Effects of multiple cracks, porosity and bidirectional thickness variation on the buckling of composite plates resting on partially supported elastic foundations

Porous functionally graded (PFG) plates with spatially varying thickness are susceptible to buckling when material imperfections, cracks, and foundation discontinuities act simultaneously. Although the individual effects of porosity, variable thickness, cracks, and elastic foundations have been extensively investigated, the stability interaction between multiple cracks, bidirectional thickness variation, and spatially partial foundation support remains insufficiently understood. This study develops a phase-field finite element framework based on higher-order shear deformation theory (HSDT) for the buckling analysis of PFG plates with bidirectional thickness variation and multiple pre-existing cracks. The formulation accounts for the through-thickness material gradation and porosity distribution, spatially varying plate thickness, crack-induced stiffness degradation, and Winkler-Pasternak foundation support. Particular attention is given to partially supported configurations in which an unsupported region may overlap the crack zone or be spatially separated from it. The critical buckling loads are obtained from an eigenvalue problem, and the formulation is validated against available benchmark solutions. Numerical results reveal that the stability response is governed not only by the individual effects of crack length, crack number, porosity, thickness variation, and foundation stiffness, but also by their spatial interaction. The most pronounced reduction in buckling resistance occurs when the unsupported region overlaps a crack located in the thinner part of the plate. Moving the unsupported region away from the crack progressively restores the foundation restraint and increases the critical buckling coefficient. These findings provide new insight into the stability sensitivity of PFG plates to the spatial coincidence of material, geometric, crack, and foundation-induced stiffness degradation.