2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
49
the axially symmetric FSDT model for nonlinear free vibrations [322] and stability
[323] of annular FG microplates. Thai and Choi [324] proposed nonlinear CPT
and FSDT models for the FG microplates. They obtained analytical formulas for
linear/nonlinear deflections, critical loads and the eigenfrequencies of the simply
supported microplates and quantified the influence of the size effect on bending,
stability and dynamic characteristics of the FG microplates. Jung et al. [325, 326]
took into consideration interaction of a plate with an elastic medium in the FSDT
model while investigating the size-dependent bending effect, vibration [325] and
stability [326] of simply supported FG microplates and nanoplates. Nonlinear models
were employed by Ansari et al. [327, 328] for the case of nonlinear vibration [327],
nonlinear bending and stability analysis [328] of the FG microplates. It was illustrated
and discussed that in the nonlinear FSDT model worked out in [328] the physically
neutral surface of the FG plate was considered, and the bending due to extension
was removed. Ansari et al. [329] used nonlinear FSDT model to investigate the size
effect on stability and frequencies of the FG microplates. Lou and He [330] also
developed nonlinear models of CPT and FSDT for the case of nonlinear analysis of
bending and free vibrations of the FG microplates. In their models, an interaction
between the plate and elastic medium, as well as physical neutral plane position of
the FG plate was taken into account.
Based on the FSDT, Zeighampour and Beni [331] and Hosseini-Hashemi et al.
[332] proposed the models of shells to analyse free vibrations of the isotropic cylindrical micro/nanoshells [331] and spherical micro/nanoshells [332]. Gholami et al.
[333] also developed the FSDT shell model which was used to study axial buckling
and dynamic stability loss of the FG microshell. Tadi Beni et al. [334] proposed the
FSDT shell model for the cylindrical FG microshells and employed it to the problems
of free vibrations. Lou et al. [335] worked out the nonlinear FSDT shell model for
estimation of the pre-buckling and buckling deformation and studied the influence
and a value of the length size parameter on the critical loads of the stability loss of the
cylindrical FG shells. Those models included a concept of a physical neutral plane
position of the FG shells.
2.3.6.3 Modified Gradient Timoshenko Theory
Wang et al. [220] belong to the first who developed the TBT model with an account
of the deformation gradient for the static bending and analysis of free vibrations of
isotropic size-dependent and simply supported microbeams. Models of TBT nonlinear gradient of deformation were developed by Ansari et al. [336] and Asghari
et al. [337] for isotropic microbeams using the von Kármán nonlinearity. It should
be noted that Ansari et al. [336] employed their model for a study of free nonlinear
vibrations, whereas Asghari et al. [337] considered in addition to nonlinear bending.
Ansari et al. [338] extended the TBT model with the deformations gradient onto
FG microbeams. Solutions in closed forms for the eigenfrequencies were obtained
of the simply supported microbeams in order to follow the influence of index of
material gradient and size-dependent length material parameter on free vibration of
49
the axially symmetric FSDT model for nonlinear free vibrations [322] and stability
[323] of annular FG microplates. Thai and Choi [324] proposed nonlinear CPT
and FSDT models for the FG microplates. They obtained analytical formulas for
linear/nonlinear deflections, critical loads and the eigenfrequencies of the simply
supported microplates and quantified the influence of the size effect on bending,
stability and dynamic characteristics of the FG microplates. Jung et al. [325, 326]
took into consideration interaction of a plate with an elastic medium in the FSDT
model while investigating the size-dependent bending effect, vibration [325] and
stability [326] of simply supported FG microplates and nanoplates. Nonlinear models
were employed by Ansari et al. [327, 328] for the case of nonlinear vibration [327],
nonlinear bending and stability analysis [328] of the FG microplates. It was illustrated
and discussed that in the nonlinear FSDT model worked out in [328] the physically
neutral surface of the FG plate was considered, and the bending due to extension
was removed. Ansari et al. [329] used nonlinear FSDT model to investigate the size
effect on stability and frequencies of the FG microplates. Lou and He [330] also
developed nonlinear models of CPT and FSDT for the case of nonlinear analysis of
bending and free vibrations of the FG microplates. In their models, an interaction
between the plate and elastic medium, as well as physical neutral plane position of
the FG plate was taken into account.
Based on the FSDT, Zeighampour and Beni [331] and Hosseini-Hashemi et al.
[332] proposed the models of shells to analyse free vibrations of the isotropic cylindrical micro/nanoshells [331] and spherical micro/nanoshells [332]. Gholami et al.
[333] also developed the FSDT shell model which was used to study axial buckling
and dynamic stability loss of the FG microshell. Tadi Beni et al. [334] proposed the
FSDT shell model for the cylindrical FG microshells and employed it to the problems
of free vibrations. Lou et al. [335] worked out the nonlinear FSDT shell model for
estimation of the pre-buckling and buckling deformation and studied the influence
and a value of the length size parameter on the critical loads of the stability loss of the
cylindrical FG shells. Those models included a concept of a physical neutral plane
position of the FG shells.
2.3.6.3 Modified Gradient Timoshenko Theory
Wang et al. [220] belong to the first who developed the TBT model with an account
of the deformation gradient for the static bending and analysis of free vibrations of
isotropic size-dependent and simply supported microbeams. Models of TBT nonlinear gradient of deformation were developed by Ansari et al. [336] and Asghari
et al. [337] for isotropic microbeams using the von Kármán nonlinearity. It should
be noted that Ansari et al. [336] employed their model for a study of free nonlinear
vibrations, whereas Asghari et al. [337] considered in addition to nonlinear bending.
Ansari et al. [338] extended the TBT model with the deformations gradient onto
FG microbeams. Solutions in closed forms for the eigenfrequencies were obtained
of the simply supported microbeams in order to follow the influence of index of
material gradient and size-dependent length material parameter on free vibration of