Structural Engineering and Mechanics

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A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory

  • Zemri, Amine (Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of Sidi Bel Abbes) ;
  • Houari, Mohammed Sid Ahmed (Departement de Genie Civil, Universite de Mascara) ;
  • Bousahla, Abdelmoumen Anis (Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of Sidi Bel Abbes) ;
  • Tounsi, Abdelouahed (Material and Hydrology Laboratory, Faculty of Technology, Civil Engineering Department, University of Sidi Bel Abbes)
  • Received : 2014.08.19
  • Accepted : 2015.02.21
  • Published : 2015.05.25
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Abstract

This paper presents a nonlocal shear deformation beam theory for bending, buckling, and vibration of functionally graded (FG) nanobeams using the nonlocal differential constitutive relations of Eringen. The developed theory account for higher-order variation of transverse shear strain through the depth of the nanobeam, and satisfy the stress-free boundary conditions on the top and bottom surfaces of the nanobeam. A shear correction factor, therefore, is not required. In addition, this nonlocal nanobeam model incorporates the length scale parameter which can capture the small scale effect and it has strong similarities with Euler-Bernoulli beam model in some aspects such as equations of motion, boundary conditions, and stress resultant expressions. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The equations of motion are derived from Hamilton's principle. Analytical solutions are presented for a simply supported FG nanobeam, and the obtained results compare well with those predicted by the nonlocal Timoshenko beam theory.

Keywords

Acknowledgement

Supported by : Algerian National Thematic Agency of Research in Science and Technology (ATRST), university of Sidi Bel Abbes (UDL SBA)

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  44. Analytical modeling of dynamic behavior of piezo-thermo-electrically affected sigmoid and power-law graded nanoscale beams vol.122, pp.9, 2016, https://doi.org/10.1007/s00339-016-0273-7
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  49. Nonlinear analysis of size-dependent and material-dependent nonlocal CNTs vol.153, 2016, https://doi.org/10.1016/j.compstruct.2016.07.013
  50. 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
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  54. 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
  55. Wave propagation analysis of quasi-3D FG nanobeams in thermal environment based on nonlocal strain gradient theory vol.122, pp.9, 2016, https://doi.org/10.1007/s00339-016-0368-1
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  57. 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
  58. Magnetic field effects on nonlocal wave dispersion characteristics of size-dependent nanobeams vol.123, pp.1, 2017, https://doi.org/10.1007/s00339-016-0646-y
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  60. 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
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  62. 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
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  65. 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
  66. Vibration analysis of embedded biaxially loaded magneto-electrically actuated inhomogeneous nanoscale plates 2017, https://doi.org/10.1177/1077546317708105
  67. Nonlinear atomic vibrations and structural phase transitions in strained carbon chains vol.138, 2017, https://doi.org/10.1016/j.commatsci.2017.07.004
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  69. 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
  70. 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
  71. Dynamic characteristics of temperature-dependent viscoelastic FG nanobeams subjected to 2D-magnetic field under periodic loading vol.123, pp.4, 2017, https://doi.org/10.1007/s00339-017-0829-1
  72. 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
  73. Wave dispersion characteristics of axially loaded magneto-electro-elastic nanobeams vol.122, pp.11, 2016, https://doi.org/10.1007/s00339-016-0465-1
  74. Mechanical analysis of functionally graded graphene oxide-reinforced composite beams based on the first-order shear deformation theory pp.1537-6532, 2020, https://doi.org/10.1080/15376494.2018.1444216
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  76. Buckling Analysis of Orthotropic Nanoscale Plates Resting on Elastic Foundations vol.55, pp.1661-9897, 2018, https://doi.org/10.4028/www.scientific.net/JNanoR.55.42
  77. 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
  78. Post-buckling analysis of piezo-magnetic nanobeams with geometrical imperfection and different piezoelectric contents pp.1432-1858, 2018, https://doi.org/10.1007/s00542-018-4241-3
  79. Nonlocal Thermal and Mechanical Buckling of Nonlinear Orthotropic Viscoelastic Nanoplates Embedded in a Visco-Pasternak Medium vol.10, pp.08, 2018, https://doi.org/10.1142/S1758825118500862
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  81. Stabilities and electronic properties of nanowires made of single atomic sulfur chains encapsulated in zigzag carbon nanotubes vol.29, pp.41, 2018, https://doi.org/10.1088/1361-6528/aad67a
  82. Axial magnetic field effects on dynamic characteristics of embedded multiphase nanocrystalline nanobeams vol.24, pp.8, 2018, https://doi.org/10.1007/s00542-018-3771-z
  83. Dynamic stability analysis of microcomposite annular sandwich plate with carbon nanotube reinforced composite facesheets based on modified strain gradient theory pp.1530-7972, 2018, https://doi.org/10.1177/1099636218782770
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  85. Smart electrical and magnetic stability analysis of exponentially graded shear deformable three-layered nanoplate based on nonlocal piezo-magneto-elasticity theory pp.1530-7972, 2018, https://doi.org/10.1177/1099636218760667
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  90. Modal participation of fixed–fixed single-walled carbon nanotube with vacancies pp.2008-6695, 2019, https://doi.org/10.1007/s40091-019-0222-8
  91. 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
  92. 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
  93. 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
  94. 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
  95. 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
  96. 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
  97. 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
  98. Hygro-thermo-mechanical vibration and buckling of exponentially graded nanoplates resting on elastic foundations via nonlocal elasticity theory vol.63, pp.3, 2015, https://doi.org/10.12989/sem.2017.63.3.401
  99. 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
  100. 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
  101. Vibration analysis of FG nanobeams based on third-order shear deformation theory under various boundary conditions vol.25, pp.1, 2017, https://doi.org/10.12989/scs.2017.25.1.067
  102. 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
  103. 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
  104. 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
  105. Surface effects on vibration and buckling behavior of embedded nanoarches vol.64, pp.1, 2017, https://doi.org/10.12989/sem.2017.64.1.001
  106. Dynamic characteristics of curved inhomogeneous nonlocal porous beams in thermal environment vol.64, pp.1, 2015, https://doi.org/10.12989/sem.2017.64.1.121
  107. 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
  108. 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
  109. An analytical solution for bending and vibration responses of functionally graded beams with porosities vol.25, pp.4, 2015, https://doi.org/10.12989/was.2017.25.4.329
  110. 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
  111. 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
  112. 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
  113. 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
  114. Vibration analysis of micro composite thin beam based on modified couple stress vol.64, pp.4, 2017, https://doi.org/10.12989/sem.2017.64.4.403
  115. 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
  116. 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
  117. 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
  118. 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
  119. 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
  120. Post-buckling responses of a laminated composite beam vol.26, pp.6, 2015, https://doi.org/10.12989/scs.2018.26.6.733
  121. Wave dispersion characteristics of nonlocal strain gradient double-layered graphene sheets in hygro-thermal environments vol.65, pp.6, 2018, https://doi.org/10.12989/sem.2018.65.6.645
  122. 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
  123. 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
  124. 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
  125. 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
  126. 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
  127. 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
  128. 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
  129. 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
  130. 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
  131. A unified formulation for modeling of inhomogeneous nonlocal beams vol.66, pp.3, 2015, https://doi.org/10.12989/sem.2018.66.3.369
  132. 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
  133. 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
  134. 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
  135. 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
  136. 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
  137. 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
  138. 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
  139. 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
  140. 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
  141. 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
  142. 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
  143. 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
  144. Analytical wave dispersion modeling in advanced piezoelectric double-layered nanobeam systems vol.67, pp.2, 2015, https://doi.org/10.12989/sem.2018.67.2.175
  145. 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
  146. 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
  147. 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
  148. 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
  149. 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
  150. 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
  151. Bending of a cracked functionally graded nanobeam vol.6, pp.3, 2015, https://doi.org/10.12989/anr.2018.6.3.219
  152. Dynamic analysis of higher order shear-deformable nanobeams resting on elastic foundation based on nonlocal strain gradient theory vol.6, pp.3, 2018, https://doi.org/10.12989/anr.2018.6.3.279
  153. 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
  154. Nonlinear finite element solutions of thermoelastic flexural strength and stress values of temperature dependent graded CNT-reinforced sandwich shallow shell structure vol.67, pp.6, 2018, https://doi.org/10.12989/sem.2018.67.6.565
  155. 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
  156. 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
  157. 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
  158. 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
  159. 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
  160. 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
  161. Critical buckling loads of carbon nanotube embedded in Kerr's medium vol.6, pp.4, 2015, https://doi.org/10.12989/anr.2018.6.4.339
  162. 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
  163. Small-scale effect on the forced vibration of a nano beam embedded an elastic medium using nonlocal elasticity theory vol.6, pp.1, 2019, https://doi.org/10.12989/aas.2019.6.1.001
  164. 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
  165. 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
  166. 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
  167. Analyzing post-buckling behavior of continuously graded FG nanobeams with geometrical imperfections vol.17, pp.2, 2019, https://doi.org/10.12989/gae.2019.17.2.175
  168. 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
  169. 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
  170. Effect of the micromechanical models on the bending of FGM beam using a new hyperbolic shear deformation theory vol.16, pp.2, 2019, https://doi.org/10.12989/eas.2019.16.2.177
  171. 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
  172. Application of the nonlocal strain gradient elasticity on the wave dispersion behaviors of inhomogeneous nanosize beams vol.134, pp.3, 2015, https://doi.org/10.1140/epjp/i2019-12464-x
  173. 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
  174. 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
  175. Free vibration of an annular sandwich plate with CNTRC facesheets and FG porous cores using Ritz method vol.7, pp.2, 2015, https://doi.org/10.12989/anr.2019.7.2.109
  176. 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
  177. 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
  178. 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
  179. 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
  180. Vibration analysis of graphene oxide powder-/carbon fiber-reinforced multi-scale porous nanocomposite beams: A finite-element study vol.134, pp.5, 2015, https://doi.org/10.1140/epjp/i2019-12594-1
  181. 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
  182. 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
  183. 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
  184. 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
  185. 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
  186. 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
  187. Finite element formulation and vibration of nonlocal refined metal foam beams with symmetric and non-symmetric porosities vol.6, pp.2, 2015, https://doi.org/10.12989/smm.2019.6.2.147
  188. Conformable solution of fractional vibration problem of plate subjected to in-plane loads vol.28, pp.6, 2019, https://doi.org/10.12989/was.2019.28.6.347
  189. A novel meshless particle method for nonlocal analysis of two-directional functionally graded nanobeams vol.41, pp.7, 2015, https://doi.org/10.1007/s40430-019-1799-3
  190. 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
  191. 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
  192. Study on shear force distribution in structural design of BRBF structure with high β value vol.193, pp.None, 2015, https://doi.org/10.1016/j.engstruct.2019.05.017
  193. Analyzing large-amplitude vibration of nonlocal beams made of different piezo-electric materials in thermal environment vol.8, pp.3, 2019, https://doi.org/10.12989/amr.2019.8.3.237
  194. 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
  195. Static analysis of monoclinic plates via a three-dimensional model using differential quadrature method vol.72, pp.1, 2019, https://doi.org/10.12989/sem.2019.72.1.131
  196. 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
  197. 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
  198. Nonlocal effect on the vibration of armchair and zigzag SWCNTs with bending rigidity vol.7, pp.6, 2015, https://doi.org/10.12989/anr.2019.7.6.431
  199. 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
  200. Dynamic modeling of a multi-scale sandwich composite panel containing flexible core and MR smart layer vol.134, pp.12, 2015, https://doi.org/10.1140/epjp/i2019-12662-6
  201. 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
  202. Flexoelectric effects on dynamic response characteristics of nonlocal piezoelectric material beam vol.8, pp.4, 2015, https://doi.org/10.12989/amr.2019.8.4.259
  203. 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
  204. Finite element forced vibration analysis of refined shear deformable nanocomposite graphene platelet-reinforced beams vol.42, pp.1, 2015, https://doi.org/10.1007/s40430-019-2118-8
  205. 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
  206. 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
  207. Transfer matrix formulations and single variable shear deformation theory for crack detection in beam-like structures vol.73, pp.2, 2020, https://doi.org/10.12989/sem.2020.73.2.109
  208. Free vibration analysis of FG nanoplate with poriferous imperfection in hygrothermal environment vol.73, pp.2, 2020, https://doi.org/10.12989/sem.2020.73.2.191
  209. Free vibration analysis of sandwich FGM shells using isogeometric B-spline finite strip method vol.34, pp.3, 2020, https://doi.org/10.12989/scs.2020.34.3.361
  210. Nonlocal vibration of DWCNTs based on Flügge shell model using wave propagation approach vol.34, pp.4, 2020, https://doi.org/10.12989/scs.2020.34.4.599
  211. 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
  212. Transient response of porous inhomogeneous nanobeams due to various impulsive loads based on nonlocal strain gradient elasticity vol.16, pp.1, 2015, https://doi.org/10.1007/s10999-019-09452-2
  213. 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
  214. 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
  215. Analysis of post-buckling of higher-order graphene oxide reinforced concrete plates with geometrical imperfection vol.9, pp.4, 2015, https://doi.org/10.12989/acc.2020.9.4.397
  216. Finite element based modeling and thermal dynamic analysis of functionally graded graphene reinforced beams vol.5, pp.2, 2015, https://doi.org/10.12989/acd.2020.5.2.177
  217. 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
  218. 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
  219. 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
  220. Theoretical impact of Kelvin's theory for vibration of double walled carbon nanotubes vol.8, pp.4, 2015, https://doi.org/10.12989/anr.2020.8.4.307
  221. Nonlinear vibration of smart nonlocal magneto-electro-elastic beams resting on nonlinear elastic substrate with geometrical imperfection and various piezoelectric effects vol.25, pp.5, 2015, https://doi.org/10.12989/sss.2020.25.5.619
  222. 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
  223. Effect of chiral structure for free vibration of DWCNTs: Modal analysis vol.9, pp.6, 2020, https://doi.org/10.12989/acc.2020.9.6.577
  224. 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
  225. Elastic wave characteristics of graphene nanoplatelets reinforced composite nanoplates vol.74, pp.6, 2020, https://doi.org/10.12989/sem.2020.74.6.809
  226. Nonlinear stability of smart nonlocal magneto-electro-thermo-elastic beams with geometric imperfection and piezoelectric phase effects vol.25, pp.6, 2020, https://doi.org/10.12989/sss.2020.25.6.707
  227. 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
  228. Analyzing exact nonlinear forced vibrations of two-phase magneto-electro-elastic nanobeams under an elliptic-type force vol.9, pp.1, 2015, https://doi.org/10.12989/anr.2020.9.1.047
  229. Post-buckling analysis of geometrically imperfect nanoparticle reinforced annular sector plates under radial compression vol.26, pp.1, 2020, https://doi.org/10.12989/cac.2020.26.1.021
  230. 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
  231. On scale-dependent stability analysis of functionally graded magneto-electro-thermo-elastic cylindrical nanoshells vol.75, pp.6, 2015, https://doi.org/10.12989/sem.2020.75.6.659
  232. Wave dispersion characteristics of fluid-conveying magneto-electro-elastic nanotubes vol.36, pp.4, 2015, https://doi.org/10.1007/s00366-019-00790-5
  233. Nonlinear vibration and stability of FG nanotubes conveying fluid via nonlocal strain gradient theory vol.78, pp.1, 2021, https://doi.org/10.12989/sem.2021.78.1.103
  234. Wave dispersion of nanobeams incorporating stretching effect vol.31, pp.4, 2015, https://doi.org/10.1080/17455030.2019.1607623
  235. Mechanical analysis of bi-functionally graded sandwich nanobeams vol.11, pp.1, 2015, https://doi.org/10.12989/anr.2021.11.1.055
  236. 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
  237. Propagation of waves with nonlocal effects for vibration response of armchair double-walled CNTs vol.11, pp.2, 2015, https://doi.org/10.12989/anr.2021.11.2.183