The Effects of the Elastic Foundations on the Frequency of the Viscoelastic FG Porous Doubly-Curved Macro/nanosize Shell Subjected to Hygro-Thermo-magnetic Fields Using a Novel Quasi-3D Shell Theory

This article investigates the size-dependent dynamic behavior of the viscoelastic functionally graded (FG) porous doubly-curved macro/nanosize shells rested on various foundations based on the nonlocal strain gradient theory (NSGT). Also, a parametric study was conducted to describe the impact of small-scale parameters, magneto-hygro-thermal environments, as well as Winkler, Pasternak, and Kerr foundations on the vibration characteristics of the system. A novel quasi-3D shell model is developed to scrutinize the dynamics of the structure by dividing the transverse displacement into deflection and thickness stretching components. For porosity-dependent dynamic analysis, two different types of porosity distributions are considered. A novel higher-order shear deformation theory (HSDT) is employed to reflect the shear effects along the system thickness. Hamilton’s principle and Kelvin–Voigt damping effects are utilized to obtain the governing equations of motion. The Navier solution approach is applied to acquire the dynamic response of the panel with simply-supported boundary conditions. It is indicated that the vibration frequency of the considered system is substantially affected by the porosity volume fraction, FG index, porosity distribution, foundation parameters, feedback control gain, viscoelastic structural damping, and length-to-thickness ratio of the structure.