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Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect

  • Chaht, Fouzia Larbi (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Kaci, Abdelhakim (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Houari, Mohammed Sid Ahmed (Laboratoire des Structures et Materiaux Avances dans le Genie Civil et Travaux Publics, Universite de Sidi Bel Abbes, Faculte de Technologie, Departement de genie civil) ;
  • Tounsi, Abdelouahed (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department) ;
  • Beg, O. Anwar (Gort Engovation-Propulsion, Nanomechanics and Biophysics) ;
  • Mahmoud, S.R. (Department of Mathematics, Faculty of Science, King Abdulaziz University)
  • Received : 2014.04.30
  • Accepted : 2014.07.27
  • Published : 2015.02.25
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Abstract

This paper addresses theoretically the bending and buckling behaviors of size-dependent nanobeams made of functionally graded materials (FGMs) including the thickness stretching effect. The size-dependent FGM nanobeam is investigated on the basis of the nonlocal continuum model. The nonlocal elastic behavior is described by the differential constitutive model of Eringen, which enables the present model to become effective in the analysis and design of nanostructures. The present model incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect, and furthermore accounts for both shear deformation and thickness stretching effects by virtue of a sinusoidal variation of all displacements through the thickness without using shear correction factor. The material properties of FGM nanobeams are assumed to vary through the thickness according to a power law. The governing equations and the related boundary conditions are derived using the principal of minimum total potential energy. A Navier-type solution is developed for simply-supported boundary conditions, and exact expressions are proposed for the deflections and the buckling load. The effects of nonlocal parameter, aspect ratio and various material compositions on the static and stability responses of the FGM nanobeam are discussed in detail. The study is relevant to nanotechnology deployment in for example aircraft structures.

Keywords

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  8. Coupled twist–bending static and dynamic behavior of a curved single-walled carbon nanotube based on nonlocal theory vol.23, pp.7, 2017, https://doi.org/10.1007/s00542-016-2933-0
  9. A general bi-Helmholtz nonlocal strain-gradient elasticity for wave propagation in nanoporous graded double-nanobeam systems on elastic substrate vol.168, 2017, https://doi.org/10.1016/j.compstruct.2017.02.090
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  12. Post-buckling responses of elastoplastic FGM beams on nonlinear elastic foundation vol.58, pp.3, 2016, https://doi.org/10.12989/sem.2016.58.3.515
  13. Buckling analysis of functionally graded rectangular nano-plate based on nonlocal exponential shear deformation theory vol.113, 2016, https://doi.org/10.1016/j.ijmecsci.2016.04.014
  14. On nonlocal characteristics of curved inhomogeneous Euler–Bernoulli nanobeams under different temperature distributions vol.122, pp.10, 2016, https://doi.org/10.1007/s00339-016-0399-7
  15. Thermo-mechanical postbuckling of symmetric S-FGM plates resting on Pasternak elastic foundations using hyperbolic shear deformation theory vol.57, pp.4, 2016, https://doi.org/10.12989/sem.2016.57.4.617
  16. On bending, buckling and vibration responses of functionally graded carbon nanotube-reinforced composite beams vol.19, pp.5, 2015, https://doi.org/10.12989/scs.2015.19.5.1259
  17. Dynamic modeling of smart shear-deformable heterogeneous piezoelectric nanobeams resting on Winkler–Pasternak foundation vol.122, pp.11, 2016, https://doi.org/10.1007/s00339-016-0466-0
  18. Size-dependent electro-magneto-elastic bending analyses of the shear-deformable axisymmetric functionally graded circular nanoplates vol.132, pp.10, 2017, https://doi.org/10.1140/epjp/i2017-11666-6
  19. On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable model vol.18, pp.4, 2015, https://doi.org/10.12989/scs.2015.18.4.1063
  20. Free vibration analysis of chiral double-walled carbon nanotube using non-local elasticity theory vol.4, pp.1, 2016, https://doi.org/10.12989/anr.2016.4.1.031
  21. Size-dependent nonlinear vibration of beam-type porous materials with an initial geometrical curvature vol.184, 2018, https://doi.org/10.1016/j.compstruct.2017.10.052
  22. A novel four variable refined plate theory for bending, buckling, and vibration of functionally graded plates vol.22, pp.3, 2016, https://doi.org/10.12989/scs.2016.22.3.473
  23. Wave dispersion characteristics of axially loaded magneto-electro-elastic nanobeams vol.122, pp.11, 2016, https://doi.org/10.1007/s00339-016-0465-1
  24. Bending analysis of FGM plates using a sinusoidal shear deformation theory vol.23, pp.6, 2016, https://doi.org/10.12989/was.2016.23.6.543
  25. Nonlocal strain gradient theory calibration using molecular dynamics simulation based on small scale vibration of nanotubes vol.514, 2017, https://doi.org/10.1016/j.physb.2017.03.030
  26. Vibration analysis of viscoelastic inhomogeneous nanobeams incorporating surface and thermal effects vol.123, pp.1, 2017, https://doi.org/10.1007/s00339-016-0511-z
  27. Nonlocal vibration analysis of FG nano beams with different boundary conditions vol.4, pp.2, 2016, https://doi.org/10.12989/anr.2016.4.2.085
  28. Critical buckling load of chiral double-walled carbon nanotube using non-local theory elasticity vol.3, pp.4, 2015, https://doi.org/10.12989/anr.2015.3.4.193
  29. Free vibration and critical angular velocity of a rotating variable thickness two-directional FG circular microplate 2018, https://doi.org/10.1007/s00542-017-3557-8
  30. A new 3-unknowns non-polynomial plate theory for buckling and vibration of functionally graded sandwich plate vol.60, pp.4, 2016, https://doi.org/10.12989/sem.2016.60.4.547
  31. A computational shear displacement model for vibrational analysis of functionally graded beams with porosities vol.19, pp.2, 2015, https://doi.org/10.12989/scs.2015.19.2.369
  32. A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation vol.20, pp.2, 2016, https://doi.org/10.12989/scs.2016.20.2.227
  33. Hygro-thermo-mechanical bending of S-FGM plates resting on variable elastic foundations using a four-variable trigonometric plate theory vol.18, pp.4, 2016, https://doi.org/10.12989/sss.2016.18.4.755
  34. Thermomechanical effects on the bending of antisymmetric cross-ply composite plates using a four variable sinusoidal theory vol.19, pp.1, 2015, https://doi.org/10.12989/scs.2015.19.1.093
  35. Peristaltic transport of bi-viscosity fluids through a curved tube: A mathematical model for intestinal flow vol.230, pp.9, 2016, https://doi.org/10.1177/0954411916658318
  36. Size-dependent hygro–thermo–electro–mechanical vibration analysis of functionally graded piezoelectric nanobeams resting on Winkler–Pasternak foundation undergoing preload and magnetic field 2017, https://doi.org/10.1007/s00542-017-3545-z
  37. Wave propagation analysis of size-dependent rotating inhomogeneous nanobeams based on nonlocal elasticity theory 2017, https://doi.org/10.1177/1077546317711537
  38. Thermal post-buckling behavior of imperfect temperature-dependent sandwich FGM plates resting on Pasternak elastic foundation vol.22, pp.1, 2016, https://doi.org/10.12989/scs.2016.22.1.091
  39. Influence of thermal and surface effects on vibration behavior of nonlocal rotating Timoshenko nanobeam vol.122, pp.7, 2016, https://doi.org/10.1007/s00339-016-0196-3
  40. A refined theory with stretching effect for the flexure analysis of laminated composite plates vol.11, pp.5, 2016, https://doi.org/10.12989/gae.2016.11.5.671
  41. A review of continuum mechanics models for size-dependent analysis of beams and plates vol.177, 2017, https://doi.org/10.1016/j.compstruct.2017.06.040
  42. Nonlinear thermal buckling of axially functionally graded micro and nanobeams vol.168, 2017, https://doi.org/10.1016/j.compstruct.2017.02.048
  43. Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects vol.120, 2017, https://doi.org/10.1016/j.ijmecsci.2016.11.025
  44. Thermal effects on nonlocal vibrational characteristics of nanobeams with non-ideal boundary conditions vol.18, pp.6, 2016, https://doi.org/10.12989/sss.2016.18.6.1087
  45. On the bending and stability of nanowire using various HSDTs vol.3, pp.4, 2015, https://doi.org/10.12989/anr.2015.3.4.177
  46. Size-dependent mechanical behavior of functionally graded trigonometric shear deformable nanobeams including neutral surface position concept vol.20, pp.5, 2016, https://doi.org/10.12989/scs.2016.20.5.963
  47. Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs vol.153, 2016, https://doi.org/10.1016/j.compstruct.2016.07.013
  48. Forced vibration analysis of viscoelastic nanobeams embedded in an elastic medium vol.18, pp.6, 2016, https://doi.org/10.12989/sss.2016.18.6.1125
  49. Buckling analysis of tapered nanobeams using nonlocal strain gradient theory and a generalized differential quadrature method vol.4, pp.6, 2017, https://doi.org/10.1088/2053-1591/aa7111
  50. An efficient shear deformation theory for wave propagation of functionally graded material plates vol.57, pp.5, 2016, https://doi.org/10.12989/sem.2016.57.5.837
  51. Effect of three-parameter viscoelastic medium on vibration behavior of temperature-dependent non-homogeneous viscoelastic nanobeams in a hygro-thermal environment vol.25, pp.5, 2018, https://doi.org/10.1080/15376494.2016.1255831
  52. On thermal stability of plates with functionally graded coefficient of thermal expansion vol.60, pp.2, 2016, https://doi.org/10.12989/sem.2016.60.2.313
  53. Buckling analysis of nonuniform nonlocal strain gradient beams using generalized differential quadrature method 2018, https://doi.org/10.1016/j.aej.2017.06.001
  54. Effect of porosity on vibrational characteristics of non-homogeneous plates using hyperbolic shear deformation theory vol.22, pp.4, 2016, https://doi.org/10.12989/was.2016.22.4.429
  55. Dynamic modeling of porous heterogeneous micro/nanobeams vol.132, pp.12, 2017, https://doi.org/10.1140/epjp/i2017-11754-7
  56. A new nonlocal hyperbolic shear deformation theory for nanobeams embedded in an elastic medium vol.55, pp.4, 2015, https://doi.org/10.12989/sem.2015.55.4.743
  57. Effect of various thermal loadings on buckling and vibrational characteristics of nonlocal temperature-dependent functionally graded nanobeams vol.23, pp.12, 2016, https://doi.org/10.1080/15376494.2015.1091524
  58. Thermo-mechanical vibration of rotating axially functionally graded nonlocal Timoshenko beam vol.123, pp.1, 2017, https://doi.org/10.1007/s00339-016-0712-5
  59. Static stability analysis of smart magneto-electro-elastic heterogeneous nanoplates embedded in an elastic medium based on a four-variable refined plate theory vol.25, pp.10, 2016, https://doi.org/10.1088/0964-1726/25/10/105014
  60. Analytical solution for nonlocal buckling characteristics of higher-order inhomogeneous nanosize beams embedded in elastic medium vol.4, pp.3, 2016, https://doi.org/10.12989/anr.2016.4.3.229
  61. A new refined nonlocal beam theory accounting for effect of thickness stretching in nanoscale beams vol.4, pp.4, 2016, https://doi.org/10.12989/anr.2016.4.4.251
  62. Electro-magnetic effects on nonlocal dynamic behavior of embedded piezoelectric nanoscale beams vol.28, pp.15, 2017, https://doi.org/10.1177/1045389X16682850
  63. Parametric excitation analysis of a piezoelectric-nanotube conveying fluid under multi-physics field 2017, https://doi.org/10.1007/s00542-017-3670-8
  64. Application of nonlocal strain gradient theory and various shear deformation theories to nonlinear vibration analysis of sandwich nano-beam with FG-CNTRCs face-sheets in electro-thermal environment vol.123, pp.5, 2017, https://doi.org/10.1007/s00339-017-0922-5
  65. A new higher order shear and normal deformation theory for functionally graded beams vol.18, pp.3, 2015, https://doi.org/10.12989/scs.2015.18.3.793
  66. Bending, buckling and vibration analyses of MSGT microcomposite circular-annular sandwich plate under hydro-thermo-magneto-mechanical loadings using DQM 2017, https://doi.org/10.1080/19475411.2017.1377312
  67. Influence of the porosities on the free vibration of FGM beams vol.21, pp.3, 2015, https://doi.org/10.12989/was.2015.21.3.273
  68. Dynamic response of a single-walled carbon nanotube under a moving harmonic load by considering modified nonlocal elasticity theory vol.133, pp.2, 2018, https://doi.org/10.1140/epjp/i2018-11868-4
  69. A nonlocal quasi-3D trigonometric plate model for free vibration behaviour of micro/nanoscale plates vol.56, pp.2, 2015, https://doi.org/10.12989/sem.2015.56.2.223
  70. Free vibration investigation of nano mass sensor using differential transformation method vol.123, pp.3, 2017, https://doi.org/10.1007/s00339-017-0796-6
  71. Thermo-mechanical analysis of FG nanobeam with attached tip mass: an exact solution vol.122, pp.12, 2016, https://doi.org/10.1007/s00339-016-0542-5
  72. A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates vol.22, pp.2, 2016, https://doi.org/10.12989/scs.2016.22.2.257
  73. Analytical solution for bending analysis of functionally graded beam vol.19, pp.4, 2015, https://doi.org/10.12989/scs.2015.19.4.829
  74. Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory vol.159, 2017, https://doi.org/10.1016/j.compstruct.2016.09.092
  75. Vibration analysis of bonded double-FGM viscoelastic nanoplate systems based on a modified strain gradient theory incorporating surface effects vol.123, pp.3, 2017, https://doi.org/10.1007/s00339-017-0784-x
  76. An efficient and simple shear deformation theory for free vibration of functionally graded rectangular plates on Winkler-Pasternak elastic foundations vol.22, pp.3, 2016, https://doi.org/10.12989/was.2016.22.3.329
  77. Effect of Longitudinal Magnetic Field on Vibration Characteristics of Single-Walled Carbon Nanotubes in a Viscoelastic Medium vol.47, pp.6, 2017, https://doi.org/10.1007/s13538-017-0524-x
  78. Nonlinear electroelastic vibration analysis of NEMS consisting of double-viscoelastic nanoplates vol.122, pp.10, 2016, https://doi.org/10.1007/s00339-016-0452-6
  79. Buckling analysis of isotropic and orthotropic plates using a novel four variable refined plate theory vol.21, pp.6, 2016, https://doi.org/10.12989/scs.2016.21.6.1287
  80. A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory vol.54, pp.4, 2015, https://doi.org/10.12989/sem.2015.54.4.693
  81. Buckling analysis of piezoelectrically actuated smart nanoscale plates subjected to magnetic field vol.28, pp.11, 2017, https://doi.org/10.1177/1045389X16672569
  82. A simple hyperbolic shear deformation theory for vibration analysis of thick functionally graded rectangular plates resting on elastic foundations vol.11, pp.2, 2016, https://doi.org/10.12989/gae.2016.11.2.289
  83. Nonlocal thermo-elastic wave propagation in temperature-dependent embedded small-scaled nonhomogeneous beams vol.131, pp.11, 2016, https://doi.org/10.1140/epjp/i2016-16383-0
  84. A modified nonlocal couple stress-based beam model for vibration analysis of higher-order FG nanobeams 2017, https://doi.org/10.1080/15376494.2017.1365979
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  86. Critical Buckling Load of Chiral Double-Walled Carbon Nanotubes Embedded in an Elastic Medium vol.53, pp.6, 2018, https://doi.org/10.1007/s11029-018-9708-x
  87. A review on nonlocal elastic models for bending, buckling, vibrations, and wave propagation of nanoscale beams vol.40, pp.5-6, 2016, https://doi.org/10.1016/j.apm.2015.11.026
  88. Free vibration of symmetric and sigmoid functionally graded nanobeams vol.122, pp.9, 2016, https://doi.org/10.1007/s00339-016-0324-0
  89. Buckling analysis on the bi-dimensional functionally graded porous tapered nano-/micro-scale beams vol.66, 2017, https://doi.org/10.1016/j.ast.2017.02.019
  90. Influence of size effect on flapwise vibration behavior of rotary microbeam and its analysis through spectral meshless radial point interpolation vol.123, pp.5, 2017, https://doi.org/10.1007/s00339-017-0955-9
  91. Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory pp.2041-2983, 2019, https://doi.org/10.1177/0954406218756451
  92. On modeling of wave propagation in a thermally affected GNP-reinforced imperfect nanocomposite shell pp.1435-5663, 2019, https://doi.org/10.1007/s00366-018-0669-4
  93. Buckling of magneto-electro-hygro-thermal piezoelectric nanoplates system embedded in a visco-Pasternak medium based on nonlocal theory pp.1432-1858, 2018, https://doi.org/10.1007/s00542-018-4082-0
  94. Vibration Analysis of Nano Beam Using Differential Transform Method Including Thermal Effect vol.54, pp.1661-9897, 2018, https://doi.org/10.4028/www.scientific.net/JNanoR.54.1
  95. Forced vibration analysis of cracked nanobeams vol.40, pp.8, 2018, https://doi.org/10.1007/s40430-018-1315-1
  96. Free vibration analysis of a piezoelectric curved sandwich nano-beam with FG-CNTRCs face-sheets based on various high-order shear deformation and nonlocal elasticity theories vol.133, pp.5, 2018, https://doi.org/10.1140/epjp/i2018-12015-1
  97. Thermal and Small-Scale Effects on Vibration of Embedded Armchair Single-Walled Carbon Nanotubes vol.51, pp.1661-9897, 2018, https://doi.org/10.4028/www.scientific.net/JNanoR.51.24
  98. Nonlocal strain gradient theory for damping vibration analysis of viscoelastic inhomogeneous nano-scale beams embedded in visco-Pasternak foundation vol.24, pp.10, 2018, https://doi.org/10.1177/1077546316678511
  99. A novel approach for nonlinear bending response of macro- and nanoplates with irregular variable thickness under nonuniform loading in thermal environment pp.1539-7742, 2019, https://doi.org/10.1080/15397734.2018.1557529
  100. Modal participation of fixed–fixed single-walled carbon nanotube with vacancies pp.2008-6695, 2019, https://doi.org/10.1007/s40091-019-0222-8
  101. A new five unknown quasi-3D type HSDT for thermomechanical bending analysis of FGM sandwich plates vol.22, pp.5, 2015, https://doi.org/10.12989/scs.2016.22.5.975
  102. Buckling analysis of functionally graded truncated conical shells under external displacement-dependent pressure vol.23, pp.1, 2017, https://doi.org/10.12989/scs.2017.23.1.001
  103. A novel quasi-3D hyperbolic shear deformation theory for functionally graded thick rectangular plates on elastic foundation vol.12, pp.1, 2015, https://doi.org/10.12989/gae.2017.12.1.009
  104. A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams vol.19, pp.2, 2017, https://doi.org/10.12989/sss.2017.19.2.115
  105. A non-polynomial four variable refined plate theory for free vibration of functionally graded thick rectangular plates on elastic foundation vol.23, pp.3, 2015, https://doi.org/10.12989/scs.2017.23.3.317
  106. Bending and stability analysis of size-dependent compositionally graded Timoshenko nanobeams with porosities vol.6, pp.1, 2017, https://doi.org/10.12989/amr.2017.6.1.045
  107. Wave propagation in functionally graded beams using various higher-order shear deformation beams theories vol.62, pp.2, 2015, https://doi.org/10.12989/sem.2017.62.2.143
  108. Analysis of functionally graded plates using a sinusoidal shear deformation theory vol.19, pp.4, 2017, https://doi.org/10.12989/sss.2017.19.4.441
  109. A novel and simple HSDT for thermal buckling response of functionally graded sandwich plates vol.62, pp.4, 2017, https://doi.org/10.12989/sem.2017.62.4.401
  110. A nonlocal zeroth-order shear deformation theory for nonlinear postbuckling of nanobeams vol.62, pp.6, 2017, https://doi.org/10.12989/sem.2017.62.6.695
  111. Free vibration analysis of embedded nanosize FG plates using a new nonlocal trigonometric shear deformation theory vol.19, pp.6, 2015, https://doi.org/10.12989/sss.2017.19.6.601
  112. A new shear deformation plate theory with stretching effect for buckling analysis of functionally graded sandwich plates vol.24, pp.5, 2017, https://doi.org/10.12989/scs.2017.24.5.569
  113. Free vibrations of laminated composite plates using a novel four variable refined plate theory vol.24, pp.5, 2017, https://doi.org/10.12989/scs.2017.24.5.603
  114. Dynamic bending response of SWCNT reinforced composite plates subjected to hygro-thermo-mechanical loading vol.20, pp.2, 2015, https://doi.org/10.12989/cac.2017.20.2.229
  115. A simple analytical approach for thermal buckling of thick functionally graded sandwich plates vol.63, pp.5, 2015, https://doi.org/10.12989/sem.2017.63.5.585
  116. An efficient shear deformation theory for wave propagation in functionally graded material beams with porosities vol.13, pp.3, 2015, https://doi.org/10.12989/eas.2017.13.3.255
  117. A four variable refined nth-order shear deformation theory for mechanical and thermal buckling analysis of functionally graded plates vol.13, pp.3, 2015, https://doi.org/10.12989/gae.2017.13.3.385
  118. Vibration analysis of nonlocal advanced nanobeams in hygro-thermal environment using a new two-unknown trigonometric shear deformation beam theory vol.20, pp.3, 2015, https://doi.org/10.12989/sss.2017.20.3.369
  119. A new and simple HSDT for thermal stability analysis of FG sandwich plates vol.25, pp.2, 2015, https://doi.org/10.12989/scs.2017.25.2.157
  120. A novel simple two-unknown hyperbolic shear deformation theory for functionally graded beams vol.64, pp.2, 2015, https://doi.org/10.12989/sem.2017.64.2.145
  121. Free vibration of functionally graded plates resting on elastic foundations based on quasi-3D hybrid-type higher order shear deformation theory vol.20, pp.4, 2017, https://doi.org/10.12989/sss.2017.20.4.509
  122. An efficient and simple four variable refined plate theory for buckling analysis of functionally graded plates vol.25, pp.3, 2015, https://doi.org/10.12989/scs.2017.25.3.257
  123. Effects of triaxial magnetic field on the anisotropic nanoplates vol.25, pp.3, 2017, https://doi.org/10.12989/scs.2017.25.3.361
  124. A novel and simple higher order shear deformation theory for stability and vibration of functionally graded sandwich plate vol.25, pp.4, 2017, https://doi.org/10.12989/scs.2017.25.4.389
  125. A simple quasi-3D sinusoidal shear deformation theory with stretching effect for carbon nanotube-reinforced composite beams resting on elastic foundation vol.13, pp.5, 2015, https://doi.org/10.12989/eas.2017.13.5.509
  126. A new nonlocal trigonometric shear deformation theory for thermal buckling analysis of embedded nanosize FG plates vol.64, pp.4, 2015, https://doi.org/10.12989/sem.2017.64.4.391
  127. Vibration analysis of FG nanoplates with nanovoids on viscoelastic substrate under hygro-thermo-mechanical loading using nonlocal strain gradient theory vol.64, pp.6, 2015, https://doi.org/10.12989/sem.2017.64.6.683
  128. A new quasi-3D HSDT for buckling and vibration of FG plate vol.64, pp.6, 2015, https://doi.org/10.12989/sem.2017.64.6.737
  129. An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of FGM sandwich plates with various boundary conditions vol.25, pp.6, 2015, https://doi.org/10.12989/scs.2017.25.6.693
  130. A new simple three-unknown shear deformation theory for bending analysis of FG plates resting on elastic foundations vol.25, pp.6, 2015, https://doi.org/10.12989/scs.2017.25.6.717
  131. An original HSDT for free vibration analysis of functionally graded plates vol.25, pp.6, 2015, https://doi.org/10.12989/scs.2017.25.6.735
  132. Vibration analysis of thick orthotropic plates using quasi 3D sinusoidal shear deformation theory vol.16, pp.2, 2015, https://doi.org/10.12989/gae.2018.16.2.141
  133. A nonlocal strain gradient theory for nonlinear free and forced vibration of embedded thick FG double layered nanoplates vol.68, pp.1, 2018, https://doi.org/10.12989/sem.2018.68.1.103
  134. Post-buckling analysis of shear-deformable composite beams using a novel simple two-unknown beam theory vol.65, pp.5, 2015, https://doi.org/10.12989/sem.2018.65.5.621
  135. Thermal stability analysis of temperature dependent inhomogeneous size-dependent nano-scale beams vol.7, pp.1, 2015, https://doi.org/10.12989/amr.2018.7.1.001
  136. Forced vibration analysis of cracked functionally graded microbeams vol.6, pp.1, 2015, https://doi.org/10.12989/anr.2018.6.1.039
  137. Post-buckling responses of a laminated composite beam vol.26, pp.6, 2015, https://doi.org/10.12989/scs.2018.26.6.733
  138. A novel four variable refined plate theory for wave propagation in functionally graded material plates vol.27, pp.1, 2018, https://doi.org/10.12989/scs.2018.27.1.109
  139. Improved HSDT accounting for effect of thickness stretching in advanced composite plates vol.66, pp.1, 2015, https://doi.org/10.12989/sem.2018.66.1.061
  140. Vibration and instability analysis of pipes reinforced by SiO2 nanoparticles considering agglomeration effects vol.21, pp.4, 2018, https://doi.org/10.12989/cac.2018.21.4.431
  141. Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles vol.27, pp.2, 2015, https://doi.org/10.12989/scs.2018.27.2.201
  142. Three dimensional dynamic response of functionally graded nanoplates under a moving load vol.66, pp.2, 2015, https://doi.org/10.12989/sem.2018.66.2.249
  143. A novel shear deformation theory for buckling analysis of single layer graphene sheet based on nonlocal elasticity theory vol.21, pp.4, 2015, https://doi.org/10.12989/sss.2018.21.4.397
  144. Novel quasi-3D and 2D shear deformation theories for bending and free vibration analysis of FGM plates vol.14, pp.6, 2015, https://doi.org/10.12989/gae.2018.14.6.519
  145. Bending of FGM rectangular plates resting on non-uniform elastic foundations in thermal environment using an accurate theory vol.27, pp.3, 2018, https://doi.org/10.12989/scs.2018.27.3.311
  146. Size dependent bending analysis of micro/nano sandwich structures based on a nonlocal high order theory vol.27, pp.3, 2018, https://doi.org/10.12989/scs.2018.27.3.371
  147. Free vibration of FGM plates with porosity by a shear deformation theory with four variables vol.66, pp.3, 2015, https://doi.org/10.12989/sem.2018.66.3.353
  148. Free vibration and buckling analysis of orthotropic plates using a new two variable refined plate theory vol.15, pp.1, 2015, https://doi.org/10.12989/gae.2018.15.1.711
  149. Vibration and instability of nanocomposite pipes conveying fluid mixed by nanoparticles resting on viscoelastic foundation vol.21, pp.5, 2018, https://doi.org/10.12989/cac.2018.21.5.569
  150. Mathematical modeling of smart nanoparticles-reinforced concrete foundations: Vibration analysis vol.27, pp.4, 2015, https://doi.org/10.12989/scs.2018.27.4.465
  151. Nonlocal free vibration analysis of a doubly curved piezoelectric nano shell vol.27, pp.4, 2018, https://doi.org/10.12989/scs.2018.27.4.479
  152. Three dimensional finite elements modeling of FGM plate bending using UMAT vol.66, pp.4, 2018, https://doi.org/10.12989/sem.2018.66.4.487
  153. A novel four-unknown quasi-3D shear deformation theory for functionally graded plates vol.27, pp.5, 2015, https://doi.org/10.12989/scs.2018.27.5.599
  154. A new nonlocal HSDT for analysis of stability of single layer graphene sheet vol.6, pp.2, 2015, https://doi.org/10.12989/anr.2018.6.2.147
  155. Mechanical buckling analysis of hybrid laminated composite plates under different boundary conditions vol.66, pp.6, 2018, https://doi.org/10.12989/sem.2018.66.6.761
  156. A new quasi-3D higher shear deformation theory for vibration of functionally graded carbon nanotube-reinforced composite beams resting on elastic foundation vol.66, pp.6, 2018, https://doi.org/10.12989/sem.2018.66.6.771
  157. Buckling analysis of new quasi-3D FG nanobeams based on nonlocal strain gradient elasticity theory and variable length scale parameter vol.28, pp.1, 2015, https://doi.org/10.12989/scs.2018.28.1.013
  158. A size-dependent quasi-3D model for wave dispersion analysis of FG nanoplates vol.28, pp.1, 2015, https://doi.org/10.12989/scs.2018.28.1.099
  159. Dynamic stability of nanocomposite Mindlin pipes conveying pulsating fluid flow subjected to magnetic field vol.67, pp.1, 2015, https://doi.org/10.12989/sem.2018.67.1.021
  160. Technical and economical assessment of applying silica nanoparticles for construction of concrete structures vol.22, pp.1, 2018, https://doi.org/10.12989/cac.2018.22.1.117
  161. Size-dependent free vibration and dynamic analyses of a sandwich microbeam based on higher-order sinusoidal shear deformation theory and strain gradient theory vol.22, pp.1, 2015, https://doi.org/10.12989/sss.2018.22.1.027
  162. Size-dependent free vibration and dynamic analyses of a sandwich microbeam based on higher-order sinusoidal shear deformation theory and strain gradient theory vol.22, pp.1, 2015, https://doi.org/10.12989/sss.2018.22.1.027
  163. Forced vibration response in nanocomposite cylindrical shells - Based on strain gradient beam theory vol.28, pp.3, 2015, https://doi.org/10.12989/scs.2018.28.3.381
  164. Single variable shear deformation model for bending analysis of thick beams vol.67, pp.3, 2015, https://doi.org/10.12989/sem.2018.67.3.291
  165. Numerical study for vibration response of concrete beams reinforced by nanoparticles vol.67, pp.3, 2018, https://doi.org/10.12989/sem.2018.67.3.311
  166. Analysis of boundary conditions effects on vibration of nanobeam in a polymeric matrix vol.67, pp.5, 2015, https://doi.org/10.12989/sem.2018.67.5.517
  167. Effect of homogenization models on stress analysis of functionally graded plates vol.67, pp.5, 2018, https://doi.org/10.12989/sem.2018.67.5.527
  168. Seismic analysis of AL2O3 nanoparticles-reinforced concrete plates based on sinusoidal shear deformation theory vol.15, pp.3, 2015, https://doi.org/10.12989/eas.2018.15.3.285
  169. Free vibration analysis of functionally graded cylindrical shells with different shell theories using semi-analytical method vol.28, pp.6, 2015, https://doi.org/10.12989/scs.2018.28.6.735
  170. A novel quasi-3D hyperbolic shear deformation theory for vibration analysis of simply supported functionally graded plates vol.22, pp.3, 2015, https://doi.org/10.12989/sss.2018.22.3.303
  171. Dynamic buckling of smart sandwich beam subjected to electric field based on hyperbolic piezoelasticity theory vol.22, pp.3, 2015, https://doi.org/10.12989/sss.2018.22.3.327
  172. Dynamic analysis of immersion concrete pipes in water subjected to earthquake load using mathematical methods vol.15, pp.4, 2015, https://doi.org/10.12989/eas.2018.15.4.361
  173. Analysis of wave propagation and free vibration of functionally graded porous material beam with a novel four variable refined theory vol.15, pp.4, 2018, https://doi.org/10.12989/eas.2018.15.4.369
  174. An analytical solution for free vibration of functionally graded beam using a simple first-order shear deformation theory vol.27, pp.4, 2015, https://doi.org/10.12989/was.2018.27.4.247
  175. A refined quasi-3D hybrid-type higher order shear deformation theory for bending and Free vibration analysis of advanced composites beams vol.27, pp.4, 2015, https://doi.org/10.12989/was.2018.27.4.269
  176. Size-dependent forced vibration response of embedded micro cylindrical shells reinforced with agglomerated CNTs using strain gradient theory vol.22, pp.5, 2015, https://doi.org/10.12989/sss.2018.22.5.527
  177. Dynamic and bending analysis of carbon nanotube-reinforced composite plates with elastic foundation vol.27, pp.5, 2018, https://doi.org/10.12989/was.2018.27.5.311
  178. A layerwise theory for buckling analysis of truncated conical shells reinforced by CNTs and carbon fibers integrated with piezoelectric layers in hygrothermal environment vol.6, pp.4, 2018, https://doi.org/10.12989/anr.2018.6.4.299
  179. Finite strain nonlinear longitudinal vibration of nanorods vol.6, pp.4, 2018, https://doi.org/10.12989/anr.2018.6.4.323
  180. A novel hyperbolic shear deformation theory for the mechanical buckling analysis of advanced composite plates resting on elastic foundations vol.30, pp.1, 2015, https://doi.org/10.12989/scs.2019.30.1.013
  181. Finite element solution of stress and flexural strength of functionally graded doubly curved sandwich shell panel vol.16, pp.1, 2015, https://doi.org/10.12989/eas.2019.16.1.055
  182. Nonlinear vibration of functionally graded nano-tubes using nonlocal strain gradient theory and a two-steps perturbation method vol.69, pp.2, 2015, https://doi.org/10.12989/sem.2019.69.2.205
  183. Dynamic investigation of porous functionally graded beam using a sinusoidal shear deformation theory vol.28, pp.1, 2015, https://doi.org/10.12989/was.2019.28.1.019
  184. Dynamic and wave propagation investigation of FGM plates with porosities using a four variable plate theory vol.28, pp.1, 2015, https://doi.org/10.12989/was.2019.28.1.049
  185. A novel refined shear deformation theory for the buckling analysis of thick isotropic plates vol.69, pp.3, 2019, https://doi.org/10.12989/sem.2019.69.3.335
  186. Dynamic analysis of concrete column reinforced with Sio2 nanoparticles subjected to blast load vol.7, pp.1, 2015, https://doi.org/10.12989/acc.2019.7.1.051
  187. Application of nonlocal elasticity theory on the wave propagation of flexoelectric functionally graded (FG) timoshenko nano-beams considering surface effects and residual surface stress vol.23, pp.2, 2015, https://doi.org/10.12989/sss.2019.23.2.141
  188. Vibration response and wave propagation in FG plates resting on elastic foundations using HSDT vol.69, pp.5, 2015, https://doi.org/10.12989/sem.2019.69.5.511
  189. Thermal buckling analysis of SWBNNT on Winkler foundation by non local FSDT vol.7, pp.2, 2015, https://doi.org/10.12989/anr.2019.7.2.089
  190. Vibration analysis of different material distributions of functionally graded microbeam vol.69, pp.6, 2015, https://doi.org/10.12989/sem.2019.69.6.637
  191. Nonlocal strain gradient model for thermal stability of FG nanoplates integrated with piezoelectric layers vol.23, pp.3, 2015, https://doi.org/10.12989/sss.2019.23.3.215
  192. Static and Dynamic Behavior of Nanotubes-Reinforced Sandwich Plates Using (FSDT) vol.57, pp.None, 2015, https://doi.org/10.4028/www.scientific.net/jnanor.57.117
  193. Postbuckling of Curved Carbon Nanotubes Using Energy Equivalent Model vol.57, pp.None, 2015, https://doi.org/10.4028/www.scientific.net/jnanor.57.136
  194. Participation Factor and Vibration of Carbon Nanotube with Vacancies vol.57, pp.None, 2015, https://doi.org/10.4028/www.scientific.net/jnanor.57.158
  195. Dynamic response of metal foam FG porous cylindrical micro-shells due to moving loads with strain gradient size-dependency vol.134, pp.5, 2015, https://doi.org/10.1140/epjp/i2019-12540-3
  196. Theoretical analysis of chirality and scale effects on critical buckling load of zigzag triple walled carbon nanotubes under axial compression embedded in polymeric matrix vol.70, pp.3, 2019, https://doi.org/10.12989/sem.2019.70.3.269
  197. A simple HSDT for bending, buckling and dynamic behavior of laminated composite plates vol.70, pp.3, 2019, https://doi.org/10.12989/sem.2019.70.3.325
  198. Improvement of thermal buckling response of FG-CNT reinforced composite beams with temperature-dependent material properties resting on elastic foundations vol.6, pp.3, 2019, https://doi.org/10.12989/aas.2019.6.3.207
  199. Dynamic analysis of nanosize FG rectangular plates based on simple nonlocal quasi 3D HSDT vol.7, pp.3, 2015, https://doi.org/10.12989/anr.2019.7.3.191
  200. Influence of shear preload on wave propagation in small-scale plates with nanofibers vol.70, pp.4, 2015, https://doi.org/10.12989/sem.2019.70.4.407
  201. The effect of parameters of visco-Pasternak foundation on the bending and vibration properties of a thick FG plate vol.18, pp.2, 2015, https://doi.org/10.12989/gae.2019.18.2.161
  202. A simple quasi-3D HSDT for the dynamics analysis of FG thick plate on elastic foundation vol.31, pp.5, 2015, https://doi.org/10.12989/scs.2019.31.5.503
  203. Stability analysis of embedded graphene platelets reinforced composite plates in thermal environment vol.134, pp.7, 2019, https://doi.org/10.1140/epjp/i2019-12581-6
  204. Dynamic analysis of multi-layered composite beams reinforced with graphene platelets resting on two-parameter viscoelastic foundation vol.134, pp.7, 2015, https://doi.org/10.1140/epjp/i2019-12739-2
  205. Vibration characteristics of zigzag and chiral functionally graded material rotating carbon nanotubes sandwich with ring supports vol.233, pp.16, 2015, https://doi.org/10.1177/0954406219855095
  206. Vibration analysis of nonlocal porous nanobeams made of functionally graded material vol.7, pp.5, 2019, https://doi.org/10.12989/anr.2019.7.5.351
  207. Nonlinear forced vibrations of sandwich smart nanobeams with two-phase piezo-magnetic face sheets vol.134, pp.10, 2019, https://doi.org/10.1140/epjp/i2019-12806-8
  208. Frequency response of initially deflected nanotubes conveying fluid via a nonlinear NSGT model vol.72, pp.1, 2015, https://doi.org/10.12989/sem.2019.72.1.071
  209. Influences of porosity on dynamic response of FG plates resting on Winkler/Pasternak/Kerr foundation using quasi 3D HSDT vol.24, pp.4, 2015, https://doi.org/10.12989/cac.2019.24.4.347
  210. A Non-Linear Spring Model for Predicting Modal Behavior of Oscillators Built from Double Walled Carbon Nanotubes vol.60, pp.None, 2015, https://doi.org/10.4028/www.scientific.net/jnanor.60.21
  211. The nano scale bending and dynamic properties of isolated protein microtubules based on modified strain gradient theory vol.7, pp.6, 2015, https://doi.org/10.12989/anr.2019.7.6.443
  212. A new higher-order shear and normal deformation theory for the buckling analysis of new type of FGM sandwich plates vol.72, pp.5, 2019, https://doi.org/10.12989/sem.2019.72.5.653
  213. On the modeling of dynamic behavior of composite plates using a simple nth-HSDT vol.29, pp.6, 2015, https://doi.org/10.12989/was.2019.29.6.371
  214. A refined of trigonometric shear deformation plate theory based on neutral surface position is proposed for static analysis of FGM plate. vol.26, pp.None, 2015, https://doi.org/10.1016/j.prostr.2020.06.016
  215. Porosity-dependent free vibration analysis of FG nanobeam using non-local shear deformation and energy principle vol.8, pp.1, 2020, https://doi.org/10.12989/anr.2020.8.1.037
  216. On transient hygrothermal vibration of embedded viscoelastic flexoelectric/piezoelectric nanobeams under magnetic loading vol.8, pp.1, 2015, https://doi.org/10.12989/anr.2020.8.1.049
  217. Hygrothermal postbuckling analysis of smart multiscale piezoelectric composite shells vol.135, pp.2, 2015, https://doi.org/10.1140/epjp/s13360-020-00137-w
  218. A simple nth-order shear deformation theory for thermomechanical bending analysis of different configurations of FG sandwich plates vol.25, pp.2, 2020, https://doi.org/10.12989/sss.2020.25.2.197
  219. Mechanical-hygro-thermal vibrations of functionally graded porous plates with nonlocal and strain gradient effects vol.7, pp.2, 2015, https://doi.org/10.12989/aas.2020.7.2.169
  220. A review of effects of partial dynamic loading on dynamic response of nonlocal functionally graded material beams vol.9, pp.1, 2015, https://doi.org/10.12989/amr.2020.9.1.033
  221. A numerical method for dynamic characteristics of nonlocal porous metal-ceramic plates under periodic dynamic loads vol.7, pp.1, 2020, https://doi.org/10.12989/smm.2020.7.1.027
  222. Parametrically excited nonlinear dynamics and instability of double-walled nanobeams under thermo-magneto-mechanical loads vol.26, pp.4, 2015, https://doi.org/10.1007/s00542-019-04638-2
  223. Computation of Non-isothermal Thermo-convective Micropolar Fluid Dynamics in a Hall MHD Generator System with Non-linear Distending Wall vol.6, pp.2, 2020, https://doi.org/10.1007/s40819-020-0792-y
  224. An inclined FGM beam under a moving mass considering Coriolis and centrifugal accelerations vol.35, pp.1, 2020, https://doi.org/10.12989/scs.2020.35.1.061
  225. A refined HSDT for bending and dynamic analysis of FGM plates vol.74, pp.1, 2020, https://doi.org/10.12989/sem.2020.74.1.105
  226. Investigation of microstructure and surface effects on vibrational characteristics of nanobeams based on nonlocal couple stress theory vol.8, pp.3, 2015, https://doi.org/10.12989/anr.2020.8.3.191
  227. Bending analysis of magneto-electro piezoelectric nanobeams system under hygro-thermal loading vol.8, pp.3, 2015, https://doi.org/10.12989/anr.2020.8.3.203
  228. Buckling and free vibration analyses of nanobeams with surface effects via various higher-order shear deformation theories vol.74, pp.2, 2020, https://doi.org/10.12989/sem.2020.74.2.175
  229. Thermal flexural analysis of anti-symmetric cross-ply laminated plates using a four variable refined theory vol.25, pp.4, 2015, https://doi.org/10.12989/sss.2020.25.4.409
  230. Stability of perforated nanobeams incorporating surface energy effects vol.35, pp.4, 2020, https://doi.org/10.12989/scs.2020.35.4.555
  231. Buckling of carbon nanotube (CNT)-reinforced composite skew plates by the discrete singular convolution method vol.231, pp.6, 2015, https://doi.org/10.1007/s00707-020-02653-3
  232. A comprehensive review on the modeling of smart piezoelectric nanostructures vol.74, pp.5, 2015, https://doi.org/10.12989/sem.2020.74.5.611
  233. Vibration analysis of nonlocal strain gradient porous FG composite plates coupled by visco-elastic foundation based on DQM vol.9, pp.3, 2020, https://doi.org/10.12989/csm.2020.9.3.201
  234. Mixture rule for studding the environmental pollution reduction in concrete structures containing nanoparticles vol.9, pp.3, 2015, https://doi.org/10.12989/csm.2020.9.3.281
  235. Application of Chebyshev-Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures vol.36, pp.3, 2015, https://doi.org/10.1007/s00366-019-00742-z
  236. Torsional vibration of functionally graded nano-rod under magnetic field supported by a generalized torsional foundation based on nonlocal elasticity theory vol.48, pp.4, 2015, https://doi.org/10.1080/15397734.2019.1642766
  237. Size-dependent free vibration and buckling analysis of sigmoid and power law functionally graded sandwich nanobeams with microstructural defects vol.234, pp.18, 2015, https://doi.org/10.1177/0954406220916481
  238. Static analysis of multilayer nonlocal strain gradient nanobeam reinforced by carbon nanotubes vol.36, pp.6, 2015, https://doi.org/10.12989/scs.2020.36.6.643
  239. Wave dispersion characteristics of fluid-conveying magneto-electro-elastic nanotubes vol.36, pp.4, 2015, https://doi.org/10.1007/s00366-019-00790-5
  240. Buckling treatment of piezoelectric functionally graded graphene platelets micro plates vol.38, pp.3, 2021, https://doi.org/10.12989/scs.2021.38.3.337
  241. Nonlinear free vibration analysis of a bi-directional functionally graded microbeam on nonlinear elastic foundation using modified couple stress theory vol.10, pp.1, 2015, https://doi.org/10.1142/s2047684121500019
  242. Nonlocal free vibration analysis of porous FG nanobeams using hyperbolic shear deformation beam theory vol.10, pp.3, 2015, https://doi.org/10.12989/anr.2021.10.3.281
  243. Buckling Analysis of CNTRC Curved Sandwich Nanobeams in Thermal Environment vol.11, pp.7, 2015, https://doi.org/10.3390/app11073250
  244. A nonlocal strain gradient theory for vibration and flutter instability analysis in rotary SWCNT with conveying viscous fluid vol.31, pp.2, 2021, https://doi.org/10.1080/17455030.2019.1584420
  245. Closed-form expressions for bending and buckling of functionally graded nanobeams by the Laplace transform vol.10, pp.2, 2015, https://doi.org/10.1142/s2047684121500123
  246. Applying Eringen’s nonlocal elasticity theory for analyzing the nonlinear free vibration of bidirectional functionally graded Euler-Bernoulli nanobeams vol.91, pp.7, 2015, https://doi.org/10.1007/s00419-021-01939-9
  247. Wave dispersion of nanobeams incorporating stretching effect vol.31, pp.4, 2015, https://doi.org/10.1080/17455030.2019.1607623
  248. Mechanical analysis of bi-functionally graded sandwich nanobeams vol.11, pp.1, 2015, https://doi.org/10.12989/anr.2021.11.1.055
  249. 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
  250. On static buckling of multilayered carbon nanotubes reinforced composite nanobeams supported on non-linear elastic foundations vol.40, pp.3, 2021, https://doi.org/10.12989/scs.2021.40.3.389
  251. Bending and buckling behaviors of heterogeneous temperature-dependent micro annular/circular porous sandwich plates integrated by FGPEM nano-Composite layers vol.23, pp.8, 2021, https://doi.org/10.1177/1099636220955027
  252. Nonlinear bending and free vibration analyses of metal-ceramic functionally graded plates by 2-D natural element method vol.35, pp.12, 2015, https://doi.org/10.1007/s12206-021-1130-y
  253. A review of size-dependent continuum mechanics models for micro- and nano-structures vol.170, pp.None, 2022, https://doi.org/10.1016/j.tws.2021.108562