1.1 Introduction
15
Barati [88] developed the free vibrational behaviour of nanoshells of porous
nanocrystalline silicon applying the hypothesis of strain gradients. Nanocrystalline
materials based on multi-phase composites in which nanopores, nanograins and interface phases contribute. The nanoshell was constructed by strain gradient hypothesis
due to the experimental analysis of strain gradients close to the interface phase.
To integrate the dimension of nanograins/nanopores and their surface energies, a
micromechanical solution consisting of the Mori-Tanaka scheme was used. The
nanoshell was modelled by the hypothesis of first-order shear deformation, and the
technique of Galerkin was illustrated to acquire vibration frequencies. The vibrational
behaviour of the nanoshell affected by the porosity proportion of nanograin size, the
strain gradient ratio, boundary conditions and the nanograin/nanopore surface phase.
Razavi [89] studied free vibration of a simply supported magneto-electro-elastic
doubly curved nanoshell in the appearance of a rotating inertia effect on the basis
of the first-order shear deformation hypothesis. Gauss’ electrostatic and magnetostatic principles were applied to model the electric and magnetic behaviours of the
nanoshell. An analytical relationship was then acquired for the natural frequency of
the double-curved magneto-electro-elastic nanoshell. In addition, the consequences
of the electric and magnetic potential, the increase in temperature, nonlocal parameters, Pasternak foundation parameters and the geometry of the nanoshell on the natural frequencies of double-curved magneto-electro-elastic nanoshells were examined.
It has been proved that natural frequency of described nanoshell reduces with rising
temperature and electrical potential or reducing magnetic potential.
Shojaeefard et al. [90] analysed free vibration of a functionally graded piezomagnetic material cylindrical nanoshell. The nanoshell was nested in viscoelastic media
under external electric, rotational and magnetic loadings. The nanoshell’s equations
were derived from the nonlocal hypothesis of Eringen. The structure’s magnetic and
piezoelectric properties exponentially changed in thickness. Due to the initial hoop
tension, the rotational loading was measured. The outcomes were achieved by applying a generalized technique of differential quadrature to the governing equations and
related boundary conditions. Angular velocity, external amperage and voltage parameter, viscoelastic media parameters and functionally graded power index impacted
to the free vibration characteristics of nanoshell.
Karami et al. [91] investigated the porous nanoshell’s wave propagation. In accordance with a higher order shear deformation shell hypothesis, the Bi-Helmholtz
nonlocal strain gradient principle was used to involve size-dependent influence. The
nanoshells were created of a functionally graded material (P-FGM), which varies
continuously in the direction of thickness. The sensitivity of the wave reaction was
studied for different fractions of the porosity volume, material features, nonlocal
parameters, humidity, temperature, length scales of the strain gradient and wavenumbers. Depending on the outcome, the reaction size dependence was confirmed to be
nearly the same as that of beams, plates and tubes.
15
Barati [88] developed the free vibrational behaviour of nanoshells of porous
nanocrystalline silicon applying the hypothesis of strain gradients. Nanocrystalline
materials based on multi-phase composites in which nanopores, nanograins and interface phases contribute. The nanoshell was constructed by strain gradient hypothesis
due to the experimental analysis of strain gradients close to the interface phase.
To integrate the dimension of nanograins/nanopores and their surface energies, a
micromechanical solution consisting of the Mori-Tanaka scheme was used. The
nanoshell was modelled by the hypothesis of first-order shear deformation, and the
technique of Galerkin was illustrated to acquire vibration frequencies. The vibrational
behaviour of the nanoshell affected by the porosity proportion of nanograin size, the
strain gradient ratio, boundary conditions and the nanograin/nanopore surface phase.
Razavi [89] studied free vibration of a simply supported magneto-electro-elastic
doubly curved nanoshell in the appearance of a rotating inertia effect on the basis
of the first-order shear deformation hypothesis. Gauss’ electrostatic and magnetostatic principles were applied to model the electric and magnetic behaviours of the
nanoshell. An analytical relationship was then acquired for the natural frequency of
the double-curved magneto-electro-elastic nanoshell. In addition, the consequences
of the electric and magnetic potential, the increase in temperature, nonlocal parameters, Pasternak foundation parameters and the geometry of the nanoshell on the natural frequencies of double-curved magneto-electro-elastic nanoshells were examined.
It has been proved that natural frequency of described nanoshell reduces with rising
temperature and electrical potential or reducing magnetic potential.
Shojaeefard et al. [90] analysed free vibration of a functionally graded piezomagnetic material cylindrical nanoshell. The nanoshell was nested in viscoelastic media
under external electric, rotational and magnetic loadings. The nanoshell’s equations
were derived from the nonlocal hypothesis of Eringen. The structure’s magnetic and
piezoelectric properties exponentially changed in thickness. Due to the initial hoop
tension, the rotational loading was measured. The outcomes were achieved by applying a generalized technique of differential quadrature to the governing equations and
related boundary conditions. Angular velocity, external amperage and voltage parameter, viscoelastic media parameters and functionally graded power index impacted
to the free vibration characteristics of nanoshell.
Karami et al. [91] investigated the porous nanoshell’s wave propagation. In accordance with a higher order shear deformation shell hypothesis, the Bi-Helmholtz
nonlocal strain gradient principle was used to involve size-dependent influence. The
nanoshells were created of a functionally graded material (P-FGM), which varies
continuously in the direction of thickness. The sensitivity of the wave reaction was
studied for different fractions of the porosity volume, material features, nonlocal
parameters, humidity, temperature, length scales of the strain gradient and wavenumbers. Depending on the outcome, the reaction size dependence was confirmed to be
nearly the same as that of beams, plates and tubes.