Dynamic analysis of a porous microbeam model based on refined beam strain gradient theory via differential quadrature hierarchical finite element method

In this paper, a size-dependent dynamic investigation of a porous metal foams microbeams is presented. The novelty of this study is to use a metal foam microbeam that contain porosities based on the refined high order shear deformation beam model, with sinusoidal shear strain function, and the modified strain gradient theory (MSGT) for the first time. The Lagrange's principle combined with differential quadrature hierarchical finite element method (DQHFEM) are used to obtain the porous microbeam governing equations. The solutions are presented for the natural frequencies of the porous and homogeneous type microbeam. The obtained results are validated with the analytical methods found in the literature, in order to confirm the accuracy of the presented resolution method. The influences of the shape of porosity distribution, slenderness ratio, microbeam thickness, and porosity coefficient on the free vibration of the porous microbeams are explored in detail. The results of this paper can be used in various design for metallic foam micro-structures in engineering.