46
2 Size-Dependent Theories of Beams, Plates and Shells
The compatibility nonlocal model was worked out also by Reddy and Pang [264]
who changed EBT and TBT with the use of nonlocal governing Eringen’s equations.
They presented solutions for deflections, critical loads and eigenfrequencies for the
nanobeams with four different boundary conditions. It should be mentioned that
closed-form solutions obtained by Reddy and Pang [87] differ from those proposed
by Wang et al. [259–262], since they were based on various TBT models.
The compatibility nonlocal TBT model has been widely employed to study nonlocal effects in CNT. For instance, Murmu and Pradhan [265] investigated the influence
of the nonlocal parameter and transverse shear buckling OCNT embedded in an elastic medium. The latter work was extended by Ansari et al. [266], where the influence
of high temperature was included. Pradhan and Murmu [267] investigated nonlocal
TBT models based on the DQ method. Numerical solutions to the carbon nanotubes
using nonlocal TBT and small-scale effects on vibration analysis of TBT nanomodel
were presented by Roque et al. [268] based on meshless method with the global
and local collocation technique and the use of radial basic functions. Vibrations of
OCNT were also investigated by Wu and Lai [269] using RMVT-based nonlocal
TBT models worked out on the basis of the variational Reissner’s principle as well
the principle of virtual displacements. Amirian et al. [270] and Zidour et al. [271]
had taken into account the influence of temperature of the OCNT vibrations, while
Ansari et al. [272] investigated the influence of thermal environment on stability of
MWCNT. Ansari et al. [273] worked out a nonlocal TBT model for nonlinear excited
vibrations of the magneto-electro-thermo-elastic nanobeams.
There were also proposals for nonlocal TBT model for FG nanobeams. Simsek
and Yurtcu [274] proposed the nonlocal EBT models and TBT models to study bending and stability of FG beams. The compatibility nonlocal TBT model was extended
by Rahmani and Pedram [275] for analysis of simply supported FG nanobeams.
Ebrahimi and Salari [276, 277] also developed nonlocal TBT model to study the
stability and free vibrations of FG nanobeams where thermal effects were taken into
account. The first proposal for nonlocal first-order shear deformation theory (FSDT)
model was proposed by Lu et al. [132] for the isotropic nanoplates. Then, it was
employed to study the size effect in the problems of bending and problems of eigenfrequencies of simply supported isotropic nanoplates. Pradhan and Phadikar [278,
279] proposed nonlocal CPT and FSDT models of free vibrations [278] and analysed stability loss of [279] SLGS and MLGS. In the MLGS models, the interaction
between two graphene sheets was modelled using the Winkler model. Influence of
the size dependence, shear deformations, elasticity modulus and stiffness of the Winkler foundation on the eigenfrequencies and critical loads of the simply supported
graphene sheets were carried out by Kananipour [280]. Ansari et al. [281, 282]
investigated vibration of SLGS [281] and MLGS [282] with an account of different
boundary conditions and using the nonlocal FSDT and the DQ method. Nonlocal
FSDT was also employed by Samaei et al. [283] and Bedroud et al. [284] in order to
study buckling of single SLGS [283] and circular nanoplates [284]. Arani et al. [285]
considered electro-thermo-torsional buckling of nanotubes with double walls based
on the nonlocal FSDT shell model. Naderi and Saidi [286] modified the nonlocal
FSDT model to study the stability of nanoplates without the nonlocal effects for the
2 Size-Dependent Theories of Beams, Plates and Shells
The compatibility nonlocal model was worked out also by Reddy and Pang [264]
who changed EBT and TBT with the use of nonlocal governing Eringen’s equations.
They presented solutions for deflections, critical loads and eigenfrequencies for the
nanobeams with four different boundary conditions. It should be mentioned that
closed-form solutions obtained by Reddy and Pang [87] differ from those proposed
by Wang et al. [259–262], since they were based on various TBT models.
The compatibility nonlocal TBT model has been widely employed to study nonlocal effects in CNT. For instance, Murmu and Pradhan [265] investigated the influence
of the nonlocal parameter and transverse shear buckling OCNT embedded in an elastic medium. The latter work was extended by Ansari et al. [266], where the influence
of high temperature was included. Pradhan and Murmu [267] investigated nonlocal
TBT models based on the DQ method. Numerical solutions to the carbon nanotubes
using nonlocal TBT and small-scale effects on vibration analysis of TBT nanomodel
were presented by Roque et al. [268] based on meshless method with the global
and local collocation technique and the use of radial basic functions. Vibrations of
OCNT were also investigated by Wu and Lai [269] using RMVT-based nonlocal
TBT models worked out on the basis of the variational Reissner’s principle as well
the principle of virtual displacements. Amirian et al. [270] and Zidour et al. [271]
had taken into account the influence of temperature of the OCNT vibrations, while
Ansari et al. [272] investigated the influence of thermal environment on stability of
MWCNT. Ansari et al. [273] worked out a nonlocal TBT model for nonlinear excited
vibrations of the magneto-electro-thermo-elastic nanobeams.
There were also proposals for nonlocal TBT model for FG nanobeams. Simsek
and Yurtcu [274] proposed the nonlocal EBT models and TBT models to study bending and stability of FG beams. The compatibility nonlocal TBT model was extended
by Rahmani and Pedram [275] for analysis of simply supported FG nanobeams.
Ebrahimi and Salari [276, 277] also developed nonlocal TBT model to study the
stability and free vibrations of FG nanobeams where thermal effects were taken into
account. The first proposal for nonlocal first-order shear deformation theory (FSDT)
model was proposed by Lu et al. [132] for the isotropic nanoplates. Then, it was
employed to study the size effect in the problems of bending and problems of eigenfrequencies of simply supported isotropic nanoplates. Pradhan and Phadikar [278,
279] proposed nonlocal CPT and FSDT models of free vibrations [278] and analysed stability loss of [279] SLGS and MLGS. In the MLGS models, the interaction
between two graphene sheets was modelled using the Winkler model. Influence of
the size dependence, shear deformations, elasticity modulus and stiffness of the Winkler foundation on the eigenfrequencies and critical loads of the simply supported
graphene sheets were carried out by Kananipour [280]. Ansari et al. [281, 282]
investigated vibration of SLGS [281] and MLGS [282] with an account of different
boundary conditions and using the nonlocal FSDT and the DQ method. Nonlocal
FSDT was also employed by Samaei et al. [283] and Bedroud et al. [284] in order to
study buckling of single SLGS [283] and circular nanoplates [284]. Arani et al. [285]
considered electro-thermo-torsional buckling of nanotubes with double walls based
on the nonlocal FSDT shell model. Naderi and Saidi [286] modified the nonlocal
FSDT model to study the stability of nanoplates without the nonlocal effects for the