On the Reconstruction of a Two-Dimensional Density of a Functionally Graded Elastic Plate

In this work, based on the general formulation of the problem of steady-state vibrations of an inhomogeneous elastic isotropic body, a direct problem of planar vibrations of a rectangular plate within the framework of a plane stress state is formulated. The left side of the plate is rigidly fixed, vibrations are forced by tensile load applied at the right side. The properties of the functionally graded material are described by two-dimensional laws of change in Young’s modulus, Poisson’s ratio and density. For generality of consideration, a dimensionless formulation of the problem is given. The solution to the direct problem of determining the displacement field was obtained using the finite element method. The effect of material characteristics on the displacement field and the value of the first resonant frequency are shown. An analysis of the obtained results was carried out. The inverse problem of determining the law of density from data on the values of the displacement field components at a fixed frequency is considered. To reduce the error in calculating derivatives of table functions of two variables, an approach based on spline approximation and a locally weighted regression algorithm is proposed. Reconstruction examples of different laws are presented to demonstrate the possibility of using this approach.