Highly Accurate Formulas for Rapid Calculation of the Key Geometrical Parameters of Elliptic Hertzian Contacts

The ellipticity ratio (κ), the complete elliptic intergrals of the first and second kinds (F and E, respectively), and the effective radius ratio (γ) for an elliptic Hertzian contact are related by a transcendental equation whose solution (or uncoupling) one would rather prefer to avoid. In this paper the currently available empirical formulas [1–3] relating each of κ, F, and E, separately, to γ are reappraised in terms of their quality of fit, then the work leading to new, more accurate, formulas is described. Two sets of such formulas are proposed which reduce the relative errors of fit, compared to the best currently available formulas, by up to well over two orders of magnitude for the range 1 ≤ γ ≤ 100. The paper ends with a brief discussion of the way the new equations presented here may also be utilized in the case of γ < 1, and of the implications of the new formulas with respect to simplified calculations of contact stresses and deformations, such as those proposed by Hamrock and Brewe [3].