  We report a new theory of dissipative forces acting between colliding
viscoelastic bodies. The impact velocity is assumed not to be large, to avoid
plastic deformations and fragmentation at the impact. The bodies may be of an
arbitrary convex shape and of different materials. We develop a mathematically
rigorous perturbation scheme to solve the continuum mechanics equation that
deals with both displacement and displacement rate fields and accounts for the
dissipation in the bulk of the material. The perturbative solution of this
equation allows to go beyond the previously used quasi-static approximation and
obtain the dissipative force. This force does not suffer from the physical
inconsistencies of the latter approximation and depends on particle deformation
and deformation rate.
