Indentation of rough-surfaced spherical punch into elastic transversely isotropic half-space with functionally-graded coating

An axisymmetric contact problem of the elasticity theory of the rigid rough-surfaced spherical punch indentation into an elastic transversely isotropic half-space with the functionally-graded transversely isotropic coating is considered. Elastic moduli of the coating vary with depth according to the arbitrary continuous or piecewise constant independent functions. The technique based on the integral transformations is used to reduce the problem to the integral equation. The punch roughness is modeled by the Fourier-Bessel series. Special approximation for the kernel transform is used to obtain the approximated analytical solution to the integral equation. The resulting solution is asymptotically exact for both small and large values of the relative thickness of the coating. A method of construction of the compliance functions is presented for the case of the simultaneous action of the arbitrary axisymmetric normal and tangential loadings.