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This paper proposes a finite element method (FEM) for the free vibration analysis of sandwich nanoplates with an auxetic honeycomb core. The proposed method uses a third-order shear deformation theory accounting for both shear deformation and stretching effects without any need for shear correction factors. The size-dependent effect is solved using the nonlocal elasticity theory. The auxetic sandwich nanoplate with negative Poisson’s ratio is applied to achieve ultra-light features and high strength. The obtained numerical results by the proposed method are compared with other published works to demonstrate the accuracy and reliability. Moreover, the influence of the nonlocal factor, geometrics parameters, and material properties (especially the auxetic honeycomb parameters) on the free vibration behavior of sandwich nanoplates is also examined in the numerical examples.

A finite element model using four-unknown shear deformation theory integrated with the nonlocal theory is proposed for the bending and free vibration analysis of functionally graded (FG) nanoplates resting on elastic foundations. The present study developed the four-node quadrilateral element using Lagrangian and Hermitian interpolation functions for analysis of the membrane and bending displacement fields of FG nanoplates. Such a finite element formulation is suitable to investigate for the FG nanoplates resting on the elastic medium foundation with the stiffness matrices, the mass matrices and the load vectors using the second derivatives. The material properties of FG nanoplates are assumed to vary through the thickness direction by a power rule distribution of volume-fractions of the constituents. The equation of motion for FG nanoplates resting on the elastic foundation is obtained through Hamilton’s principle. Several numerical results are presented to demonstrate the accuracy and reliability of the present approach in comparison with other existing methods. In addition, the effects of geometrical parameters, material parameters, nonlocal parameters on the static bending and the free vibration responses of the nanoplates is also investigated in detail.

An edge-based smoothed finite element method (ES-FEM) combined with the mixed interpolation of tensorial components technique (MITC) for triangular elements, named as ES-MITC3, was recently proposed to enhance the accuracy of the original MITC3 for analysis of shells. In this study, the ES-MITC3 is extended to the geometrically nonlinear analysis of functionally graded shells. In the ES-MITC3, the stiffness matrices are obtained using the strain smoothing technique over the smoothing domains that formed by two adjacent MITC3 triangular shell elements sharing an edge. The material properties of functionally graded (FG) shells are assumed to vary through the thickness direction by a power rule distribution of volume fractions of the constituents. The nonlinear finite element formulation based on the first-order shear deformation theory with the von Karman’s large deflection assumption is used to describe the large deformations of the FG shells. Several numerical examples are given to demonstrate the performance of the present approach in comparison with other existing methods.