1.1 Introduction
13
solution was acquired using the classical separation of variables. Compared to their
local counterparts, the impact of the nonlocal parameter on natural frequencies and
modal forms was debated. The carried out research may be helpful in biomedical
and bioengineering-oriented nanotechnology fields.
Ke et al. [81] investigated the thermo-electro-mechanical vibration of piezoelectric cylindrical nanoshells by applying nonlocal hypothesis and the concept of the
thin shell of Love. From the principle of Hamilton, the boundary conditions and
governing equations were derived. For the simply supported piezoelectric nanoshell,
an analytical solution was provided by employing the double Fourier series of the
displacement components. The method of differential quadrature (DQ) was used to
acquire numerical solutions for piezoelectric nanoshells under different boundary
conditions. The effect of the nonlocal parameter, increase in temperature, external
electric voltage, radius-to-thickness and radius-to-radius ratio on the natural frequencies of piezoelectric nanoshells were extensively analysed. The nonlocal effect
and thermoelectric loading were discovered to have a major impact on the natural
frequencies of piezoelectric nanoshells.
Rouhi et al. [82] observed that when dimensions were in the order of nanometer,
the surface stress effect played a key role in the mechanical behaviour of nanostructures. The free vibration behaviour of simply supported cylindrical nanoshells was
investigated in the context of the surface elasticity principle, taking into account the
aforementioned effect using an exact solution method. Energy-based approach was
employed to receive the governing equations of motion and boundary conditions.
In compliance with the Gurtin-Murdoch hypothesis, the effect of surface stress was
estimated. The nanoshell was modelled using the principle of shell deformation of
the first-order shear. The problem of free vibration was resolved due to the dimensionless form of governing equations and then explained using a Navier-type solution
under the simply supported boundary conditions. Chosen numerical outcomes on the
effects of surface stress and surface material features on the natural frequencies of
different radii and lengths of nanoshells were illustrated. The findings indicated that
the surface energies have a major impact on the vibrational behaviour of nanoshells
with small thickness magnitudes. It was noted that the nanoshell’s natural frequency
depends on the properties of the surface material.
Mehralian et al. [83] investigated the size-dependent conical shell formulation. It
was derived on the basis of the modified couple stress theory and the first-order shear
deformation model in order to confirm the free vibration of functionally graded conical shell inserted in an elastic Pasternak medium and subjected to thermal conditions.
The material properties were deemed temperature-dependent and classified according to the distribution of power law in the thickness direction. By using the principle
of Hamilton, the boundary conditions and governing equations were derived. The
size effect was assessed applying the modified hypothesis of couple stress, and the
free vibration of simply supported conical nanoshell truncated by FG was examined
as a special case. The consequences of various parameters such as dimensionless
length parameter or change in temperature were studied on the basis of the modified
couple stress and classical continuum theories.
13
solution was acquired using the classical separation of variables. Compared to their
local counterparts, the impact of the nonlocal parameter on natural frequencies and
modal forms was debated. The carried out research may be helpful in biomedical
and bioengineering-oriented nanotechnology fields.
Ke et al. [81] investigated the thermo-electro-mechanical vibration of piezoelectric cylindrical nanoshells by applying nonlocal hypothesis and the concept of the
thin shell of Love. From the principle of Hamilton, the boundary conditions and
governing equations were derived. For the simply supported piezoelectric nanoshell,
an analytical solution was provided by employing the double Fourier series of the
displacement components. The method of differential quadrature (DQ) was used to
acquire numerical solutions for piezoelectric nanoshells under different boundary
conditions. The effect of the nonlocal parameter, increase in temperature, external
electric voltage, radius-to-thickness and radius-to-radius ratio on the natural frequencies of piezoelectric nanoshells were extensively analysed. The nonlocal effect
and thermoelectric loading were discovered to have a major impact on the natural
frequencies of piezoelectric nanoshells.
Rouhi et al. [82] observed that when dimensions were in the order of nanometer,
the surface stress effect played a key role in the mechanical behaviour of nanostructures. The free vibration behaviour of simply supported cylindrical nanoshells was
investigated in the context of the surface elasticity principle, taking into account the
aforementioned effect using an exact solution method. Energy-based approach was
employed to receive the governing equations of motion and boundary conditions.
In compliance with the Gurtin-Murdoch hypothesis, the effect of surface stress was
estimated. The nanoshell was modelled using the principle of shell deformation of
the first-order shear. The problem of free vibration was resolved due to the dimensionless form of governing equations and then explained using a Navier-type solution
under the simply supported boundary conditions. Chosen numerical outcomes on the
effects of surface stress and surface material features on the natural frequencies of
different radii and lengths of nanoshells were illustrated. The findings indicated that
the surface energies have a major impact on the vibrational behaviour of nanoshells
with small thickness magnitudes. It was noted that the nanoshell’s natural frequency
depends on the properties of the surface material.
Mehralian et al. [83] investigated the size-dependent conical shell formulation. It
was derived on the basis of the modified couple stress theory and the first-order shear
deformation model in order to confirm the free vibration of functionally graded conical shell inserted in an elastic Pasternak medium and subjected to thermal conditions.
The material properties were deemed temperature-dependent and classified according to the distribution of power law in the thickness direction. By using the principle
of Hamilton, the boundary conditions and governing equations were derived. The
size effect was assessed applying the modified hypothesis of couple stress, and the
free vibration of simply supported conical nanoshell truncated by FG was examined
as a special case. The consequences of various parameters such as dimensionless
length parameter or change in temperature were studied on the basis of the modified
couple stress and classical continuum theories.