2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
39
of free vibrations of isotropic microplates. Askari and Tahani [176] obtained analytical solutions for the eigenfrequencies of the clamped CPT microplates with the use
of the extended Kantorovich method. Simsek et al. [177] adapted the CPT model to
study the influence of the size on the excited vibrations of isotropic microplates under
action of moving load. The dynamic behaviour of microplates for various boundary
conditions was quantified with the use of the Newmark implicit integration method.
Zhou et al. [178] developed the modified couple stress model for free vibrations
of isotropic microbeams based on the classical theory of shells. It was found that
the size effect plays an important role when the size of the characteristic radius is
compared with the length scale material parameter [178].
Asghari [179] proposed the CPT model and the nonlinear modified couple stress
theory for the geometric nonlinear analysis of microplates of an arbitrary form.
Wang et al. [180, 181] worked out the CPT model within the nonlinear modified
couple stress theory in order to investigate the size effects influenced on nonlinear
free vibrations [180] and nonlinear deflections [181] of circle microplates. Farokhi
and Ghayesh [182] also employed CPT model and nonlinear modified couple stress
theory to carry out nonlinear dynamic analysis of microplates with geometric imperfections.
The similar approaches were employed also to the FG microplates. Ke et al.
[183] studied the influence of size effects on the deflections, critical loads and eigenfrequencies of ring FG microplates for different boundary conditions. Asghari and
Taati [184] investigated free vibrations of the FG microbeams of the arbitrary form.
Ashoori and Sadough Vanini [185] presented the modified CPT model to study stability loss of the FG microplates including thermal effects and interaction between
plate and elastic matter. Ashoori and Sadough Vanini [186] extended their earlier
work [185] in order to include the geometric nonlinearity under thermal buckling of
circle FG microplates. Taati [187] obtained analytical formulas of the CPT model
with the help of modified couple stress theory for stability of the FG microplates with
various boundary conditions under shear in plane, two-side compression and uniform
transversal load. Based on the classical theory of KLT shells, Beni et al. [188] developed the modified couple stress theory for the investigation of influence of the size
effects on free frequencies of the simply supported cylindrical FG microplates. Tsiatas and Yiotis [189] also employed the modified couple stress theory of KLT model
in order to investigate the size-dependent influence on the state deflections and characteristics of longitudinal and free vibrations of the microplates. In comparison with
nonlocal KLT regimes, it was shown that the influence of scale length parameter on
the critical loads and eigenfrequencies stands in contrary to the nonlocal models.
2.3.5.3 Beams (EBT), Plates (CPT) and Shells (KLT) Models Based on
the Modified Gradient Theory
Kong et al. [190] proposed one of the earlier EBT models with gradient of deformations in order to study the size-dependent influence on deflections and eigenfrequency of isotropic cantilever microbeams. Accuracy of results yielded by the theory
39
of free vibrations of isotropic microplates. Askari and Tahani [176] obtained analytical solutions for the eigenfrequencies of the clamped CPT microplates with the use
of the extended Kantorovich method. Simsek et al. [177] adapted the CPT model to
study the influence of the size on the excited vibrations of isotropic microplates under
action of moving load. The dynamic behaviour of microplates for various boundary
conditions was quantified with the use of the Newmark implicit integration method.
Zhou et al. [178] developed the modified couple stress model for free vibrations
of isotropic microbeams based on the classical theory of shells. It was found that
the size effect plays an important role when the size of the characteristic radius is
compared with the length scale material parameter [178].
Asghari [179] proposed the CPT model and the nonlinear modified couple stress
theory for the geometric nonlinear analysis of microplates of an arbitrary form.
Wang et al. [180, 181] worked out the CPT model within the nonlinear modified
couple stress theory in order to investigate the size effects influenced on nonlinear
free vibrations [180] and nonlinear deflections [181] of circle microplates. Farokhi
and Ghayesh [182] also employed CPT model and nonlinear modified couple stress
theory to carry out nonlinear dynamic analysis of microplates with geometric imperfections.
The similar approaches were employed also to the FG microplates. Ke et al.
[183] studied the influence of size effects on the deflections, critical loads and eigenfrequencies of ring FG microplates for different boundary conditions. Asghari and
Taati [184] investigated free vibrations of the FG microbeams of the arbitrary form.
Ashoori and Sadough Vanini [185] presented the modified CPT model to study stability loss of the FG microplates including thermal effects and interaction between
plate and elastic matter. Ashoori and Sadough Vanini [186] extended their earlier
work [185] in order to include the geometric nonlinearity under thermal buckling of
circle FG microplates. Taati [187] obtained analytical formulas of the CPT model
with the help of modified couple stress theory for stability of the FG microplates with
various boundary conditions under shear in plane, two-side compression and uniform
transversal load. Based on the classical theory of KLT shells, Beni et al. [188] developed the modified couple stress theory for the investigation of influence of the size
effects on free frequencies of the simply supported cylindrical FG microplates. Tsiatas and Yiotis [189] also employed the modified couple stress theory of KLT model
in order to investigate the size-dependent influence on the state deflections and characteristics of longitudinal and free vibrations of the microplates. In comparison with
nonlocal KLT regimes, it was shown that the influence of scale length parameter on
the critical loads and eigenfrequencies stands in contrary to the nonlocal models.
2.3.5.3 Beams (EBT), Plates (CPT) and Shells (KLT) Models Based on
the Modified Gradient Theory
Kong et al. [190] proposed one of the earlier EBT models with gradient of deformations in order to study the size-dependent influence on deflections and eigenfrequency of isotropic cantilever microbeams. Accuracy of results yielded by the theory