Extensive Theory of Force-Approach Relations of Elastic Spheres in Compression and in Impact

The prevailing Hertz theory in contact and in impact is based on the total compressive displacement of a semi-infinite elastic body. This paper considers displacements of finite elastic medium in each of elastic spheres and presents analytically extensive force-approach relations of the Hertz theory for two elastic spheres in statical compression and in impact. In the statical conditions, expansive displacements of the mutual surf ace of contact due to compressive displacements by the reactions, which act on the opposite surfaces in a distance equal to each diameter, are considered analytically in two approximate cases. The force-approach relations obtained here are much closer than the Hertz’s one in a wide range of deformations to one experimental result carried out for one rubber sphere. In impact, it is considered that relative position of each center of mass of the impacting spheres accompanying asymmetrical deformations is shifted from the initial position. The force-approach relation has another extensive term different from the Hertz’s relation and from the above relations in the statical conditions. In the case of very small deformations for hard spheres, the extensive terms can be neglected and the Hertz theory is valid in compression and in impact. The present force-approach relations can be applicable to the cases of large deformations in compression and in impact.