2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
55
momechanical buckling behaviour of the FG microbeams. The results showed that
the influence of temperature on the bending loads of the FG microbeams becomes
essential when a ratio of the length scale parameter to material thickness increases
[424]. Al-Basyouni et al. [425] also proposed a modified couple stress sinusoidal
model for the FG microbeams. However, it was based on the simple sinusoidal theory proposed by Thai and Vo [403] and included the physical plane position surface
of the FG beams.
A TSDT model based on modified couple stress theory was first developed by
Gao et al. [426] to study isotropic plates. The model was used for investigation of
the size of simply supported microplates on their eigenfrequencies. The model was
validated by Thai and Kim [420] and Chen et al. [427] for the FG microplates and the
laminated composite materials, respectively. Jung and Han [428] also proposed the
TSDT model for the FG microplates, but the authors used other law for computation
of the equivalent mechanical properties of the FG microplates. Eshraghi et al. [429]
worked out the TSDT model for the FG microplates with annular and circular forms.
Field of displacements was unified with the inclusion of three different plate theories,
i.e. CPT, FSDT and TSDT. In order to solve the problem of static bending and free
vibrations, the DQ method was used by Eshraghi et al. [430]. Ghayesh and Farokhi
[431] employed their previous work [429] and developed the nonlinear TSDT model
to study nonlinear characteristics of vibrations of isotropic microplates. Based on the
TSDT, Sahmani et al. [432] worked out the modified couple stress model of shells to
analyse the dynamic instability of cylindrical FG microshells. They obtained closedform solutions for the simply supported cylindrical microshells.
Thai and Vo [433] proposed the HSDT model based on the modified couple stress
theory of the FG microplates. The field of displacements was based on the sinusoidal
theory of Touratier [371]. For the case of simply supported microplates, there were
obtained solutions of deflections and eigenfrequencies. Darijani and Shahdadi [434]
proposed a simple HSDT model for the isotropic microplates via separating shear
and bending components. Function of the form of shear component in plane was
obtained from a condition of zeroth value of the transversal shear and of a higher
order couple stresses on the upper/lower plate boundary. He et al. [435] reformulated
theory of plates of Shimpi [367] for account of size effects in the FG microplates
with the use of the modified couple stress theory. Though the work of Lou et al.
[436] is analogous to that of He et al. [435], however, Lou et al. [437] used various
forms of the Thai and Choi [379]. Lou et al. [436] more recently validated the work
of He et al. [435] and included a geometric nonlinearity, and took into account the
interaction between the plate and an elastic medium. Trinh et al. [438] also presented
the unified and modified couple stress model for the FG microplates with mechanical
and thermal loads based on the quasi-3D theory.
Reddy and Kim [439] worked out the nonlinear quasi-3D model for the FG
microplates with an account of thermal effects. The latter was based on the von
Kármán nonlinearity and a general quasi-3D theory an account cubic and square
variations of plane and transversal displacement along thickness. The field of displacement of the general quasi-3D theory contained 11 unknowns. The CPT, FSDT
and TSDT are yielded by the general theory as a particular case. The analytical for-
55
momechanical buckling behaviour of the FG microbeams. The results showed that
the influence of temperature on the bending loads of the FG microbeams becomes
essential when a ratio of the length scale parameter to material thickness increases
[424]. Al-Basyouni et al. [425] also proposed a modified couple stress sinusoidal
model for the FG microbeams. However, it was based on the simple sinusoidal theory proposed by Thai and Vo [403] and included the physical plane position surface
of the FG beams.
A TSDT model based on modified couple stress theory was first developed by
Gao et al. [426] to study isotropic plates. The model was used for investigation of
the size of simply supported microplates on their eigenfrequencies. The model was
validated by Thai and Kim [420] and Chen et al. [427] for the FG microplates and the
laminated composite materials, respectively. Jung and Han [428] also proposed the
TSDT model for the FG microplates, but the authors used other law for computation
of the equivalent mechanical properties of the FG microplates. Eshraghi et al. [429]
worked out the TSDT model for the FG microplates with annular and circular forms.
Field of displacements was unified with the inclusion of three different plate theories,
i.e. CPT, FSDT and TSDT. In order to solve the problem of static bending and free
vibrations, the DQ method was used by Eshraghi et al. [430]. Ghayesh and Farokhi
[431] employed their previous work [429] and developed the nonlinear TSDT model
to study nonlinear characteristics of vibrations of isotropic microplates. Based on the
TSDT, Sahmani et al. [432] worked out the modified couple stress model of shells to
analyse the dynamic instability of cylindrical FG microshells. They obtained closedform solutions for the simply supported cylindrical microshells.
Thai and Vo [433] proposed the HSDT model based on the modified couple stress
theory of the FG microplates. The field of displacements was based on the sinusoidal
theory of Touratier [371]. For the case of simply supported microplates, there were
obtained solutions of deflections and eigenfrequencies. Darijani and Shahdadi [434]
proposed a simple HSDT model for the isotropic microplates via separating shear
and bending components. Function of the form of shear component in plane was
obtained from a condition of zeroth value of the transversal shear and of a higher
order couple stresses on the upper/lower plate boundary. He et al. [435] reformulated
theory of plates of Shimpi [367] for account of size effects in the FG microplates
with the use of the modified couple stress theory. Though the work of Lou et al.
[436] is analogous to that of He et al. [435], however, Lou et al. [437] used various
forms of the Thai and Choi [379]. Lou et al. [436] more recently validated the work
of He et al. [435] and included a geometric nonlinearity, and took into account the
interaction between the plate and an elastic medium. Trinh et al. [438] also presented
the unified and modified couple stress model for the FG microplates with mechanical
and thermal loads based on the quasi-3D theory.
Reddy and Kim [439] worked out the nonlinear quasi-3D model for the FG
microplates with an account of thermal effects. The latter was based on the von
Kármán nonlinearity and a general quasi-3D theory an account cubic and square
variations of plane and transversal displacement along thickness. The field of displacement of the general quasi-3D theory contained 11 unknowns. The CPT, FSDT
and TSDT are yielded by the general theory as a particular case. The analytical for-