Nonlinear axisymmetric free vibration of moderately thick FGM shallow spherical caps including porosities and meridional periphery constraint resting on nonlinear elastic foundations

The main aim of this paper is to investigate the combined influences of porosity, initial geometric imperfection, nonlinear elastic foundations, tangential elastic constraint of boundary edge and elevated temperature on the nonlinear axisymmetric free vibration of functionally graded material (FGM) shallow spherical caps. To this goal, a semi–analytical method is used. Unlike previous studies, the effects of imperfections in material, geometry and meridional periphery constraint on the nonlinear vibration of shear deformable FGM shallow spherical caps are included. The properties of constituent materials are considered to be temperature–dependent and the effective properties of porous FGM are determined using a modified rule of mixture. Motion and compatibility equations in terms of deflection and stress function are established within the framework of first order shear deformation theory (FSDT) incorporating initial geometric imperfection and interactive pressure from three-parameter elastic foundation. The derived equations are treated employing analytical solutions and Galerkin method for clamped boundary condition to obtain a time–dependent nonlinear ordinary differential equation. This equation is solved by means of fourth-order Runge–Kutta numerical scheme to determine the frequencies of nonlinear free vibration. It is found that curvature ratio, tangential constraint of periphery, initial geometric imperfection and elastic foundations have significant effects on the frequency–amplitude response of porous FGM spherical caps. Additionally, it is revealed that the nonlinear free vibration of porous FGM spherical caps exhibits the softening type when the shell is sufficiently curved and meridional periphery constraint is sufficiently rigorous. The results detect that geometric imperfection has prominent influence on the intensity of softening–type response.