56
2 Size-Dependent Theories of Beams, Plates and Shells
mulas of this model were obtained by Kim and Reddy [440] for the simply supported
plates. Lei et al. [441] proposed a simple quasi-3D theory to study static bending and
free vibrations of the FG microplates which include only five unknowns and cubic
form nonlinearity.
2.3.8.2 Modified Gradient Theories of RBT, TSDT, NSDT, HSDT
Based on a strain gradient elasticity theory of Lam et al. [18], Wang et al. [442]
formulated the RBT model for the account of influence of the size on static bending
and free vibrations of the isotropic microbeams. Sahmani and Ansari [443] improved
the model RBT of the deformations gradient by inclusion of the thermal effects of the
FG materials to study the stability of FG microbeams. Ansari et al. [444] investigated
the influence of size-dependent parameter on free vibrations of the simply supported
FG beams. Zhang et al. [445] proposed the RBT model for the FG beams based
on the improved RBT model of Shi [446]. Sahmani et al. [447] proposed the RBT
nonlinear model on the basis of the modified gradient theory of free vibrations of
the FG microbeams. In addition to the RBT, the HSDT models were also proposed
for the microbeams on the basis of the modified gradient strain theories such as
the sinusoidal theory of Touratier [371], hyperbolic theory of Soldatos [405] and
theory of shear n-th order deformation proposed by Xiang et al. [448]. For instance,
Akgoz and Civalek [449] and Lei et al. [450] proposed the sinusoidal models of the
deformation gradient for the analysis of bending and free vibrations of microplates
made from isotropic materials [449] and FG materials [450] on the basis of Touratier
theory [371]. Akgoz and Civalek [451] employed their previous theory [449] to study
problems of buckling of isotropic microbeams. Akgoz and Civalek [452] also worked
out the sinusoidal model of deformation gradient for the FG microbeams, as it was
done by Lei et al. [450]. They also presented a new equation useful for computation
of the shear correction coefficient of TBT model. In the mentioned equation, the
coefficient of the shear correction stands for the function of the scale length material
parameter. Akgoz and Civalek [453] extended their previous work [452] in order to
clarify the interaction of FG microbeam and the Winkler elastic foundation. Based
on the hyperbolic theory of Soldatos [405], Akgoz and Civalek [454] proposed the
hyperbolic model of the deformation gradient to study bending and stability of the
microbeams. Akgoz and Civalek [455] proposed unified HSDT model for analysis of
bending of simply supported carbon nanotubes resting on an elastic foundation. The
field of displacements followed the work of Simsek and Reddy [419], which accounts
of theories of beams including EBT, TBT, RBT, sinusoidal theory of Touratier [371],
the hyperbolic theory of Soldatos [420], exponential theory of Karama et al. [365]
and the general exponential theory proposed by Aydogdu [364]. Zhang et al. [456]
proposed the HSDT model to study bending and vibrations of curved FG microbeams
based on the theory of shear deformations of n-th order (see [448]).
Sahmani and Ansari [347] developed the TSDT model with a gradient of deformations for free vibrations analysis of the FG microplates. In the case of simply
supported plates, there were also presented closed solutions for the eigenfrequen-
2 Size-Dependent Theories of Beams, Plates and Shells
mulas of this model were obtained by Kim and Reddy [440] for the simply supported
plates. Lei et al. [441] proposed a simple quasi-3D theory to study static bending and
free vibrations of the FG microplates which include only five unknowns and cubic
form nonlinearity.
2.3.8.2 Modified Gradient Theories of RBT, TSDT, NSDT, HSDT
Based on a strain gradient elasticity theory of Lam et al. [18], Wang et al. [442]
formulated the RBT model for the account of influence of the size on static bending
and free vibrations of the isotropic microbeams. Sahmani and Ansari [443] improved
the model RBT of the deformations gradient by inclusion of the thermal effects of the
FG materials to study the stability of FG microbeams. Ansari et al. [444] investigated
the influence of size-dependent parameter on free vibrations of the simply supported
FG beams. Zhang et al. [445] proposed the RBT model for the FG beams based
on the improved RBT model of Shi [446]. Sahmani et al. [447] proposed the RBT
nonlinear model on the basis of the modified gradient theory of free vibrations of
the FG microbeams. In addition to the RBT, the HSDT models were also proposed
for the microbeams on the basis of the modified gradient strain theories such as
the sinusoidal theory of Touratier [371], hyperbolic theory of Soldatos [405] and
theory of shear n-th order deformation proposed by Xiang et al. [448]. For instance,
Akgoz and Civalek [449] and Lei et al. [450] proposed the sinusoidal models of the
deformation gradient for the analysis of bending and free vibrations of microplates
made from isotropic materials [449] and FG materials [450] on the basis of Touratier
theory [371]. Akgoz and Civalek [451] employed their previous theory [449] to study
problems of buckling of isotropic microbeams. Akgoz and Civalek [452] also worked
out the sinusoidal model of deformation gradient for the FG microbeams, as it was
done by Lei et al. [450]. They also presented a new equation useful for computation
of the shear correction coefficient of TBT model. In the mentioned equation, the
coefficient of the shear correction stands for the function of the scale length material
parameter. Akgoz and Civalek [453] extended their previous work [452] in order to
clarify the interaction of FG microbeam and the Winkler elastic foundation. Based
on the hyperbolic theory of Soldatos [405], Akgoz and Civalek [454] proposed the
hyperbolic model of the deformation gradient to study bending and stability of the
microbeams. Akgoz and Civalek [455] proposed unified HSDT model for analysis of
bending of simply supported carbon nanotubes resting on an elastic foundation. The
field of displacements followed the work of Simsek and Reddy [419], which accounts
of theories of beams including EBT, TBT, RBT, sinusoidal theory of Touratier [371],
the hyperbolic theory of Soldatos [420], exponential theory of Karama et al. [365]
and the general exponential theory proposed by Aydogdu [364]. Zhang et al. [456]
proposed the HSDT model to study bending and vibrations of curved FG microbeams
based on the theory of shear deformations of n-th order (see [448]).
Sahmani and Ansari [347] developed the TSDT model with a gradient of deformations for free vibrations analysis of the FG microplates. In the case of simply
supported plates, there were also presented closed solutions for the eigenfrequen-