A homogenization procedure for geometrically non-linear free vibration analysis of functionally graded circular plates involving the coupling between transverse and in-plane displacements

Abstract

A non-linear free axisymmetric vibration of functionally graded thin circular plates denoted by FGCP whose properties vary through the thickness subjected to the coupling between transverse and in-plane displacements is investigated. The equations of motion are derived using the energy method and a multimode approach. A homogenization procedure has been employed to reduce the problem under consideration to that of an isotropic homogeneous circular plate. The inhomogeneity of the plate is characterized by a power law variation of Young's modulus and mass density of the material along the thickness direction whereas Poisson's ratio is assumed to be constant. This variation in material properties of the plate introduces a coupling between the in-plane and transverse displacements. The problem is solved by a numerical iterative method. The formulations are validated by comparing the results with the available solutions in the literature for FG circular plates. The non-linear to linear frequency ratios are presented for various volume fraction index n. The effects of the coupling between the in-plane and transverse displacements on the frequency parameters are proved to be significant. The distributions of the associated bending, membrane and total stresses are also given for various vibration amplitudes with different values of the volume fraction index n and compared with those predicted by the linear theory.