Finite element buckling analysis for micro structures based on the couple stress theory

Mechanical behavior at the micro- and nano-scale exhibits size-dependent effects, such as increased stiffness, which are not captured by classical continuum mechanics. These effects become particularly important in thin structures, where buckling is a critical failure mode; however, previous studies have been limited to simple geometries due to the complexity of the governing equations. This study develops a finite element framework for buckling analysis based on the modified couple stress theory (MCST), applicable to arbitrary geometries. Beam and shell elements are formulated to incorporate size effects, enabling buckling analysis of general structures. The framework is demonstrated on various configurations, including thin membranes under residual stress, stiffened plates, and tensile bars, highlighting its versatility. Numerical results show that accounting for size effects consistently increases the predicted critical buckling loads compared to classical continuum mechanics. The proposed approach offers a broadly applicable tool for stability assessment in micro-electromechanical systems (MEMS), flexible electronics, and thin-film micro sensors and actuators.