A Theory for Plane Strain Tangential Contacts of Functionally Graded Elastic Solids with Application to Fretting

Due to their superior tribological properties compared to conventional materials, the use of functionally graded materials (FGMs) has long become indispensable in mechanical engineering. The wide variety of in-depth gradings means that solving contact problems requires specific, complex numerical analysis. In many cases, however, the spatial change in Young’s modulus can be approximated by a power law, which allows closed-form analytical solutions. In the present work, integral equations for solving tangentially loaded power-law graded elastic half-planes are derived by using the Mossakovskii–Jäger procedure. In this way, the application of highly complicated singular integrals arising from a superposition of fundamental solutions is avoided. A distinction is made between different mixed boundary conditions. The easy tractability of the novel equations is substantiated by solving the plane strain fretting contact of a rigid parabolic cylinder and a power-law graded (PLG) elastic half-space. The effect of the type of in-depth grading on the dissipated energy density and the total energy lost per cycle is investigated in detail. A comparison of the total dissipated energy per cycle shows that, for very thin stiff layers on soft substrates, the total dissipated energy exceeds that of a homogeneous material. The same trend is observed for thick layers of a functionally graded material whose Young’s modulus gradually increases with depth, matching that of the underlying substrate at the bonded interface. In addition, a closed-form analytical solution for the total dissipated energy per cycle for plane strain parabolic contact of elastically homogeneous material is presented for the first time.