Nonlinear axisymmetric free vibration of thin FGM shallow spherical caps with porosity and tangentially restrained edges

Shallow spherical cap (SSC) is a shell-type component extensively used in various fields, especially in ship and aerospace structures. In spite of practical importance, investigation on the nonlinear free vibration of functionally graded material (FGM) spherical caps with elastically restrained edges is evidently absent. As a first attempt, this paper deals with the nonlinear free vibration of FGM SSCs exposed to thermal environments, including the effects of porosity and tangential constraints of boundary edges. Pores are presented in FGM through even and uneven distributions. The effective properties of porous FGM are determined using a modified rule of mixture. Governing equations are derived using classical shell theory, taking into account von Kármán–Donnell nonlinearity, geometric imperfection, and interactive pressure from elastic foundations. Analytical solutions are assumed to satisfy clamped boundary conditions, and the Galerkin method is applied to obtain a nonlinear ordinary differential equation. This equation is numerically solved adopting fourth-order Runge–Kutta integration scheme to determine the frequencies of nonlinear free vibrations. Numerous influences on both natural and nonlinear frequencies of porous FGM SSCs are analyzed through parametric studies. It is found that the geometry ratios, imperfection, and tangential constraint of edges dramatically affect the frequencies and type of nonlinear free vibration behavior of porous FGM SSCs.