On Prandtl's lifting equation arising in wear mechanics

Abstract

A sliding wear contact between a rigid punch and an elastic half plane in presence of a thin aggregate film composed of solid debris and a lubricant fluid is studied. The model is based on any wear criterion and constitutive law of the film suggested by micromechanics approximation. The mechanical system is governed by the evolution of the volume fraction of debris, considered as the internal state variable. The key step of iterative computations for solving the nonlinear system of equations is based on the solution of the fundamental linear integro-differential equation for the compressive normal stress (the W-equation). The uniqueness of the solution of the integro-differential equation is then proved. It is shown that there is a profound relationship between the latter equation and the Prandtl's lifting equation in aerodynamics: both equations can be solved numerically by Chebyshev's series, and experimentally by similar electrical setups. Mathematically, it is found that both equations are related to real and imaginary components of some complex potential respectively and to weakly adjoint integro-differential operators.