Vibration analysis of a strain gradient plate model via a mesh-free moving Kriging Interpolation Method

The strain gradient elasticity theory is adopted to capture the small-scale effect of micro/nanostructures to derive the governing equation by using Hamilton's principle. A mesh-free method on the basis of moving Kriging interpolation, of which shape function possesses C² continuum, is developed to investigate the vibration of the strain gradient plate. Owing to the inherent Kronecker delta function property of the moving Kriging interpolation, essential boundary conditions can be directly implemented. One superiority of the moving Kriging interpolation over the traditional moving least square approximation is that it can ensure the continuity and stability at the third order derivatives, which is confirmed by comparison with analytic solutions. Parametric studies in terms of small-scale parameters, boundary conditions and side length on the frequencies and mode shapes are carried out. Natural frequencies evaluated by strain gradient elasticity theory are found to be lower than those of classical plate model. Some lower natural frequencies corresponding to ultrahigh-order mode shapes appear. This demonstrates that the mesh-free moving Kriging interpolation method has high precision and good stability in the construction of the C² continuum shape functions.