Hertzian contact problem: Numerical reduction and volumetric modification

A technique for the analytical formulation and numerical implementation of an elastic contact model for rigid bodies in the framework of the Hertzian contact problem is described. The normal elastic force and the semiaxes of the contact area are computed so that the problem is sequentially reduced to a scalar transcendental equation depending on complete elliptic integrals of the first and second kinds. Based on the classical solution to the Hertzian contact problem, an invariant volumetric force function is proposed that depends on the geometric characteristics of interpenetration of two undeformed bodies. The normal forces computed using the force function agree with results obtained previously for non-Hertzian contact of elastic bodies. As an example, a ball bearing is used to compare the contact dynamics of elastic bodies simulated in the classical Hertzian model and its volumetric modification.