Nonlinear vibration characteristics of multi-directionally graded porous material beams resting partly on elastic foundations

This article investigates the nonlinear vibration behavior of sandwich beams with mechanical characteristics and porosities that change in multiple directions. In which, the whole beam structure rests partially on an elastic foundation. The mechanical characteristics of the beam demonstrate a constant fluctuation throughout its length and height, following a power law relationship. Moreover, the arrangement of porosity in the beam aligns with the vertical dimension. The finite element method (FEM) together with Von Kármán nonlinear relationships and high-order shear deformation theory via three implicit functions are used to analyze the mechanical behavior relationships in beams. The derivation of the beam’s system of nonlinear motion equations is based on Hamilton’s principle. The approach used to determine the nonlinear frequency of the beam is iterative and based on the displacement control technique. Utilize published findings to evaluate and compare the reliability of the research methodologies employed in the article. Beams’ nonlinear vibration is thoroughly examined and analyzed with respect to various material properties, porosity distribution, and geometry. The findings of the research serve as a crucial foundation for addressing several future intricate issues associated with multilayer sandwich beams composed of functionally graded materials, including stability analysis and dynamic problems, among others.