Axisymmetric indentation of an electroelastic piezoelectric half-space with functionally graded piezoelectric coating by a circular punch

A piezoelectric electroelastic half-space with functionally graded piezoelectric coating is considered. The elastic moduli, piezoelectric moduli and dielectric permeabilities of the coating vary independently with depth according to arbitrary functions. A rigid circular punch interacts with the surface of the half-space under the action of normal centrally applied force. It is assumed that the punch is a perfect electrical conductor having constant value of electric charge. Using the integral transformation technique, the problem is reduced to the solution of a system of two dual integral equations over images of functions describing normal pressure and electric induction on the surface. An approximated analytical solution of the problem is constructed. For this purpose, the bilateral asymptotic method was extended to the case of system of dual integral equation. The solution is asymptotically exact both for small and large values of relative coating thickness and has high accuracy for intermediate values.