36
2 Size-Dependent Theories of Beams, Plates and Shells
and Nejad et al. [120] investigated bending and stability loss of FG nanobeams with
an account of changes of the elasticity modulus in the axial and transverse directions.
The governing equations were solved by the DQ method. Nonlinear free vibrations
of the FG nanobeams were investigated by Nazemnezhad and Hosseini-Hashemi
[121] with the help of nonlinear theory of von Kármán and using the nonlocal EBT
model. Multiple scales method yielded solutions for the case of eigenfrequencies
of nonlinear nonlocal EBT model and for the case of free and excited vibrations
of the simply supported beams. El-Borgi et al. [122] proposed the nonlocal EBT
model for nonlinear free and excited vibrations of the FG nanobeams. The multiplescale method was employed to obtain nonlinear frequencies of the simply supported
beams. Shafiei et al. [123] also worked out the nonlinear nonlocal EBT model for
investigation of nonlinear free vibrations of FG beams with changeable transversal
cross sections. Nonlinear frequency of beams was obtained for various boundary
conditions with the help of generalized DQ methodology.
Zhang et al. [124] developed one of the earlier nonlocal shell models for analysis
of stability loss of MWCNT under axial compression on the basis of classical theory
of shells. In order to solve the problems with an account of the nonlocal effect of
the axial buckling of simply supported DWCNT, the solutions were derived in the
closed form. Li and Kardomateas [125, 126] derived non-classical nonlocal models
of shells for a study of thermal bending and free vibration of MWCNT. The model of
nonlocal classical shell was proposed by Wang and Varadan [127] and Hu et al. [128]
and employed to investigate the wave propagation in CNTs. The accuracy of nonlocal
classical shell model under prognosis of buckling of axially loaded SWCNTs was
quantified by Zhang et al. [129] in comparison with the results obtained by MD
modelling. It was shown that for the long SWCNT the local KLT (Kirchhoff-Love
theory) model can yield the results compared with the results obtained with the help of
nonlocal KLT model and MD modelling. However, for short SWCNT, only nonlocal
shell model may guarantee the results compared with MD. Rouhi and Ansari [130]
also proposed the nonlocal classical shell model for the case of axial compression
of DWCNT validated for different boundary conditions. More recently, Sarvestani
[131] proposed the nonlocal classical shell model to analyse stability loss of the
bounded MWCNT under axial comparison.
One of the most earlier models of the nonlocal CPT plate was given by Lu et
al. [132] based on classical CPT. The latter model was employed for the study
of influence of the size-dependent bending behaviour and stability of the isotropic
nanoplates. Duan and Wang [133] derived exact solutions of nonlocal CPT model
while analysing axially symmetric of circle nanoplates under arbitrary load. The
influence of the nonlocal parameter regarding deviation, radial moment, etc. has
been investigated.
Aksencer and Aydogdu [134] obtained Levy’s solution for the nonlocal CPT
model in the stability problems and free vibrations of the rectangular nanoplates
with two edges simply supported and two arbitrary supported. Shakouri et al. [135]
employed Galerkin’s approach for solving the resolving equations of the nonlocal
CPT model for estimation of the eigenfrequencies of the isotropic nanoplates with
various boundary conditions.
2 Size-Dependent Theories of Beams, Plates and Shells
and Nejad et al. [120] investigated bending and stability loss of FG nanobeams with
an account of changes of the elasticity modulus in the axial and transverse directions.
The governing equations were solved by the DQ method. Nonlinear free vibrations
of the FG nanobeams were investigated by Nazemnezhad and Hosseini-Hashemi
[121] with the help of nonlinear theory of von Kármán and using the nonlocal EBT
model. Multiple scales method yielded solutions for the case of eigenfrequencies
of nonlinear nonlocal EBT model and for the case of free and excited vibrations
of the simply supported beams. El-Borgi et al. [122] proposed the nonlocal EBT
model for nonlinear free and excited vibrations of the FG nanobeams. The multiplescale method was employed to obtain nonlinear frequencies of the simply supported
beams. Shafiei et al. [123] also worked out the nonlinear nonlocal EBT model for
investigation of nonlinear free vibrations of FG beams with changeable transversal
cross sections. Nonlinear frequency of beams was obtained for various boundary
conditions with the help of generalized DQ methodology.
Zhang et al. [124] developed one of the earlier nonlocal shell models for analysis
of stability loss of MWCNT under axial compression on the basis of classical theory
of shells. In order to solve the problems with an account of the nonlocal effect of
the axial buckling of simply supported DWCNT, the solutions were derived in the
closed form. Li and Kardomateas [125, 126] derived non-classical nonlocal models
of shells for a study of thermal bending and free vibration of MWCNT. The model of
nonlocal classical shell was proposed by Wang and Varadan [127] and Hu et al. [128]
and employed to investigate the wave propagation in CNTs. The accuracy of nonlocal
classical shell model under prognosis of buckling of axially loaded SWCNTs was
quantified by Zhang et al. [129] in comparison with the results obtained by MD
modelling. It was shown that for the long SWCNT the local KLT (Kirchhoff-Love
theory) model can yield the results compared with the results obtained with the help of
nonlocal KLT model and MD modelling. However, for short SWCNT, only nonlocal
shell model may guarantee the results compared with MD. Rouhi and Ansari [130]
also proposed the nonlocal classical shell model for the case of axial compression
of DWCNT validated for different boundary conditions. More recently, Sarvestani
[131] proposed the nonlocal classical shell model to analyse stability loss of the
bounded MWCNT under axial comparison.
One of the most earlier models of the nonlocal CPT plate was given by Lu et
al. [132] based on classical CPT. The latter model was employed for the study
of influence of the size-dependent bending behaviour and stability of the isotropic
nanoplates. Duan and Wang [133] derived exact solutions of nonlocal CPT model
while analysing axially symmetric of circle nanoplates under arbitrary load. The
influence of the nonlocal parameter regarding deviation, radial moment, etc. has
been investigated.
Aksencer and Aydogdu [134] obtained Levy’s solution for the nonlocal CPT
model in the stability problems and free vibrations of the rectangular nanoplates
with two edges simply supported and two arbitrary supported. Shakouri et al. [135]
employed Galerkin’s approach for solving the resolving equations of the nonlocal
CPT model for estimation of the eigenfrequencies of the isotropic nanoplates with
various boundary conditions.