Free vibration, buckling and dynamic stability of bi-directional FG microbeam with a variable length scale parameter embedded in elastic medium

Abstract

• Material properties of microbeam vary along both thickness and axial directions . • Material length scale parameter is a function of spatial coordinates and varies with the gradient parameters. • It shows that the variable scale parameter has significant influence on the static and dynamic characters of microbeam. In this paper, size dependent free vibration , buckling and dynamic stability of bi-directional functionally graded (BDFG) microbeam embedded in elastic medium are investigated. The material properties vary along both thickness and axial directions. In particular, the material length scale parameter of microbeam is considered as a function of spatial coordinates and varies with the material gradient parameters. The system of differential equations with variable coefficients governing the motion of BDFG microbeam is derived employing Hamilton’s principle, the modified couple stress theory and third-order shear deformation beam theory. The differential quadrature method (DQM) is utilized to solve the static and dynamic problem. Three different models evaluating the material length scale parameter of BDFG microbeam are presented for comparison. Parametric studies are carried out to show the influence of gradient parameters, size effect, stiffness of elastic medium on the free vibration, buckling and dynamic stability characteristic of BDFG microbeam. Results show that the variation of material length scale parameter should be considered in the analysis of BDFG microbeam.