High-order finite element investigation of nonlinear buckling and post-buckling behavior of porous FGM plates on elastic foundations

This study extensively examines the stability and post-buckling characteristics of porous functionally graded material (FGM) plates supported on a geometrically nonlinear Winkler/Pasternak elastic foundation. The analysis utilizes the high-order continuation finite element approach (FE-HOCA), which incorporates high-order shear deformation theory (HOSDT) and employs the numerical asymptotic method. The research explicitly explores the impacts of both even and uneven porosity distributions and the effects of foundation parameters on the buckling and post-buckling behavior of FGM plates. A notable aspect of the FE-HOCA is its adaptive step size, which adjusts to local nonlinearities within the solution, thereby improving the curve-tracing process and aiding in identifying bifurcation points. By utilizing a Taylor series expansion of the governing equilibrium equations, the method reformulates them into a recursive sequence of linear subproblems for each order. Discretization is performed using an eight-node quadrilateral finite element, with nine degrees of freedom assigned to each node. Furthermore, the continuation algorithm enables tracing entire solution branches by inverting the tangent matrix only once per branch, resulting in considerable computational efficiency compared to conventional Newton–Raphson methods, which are typically more resource-intensive and solve iteratively. Various numerical examples and parametric investigations illustrate the effectiveness of the FE-HOCA, analyzing the influences of porosity distribution, foundation stiffness, power-law index, and boundary conditions on the overall structural response.