Advances in nano research

  • 제6권1호
  • /
  • Pages.57-80
  • /
  • 2018
  • /
  • 2287-237X(pISSN)
  • /
  • 2287-2388(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

Vibration analysis of carbon nanotubes with multiple cracks in thermal environment

  • Ebrahimi, Farzad (Mechanical Engineering Department, Faculty of Engineering, Imam Khomeini International University) ;
  • Mahmoodi, Fatemeh (Mechanical Engineering Department, Faculty of Engineering, Imam Khomeini International University)
  • 투고 : 2016.06.15
  • 심사 : 2017.08.29
  • 발행 : 2018.03.25
⟨ 이전 논문 다음 논문 ⟩

초록

In this study, the thermal loading effect on free vibration characteristics of carbon nanotubes (CNTs) with multiple cracks is studied. Various boundary conditions for nanotube are taken in to account. In order to take the small scale effect, the nonlocal elasticity of Eringen is employed in the framework of Euler-Bernoulli beam theory. This theory states that the stress at a reference point is a function of strains at all points in the continuum. A cracked nanotube is assumed to be consisted of two segments that are connected by a rotational spring which is located in the position of the cracked section. Hamilton's principle is used to achieve the governing equations. Influences of the nonlocal parameter, crack severity, temperature change and the number of cracks on the system frequencies are investigated. Also, it is found that at room or lower temperature the natural frequency for CNT decreases as the value of temperature change increases, while at temperature higher than room temperature the natural frequency of CNT increases as the value of temperature change increases. Various boundary conditions have been applied to the nanotube.

키워드

참고문헌

  1. Belytschko T., Xiao, S.P., Schatz, G.C. and Ruoff, R.S. (2002), "Atomistic simulations of nanotube fracture", Phys. Rev. B, 65(23), 235430. https://doi.org/10.1103/PhysRevB.65.235430
  2. 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 Appl. Phys., 41(22), 225404. https://doi.org/10.1088/0022-3727/41/22/225404
  3. 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., Int. J., 3(1), 29-37. https://doi.org/10.12989/anr.2015.3.1.029
  4. Binici, B. (2005), "Vibration of beams with multiple open cracks subjected to axial force", J. Sound Vib., 287(1-2), 277-295. https://doi.org/10.1016/j.jsv.2004.11.010
  5. 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., Int. J., 20(2), 227-249. https://doi.org/10.12989/scs.2016.20.2.227
  6. Challamel, N. and Wang, C.M. (2008), "The small length scale effect for a non-local cantilever beam: A paradox solved", Nanotechnology, 19(34), 345703. https://doi.org/10.1088/0957-4484/19/34/345703
  7. Di Sia, P. (2013), "A new theoretical Model for the dynamical Analysis of Nano-Bio-Structures", Adv. Nano Res., Int. J., 1(1), 29-34. https://doi.org/10.12989/anr.2013.1.1.029
  8. Ebrahimi, F. (2013), "Analytical investigation on vibrations and dynamic response of functionally graded plate integrated with piezoelectric layers in thermal environment", Mech. Adv. Mater. Struct., 20(10), 854-870. https://doi.org/10.1080/15376494.2012.677098
  9. Ebrahimi, F. and Barati, M.R. (2016a), "Magneto-electro-elastic buckling analysis of nonlocal curved nanobeams", Eur. Phys. J. Plus, 131(9), 346. https://doi.org/10.1140/epjp/i2016-16346-5
  10. Ebrahimi, F. and Barati, M.R. (2016b), "Static stability analysis of smart magneto-electro-elastic heterogeneous nanoplates embedded in an elastic medium based on a four-variable refined plate theory", Smart Mater. Struct., 25(10), 105014. https://doi.org/10.1088/0964-1726/25/10/105014
  11. Ebrahimi, F. and Barati, M.R. (2016c), "Temperature distribution effects on buckling behavior of smart heterogeneous nanosize plates based on nonlocal four-variable refined plate theory", Int. J. Smart Nano Mater., 7(3), 119-143. https://doi.org/10.1080/19475411.2016.1223203
  12. Ebrahimi, F. and Barati, M.R. (2016d), "An exact solution for buckling analysis of embedded piezoelectro-magnetically actuated nanoscale beams", Adv. Nano Res., Int. J., 4(2), 65-84. https://doi.org/10.12989/anr.2016.4.2.065
  13. Ebrahimi, F. and Barati, M.R. (2016e), "Buckling analysis of smart size-dependent higher order magnetoelectro-thermo-elastic functionally graded nanosize beams", J. Mech., 33(1), 23-33.
  14. Ebrahimi, F. and Barati, M.R. (2016f), "A nonlocal higher-order shear deformation beam theory for vibration analysis of size-dependent functionally graded nanobeams", Arab. J. Sci. Eng., 41(5), 1679-1690. https://doi.org/10.1007/s13369-015-1930-4
  15. Ebrahimi, F. and Barati, M.R. (2016g), "Vibration analysis of smart piezoelectrically actuated nanobeams subjected to magneto-electrical field in thermal environment", J. Vib. Control, 1077546316646239. https://doi.org/10.1177/1077546316646239
  16. Ebrahimi, F. and Barati, M.R. (2016h), "Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium", J. Brazil. Soc. Mech. Sci. Eng., 39(3), 937-952.
  17. Ebrahimi, F. and Barati, M.R. (2016i), "Small scale effects on hygro-thermo-mechanical vibration of temperature dependent nonhomogeneous nanoscale beams", Mech. Adv. Mater. Struct., 24(11), 924-936.
  18. Ebrahimi, F. and Barati, M.R. (2016j), "Dynamic modeling of a thermo-piezo-electrically actuated nanosize beam subjected to a magnetic field", Appl. Phys. A, 122(4), 1-18.
  19. Ebrahimi, F. and Barati, M.R. (2016k), "Magnetic field effects on buckling behavior of smart sizedependent graded nanoscale beams", Eur. Phys. J. Plus, 131(7), 1-14. https://doi.org/10.1140/epjp/i2016-16001-3
  20. Ebrahimi, F. and Barati, M.R. (2016l), "Vibration analysis of nonlocal beams made of functionally graded material in thermal environment", Eur. Phys. J. Plus, 131(8), 279. https://doi.org/10.1140/epjp/i2016-16279-y
  21. Ebrahimi, F. and Barati, M.R. (2016m), "A nonlocal higher-order refined magneto-electro-viscoelastic beam model for dynamic analysis of smart nanostructures", Int. J. Eng. Sci., 107, 183-196. https://doi.org/10.1016/j.ijengsci.2016.08.001
  22. Ebrahimi, F. and Barati, M.R. (2016n), "Small-scale effects on hygro-thermo-mechanical vibration of temperature-dependent nonhomogeneous nanoscale beams", Mech. Adv. Mater. Struct., 24(11), 924-936.
  23. Ebrahimi, F. and Barati, M.R. (2016o), "A unified formulation for dynamic analysis of nonlocal heterogeneous nanobeams in hygro-thermal environment", Appl. Phys. A, 122(9), 792. https://doi.org/10.1007/s00339-016-0322-2
  24. Ebrahimi, F. and Barati, M.R. (2016p), "Electromechanical buckling behavior of smart piezoelectrically actuated higher-order size-dependent graded nanoscale beams in thermal environment", Int. J. Smart Nano Mater., 7(2), 69-90. https://doi.org/10.1080/19475411.2016.1191556
  25. Ebrahimi, F. and Barati, M.R. (2016q), "Wave propagation analysis of quasi-3D FG nanobeams in thermal environment based on nonlocal strain gradient theory", Appl. Phys. A, 122(9), 843. https://doi.org/10.1007/s00339-016-0368-1
  26. Ebrahimi, F. and Barati, M.R. (2016r), "Flexural wave propagation analysis of embedded S-FGM nanobeams under longitudinal magnetic field based on nonlocal strain gradient theory", Arab. J. Sci. Eng., 42(5), 1715-1726.
  27. Ebrahimi, F. and Barati, M.R. (2016s), "On nonlocal characteristics of curved inhomogeneous Euler-Bernoulli nanobeams under different temperature distributions", Appl. Phys. A, 122(10), 880. https://doi.org/10.1007/s00339-016-0399-7
  28. Ebrahimi, F. and Barati, M.R. (2016t), "Buckling analysis of piezoelectrically actuated smart nanoscale plates subjected to magnetic field", J. Intell. Mater. Syst. Struct., 28(11), 1472-1490.
  29. Ebrahimi, F. and Barati, M.R. (2016u), "Size-dependent thermal stability analysis of graded piezomagnetic nanoplates on elastic medium subjected to various thermal environments", Appl. Phys. A, 122(10), 910. https://doi.org/10.1007/s00339-016-0441-9
  30. Ebrahimi, F. and Barati, M.R. (2016v), "Magnetic field effects on dynamic behavior of inhomogeneous thermo-piezo-electrically actuated nanoplates", J. Brazil. Soc. Mech. Sci. Eng., 39(6), 2203-2223.
  31. Ebrahimi, F. and Barati, M.R. (2017a), "Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory", Compos. Struct., 159, 433-444. https://doi.org/10.1016/j.compstruct.2016.09.092
  32. Ebrahimi, F. and Barati, M.R. (2017b), "A nonlocal strain gradient refined beam model for buckling analysis of size-dependent shear-deformable curved FG nanobeams", Compos. Struct., 159, 174-182. https://doi.org/10.1016/j.compstruct.2016.09.058
  33. Ebrahimi, F. and Daman, M. (2016), "Dynamic modeling of embedded curved nanobeams incorporating surface effects", Coupl. Syst. Mech., Int. J., 5(3), 000
  34. Ebrahimi, F. and Daman, M. (2017), "Analytical investigation of the surface effects on nonlocal vibration behavior of nanosize curved beams", Adv. Nano Res., Int. J., 5(1), 35-47. https://doi.org/10.12989/anr.2017.5.1.035
  35. Ebrahimi, F. and Hosseini, S.H.S. (2016a), "Double nanoplate-based NEMS under hydrostatic and electrostatic actuations", Eur. Phys. J. Plus, 131(5), 1-19. https://doi.org/10.1140/epjp/i2016-16001-3
  36. Ebrahimi, F. and Hosseini, S.H.S. (2016b), "Nonlinear electroelastic vibration analysis of NEMS consisting of double-viscoelastic nanoplates", Appl. Phys. A, 122(10), 922. https://doi.org/10.1007/s00339-016-0452-6
  37. Ebrahimi, F. and Hosseini, S.H.S. (2016c), "Thermal effects on nonlinear vibration behavior of viscoelastic nanosize plates", J. Therm. Stress., 39(5), 606-625. https://doi.org/10.1080/01495739.2016.1160684
  38. Ebrahimi, F. and Jafari, A. (2016), "Buckling behavior of smart MEE-FG porous plate with various boundary conditions based on refined theory", Adv. Mater. Res., Int. J., 5(4), 261-276.
  39. Ebrahimi, F. and Mokhtari, M. (2015), "Transverse vibration analysis of rotating porous beam with functionally graded microstructure using the differential transform method", J. Brazil. Soc. Mech. Sci. Eng., 37(4), 1435-1444. https://doi.org/10.1007/s40430-014-0255-7
  40. Ebrahimi, F. and Nasirzadeh, P. (2015), "A nonlocal Timoshenko beam theory for vibration analysis of thick nanobeams using differential transform method", J. Theor. Appl. Mech., 53(4), 1041-1052.
  41. Ebrahimi, F. and Rastgoo, A. (2008a), "Free vibration analysis of smart annular FGM plates integrated with piezoelectric layers", Smart Mater. Struct., 17(1), 015044. https://doi.org/10.1088/0964-1726/17/1/015044
  42. Ebrahimi, F. and Rastgoo, A. (2008b), "An analytical study on the free vibration of smart circular thin FGM plate based on classical plate theory", Thin-Wall. Struct., 46(12), 1402-1408. https://doi.org/10.1016/j.tws.2008.03.008
  43. Ebrahimi, F. and Rastgoo, A. (2008c), "Free vibration analysis of smart FGM plates", Int. J. Mech. Syst. Sci. Eng., 2(2), 94-99.
  44. Ebrahimi, F. and Salari, E (2015a), "Size-dependent thermo-electrical buckling analysis of functionally graded piezoelectric nanobeams", Smart Mater. Struct., 24(12), 125007. https://doi.org/10.1088/0964-1726/24/12/125007
  45. Ebrahimi, F. and Salari, E. (2015b), "Nonlocal thermo-mechanical vibration analysis of functionally graded nanobeams in thermal environment", Acta Astronautica, 113, 29-50. https://doi.org/10.1016/j.actaastro.2015.03.031
  46. Ebrahimi, F. and Salari, E. (2015c), "Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method", Compos. B, 79, 156-169. https://doi.org/10.1016/j.compositesb.2015.04.010
  47. Ebrahimi, F. and Salari, E. (2015d), "A semi-analytical method for vibrational and buckling analysis of functionally graded nanobeams considering the physical neutral axis position", CMES: Comput. Model. Eng. Sci., 105(2), 151-181.
  48. Ebrahimi, F. and Salari, E. (2015e), "Thermal buckling and free vibration analysis of size dependent Timoshenko FG nanobeams in thermal environments", Compos. Struct., 128, 363-380. https://doi.org/10.1016/j.compstruct.2015.03.023
  49. Ebrahimi, F. and Salari, E. (2015f), "Thermo-mechanical vibration analysis of nonlocal temperature-dependent FG nanobeams with various boundary conditions", Compos. B, 78, 272-290. https://doi.org/10.1016/j.compositesb.2015.03.068
  50. Ebrahimi, F. and Salari, E. (2016), "Effect of various thermal loadings on buckling and vibrational characteristics of nonlocal temperature-dependent functionally graded nanobeams", Mech. Adv. Mater. Struct., 23(12), 1379-1397. https://doi.org/10.1080/15376494.2015.1091524
  51. Ebrahimi, F. and Zia, M. (2015), "Large amplitude nonlinear vibration analysis of functionally graded Timoshenko beams with porosities", Acta Astronautica, 116, 117-125. https://doi.org/10.1016/j.actaastro.2015.06.014
  52. Ebrahimi, F., Rastgoo, A. and Kargarnovin, M.H. (2008), "Analytical investigation on axisymmetric free vibrations of moderately thick circular functionally graded plate integrated with piezoelectric layers", J. Mech. Sci. Technol., 22(6), 1058-1072. https://doi.org/10.1007/s12206-008-0303-2
  53. Ebrahimi F., Rastgoo, A. and Atai, A.A. (2009a), "Theoretical analysis of smart moderately thick shear deformable annular functionally graded plate", Eur. J. Mech. - A/Solids, 28(5), 962-997. https://doi.org/10.1016/j.euromechsol.2008.12.008
  54. Ebrahimi, F., Naei, M.H. and Rastgoo, A. (2009b), "Geometrically nonlinear vibration analysis of piezoelectrically actuated FGM plate with an initial large deformation", J. Mech. Sci. Technol., 23(8), 2107-2124. https://doi.org/10.1007/s12206-009-0358-8
  55. Ebrahimi, F., Ghadiri, M., Salari, E., Hoseini, S.A.H. and Shaghaghi, G.R. (2015a), "Application of the differential transformation method for nonlocal vibration analysis of functionally graded nanobeams", J. Mech. Sci. Tech., 29(3), 1207-1215. https://doi.org/10.1007/s12206-015-0234-7
  56. Ebrahimi, F., Salari, E. and Hosseini, S.A.H. (2015b), "Thermomechanical vibration behavior of FG nanobeams subjected to linear and non-linear temperature distributions", J. Therm. Stress., 38(12), 1360-1386. https://doi.org/10.1080/01495739.2015.1073980
  57. Ebrahimi, F., Salari, E. and Hosseini, S.A.H. (2016a), "In-plane thermal loading effects on vibrational characteristics of functionally graded nanobeams", Meccanica, 51(4), 951-977. https://doi.org/10.1007/s11012-015-0248-3
  58. Ebrahimi, F., Ghasemi, F. and Salari, E. (2016b), "Investigating thermal effects on vibration behavior of temperature-dependent compositionally graded Euler beams with porosities", Meccanica, 51(1), 223-249. https://doi.org/10.1007/s11012-015-0208-y
  59. Ebrahimi, F., Ehyaei, J. and Babaei, R. (2016c), "Thermal buckling of FGM nanoplates subjected to linear and nonlinear varying loads on Pasternak foundation", Adv. Mater. Res., Int. J., 5(4), 245-261.
  60. Ebrahimi, F., Barati, M.R. and Haghi, P. (2017), "Thermal effects on wave propagation characteristics of rotating strain gradient temperature-dependent functionally graded nanoscale beams", J. Therm. Stress., 40(5), 535-547. https://doi.org/10.1080/01495739.2016.1230483
  61. Eringen, A.C. (1972), "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
  62. 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
  63. Ehyaei, J., Ebrahimi, F. and Salari, E. (2016), "Nonlocal vibration analysis of FG nano beams with different boundary conditions", Adv. Nano Res., Int. J., 4(2), 85-111.
  64. Eltaher, M.A., Khater, M.E. and Emam, S.A. (2016a), "A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams", Appl. Math. Model., 40(5), 4109-4128. https://doi.org/10.1016/j.apm.2015.11.026
  65. Eltaher, M.A., Khater, M.E., Park, S., Abdel-Rahman, E. and Yavuz, M. (2016b), "On the static stability of nonlocal nanobeams using higher-order beam theories", Adv. Nano Res., Int. J., 4(1), 51-64.
  66. Elmerabet, A.H., Heireche, H., Tounsi, A. and Semmah, A. (2017), "Buckling temperature of a single-walled boron nitride nanotubes using a novel nonlocal beam model", Adv. Nano Res., Int. J., 5(1), 1-12.
  67. Falvo, M.R., Clary, G.J., Taylor, R.M., Chi, V., Brooks, F.P., Washburn, S. and Superfine, R. (1997), "Bending and buckling of carbon nanotubes under large strain", Nature, 389(6651), 582-584. https://doi.org/10.1038/39282
  68. Ghadiri, M., Soltanpour, M., Yazdi, A. and Safi, M. (2016), "Studying the influence of surface effects on vibration behavior of size-dependent cracked FG Timoshenko nanobeam considering nonlocal elasticity and elastic foundation", Appl. Phys. A, 122(5), 520. https://doi.org/10.1007/s00339-016-0036-5
  69. Ghosh, A., Bera, A. and Ghosh, M. (2017), "Optical dielectric function of impurity doped Quantum dots in presence of noise", Adv. Nano Res., Int. J., 5(1), 13-25. https://doi.org/10.12989/anr.2017.5.1.013
  70. Haghshenas A.A. and Arani, A.G. (2013), "Nonlocal vibration of a piezoelectric polymeric nanoplate carrying nanoparticle via Mindlin plate theory", Proc. Inst. Mech. Eng. C, J. Mech. Eng. Sci., 228(5), 907-920.
  71. Hasheminejad, B.S.M., Gheshlaghi, B., Mirzaei, Y. and Abbasion, S. (2011), "Free transverse vibrations of cracked nanobeams with surface effects", Thin Solid Films, 519(8), 2477-2482. https://doi.org/10.1016/j.tsf.2010.12.143
  72. Hsu, J.C., Lee, H.L. and Chang, W.J. (2011), "Longitudinal vibration of cracked nanobeams using nonlocal elasticity theory", Current Appl. Phys., 11(6), 1384-1388. https://doi.org/10.1016/j.cap.2011.04.026
  73. Karlicic, D., Kozic, P. and Pavlovic, R. (2015a), "Flexural vibration and buckling analysis of single-walled carbon nanotubes using different gradient elasticity theories based on Reddy and Huu-Tai formulations", J. Theor. Appl. Mech., 53.
  74. Karlicic, D., Cajic, M., Murmu, T. and Adhikari, S. (2015b), "Nonlocal longitudinal vibration of viscoelastic coupled double-nanorod systems", Eur. J. Mech.-A/Solids, 49, 183-196. https://doi.org/10.1016/j.euromechsol.2014.07.005
  75. Karlicic, D., Jovanovic, D., Kozic, P. and Cajic, M. (2015c), "Thermal and magnetic effects on the vibration of a cracked nanobeam embedded in an elastic medium", J. Mech. Mater. Struct., 10(1), 43-62. https://doi.org/10.2140/jomms.2015.10.43
  76. Karlicic, D., Kozic, P., Pavlovic, R. and Nesic, N. (2017), "Dynamic stability of single-walled carbon nanotube embedded in a viscoelastic medium under the influence of the axially harmonic load", Compos. Struct., 162, 227-243. https://doi.org/10.1016/j.compstruct.2016.12.003
  77. Khater, H.M. (2016), "Nano-Silica effect on the physicomechanical properties of geopolymer composites", Adv. Nano Res., Int. J., 4(3), 181-195.
  78. Kiani, K. (2012), "Magneto-thermo-elastic fields caused by an unsteady longitudinal magnetic field in a conducting nanowire accounting for eddy-current loss", Mater. Chem. Phys., 136(2-3), 589-598. https://doi.org/10.1016/j.matchemphys.2012.07.031
  79. Ke, L.-L. and Wang, Y.-S. (2012), "Thermoelectric-mechanical vibration of piezoelectric nanobeams based on the nonlocal theory", Smart Mater. Struct., 21(2), 025018. https://doi.org/10.1088/0964-1726/21/2/025018
  80. Kozic, P., Pavlovic, R. and Karlicic, D. (2014), "The flexural vibration and buckling of the elastically connected parallel-beams with a Kerr-type layer in between", Mech. Res. Commun., 56, 83-89. https://doi.org/10.1016/j.mechrescom.2013.12.003
  81. Li, L. (2001), "Vibratory characteristics of multi-step beams with an arbitrary number of cracks and concerted masses", Appl. Acoust., 62(6), 691-706. https://doi.org/10.1016/S0003-682X(00)00066-9
  82. Loya, J., Lopez-Puente, J., Zaera, R. and Fernandez-Saez, J. (2009), "Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model", J. Appl. Phys., 105(4), 044309. https://doi.org/10.1063/1.3068370
  83. Meyer, J.C., Geim, A.K., Katsnelson, M.I., Novoselov, K.S., Booth, T.J. and Roth, S. (2007), "The structure of suspended graphene sheets", Nature, 446(7131), 60-63. https://doi.org/10.1038/nature05545
  84. Mandal, S., Dey, A. and Pal, U. (2016), "Low temperature wet-chemical synthesis of spherical hydroxyapatite nanoparticles and their in situ cytotoxicity study", Adv. Nano Res., Int. J., 4(4), 295-307.
  85. Murmu, T. and Adhikari, S. (2010), "Nonlocal effects in the longitudinal vibration of double-nanorod systems", Phys. E, 43(1), 415-422. https://doi.org/10.1016/j.physe.2010.08.023
  86. Murmu, T. and Pradhan, S.C. (2009), "Thermo-mechanical vibration of a single-walled carbon nanotube embedded in an elastic medium based on nonlocal elasticity theory", Comput. Mater. Sci., 46(4), 854-859. https://doi.org/10.1016/j.commatsci.2009.04.019
  87. Murmu, T. and Pradhan, S.C. (2010), "Thermal effects on the stability of embedded carbon nanotubes", Comput. Mater. Sci. 47(3), 721-726. https://doi.org/10.1016/j.commatsci.2009.10.015
  88. Murmu, T., McCarthy, M.A. and Adhikari, S. (2013), "In-plane magnetic field affected transverse vibration of embedded single-layer graphene sheets using equivalent nonlocal elasticity approach", Compos. Struct., 96, 57-63. https://doi.org/10.1016/j.compstruct.2012.09.005
  89. Murmu, T., Adhikari, S. and McCarthy, M.A. (2014), "Axial vibration of embedded nanorods under transverse magnetic field effects via nonlocal elastic continuum theory", J. Comput. Theor. Nanosci., 11(5), 1230-1236. https://doi.org/10.1166/jctn.2014.3487
  90. Park, S.H., Kim, J.S., Park, J.H., Lee, J.S., Choi, Y.K. and Kwon, O.M. (2005), "Molecular dynamics study on size-dependent elastic properties of silicon nanocantilevers", Thin Solid Films, 492(1-2), 285-289. https://doi.org/10.1016/j.tsf.2005.06.056
  91. Phadikar, J.K. and Pradhan, S.C. (2010), "Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates", Comput. Mater. Sci., 49(3), 492-499. https://doi.org/10.1016/j.commatsci.2010.05.040
  92. Rahmani, O., Hosseini, S.A.H., Noroozi Moghaddam, M.H. and Fakhari Golpayegani, I. (2015), "Torsional vibration of cracked nanobeam based on nonlocal stress theory with various boundary conditions: An analytical study", Int. J. Appl. Mech., 7(3), 1550036. https://doi.org/10.1142/S1758825115500362
  93. Reddy, J.N. and Pang, S.D. (2008), "Nonlocal continuum theories of beams for the analysis of carbon nanotubes", J. Appl. Phys., 103(2), 023511. https://doi.org/10.1063/1.2833431
  94. Roostai, H. and Haghpanahi, M. (2014), "Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory", Appl. Math. Model., 38(3), 1159-1169. https://doi.org/10.1016/j.apm.2013.08.011
  95. Torabi, K. and Dastgerdi, J.N. (2012), "An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model", Thin Solid Films, 520(21), 6595-6602. https://doi.org/10.1016/j.tsf.2012.06.063
  96. 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
  97. Wang, Q. (2005), "Wave propagation in carbon nanotubes via nonlocal continuum mechanics", J. Appl. Phys., 98(12), 124301. https://doi.org/10.1063/1.2141648
  98. Wang, L., Ni, Q., Li, M. and Qian, Q. (2008), "The thermal effect on vibration and instability of carbon nanotubes conveying fluid", Physica E, 40(10), 3179-3182. https://doi.org/10.1016/j.physe.2008.05.009
  99. Yang, J. and Chen, Y. (2008), "Free vibration and buckling analyses of functionally graded beams with edge cracks", Compos. Struct., 83(1), 48-60. https://doi.org/10.1016/j.compstruct.2007.03.006
  100. Yao, X.H., Han, Q. (2006), "Buckling analysis of multiwalled carbon nanotubes under torsional load coupling with temperature change", J. Eng. Mater. Technol., 128(3), 419-428. https://doi.org/10.1115/1.2203102
  101. Zenkour, A.M. (2016), "Buckling of a single-layered graphene sheet embedded in visco-Pasternak's medium via nonlocal first-order theory", Adv. Nano Res., Int. J., 4(4), 309-326. https://doi.org/10.12989/anr.2016.4.4.309
  102. Zhang, Y.Q., Liu, X. and Zhao, J.H. (2008), "Influence of temperature change on column buckling of multiwalled carbon nanotubes", Phys. Lett. A, 372(10), 1676-1681. https://doi.org/10.1016/j.physleta.2007.10.033

피인용 문헌

  1. Thermal stress and magnetic effects on nonlinear vibration of nanobeams embedded in nonlinear elastic medium vol.43, pp.10, 2018, https://doi.org/10.1080/01495739.2020.1780175
  2. Size dependent vibration of embedded functionally graded nanoplate in hygrothermal environment by Rayleigh-Ritz method vol.10, pp.1, 2018, https://doi.org/10.12989/anr.2021.10.1.025