40
2 Size-Dependent Theories of Beams, Plates and Shells
of gradient deformations was compared with the accuracy of modified couple stress
theory and classical theory. The comparison results showed that the model of gradient deformations predicts the size effect in a better way than the method of modified
couple stress theory, since it takes in addition into account the additional tensor of
the dilatation gradient, deviator tensor of extension gradient and a supplement to the
tensor of rotations gradient. Akgoz and Civalek [191] extended the EBT model of
deformations gradient into analysis of stability loss of isotropic microbeams for cantilever and simply supported beams. Akgoz and Civalek [192–195] also employed
EBT model of the deformation gradient for the study of influence of the size effect
on stability of SWCNT [192], static bending SWCNT [193], stability of linearly
narrow microbeams [194] and longitudinal vibrations of microbeams [195]. Zhao et
al. [196] developed the nonlinear EBT model of deformation gradient for nonlinear
stability loss and nonlinear analysis of free vibrations of isotropic microbeams. They
pointed out the importance of geometric nonlinearity and size-dependent effects
while getting reliable results. Rajabi and Ramezani [197] also developed the model
of nonlinear gradient of EBT deformations for the isotropic microbeams but mainly
considered static bending and free vibrations. The model of nonlinear EBT beam was
considered by Mohammadi and Mahzoon [198] on a basis of deformation gradient
where also the influence of temperature on stability of isotropic microbeams was
taken into account. They derived analytical solutions of microbeams with various
boundary conditions. Vatankhah et al. [199] used the model of nonlinear gradient of
deformations of EBT for a study of nonlinear vibrations of isotropic microbeams.
Kahrobaiyan et al. [200] extended application of the EBT model with gradient of
deformations to quantify stability and free vibrations of the FG microbeams. Further
extension of this model into problems of stability of FG microbeams was carried out
by Akgoz and Civalek [201]. For FG microscopes for various boundary conditions,
the solutions regarding estimation of critical loads have been derived. Akgoz and
Civalek [202] employed the EBT model of the gradient of deformations for a study
of longitudinal vibrations of the FG microbeams. The method of Rayleigh-Ritz was
used to solve the problem regarding eigenfrequencies of the FG beams with clamped
ends and one free end. Rahaeifard et al. [203] developed the EBT model of nonlinear
gradient of deformations for study of influence of the geometric nonlinearity and
scale length material parameter on deflections and eigenfrequencies of FG simply
supported beam.
Wang et al. [204] belong to the first who developed the CPT model with an
account of the deformations gradient to predict the dependence of the characteristics
of isotropic size-dependent microplates. A comparison between the model of deformation gradient and the model based on the modified couple stress theory showed
that the first one yields better results [204]. Deflections of CPT model within theory of gradient of deformations were obtained by Ashoori and Mahmoodi [205]
for the microplates and for various boundary conditions with the help of extended
Kantorovich method. Besides, the analytical buckling solutions were obtained by
Mohammadi and Fooladi [206] and Mohammadi et al. [207] for the Levi microplates.
Wang et al. [208] derived the CPT model within gradient of deformations to analyse
bending of microplates for various boundary conditions. Zeighampour and Tadi Beni
2 Size-Dependent Theories of Beams, Plates and Shells
of gradient deformations was compared with the accuracy of modified couple stress
theory and classical theory. The comparison results showed that the model of gradient deformations predicts the size effect in a better way than the method of modified
couple stress theory, since it takes in addition into account the additional tensor of
the dilatation gradient, deviator tensor of extension gradient and a supplement to the
tensor of rotations gradient. Akgoz and Civalek [191] extended the EBT model of
deformations gradient into analysis of stability loss of isotropic microbeams for cantilever and simply supported beams. Akgoz and Civalek [192–195] also employed
EBT model of the deformation gradient for the study of influence of the size effect
on stability of SWCNT [192], static bending SWCNT [193], stability of linearly
narrow microbeams [194] and longitudinal vibrations of microbeams [195]. Zhao et
al. [196] developed the nonlinear EBT model of deformation gradient for nonlinear
stability loss and nonlinear analysis of free vibrations of isotropic microbeams. They
pointed out the importance of geometric nonlinearity and size-dependent effects
while getting reliable results. Rajabi and Ramezani [197] also developed the model
of nonlinear gradient of EBT deformations for the isotropic microbeams but mainly
considered static bending and free vibrations. The model of nonlinear EBT beam was
considered by Mohammadi and Mahzoon [198] on a basis of deformation gradient
where also the influence of temperature on stability of isotropic microbeams was
taken into account. They derived analytical solutions of microbeams with various
boundary conditions. Vatankhah et al. [199] used the model of nonlinear gradient of
deformations of EBT for a study of nonlinear vibrations of isotropic microbeams.
Kahrobaiyan et al. [200] extended application of the EBT model with gradient of
deformations to quantify stability and free vibrations of the FG microbeams. Further
extension of this model into problems of stability of FG microbeams was carried out
by Akgoz and Civalek [201]. For FG microscopes for various boundary conditions,
the solutions regarding estimation of critical loads have been derived. Akgoz and
Civalek [202] employed the EBT model of the gradient of deformations for a study
of longitudinal vibrations of the FG microbeams. The method of Rayleigh-Ritz was
used to solve the problem regarding eigenfrequencies of the FG beams with clamped
ends and one free end. Rahaeifard et al. [203] developed the EBT model of nonlinear
gradient of deformations for study of influence of the geometric nonlinearity and
scale length material parameter on deflections and eigenfrequencies of FG simply
supported beam.
Wang et al. [204] belong to the first who developed the CPT model with an
account of the deformations gradient to predict the dependence of the characteristics
of isotropic size-dependent microplates. A comparison between the model of deformation gradient and the model based on the modified couple stress theory showed
that the first one yields better results [204]. Deflections of CPT model within theory of gradient of deformations were obtained by Ashoori and Mahmoodi [205]
for the microplates and for various boundary conditions with the help of extended
Kantorovich method. Besides, the analytical buckling solutions were obtained by
Mohammadi and Fooladi [206] and Mohammadi et al. [207] for the Levi microplates.
Wang et al. [208] derived the CPT model within gradient of deformations to analyse
bending of microplates for various boundary conditions. Zeighampour and Tadi Beni