Static Bending, Buckling and Free Vibration of FG-CNTRC Microbeams Resting on Elastic Foundation Using Higher-Order Shear Deformation Beam Theories

This study investigates the static bending, buckling, and free vibration behaviors of functionally graded carbon nanotube–reinforced composite (FG-CNTRC) microbeams supported on an elastic foundation. The proposed model is formulated by coupling higher-order shear deformation beam theories (HoSDBTs) with the modified couple stress theory (MCST). Four distinct CNT distribution patterns within the polymer matrix are considered. Using Hamilton’s principle, governing equations and boundary conditions for simply-supported microbeams are derived and solved analytically. A comprehensive parametric study explores the effects of the material length scale parameter, CNT volume fraction, aspect ratio, foundation stiffness (Winkler and Pasternak models), and CNT gradation on bending stiffness, buckling loads, and natural frequencies. Results reveal that all parameters notably influence the mechanical response, with size-dependent effects and elastic foundation interactions playing key roles. The proposed MCST-enhanced HoSDBT model effectively captures size-dependent behaviors, making it suitable for the design and optimization of FG–CNTRC microdevices.