On solution of a contact problem with linear law of roughness deformation (Q2760519)

An axisymmetric problem of indentation a circular stamp \(z=g(\rho)\) into elastic half-space with rough surface is considered. The stamp surface is described by equation \(g(\rho) = a^2 \sum_{k=1}^\infty \alpha_k (\rho/a)^{2k}\). Pressure satisfies to two-dimensional integral equation with the kernel \(r(\rho,\rho') = [\rho^2 + {\rho'}^2 - 2\rho\rho'\cos(\theta - \theta')]^{1/2}\) in the polar coordinates within the contact zone. In dependence of the roughness coefficient, that enters the equation, different methods of solution of the equation are used. In the result of solution, values of pressure, contact radius and stamp penetration are found in analytic form for large and small roughnesses.