Shift and torsion contact problems for arbitrary axisymmetric normal stress distributions

In this paper, we analyze the contact interaction of axisymmetric particles subject to a subsequent application of a constant normal load and a tangential or rotational force. A rigorous solution to the frictional contact problem is given by the known Jäger theorem that presents a relationship between shear and normal stress distributions provided the latter is an exact solution to a normal contact problem. However, in the case of strong loading, when the normal displacement reaches a value of 5–10% of the spheres’ diameter, exact solutions for the normal problem are absent; some model concepts exist instead. For instance, the rod model describes strong normal loading of spheres as a sort of combination of the Hertz problem (weak loading of spheres) with a compression of a pair of confined cylinder of the same radius as the Hertz contact spot. Here we propose a method that is based on considerations similar to the Jäger theorem but is appropriate for any model (or empirical) normal stress distribution. The resulting integral representations describe both shearing and torsion of prestressed particles with axisymmetric profiles. Rolling of particles, as well as plasticity and adhesion of the particles’ material, are not considered. We also analyze the asymptotic behavior of the integral representations for weak and strong strains. The obtained general solutions allow us to use the method of memory diagrams in order to calculate the reaction of the system on arbitrarily varying tangential or rotational actions.