Indentation of a hard transversely isotropic functionally graded coating by a conical indenter

An axisymmetric contact problem on indentation of a rigid conical punch into an elastic transversely isotropic half-space with a functionally graded transversely isotropic coating is considered. Elastic moduli of the coating vary in depth according to arbitrary continuous positive functions, independent of each other. Mathematical statement of the problem is made in terms of the linear theory of elasticity. Using integral transformation technique the problem is reduced to the solution of a dual integral equation. Kernel transform of the integral equation, which is calculated numerically from a two-point boundary value problem for a system of ordinary differential equations with variable coefficients, is approximated by a product of fractional quadratic functions. Using these approximations, an approximated solution of the problem is constructed in analytical form. The solution is asymptotically exact both for small and big values of the characteristic geometrical parameter of the problem (ratio of thickness of the coating to radius of the contact area). Approximated analytical expressions relating the displacement of the punch, indentation force acting on the punch and the size of the contact area are obtained. Correlation between the contact normal stresses arising on surface of the coated half-space and on surface of the homogeneous half-space without a coating is studied. Some relations are obtained analytically using asymptotic analysis and illustrated numerically. Results on numerical simulation of an indentation of a conical punch into a hard homogeneous or functionally graded (with linearly varying elastic moduli in depth) transversely isotropic coating are provided. The materials widely used in electronics are chosen for numerical examples. Qualitative differences in process of elastic deformation of bodies with homogeneous and functionally graded coatings are illustrated.