Nonlinear forced vibrations of FGM microplates: a numerical approach in the context of Mindlin's strain gradient elasticity

Abstract

This study aims to investigate the geometrically nonlinear forced vibration behavior of micro-scale plates made of functionally graded materials (FGMs) by developing a variational numerical approach that accounts for strain gradient effects. The proposed approach, which is based on the variational differential quadrature technique, is capable of addressing the problem with arbitrary geometry (e.g. quadrilateral plate, annular sector plate, triangular plate, etc.). Besides, Mindlin's strain gradient theory is applied that leads to a formulation which encompasses the modified versions of strain gradient and couple stress theories (MSGT & MCST). An important novelty of present work is its vector-matrix presentation which can be beneficial for researchers working on numerical methods. Based on Hamilton's principle together with Mindlin's plate theory, the governing equations are derived. In the numerical results, the effects of thickness-to-material length-scale parameter on the frequency-response curves of FG plates with various shapes are analyzed. Also, comparisons are made between the predictions of MCST, MSGT as well as the classical theory. The results indicate that strain gradient terms have a pronounced influence on the nonlinear dynamic response of FGM microplates, giving rise to evident stiffening behavior and noticeable changes in the frequency–response curves. These observations highlight the importance of using higher-order continuum models to achieve reliable predictions of nonlinear forced vibration behavior at the micro-scale. © 2026 Elsevier Masson SAS