An extension of Hertz’s formula for the stiffness of conformal spherical contacts

Abstract

Hertz’s classical theory of contact requires the surfaces to be non-conformal. Despite of this, Hertzian formulas are often used also for conformal contacts as for instance for the evaluation of pivot stiffness in tilting pad journal bearings. In this paper, finite element simulations of conformal contacts between spherical elastic bodies are performed for different materials and geometry, in particular by varying the clearance. A first result is the introduction of a novel normalization which allows to calculate stiffness as a clearance-invariant function. Then, a novel model for stiffness is introduced. The model reduces back to Hertz’s theory in the non-conformal limit. The model requires fitting of three empirical parameters which depend on the boundary conditions and on the material properties. Analytical expressions for the parameters are provided for a subset of contact problems with a simple geometry and given material properties. More general formulas for the parameters will be developed in a future work.