DOI QR코드

DOI QR Code

Torsional vibration analysis of bi-directional FG nano-cone with arbitrary cross-section based on nonlocal strain gradient elasticity

  • Noroozi, Reza (School of Mechanical Engineering, College of Engineering, University of Tehran) ;
  • Barati, Abbas (Department of Mechanical Engineering, University of Guilan) ;
  • Kazemi, Amin (School of Mechanical Engineering, College of Engineering, University of Tehran) ;
  • Norouzi, Saeed (School of Mechanical Engineering, College of Engineering, University of Tehran) ;
  • Hadi, Amin (Cellular and Molecular Research Center, School of Medicine, Yasuj University of Medical Sciences)
  • 투고 : 2019.01.08
  • 심사 : 2019.12.17
  • 발행 : 2020.01.25

초록

In this paper, for the first time based on the nonlocal strain gradient theory the effect of size dependency in torsional vibration of bi-direction functionally graded (FG) nonlinear nano-cone is study. The material properties were assumed to vary according to the arbitrary function in radial and axial directions. The Navier equation and boundary conditions of the size-dependent bidirectional FG nonlinear nano-cone were derived by Hamilton's principle. These equations were solved by employing the generalized differential quadrature method (GDQM). The presented model can turn into the classical model if the material length scale parameters are taken to be zero. The effects of some parameters, such as inhomogeneity constant, cross-sectional area parameter and small-scale parameters, were studied. As an essential result of this study can be stated that an FG nano-cone model based on the nonlocal elasticity theory behaves softer and based on the strain gradient theory behaves harder.

키워드

참고문헌

  1. Abazari, A.M., Safavi, S.M., Rezazadeh, G. and Villanueva, L.G. (2015), "Size Effects on Mechanical Properties of Micro/Nano Structures", arXiv preprint arXiv:150801322.
  2. Adeli, M.M., Hadi, A., Hosseini, M. and Gorgani, H.H. (2017), "Torsional vibration of nano-cone based on nonlocal strain gradient elasticity theory", Eur. Phys. J. Plus, 132(9), 393. https://doi.org/10.1140/epjp/i2017-11688-0
  3. Akbas, S.D. (2018), "Bending of a cracked functionally graded nanobeam", Adv. Nano Res., Int. J., 6(3), 219-242. https://doi.org/10.12989/anr.2018.6.3.219
  4. Asemi, S.R. and Farajpour, A. (2014), "Vibration characteristics of double-piezoelectric-nanoplate-systems", Micro. Nano Lett., 9(4), 280-285. https://doi.org/10.1049/mnl.2013.0741
  5. Aydogdu, M., Arda, M. and Filiz, S. (2018), "Vibration of axially functionally graded nano rods and beams with a variable nonlocal parameter", Adv. Nano Res., Int. J., 6(3), 257-278. https://doi.org/10.12989/anr.2018.6.3.257
  6. Barati, M.R. (2017), "Vibration analysis of FG nanoplates with nanovoids on viscoelastic substrate under hygro-thermomechanical loading using nonlocal strain gradient theory", Struct. Eng. Mech., Int. J., 64(6), 683-693. https://doi.org/10.12989/sem.2017.64.6.683
  7. Bodaghi, M., Noroozi, R., Zolfagharian, A., Fotouhi, M. and Norouzi, S. (2019), "4D printing self-morphing structures", Materials, 2(8), 1353. https://doi.org/10.3390/ma12081353
  8. Demir, C. and Civalek, O . (2013), "Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models", Appl. Math. Modell., 37(22), 9355-9367. https://doi.org/10.1016/j.apm.2013.04.050
  9. Dowling, A.P.J.M.T. (2004), "Development of nanotechnologies", Mater. Today, 7(12), 30-35. https://doi.org/10.1016/S1369-7021(04)00628-5
  10. Ebrahimi, F. and Barati, M.R. (2017a), "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
  11. Ebrahimi, F. and Barati, M.R. (2017b), "Vibration analysis of viscoelastic inhomogeneous nanobeams incorporating surface and thermal effects", Appl. Phys. A, 123(1), 5. https://doi.org/10.1007/s00339-016-0511-z
  12. Ebrahimi, F. and Barati, M.R. (2018a), "Buckling analysis of nonlocal strain gradient axially functionally graded nanobeams resting on variable elastic medium", Proc. Inst. Mech. Eng. Pt. C J. Mechan. Eng. Sci. 232(11), 2067-2078. https://doi.org/10.1177/0954406217713518
  13. Ebrahimi, F. and Barati, M.R. (2018b), "Stability analysis of functionally graded heterogeneous piezoelectric nanobeams based on nonlocal elasticity theory", Adv. Nano Res., Int. J., 6(2), 93-112. https://doi.org/10.12989/anr.2018.6.2.093
  14. Ebrahimi, F. and Dabbagh, A. (2017), "Nonlocal strain gradient based wave dispersion behavior of smart rotating magnetoelectro-elastic nanoplates", Mater. Res. Express, 4(2), 025003. https://doi.org/10.1088/2053-1591/aa55b5
  15. Ebrahimi, F. and Fardshad, R.E. (2018), "Modeling the size effect on vibration characteristics of functionally graded piezoelectric nanobeams based on Reddy's shear deformation beam theory", Adv. Nano Res., Int. J., 6(2), 113-133. https://doi.org/10.12989/anr.2018.6.2.113
  16. Ebrahimi, F. and Haghi, P. (2017), "Wave propagation analysis of rotating thermoelastically-actuated nanobeams based on nonlocal strain gradient theory", Acta Mech. Solida Sin.. 30(6), 647-657. https://doi.org/10.1016/j.camss.2017.09.007
  17. Ebrahimi, F. and Haghi, P. (2018), "Elastic wave dispersion modelling within rotating functionally graded nanobeams in thermal environment", Adv. Nano Res., Int. J., 6(3), 201-217. https://doi.org/10.12989/anr.2018.6.3.201
  18. Ebrahimi, F. and Salari, E. (2018), "Effect of non-uniform temperature distributions on nonlocal vibration and buckling of inhomogeneous size-dependent beams", Adv. Nano Res., Int. J., 6(4), 377-397. https://doi.org/10.12989/anr.2018.6.4.377
  19. Ebrahimi, F., Barati, M.R. and Haghi, P. (2017a), "Thermal effects on wave propagation characteristics of rotating strain gradient temperature-dependent functionally graded nanoscale beams", J. Therm. Stresses, 40(5), 535-547. https://doi.org/10.1080/01495739.2016.1230483
  20. Ebrahimi, F., Daman, M. and Jafari, A. (2017b), "Nonlocal strain gradient-based vibration analysis of embedded curved porous piezoelectric nano-beams in thermal environment", Smart Struct. Syst., Int. J., 20(6), 709-728. https://doi.org/10.12989/sss.2017.20.6.709
  21. El-Borgi, S., Rajendran, P., Friswell, M., Trabelssi, M. and Reddy, J. (2018), "Torsional vibration of size-dependent viscoelastic rods using nonlocal strain and velocity gradient theory", Compos. Struct., 186, 274-292. https://doi.org/10.1016/j.compstruct.2017.12.002
  22. Eltaher, M., Fouda, N., El-midany, T., Sadoun, A.J.J.o.t.B.S.o.M.S. and Engineering (2018), "Modified porosity model in analysis of functionally graded porous nanobeams", J. Brazil. Soc. Mech. Sci. Eng., 40(3), 141. https://doi.org/10.1007/s40430-018-1065-0
  23. 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
  24. 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
  25. Eringen, A.C. (2002), Nonlocal continuum field theories, Springer Science & Business Media
  26. Farajpour, A. and Rastgoo, A. (2017), "Influence of carbon nanotubes on the buckling of microtubule bundles in viscoelastic cytoplasm using nonlocal strain gradient theory", Results Phys., 7, 1367-1375. https://doi.org/10.1016/j.rinp.2017.03.038
  27. Farajpour, M.R., Rastgoo, A., Farajpour, A. and Mohammadi, M. (2016), "Vibration of piezoelectric nanofilm-based electromechanical sensors via higher-order non-local strain gradient theory", Micro. Nano Lett., 11(6), 302-307. https://doi.org/10.1049/mnl.2016.0081
  28. Farajpour, A., Rastgoo, A. and Farajpour, M. (2017a), "Nonlinear buckling analysis of magneto-electro-elastic CNT-MT hybrid nanoshells based on the nonlocal continuum mechanics", Compos. Struct., 180, 179-191. https://doi.org/10.1016/j.compstruct.2017.07.100
  29. Farajpour, A., Rastgoo, A. and Mohammadi, M. (2017b), "Vibration, buckling and smart control of microtubules using piezoelectric nanoshells under electric voltage in thermal environment", Physica B: Condensed Matter, 509, 100-114. https://doi.org/10.1016/j.physb.2017.01.006
  30. Farajpour, M., Shahidi, A., Hadi, A. and Farajpour, A. (2018a), "Influence of initial edge displacement on the nonlinear vibration, electrical and magnetic instabilities of magnetoelectro-elastic nanofilms", Mech. Adv. Mater. Struct., 26(17), 1469-1481. https://doi.org/10.1080/15376494.2018.1432820
  31. Farajpour, M., Shahidi, A., Tabataba'i-Nasab, F. and Farajpour, A. (2018b), "Vibration of initially stressed carbon nanotubes under magneto-thermal environment for nanoparticle delivery via higher-order nonlocal strain gradient theory", Eur Phys J Plus. 133(6), 219. https://doi.org/10.1140/epjp/i2018-12039-5
  32. Farajpour, M.R., Shahidi, A. and Farajpour, A. (2018c), "Resonant frequency tuning of nanobeams by piezoelectric nanowires under thermo-electro-magnetic field: a theoretical study", Micro. Nano Lett., 13(11), 1627-1632. https://doi.org/10.1049/mnl.2018.5286
  33. Goyal, A. and Soni, P.J.M.L. (2018), "Functionally graded nanocrystalline silicon powders by mechanical alloying", 214, 111-114. https://doi.org/10.1016/j.matlet.2017.11.114
  34. Guo, S., He, Y., Liu, D., Lei, J., Shen, L. and Li, Z. (2016), "Torsional vibration of carbon nanotube with axial velocity and velocity gradient effect", Int. J. Mech. Sci., 119, 88-96. https://doi.org/10.1016/j.ijmecsci.2016.09.036
  35. Hadi, A., Nejad, M.Z. and Hosseini, M. (2018a), "Vibrations of three-dimensionally graded nanobeams", Int. J. Mech. Sci., 128, 12-23. https://doi.org/10.1016/j.ijengsci.2018.03.004
  36. Hadi, A., Rastgoo, A., Haghighipour, N. and Bolhassani, A. (2018b), "Numerical modelling of a spheroid living cell membrane under hydrostatic pressure", J. Stat. Mech. Theory Experiment., 2018(8), 083501. https://doi.org/10.1088/1742-5468/aad369
  37. Hosseini, M., Shishesaz, M., Tahan, K.N. and Hadi, A. (2016), "Stress analysis of rotating nano-disks of variable thickness made of functionally graded materials", Int. J. Eng. Sci., 109, 29-53. https://doi.org/10.1016/j.ijengsci.2016.09.002
  38. Hosseini, M., Gorgani, H.H., Shishesaz, M. and Hadi, A. (2017), "Size-dependent stress analysis of single-wall carbon nanotube based on strain gradient theory", Int. J. Appl. Mech., 9(6), 1750087. https://doi.org/10.1142/S1758825117500879
  39. Hosseini, M., Hadi, A., Malekshahi, A. and Shishesaz, M. (2018), "A review of size-dependent elasticity for nanostructures", J. Computat. Appl. Mech., 49(1), 197-211. https://doi.org/10.22059/JCAMECH.2018.259334.289
  40. Hosseini, M., Shishesaz, M. and Hadi, A. (2019), "Thermoelastic analysis of rotating functionally graded micro/nanodisks of variable thickness", Thin-Wall. Struct., 134, 508-523. https://doi.org/10.1016/j.tws.2018.10.030
  41. Kafadar, C. and Eringen, A.C. (1971a), "Micropolar media-I the classical theory", Int. J. Eng. Sci., 9(3), 271-305. https://doi.org/10.1016/0020-7225(71)90040-1
  42. Kafadar, C. and Eringen, A.C. (1971b), "Micropolar media-II the relativistic theory", Int. J. Eng. Sci., 9(3), 307-329. https://doi.org/10.1016/0020-7225(71)90041-3
  43. Karami, B., Janghorban, M. and Tounsi, A. (2018a), "Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles", Steel Compos. Struct., Int. J., 27(2), 201-216. https://doi.org/10.12989/scs.2018.27.2.201
  44. Karami, B., Shahsavari, D. and Janghorban, M. (2018b), "Wave propagation analysis in functionally graded (FG) nanoplates under in-plane magnetic field based on nonlocal strain gradient theory and four variable refined plate theory", Mech. Adv. Mater. Struct., 25(12), 1047-1057. https://doi.org/10.1080/15376494.2017.1323143
  45. Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2018c), "Wave dispersion of mounted graphene with initial stress", Thin-Wall. Struct., 122, 102-111. https://doi.org/10.1016/j.tws.2017.10.004
  46. Karami, B., Shahsavari, D. and Li, L. (2018d), "Hygrothermal wave propagation in viscoelastic graphene under in-plane magnetic field based on nonlocal strain gradient theory", Physica E, 97, 317-327. https://doi.org/10.1016/j.physe.2017.11.020
  47. Li, C. (2014), "Torsional vibration of carbon nanotubes: Comparison of two nonlocal models and a semi-continuum model", Int. J. Mech. Sci., 82, 25-31. https://doi.org/10.1016/j.ijmecsci.2014.02.023
  48. Li, L. and Hu, Y. (2016), "Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material", Int. J. Eng. Sci., 107, 77-97. https://doi.org/10.1016/j.ijengsci.2016.07.011
  49. Li, L. and Hu, Y. (2017), "Torsional vibration of bi-directional functionally graded nanotubes based on nonlocal elasticity theory", Compos. Struct., 172, 242-250. https://doi.org/10.1016/j.compstruct.2017.03.097
  50. Li, L., Hu, Y. and Ling, L. (2015), "Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory", Compos. Struct., 133, 1079-1092. https://doi.org/10.1016/j.compstruct.2015.08.014
  51. Li, L., Li, X. and Hu, Y. (2016), "Free vibration analysis of nonlocal strain gradient beams made of functionally graded material", Int. J. Eng. Sci., 102, 77-92. https://doi.org/10.1016/j.ijengsci.2016.02.010
  52. Li, L., Tang, H. and Hu, Y. (2018), "Size-dependent nonlinear vibration of beam-type porous materials with an initial geometrical curvature", Compos. Struct., 184, 1177-1188. https://doi.org/10.1016/j.compstruct.2017.10.052
  53. Lim, C., Zhang, G. and Reddy, J. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solids, 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001
  54. Lu, L., Guo, X. and Zhao, J. (2017a), "Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory", Int. J. Eng. Sci., 116, 12-24. https://doi.org/10.1016/j.ijengsci.2017.03.006
  55. Lu, L., Guo, X. and Zhao, J. (2017b), "A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms", Int. J. Eng. Sci., 119, 265-277. https://doi.org/10.1016/j.ijengsci.2017.06.024
  56. Mehralian, F., Beni, Y.T. and Zeverdejani, M.K. (2017a), "Calibration of nonlocal strain gradient shell model for buckling analysis of nanotubes using molecular dynamics simulations", Physica B, Condens. Matter, 521, 102-111. https://doi.org/10.1016/j.physb.2017.06.058
  57. Mehralian, F., Beni, Y.T. and Zeverdejani, M.K. (2017b), "Nonlocal strain gradient theory calibration using molecular dynamics simulation based on small scale vibration of nanotubes", Physica B, Condens. Matter, 514, 61-69. https://doi.org/10.1016/j.physb.2017.03.030
  58. Mindlin, R. and Eshel, N. (1968), "On first strain-gradient theories in linear elasticity", Int. J. Solids Struct., 4(1), 109-124. https://doi.org/10.1016/0020-7683(68)90036-X
  59. Miyamoto, Y., Kaysser, W., Rabin, B., Kawasaki, A. and Ford, R.G. (2013), Functionally graded materials: design, processing and applications, Springer Science & Business Media
  60. Nejad, M.Z. and Hadi, A. (2016a), "Eringen's non-local elasticity theory for bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams", Int. J. Eng. Sci., 106, 1-9. https://doi.org/10.1016/j.ijengsci.2016.05.005
  61. Nejad, M.Z. and Hadi, A. (2016b), "Non-local analysis of free vibration of bi-directional functionally graded Euler-Bernoulli nano-beams", Int. J. Eng. Sci., 105, 1-11. https://doi.org/10.1016/j.ijengsci.2016.04.011
  62. Nejad, M.Z., Hadi, A. and Rastgoo, A. (2016), "Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams based on nonlocal elasticity theory", Int. J. Eng. Sci., 103, 1-10. https://doi.org/10.1016/j.ijengsci.2016.03.001
  63. Nejad, M.Z., Alamzadeh, N. and Hadi, A. (2018), "Thermoelastoplastic analysis of FGM rotating thick cylindrical pressure vessels in linear elastic-fully plastic condition", Compos. Part B-Eng., 154, 410-422. https://doi.org/10.1016/j.compositesb.2018.09.022
  64. Noroozi, R. and Ataee, A. (2018), "Behavioral Optimization of Pseudo-Neutral Hole in Hyperelastic Membranes Using Functionally graded Cables", J. Computat. Appl. Mech., 49(2), 282-291. https://doi.org/10.22059/JCAMECH.2018.268899.338
  65. Petit, C., Montanaro, L. and Palmero, O. (2018), "Functionally graded ceramics for biomedical application: Concept, manufacturing, and properties", Int. J. Appl. Ceram. Technol., 15(4), 820-840. https://doi.org/10.1111/ijac.12878
  66. Phung-Van, P., Ferreira, A., Nguyen-Xuan, H. and Wahab, M.A. (2017a), "An isogeometric approach for size-dependent geometrically nonlinear transient analysis of functionally graded nanoplates", Compos. Part B-Eng., 118, 125-134. https://doi.org/10.1016/j.compositesb.2017.03.012
  67. Phung-Van, P., Lieu, Q.X., Nguyen-Xuan, H. and Wahab, M.A. (2017b), "Size-dependent isogeometric analysis of functionally graded carbon nanotube-reinforced composite nanoplates", Compos. Struct., 166, 120-135. https://doi.org/10.1016/j.compstruct.2017.01.049
  68. Phung-Van, P., Tran, L.V., Ferreira, A., Nguyen-Xuan, H. and Abdel-Wahab, M.J.N.D. (2017c), "Nonlinear transient isogeometric analysis of smart piezoelectric functionally graded material plates based on generalized shear deformation theory under thermo-electro-mechanical loads", Nonliear Dyn., 87(2), 879-894. https://doi.org/10.1007/s11071-016-3085-6
  69. Phung-Van, P., Thai, C.H., Nguyen-Xuan, H. and Wahab, M.A. (2019), "Porosity-dependent nonlinear transient responses of functionally graded nanoplates using isogeometric analysis", Compos. Part B-Eng., 164, 215-225. https://doi.org/10.1016/j.compositesb.2018.11.036
  70. Pompe, W., Worch, H., Epple, M., Friess, W., Gelinsky, M., Greil, P., Hempel, U., Scharnweber, D., Schulte, K.J.M.S. and A, E. (2003), "Functionally graded materials for biomedical applications", Mater. Sci. Eng. A, 362(1-2), 40-60. https://doi.org/10.1016/S0921-5093(03)00580-X
  71. Poole Jr, C.P. and Owens, F.J. (2003), Introduction to nanotechnology, John Wiley & Sons
  72. Rahmani, O., Norouzi, S., Golmohammadi, H. and Hosseini, S. (2017), "Dynamic response of a double, single-walled carbon nanotube under a moving nanoparticle based on modified nonlocal elasticity theory considering surface effects", Mech. Adv. Mater. Struct., 24(15), 1274-1291. https://doi.org/10.1080/15376494.2016.1227504
  73. Sahmani, S. and Aghdam, M. (2017), "A nonlocal strain gradient hyperbolic shear deformable shell model for radial postbuckling analysis of functionally graded multilayer GPLRC nanoshells", Compos. Struct., 178, 97-109. https://doi.org/10.1016/j.compstruct.2017.06.062
  74. She, G.-L., Yuan, F.-G., Ren, Y.-R., Liu, H.-B. and Xiao, W.-S. (2018), "Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory", Compos. Struct., 203, 614-623. https://doi.org/10.1016/j.compstruct.2018.07.063
  75. Shen, Y., Chen, Y. and Li, L. (2016), "Torsion of a functionally graded material", Int. J. Eng. Sci., 109, 14-28. https://doi.org/10.1016/j.ijengsci.2016.09.003
  76. Shishesaz, M., Hosseini, M., Tahan, K.N. and Hadi, A. (2017), "Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory", Acta Mechanica, 228(12), 4141-4168. https://doi.org/10.1007/s00707-017-1939-8
  77. Shu, C. and Chew, Y. (1998), "On the equivalence of generalized differential quadrature and highest order finite difference scheme", Comput. Method Appl. M, 155(3-4), 249-260. https://doi.org/10.1016/S0045-7825(97)00150-3
  78. Simsek, M. (2016), "Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach", Int. J. Eng. Sci., 105, 12-27. https://doi.org/10.1016/j.ijengsci.2016.04.013
  79. Thai, C.H., Ferreira, A., Wahab, M.A. and Nguyen-Xuan, H. (2018), "A moving Kriging meshfree method with naturally stabilized nodal integration for analysis of functionally graded material sandwich plates", Acta Mech., 229(7), 2997-3023. https://doi.org/10.1007/s00707-018-2156-9
  80. Thanh, C.-L., Phung-Van, P., Thai, C.H., Nguyen-Xuan, H. and Wahab, M.A. (2018), "Isogeometric analysis of functionally graded carbon nanotube reinforced composite nanoplates using modified couple stress theory", Compos. Struct., 184, 633-649. https://doi.org/10.1016/j.compstruct.2017.10.025
  81. Thanh, C.-L., Ferreira, A. and Wahab, M.A. (2019a), "A refined size-dependent couple stress theory for laminated composite micro-plates using isogeometric analysis", Thin-Wall. Struct., 145, 106427. https://doi.org/10.1016/j.tws.2019.106427
  82. Thanh, C.-L., Tran, L.V., Bui, T.Q., Nguyen, H.X. and Abdel-Wahab, M. (2019b), "Isogeometric analysis for size-dependent nonlinear thermal stability of porous FG microplates", Compos Struct., 221, 110838. https://doi.org/10.1016/j.compstruct.2019.04.010
  83. Thanh, C.-L., Tran, L.V., Vu-Huu, T. and Abdel-Wahab, M. (2019c), "The size-dependent thermal bending and buckling analyses of composite laminate microplate based on new modified couple stress theory and isogeometric analysis", Comput. Method Appl. M, 350, 337-361. https://doi.org/10.1016/j.cma.2019.02.028
  84. Thanh, C.-L., Tran, L.V., Vu-Huu, T., Nguyen-Xuan, H. and Abdel-Wahab, M. (2019d), "Size-dependent nonlinear analysis and damping responses of FG-CNTRC micro-plates", Comput. Method Appl. M, 353, 253-276. https://doi.org/10.1016/j.cma.2019.05.002
  85. Toupin, R.A. (1962), "Elastic materials with couple-stresses", Arch. Ration. Mech. Anal., 11(1), 385-414. https://doi.org/10.1007/BF00253945
  86. Xu, X.-J., Zheng, M.-L. and Wang, X.-C. (2017), "On vibrations of nonlocal rods: Boundary conditions, exact solutions and their asymptotics", Int. J. Eng. Sci., 119, 217-231. https://doi.org/10.1016/j.ijengsci.2017.06.025
  87. Yayli, M.O . (2018), "Torsional vibrations of restrained nanotubes using modified couple stress theory", Microsyst. Technol., 24(8), 3425-3435. https://doi.org/10.1007/s00542-018-3735-3
  88. Zamani Nejad, M., Jabbari, M. and Hadi, A. (2017), "A review of functionally graded thick cylindrical and conical shells", J. Computat. Appl. Mech., 48(2), 357-370. https://doi.org/10.22059/JCAMECH.2017.247963.220
  89. Zhu, X. and Li, L. (2017a), "Closed form solution for a nonlocal strain gradient rod in tension", Int. J. Eng. Sci., 119, 16-28. https://doi.org/10.1016/j.ijengsci.2017.06.019
  90. Zhu, X. and Li, L. (2017b), "On longitudinal dynamics of nanorods", Int. J. Eng. Sci., 120, 129-145. https://doi.org/10.1016/j.ijengsci.2017.08.003

피인용 문헌

  1. Free vibration of electro-magneto-thermo sandwich Timoshenko beam made of porous core and GPLRC vol.10, pp.2, 2020, https://doi.org/10.12989/anr.2021.10.2.115
  2. Elastic wave phenomenon of nanobeams including thickness stretching effect vol.10, pp.3, 2020, https://doi.org/10.12989/anr.2021.10.3.271
  3. An investigation of mechanical properties of kidney tissues by using mechanical bidomain model vol.11, pp.2, 2020, https://doi.org/10.12989/anr.2021.11.2.193
  4. Effect of autofrettage on the ultimate behavior of thick cylindrical pressure vessels vol.194, pp.no.pb, 2021, https://doi.org/10.1016/j.ijpvp.2021.104546