Wave propagation of functionally graded layers treated by recursion relations and effective boundary conditions

Wave propagation through a layer of a material that is inhomogeneous in the thickness direction, typically a functionally graded material (FGM), is investigated. The material parameters and the displacement components are expanded in power series in the thickness coordinate, leading to recursion relations among the displacement expansion functions. These can be used directly in a numerical scheme as a means to get good field representations when applying boundary conditions, and this can be done even if the layer is not thin. This gives a schema that is much more efficient than the approach of subdividing the layer into many sublayers with constant material properties. For thin layers for which the material parameters do not depend on the layer thickness the recursion relations can be used to eliminate all but the lowest order expansion functions. Employing the boundary conditions this leads to a set of effective boundary conditions relating the displacements and stresses on the two sides of the layer, thus completely replacing the layer by these effective boundary conditions. Numerical examples illustrate the convergence properties of the scheme for FG layers and the influence of different variations of the material parameters in the FG layer.