Instability of Primary Resonance Solutions and Nonlinear Vibration Behavior in FGM Doubly Curved Shallow Shells with Hybrid Auxetic Honeycomb Cores

Dynamic instability phenomena, such as amplitude jumps and multi-valued resonance states, present critical challenges to the safety of structures subjected to forced vibrations. To address these behaviors, a nonlinear dynamic model was developed for a doubly curved shallow shell made of functionally graded material (FGM) and reinforced with a hybrid re-entrant auxetic honeycomb core. The governing equations were derived through Hamilton’s principle and Reddy’s third-order shear deformation theory, incorporating von Kármán-type geometric nonlinearity. Galerkin’s method and the method of multiple scales were employed to analyze primary resonance and instability zones, and model validation against published benchmarks confirmed excellent accuracy. Numerical investigations revealed that increasing the thickness-to-width ratio enhanced stiffness, reduced resonance amplitudes, and narrowed instability regions. A fourfold increase in damping completely suppressed unstable solutions and reduced resonance amplitude to below 40% of its original value, confirming the powerful stabilizing role of damping. By contrast, variations in wall angles had negligible influence on the location of instability points, although they affected auxetic efficiency. Increasing the Winkler foundation modulus yielded only a modest 10% reduction in jump magnitudes, leaving residual instability, while cell orientation was shown to be crucial in balancing stiffness with energy absorption. These results underline the potential of combining FGMs with hybrid auxetic cores to design lightweight, smart, and dynamically stable structural systems for aerospace, defense, and protective applications.