Collision of viscoelastic bodies: Rigorous derivation of dissipative force

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.