A Novel Differential Quadrature Galerkin Method for Dynamic and Stability Behaviour of Bi-directional Functionally Graded Porous Micro Beams

Abstract

The free vibration and buckling behaviours of 2D-FG porous microbeams are explored in this paper utilizing the Quasi-3D beam deformation theory based on the modified couple stress theory and a Differential Quadrature Galerkin Method (DQGM) systematically, as a combination of the Differential Quadrature Method (DQM) and the semi-analytical Galerkin method, which has used to reduce computational cost for problems in dynamics. The governing equations are obtained using the Lagrange’s principle. The mass and stiffness matrices are calculated using the weighting coefficient matrices given by the differential quadrature (DQ) and Gauss-Lobatto quadrature rules. The matrices are expressed in a similar form to that of the Differential Quadrature Method by introducing an interpolation basis on the element boundary of the Galerkin method. The sampling points are determined by the Gauss-Lobatto node method. The influence of the thickness-to-material length scale parameter (MLSP) on the nondimensional natural frequencies and nondimensional critical buckling loads of 2D-FG porous microbeams are investigated, along with the effects of the boundary condition, aspect ratio and gradient index. The results are validated with literature to establish the accuracy of the procedure described. This work will provide a numerical basis for the design of FG microstructures in the field of micromechanics. These results can be applied to the engineering design of porous FG microstructures.