Buckling of Porous FGM Sandwich Beams: A Spectral Analysis

This research studies the mathematical modelling and buckling behaviour of sandwich beam structures featuring a homogeneous metallic core and porous, functionally graded material (FGM) face sheets. A power-law distribution is used to model the continuous variation in material properties across the thickness of the FGM layers. To evaluate the impact of manufacturing-induced micro-voids, two specific porosity topographies—uniform (even) and cosine-based (uneven) distributions—are embedded into the constitutive relations. The governing equations of equilibrium are established using Timoshenko beam theory, ensuring that transverse shear deformation is rigorously accounted for. The resulting differential equations are solved using the Chebyshev Pseudospectral Method, which converts the formulation into a robust eigenvalue problem. The CPM achieves spectral convergence, with the relative error dropping below 10⁻⁹ using as few as N = 12 collocation points. The critical buckling loads agree with exact analytical benchmarks to four decimal places (e.g., β = 9.8709 vs the exact value 9.8696 for a homogeneous simply-supported beam, a relative error of 0.013%). For the baseline FGM sandwich configuration (L/h = 10, p = 1.0, α = 0.1, λ = 0.4, P-P boundary conditions), the dimensionless buckling load is β = 25.99 under the uniform porosity model (Model I) and β = 27.45 under the cosine porosity model (Model II). A comprehensive parametric study explores the synergistic influence of the power-law exponent, porosity volume fraction, thickness-to-length ratios, and diverse boundary conditions on the critical buckling thresholds. The results indicate that, while porosity generally reduces structural stiffness, the uneven distribution model leads to a greater reduction in stability than its uniform counterpart. Additionally, a comparison with Euler-Bernoulli theory underscores the need to employ shear-deformable models for deep beams. The developed semi-analytical framework serves as a computationally efficient tool for the structural optimization of porous FGM-based sandwich systems in high-performance engineering fields.