Investigation of Free Vibrations and Stability of Functionally Graded Three-Layer Plates by Using the R-Functions Theory and Variational Methods

Abstract

We propose a numerical-analytic method aimed at the investigation of free vibrations and stability of functionally graded sandwich-type plates and based on the refined Timoshenko-type theory of the first order. We considered various schemes of arrangement of the layers: (1) the filler is a functionally graded material and the top and bottom layers are made of isotropic materials and (2) vice versa, the filler is made of an isotropic material, whereas the top and bottom layers are made of functionally graded materials. The proposed method is based on the R-functions theory and the Ritz variational method. This method and the developed software take into account the presence of an inhomogeneous subcritical state of the plate. Analytic relations are obtained for the evaluation of the elastic constants and density of the material under the assumption that Poisson’s ratios are identical, for the components of functionally graded materials. The numerical results obtained for the eigenfrequencies and the critical load are compared with the available data and their good agreement is demonstrated. To illustrate the possibilities of the proposed approach, we perform the numerical analyses of plates of complex geometric shapes loaded in the middle plane. The influence of various geometric and mechanical parameters on the dynamic behavior of the plate, critical load, and the zones of dynamic instability is investigated.