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Hygro-thermal wave propagation in functionally graded double-layered nanotubes systems

  • She, Gui-Lin (State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University) ;
  • Ren, Yi-Ru (State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University) ;
  • Yuan, Fuh-Gwo (Department of Mechanical and Aerospace Engineering, North Carolina State University)
  • Received : 2019.02.18
  • Accepted : 2019.05.11
  • Published : 2019.06.25
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Abstract

In this paper, wave propagation is studied and analyzed in double-layered nanotubes systems via the nonlocal strain gradient theory. To the author's knowledge, the present paper is the first to investigate the wave propagation characteristics of double-layered porous nanotubes systems. It is generally considered that the material properties of nanotubes are related to the porosity and hygro-thermal effects. The governing equations of the double-layered nanotubes systems are derived by using the Hamilton principle. The dispersion relations and displacement fields of wave propagation in the double nanotubes systems which experience three different types of motion are obtained and discussed. The results show that the phase velocities of the double nanotubes systems depend on porosity, humidity change, temperature change, material composition, non-local parameter, strain gradient parameter, interlayer spring, and wave number.

Keywords

References

  1. Ahouel, M., Houari, M.S.A., Bedia, E. and Tounsi, A. (2016), "Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept", Steel Compos. Struct., Int. J., 20(5), 963-981. http://dx.doi.org/10.12989/scs.2016.20.5.963
  2. Aifantis, E.C. (1992), "On the role of gradients in the localization of deformation and fracture", Int. J. Eng. Sci., 30, 1279-1299. https://doi.org/10.1016/0020-7225(92)90141-3
  3. Akgoz, B. and Civalek, O. (2012), "Investigation of size effects on static response of single-walled carbon nanotubes based on strain gradient elasticity", Int. J. Computat. Methods, 9, 1240032. https://doi.org/10.1142/S0219876212400324
  4. Akgoz, B. and Civalek, O. (2013a), "Modeling and analysis of micro-sized plates resting on elastic medium using the modified couple stress theory", Meccanica, 48(4), 863-873. https://doi.org/10.1007/s11012-012-9639-x
  5. Akgoz, B. and Civalek, O. (2013b), "Buckling analysis of linearly tapered micro-columns based on strain gradient elasticity", Struct. Eng. Mech., Int. J., 48(2), 195-205. https://doi.org/10.12989/sem.2013.48.2.195
  6. Akgoz, B. and Civalek, O. (2017a), "A size-dependent beam model for stability of axially loaded carbon nanotubes surrounded by pasternak elastic foundation", Compos. Struct., 176, 1028-1038. https://doi.org/10.1016/j.compstruct.2017.06.039
  7. Akgoz, B. and Civalek, O. (2017b), "Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams", Compos. Part B: Eng., 129, 77-87. https://doi.org/10.1016/j.compositesb.2017.07.024
  8. Amar, L.H.H., Kaci, A., Yeghnem, R. and Tounsi, A. (2018), "A new four-unknown refined theory based on modified couple stress theory for size-dependent bending and vibration analysis of functionally graded micro-plate", Steel Compos. Struct., Int. J., 26(1), 89-102. http://dx.doi.org/10.12989/scs.2018.26.1.089
  9. Amiri, A., Talebitooti, R. and Li, L. (2018), "Wave propagation in viscous-fluid-conveying piezoelectric nanotubes considering surface stress effects and Knudsen number based on nonlocal strain gradient theory", Euro. Phys. J. Plus, 133(7), 252. https://doi.org/10.1140/epjp/i2018-12077-y
  10. Apuzzo, A., Barretta, R., Faghidian, S.A., Luciano, R. and Marotti de Sciarra, F. (2018), "Free vibrations of elastic beams by modified nonlocal strain gradient theory", Int. J. Eng. Sci., 133, 99-108. https://doi.org/10.1016/j.ijengsci.2018.09.002
  11. Arefi, M. and Zenkour, A.M. (2017), "Size-dependent electromagneto-elastic bending analyses of the shear-deformable axisymmetric functionally graded circular nanoplates", Eur. Phys. J. Plus, 132(10), 423. https://doi.org/10.1140/epjp/i2017-11666-6
  12. Arefi, M. and Zenkour, A.M. (2018), "Size-dependent thermoelastic analysis of a functionally graded nanoshell", Modern Phys. Lett. B, 32(3), 1850033. https://doi.org/10.1142/S0217984918500331
  13. Aria, A.I. and Biglari, H. (2018), "Computational vibration and buckling analysis of microtubule bundles based on nonlocal strain gradient theory", Appl. Math. Comput., 321, 313-332. https://doi.org/10.1016/j.amc.2017.10.050
  14. Arioui, O., Belakhdar, K., Kaci, A. and Tounsi, A. (2018), "Thermal buckling of FGM beams having parabolic thickness variation and temperature dependent materials", Steel Compos. Struct., Int. J., 27(6), 777-788. http://dx.doi.org/10.12989/scs.2018.27.6.777
  15. Attia, M.A. (2017), "On the mechanics of functionally graded nanobeams with the account of surface elasticity", Int. J. Eng. Sci., 115, 73-101. https://doi.org/10.1016/j.ijengsci.2017.03.011
  16. Attia, M.A. and Rahman, A.A.A. (2018), "On vibrations of functionally graded viscoelastic nanobeams with surface effects", Int. J. Eng. Sci., 127, 1-32. https://doi.org/10.1016/j.ijengsci.2018.02.005
  17. Babaei, H., Kiani, Y. and Eslami, M.R. (2019a), "Thermally induced large deflection analysis of shear deformable FGM shallow curved tubes using perturbation method", ZAMMZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK, 99(2), e201800148. https://doi.org/10.1002/zamm.201800148
  18. Babaei, H., Kiani, Y. and Eslami, M.R. (2019b), "Thermal buckling and post-buckling analysis of geometrically imperfect FGM clamped tubes on nonlinear elastic foundation", Appl. Math. Model., 71, 12-30. https://doi.org/10.1016/j.apm.2019.02.009
  19. Barati, M.R. (2017), "On wave propagation in nanoporous materials", Int. J. Eng. Sci., 116, 1-11. https://doi.org/10.1016/j.ijengsci.2017.03.007
  20. Barretta, R. and Sciarra, F.M.D. (2018), "Constitutive boundary conditions for nonlocal strain gradient elastic nano-beams", Int. J. Eng. Sci., 130, 187-198. https://doi.org/10.1016/j.ijengsci.2018.05.009
  21. Barretta, R., Canadija, M., Luciano, R. and de Sciarra, F.M. (2018), "Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams", Int. J. Eng. Sci., 126, 53-67. https://doi.org/10.1016/j.ijengsci.2018.02.012
  22. Barretta, R. Ali Faghidian, S. and Marotti de Sciarra, F. (2019), "Stress-driven nonlocal integral elasticity for axisymmetric nano-plates", Int. J. Eng. Sci., 136, 38-52. https://doi.org/10.1016/j.ijengsci.2019.01.003
  23. Bellifa, H., Bakora, A., Tounsi, A., Bousahla, A.A. and Mahmoud, S.R. (2017), "An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates", Steel Compos. Struct., Int. J., 25(3), 257-270. http://dx.doi.org/10.12989/scs.2017.25.3.257
  24. Benlahcen, F., Belakhdar, K., Sellami, M. and Tounsi, A. (2018), "Thermal buckling resistance of simply supported FGM plates with parabolic-concave thickness variation", Steel Compos. Struct., Int. J., 29(5), 591-602. http://dx.doi.org/10.12989/scs.2018.29.5.591
  25. Dehrouyeh-Semnani, A.M. (2017), "On boundary conditions for thermally loaded fg beams", Int. J. Eng. Sci., 119, 109-127. https://doi.org/10.1016/j.ijengsci.2017.06.017
  26. Dehrouyeh-Semnani, A.M. (2018), "On the thermally induced non-linear response of functionally graded beams", Int. J. Eng. Sci., 125, 53-74. https://doi.org/10.1016/j.ijengsci.2017.12.001
  27. Dehrouyeh-Semnani, A.M., Mostafaei, H., Dehrouyeh, M. and Nikkhah-Bahrami, M. (2017), "Thermal pre- and post-snapthrough buckling of a geometrically imperfect doubly-clamped microbeam made of temperature-dependent functionally graded materials", Compos. Struct., 170, 122-134. https://doi.org/10.1016/j.compstruct.2017.03.003
  28. Ebrahimi, F. and Dabbagh, A. (2018), "NSGT-based acoustical wave dispersion characteristics of thermo-magnetically actuated double-nanobeam systems", Struct. Eng. Mech., Int. J., 68(6), 701-711. http://dx.doi.org/10.12989/sem.2018.68.6.701
  29. Ebrahimi, F. and Barati, M.R. (2018), "Wave propagation analysis of smart strain gradient piezo-magneto-elastic nonlocal beams", Struct. Eng. Mech., Int. J., 66(2), 237-248. http://dx.doi.org/10.12989/sem.2018.66.2.237
  30. Ebrahimi, F. and Farazmandnia, N. (2018), "Thermal buckling analysis of functionally graded carbon nanotube-reinforced composite sandwich beams", Steel Compos. Struct., Int. J., 27(2), 149-159. http://dx.doi.org/10.12989/scs.2018.27.2.149
  31. Ebrahimi, F. and Habibi, S. (2016), "Deflection and vibration analysis of higher-order shear deformable compositionally graded porous plate", Steel Compos. Struct., Int. J., 20(1), 205-225. http://dx.doi.org/10.12989/scs.2016.20.1.205
  32. 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. http://dx.doi.org/10.12989/sem.2018.66.1.097
  33. Eltaher, M.A., Fouda, N., El-Midany, T. and Sadoun, A.M. (2018b), "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
  34. 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
  35. Eltaher, M.A., Almalki, T.A., Ahmed, K.I. and Almitani, K.H. (2019a), "Characterization and behaviors of single walled carbon nanotube by equivalent continuum mechanics approach", Adv. Nano. Res., Int. J., 7(1), 39-49. http://dx.doi.org/10.12989/anr.2019.7.1.039
  36. Eltaher, M.A., Almalki, T.A., Almitani, K.H., Ahmed, K.I.E. and Abdraboh, A.M. (2019b), "Modal participation of fixed-fixed single-walled carbon nanotube with vacancies", Int. J.of Adv. Struct. Eng., 1-13. https://doi.org/10.1007/s40091-019-0222-8
  37. Eringen, A.C. (1998), "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
  38. Faleh, N.M., Ahmed, R.A. and Fenjan, R.M. (2018), "On vibrations of porous FG nanoshells", Int. J. Eng. Sci., 133, 1-14. https://doi.org/10.1016/j.ijengsci.2018.08.007
  39. Fourn, H., Atmane, H.A., Bourada, M., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2018), "A novel four variable refined plate theory for wave propagation in functionally graded material plates", Steel Compos. Struct., Int. J., 27(1), 109-122. http://dx.doi.org/10.12989/scs.2018.27.1.109
  40. Ghayesh, M.H. (2018), "Dynamics of functionally graded viscoelastic microbeams", Int. J. Eng. Sci., 124, 115-131. https://doi.org/10.1016/j.ijengsci.2017.11.004
  41. Ghayesh, M.H. (2019), "Viscoelastic dynamics of axially FG microbeams", Int. J. Eng. Sci., 135, 75-85. https://doi.org/10.1016/j.ijengsci.2018.10.005
  42. Ghayesh, M.H. and Farajpour, A. (2018), "Nonlinear mechanics of nanoscale tubes via nonlocal strain gradient theory", Int. J. Eng. Sci., 129, 84-95. https://doi.org/10.1016/j.ijengsci.2018.04.003
  43. Ghayesh, M.H., Farokhi, H., Gholipour, A. and Tavallaeinejad, M. (2018), "Nonlinear oscillations of functionally graded microplates", Int. J. Eng. Sci., 122, 56-72. https://doi.org/10.1016/j.ijengsci.2017.03.014
  44. Hachemi, H., Kaci, A., Houari, M.S.A., Bourada, M., Tounsi, A. and Mahmoud, S.R. (2017), "A new simple three-unknown shear deformation theory for bending analysis of FG plates resting on elastic foundations", Steel Compos. Struct., Int. J., 25(6), 717-726. http://dx.doi.org/10.12989/scs.2017.25.6.717
  45. Hebbar, N., Bourada, M., Sekkal, M., Tounsi, A. and Mahmoud, S.R. (2018), "A novel four-unknown quasi-3D shear deformation theory for functionally graded plates", Steel Compos. Struct., Int. J., 27(5), 599-611. http://dx.doi.org/10.12989/scs.2018.27.5.599
  46. Heydari, A. (2018a), "Size-dependent damped vibration and buckling analyses of bidirectional functionally graded solid circular nano-plate with arbitrary thickness variation", Struct. Eng. Mech., Int. J., 68(2), 171-182. http://dx.doi.org/10.12989/sem.2018.68.2.171
  47. Heydari, A. (2018b), "Exact vibration and buckling analyses of arbitrary gradation of nano-higher order rectangular beam", Steel Compos. Struct., Int. J., 28(5),589-606. http://dx.doi.org/10.12989/scs.2018.28.5.589
  48. Heydari, A. and Shariati, M. (2018), "Buckling analysis of tapered bdfgm nano-beam under variable axial compression resting on elastic medium", Struct. Eng. Mech., Int. J., 66(6), 737-748. http://dx.doi.org/10.12989/sem.2018.66.6.737
  49. Houari, M.S.A., Bessaim, A., Bernard, F., Tounsi, A. and Mahmoud, S.R. (2018), "Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter", Steel Compos. Struct., Int. J., 28, 13-24. http://dx.doi.org/10.12989/scs.2018.28.1.013
  50. Karami, B. and Janghorban, M. (2019), "On the dynamics of porous nanotubes with variable material properties and variable thickness", Int. J. Eng. Sci., 136, 53-66. https://doi.org/10.1016/j.ijengsci.2019.01.002
  51. 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. http://dx.doi.org/10.12989/scs.2018.27.2.201
  52. Karami, B., Janghorban, M. and Tounsi, A. (2018b), "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
  53. Karami, B., Shahsavari, D., Nazemosadat, S.M.R., Li, L. and Ebrahimi, A. (2018c), "Thermal buckling of smart porous functionally graded nanobeam rested on Kerr foundation", Steel Compos. Struct., Int. J., 29(3), 349-362. http://dx.doi.org/10.12989/scs.2018.29.3.349
  54. Karlicic, D., Kozic, P. and Pavlovic, R. (2016), "Nonlocal vibration and stability of a multiple-nanobeam system coupled by the Winkler elastic medium", Appl. Math. Model., 40(2), 1599-1614. https://doi.org/10.1016/j.apm.2015.06.036
  55. Khaniki, H.B. (2018), "On vibrations of nanobeam systems", Int. J. Eng. Sci., 124, 85-103. https://doi.org/10.1016/j.ijengsci.2017.12.010
  56. 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
  57. 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
  58. Lu, L., Guo, X. and Zhao, J. (2017), "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
  59. Lu, L., Guo, X. and Zhao, J. (2019), "A unified size-dependent plate model based on nonlocal strain gradient theory including surface effects", Appl. Math. Model., 68, 583-602. https://doi.org/10.1016/j.apm.2018.11.023
  60. Malikan, M. and Nguyen, V.B. (2018), "Buckling analysis of piezo-magnetoelectric nanoplates in hygrothermal environment based on a novel one variable plate theory combining with higher-order nonlocal strain gradient theory", Physica E, 102, 8-28. https://doi.org/10.1016/j.physe.2018.04.018
  61. Malikan, M., Nguyen, V.B. and Tornabene, F. (2018a), "Electromagnetic forced vibrations of composite nanoplates using nonlocal strain gradient theory", Mater. Res. Express, 5(75031), 1-20.
  62. Malikan, M., Tornabene, F. and Dimitri, R. (2018b), "Nonlocal three-dimensional theory of elasticity for buckling behavior of functionally graded porous nanoplates using volume integrals", Mater. Res. Express, 5(9), 095006. https://doi.org/10.1088/2053-1591/aad4c3
  63. Malikan, M., Tornabene, F. and Dimitri, R. (2019), "Transient response of oscillated carbon nanotubes with an internal and external damping", Compos. Part B: Eng., 158, 198-205. https://doi.org/10.1016/j.compositesb.2018.09.092
  64. Meftah, A., Bakora, A., Zaoui, F.Z., Tounsi, A. and Bedia, E.A.A. (2017), "A non-polynomial four variable refined plate theory for free vibration of functionally graded thick rectangular plates on elastic foundation", Steel Compos. Struct., Int. J., 23(3), 317-330. http://dx.doi.org/10.12989/scs.2017.23.3.317
  65. Mehar, K. and Panda, S.K. (2019), "Theoretical deflection analysis of multi-walled carbon nanotube reinforced sandwich panel and experimental verification", Compos. Part B: Eng., 167, 317-328. https://doi.org/10.1016/j.compositesb.2018.12.058
  66. Mindlin, R.D. (1965), "Second gradient of strain and surfacetension in linear elasticity", Int. J. Solid Struct., 1(4), 417-438. https://doi.org/10.1016/0020-7683(65)90006-5
  67. Murmu, T. and Adhikari, S. (2010), "Nonlocal transverse vibration of double-nanobeam-systems", J. Appl. Phys., 108(8), 147. https://doi.org/10.1063/1.3496627
  68. Murmu, T. and Adhikari, S. (2011), "Axial instability of doublenanobeam-systems", Phys. Lett. A, 375(3), 601-608. https://doi.org/10.1016/j.physleta.2010.11.007
  69. Numanoglu, H.M., Akgoz, B. and Civalek, O. (2018), "On dynamic analysis of nanorods", Int. J. Eng. Sci., 130, 33-50. https://doi.org/10.1016/j.ijengsci.2018.05.001
  70. Radic, N. (2018), "On buckling of porous double-layered FG nanoplates in the Pasternak elastic foundation based on nonlocal strain gradient elasticity", Compos. Part B-Eng., 153, 465-479. https://doi.org/10.1016/j.compositesb.2018.09.014
  71. Rajasekaran, S. and Khaniki, H.B. (2017), "Bending, buckling and vibration of small-scale tapered beams", Int. J. Eng. Sci., 120, 172-188. https://doi.org/10.1016/j.ijengsci.2017.08.005
  72. Reddy, J.N. and Chin, C.D. (1998), "Thermomechanical analysis of functionally graded cylinders and plates", J. Thermal Stresses, 21(6), 593-626. https://doi.org/10.1080/01495739808956165
  73. Romano, G. and Barretta, R. (2017), "Nonlocal elasticity in nanobeams: the stress-driven integral model", Int. J. Eng. Sci., 115, 14-27. https://doi.org/10.1016/j.ijengsci.2017.03.002
  74. Sahmani, S. and Aghdam, M.M. (2017), "Nonlocal strain gradient beam model for nonlinear vibration of prebuckled and postbuckled multilayer functionally graded GPLRC nanobeams", Compos. Struct., 179, 77-88. https://doi.org/10.1016/j.compstruct.2017.07.064
  75. Shaat, M. and Abdelkefi, A. (2017), "New insights on the applicability of Eringen's, nonlocal theory", Int. J. Mech. Sci. 121, 67-75. https://doi.org/10.1016/j.ijmecsci.2016.12.013
  76. Shahverdi, H. and Barati, M.R. (2017), "Vibration analysis of porous functionally graded nanoplates", Int. J. Eng. Sci., 120, 82-99. https://doi.org/10.1016/j.ijengsci.2017.06.008
  77. She, G.L., Yuan, F.G. and Ren, Y.R. (2017a), "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
  78. She, G.L., Yuan, F.G., Ren, Y.R. and Xiao, W.S. (2017b), "On buckling and postbuckling behavior of nanotubes", Int. J. Eng. Sci., 121, 130-142. https://doi.org/10.1016/j.ijengsci.2017.09.005
  79. She, G.L., Yan, K.M., Zhang, Y.L., Liu, H.B. and Ren, Y.R. (2018a), "Wave propagation of functionally graded porous nanobeams based on non-local strain gradient theory", Euro. Phys. J. Plus, 133(9), 368. https://doi.org/10.1140/epjp/i2018-12196-5
  80. She, G.L., Ren, Y.R., Yuan, F.G. and Xiao, W.S. (2018b), "On vibrations of porous nanotubes", Int. J. Eng. Sci., 125, 23-35. https://doi.org/10.1016/j.ijengsci.2017.12.009
  81. She, G.L., Yuan, F.G., Ren, Y.R., Liu, H., Xiao, W.S. (2018c), "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
  82. She, G.L., Yuan, F.G. and Ren, Y.R. (2018d), "On wave propagation of porous nanotubes", Int. J. Eng. Sci., 130, 62-74. https://doi.org/10.1016/j.ijengsci.2018.05.002
  83. She, G.L., Ren, Y.R. and Yan, K.M. (2019), "On snap-buckling of porous FG curved nanobeams", Acta Astronautica, 161, 475-484. https://doi.org/10.1016/j.actaastro.2019.04.010
  84. Sidhoum, I.A., Boutchicha, D., Benyoucef, S. and Tounsi, A. (2017), "An original HSDT for free vibration analysis of functionally graded plates", Steel Compos. Struct., Int. J., 25(6), 735-745. http://dx.doi.org/10.12989/scs.2017.25.6.735
  85. Tang, H., Li, L. and Hu, Y. (2019), "Coupling effect of thickness and shear deformation on size-dependent bending of micro/nano-scale porous beams", Appl. Math. Model., 66, 527-547. https://doi.org/10.1016/j.apm.2018.09.027
  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. Zenkour, A.M. (2018), "A quasi-3D refined theory for functionally graded single-layered and sandwich plates with porosities", Compos. Struct., 201, 38-48. https://doi.org/10.1016/j.compstruct.2018.05.147
  88. Zenkour, A.M. and Radwan, A.F. (2019), "Bending response of FG plates resting on elastic foundations in hygrothermal environment with porosities", Compos. Struct., 213, 133-143. https://doi.org/10.1016/j.compstruct.2019.01.065
  89. Zhang, P. and Fu, Y. (2013), "A higher-order beam model for tubes", Eur. J. Mech. A-Solid., 38(3), 12-19. https://doi.org/10.1016/j.euromechsol.2012.09.009
  90. Zhong, J., Fu, Y., Wan, D. and Li, Y. (2016), "Nonlinear bending and vibration of functionally graded tubes resting on elastic foundations in thermal environment based on a refined beam model", Appl. Math. Model., 40(17-18), 7601-7614. https://doi.org/10.1016/j.apm.2016.03.031

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