2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
47
transverse shear stresses. Nonlinear nonlocal CPT and FSDT models were worked
out by Reddy [287] for nonlinear analysis of bending of isotropic nanoplates on a
basis of von Kármán nonlinearity with the help of variational approaches.
Nonlocal FSDT models were also proposed for nanoplates made from FG and
orthotropic materials. Hosseini-Hashemi et al. [288] proposed nonlocal FSDT for
the circular/annular FG nanoplates. The closed-form solutions for the eigenfrequencies of circular nanoplates for various boundary conditions were also reported.
Anjomshoa and Tahani [289] developed nonlocal FSDT model for free vibrations
of orthotropic circular and elliptical SLGS embedded into an elastic medium. Golmakani and Rezatalab [290] proposed a nonlocal nonlinear FSDT model for nonlinear
analysis of bending of orthotropic SLGS with the use of the von Kármán nonlinear
deformations. Dastjerdi et al. [291] and Dastjerdi and Jabbarzadeh [292] developed
nonlinear nonlocal model for geometric nonlinear analysis of the annular/circular
orthotropic SLGS [291] and MLGS [292], where the temperature influence was
taken into account.
2.3.6.2 Modified Couple Stress Timoshenko Theory
Ma et al. [293] developed the modified couple stress TBT model [155] in order
to the account of the shear deformation effect. The model was employed to the
investigation of the influence of length scale parameter and shear deformation on
deflections and eigenfrequencies of simply supported isotropic microbeams. The
closed form of solutions of the TBT model was proposed by Asghari et al. [294]
for beam bending for various boundary conditions, whereas Dos Santos and Reddy
[295] employed the Ritz method for the input loads and eigenfrequencies of the
beams for various boundary conditions. Dehrouyeh-Semnani and Nikkhah-Bahrami
[296] employed the EBT and TBT models for a study of the Poisson’s effect in the
isotropic microbeams. It was shown that inclusion of Poisson’s effect into the model
of the modified couple stress theory yields under estimated deflection of epoxidal
cantilever. Liu and Reddy [297] worked out the modified coupled stress theory of the
TBT model for the isotropic bended microbeams, and employed it to the problems
of bending and free vibrations of the simply supported beams. Taati et al. [298]
also worked out the TBT model for investigation of heat effects in the isotropic
microbeams. Asghari et al. [299] proposed a nonlinear TBT model for analysis of
deflections and free vibrations of isotropic microbeams. Ghayesh et al. [300, 301]
also proposed a nonlinear TBT model for nonlinear dynamic problem of isotropic
microbeams.
The TBT model was employed to study FG microbeams made from laminated
composite materials. Reddy [302] developed the EBT and TBT models for FG
microbeams with an account of a geometric nonlinearity. There were also derived
analytical formulas for the critical loads and eigenfrequencies of the simply supported microbeams. Ke et al. [303] employed the nonlinear TBT model for a study
of the influence of a size length parameter on nonlinear characteristics of vibrating
FG microbeams. Asghari et al. [304] proposed the TBT model to investigate an influ-
47
transverse shear stresses. Nonlinear nonlocal CPT and FSDT models were worked
out by Reddy [287] for nonlinear analysis of bending of isotropic nanoplates on a
basis of von Kármán nonlinearity with the help of variational approaches.
Nonlocal FSDT models were also proposed for nanoplates made from FG and
orthotropic materials. Hosseini-Hashemi et al. [288] proposed nonlocal FSDT for
the circular/annular FG nanoplates. The closed-form solutions for the eigenfrequencies of circular nanoplates for various boundary conditions were also reported.
Anjomshoa and Tahani [289] developed nonlocal FSDT model for free vibrations
of orthotropic circular and elliptical SLGS embedded into an elastic medium. Golmakani and Rezatalab [290] proposed a nonlocal nonlinear FSDT model for nonlinear
analysis of bending of orthotropic SLGS with the use of the von Kármán nonlinear
deformations. Dastjerdi et al. [291] and Dastjerdi and Jabbarzadeh [292] developed
nonlinear nonlocal model for geometric nonlinear analysis of the annular/circular
orthotropic SLGS [291] and MLGS [292], where the temperature influence was
taken into account.
2.3.6.2 Modified Couple Stress Timoshenko Theory
Ma et al. [293] developed the modified couple stress TBT model [155] in order
to the account of the shear deformation effect. The model was employed to the
investigation of the influence of length scale parameter and shear deformation on
deflections and eigenfrequencies of simply supported isotropic microbeams. The
closed form of solutions of the TBT model was proposed by Asghari et al. [294]
for beam bending for various boundary conditions, whereas Dos Santos and Reddy
[295] employed the Ritz method for the input loads and eigenfrequencies of the
beams for various boundary conditions. Dehrouyeh-Semnani and Nikkhah-Bahrami
[296] employed the EBT and TBT models for a study of the Poisson’s effect in the
isotropic microbeams. It was shown that inclusion of Poisson’s effect into the model
of the modified couple stress theory yields under estimated deflection of epoxidal
cantilever. Liu and Reddy [297] worked out the modified coupled stress theory of the
TBT model for the isotropic bended microbeams, and employed it to the problems
of bending and free vibrations of the simply supported beams. Taati et al. [298]
also worked out the TBT model for investigation of heat effects in the isotropic
microbeams. Asghari et al. [299] proposed a nonlinear TBT model for analysis of
deflections and free vibrations of isotropic microbeams. Ghayesh et al. [300, 301]
also proposed a nonlinear TBT model for nonlinear dynamic problem of isotropic
microbeams.
The TBT model was employed to study FG microbeams made from laminated
composite materials. Reddy [302] developed the EBT and TBT models for FG
microbeams with an account of a geometric nonlinearity. There were also derived
analytical formulas for the critical loads and eigenfrequencies of the simply supported microbeams. Ke et al. [303] employed the nonlinear TBT model for a study
of the influence of a size length parameter on nonlinear characteristics of vibrating
FG microbeams. Asghari et al. [304] proposed the TBT model to investigate an influ-