Approximated analytical solution of contact problem on indentation of elastic half-space with coating reinforced with inhomogeneous interlayer

Axisymmetric contact problem on indentation of linearly elastic half-space with coating reinforced with inhomogeneous in depth interlayer is considered. Elastic moduli of the interlayer vary with depth according to arbitrary continuously differentiable independent functions. Construction of the compliance functions is reduced to the solution of Cauchy problems for a system of ordinary differential equations with variable coefficients. Contact problem is reduced to the solution of an integral equation which is solved using the bilateral asymptotic method. Approximated analytical expressions for contact stresses and indentation force are provided. Stresses and displacements inside the half-space and coating are obtained in the form of quadratures.