Nonlinear vibration of FGM sandwich beams with porosity and elastically restrained ends resting on Winkler/Pasternak/Kerr foundations in thermal environments

The combined influences of porosity, elasticity of tangential constraints of ends, initial geometric imperfection, elastic foundations, and elevated temperatures on the nonlinear free vibration of sandwich beams made of functionally graded material (FGM) and homogeneous layers are investigated in this paper. Two sandwich models corresponding to FGM face sheets and core layer are considered. The pores are assumed to distribute into FGM layers according to even and uneven types. The effective properties of porous FGM are determined using a modified rule of mixture. The motion equations are established based on the Timoshenko beam theory including von Kármán nonlinear terms, initial geometric imperfection and interaction from elastic foundations. These equations are solved by analytical solutions in combination with Galerkin method to obtain a nonlinear ordinary differential equation. Fourth–order Runge–Kutta scheme is applied to solve nonlinear differential equation and determine the nonlinear frequencies. Parametric studies are carried out to assess various effects on both natural frequencies and frequency ratios of FGM sandwich beams. The results demonstrate that the support of elastic foundations makes the natural frequencies higher and frequency nonlinearity less significant. The study also find that the existence of geometric imperfection increases the natural frequencies and weakens the frequency nonlinearity in the deep region of dynamic deflection of the sandwich beams.