  A rigorous theory for solving tangential contacts between three-dimensional
power-law graded elastic solids of arbitrary geometry is presented. For
multiple contacts such as those occurring between two nominally flat but rough
half-spaces, the well-known Ciavarella-J\"ager theorem is established
accompanied by a discussion of tangential coupling. Nevertheless, the focus of
the work is on axisymmetric single contacts under arbitrary unidirectional
tangential loading, for which closed-form analytical solutions are derived
based on the Mossakovskii-J\"ager procedure. In comparison to the results of
common approximate methods, the solutions include the non-axisymmetric
components of tangential displacements, which are indispensable for the
accurate determination of the relative slip components and thus the surface
density of frictional energy dissipation in the partial slip regime. Although a
simplified approach is used for the calculation of the dissipated energy
density, the results in the limiting case of homogeneous material are in
excellent agreement with those from a full numerical computation. As an
application example, the complete solutions for the tangential contact of
parabolically shaped power-law graded elastic solids in the partial slip regime
are derived and the influence of the material gradient as well as Poisson's
ratio on the surface density of dissipated energy is investigated.
