Eigenbuckling of Variable-Thickness Microplates with Strain Gradient Effects: Theoretical Modeling and DQFE Simulation

This study develops an eigenbuckling model for variable-thickness microplates based on the modified strain gradient theory (MSGT) and Kirchhoff–Love assumptions. The governing differential equations and corresponding boundary conditions are systematically derived for both in-plane and out-of-plane deformations of the microplates. Both uniformly and non-uniformly distributed in-plane loading cases are considered. Under constant-volume constraints, three longitudinal thickness variation profiles — linear, parabolic, and cubic — are investigated. Both [Formula: see text]-and [Formula: see text]-continuous nine-node differential quadrature finite elements (DQFEs) are applied to solve the developed eigenbuckling model. The accuracy and reliability of the derived formulations are established through a series of benchmark examples. Extensive parametric studies are conducted to show the effects of thickness variation, geometry, material length scale parameter (MLSP), in-plane loading pattern, and boundary condition on the eigenbuckling behavior of the microplates. Mode localization under various influencing factors is quantitatively assessed using the relative value of the direction cosine of the mode vector. The results demonstrate that both the taper ratio and dimensionless MLSP have a significant effect on the eigenbuckling loads and their associated mode contours. This influence is further enhanced in higher-order eigenbuckling modes, where the size effect is most evident. As the strain gradient effect intensifies, the eigenbuckling mode localization induced by the taper ratio is correspondingly mitigated. Moreover, mode transition phenomena are identified in symmetrically constrained microplates with uniform mass distribution under distributed in-plane compressive loading. These findings offer important insights for the design of tunable microscale structural systems.