cac.2017.20.3.361
  65. Motezaker, M., and Kolahchi, R. (2017b), "Seismic response of SiO 2 nanoparticles-reinforced concrete pipes based on DQ and newmark methods", Computers and Concrete, 19(6), 745-753. https://doi.org/10.12989/cac.2017.19.6.745
  66. Motezaker, M., Jamali, M., and Kolahchi, R. (2020), Application of differential cubature method for nonlocal vibration, buckling and bending response of annular nanoplates integrated by piezoelectric layers based on surface-higher order nonlocal-piezoelasticity theory. Journal of Computational and Applied Mathematics, 369, 112625. https://doi.org/10.1016/j.cam.2019.112625
  67. Narendar, S. (2011), "Terahertz wave propagation in uniform nanorods: A nonlocal continuum mechanics formulation including the effect of lateral inertia", Physica E, 43, 1015-1020. https://doi.org/10.1016/j.physe.2010.12.004
  68. Narwariya, M., Choudhury, A. and Sharma, A.K (2018), "Harmonic analysis of moderately thick symmetric cross-ply laminated composite plate using FEM", Adv. Comput. Des., 3(2), 113-132 https://doi.org/10.12989/ACD.2018.3.2.113
  69. Natsuki T, Qing. QN., and Morinobu, E. (2007), "Wave propagation in single-walled and double-walled carbon nanotubes filled with fluids", J. Appl Phys., 101(3), 034319-034319-5. https://doi.org/10.1063/1.2432025.
  70. O'connell, M. J. (2006), Carbon nanotubes: properties and applications. CRC press.
  71. Paliwal, D.N., Kanagasabapathy, H., and Gupta, K.M. (1995), "The large deflection of an orthotropic cylindrical shell on a Pasternak foundation", Compos. Struct., 31(1), 31-37. https://doi.org/10.1016/0263-8223(94)00068-9.
  72. Peddieson, J., Buchanan, G.R., and McNitt, R.P. (2003), "Application of Nonlocal Continuum Models to Nanotechnology", Int. J. Eng. Sei., 41, 305-312. https://doi.org/10.1016/S0020-7225(02)00210-0.
  73. Rouhi, H., Ansari, R., Arash, B. (2012), "Vibration Analysis of double-walled carbon nanotubes based on the non-local donnell shell via a new numerical approach", Int J. Mech. Sei., 37, 91-105.
  74. Ru, C. (2004), "Elastic models for carbon nanotubes", Encyclopedia of Nanoscience and Nanotechnology, 2(744), American Scientific Publishers, USA. 731-744.
  75. Sadoughifar, A., Farhatnia, F., Izadinia, M., and Talaeetaba, S. B. (2020), "Size-dependent buckling behaviour of FG annular/circular thick nanoplates with porosities resting on Kerr foundation based on new hyperbolic shear deformation theory", Struct. Eng. Mech., 73(3), 225. https://doi.org/10.12989/sem.2020.73.3.225
  76. Safeer, M., Taj, M. and Abbas, S.S. (2019), "Effect of viscoelastic medium on wave propagation along protein microtubules", AIP Advances, 9(4), https://doi.org/10.1016/0263-8223(94)00068-9.
  77. Salah, F., Boucham, B., Bourada, F., Benzair, A., Bousahla, A.A. and Tounsi, A. (2019), "Investigation of thermal buckling properties of ceramic-metal FGM sandwich plates using 2D integral plate model", Steel Compos. Struct., 32(5), 595-610. https://doi.org/10.12989/scs.2019.33.6.805.
  78. Sanchez-Portal, D., Artacho, E., Soler, J. M., Rubio, A., and Ordejon, P. (1999), "Ab initio structural, elastic, and vibrational properties of carbon nanotubes", Physical Review B, 59(19), 12678. https://doi.org/10.1103/PhysRevB.59.12678
  79. Selmi, A. (2019), "Effectiveness of SWNT in reducing the crack effect on the dynamic behavior of aluminium alloy", Adv. Nano Res., 7(5), 365-377. https://doi.org/10.12989/anr.2019.7.5.365
  80. Selmi, A. and Bisharat, A. (2018), "Free vibration of functionally graded SWNT reinforced aluminum alloy beam", J. Vibroeng., 20(5), 2151-2164. https://doi.org/10.21595/jve.2018.19445.
  81. Shamshirsaz, M., Sharafi, S., Rahmatian, J., Rahmatian, S., and Sepehry, N. (2020), "A semi-analytical mesh-free method for 3D free vibration analysis of bi-directional FGP circular structures subjected to temperature variation", Struct. Eng. Mech., 73(4), 407. https://doi.org/10.12989/sem.2020.73.4.407
  82. Sharma, P., Singh, R., Hussain, M. (2019), "On modal analysis of axially functionally graded material beam under hygrothermal effect", Proceedings of the Institution of Mechanical Engineers, Part C: J. Mech. Eng. Sci., https://doi.org/10.1177/0954406219888234.
  83. Shen, (2009), H.S. "Nonlinear bending of functionally graded carbon nanotube-reinforced composite plates in thermal environments", Compos. Struct., 91, 9-19. https://doi.org/10.1016/j.compstruct.2009.04.026
  84. Simsek, M. (2010), "Vibration analysis of a single-walled carbon nanotube under action of a moving harmonic load based on nonlocal elasticity theory", Physica E, 43, 182-191. https://doi.org/10.12989/scs.2011.11.1.059
  85. Simsek, M. (2011), "Nonlocal effects in the forced vibration of an elastically connected double-carbon nanotube system under a moving nanoparticle", Comput. Mater. Sci., 50(7), 2112-2123. https://doi.org/10.1016/j.commatsci.2011.02.017
  86. Swain, A., Roy, T., and Nanda, B.K. (2013), "Vibration behavior of single-walled carbon nanotube using finite element", Int. J. Theor. And Appl. Res. in Mech. Eng., 2, 129-133.
  87. Torabi, J., and Ansari, R. (2018), "Thermally induced mechanical analysis of temperature-dependent FG-CNTRC conical shells", Struct. Eng. Mech., 68(3), 313-323. https://doi.org/10.12989/sem.2018.68.3.313
  88. Usuki, T. and Yogo, K. (2009), "Beam equations for multi-walled carbon nanotubes derived from Flugge shell theory", Proceedings of Royal Society A, 465. https://doi.org/10.1098/rspa.2008.0394
  89. Wang, J., and Gao, Y. (2016), "Nonlocal orthotropic shell model applied on wave propagation in microtubules", Appl. Math. Model., 40(11-12), 5731-5744. https://doi.org/10.1016/j.apm.2016.01.013.
  90. Wang, Q., and Varadan, V.K. (2006), "Vibration of carbon nanotubes studied using nonlocal continuum mechanics", Smart Mater. Struct., 15(2), 659. https://doi.org/10.1088/0964-1726/16/1/022.
  91. Xu, K.U., Aifantis, E.C. and Yan, Y.H. (2008), "Vibrations of double-walled carbon nanotubes with different boundary conditions between inner and outer tubes", J. Appl. Mech., 75(2), 021013-1. 10.1115/1.2793133.
  92. Yang, J., Ke, L. L.,and Kitipornchai, S. (2010), "Nonlinear free vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory", Physica E, 42(5), 1727-1735. https://doi.org/10.1016/j.physe.2010.01.035.
  93. 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., 21(1), 15-25. https://doi.org/10.12989/sss.2018.21.1.015.
  94. Yoon, J., Ru, C.Q., and Mioduchowski, A. (2002), "Noncoaxial resonance of an isolated multiwall carbon nanotube", Physical Review B., 66(23), 2334021-2334024. https://doi.org/10.1103/PhysRevB.66.233402.
  95. Zamanian M, Kolahchi, R, and Bidgoli, M.R. (2017), "Agglomeration effects on the buckling behaviour of embedded concrete columns reinforced with SiO2 nano-particles", Wind Struct, 24(1), 43-57. https://doi.org/10.12989/was.2017.24.1.043
  96. Zamanian, M., Kolahchi, R., and Bidgoli, M. R. (2017), "Agglomeration effects on the buckling behaviour of embedded concrete columns reinforced with SiO2 nano-particles", Wind Struct, 24(1), 43-57. https://doi.org/10.12989/was.2017.24.1.043
  97. Zhang, J. F., Liu, Q. S., Ge, Y. J., and Zhao, L. (2019), "Studies on the influence factors of wind dynamic responses on hyperbolic cooling tower shells", Struct. Eng. Mech., 72(5), 541. https://doi.org/10.12989/sem.2019.72.5.541
  98. Zou, R.D., and Foster, C.G. (1995), "Simple solution for buckling of orthotropic circular cylindrical shells", Thin-Walled Struct., 22(3), 143-158. https://doi.org/10.1016/0263-8231(94)00026-V.