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.

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