54
2 Size-Dependent Theories of Beams, Plates and Shells
2.3.8.1 Modified Coupled Stress Theory RBT, TSDT, NSDT, HSDT
Model of the modified RBT couple stress theory was first proposed by Ma et al. [409]
for the isotropic microbeams. It was used for investigation of the influence of size
effects on static deflections and free vibrations of the simply supported microbeams.
Application of the RBT models was extended by Mohammad-Abadi and Daneshmehr [410] for investigation of the size material length parameter on stability of
microbeams based on the models of EBT and TBT. In order to compare the obtained
results, analytical solutions for the simply supported beams were developed. Salamattalab et al. [411] extended the application of the RBT model into the FG simply
supported microbeams. Nateghi et al. [412] and Aghazadeh et al. [413] proposed
the unified model of stability [412] of deflection and free vibrations [413] of the FG
beams. Their model includes three different theories of beams: EBT, FBT and TBT.
The DQ method yielded the values of critical load, deflections and eigenfrequencies
of the FG microbeams for various boundary conditions. Chen et al. [414] worked
out the model RBT for laminated composite microbeams on the basis of novel introduced relations for anisotropic materials. The model was used for investigation of
the size effects on deflection of the simply supported microbeams under uniformed
load. Mohammad-Abadi and Daneshmehr [415] and Mohammad-Abadi et al. [416]
extended their isotropic model presented in [229] for a study of the free vibrations
[232] and thermal bending [417] for the laminated microbeams for various boundary
conditions.
Darijani and Mohammadabadi [417] proposed the modified couple stress HSDT
model for the isotropic microbeams by the separation of axial and transverse displacements into two parts related to shear and bending. The function of the form of
shear part is based on observation that the transversal shear stress and couple stress of
a higher order vanish on the upper and lower beam surfaces. More recently, Noori et
al. [418] proposed the HSDT model for free vibrations of the isotropic microbeams
on a basis of description of the axial displacement along thickness by a polynomial of
fifth order. In order to define the eigenfrequencies of nanobeams for various boundary conditions, the DQ method was employed. Simsek and Reddy [419] developed
the unified HSDT model for the FG microbeams by spanning seven various theories
of beams including EBT, TBT and RBT. Touratier [371] sinusoidal theory, hyperbolic theory of Soldatos [420], exponential theory of Karama et al. [365] and general
exponential theory of Aydogdu [364]. This model was employed to the problems of
bending and free vibrations of simply supported FG microbeams. That model was
extended by Simsek and Reddy [419] and Akbarzadeh et al. [421] for the problem
FG microbeams and the problem of post-buckling stability of the FG microbeams
with general boundary conditions [421]. Trinh et al. [422] also proposed the modified
couple stress FG microbeam model based on both HSDT and the quasi-2D theory of
beams. The field of displacement of their model was reconsidered by Thai et al. [423]
based on the assumption of separating component of bending, shear and thickness
change.
Based on the sinusoidal theory of Touratier [371], Akgoz and Civalek [424]
worked out the modified couple stress sinusoidal model for investigation of ther-
2 Size-Dependent Theories of Beams, Plates and Shells
2.3.8.1 Modified Coupled Stress Theory RBT, TSDT, NSDT, HSDT
Model of the modified RBT couple stress theory was first proposed by Ma et al. [409]
for the isotropic microbeams. It was used for investigation of the influence of size
effects on static deflections and free vibrations of the simply supported microbeams.
Application of the RBT models was extended by Mohammad-Abadi and Daneshmehr [410] for investigation of the size material length parameter on stability of
microbeams based on the models of EBT and TBT. In order to compare the obtained
results, analytical solutions for the simply supported beams were developed. Salamattalab et al. [411] extended the application of the RBT model into the FG simply
supported microbeams. Nateghi et al. [412] and Aghazadeh et al. [413] proposed
the unified model of stability [412] of deflection and free vibrations [413] of the FG
beams. Their model includes three different theories of beams: EBT, FBT and TBT.
The DQ method yielded the values of critical load, deflections and eigenfrequencies
of the FG microbeams for various boundary conditions. Chen et al. [414] worked
out the model RBT for laminated composite microbeams on the basis of novel introduced relations for anisotropic materials. The model was used for investigation of
the size effects on deflection of the simply supported microbeams under uniformed
load. Mohammad-Abadi and Daneshmehr [415] and Mohammad-Abadi et al. [416]
extended their isotropic model presented in [229] for a study of the free vibrations
[232] and thermal bending [417] for the laminated microbeams for various boundary
conditions.
Darijani and Mohammadabadi [417] proposed the modified couple stress HSDT
model for the isotropic microbeams by the separation of axial and transverse displacements into two parts related to shear and bending. The function of the form of
shear part is based on observation that the transversal shear stress and couple stress of
a higher order vanish on the upper and lower beam surfaces. More recently, Noori et
al. [418] proposed the HSDT model for free vibrations of the isotropic microbeams
on a basis of description of the axial displacement along thickness by a polynomial of
fifth order. In order to define the eigenfrequencies of nanobeams for various boundary conditions, the DQ method was employed. Simsek and Reddy [419] developed
the unified HSDT model for the FG microbeams by spanning seven various theories
of beams including EBT, TBT and RBT. Touratier [371] sinusoidal theory, hyperbolic theory of Soldatos [420], exponential theory of Karama et al. [365] and general
exponential theory of Aydogdu [364]. This model was employed to the problems of
bending and free vibrations of simply supported FG microbeams. That model was
extended by Simsek and Reddy [419] and Akbarzadeh et al. [421] for the problem
FG microbeams and the problem of post-buckling stability of the FG microbeams
with general boundary conditions [421]. Trinh et al. [422] also proposed the modified
couple stress FG microbeam model based on both HSDT and the quasi-2D theory of
beams. The field of displacement of their model was reconsidered by Thai et al. [423]
based on the assumption of separating component of bending, shear and thickness
change.
Based on the sinusoidal theory of Touratier [371], Akgoz and Civalek [424]
worked out the modified couple stress sinusoidal model for investigation of ther-