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In this article, the buckling and vibration analysis of a double-bonded nanocomposite piezoelectric plate reinforced by a boron nitride nanotube based on the Eshelby-Mori-Tanaka approach is developed using modified couple stress theory under electro-thermo-mechanical loadings surrounded by an elastic foundation. Using Hamilton's principle, the governing equations of motion are obtained by applying a modified couple stress theory and the Eshelby-Mori-Tanaka approach for piezoelectric material and Kirchhoff plate. These equations are coupled for the double-layer plate using the Pasternak foundation and solved using Navier’s type solution. Then the dimensionless frequencies and critical buckling load for simply-supported boundary conditions are obtained. The effects of material length scale parameter, elastic foundation coefficients, aspect ratio ( a/b), length to thickness ratio ( a/h), transverse and longitudinal wave numbers on the dimensionless natural are investigated. The dimensionless frequency of a double-bonded nanocomposite piezoelectric plate increases with increasing length to thickness ratio and decreases with increasing aspect ratio. In addition, the effect of the elastic foundation on the dimensionless frequency of double-bonded nanocomposite piezoelectric plates is more considerable for higher elastic medium parameters. The critical buckling load also decreases with an increase in the dimensionless material length scale parameter.

In this paper, the free vibration of an orthotropic beam undergoing finite strain are studied. The second Piola-Kirchhoff stress tensor and Green-Lagrange strain tensor according to finite strain assumption were used to obtain Euler-Bernoulli beam governing equations. The Galerkin method and Generalized Differential Quadrature method were employed for solving the governing equations and boundary condition. The effect of beam thickness and different boundary conditions were considered in finite strain formulation of the beam equations. Natural frequencies of different composite materials are obtained and compared. The results revealed that by increasing the beams thickness, the difference between maximum vibration amplitude increased between von Karman and finite strain formulations. Also, in a beam with simply- simply supports, differences between linear and non linear mode shapes was remarkable.

In this article, free vibration of rotating fiber–metal laminate thin circular cylindrical shells has been analyzed. Strain–displacement relations have been obtained based on Love’s first approximation shell theory. The variations of frequencies of the fiber–metal laminate cylindrical shell with rotational speeds for different axial and circumferential wave numbers, L/R ratios, metal thicknesses and volume fractions of metal have been presented. Also, free vibrations of the rotating fiber–metal laminate shell have been studied for carbon/epoxy, glass/epoxy and aramid/epoxy composite materials combining thin aluminum layers. The results showed that with increasing rotating speed, the gap between backward and forward waves frequencies increased.