2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
53
like CPT, FSDT and TSDT. Hosseini-Hashemi et al. [387] obtained Levi’s solutions
for the critical loads and eigenfrequencies of the isotropic nanoplates. Daneshmehr
et al. [388, 389] extended employment of the nonlocal TSDT into stability [388] and
analysis of free vibrations [389] of the FG nanoplates.
2.3.8 Nonlocal HSDT-Based Models
Narendar [390] proposed a nonlocal HSDT model for the analysis of stability loss
for isotropic micro/nanoscale plates based on the improved theory of plates proposed
by Shimpi [367] (two-variable refined plate theory). The latter model was extended
by Malekzadeh and Shojaee [391], Narendar and Gopalakrishnan [392] to study
free vibrations of the nanoplates [391] and to analyse stability loss of orthotropic
nanoplates [392]. This model was also employed by Sobhy [393] to study free vibrations of the orthotropic DLGS under hygrothermal deformations. Sobhy [394] proposed the general model of HSDT and MLGS based on the simple HSDT proposed
by Thai and Choi [379]. Analytical solutions with regard to eigenfrequencies, critical
loads, temperature and stability loss were also obtained for MLGS for various boundary conditions. Levi’s solutions of the nonlocal HSDT model introduced by Narendar
[390] were also obtained by Sobhy [395, 396] while analysing bending of isotropic
SLGS in the thermal environment [395] and orthotropic nanoplates in hygrothermal
environment [396]. Zenkour and Sobhy [397], Alzahrani et al. [398], Thai et al.
[399] and Sobhy [400, 401] developed the nonlocal sinusoidal models of thermal
buckling of nanoplates [397], hygro-thermomechanical bending of nanoplates [398],
isotropic nanoplates [399], thermomechanical bending of SLGS [400] and buckling
of orthotropic nanoplates [401] based on the sinusoidal theory of Touratier [371].
It should be noted that the nonlocal sinusoidal model proposed by Sobhy [402] for
the FGM nanoplates was based on the simple theory of Thai and Vo [403], and it is
more simple in comparison to nonlocal sinusoidal models proposed in [397–401].
Belkorissat et al. [404] also worked out a simple nonlocal HSDT model for the FG
nanoplates based on the work of Sobhy [402], but it was based on the hyperbolic
function proposed by Soldatos [405]. Khorshidi and Fallah [406] reformulated the
exponential theory of Karama et al. [365] for the nanoplates. Bessaim et al. [407]
worked out the nonlocal quasi-3D model for the analysis of free vibrations of the
isotropic nanoplates based on the quasi-3D sinusoidal theory of Thai and Kim [373],
which is spanned on five unknowns. In the last years, Sobhy and Radwan [408] also
developed a nonlocal quasi-3D theory for free vibrations and bending of the FG
nanoplates. The latter consists of five unknown and is analogous to that proposed in
[407] but being based on the new hyperbolic function.
53
like CPT, FSDT and TSDT. Hosseini-Hashemi et al. [387] obtained Levi’s solutions
for the critical loads and eigenfrequencies of the isotropic nanoplates. Daneshmehr
et al. [388, 389] extended employment of the nonlocal TSDT into stability [388] and
analysis of free vibrations [389] of the FG nanoplates.
2.3.8 Nonlocal HSDT-Based Models
Narendar [390] proposed a nonlocal HSDT model for the analysis of stability loss
for isotropic micro/nanoscale plates based on the improved theory of plates proposed
by Shimpi [367] (two-variable refined plate theory). The latter model was extended
by Malekzadeh and Shojaee [391], Narendar and Gopalakrishnan [392] to study
free vibrations of the nanoplates [391] and to analyse stability loss of orthotropic
nanoplates [392]. This model was also employed by Sobhy [393] to study free vibrations of the orthotropic DLGS under hygrothermal deformations. Sobhy [394] proposed the general model of HSDT and MLGS based on the simple HSDT proposed
by Thai and Choi [379]. Analytical solutions with regard to eigenfrequencies, critical
loads, temperature and stability loss were also obtained for MLGS for various boundary conditions. Levi’s solutions of the nonlocal HSDT model introduced by Narendar
[390] were also obtained by Sobhy [395, 396] while analysing bending of isotropic
SLGS in the thermal environment [395] and orthotropic nanoplates in hygrothermal
environment [396]. Zenkour and Sobhy [397], Alzahrani et al. [398], Thai et al.
[399] and Sobhy [400, 401] developed the nonlocal sinusoidal models of thermal
buckling of nanoplates [397], hygro-thermomechanical bending of nanoplates [398],
isotropic nanoplates [399], thermomechanical bending of SLGS [400] and buckling
of orthotropic nanoplates [401] based on the sinusoidal theory of Touratier [371].
It should be noted that the nonlocal sinusoidal model proposed by Sobhy [402] for
the FGM nanoplates was based on the simple theory of Thai and Vo [403], and it is
more simple in comparison to nonlocal sinusoidal models proposed in [397–401].
Belkorissat et al. [404] also worked out a simple nonlocal HSDT model for the FG
nanoplates based on the work of Sobhy [402], but it was based on the hyperbolic
function proposed by Soldatos [405]. Khorshidi and Fallah [406] reformulated the
exponential theory of Karama et al. [365] for the nanoplates. Bessaim et al. [407]
worked out the nonlocal quasi-3D model for the analysis of free vibrations of the
isotropic nanoplates based on the quasi-3D sinusoidal theory of Thai and Kim [373],
which is spanned on five unknowns. In the last years, Sobhy and Radwan [408] also
developed a nonlocal quasi-3D theory for free vibrations and bending of the FG
nanoplates. The latter consists of five unknown and is analogous to that proposed in
[407] but being based on the new hyperbolic function.