Effect of surface stresses on the dynamic behavior of bi-directional functionally graded nonlocal strain gradient nanobeams via generalized differential quadrature rule

A mathematical model for bi-directional functionally graded Euler-Bernoulli nanobeams has been developed based on nonlocal strain gradient theory intervened with Gurtin-Murdoch surface elasticity theory. This model estimates the combined effect of nonlocal, strain gradient and surface stresses on natural frequencies of the nanobeams with varying material properties in axial and transverse directions both as per the power-law functions. The governing differential equation of motion and nonclassical boundary conditions for considered nanobeams have been derived with the help of Hamilton's energy principle. For the first time, the problem of bi-directional functionally graded nonlocal strain gradient nanobeam has been well-posed by considering higher-order nonclassical boundary conditions. Differential quadrature rule has been used to solve the governing differential equations for frequency parameters. A parametric study has been performed to analyze the significance of considering the surface effect on the vibration behavior of nonlocal strain gradient nanobeams with varying values of nonlocal parameter, length scale parameter, gradient indices and slenderness ratio for three different boundary conditions. The results reveal that the surface effect has a considerable increase in the frequencies of bi-directional functionally graded nanobeam, especially at lower thicknesses. An additional increase in the effect of material length scale parameter and nonlocal parameter has been noticed in the presence of surface parameters. •Effect of surface stresses has been accounted using Gurtin-Murdoch surface elasticity theory.•Bi-directional variation of material properties has been considered.•The material properties of bulk part are taken to be different from the surface layers.•Analysis of vibration characteristic has been presented using nonlocal strain gradient theory.