Elastic foundation effect on the small-scale analysis of functionally graded porous microbeams using a modified strain gradient theory

The impact of foundation properties on the mechanical behaviour of microstructures is an essential and compelling area of research in micro/nano-electro-mechanical systems. This study examines the foundation's influence on the buckling, bending, and free vibration responses of functionally graded (FG) porous microbeams. The beam model is based on a modified strain gradient theory and third-order shear deformation theory. A Ritz solution using Legendre functions is developed to address the governing equations of motion. FG porous microbeams with symmetric (D1) and asymmetric (D2) porosity distribution patterns and three boundary conditions (clamped-clamped, clamped-free, and simply-supported) are thoroughly investigated. A comprehensive analysis scrutinises the effects of elastic foundation, porosity ratio, porosity distribution, boundary condition, and geometry on FG porous microbeams' buckling, bending, and vibration responses. The findings of this study suggest that the foundation effect is particularly significant for clamped-free beams and D2 beams, and it becomes more pronounced with an increase in the thickness-to-material length scale parameter ratio. This research provides valuable insights into FG porous microbeams on foundations using the modified strain gradient theory, thereby establishing a basis for future investigations. Furthermore, the present results have implications for the design of micro-structured devices.