Static and dynamic analysis of the postbuckling of bi-directional functionally graded material microbeams

摘要

In this paper, the static and dynamic responses of bi-directional functionally graded (BDFG) microbeams are investigated. The material properties vary along both thickness and axial directions. Employing Hamilton's principle, the differential equations are derived based on von-Karman geometric nonlinearity and third-order shear deformation beam theory. The modified couple stress theory is adopted to capture the size effects. The material length scale parameter of microbeam is considered as a function of spatial coordinates and varies with the material gradient parameters. The differential equations and boundary conditions are discretized using differential quadrature method. Subsequently, static bifurcation of microbeams is calculated utilizing pseudo-arc length continuation technique. The free vibration around the postbuckling configuration of microbeams is studied by solving the associated linear eigenvalue problem. The influences of the functional gradient parameters, dimensionless length scale parameter and aspect ratio on the static response and vibration characteristics of budded BDFG microbeams with various boundary conditions are investigated. It is demonstrated that the buckling of BDFG microbeam occurs through a transcritical bifurcation. The static stable responses of BDFG microbeam are asymmetric due to the stretching-bending coupling that results from the asymmetry of the material distribution in thickness direction of beam. The mode veering phenomenon is detected in postbuckling domain of microbeams. It is examined that the longitudinal dynamic displacement plays a pivotal role in the mode veering phenomenon.