Nonlinear free vibration of geometrically imperfect porous FGM shell panels on nonlinear foundations including elastic edge restraints and elevated temperatures

The combined influences of porosity, geometric imperfection, nonlinear elastic foundations, elevated temperature and tangentially elastic restraints of edges on the nonlinear free vibration of functionally graded material (FGM) doubly curved shell panels are investigated in this paper. Motion and compatibility equations of geometrically imperfect shell panels are established within the framework of first-order shear deformation shell theory including von Kármán–Donnell nonlinearity and interactive pressure from three-parameter nonlinear foundations. Analytical solutions are assumed to satisfy simply supported boundary conditions, and Galerkin method is applied to derive a time-variable ordinary differential equation including both quadratic and cubic nonlinear terms. This equation is numerically solved employing fourth-order Runge–Kutta integration scheme to determine the frequencies of nonlinear free vibration. Parametric studies are carried out to assess various influences on both natural and nonlinear frequencies. It is found that tangential constraints of edges, size of geometric imperfection and elastic foundations have dramatic influences on the nonlinear vibration of porous FGM shell panels. It is also revealed that nonlinear vibration behavior can be of the softening type when panels are more curved, edges are more rigorously restrained, temperature is more elevated, geometric imperfection is more outward and nonlinear foundation is of the softening type.