This paper presents one general theory of large elastic deformations of a rubber sphere in simple compression, as the removal of restrictions of the constant Young modulus and small deformation in the prevailing Hertzian theory in contact of elastic bodies. It derives a set of five equations associated with approach, radii of contact surface without and with lateral extension of free surface, the lateral extensive displacement on the contact surface and the position of the contact surface in a very large range of applied forces, on the basis of the Hertz theory (half-space elastic body model) with an extensive term, in consideration of the rubber-elastic nonlinear elasticity, the lateral extension and the symmetry of the deformed shape of the rubber sphere. In Part 2 it is shown that results calculated by the set of the equations fit experimental data for a rubber sphere.

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.

This paper presents experimental results of simple compression of a soft rubber sphere in a very large range of forces attaining at 5000 N, presenting calculational results by a set of five equations presented in Part 1. The calculational values of approach, the radius of contact surface, and lateral expansion agree well with the experimental data in the large range of deformations. It is thus verified experimentally that the set of the equations (12), (13), (31), (40), and (43) in Part 1 is approximately valid in large deformations for rubber sphere. Program using a personal computer in calculating five quantities from the five nonlinear equations associated with the five quantities is noted.

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