Axisymmetric contact of two different power-law graded elastic bodies and an integral equation with two Weber-Schafheitlin kernels

This article analyzes the axisymmetric contact problem of two elastic inhomogeneous bodies whose Young moduli are power functions of depth and the exponents are not necessarily the same. It is shown that the model problem is equivalent to an integral equation with respect to the pressure distribution whose kernel is a linear combination of two Weber-Schafheitlin integrals. The pressure is expanded in terms of the Jacobi polynomials, and the expansion coefficients are recovered by solving an infinite system of linear algebraic equations of the second kind. The coefficients of the system are represented through Mellin convolution integrals and computed explicitly. The Hertzian and Johnson-Kendall-Robertson adhesive models are employed to determine the contact radius, the displacement of distant points of the contacting bodies, the pressure distribution and the elastic normal displacement of surface points outside the contact circular zone. The effects of the exponents of the Young moduli and the surface energy density on the pressure distribution and the displacements are numerically analyzed.