8
1 Nanostructural Members in Various Fields: A Literature Review
based on the Hamilton principle, and both natural frequencies and critical electric
voltages were estimated.
Salehipour et al. [53] used converted couple stress and three-dimensional elasticity conception to derive a model for static and vibration of functionally graded
microplates and nanoplates. Developed model included small size effect. Hamilton’s principle was used to create the equations of motion and boundary conditions.
For in-plane and out-of-plane free vibrations of simply supported plates, analytical
closed-form solutions were described. In order to achieve the analytical solutions,
the elasticity modulus and mass density were assumed to differ exponentially by the
thickness of the plate.
Jafari et al. [54] used nonclassical constitutive equations consisting of the
first/second-order strain gradients. Navier and Galerkin methods were employed to
solve the governing PDEs and obtain the approximate system outputs, respectively.
The author studied the influence of different parameters, boundary conditions and
the plate size on the natural frequencies of the nanoplates.
Ghassabi et al. [55] derived governing PDEs and the associated boundary conditions with an account of a nonlocal parameter based on Hamilton’s principle. The
Kirchhoff, Mindlin and the third-order shear deformation theories were employed.
The analysed case studies included simply supported and cantilever nanoplates with
an emphasis put to demonstrate a role of the dimensionless plate length, plate theory,
power-low index and nonlocal parameter ratio on the vibration of the functionally
graded rectangular nanoplates.
Mehdi Zarei et al. [56] studied free vibration and buckling of a round tapered
nanoplate subjected to in-plane forces. Assumption of nonlocal resilience was used to
demonstrate effects depending on the size. In order to obtain frequency equations for
simply supported and clamped nanoplates, the Raleigh-Ritz method and differential
transformation approach were used. The final results showed that incrementing the
parameter of taper yielded rise of the buckling load and natural frequencies.
Shahrbabaki [57] employed the Ritz and Galerkin methods to study 3D nonlocal elasticity of rectangular nanoplate. Two simple cases of 3D free vibrations of
simply supported nanoplate and wave propagation in 3D infinite nonlocal solid were
studied. In particular, the author utilized novel trigonometric series as approximating
functions while using Galerkin approach. Effects of length to thickness ratio, aspect
ratio, nonlocal parameter and different boundary conditions influence on the natural
frequencies of the nanoplate vibrations were analysed.
Despotovic [58] investigated the problem of stability and vibrations of square
single-layer graphene sheet employing Eringen’s approach. Natural frequencies of
transverse vibrations versus the body length and nonlocality features were estimated
based on the Galerkin method and the classical and nonlocal elasticity theories.
Critical values of the body load parameter and the mode shapes were determined.
Singh et al. [59] carried out the vibration study of a nanoplate supported by
Winkler foundations in the framework of the classical/ Eringen’s elasticity theory.
Effects of various nanoplates’ parameters on the non-dimensional frequencies were
illustrated.
1 Nanostructural Members in Various Fields: A Literature Review
based on the Hamilton principle, and both natural frequencies and critical electric
voltages were estimated.
Salehipour et al. [53] used converted couple stress and three-dimensional elasticity conception to derive a model for static and vibration of functionally graded
microplates and nanoplates. Developed model included small size effect. Hamilton’s principle was used to create the equations of motion and boundary conditions.
For in-plane and out-of-plane free vibrations of simply supported plates, analytical
closed-form solutions were described. In order to achieve the analytical solutions,
the elasticity modulus and mass density were assumed to differ exponentially by the
thickness of the plate.
Jafari et al. [54] used nonclassical constitutive equations consisting of the
first/second-order strain gradients. Navier and Galerkin methods were employed to
solve the governing PDEs and obtain the approximate system outputs, respectively.
The author studied the influence of different parameters, boundary conditions and
the plate size on the natural frequencies of the nanoplates.
Ghassabi et al. [55] derived governing PDEs and the associated boundary conditions with an account of a nonlocal parameter based on Hamilton’s principle. The
Kirchhoff, Mindlin and the third-order shear deformation theories were employed.
The analysed case studies included simply supported and cantilever nanoplates with
an emphasis put to demonstrate a role of the dimensionless plate length, plate theory,
power-low index and nonlocal parameter ratio on the vibration of the functionally
graded rectangular nanoplates.
Mehdi Zarei et al. [56] studied free vibration and buckling of a round tapered
nanoplate subjected to in-plane forces. Assumption of nonlocal resilience was used to
demonstrate effects depending on the size. In order to obtain frequency equations for
simply supported and clamped nanoplates, the Raleigh-Ritz method and differential
transformation approach were used. The final results showed that incrementing the
parameter of taper yielded rise of the buckling load and natural frequencies.
Shahrbabaki [57] employed the Ritz and Galerkin methods to study 3D nonlocal elasticity of rectangular nanoplate. Two simple cases of 3D free vibrations of
simply supported nanoplate and wave propagation in 3D infinite nonlocal solid were
studied. In particular, the author utilized novel trigonometric series as approximating
functions while using Galerkin approach. Effects of length to thickness ratio, aspect
ratio, nonlocal parameter and different boundary conditions influence on the natural
frequencies of the nanoplate vibrations were analysed.
Despotovic [58] investigated the problem of stability and vibrations of square
single-layer graphene sheet employing Eringen’s approach. Natural frequencies of
transverse vibrations versus the body length and nonlocality features were estimated
based on the Galerkin method and the classical and nonlocal elasticity theories.
Critical values of the body load parameter and the mode shapes were determined.
Singh et al. [59] carried out the vibration study of a nanoplate supported by
Winkler foundations in the framework of the classical/ Eringen’s elasticity theory.
Effects of various nanoplates’ parameters on the non-dimensional frequencies were
illustrated.