Аннотация:A complete contact of a composite surface with a viscoelastic half-space has been considered. The 2-D problem was solved under assumption of a steady-state wear regime, and analytical relationships for a shape of the worn surface, for a contact pressure distribution and for an effective wear coefficient have been derived. Based on the analysis of the obtained wear contact problem solution, it was shown that the viscoelastic properties of the counterbody material in combination with various speed regimes have a significant effect on the worn composite surface shape. It was shown that the worn surface had the asymmetric wavy relief. A range of relative sliding velocities is determined, at which the most significant asymmetry of the formed relief is observed. Moreover, the solution obtained at relatively small sliding velocities asymptotically tends to the solution for the case of an elastic counterbody. The influence of the structure of the composite material and of the mechanical characteristics of the counterbody on the shape of the composite worn surface has been analyzed.