In this paper a N th order nanoplate model is developed for the b ending and buckling analysis of a graphene nanoplate based on a Modified couple stress theory. The strain energy, external work and buckling equations are solved Also using Hamilton’ principle, main and auxiliary equations of nano plate are obtained. The bending rates and dimensionless bending values under uniform surface traction and sinusoidal load, the dimensionless critical force under a uni axial surface force in x direction are all obtained for various plate's dimensional ratios and material length scale to thickness ratios. The governing equations are numerically solved. The effect of material length scale, length, width and thickness of the nanoplate on the bending and buckling ratio s are investigated and the results are presented and discussed in details.

In this paper bending and buckling characteristics of third-order shear, and deformation nanoplates were investigated using the Modified couple stress theory and Navier type solution. It can be useful for designing and manufacturing micro-electromechanical and nano-electromechanical systems. The Modified couple stress theory was applied to provide the possibility of considering the effects of small scales that have only one material length scale parameter. In this theory, the strain energy density is a function of the strain tensor components, curvature tensor, stress tensor, and the symmetric part of the couple stress tensor. After obtaining the strain energy, external work, and buckling equations, the Hamilton principle is employed to derive the governing equations. Furthermore, by applying boundary and loading conditions in the governing equations, the bending and buckling of a third-order shear deformation nanoplate with simply-supported bearings are obtained and the Navier’, s solution is used to solve the equations. The results indicate that the third-order nanoplate subjected to sinusoidal loading yields smaller values of dimensionless bending than it does while subjected to uniform surface traction. It was also found that by increasing the length to thickness ratio, the value of the dimensionless bending of nanoplate decreases but by increasing the aspect ratio of the plate, this value increases. Furthermore, it was shown that the critical buckling load of the third-order nanoplate under uniaxial loading increases by increasing the ratio of the length scale parameter to the thickness of the nanoplate but it decreases by increasing the length to thickness ratio of the nanoplate.

In this paper, the free vibrations of a three-layer sandwich plate with magneto-rheological fluid (MR) core as a smart structure using Trigonometric shear deformation theory (TSDPT) are investigated. The equations of motion are obtained using the Hamilton principle and solved using the Galerkin residual weight method. The complex shear modulus of the MR material in the pre-yield region was described by complex modulus approach as a function of magnetic field intensity. Primary attention is focused on the effects of magnetic field magnitude, geometric aspect ratio, and MR core layer thickness on the dynamic characteristics of the sandwich plate. When an electric field is applied, the damping of the system is more effective. After validation of the present study with the available results in the literature, the effects of the natural frequencies and loss factors on the dynamic behavior of the sandwich plate are examined and discussed. The results show that increasing the intensity of the magnetic field increases the frequency and depreciation coefficient of each mode. Furthermore, increasing the thickness of the fluid has a direct effect on increasing the depreciation coefficient and decreasing the frequency. With the increasing use of smart structures, it is hoped that the findings of this study will make engineering applications more effective.

The aim of this paper is to study the buckling analysis of power law functionally graded rectangular microplates. The Modified couple stress theory based on the exponential shear deformation theory has been used to obtain the dimensionless critical buckling load of the functionally graded microplate. In exponential shear deformation theory, exponential functions are used in term of thickness coordinate to include the effect of transverse shear deformation and rotary inertia. To obtain the critical buckling loads for all boundary conditions, the equations of motion are obtained using Rayleigh– Ritz method based on the Modified couple stress theory that the this theory contains only one material length scale parameter. The temperature is assumed to be constant in the plane of the plate and to vary in the thickness direction. Material properties are assumed to be temperature dependent and vary continuously through the thickness according to a power law distribution in term of the volume fraction of the constituents. Finally, the effect of various parameters such as n Power Law indexes, aspect ratio (a/b), length to thickness ratio and the length scale parameter on the non-dimensional critical buckling load of rectangular FG micro nano-plates are presented.

In this paper a variationally consistent trigonometric shear deformation theory is presented for the free vibration of thick isotropic square and rectangular plate. In this displacement based theory, the in-plane displacement field uses sinusoidal function in terms of thickness coordinate to include the shear deformation effect. The cosine function in terms of thickness coordinate is used in transverse displacement to include the effect of transverse normal strain. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Results of frequency of bending mode, thickness-shear mode and thickness-stretch mode are obtained from free vibration of simply supported isotropic square and rectangular plates and compared with those of other refined theories and frequencies from exact theory. Present theory yields exact dynamic shear correction factor p2/12 from thickness shear motion of the plate.

An analytical study on fluid induced dynamics of cantilever microbeam is presented in this paper. Modified strain gradient theory has been used to consider the effect of small sizes. By considering the interaction between structure and fluid, the governing equations of motion are derived from hyperbolic beam theory. Governing equations of motion are discretized with the Galerkin method, and then the solution is found numerically. The dynamic response of the system and the amplitude-velocity curves of the fluid flow at different values of small size parameters and fluid flow velocity are determined and the effects of these parameters are examined. The results show that the hyperbolic beam theory provides more accurate results than classical Euler-Bernoulli and Timoshenko beam theories. Each of the three theories exhibits different lock-in regions and maximum amplitudes of the microbeam. It is also relevant to note that Euler-Bernoulli's theory predicts natural frequencies more than the other two theories, which ignores the rotational inertia of the beam's cross-section. Timoshenko beam theory predicts higher oscillation frequencies than hyperbolic beam theory, however, when the length is increased, the natural frequencies for the two theories are almost identical.

In this study, the dynamic buckling of the embedded laminated nanocomposite plates is investigated. Theplates are reinforced with the single-walled carbon nanotubes (SWCNTs), and the Mori-Tanaka model isapplied to obtain the equivalent material properties of them. Based on the sinusoidal shear deformationtheory (SSDT), the motion equations are derived using the energy method and Hamilton's principle. TheNavier’ s method is used in conjunction with the Bolotin's method for obtaining the dynamic instabilityregion (DIR) of the structure. The effects of different parameters such as the volume percentage ofSWCNTs, the number and orientation angle of the layers, the elastic medium, and the geometricalparameters of the plates are shown on DIR of the structure. Results indicate that by increasing the volumepercentage of SWCNTs the resonance frequency increases, and DIR shifts to right. Moreover, it is foundthat the present results are in good agreement with the previous researches.

A Trigonometric shear deformation theory (TSDT) for the analysis of isotropic plate, taking into account transverse shear deformation effect as well as transverse normal strain effect, is presented. The theory presented herein is built upon the classical plate theory. In this displacement-based, trigonometric shear deformation theory, the in-plane displacement field uses sinusoidal function in terms of thickness coordinate to include the shear deformation effect. The cosine function in terms of thickness coordinate is used in transverse displacement to include the effect of transverse normal strain. It accounts for realistic variation of the transverse shear stress through the thickness and satisfies the shear stress free surface conditions at the top and bottom surfaces of the plate. The theory obviates the need of shear correction factor like other higher order or equivalent shear deformation theories. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Results obtained for static flexural analysis of simply supported thick isotropic plates for various loading cases are compared with those of other refined theories and exact solution from theory of elasticity.