2.3 Non-classical (Size-Dependent) Models of Beams, Plates and Shells
37
The nonlocal CPT model was also used for computation of the size-dependent
SLGS and MLGS behaviours. For instance, Pradhan and Murmu [136] and Pradhan
and Kumar [137] investigated the nonlocal effect influenced on the SLGS stability
using the DQ method, whereas Babaei and Shahidi [138] and Farajpour et al. [139]
considered the influence of size dependence on bending of rectangular SLGS [138]
and SLGS with variable thickness [139] employing the Galerkin method. Free vibrations of MLGS embedded into a polymer matrix were investigated by Pradhan and
Phadikar [140]. It was shown how the size effect increases with the increase of the
layer number. Shen et al. [141] extended application of the nonlocal CPT model for
investigation of vibrations of the simple support mass sensor of SLGS. Ansari et
al. [142] derived analytical formulas for the SLGS eigenfrequencies with arbitrary
boundary conditions based on construction of the atomic potential while estimating elastic properties of SLGS. More recently, Zhang et al. [143–145] employed
equations of the nonlocal CPT model for derivation of the eigenfrequencies [143],
nonlinear bending [144] and critical loads [145] of SLGS for various boundary conditions.
Employment of the nonlocal CPT in the above-mentioned investigations was
limited to consideration of graphene sheets made from isotropic materials. However,
numerical results of MD modelling carried out by Ni et al. [146] showed that mechanical properties of the graphene sheets are anisotropic due to hexagonal structure of
the elementary graphene cells [146]. Therefore, nonlocal orthotropic CPT models
were carried out to account of influence of the anisotropic mechanical properties of
the graphene sheets. Pouresmaeeli et al. [147] proposed the nonlocal CPT model for
vibrations of the orthotropic DLGS, embedded into an elastic matter. Mohammadi et
al.[148, 149] worked out the nonlocal CPT model for free vibrations of orthotropic
SLGS embedded into the temperature field. Navier and Levy solutions of the eigenfrequencies of the rectangular SLGS were obtained. Sari and Al-Kouz [150] developed the nonlocal CPT model to analyse free vibrations of orthotropic SLGS with an
account of the changeable thickness. Anjomshoa [151] and Anjomshoa et al. [152]
proposed nonlocal CPT models for study stability loss [151] and free vibrations [152]
of orthotropic circled and elliptic SLGS embedded into elastic matter. The nonlocal
CPT model was also proposed by Mohammadi et al. [153] for the investigation of
stability loss and shear effects of orthotropic SLGS inclusions subjected to temperature fields. Ashoori et al. [154] worked out the nonlocal CPT model suitable for
prognosis of thermal stability of ring-type FG nanoplates under various kinds of
thermal loads. Exact solutions for the critical temperature responsible for stability
loss were also obtained for ring-clamped FG nanoplates.
2.3.5.2 Beam (EBT), Plate (CPT) and Shell (KLT) Models Based on the
Modified Couple Stress Theory
Park and Gao [155] belong to the first who presented the modified couple stress
EBT model for isotropic microbeams. They employed the developed model for the
investigation of influence of the length parameter on deflections and bending stiffness
37
The nonlocal CPT model was also used for computation of the size-dependent
SLGS and MLGS behaviours. For instance, Pradhan and Murmu [136] and Pradhan
and Kumar [137] investigated the nonlocal effect influenced on the SLGS stability
using the DQ method, whereas Babaei and Shahidi [138] and Farajpour et al. [139]
considered the influence of size dependence on bending of rectangular SLGS [138]
and SLGS with variable thickness [139] employing the Galerkin method. Free vibrations of MLGS embedded into a polymer matrix were investigated by Pradhan and
Phadikar [140]. It was shown how the size effect increases with the increase of the
layer number. Shen et al. [141] extended application of the nonlocal CPT model for
investigation of vibrations of the simple support mass sensor of SLGS. Ansari et
al. [142] derived analytical formulas for the SLGS eigenfrequencies with arbitrary
boundary conditions based on construction of the atomic potential while estimating elastic properties of SLGS. More recently, Zhang et al. [143–145] employed
equations of the nonlocal CPT model for derivation of the eigenfrequencies [143],
nonlinear bending [144] and critical loads [145] of SLGS for various boundary conditions.
Employment of the nonlocal CPT in the above-mentioned investigations was
limited to consideration of graphene sheets made from isotropic materials. However,
numerical results of MD modelling carried out by Ni et al. [146] showed that mechanical properties of the graphene sheets are anisotropic due to hexagonal structure of
the elementary graphene cells [146]. Therefore, nonlocal orthotropic CPT models
were carried out to account of influence of the anisotropic mechanical properties of
the graphene sheets. Pouresmaeeli et al. [147] proposed the nonlocal CPT model for
vibrations of the orthotropic DLGS, embedded into an elastic matter. Mohammadi et
al.[148, 149] worked out the nonlocal CPT model for free vibrations of orthotropic
SLGS embedded into the temperature field. Navier and Levy solutions of the eigenfrequencies of the rectangular SLGS were obtained. Sari and Al-Kouz [150] developed the nonlocal CPT model to analyse free vibrations of orthotropic SLGS with an
account of the changeable thickness. Anjomshoa [151] and Anjomshoa et al. [152]
proposed nonlocal CPT models for study stability loss [151] and free vibrations [152]
of orthotropic circled and elliptic SLGS embedded into elastic matter. The nonlocal
CPT model was also proposed by Mohammadi et al. [153] for the investigation of
stability loss and shear effects of orthotropic SLGS inclusions subjected to temperature fields. Ashoori et al. [154] worked out the nonlocal CPT model suitable for
prognosis of thermal stability of ring-type FG nanoplates under various kinds of
thermal loads. Exact solutions for the critical temperature responsible for stability
loss were also obtained for ring-clamped FG nanoplates.
2.3.5.2 Beam (EBT), Plate (CPT) and Shell (KLT) Models Based on the
Modified Couple Stress Theory
Park and Gao [155] belong to the first who presented the modified couple stress
EBT model for isotropic microbeams. They employed the developed model for the
investigation of influence of the length parameter on deflections and bending stiffness