A Refined Four-Variable First-Order Shear Deformation Theory for Free Vibration Analysis of FG Doubly Curved Nanoshells

The free vibration behaviour of functionally graded (FG) doubly curved nanoshells is explored by adopting a refined first-order shear deformation theory (FSDT) formulated with only four displacement variables. The presented kinematic model, which decomposes the transverse displacement to bending and shear components, provides an efficient and accurate framework for capturing structural response while requiring substantially lower computational effort than traditional higher-order theories. By utilising a power-law pattern, the nanoshell’s material properties are changing continuously within the thickness. Eringen’s nonlocal elasticity theory is implemented, which considers the size-dependent impact that occurs at the nanoscale. The governing equations of motion are constructed via the application of Hamilton’s principle and solved analytically by Navier’s method for simply supported boundary conditions. The current model’s accuracy and dependability are validated by comparisons with published results for various limiting cases such as spherical, cylindrical, and hyperbolic paraboloidal shells. A thorough parametric study is then carried out to examine the effects of the nonlocal parameter, power-law index, side-to-thickness ratio, curvature ratio, and aspect ratio on natural frequencies.