In this study, the vibration behavior of functional graded (FG) circular and annular nanoplate embedded in a Visco-Pasternak foundation and coupled with temperature change is studied. It is assumed that a uniform radial magnetic field acts on the top surface of the plate and the magnetic permeability coefficient of the plate along its thickness are assumed to vary according to the volume distribution function. The effect of in-plane pre-load and temperature change are investigated on the vibration frequencies of FG circular and annular nanoplate. To obtain the vibration frequencies of the FG circular and annular nanoplate, two different size dependent theories are utilized. The material properties of the FGM nanoplates are assumed to vary in the thickness direction and are estimated through the Mori– Tanaka homogenization technique. The FG circular and annular nanoplate is coupled by an enclosing viscoelastic medium which is simulated as a Visco-Pasternak foundation. By using the modified strain gradient theory (MSGT) and the modified couple stress theory (MCST), the governing equation is derived for FG circular and annular nanoplate. The differential quadrature method (DQM) and the Galerkin method (GM) are utilized to solve the governing equation to obtain the frequency vibration of FG circular and annular nanoplate. The results are subsequently compared with valid result reported in the literature. The effects of the size dependent, the in-plane pre-load, the temperature change, the magnetic field, the power index parameter, the elastic medium and the boundary conditions on the natural frequencies are investigated. The results show that the size dependent parameter has an increasing effect on the vibration response of circular and annular nanoplate. Also, the temperature change play important role in the mechanical behavior of the FG circular and annular nanoplate. According to the results, the application of radial magnetic field to the top surface of the plate changes the state of stress in both tangential and radial direction. The present analysis results can be used for the design of the next generation of nano-devices that make use of the thermal vibration properties of the nanoplate.

In present study, the third-order shear deformation theory has been developed to investigate vibration analysis of FG Nano-plates based on Eringen nonlocal elasticity theory. The materials distribution regarding to the thickness of Nano-plate has been considered based on two different models of power function and exponential function. All equations governing on the vibration of FG Nano-plate have been derived from Hamilton’s principle. It has been also obtained the analytical solution for natural frequencies and corresponding mode shapes of simply supported FG Nano-plates. In addition, the general form of stiffness and mass matrix elements has been expressed based on this theory. The effect of different parameters such as power and exponential indexes of targeted function, nonlocal parameter of Nano-plate, aspect ratio and thickness to length ratio of Nano-plate on non-dimensional natural frequencies of free vibration responses have been investigated. The obtained analytical results show an excellent agreement with other available solutions of previous studies. The formulation and analytical results obtained from proposed method can be used as a benchmark for further studies to develop this area of research.

In this study, the vibration behavior of functional graded (FG) circular and annular nanoplate embedded in a Visco-Pasternak foundation and coupled with temperature change is studied. The effect of in-plane pre-load and temperature change are investigated on the vibration frequencies of FG circular and annular nanoplate. To obtain the vibration frequencies of the FG circular and annular nanoplate, two different size dependent theories are utilized. The material properties of the FGM nanoplates are assumed to vary in the thickness direction and are estimated through the Mori–Tanaka homogenization technique. The FG circular and annular nanoplate is coupled by an enclosing viscoelastic medium which is simulated as a Visco- Pasternak foundation. By using the modified strain gradient theory (MSGT) and the modified couple stress theory (MCST), the governing equation is derived for FG circular and annular nanoplate. The differential quadrature method (DQM) and the Galerkin method (GM) are utilized to solve the governing equation to obtain the frequency vibration of FG circular and annular nanoplate. The results are subsequently compared with valid result reported in the literature. The effects of the size dependent, the in-plane pre-load, the temperature change, the power index parameter, the elastic medium and the boundary conditions on the natural frequencies are investigated. The results show that the size dependent parameter has an increasing effect on the vibration response of circular and annular nanoplate. The temperature change also play an important role in the mechanical behavior of the FG circular and annular nanoplate. The present analysis results can be used for the design of the next generation of nanodevices that make use of the thermal vibration properties of the nanoplate.

In this article, the supersonic flutter analysis of a truncated conical shell made of polymer enriched with graphene nanoplatelets (GNPs) exposed to supersonic fluid flow is discussed. It is assumed that the mass fraction of the GNPs is functionally graded (FG) along thickness and length directions according to different dispersion patterns. Modeling of the shell is done using the first-order shear deformation theory (FSDT), the mechanical properties are computed according to the Halpin-Tsai model alongside the rule of the mixture, and the aerodynamic pressure is computed utilizing the piston theory. Utilizing Hamilton’s principle, the boundary conditions and the governing equations are achieved. Harmonic trigonometric functions are used to provide an analytical solution in the circumferential direction and an approximate solution is presented in the meridional direction using the differential quadrature method (DQM). The efficacy of various parameters on the aeroelastic stability are discussed such as the percentage and dispersion pattern of the GNPs and gradient indices. It is observed that to achieve higher aeroelastic stability in the GNP-enriched truncated conical shells, it is better to dispense the GNPs near the small radius and the inner surface of the shell.

In this paper, an analysis of free vibration in functionally graded nanoplate is presented. Third-order shear deformation plate theory is used to reach more accuracy in results. Small-scale effects are investigated using Eringen`s nonlocal theory. The governing equations of motion are obtained by Hamilton`s principle. It is assumed that the properties of nanoplates vary through their thicknesses according to a volume fraction power law distribution. The finite element method (FEM) is presented to model the functionally graded nanoplate and solve mathematical equations accurately. The finite element formulation for HSDT nanoplate is also presented. Natural frequencies of FG nanoplate with various boundary conditions are compared with available results in the literature. At the end some numerical results are presented to evaluate the influence of different parameters, such as power law index, nonlocal parameter, aspect ratio and aspect of length to thickness of nanoplate. In addition, all combinations of simply supported and clamped boundary conditions are considered.