Free vibration behavior of FG-CNT reinforced metal foam doubly-curved shells resting on a Kerr elastic foundation: Potential application to high-performance sport structures

Lightweight doubly-curved structures require an appropriate balance between structural mass and dynamic stiffness, particularly in vibration-sensitive engineering applications. This study investigates the free vibration of simply supported porous metal-foam doubly-curved shells reinforced with functionally graded carbon nanotubes (FG-CNTs) and supported by a three-parameter Kerr elastic foundation. The effective material properties are evaluated using the Halpin–Tsai micromechanical model and the extended rule of mixtures, while the governing equations are derived within first-order shear deformation theory (FSDT) using Hamilton's principle and solved through a closed-form Fourier-series procedure. The effects of porosity distribution, CNT grading and content, CNT geometric parameters, Kerr-foundation stiffnesses, shell curvature, and slenderness are examined. The results show that increasing porosity generally reduces the natural frequencies because the associated stiffness degradation dominates the reduction in inertia, whereas CNT reinforcement and increased foundation stiffness enhance the dynamic stiffness and raise the frequencies. The CNT distribution and geometric parameters also significantly influence the vibration response, while reduced curvature and increased shell slenderness lead to lower frequencies. The proposed framework provides an efficient parametric tool for understanding and tailoring the free-vibration behavior of porous FG-CNT-reinforced curved structures, including potential sport-related applications.