Nowadays, Finite Elements softwares allow the user to model efficiently the contact between two solids. It is though necessary to make numerous assumptions in order to compute a contact in a reasonable time: generally surfaces are smooth and the mesh is too coarse to accurately study transient phenomena. Moreover, friction on the surface is rarely taken into account since models are usually axi-symmetric. This paper presents a semi-analytical formulation that allows computing the contact between two elastic-plastic solids with rough surfaces. The numerical methods used, i.e. the conjugate gradient and the fast Fourier transform allow to refine the mesh drastically. The main advantage of this formulation over Finite Elements method is the much shorter computation times. This model takes into consideration the hardening of material as well as friction and frictional heating at the interface. Loading can be vertical or rolling/sliding. This formulation allows realizing cyclic loadings in order to model numerous engineering problems as running-in, fretting, asperities tugging, as well as electro-mechanical micro-contacts. A comparison is made between the vertical contact (static loading) and the rolling/sliding contact (transient loading) in the case of an adiabatic elastic sphere rotating and pressed against a rough and stationary elastic-plastic surface. The influence of the friction coefficient is underlined.
