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 ξ/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. Rolling friction is one of the basic phenomenon men encounters in his everyday life since antediluvian times when the wheel was invented. The phenomenon of rolling friction has been interesting to scientist for a long time. Scientific publications on this subject range back to (at least) 1785 when Vince described systematic experiments to determine the nature of friction laws [1], and important scientists dealt with this problem among them O. Reynolds [2]. The rolling friction is of great importance in engineering and science. From the speed of landslides Huang and Wang [3] argued that the rolling friction plays an important rôle even in geological processes, for a theoretical consideration see [4]. For its major importance the phenomenon has been studied intensively by engineers and physicists (e.g. [5]), however, surprisingly few is known about its basic mechanisms. To our knowledge there is still no “first-principle” expression for the rolling friction coefficient available which relates this coefficient only to the material constants of the rolling body and does not contain empirical parameters. It has been shown that surface effects like adhesion [6], electrostatic interaction [7], and other surface properties [8] might influence the value of the rolling friction coefficient. Theoretically this problem was studied in Ref. [9] where the authors propose a model of a surface with asperities to mimic friction (see also [10]). In other studies [11, 12] it was argued that for (∗) email [email protected] (∗∗) email [email protected] Typeset using EURO-TEX 2 EUROPHYSICS LETTERS viscoelastic materials “the rolling friction is due very little to surface interactions: the major part is due to deformation losses within the bulk of the material” [11]. Based on this concept the rolling friction coefficient was calculated in [11] where the deformation in the bulk was assumed to be completely plastic; then an empirical coefficient was introduced to account for the retarded recover of the material. In the present letter we also consider the rolling friction as a phenomenon appearing due to viscous processes in the bulk. We assume that energy losses due to surface effects may be neglected, compared to the viscous dissipation in the bulk. Thus we attribute the effect of rolling friction to viscous dissipation in the material due to time-dependent deformation. We assume that the only rôle of surface forces is to keep the rolling body from sliding. We use a quasi-static approach [13] and obtain the rolling friction coefficient.