Nonlinear chaotic responses of obliquely stiffened porous FG doubly curved shallow shells under primary and internal resonances

This study investigates the nonlinear dynamic (ND) behavior of obliquely stiffened porous functionally graded (PFG) doubly curved shallow shells (DCSSs) exposed to external excitation, with particular emphasis on primary resonance (PR) and 1:2 internal resonance (IR). Both the shell and the oblique stiffeners (OSs) are made of PFG materials. Two PFG layouts are examined: (1) shells graded from ceramic-rich at the surfaces toward metal-rich, with uniform porosity through the thickness; and (2) the same grading is used, but the porosity varies through the thickness. In both layouts, the stiffeners are functionally graded (FG) members with uniform porosity through their thickness. A mechanical model is developed for PFG shells reinforced by two crossed families of OSs, where the stiffener angles can be identical or different. Based on first-order shear deformation theory (FSDT), von Kármán kinematics, and Hooke’s law, the stress–strain relationships are established, and the governing partial differential equations are obtained via Hamilton’s principle. These equations are then converted into a two-degree-of-freedom nonlinear ordinary differential system using the Galerkin method. After that, the method of multiple scales (MMSs) is used to derive a four-dimensional system of nonlinear averaged equations. The MMS-based analytical formulation is therefore established within a weakly nonlinear framework, in which small-amplitude responses, weak external excitation, and light damping are assumed. Finally, numerical simulations are conducted to explore key response features, such as phase portraits, Poincaré sections, and time histories, showing how material gradation patterns and stiffener orientation affect the shells’ nonlinear dynamics.