Rolling friction of a viscous sphere on a hard plane

A first-principle continuum-mechanics expression for the rolling friction coefficient is obtained for the rolling motion of a viscoelastic sphere on a hard plane. It relates the friction coefficient to the viscous and elastic constants of the sphere material. The relation obtained refers to the case when the deformation of the sphere ξ is small, the velocity of the sphere V is much less than the speed of sound in the material and when the characteristic time $\xi/V$ is much larger than the dissipative relaxation times of the viscoelastic material. To our knowledge this is the first “first-principle” expression of the rolling friction coefficient which does not contain empirical parameters.