Advances in nano research

  • 제7권4호
  • /
  • Pages.249-263
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  • 2019
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  • 2287-237X(pISSN)
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  • 2287-2388(eISSN)
테크노프레스 (Techno-Press)
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DOI QR코드

DOI QR Code

Thermo-mechanical vibration analysis of curved imperfect nano-beams based on nonlocal strain gradient theory

  • Ebrahimi, Farzad (Mechanical Engineering Department, Faculty of Engineering, Imam Khomeini International University) ;
  • Daman, Mohsen (Mechanical Engineering Department, Faculty of Engineering, Imam Khomeini International University) ;
  • Mahesh, Vinyas (Department of Mechanical Engineering, Nitte Meenakshi Institute of Technology)
  • 투고 : 2018.08.05
  • 심사 : 2019.04.25
  • 발행 : 2019.07.25
⟨ 이전 논문 다음 논문 ⟩

초록

In the current paper, an exact solution method is carried out for analyzing the thermo-mechanical vibration of curved FG nano-beams subjected to uniform thermal environmental conditions, by considering porosity distribution via nonlocal strain gradient beam theory for the first time. Nonlocal strain gradient elasticity theory is adopted to consider the size effects in which the stress for not only the nonlocal stress field but also the strain gradients stress field is considered. It is perceived that during manufacturing of functionally graded materials (FGMs) porosities and micro-voids can be occurred inside the material. Material properties of curved porous FG nanobeam are assumed to be temperature-dependent and are supposed to vary through the thickness direction of beam which modeled via modified power-law rule. Since variation of pores along the thickness direction influences the mechanical and physical properties, porosity play a key role in the mechanical response of curved FG nano-structures. The governing equations and related boundary condition of curved porous FG nanobeam under temperature field are derived via the energy method based on Timoshenko beam theory. An analytical Navier solution procedure is utilized to achieve the natural frequencies of porous FG curved nanobeam supposed to thermal loading. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality parameter, porosity volume fractions, thermal effect, gradient index, opening angle and aspect ratio on the natural frequency of curved FG porous nanobeam are successfully discussed. It is concluded that these parameters play key roles on the dynamic behavior of porous FG curved nanobeam. Presented numerical results can serve as benchmarks for future analyses of curve FG nanobeam with porosity phases.

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참고문헌

  1. Agwa, M.A. and Eltaher, M.A. (2016), "Vibration of a carbyne nanomechanical mass sensor with surface effect", Appl. Phys. A, 122(4), 1-8. https://doi.org/10.1007/s00339-016-9934-9
  2. Akbas, S.D. (2018), "Forced vibration analysis of cracked functionally graded microbeams", Adv. Nano. Res., Int. J., 6(1), 39-55. https://doi.org/10.12989/anr.2018.6.1.039
  3. Altabey, W.A. (2017), "An exact solution for mechanical behavior of BFRP Nano-thin films embedded in NEMS", Adv. Nano Res., Int. J., 5(4), 337-357. https://doi.org/10.12989/anr.2017.5.4.337
  4. Ansari, R., Gholami, R. and Sahmani, S. (2013), "Size-dependent vibration of functionally graded curved microbeams based on the modified strain gradient elasticity theory", Arch. Appl. Mech. 83(10), 1439-1449. https://doi.org/10.1007/s00419-013-0756-3
  5. Attia, M.A., Eltaher, M.A., Soliman, A., Abdelrahman, A. and Alshorbagy, A.E. (2018), "Thermoelastic Crack Analysis in Functinally Graded Piplines Conveying Natural Gas by a FEM", Int. J. Appl. Mech., 10(4), 1850036. https://doi.org/10.1142/S1758825118500369
  6. Barati, M.R. (2017), "Nonlocal-strain gradient forced vibration analysis of metal foam nanoplates with uniform and graded porosities", Adv. Nano. Res., Int. J., 5(4), 393-414. https://doi.org/10.12989/anr.2017.5.4.393
  7. 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. http://dx.doi.org/10.12989/anr.2015.3.1.029
  8. 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., Int. J., 6(2), 147-162. https://doi.org/10.12989/anr.2018.6.2.147
  9. Castrucci, P. (2014), "Carbon nanotube/silicon hybrid heterojunctions for photovoltaic devices", Adv. Nano. Res., Int. J., 2(1), 23-56. http://dx.doi.org/10.12989/anr.2014.2.1.023
  10. Chemi, A., Heireche, H., Zidour, M., Rakrak, K. and Bousahla, A.A. (2015), "Critical buckling load of chiral double-walled carbon nanotube using non-local theory elasticity", Adv. Nano. Res., Int. J., 3(4), 193-206. http://dx.doi.org/10.12989/anr.2015.3.4.193
  11. Daulton, T.L., Bondi, K.S. and Kelton, K.F. (2010), "Nanobeam diffraction fluctuation electron microscopy technique for structural characterization of disordered materials-Application to Al 88- x Y 7 Fe 5 Ti x metallic glasses", Ultramicroscopy, 110(10), 1279-1289. https://doi.org/10.1016/j.ultramic.2010.05.010
  12. Ebrahimi, F. and Barati, M.R. (2016a), "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. https://doi.org/10.1007/s40430-016-0551-5
  13. Ebrahimi, F. and Barati, M.R. (2016b), "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
  14. Ebrahimi, F. and Barati, M.R. (2016c), "A nonlocal higher-order refined magneto-electro-viscoelastic beam model for dynamic analysis of smart nanostructures", Int. J. Eng. Science, 107, 183-196. https://doi.org/10.1016/j.ijengsci.2016.08.001
  15. Ebrahimi, F. and Barati, M.R. (2016d), "Size-dependent dynamic modeling of inhomogeneous curved nanobeams embedded in elastic medium based on nonlocal strain gradient theory", Proceedings of the Institution of Mechanical Engineers, Part C: J. Mech. Eng. Sci., 0954406216668912. https://doi.org/10.1177/0954406216668912
  16. Ebrahimi, F. and Barati, M.R. (2016e), "Thermal environment effects on wave dispersion behavior of inhomogeneous strain gradient nanobeams based on higher order refined beam theory", J. Thermal Stress, 39(12), 1560-1570. https://doi.org/10.1080/01495739.2016.1219243
  17. 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
  18. 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
  19. Ebrahimi, F. and Daman, M. (2016a), "Investigating Surface Effects on Thermomechanical Behavior of Embedded Circular Curved Nanosize Beams", J. Eng., 2016. http://dx.doi.org/10.1155/2016/9848343
  20. Ebrahimi, F. and Daman, M. (2016b), "An Investigation of Radial Vibration Modes of Embedded Double-Curved-Nanobeam-Systems", Cankaya Univ. J. Sci. Eng., 13, 58-79.
  21. Ebrahimi, F. and Daman, M. (2017a), "Nonlocal thermoelectro-mechanical vibration analysis of smart curved FG piezoelectric Timoshenko nanobeam", Smart Struct. Syst., Int. J., 20(3), 351-368. http://dx.doi.org/10.12989/sss.2017.20.3.351
  22. Ebrahimi, F. and Daman, M. (2017b), "Dynamic characteristics of curved inhomogeneous nonlocal porous beams in thermal environment", Struct. Eng. Mech., Int. J., 64(1), 121-133. http://dx.doi.org/10.12989/sem.2017.64.1.121
  23. Ebrahimi, F. and Jafari, A. (2016), "Thermo-mechanical vibration analysis of temperature-dependent porous FG beams based on Timoshenko beam theory", Struct. Eng. Mech., Int. J., 59(2), 343-371. http://dx.doi.org/10.12989/sem.2016.59.2.343
  24. Ebrahimi, F. and Salari, E. (2015), "Size-dependent free flexural vibrational behavior of functionally graded nanobeams using semi-analytical differential transform method", Compos. Part B: Eng., 79, 156-169. https://doi.org/10.1016/j.compositesb.2015.04.010
  25. 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
  26. Ebrahimi, F., Ghasemi, F. and Salari, E. (2016), "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
  27. Ebrahimi, F., Daman, M. and Jafari, A. (2017), "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. http://dx.doi.org/10.12989/sss.2017.20.6.709
  28. 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. https://doi.org/10.12989/anr.2016.4.2.085
  29. Eltaher, M.A., Emam, S.A. and Mahmoud, F.F. (2012), "Free vibration analysis of functionally graded size-dependent nanobeams", Appl. Math. Computat., 218(14), 7406-7420. https://doi.org/10.1016/j.amc.2011.12.090
  30. Eltaher, M.A., Alshorbagy, A.E. and Mahmoud, F.F. (2013), "Vibration analysis of Euler-Bernoulli nanobeams by using finite element method", Appl. Math. Model., 37(7), 4787-4797. https://doi.org/10.1016/j.amc.2011.12.090
  31. Eltaher, M.A., Abdelrahman, A.A., Al-Nabawy, A., Khater, M. and Mansour, A. (2014a), "Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position", Appl. Math. Computat., 235, 512-529. https://doi.org/10.1016/j.amc.2014.03.028
  32. Eltaher, M.A., Hamed, M.A., Sadoun, A.M. and Mansour, A. (2014b), "Mechanical analysis of higher order gradient nanobeams", Appl. Math. Computat., 229, 260-272. https://doi.org/10.1016/j.amc.2013.12.076
  33. Eltaher, M.A., El-Borgi S. and Reddy J.N. (2016a), "Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs", Compos. Struct., 153, 902-913. https://doi.org/10.1016/j.compstruct.2016.07.013
  34. Eltaher, M.A., Khater, M.E. and Emam, S.A. (2016b), "A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams", Appl. Math. Model., 40(5-6), 4109-4128. https://doi.org/10.1016/j.apm.2015.11.026
  35. Eltaher, M.A., Attia, M.A., Soliman, A.E. and Alshorbagy, A.E. (2018a), "Analysis of crack occurs under unsteady pressure and temperature in a natural gas facility by applying FGM", Struct. Eng. Mech., Int. J., 66(1), 97-111. https://doi.org/10.12989/sem.2018.66.1.097
  36. Eltaher, M.A., Fatema-Alzahraa Omar, Abdalla W.S. and E.H. Gad (2018b), "Bending and vibrational behaviors of piezoelectric nonlocal nanobeam including surface elasticity", Waves Random Complex Media, 29(2), 264-280. https://doi.org/10.1080/17455030.2018.1429693
  37. Eltaher, M.A., Agwa, M. and Kabeel, A. (2018c), "Vibration analysis of material size-dependent CNTs using energy equivalent model", J. Appl. Computat. Mech., 4(2), 75-86. https://doi.org/10.22055/JACM.2017.22579.1136
  38. Eltaher, M.A., Fouda, N., El-midany, T. and Sadoun, A.M. (2018d), "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
  39. Eringen, A.C. (1972a), "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
  40. Eringen, A.C. (1972b), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10(1), 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  41. 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
  42. Eringen, A.C. (2002), Nonlocal Continuum Field Theories, Springer Science & Business Media.
  43. Fleck, N. and Hutchinson, J. (1993), "A phenomenological theory for strain gradient effects in plasticity", J. Mech. Phys. Solids, 41(12), 1825-1857. https://doi.org/10.1016/0022-5096(93)90072-N
  44. Hamed, M.A. and Eltaher, M.A., Sadoun, A.M. and Almitani, K.H. (2016), "Free vibration of symmetric and sigmoid functionally graded nanobeams", Appl. Phys. A, 122(9), 829. https://doi.org/10.1007/s00339-016-0324-0
  45. Hosseini, S. and Rahmani, O. (2016), "Free vibration of shallow and deep curved FG nanobeam via nonlocal Timoshenko curved beam model", Appl. Phys. A, 122(3), 1-11. https://doi.org/10.1007/s00339-016-9696-4
  46. Hu, B., Ding, Y., Chen, W., Kulkarni, D., Shen, Y., Tsukruk, V.V. and Wang, Z.L. (2010), "External-Strain Induced Insulating Phase Transition in VO2 Nanobeam and Its Application as Flexible Strain Sensor", Adv. Mater., 22(45), 5134-5139. https://doi.org/10.1002/adma.201002868
  47. Kananipour, H., Ahmadi, M. and Chavoshi, H. (2014), "Application of nonlocal elasticity and DQM to dynamic analysis of curved nanobeams", Latin Am. J. Solids Struct., 11(5), 848-853. http://dx.doi.org/10.1590/S1679-78252014000500007
  48. Koizumi, M. and Niino, M. (1995), "Overview of FGM Research in Japan", Mrs Bulletin, 20(1), 19-21. https://doi.org/10.1557/S0883769400048867
  49. Kumar, B.R. (2018), "Investigation on mechanical vibration of double-walled carbon nanotubes with inter-tube Van der waals forces", Adv. Nano. Res., Int. J., 6(2), 135-145. https://doi.org/10.12989/anr.2018.6.2.135
  50. Lam, D.C., Yang, F., Chong, A.C.M., 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
  51. 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
  52. Li, L. and Hu, Y. (2016), "Nonlinear bending and free vibration analysis of nonlocal strain gradient beams made of functionally graded material", Int. J. Eng. Sci., 102, 77-97. https://doi.org/10.1016/j.ijengsci.2016.07.011
  53. 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
  54. Lim, C.W., Zhang, G. and Reddy, J.N. (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
  55. Malekzadeh, P., Haghighi, M.G. and Atashi, M.M. (2010), "Outof-plane free vibration of functionally graded circular curved beams in thermal environment", Compos. Struct., 92(2), 541-552. https://doi.org/10.1016/j.compstruct.2009.08.040
  56. Marani, R. and Perri, A.G. (2017), "An approach to model the temperature effects on IV characteristics of CNTFETs", Adv. Nano. Res., Int. J., 5(1), 61-67. https://doi.org/10.12989/anr.2017.5.1.061
  57. Mechab, I., Mechab, B., Benaissa, S., Serier, B. and Bouiadjra, B.B. (2016), "Free vibration analysis of FGM nanoplate with porosities resting on Winkler Pasternak elastic foundations based on two-variable refined plate theories", J. Brazil. Soc. Mech. Sci. Eng., 38(8), 2193-2211. https://doi.org/10.1007/s40430-015-0482-6
  58. Miyamoto, Y., Kaysser, W.A., Rabin, B.H., Kawasaki, A. and Ford, R.G. eds. (2013), Functionally Graded Materials: Design, Processing and Applications, Springer Science & Business Media.
  59. Mohamed, N., Eltaher, M.A., Mohamed, S.A. and Seddek, L.F. (2018), "Numerical analysis of nonlinear free and forced vibrations of buckled curved beams resting on nonlinear elastic foundations", Int. J. Non-Linear Mech., 101, 157-173. https://doi.org/10.1016/j.ijnonlinmec.2018.02.014
  60. Mortensen, A. and Suresh, S. (2013), "Functionally graded metals and metal-ceramic composites: Part 1 Processing", Int. Mater. Rev., 40(6), 239-265. https://doi.org/10.1179/imr.1995.40.6.239
  61. Murmu, T. and Adhikari, S. (2010), "Nonlocal transverse vibration of double-nanobeam-systems", J. Appl. Phys., 108(8), 083514. https://doi.org/10.1063/1.3496627
  62. Nazemnezhad, R. and Hosseini-Hashemi, S. (2014), "Nonlocal nonlinear free vibration of functionally graded nanobeams", Compos. Struct., 110, 192-199. https://doi.org/10.1016/j.compstruct.2013.12.006
  63. Pompe, W., Worch, H., Epple, M., Friess, W., Gelinsky, M., Greil, P., Hempel, U., Scharnweber, D. and Schulte, K. (2003), "Functionally graded materials for biomedical applications", Mater. Sci. Eng.: A, 362(1), 40-60. https://doi.org/10.1016/S0921-5093(03)00580-X
  64. Rahmani, O. and Pedram, O. (2014), "Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory", Int. J. Eng. Sci., 77, 55-70. https://doi.org/10.1016/j.ijengsci.2013.12.003
  65. Setoodeh, A., Derahaki, M. and Bavi, N. (2015), "DQ thermal buckling analysis of embedded curved carbon nanotubes based on nonlocal elasticity theory", Latin Am. J. Solids Struct., 12(10), 1901-1917. http://dx.doi.org/10.1590/1679-78251894
  66. Shafiei, N. and She, G.L. (2018), "On vibration of functionally graded nano-tubes in the thermal environment", Int. J. Eng. Sci., 133, 84-98. https://doi.org/10.1016/j.ijengsci.2018.08.004
  67. She, G.L., Shu, X. and Ren, Y.R. (2017a), "Thermal buckling and postbuckling analysis of piezoelectric FGM beams based on high-order shear deformation theory", J. Thermal Stress., 40(6), 783-797. https://doi.org/10.1080/01495739.2016.1261009
  68. She, G.L., Yuan, F.G. and Ren, Y.R. (2017b), "Research on nonlinear bending behaviors of FGM infinite cylindrical shallow shells resting on elastic foundations in thermal environments", Compos. Struct., 170, 111-121. https://doi.org/10.1016/j.compstruct.2017.03.010
  69. She, G.L., Yuan, F.G. and Ren, Y.R. (2017c), "Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory", Appl. Math. Model., 47, 340-357. https://doi.org/10.1016/j.apm.2017.03.014
  70. She, G.L., Yuan, F.G., Ren, Y.R. and Xiao, W.S. (2017d), "On buckling and postbuckling behavior of nanotubes", Int. J. Eng. Sci., 121, 130-142. https://doi.org/10.1016/j.ijengsci.2017.09.005
  71. She, G.L., Ren, Y.R., Yuan, F.G. and Xiao, W.S. (2018a), "On vibrations of porous nanotubes", Int. J. Eng. Sci., 125, 23-35. https://doi.org/10.1016/j.ijengsci.2017.12.009
  72. She, G.L., Yuan, F.G. and Ren, Y.R. (2018b), "On wave propagation of porous nanotubes", Int. J. Eng. Sci., 130, 62-74. https://doi.org/10.1016/j.ijengsci.2018.05.002
  73. She, G.L., Yan, K.M., Zhang, Y.L., Liu, H.B. and Ren, Y.R. (2018c), "Wave propagation of functionally graded porous nanobeams based on non-local strain gradient theory", Eur. Phys. J. Plus, 133, 368. https://doi.org/10.1140/epjp/i2018-12196-5
  74. She, G.L., Yuan, F.G., Ren, Y.R., Liu, H.B. and Xiao, W.S. (2018d), "Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory", Composite Structures, 203, 614-623. https://doi.org/10.1016/j.compstruct.2018.07.063
  75. She, G.L., Yuan, F.G., Karami, B., Ren, Y.R. and Xiao, W.S. (2019), "On nonlinear bending behavior of FG porous curved nanotubes", Int. J. Eng. Sci., 135, 58-74. https://doi.org/10.1016/j.ijengsci.2018.11.005
  76. Shen, H.-S. (2016), Functionally Graded Materials: Nonlinear Analysis of Plates and Shells, CRC press.
  77. Soliman, A.E., Eltaher, M.A., Attia, M.A. and Alshorbagy, A.E. (2018), "Nonlinear transient analysis of FG pipe subjected to internal pressure and unsteady temperature in a natural gas facility", Struct. Eng. Mech., Int. J.., 66(1), 85-96. https://doi.org/10.12989/sem.2018.66.1.085
  78. Stolken, J. and Evans, A. (1998), "A microbend test method for measuring the plasticity length scale", Acta Materialia, 46(14), 5109-5115. https://doi.org/10.1016/S1359-6454(98)00153-0
  79. 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
  80. Touloukian, Y.S. (1966), "Thermophysical Propertiesof High Temperature Solid Materials" 6. Intermetallics, Ceramets, Polymers, andCompositeSystems. Part II. Ceramets, Polymers, CompositeSystems, DTIC Document.
  81. Wang, C.M. and Duan, W. (2008), "Free vibration of nanorings/arches based on nonlocal elasticity", J. Appl. Phys., 104(1), 014303. https://doi.org/10.1063/1.2951642
  82. Wattanasakulpong, N. and Chaikittiratana, A. (2015), "Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method", Meccanica, 50(5), 1331-1342. https://doi.org/10.1007/s11012-014-0094-8
  83. Wattanasakulpong, N. and Ungbhakorn, V. (2014), "Linear and nonlinear vibration analysis of elastically restrained ends FGM beams with porosities", Aerosp. Sci. Technol., 32(1), 111-120. https://doi.org/10.1016/j.ast.2013.12.002
  84. Yahia, S.A., Atmane, H.A., Houari, M.S.A. and Tounsi, A. (2015), "Wave propagation in functionally graded plates with porosities using various higher-order shear deformation plate theories", Struct. Eng. Mech., Int. J., 53(6), 1143-1165. http://dx.doi.org/10.12989/sem.2015.53.6.1143
  85. Yan, Z. and Jiang, L. (2011), "Electromechanical response of a curved piezoelectric nanobeam with the consideration of surface effects", J. Phys. D: Appl. Phys., 44(36), 365301. https://doi.org/10.1088/0022-3727/44/36/365301
  86. Youcef, D.O., Kaci, A., Houari, M.S.A., Tounsi, A., Benzair, A. and Heireche, H. (2015), "On the bending and stability of nanowire using various HSDTs", Adv. Nano. Res., Int. J., 3(4), 177-191. https://doi.org/10.12989/anr.2015.3.4.177
  87. Zhang, Y.Y., Wang, C.M. and Challamel, N. (2009), "Bending, buckling, and vibration of micro/nanobeams by hybrid nonlocal beam model", J. Eng. Mech., 136(5), 562-574. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000107
  88. Zhu, J., Lai, Z., Yin, Z., Jeon, J. and Lee, S. (2001), "Fabrication of ZrO 2-NiCr functionally graded material by powder metallurgy", Mater. Chemi. Phys., 68(1), 130-135. https://doi.org/10.1016/S0254-0584(00)00355-2