A Galerkin weak formulation of micropolar elasticity theory for analyzing the 3D static bending and free vibration characteristics of homogeneous and functionally graded micropolar plates

Abstract

Applying Hamilton’s principle, we derive a Galerkin weak formulation of micropolar elasticity theory (MPET). Subsequently, based on this formulation, we further develop a size-dependent finite layer method (FLM) to investigate the static bending and free vibration characteristics of a simply supported, functionally graded (FG) micropolar plate. In the formulation, the micropolar plate is artificially divided into n_l layers, with equal or unequal thicknesses. Six degrees of freedom, including three displacements and three microrotations, are selected to be the primary variables for each nodal surface. MPET’s applicability to aluminum (Al), epoxy (Ep), syntactic foam (SyF), and polyurethane foam (PUF) micropolar plates is examined, revealing that microrotation effects are significant for SyF and PUF micropolar plates and nearly no impact for Al and Ep ones. After validating the MPET-based size-dependent FLM against relevant exact solutions reported in the literature, we conduct a parametric study of FG SyF and PUF micropolar plates to assess the influence of key factors on their static bending and free vibration characteristics. These factors include the inhomogeneity index, length-to-thickness ratio, aspect ratio, and microrotation, which are identified as significant. The results for different Lagrange polynomial orders of modal displacements and microrotations are compared and discussed. Considering accuracy and computational efficiency, we recommend the optimal orders for in-plane displacement (n₁), out-of-plane displacement (n₂), in-plane microrotations (n₃), and out-of-plane microrotation (n₄) are n₁ = 3, n₂ = 2, n₃ = 2, n₄ = 3.