52
2 Size-Dependent Theories of Beams, Plates and Shells
One of the earlier nonlocal HSDT (higher order shear deformation theory) models
was proposed by Aydogdu [363, 364] for isotropic nanobeams based on general exponential theory of shear deformations. The latter theory generalized the exponential
shear theory proposed by Karama et al. [365].
Thai [366] also proposed a nonlocal HSDT model for the isotropic nanobeams
but it was based on the refined theory of plates developed by Shimpi [367]. In this
theory, the field of displacement was separated into shear and bending parts. Tounsi
et al. [368] and Zemri et al. [369] extended the nonlocal HSDT model by inclusion
of thermal effects [368] and non-homogeneous FG material distribution [369]. Thai
and Vo [370] developed the nonlocal HSDT model for isotropic nanobeams based
on the sinusoidal theory of shear deformations. Touratier [371] and Tounsi et al.
[372] proposed the nonlocal quasi-3D-model for isotropic nanobeams based on the
quasi-3D sinusoidal theory proposed by Thai and Kim [373].
It should be noted that in contrary to the HSDT the quasi-3D-model enables to
describe an effect of thickness change which plays a crucial role in thick and short
beams. The general sinusoidal model [370] and quasi-3D sinusoidal model [372]
for the FG nanobeams were more extended by Ahouel et al. [374] and Chaht et
al. [375], respectively. The Thai and Vo [370] model was also revised by Pour et
al. [376] and Sadatshojaei and Sadatshojaie [377] for the prognosis of nonlinear
dynamic reactions of SWCNTs of carbon and boron nitride nanotube. Berrabah et
al. [378] compared the accuracy of various nonlocal HSDT models for the prognosis
of deflections, critical loads and eigenfrequency of nanobeams.
The fields of displacements of the nonlocal HSDT models were developed based
on the simple HSDT proposed by Thai and Choi [379], where tangential and transverse displacements have influenced the bending and shear components. Ebrahimi
and Barati [380] worked out the unified nonlocal HSDT model for heterogeneous FG
beams based on the simple HSDT model introduced by Thai and Choi [379]. This
model was employed to study the influence of temperature on vibration characteristics of the FG nanobeams. Mashat et al. [381] investigated vibrations and thermal
buckling of nanobeams embedded in an elastic medium for various boundary conditions with the use of a unified nonlocal HSDT model including EBT, TBT, RBT and
the sinusoidal theory. More recently, Thai et al. [382] proposed the simple nonlocal
HSDT model for isotropic nanobeams which consisted of only one unknown. In the
case of deflections and eigenfrequencies, there were obtained closed-form solutions
for the studied boundary conditions. The carried out simulations showed that accuracy of the mentioned theory is similar to the TBT nonlocal model though it is built
only on one unknown.
Nonlocal TSDT model was first proposed by Aghababaei and Reddy [383] for
isotropic nanoplates by transforming the TSDT proposed by Reddy [71] with the use
of nonlocal Eringen’s relations. In the case of freely supported nanoplates, there were
also obtained solutions in closed forms for the deflections and eigenfrequencies. The
last model was employed by Pradhan [384] and Pradhan and Sahu [385] to study
nonlocal effect under stability loss [384] and estimation of the eigenfrequencies [385]
of simply supported SLGS. The bending of SLGS was also considered by Ansari and
Sahmani [386] with the use of the unified nonlocal modes including various theories
2 Size-Dependent Theories of Beams, Plates and Shells
One of the earlier nonlocal HSDT (higher order shear deformation theory) models
was proposed by Aydogdu [363, 364] for isotropic nanobeams based on general exponential theory of shear deformations. The latter theory generalized the exponential
shear theory proposed by Karama et al. [365].
Thai [366] also proposed a nonlocal HSDT model for the isotropic nanobeams
but it was based on the refined theory of plates developed by Shimpi [367]. In this
theory, the field of displacement was separated into shear and bending parts. Tounsi
et al. [368] and Zemri et al. [369] extended the nonlocal HSDT model by inclusion
of thermal effects [368] and non-homogeneous FG material distribution [369]. Thai
and Vo [370] developed the nonlocal HSDT model for isotropic nanobeams based
on the sinusoidal theory of shear deformations. Touratier [371] and Tounsi et al.
[372] proposed the nonlocal quasi-3D-model for isotropic nanobeams based on the
quasi-3D sinusoidal theory proposed by Thai and Kim [373].
It should be noted that in contrary to the HSDT the quasi-3D-model enables to
describe an effect of thickness change which plays a crucial role in thick and short
beams. The general sinusoidal model [370] and quasi-3D sinusoidal model [372]
for the FG nanobeams were more extended by Ahouel et al. [374] and Chaht et
al. [375], respectively. The Thai and Vo [370] model was also revised by Pour et
al. [376] and Sadatshojaei and Sadatshojaie [377] for the prognosis of nonlinear
dynamic reactions of SWCNTs of carbon and boron nitride nanotube. Berrabah et
al. [378] compared the accuracy of various nonlocal HSDT models for the prognosis
of deflections, critical loads and eigenfrequency of nanobeams.
The fields of displacements of the nonlocal HSDT models were developed based
on the simple HSDT proposed by Thai and Choi [379], where tangential and transverse displacements have influenced the bending and shear components. Ebrahimi
and Barati [380] worked out the unified nonlocal HSDT model for heterogeneous FG
beams based on the simple HSDT model introduced by Thai and Choi [379]. This
model was employed to study the influence of temperature on vibration characteristics of the FG nanobeams. Mashat et al. [381] investigated vibrations and thermal
buckling of nanobeams embedded in an elastic medium for various boundary conditions with the use of a unified nonlocal HSDT model including EBT, TBT, RBT and
the sinusoidal theory. More recently, Thai et al. [382] proposed the simple nonlocal
HSDT model for isotropic nanobeams which consisted of only one unknown. In the
case of deflections and eigenfrequencies, there were obtained closed-form solutions
for the studied boundary conditions. The carried out simulations showed that accuracy of the mentioned theory is similar to the TBT nonlocal model though it is built
only on one unknown.
Nonlocal TSDT model was first proposed by Aghababaei and Reddy [383] for
isotropic nanoplates by transforming the TSDT proposed by Reddy [71] with the use
of nonlocal Eringen’s relations. In the case of freely supported nanoplates, there were
also obtained solutions in closed forms for the deflections and eigenfrequencies. The
last model was employed by Pradhan [384] and Pradhan and Sahu [385] to study
nonlocal effect under stability loss [384] and estimation of the eigenfrequencies [385]
of simply supported SLGS. The bending of SLGS was also considered by Ansari and
Sahmani [386] with the use of the unified nonlocal modes including various theories