Responses of piezoelectric, transversely isotropic, functionally graded, and multilayered half spaces to uniform circular surface loadings

In this paper, an analytical solution is presented to study the response of piezoelectric, transversely isotropic, functionally graded, and multilayered half spaces to uniform circular surface loadings (pressure or negative electric charge). The inhomogeneous material is exponentially graded in the vertical direction and can have multiple discrete layers. The propagator matrix method and cylindrical system of vector functions are used to first derive the solution in the transformed domain. In order to find the responses in the physical-domain, which are expressed in one-dimensional infinite integrals of the Bessel function products, we introduced and adopted an adaptive Gauss quadrature. Two piezoelectric functionally graded half-space models are analyzed numerically: One is a functionally graded PZT-4 half space, and the other a multilayered functionally graded half space with two different piezoelectric materials (PZT-4 and PZT-6B). The effect of different exponential factors of the functionally graded material on the field responses is clearly demonstrated. The difference of the responses between the two surface loading cases is also discussed via the numerical examples. The results should be particularly useful in the characterization of material properties using indentation tests, and could indirectly contribute to the design and manufacturing of piezoelectric functionally graded structures.