DOI QR코드

DOI QR Code

Nonlocal vibration of DWCNTs based on Flügge shell model using wave propagation approach

  • Asghar, Sehar (Department of Mathematics, Govt. College University Faisalabad) ;
  • Naeem, Muhammad N. (Department of Mathematics, Govt. College University Faisalabad) ;
  • Hussain, Muzamal (Department of Mathematics, Govt. College University Faisalabad) ;
  • Tounsi, Abdelouahed (Faculty of Technology Civil Engineering Department, Materials and Hydrology Laboratory University of Sidi Bel Abbes)
  • 투고 : 2019.08.01
  • 심사 : 2019.10.19
  • 발행 : 2020.02.25

초록

In this article, free vibration attributes of double-walled carbon nanotubes based on nonlocal elastic shell model have been investigated. For this purpose, a nonlocal Flügge shell model is established to observe the small scale effect. The wave propagation is employed to frame the governing equations as eigenvalue system. The influence of nonlocal parameter subjected to different end supports has been overtly examined. A suitable choice of material properties and nonlocal parameter been focused to analyze the vibration characteristics. The new set of inner and outer tubes radii investigated in detail against aspect ratio and length. The dominance of boundary conditions via nonlocal parameter is shown graphically. The results generated furnish the evidence regarding applicability of nonlocal shell model and also verified by earlier published literature.

키워드

과제정보

The author(s) received no financial support for the research, authorship, and/or publication of this article.

참고문헌

  1. Adela, I., (2018), Computational Fluid Dynamics, Romania.
  2. Akbas, S.D. (2015), "Wave propagation of a functionally graded beam in thermal environments", Steel Compos. Struct., 19(6), 1421-1447. http://dx.doi.org/10.12989/scs.2015.19.6.1421.
  3. Amara, K., Tounsi, A., Mechab, I. and Adda-Bedia, E.A. (2010), "Nonlocal elasticity effect on column buckling of multiwalled carbon nanotubes under temperature field", Appl. Math. Model., 34(12), 3933-3942. https://doi.org/10.1016/j.apm.2010.03.029
  4. Amnieh, H.B., Zamzam, M.S. and Kolahchi, R. (2018), "Dynamic analysis of non-homogeneous concrete blocks mixed by SiO2 nanoparticles subjected to blast load experimentally and theoretically", Constr. Build. Mater., 174, 633-644. https://doi.org/10.1016/j.conbuildmat.2018.04.140.
  5. 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, 021006-1. https://doi.org/10.1142/S179329201250018X.
  6. Ansari, R., Hemmatnezhad, M. and Rezapour, J. (2011), "The thermal effect on nonlinear oscillations of carbon nanotubes with arbitrary boundary conditions", Curr. Appl. Phys., 11(3), 692-697. https://doi.org/10.1016/j.cap.2010.11.034.
  7. Ansari, R. and Rouhi, H. (2012), "Nonlocal analytical Flugge shell model for the axial buckling of double-walled carbon nanotubes with different end conditions", Int. J. Nano, 7, 1250081. https://doi.org/10.1142/S179329201250018X.
  8. Ansari, R., Sahmani, S. and Arash, B. (2010), "Nonlocal plate model for free vibrations of single-layered graphene sheets", Phys. Lett. A., 375(1), 53-62. https://doi.org/10.1016/j.physleta.2010.10.028
  9. Arani, Jafarian A. and Kolahchi, R. (2016), "Buckling analysis of embedded concrete columns armed with carbon nanotubes", Comput Concret, 17(5), 567-578. http://dx.doi.org/10.12989/cac.2016.17.5.567.
  10. Arefi, M., Mohammadi, M., Tabatabaeian, A., Dimitri, R. and Tornabene, F. (2018), "Two-dimensional thermo-elastic analysis of FG-CNTRC cylindrical pressure vessels", Steel Compos. Struct., 27(4), 525-536. http://dx.doi.org/10.12989/scs.2018.27.4.525.
  11. Asghar, S. Hussain M. and Naeem, M.N. (2019), "Non-local effect on the vibration analysis of double walled carbon nanotubes based on Donnell shell theory", J. Physica E: Low-dimensional Syst. Nanostruct., 116(2010), https://doi.org/10.1016/j.physe.2019.113726.
  12. Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Compos. Struct., 30(6), 603-615. http://dx.doi.org/10.12989/scs.2019.30.6.603.
  13. Azmi, M, Kolahchi, R and Bidgoli, M.R. (2019), "Dynamic analysis of concrete column reinforced with Sio 2 nanoparticles subjected to blast load", Adv. Concrete Constr., 7(1), 51-63. https://doi.org/10.12989/acc.2019.7.1.051.
  14. Benguediab, S., Tounsi, A., Ziadour, and Semmah, A. (2014), "Chirality and scale effects on mechanical and buckling properties of zigzag double-walled carbon nanotubes", Compos. Part B, 57, 21-24. https://doi.org/10.1016/j.compositesb.2013.08.020.
  15. 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.
  16. Bouadi, A., Bousahla, A.A., Houari, M.S.A., Heireche, H. and Tounsi, A. (2018), "A new nonlocal HSDT for analysis of stability of single layer graphene sheet", Adv. Nano Res., 6(2), 147-162. https://doi.org/10.12989/anr.2018.6.2.147.
  17. Brischotto, S. (2015), "A continuum shell model including van der Waals interaction for free vibrations of double-walled carbon nanotubes", CMES, 104, 305-327.
  18. Do, Q.C., Pham, D.N., Vu, D.Q., Vu, T.T.A. and Nguyen, D.D. (2019), "Nonlinear buckling and post-buckling of functionally graded CNTs reinforced composite truncated conical shells subjected to axial load", Steel Compos. Struct., 31.
  19. Duan, W.H., Wang, C.M. and Zhang, Y.Y. (2007), "Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics", J. Appl. Phys., 101(2), 024305. https://doi.org/10.1063/1.2423140.
  20. Ehyaei, J. and Daman, M. (2017), "Free vibration analysis of double walled carbon nanotubes embedded in an elastic medium with initial imperfection", Adv. Nano Res., 5(2), 179-192. https://doi.org/10.12989/anr.2017.5.2.179.
  21. 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.
  22. Eringen, A.C. (2002), Nonlocal continuum field theories, Science and Business Media, New York.
  23. Fakhar, A. and Kolahchi, R. (2018), "Dynamic buckling of magnetorheological fluid integrated by visco-piezo-GPL reinforced plates", Int. J. Mech. Sci., 144, 788-799. https://doi.org/10.1016/j.ijmecsci.2018.06.036.
  24. Fakhrabadi, M.M.S., Rastgoo, A. and Ahmadian, M.T. (2015), "Application of electrostatically actuated carbon nanotubes in nanofluidic and bio-nanofluidic sensors and actuators", Measurement, 73, 127-136. https://doi.org/10.1016/j.measurement.2015.05.009
  25. Farahani, H. and Barati, F. (2015), "Vibration of sumberged functionally graded cylindrical shell based on first order shear deformation theory using wave propagation method", Struct. Eng. Mech., 53(3), 575-587. : http://dx.doi.org/10.12989/sem.2015.53.3.575.
  26. Fatahi-Vajari, A., Azimzadeh, Z. and Hussain. M. (2019), "Nonlinear coupled axial-torsional vibration of single-walled carbon nanotubes using Galerkin and Homotopy perturbation method", Micro and Nano Letter, Accepted Sep. 2019.
  27. Flugge, W. (1962), Statik und Dynamik der Scahlen, Springer, Berlin, Germany.
  28. Fu, Y.M., Hong, J.W. and Wang, X.Q. (2006), "Analysis of nonlinear vibration for embedded carbon nanotubes", J. Sound Vib., 296(4-5), 746-756. https://doi.org/10.1016/j.jsv.2006.02.024.
  29. Golabchi, H,, Kolahchi, R. and Bidgoli, M.R. (2018), "Vibration and instability analysis of pipes reinforced by SiO 2 nanoparticles considering agglomeration effects", Comput. Concrete, 21(4), 431-440. https://doi.org/10.12989/CAC.2018.21.4.431
  30. Hajmohammad, H.M., Farrokhian, A. and Kolahchi, R. (2018a), "Smart control and vibration of viscoelastic actuator-multiphase nanocomposite conical shells-sensor considering hygrothermal load based on layerwise theory", Aerosp. Sci. Technol., 78, 260-270 https://doi.org/10.1016/j.ast.2018.04.030
  31. Hajmohammad, H.M., et al. (2018), "Earthquake induced dynamic deflection of submerged viscoelastic cylindrical shell reinforced by agglomerated CNTs considering thermal and moisture effects", Compos. Struct., 187, 498-508. https://doi.org/10.1016/j.compstruct.2017.12.004.
  32. Hajmohammad, H.M., et al. (2017), "Dynamic buckling of sensor/functionally graded-carbon nanotube-reinforced laminated plates/actuator based on sinusoidal-viscopiezoelasticity theories", J. Sandw. Struct. Mater., 1099636217720373.
  33. Hajmohammad, H.M., Maleki, M. and Kolahchi R. (2018), "Seismic response of underwater concrete pipes conveying fluid covered with nano-fiber reinforced polymer layer", Soil Dynam. Earthq. Eng., 110, 18-27. https://doi.org/10.1016/j.soildyn.2018.04.002.
  34. Hajmohammad, M.H., et al. (2019), "Dynamic response of auxetic honeycomb plates integrated with agglomerated CNTreinforced face sheets subjected to blast load based on viscosinusoidal theory", Int. J. Mech. Sci., 153-154, 391-401. https://doi.org/10.1016/j.ijmecsci.2019.02.008.
  35. Hao, M.J., Guo, X.M. and Wang, Q. (2010), "Small-scale effect on torsional buckling of multi-walled carbon nanotubes", Eur. J. Mech. A-Solids, 29(1), 49-55. https://doi.org/10.1016/j.euromechsol.2009.05.008
  36. Hernandez, E., Goze, C., Bemier, P. and Rubio, A. (1998), "Elastic properties of C and BxCyNz composite nanotubes", Phys. Rev. Lett., 80, 4502-4505. https://doi.org/10.1103/PhysRevLett.80.4502.
  37. 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.
  38. Hosseini, H. and Kolahchi, R. (2018), "Seismic response of functionally graded-carbon nanotubes-reinforced submerged viscoelastic cylindrical shell in hygrothermal environment", Physica E: Low-dimensional Syst. Nanostruct., 102, 101-109. https://doi.org/10.1016/j.physe.2018.04.037.
  39. Hsu, J.C., Chang, R.P. and Chang, W.J. (2008), "Resonance frequency of chiral single-walled carbon nanotubes using Timoshenko beam theory", Phys. Lett. A, 372(16), 2757-2759. https://doi.org/10.1016/j.physleta.2008.01.007
  40. 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(12), 3475-3485. https://doi.org/10.1016/j.jmps.2008.08.010.
  41. Hussain, M. and Naeem, M. (2019c), "Rotating response on the vibrations of functionally graded zigzag and chiral single walled carbon nanotubes", Appl. Math. Model., 75, 506-520. https://doi.org/10.1016/j.apm.2019.05.039.
  42. Hussain, M. and Naeem, M. (2018b), "Vibration of single-walled carbon nanotubes based on Donnell shell theory using wave propagation approach", Chapter, Intechopen, Novel Nanomaterials - Synthesis and Applications, ISBN 978-953-51-5896-7, 10.5772 /intechopen.73503.
  43. Hussain, M. and Naeem, M.N. (2018a), "Effect of various edge conditions on free vibration characteristics of rectangular plates", Chapter, Intechopen, Advance Testing and Engineering, ISBN 978-953-51-6706-8, Intechopen.
  44. Hussain, M. and Naeem, M.N. (2019a), "Effects of ring supports on vibration of armchair and zigzag FGM rotating carbon nanotubes using Galerkin's method", Compos.: Part B. Eng., 163, 548-561. https://doi.org/10.1016/j.compositesb.2018.12.144.
  45. Hussain, M. and Naeem, M.N. (2019b), "Vibration characteristics of zigzag and chiral FGM rotating carbon nanotubes sandwich with ring supports", J. Mech. Eng. Sci., Part C., 233(16), 5763-5780. https://doi.org/10.1177/0954406219855095.
  46. Hussain, M. and Naeem, M.N. (2019d), "Axial vibration of zigzag and chiral SWCNTs based on nonlocal Donnell shell model: An analytical approach", Appl. Math. Model., In revisions, Aug, 2019.
  47. Hussain, M., Naeem, M., Shahzad, A. and He, M. (2018a), "Vibration characteristics of fluid-filled functionally graded cylindrical material with ring supports", Chapter, Intechopen, Computational Fluid Dynamics, ISBN 978-953-51-5706-9, DOI:10.5772 /intechopen.72172.
  48. Hussain, M., Naeem, M.N. and Isvandzibaei, M. (2018c), "Effect of Winkler and Pasternak elastic foundation on the vibration of rotating functionally graded material cylindrical shell", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 232(24), 4564-4577. https://doi.org/10.1177/0954406217753459
  49. Hussain, M., Naeem, M.N. and Taj, M. (2019b), "Effect of length and thickness variations on the vibration of SWCNTs based on Flugge's shell model", Micro & Nano Letters, In revisions, Aug, 2019.
  50. Hussain, M., Naeem, M.N., Shahzad A, He, M. and Habib, S. (2018b), "Vibrations of rotating cylindrical shells with FGM using wave propagation approach", IMechE Part C: J Mech. Eng. Sci., 232(23), 4342-4356. https://doi.org/10.1177/0954406218802320
  51. Hussain, M., Naeem, M.N., Tounsi A. and Taj. M. (2019a), "Nonlocal effect on the vibration of armchair and zigzag SWCNTs with bending rigidity", Adv. Nano Res., 7(6), 431-442. https://doi.org/10.12989/anr.2019.7.6.431.
  52. Hussain, M. and Naeem., M.N. (2017), "Vibration analysis of single-walled carbon nanotubes using wave propagation approach", Mechanical Sciences, 8(1), 155-164. https://doi.org/10.5194/ms-8-155-2017
  53. 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), 045114. https://doi.org/10.1063/1.4979112.
  54. Jassas, M.R., Bidgoli, M.R. and Kolahchi, R. (2019), "Forced vibration analysis of concrete slabs reinforced by agglomerated SiO2 nanoparticles based on numerical methods", Constr. Build. Mater., 211, 796-806. https://doi.org/10.1016/j.conbuildmat.2019.03.263.
  55. Kolahchi, R. and Cheraghbak, A. (2017b), "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.
  56. Kolahchi, R., et al. (2017a), "Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods", Thin-Wall. Struct., 113, 162-169. https://doi.org/10.1016/j.tws.2017.01.016.
  57. Kolahchi, R., et al. (2017c), "Wave propagation of embedded viscoelastic FG-CNT-reinforced sandwich plates integrated with sensor and actuator based on refined zigzag theory", Int. J. Mech. Sci., 130, 534-545. https://doi.org/10.1016/j.ijmecsci.2017.06.039.
  58. 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.
  59. Kolahchi, R., Keshtegar, B. and Fakhar, M.H. (2017d), "Optimization of dynamic buckling for sandwich nanocomposite plates with sensor and actuator layer based on sinusoidal-visco-piezoelasticity theories using Grey Wolf algorithm", J. Sandw. Struct. Mater., 1099636217731071.
  60. 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.
  61. 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.
  62. Kolahchi, R., Hosseini, H., Fakhar, M.H., Taherifar, R. and Mahmoudi, M. (2019), "A numerical method for magnetohygro-thermal postbuckling analysis of defective quadrilateral graphene sheets using higher order nonlocal strain gradient theory with different movable boundary conditions", Computers & Mathematics with Applications.
  63. Kolahchi, R. and Moniri 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.
  64. Kumar, B.R. (2018), "Investigation on mechanical vibration of double-walled carbon nanotubes with inter-tube Van der waals forces", Adv. Nano Res., 6(2), 135-145. https://doi.org/10.12989/anr.2018.6.2.135.
  65. Lei, Z. and Zhang, Y. (2018), "Characterizing buckling behavior of matrix-cracked hybrid plates containing CNTR-FG layers", Steel Compos. Struct., 28(4), 495-508. https://doi.org/10.12989/scs.2018.28.4.495.
  66. Madani, H., Hosseini, H. and Shokravi, M. (2016), "Differential cubature method for vibration analysis of embedded FG-CNTreinforced piezoelectric cylindrical shells subjected to uniform and non-uniform temperature distributions", Steel Compos. Struct., 22(4), 889-913. https://doi.org/10.12989/scs.2018.22.4.889.
  67. Moradi-Dastjerdi, R. (2016), "Wave propagation in functionally graded composite cylinders reinforced by aggregated carbon nanotube", Struct. Eng. Mech., 57(3), 441-456. https://doi.org/10.12989/sem.2016.57.3.441.
  68. Moradi-Dastjerdi, R. and Payganeh, G. (2017), "Transient heat transfer analysis of functionally graded CNT reinforced cylinders with various boundary conditions", Steel Compos Struct., 24(3), 359-367. https://doi.org/10.12989/scs.2017.24.3.359.
  69. Natsuki, T., Qing, Q.N. 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. Nikkar, A., Rouhi, S. and Ansari, R. (2017), "Finite element modeling of the vibrational behavior of multi-walled nested silicon-carbide and carbon nanotubes", Struct. Eng. Mech., 64(3), 329-337. https://doi.org/10.12989/sem.2017.64.3.329.
  71. 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.
  72. Pradhan, S.C. and Phadikar, J.K. (2009), "Small scale effect on vibration of embedded multilayered graphene sheets based on nonlocal continuum models", Phys. Lett. A., 373(11), 1062-9. https://doi.org/10.1016/j.physleta.2009.01.030.
  73. 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.
  74. Rouhi, H., Ansari, R. and Arash, B. (2013), "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.
  75. Rouhi, H., BazdidVahdati, M. and Ansari, R. (2015), "Rayleigh-Rits vibrational analysis of multi-walled carbon nanotubes based on the non-local Flugge shell theory", J. Comp., 750392. https://doi.org/10.1155/2015/750392
  76. Sanchez-Portal, D., Artacho, E., Soler, J.M., Rubio, A. and Ordejon, P. (1999), "Ab-initio structural, elastic, and Vibrational Properties of Carbon Nanotubes", Phys. Rev. B, 59, 12678-2688. http://dx.doi.org/10.1103/PhysRevB.59.12678.
  77. 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.
  78. Semmah, A., Heireche, H., Bousahla, A.A. and Tounsi, A. (2019), "Thermal buckling analysis of SWBNNT on Winkler foundation by non local FSDT", Adv. Nano Res., 7(2), 89-95. https://doi.org/10.12989/anr.2019.7.2.089.
  79. Shafiei, H. and Setoodeh, A.R. (2017), "Nonlinear free vibration and post-buckling of FG-CNTRC beams on nonlinear foundation", Steel Compos. Struct., 24(1), 65-77. https://doi.org/10.12989/scs.2017.24.1.065
  80. Sharma, P., Singh, R. and Hussain, M. (2019), "On modal analysis of axially functionally graded material beam under hygrothermal effect", Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, In revisions, Aug, 2019.
  81. She, G.L., Ren, Y.R. and Yuan, F.G. (2019), "Hygro-thermal wave propagation in functionally graded double-layered nanotubes systems", Steel Compos. Struct., 31(6), 641-653. https://doi.org/10.12989/SCS.2019.31.6.641
  82. Shen, H.S. and Zhang, C.L. (2010), "Torsional buckling and post buckling of double-walled carbon nanotubes by nonlocal shear deformable shell model", Compos. Struct., 92(5), 1073-1084. https://doi.org/10.1016/j.compstruct.2009.10.002.
  83. Sofiyev, A.H., Avcar, M., Ozyigit, P. and Adigozel, S. (2009), "The free vibration of non-homogeneous truncated conical shells on a winkler foundation", Int. J. Eng. Appl. Sci., 1(1), 34-41.
  84. Soldano, C. (2015), "Hybrid metal-based carbon nanotubes", "Novel platform for multifunctional applications", Progress in Materials Science, 69, 183-212. https://doi.org/10.1016/j.pmatsci.2014.11.001.
  85. Sosa, E.D., Darlington, T.K., Hanos, B.A. and O'Rourke, M.J.E. (2014), "Multifunctional thermally remendable nanocomposites", J. Comp., Article ID 705687, 12 pages. http://dx.doi.org/10.1155/2014/705687
  86. Sudak, L.J. (2003), "Column buckling of multi-walled carbon nanotubes using nonlocal continuum mechanics", J. Appl. Phys., 94, 7281-7287. https://doi.org/10.1063/1.1625437.
  87. Sun, C.T. and Zhang, H. (2002), "Size-dependent elastic moduli of plate like nanomaterials", J. Appl. Phys., 93, 212-1218. https://doi.org/10.1063/1.1530365.
  88. Tahouneh, V. (2017), "Effects of CNTs waviness and aspect ratio on vibrational response of FG-sector plate", Steel Compos. Struct., 25(6), 649-661. https://doi.org/10.12989/scs.2017.25.6.649.
  89. Tlidji, Y., Zidour, M., Draiche, K., Safa, A., Bourada, M., Tounsi, A. and Mahmoud, S.R. (2019), "Vibration analysis of different material distributions of functionally graded microbeam", Struct. Eng. Mech., 69(6), 637-649. https://doi.org/10.12989/SEM.2019.69.6.637
  90. Usuki, T. and Yogo, K. (2009), "Beam equations for multi-walled carbon nanotubes derived from Flugge shell theory", Proceedings of Royal Society A., 465(2104). https://doi.org/10.1098/rspa.2008.0394
  91. Vodenitcharova, T. and Zhang, L.C. (2003), "Effective wall thickness of single walled carbon nanotubes", Phys. Rev. B., 68, 165401. https://doi.org/10.1103/PhysRevB.68.165401.
  92. Wang, C.Y. and Zhang, L.C. (2007), 5th Australasian Congress on Applied Mechanics, ACAM, Brisbane, Australia.
  93. Wang, Q. Varadan, V.K. and Quek, S.T. (2006), "Small scale effect on elastic buckling of carbon nanotubes with nonlocal continuum models", Phys. Lett. A., 357(2), 130-135. https://doi.org/10.1016/j.physleta.2006.04.026.
  94. Wang, Q., Zhou, G.Y. and Lin, K.C. (2006), "Scale effect on wave propagation of double-walled carbon nanotubes", Int. J. Solid. Struct., 43, 6071-6084. https://doi.org/10.1016/j.ijsolstr.2005.11.005.
  95. Xiaobin, L., Shuangxi, X., Weiguo, W. and Jun, L. (2014), "An exact dynamic stiffness matrix for axially loaded double-beam systems", Sadhana, 39(3), 607-623. https://doi.org/10.1007/s12046-013-0214-5
  96. 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. DOI:10.1115/1.2793133.
  97. Yakobson, B.I., Brabec, C.J. and Bernholc, J. (1996), "Nanomechanics of carbon tubes: instabilities beyond linear response", Phys. Rev. Lett., 76, 2511-2514. https://doi.org/10.1103/PhysRevLett.76.2511.
  98. Yakobson, B.I., Campbell, M.P., Brabec, C.J. and Bemholc J. (1997), "High strain rate fracture and C-chain unravelling in carbon nanotubes", Comput. Mater. Sei., 8(4), 341-348. https://doi.org/10.1016/S0927-0256(97)00047-5.
  99. Yoon, J., Ru, C.Q. and Mioduchowski. A. (2003), "Vibration of an embedded multiwall carbon nanotube", Compos. Sei. Tech., 63(11), 1533-1542. https://doi.org/10.1016/S0266-3538(03)00058-7.
  100. 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., 21(1), 65-74. https://doi.org/10.12989/SSS.2018.21.1.065
  101. 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.
  102. Zarei, Sharif M, et al. (2017), "Seismic response of underwater fluid-conveying concrete pipes reinforced with SiO2 nanoparticles and fiber reinforced polymer (FRP) layer", Soil Dynam. Earthq. Eng., 103, 76-85. https://doi.org/10.1016/j.soildyn.2017.09.009.
  103. 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. and Mech., 54(4), 693-710. http://dx.doi.org/10.12989/sem.2015.54.4.693.
  104. Zhang, Y.Y., Wang, C.M. and Tan, V.B.C. (2009), "Assessment of Timoshenko beam models for vibrational behavior of singlewalled carbon nanotubes using molecular dynamics", Adv. Appl. Math. Mech., 1, 89-106.

피인용 문헌

  1. Mechanical and thermal buckling analysis of laminated composite plates vol.40, pp.5, 2020, https://doi.org/10.12989/scs.2021.40.5.697