Advances in nano research

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Free vibrations analysis of arbitrary three-dimensionally FGM nanoplates

  • Dehshahri, Kasra (Department of Mechanical Engineering, Yasouj University) ;
  • Nejad, Mohammad Zamani (Department of Mechanical Engineering, Yasouj University) ;
  • Ziaee, Sima (Department of Mechanical Engineering, Yasouj University) ;
  • Niknejad, Abbas (Department of Mechanical Engineering, Yasouj University) ;
  • Hadi, Amin (Cellular and Molecular Research Center, School of Medicine, Yasuj University of Medical Sciences)
  • Received : 2019.07.21
  • Accepted : 2019.10.11
  • Published : 2020.02.25
⟨ Previous Next ⟩

Abstract

In this paper, the free vibrations analysis of the nanoplates made of three-directional functionally graded material (TDFGM) with small scale effects is presented. To study the small-scale effects on natural frequency, modified strain gradient theory (MSGT) has been used. Material properties of the nanoplate follow an arbitrary function that changes in three directions along the length, width and thickness of the plate. The equilibrium equations and boundary conditions of nanoplate are obtained using the Hamilton's principle. The generalized differential quadrature method (GDQM) is used to solve the governing equations and different boundary conditions for obtaining the natural frequency of nanoplate made of three-directional functionally graded material. The present model can be transformed into a couple stress plate model or a classic plate model if two or all parameters of the length scales set to zero. Finally, numerical results are presented to study the small-scale effect and heterogeneity constants and the aspect ratio with different boundary conditions on the free vibrations of nanoplates. To the best of the researchers' knowledge, in the literature, there is no study carried out into MSGT for free vibration analysis of FGM nanoplate with arbitrary functions.

Keywords

References

  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:1508.01322.
  2. Abouelregal, A. (2019), "Rotating magneto-thermoelastic rod with finite length due to moving heat sources via Eringen's nonlocal model", J. Computat. Appl. Mech., 50(1), 118-126. https://doi.org/10.22059/JCAMECH.2019.275893.360
  3. Abouelregal, A. and Zenkour, A. (2019), "Vibration of FG viscoelastic nanobeams due to a periodic heat flux via fractional derivative model", J. Computat. Appl. Mech., 50(1), 148-156. https://doi.org/10.22059/JCAMECH.2019.277115.367
  4. 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
  5. Afshin, A., Nejad, M.Z. and Dastani, K. (2017), "Transient thermoelastic analysis of FGM rotating thick cylindrical pressure vessels under arbitrary boundary and initial conditions", J. Computat. Appl. Mech., 48(1), 15-26. https://doi.org/10.22059/JCAMECH.2017.233643.144
  6. Ajri, M. and Fakhrabadi, M.M.S. (2018), "Nonlinear free vibration of viscoelastic nanoplates based on modified couple stress theory", J. Computat. Appl. Mech., 49(1), 44-53. https://doi.org/10.22059/JCAMECH.2018.228477.129
  7. Akgoz, B. and Civalek, O. (2013a), "Free vibration analysis of axially functionally graded tapered Bernoulli-Euler microbeams based on the modified couple stress theory", Compos. Struct., 98, 314-322. https://doi.org/10.1016/j.compstruct.2012.11.020
  8. Akgoz, B. and Civalek, O. (2013b), "Longitudinal vibration analysis of strain gradient bars made of functionally graded materials (FGM)", Compos. Part B: Eng., 55, 263-268. https://doi.org/10.1016/j.compositesb.2013.06.035
  9. Alizada, A. and Sofiyev, A. (2011), "On the mechanics of deformation and stability of the beam with a nanocoating", J. Reinf. Plast. Compos., 30(18), 1583-1595. https://doi.org/10.1177/0731684411428382
  10. Ansari, R., Gholami, R., Shojaei, M.F., Mohammadi, V. and Sahmani, S. (2015), "Bending, buckling and free vibration analysis of size-dependent functionally graded circular/annular microplates based on the modified strain gradient elasticity theory", Eur. J. Mech.-A/Solids, 49, 251-267. https://doi.org/10.1016/j.euromechsol.2014.07.014
  11. Ansari, R., Hasrati, E., Faghih Shojaei, M., Gholami, R., Mohammadi, V. and Shahabodini, A. (2016), "Size-dependent bending, buckling and free vibration analyses of microscale functionally graded mindlin plates based on the strain gradient elasticity theory", Latin Am. J. Solids Struct., 13(4), 632-664. https://doi.org/10.1590/1679-78252322
  12. Azimi, M., Mirjavadi, S.S., Shafiei, N., Hamouda, A. and Davari, E. (2018), "Vibration of rotating functionally graded Timoshenko nano-beams with nonlinear thermal distribution", Mech. Adv. Mater. Struct., 25(6), 467-480. https://doi.org/10.1080/15376494.2017.1285455
  13. Cho, J. and Oden, J.T. (2000), "Functionally graded material: a parametric study on thermal-stress characteristics using the Crank-Nicolson-Galerkin scheme", Comput. Methods Appl. Mech. Eng., 188(1-3), 17-38. https://doi.org/10.1016/S0045-7825(99)00289-3
  14. Dehghan, M., Nejad, M.Z. and Moosaie, A. (2016), "Thermo-electro-elastic analysis of functionally graded piezoelectric shells of revolution: Governing equations and solutions for some simple cases", Int. J. Eng. Sci., 104, 34-61. https://doi.org/10.1016/j.ijengsci.2016.04.007
  15. Deng, H. and Cheng, W. (2016), "Dynamic characteristics analysis of bi-directional functionally graded Timoshenko beams", Compos. Struct., 141, 253-263. https://doi.org/10.1016/j.compstruct.2016.01.051
  16. Ebrahimi, F. and Barati, M.R. (2017), "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
  17. Ebrahimi, F. and Dabbagh, A. (2017a), "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
  18. Ebrahimi, F. and Dabbagh, A. (2017b), "On flexural wave propagation responses of smart FG magneto-electro-elastic nanoplates via nonlocal strain gradient theory", Compos. Struct., 162, 281-293. https://doi.org/10.1016/j.compstruct.2016.11.058
  19. Ebrahimi, F. and Dabbagh, A. (2019), "A comprehensive review on modeling of nanocomposite materials and structures", J. Computat. Appl. Mech., 50(1), 197-209. https://doi.org/10.22059/JCAMECH.2019.282388.405
  20. Ebrahimi, F., Barati, M.R. and Dabbagh, A. (2016), "A nonlocal strain gradient theory for wave propagation analysis in temperature-dependent inhomogeneous nanoplates", Int. J. Eng. Sci., 107, 169-182. https://doi.org/10.1016/j.ijengsci.2016.07.008
  21. 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
  22. Farajpour, M., Shahidi, A., Hadi, A. and Farajpour, A. (2018), "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
  23. Fatehi, P. and Nejad, M.Z. (2014), "Effects of material gradients on onset of yield in FGM rotating thick cylindrical shells", Int. J. Appl. Mech., 6(4), 1450038. https://doi.org/10.1142/S1758825114500380
  24. Ghannad, M. and Nejad, M.Z. (2010), "Elastic analysis of pressurized thick hollow cylindrical shells with clampedclamped ends", Mech., 85(5), 11-18.
  25. Ghannad, M., Nejad, M.Z. and Rahimi, G. (2009), "Elastic solution of axisymmetric thick truncated conical shells based on first-order shear deformation theory", Mech., 79(5), 13-20.
  26. Ghannad, M., Nejad, M.Z., Rahimi, G. and Sabouri, H. (2012), "Elastic analysis of pressurized thick truncated conical shells made of functionally graded materials", Struct. Eng. Mech., Int. J., 43(1), 105-126. https://doi.org/10.12989/sem.2012.43.1.105
  27. Ghannad, M., Rahimi, G.H. and Nejad, M.Z. (2013), "Elastic analysis of pressurized thick cylindrical shells with variable thickness made of functionally graded materials", Compos. Part B: Eng., 45(1), 388-396. https://doi.org/10.1016/j.compositesb.2012.09.043
  28. Gharibi, M., Nejad, M.Z. and Hadi, A. (2017), "Elastic analysis of functionally graded rotating thick cylindrical pressure vessels with exponentially-varying properties using power series method of Frobenius", J. Computat. Appl. Mech., 48(1), 89-98. https://doi.org/10.22059/JCAMECH.2017.233633.143
  29. Ghorbani Shenas, A., Ziaee, S. and Malekzadeh, P. (2017), "Nonlinear vibration analysis of pre-twisted functionally graded microbeams in thermal environment", Thin-Wall. Struct., 118, 87-104. https://doi.org/10.1016/j.tws.2017.05.003
  30. Haciyev, V., Sofiyev, A. and Kuruoglu, N. (2018), "Free bending vibration analysis of thin bidirectionally exponentially graded orthotropic rectangular plates resting on two-parameter elastic foundations", Compos. Struct., 184, 372-377. https://doi.org/10.1016/j.compstruct.2017.10.014
  31. Haciyev, V., Sofiyev, A. and Kuruoglu, N. (2019), "On the free vibration of orthotropic and inhomogeneous with spatial coordinates plates resting on the inhomogeneous viscoelastic foundation", Mech. Adv. Mater. Struct., 26(10), 886-897. https://doi.org/10.1080/15376494.2018.1430271
  32. Hadi, A., Nejad, M.Z. and Hosseini, M. (2018a), "Vibrations of three-dimensionally graded nanobeams", Int. J. Eng. Sci., 128, 12-23. https://doi.org/10.1016/j.ijengsci.2018.03.004
  33. Hadi, A., Nejad, M.Z., Rastgoo, A. and Hosseini, M. (2018b), "Buckling analysis of FGM Euler-Bernoulli nano-beams with 3D-varying properties based on consistent couple-stress theory", Steel Compos. Struct., Int. J., 26(6), 663-672. https://doi.org/10.12989/scs.2018.26.6.663
  34. Hadi, A., Rastgoo, A., Haghighipour, N. and Bolhassani, A. (2018c), "Numerical modelling of a spheroid living cell membrane under hydrostatic pressure", J. Statist. Mech.: Theory Experim., 2018(8), 083501. https://doi.org/10.1088/1742-5468/aad369
  35. Hosseini-Hashemi, S. and Nazemnezhad, R. (2013), "An analytical study on the nonlinear free vibration of functionally graded nanobeams incorporating surface effects", Compos. Part B: Eng., 52, 199-206. https://doi.org/10.1016/j.compositesb.2013.04.023
  36. Hosseini-Hashemi, S., Fadaee, M. and Atashipour, S.R. (2011), "A new exact analytical approach for free vibration of Reissner- Mindlin functionally graded rectangular plates", Int. J. Mech. Sci., 53(1), 11-22. https://doi.org/10.1016/j.ijmecsci.2010.10.002
  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. 10.22059/JCAMECH.2018.259334.289
  40. Huang, Y., Yang, L.-E. and Luo, Q.-Z. (2013), "Free vibration of axially functionally graded Timoshenko beams with non-uniform cross-section", Compos. Part B: Eng., 45(1), 1493-1498. https://doi.org/10.1016/j.compositesb.2012.09.015
  41. Jabbari, M., Nejad, M.Z. and Ghannad, M. (2015), "Thermoelastic analysis of axially functionally graded rotating thick cylindrical pressure vessels with variable thickness under mechanical loading", Int. J. Eng. Sci., 96, 1-18. https://doi.org/10.1016/j.ijengsci.2015.07.005
  42. Jabbari, M., Nejad, M.Z. and Ghannad, M. (2016), "Thermoelastic analysis of axially functionally graded rotating thick truncated conical shells with varying thickness", Compos. Part B: Eng., 96, 20-34. https://doi.org/10.1016/j.compositesb.2016.04.026
  43. Javidi, R., Haghshenas Gorgani, H. and Mahdavi Adeli, M. (2019), "Size-dependent on vibration and flexural sensitivity of atomic force microscope", J. Computat. Appl. Mech., 50(1), 191-196. https://doi.org/10.22059/JCAMECH.2018.250335.233
  44. Karami, B., Janghorban, M. and Tounsi, A. (2018), "Variational approach for wave dispersion in anisotropic doubly-curved nanoshells based on a new nonlocal strain gradient higher order shell theory", Thin-Wall. Struct., 129, 251-264. https://doi.org/10.1016/j.tws.2018.02.025
  45. Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2019), "On the resonance of functionally graded nanoplates using bi-Helmholtz nonlocal strain gradient theory", Int. J. Eng. Sci., 144, 103143. https://doi.org/10.1016/j.ijengsci.2019.103143
  46. Kashkoli, M.D. and Nejad, M.Z. (2018), "Time-dependent creep analysis and life assessment of 304 L austenitic stainless steel thick pressurized truncated conical shells", Steel Compos. Struct., Int. J., 28(3), 349-362. https://doi.org/10.12989/scs.2018.28.3.349
  47. Kashkoli, M.D., Tahan, K.N. and Nejad, M.Z. (2017), "Timedependent thermomechanical creep behavior of FGM thick hollow cylindrical shells under non-uniform internal pressure", Int. J. Appl. Mech., 9(6), 1750086. https://doi.org/10.1142/S1758825117500867
  48. Kashkoli, M.D., Tahan, K.N. and Nejad, M.Z. (2018), "Thermomechanical creep analysis of FGM thick cylindrical pressure vessels with variable thickness", Int. J. Appl. Mech., 10(1), 1850008. https://doi.org/10.1142/S1758825118500084
  49. Kolahchi, R., Bidgoli, A.M.M. and Heydari, M.M. (2015), "Sizedependent bending analysis of FGM nano-sinusoidal plates resting on orthotropic elastic medium", Struct. Eng. Mech., Int. J., 55(5), 1001-1014. https://doi.org/10.12989/sem.2015.55.5.1001
  50. Kolahchi, R., Zarei, M.S., Hajmohammad, M.H. and Naddaf Oskouei, A. (2017), "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
  51. Kong, S., Zhou, S., Nie, Z. and Wang, K. (2009), "Static and dynamic analysis of micro beams based on strain gradient elasticity theory", Int. J. Eng. Sci., 47(4), 487-498. https://doi.org/10.1016/j.ijengsci.2008.08.008
  52. Lam, D.C., Yang, F., Chong, A., Wang, J. and Tong, P. (2003), "Experiments and theory in strain gradient elasticity", J. Mech. Phys. Solids, 51(8), 1477-1508. https://doi.org/10.1016/S0022-5096(03)00053-X
  53. Lazopoulos, K. (2004), "On the gradient strain elasticity theory of plates", Eur. J. Mech-A/Solids, 23(5), 843-852. https://doi.org/10.1016/j.euromechsol.2004.04.005
  54. Lei, J., He, Y., Guo, S., Li, Z. and Liu, D. (2016), "Size-dependent vibration of nickel cantilever microbeams: experiment and gradient elasticity", Aip Advances, 6(10), 105202. https://doi.org/10.1063/1.4964660
  55. Li, L. and Hu, Y. (2015), "Buckling analysis of size-dependent nonlinear beams based on a nonlocal strain gradient theory", Int. J. Eng. Sci., 97, 84-94. https://doi.org/10.1016/j.ijengsci.2015.08.013
  56. Li, L. and Hu, Y. (2016a), "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
  57. Li, L. and Hu, Y. (2016b), "Wave propagation in fluid-conveying viscoelastic carbon nanotubes based on nonlocal strain gradient theory", Computat. Mater. Sci., 112, 282-288. https://doi.org/10.1016/j.commatsci.2015.10.044
  58. Li, L. and Hu, Y. (2017a), "Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects", Int. J. Mech. Sci., 120, 159-170. https://doi.org/10.1016/j.ijmecsci.2016.11.025
  59. Li, L. and Hu, Y. (2017b), "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
  60. Li, X. and Luo, Y. (2017), "Size-dependent postbuckling of piezoelectric microbeams based on a modified couple stress theory", Int. J. Appl. Mech., 9(4), 1750053. https://doi.org/10.1142/S1758825117500533
  61. Li, A., Zhou, S., Zhou, S. and Wang, B. (2014), "A size-dependent model for bi-layered Kirchhoff micro-plate based on strain gradient elasticity theory", Compos. Struct., 113, 272-280. https://doi.org/10.1016/j.compstruct.2014.03.028
  62. 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
  63. 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
  64. Li, X., Li, L., Hu, Y., Ding, Z. and Deng, W. (2017), "Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory", Compos. Struct., 165, 250-265. https://doi.org/10.1016/j.compstruct.2017.01.032
  65. Lu, C., Chen, W., Xu, R. and Lim, C.W. (2008), "Semi-analytical elasticity solutions for bi-directional functionally graded beams", Int. J. Solids Struct., 45(1), 258-275. https://doi.org/10.1016/j.ijsolstr.2007.07.018
  66. Lu, C.F., Lim, C.W. and Chen, W.Q. (2009), "Semi-analytical analysis for multi-directional functionally graded plates: 3-D elasticity solutions", Int. J. Numer. Methods Eng., 79(1), 25-44. https://doi.org/10.1002/nme.2555
  67. Luttge, R. (2016), Chapter 4 - Nanotechnology, William Andrew Publishing.
  68. Malekzadeh, P., Golbahar Haghighi, M.R. and Shojaee, M. (2014), "Nonlinear free vibration of skew nanoplates with surface and small scale effects", Thin-Wall. Struct., 78, 48-56. https://doi.org/10.1016/j.tws.2013.10.027
  69. Mazarei, Z., Nejad, M.Z. and Hadi, A. (2016), "Thermo-elastoplastic analysis of thick-walled spherical pressure vessels made of functionally graded materials", Int. J. Appl. Mech., 8(4), 1650054. https://doi.org/10.1142/S175882511650054X
  70. Mehralian, F. and Tadi Beni, Y. (2018), "Buckling of bimorph functionally graded piezoelectric cylindrical nanoshell", Proceedings of the Institution of Mechanical Engineers, Part C: J. Mech. Eng. Sci., 232(19), 3538-3550. https://doi.org/10.1177/0954406217738033
  71. Mindlin, R.D. (1965), "Second gradient of strain and surfacetension in linear elasticity", Int. J. Solids Struct., 1(4), 417-438. https://doi.org/10.1016/0020-7683(65)90006-5
  72. Mirsalehi, M., Azhari, M. and Amoushahi, H. (2017), "Buckling and free vibration of the FGM thin micro-plate based on the modified strain gradient theory and the spline finite strip method", Eur. J. Mech.-A/Solids, 61, 1-13. https://doi.org/10.1016/j.euromechsol.2016.08.008
  73. ohammadi, M., Hosseini, M., Shishesaz, M., Hadi, A. and Rastgoo, A. (2019), "Primary and secondary resonance analysis of porous functionally graded nanobeam resting on a nonlinear foundation subjected to mechanical and electrical loads", Eur. J. Mech.-A/Solids, 77, 103793. https://doi.org/10.1016/j.euromechsol.2019.05.008
  74. Nejad, M.Z. and Fatehi, P. (2015), "Exact elasto-plastic analysis of rotating thick-walled cylindrical pressure vessels made of functionally graded materials", Int. J. Eng. Sci., 86, 26-43. https://doi.org/10.1016/j.ijengsci.2014.10.002
  75. 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
  76. 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
  77. Nejad, M.Z. and Kashkoli, M.D. (2014), "Time-dependent thermocreep analysis of rotating FGM thick-walled cylindrical pressure vessels under heat flux", Int. J. Eng. Sci., 82, 222-237. https://doi.org/10.1016/j.ijengsci.2014.06.006
  78. Nejad, M.Z. and Rahimi, G. (2009), "Deformations and stresses in rotating FGM pressurized thick hollow cylinder under thermal load", Sci. Res. Essays, 4(3), 131-140.
  79. Nejad, M.Z. and Rahimi, G.H. (2010), "Elastic analysis of FGM rotating cylindrical pressure vessels", J. Chinese Inst. Engr., 33(4), 525-530. https://doi.org/10.1080/02533839.2010.9671640
  80. Nejad, M.Z., Rahimi, G. and Ghannad, M. (2009), "Set of field equations for thick shell of revolution made of functionally graded materials in curvilinear coordinate system", Mech., 77(3), 18-26.
  81. Nejad, M.Z., Rastgoo, A. and Hadi, A. (2014a), "Effect of exponentially-varying properties on displacements and stresses in pressurized functionally graded thick spherical shells with using iterative technique", J. Solid Mech., 6(4), 366-377.
  82. Nejad, M.Z., Rastgoo, A. and Hadi, A. (2014b), "Exact elastoplastic analysis of rotating disks made of functionally graded materials", Int. J. Eng. Sci., 85, 47-57. https://doi.org/10.1016/j.ijengsci.2014.07.009
  83. Nejad, M.Z., Jabbari, M. and Ghannad, M. (2015a), "Elastic analysis of axially functionally graded rotating thick cylinder with variable thickness under non-uniform arbitrarily pressure loading", Int. J. Eng. Sci., 89, 86-99. https://doi.org/10.1016/j.ijengsci.2014.12.004
  84. Nejad, M.Z., Jabbari, M. and Ghannad, M. (2015b), "Elastic analysis of FGM rotating thick truncated conical shells with axially-varying properties under non-uniform pressure loading", Compos. Struct., 122, 561-569. https://doi.org/10.1016/j.compstruct.2014.12.028
  85. 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
  86. Nejad, M.Z., Jabbari, M. and Ghannad, M. (2017a), "A general disk form formulation for thermo-elastic analysis of functionally graded thick shells of revolution with arbitrary curvature and variable thickness", Acta Mechanica, 228(1), 215-231. https://doi.org/10.1007/s00707-016-1709-z
  87. Nejad, M.Z., Jabbari, M. and Hadi, A. (2017b), "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
  88. Nejad, M.Z., Hadi, A. and Farajpour, A. (2017c), "Consistent couple-stress theory for free vibration analysis of Euler-Bernoulli nano-beams made of arbitrary bi-directional functionally graded materials", Struct. Eng. Mech., Int. J., 63(2), 161-169. https://doi.org/10.12989/sem.2017.63.2.161
  89. Nejad, M.Z., Taghizadeh, T., Mehrabadi, S.J. and Herasati, S. (2017d), "Elastic analysis of carbon nanotube-reinforced composite plates with piezoelectric layers using shear deformation theory", Int. J. Appl. Mech., 9(1), 1750011. https://doi.org/10.1142/S1758825117500119
  90. Nejad, M.Z., Alamzadeh, N. and Hadi, A. (2018a), "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
  91. Nejad, M.Z., Hadi, A., Omidvari, A. and Rastgoo, A. (2018b), "Bending analysis of bi-directional functionally graded Euler-Bernoulli nano-beams using integral form of Eringen's non-local elasticity theory", Struct. Eng. Mech., Int. J., 67(4), 417-425. https://doi.org/10.12989/sem.2018.67.4.417
  92. Nguyen, T.-T. and Lee, J. (2018), "Interactive geometric interpretation and static analysis of thin-walled bi-directional functionally graded beams", Compos. Struct., 191, 1-11. https://doi.org/10.1016/j.compstruct.2018.01.064
  93. Nie, G. and Zhong, Z. (2010), "Dynamic analysis of multidirectional functionally graded annular plates", Appl. Mathe. Model., 34(3), 608-616. https://doi.org/10.1016/j.apm.2009.06.009
  94. Norouzzadeh, A. and Ansari, R. (2018), "Isogeometric vibration analysis of functionally graded nanoplates with the consideration of nonlocal and surface effects", Thin-Wall. Struct., 127, 354-372. https://doi.org/10.1016/j.tws.2017.11.040
  95. Papargyri-Beskou, S., Giannakopoulos, A. and Beskos, D. (2010), "Variational analysis of gradient elastic flexural plates under static loading", Int. J. Solids Struct., 47(20), 2755-2766. https://doi.org/10.1016/j.ijsolstr.2010.06.003
  96. Preethi, K., Raghu, P., Rajagopal, A. and Reddy, J. (2018), "Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam", Mech. Adv. Mater. Struct., 25(5), 439-450. https://doi.org/10.1080/15376494.2016.1278062
  97. Pydah, A. and Batra, R. (2017), "Shear deformation theory using logarithmic function for thick circular beams and analytical solution for bi-directional functionally graded circular beams", Compos. Struct., 172, 45-60. https://doi.org/10.1016/j.compstruct.2017.03.072
  98. Pydah, A. and Sabale, A. (2017), "Static analysis of bi-directional functionally graded curved beams", Compos. Struct., 160, 867-876. https://doi.org/10.1016/j.compstruct.2016.10.120
  99. Rajasekaran, S. and Khaniki, H.B. (2018), "Free vibration analysis of bi-directional functionally graded single/multi-cracked beams", Int. J. Mech. Sci., 144, 341-356. https://doi.org/10.1016/j.ijmecsci.2018.06.004
  100. Ramsden, J.J. (2016), Chapter 1 - What is nanotechnology?, William Andrew Publishing, Oxford, UK.
  101. Reddy, J.N. (2006), Theory and Analysis of Elastic Plates and Shells, CRC press.
  102. Rohani Rad, E. and Farajpour, M.R. (2019), "Influence of taxol and CNTs on the stability analysis of protein microtubules", J. Computat. Appl. Mech., 50(1), 140-147. 10.22059/JCAMECH.2019.277874.369
  103. Seyyed Nosrati, A., Parvizi, A., Afzal, S.A. and Alimirzaloo, V. (2019), "Elasto-plastic solution for thick-walled spherical vessels with an inner FGM layer", J. Computat. Appl. Mech., 50(1), 1-13. 10.22059/JCAMECH.2017.239277.173
  104. She, G.-L., Yuan, F.-G., Ren, Y.-R. and Xiao, W.-S. (2017), "On buckling and postbuckling behavior of nanotubes", Int. J. Eng. Sci., 121, 130-142. https://doi.org/10.1016/j.ijengsci.2017.09.005
  105. She, G.-L., Yuan, F.-G. and Ren, Y.-R. (2018a), "On wave propagation of porous nanotubes", Int. J. Eng. Sci., 130, 62-74. https://doi.org/10.1016/j.ijengsci.2018.05.002
  106. She, G.-L., Yuan, F.-G., Ren, Y.-R., Liu, H.-B. and Xiao, W.-S. (2018b), "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
  107. She, G.-L., Ren, Y.-R., Yuan, F.-G. and Xiao, W.-S. (2018c), "On vibrations of porous nanotubes", Int. J. Eng. Sci., 125, 23-35. https://doi.org/10.1016/j.ijengsci.2017.12.009
  108. She, G.-L., Yan, K.-M., Zhang, Y.-L., Liu, H.-B. and Ren, Y.-R. (2018d), "Wave propagation of functionally graded porous nanobeams based on non-local strain gradient theory", Eur. Phys. J. Plus, 133(9), 368. https://doi.org/10.1140/epjp/i2018-12196-5
  109. Shenas, A.G. and Malekzadeh, P. (2016), "Free vibration of functionally graded quadrilateral microplates in thermal environment", Thin-Wall. Struct., 106, 294-315. https://doi.org/10.1016/j.tws.2016.05.001
  110. Shenas, A.G., Malekzadeh, P. and Ziaee, S. (2017), "Thermal buckling of rotating pre-twisted functionally graded microbeams with temperature-dependent material properties", Acta Mechanica, 228(3), 1115-1133. https://doi.org/10.1007/s00707-016-1759-2
  111. 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
  112. Simsek, M. (2012), "Nonlocal effects in the free longitudinal vibration of axially functionally graded tapered nanorods", Computat. Mater. Sci., 61, 257-265. https://doi.org/10.1016/j.commatsci.2012.04.001
  113. Simsek, M. (2015), "Bi-directional functionally graded materials (BDFGMs) for free and forced vibration of Timoshenko beams with various boundary conditions", Compos. Struct., 133, 968-978. https://doi.org/10.1016/j.compstruct.2015.08.021
  114. Steinberg, M.A. (1986), "Materials for aerospace", Scientific American, 255(4), 66-73. https://doi.org/10.1038/scientificamerican1086-66
  115. Tadi Beni, Y. (2016), "Size-dependent electromechanical bending, buckling, and free vibration analysis of functionally graded piezoelectric nanobeams", J. Intel. Mater. Syst. Struct., 27(16), 2199-2215. https://doi.org/10.1177/1045389X15624798
  116. Thai, H.-T. and Choi, D.-H. (2013), "Size-dependent functionally graded Kirchhoff and Mindlin plate models based on a modified couple stress theory", Compos. Struct., 95, 142-153. https://doi.org/10.1016/j.compstruct.2012.08.023
  117. Thai, S., Thai, H.-T., Vo, T.P. and Patel, V.I. (2017), "Sizedependant behaviour of functionally graded microplates based on the modified strain gradient elasticity theory and isogeometric analysis", Comput. Struct., 190, 219-241. https://doi.org/10.1016/j.compstruc.2017.05.014
  118. Trinh, L.C., Vo, T.P., Thai, H.-T. and Nguyen, T.-K. (2018), "Sizedependent vibration of bi-directional functionally graded microbeams with arbitrary boundary conditions", Compos. Part B: Eng., 134, 225-245. https://doi.org/10.1016/j.compositesb.2017.09.054
  119. Udupa, G., Rao, S.S. and Gangadharan, K. (2014), "Functionally graded composite materials: an overview", Procedia Materials Science, 5, 1291-1299. https://doi.org/10.1016/j.mspro.2014.07.442
  120. Van Do, T., Nguyen, D.K., Duc, N.D., Doan, D.H. and Bui, T.Q. (2017), "Analysis of bi-directional functionally graded plates by FEM and a new third-order shear deformation plate theory", Thin-Wall. Struct., 119, 687-699. https://doi.org/10.1016/j.tws.2017.07.022
  121. Wang, B., Zhao, J. and Zhou, S. (2010), "A micro scale Timoshenko beam model based on strain gradient elasticity theory", Eur. J. Mech.-A/Solids, 29(4), 591-599. https://doi.org/10.1016/j.euromechsol.2009.12.005
  122. Wang, B., Zhou, S., Zhao, J. and Chen, X. (2011), "A sizedependent Kirchhoff micro-plate model based on strain gradient elasticity theory", Eur. J. Mech.-A/Solids, 30(4), 517-524. https://doi.org/10.1016/j.euromechsol.2011.04.001
  123. Wang, B., Huang, S., Zhao, J. and Zhou, S. (2016a), "Reconsiderations on boundary conditions of Kirchhoff microplate model based on a strain gradient elasticity theory", Appl. Mathe. Model., 40(15-16), 7303-7317. https://doi.org/10.1016/j.apm.2016.03.014
  124. Wang, Z.-h., Wang, X.-h., Xu, G.-d., Cheng, S. and Zeng, T. (2016b), "Free vibration of two-directional functionally graded beams", Compos. Struct., 135, 191-198. https://doi.org/10.1016/j.compstruct.2015.09.013
  125. Yang, F., Chong, A., Lam, D.C.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  126. Yang, T., Tang, Y., Li, Q. and Yang, X.-D. (2018), "Nonlinear bending, buckling and vibration of bi-directional functionally graded nanobeams", Compos. Struct., 204, 313-319. https://doi.org/10.1016/j.compstruct.2018.07.045
  127. Zarezadeh, E., Hosseini, V. and Hadi, A. (2019), "Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory", Mech. Based Des. Struct. Mach., 1-16. https://doi.org/10.1080/15397734.2019.1642766
  128. Zargaripoor, A. and Bahrami, A. (2018), "Free vibration and buckling analysis of third-order shear deformation plate theory using exact wave propagation approach", J. Computat. Appl. Mech., 49(1), 102-124. https://doi.org/10.22059/JCAMECH.2018.249468.227
  129. Zargaripoor, A., Daneshmehr, A., Isaac Hosseini, I. and Rajabpoor, A. (2018), "Free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory using finite element method", J. Computat. Appl. Mech., 49(1), 86-101. https://doi.org/10.22059/JCAMECH.2018.248906.223
  130. Zeighampour, H. and Tadi Beni, Y. (2014a), "Analysis of conical shells in the framework of coupled stresses theory", Int. J. Eng. Sci., 81, 107-122. https://doi.org/10.1016/j.ijengsci.2014.04.008
  131. Zeighampour, H. and Tadi Beni, Y. (2014b), "Cylindrical thin-shell model based on modified strain gradient theory", Int. J. Eng. Sci., 78, 27-47. https://doi.org/10.1016/j.ijengsci.2014.01.004
  132. Zeighampour, H. and Tadi Beni, Y. (2015), "Free vibration analysis of axially functionally graded nanobeam with radius varies along the length based on strain gradient theory", Appl. Mathe. Model., 39(18), 5354-5369. https://doi.org/10.1016/j.apm.2015.01.015
  133. Zeighampour, H., Tadi Beni, Y. and Botshekanan Dehkordi, M. (2018), "Wave propagation in viscoelastic thin cylindrical nanoshell resting on a visco-Pasternak foundation based on nonlocal strain gradient theory", Thin-Wall. Struct., 122, 378-386. https://doi.org/10.1016/j.tws.2017.10.037
  134. Zhang, Y.Q., Liu, G.R. and Wang, J.S. (2004), "Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression", Phys. Rev. B, 70(20), 205430. https://doi.org/10.1103/PhysRevB.70.205430
  135. Zhao, L., Zhu, J. and Wen, X.D. (2016), "Exact analysis of bidirectional functionally graded beams with arbitrary boundary conditions via the symplectic approach", Struct. Eng. Mech., Int. J., 59(1), 101-122. https://doi.org/10.12989/sem.2016.59.1.101
  136. Ziegler, T. and Kraft, T. (2014), "Functionally graded materials with a soft surface for improved indentation resistance: Layout and corresponding design principles", Computat. Mater. Sci., 86, 88-92. https://doi.org/10.1016/j.commatsci.2014.01.032

Cited by

  1. Torsional vibration of irregular single-walled carbon nanotube incorporating compressive initial stress effects vol.37, 2021, https://doi.org/10.1093/jom/ufab002
  2. Frequency and thermal buckling information of laminated composite doubly curved open nanoshell vol.10, pp.1, 2020, https://doi.org/10.12989/anr.2021.10.1.001
  3. Size dependent vibration of embedded functionally graded nanoplate in hygrothermal environment by Rayleigh-Ritz method vol.10, pp.1, 2020, https://doi.org/10.12989/anr.2021.10.1.025
  4. Thermoelastoplastic response of FGM linearly hardening rotating thick cylindrical pressure vessels vol.38, pp.2, 2021, https://doi.org/10.12989/scs.2021.38.2.189
  5. Investigation on the dynamic response of porous FGM beams resting on variable foundation using a new higher order shear deformation theory vol.39, pp.1, 2020, https://doi.org/10.12989/scs.2021.39.1.095
  6. Free vibration analysis of open-cell FG porous beams: analytical, numerical and ANN approaches vol.40, pp.2, 2021, https://doi.org/10.12989/scs.2021.40.2.157
  7. Computer modeling for frequency performance of viscoelastic magneto-electro-elastic annular micro/nanosystem via adaptive tuned deep learning neural network optimization vol.11, pp.2, 2020, https://doi.org/10.12989/anr.2021.11.2.203