The nonlinear static behavior of multi-dimensional functionally graded porous sandwich microbeams supported by partially elastic foundations and elastic boundary restraints

This study investigates the nonlinear static response of multidirectional functionally graded porous sandwich microbeams (m-FGPS microbeams) resting on partially elastic foundations (pEF) with elastic boundary restraints (eBR). Material properties vary along the axial and thickness directions according to power-law distributions, incorporating porosity effects. Size-dependent behavior is captured using the modified couple stress theory (MCST), while a refined beam theory (RBT) accounts for transverse shear deformation. Geometric nonlinearity is described by the von Kármán strain–displacement relations. Hamilton’s principle is used to derive the governing equations, which are solved using the finite element method (FEM) and an iterative displacement-controlled scheme. Parametric studies demonstrate that material gradation, porosity distribution, size effects, foundation stiffness, and boundary flexibility significantly influence the nonlinear load–deflection response. The interaction between size effects and partial elastic support further modifies structural stiffness. The proposed framework provides an effective tool for analyzing and designing microscale sandwich beams for MEMS/NEMS applications.