## Abstract

Previous work on the mechanism of rolling friction has shown that it is mainly due to elastic hysteresis losses in the rolling elements. Under conditions of uniform tension or torsion it is generally assumed that the energy dissipated by hysteresis is a constant fraction (the hysteresis loss factor) of the elastic energy introduced during the cycle. This elastic input energy has been calculated for a hard cylinder or sphere rolling on rubber, and with the above assumption expressions have been obtained for the rolling friction. These expressions predict correctly the dependence of rolling friction on load, ball or cylinder diameter and elastic constants of the rubber. However, the absolute values of the frictional force are too small. The experimental values are two to three times the theoretical estimates. This suggests that it is invalid to use the hysteresis loss factor, as measured in a simple deformation cycle, for more complex cycles such as those involved in rolling. For a thin-walled rubber tube to which tensional and torsional stresses could be applied, the hysteresis energy losses for complex stress cycles were measured. In such a system there are two relevant shear stresses which contribute to hysteresis losses; if these are plotted graphically, a deformation cycle can be expressed by a stress diagram which, in general, will form a closed loop. A hypothesis is put forward which suggests that the losses in the cycle are proportional to the square of the length of this loop. An investigation of different types of stress cycle show that this hypothesis may be used fairly reliably to estimate energy losses in complex stress cycles. In particular it may be shown that if the rubber is deformed in such a way that its elastic strain energy is constant throughout the cycle, marked hysteresis losses occur. The earlier concept would predict no energy loss. This result is of general importance and explains the earlier discrepancies between theoretical and experimental values for the friction of rolling cylinders and spheres. Although this hypothesis has limitations (these and alternative hypotheses are discussed) it is applicable to the deformation of material by a hard roller and an analysis is given for a long cylinder rolling on rubber. It is shown that the losses are over three times as great as those deduced from the earlier concept and for a spherical roller the loss is about twice as great as earlier estimates. This is in good agreement with experiment. The analysis is extended to the case of rolling over short distances and it is shown that the resistance increases linearly from zero as a cylinder, starting from rest, commences to roll. An experimental study indicates that this is approximately true. Only when each element has passed under the roller and out of the contact region does the hysteresis loss reach its maximum steady-state value. For minute displacements of a roller, there is evidence that adhesion may not be entirely destroyed, and though, in general, rolling friction is dominated by hysteresis losses, it is suggested that under these limiting conditions the effect of adhesion may not be negligible.