An Approximate Solution for the Contact Problem of Profiles Slightly Deviating from Axial Symmetry

An approximate solution for a contact problem of profiles which are not axially symmetrical but deviate only slightly from the axial symmetry is found in a closed explicit analytical form. The solution is based on Betti's reciprocity theorem, first applied to contact problems by R.T. Shield in 1967, in connection with the extremal principle for the contact force found by J.R. Barber in 1974 and Fabrikant's approximation (1986) for the pressure distribution under a flat punch with arbitrary cross-section. The general solution is validated by comparison with the Hertzian solution for the contact of ellipsoids with small eccentricity and with numerical solutions for conical shapes with polygonal cross-sections. The solution provides the dependencies of the force on the indentation, the size and the shape of the contact area as well as the pressure distribution in the contact area. The approach is illustrated by linear (conical) and quadratic profiles with arbitrary cross-sections as well as for separable shapes, which can be represented as a product of a power-law function of the radius with an arbitrary exponent and an arbitrary function of the polar angle. A generalization of the Method of Dimensionality Reduction to non-axisymmetric profiles is formulated.

Automated summary

This study presents an approximate solution for contact problems of profiles that deviate slightly from axial symmetry. The solution is derived through Betti's reciprocity theorem and validated through comparisons with other analytical and numerical solutions. The solution provides insights into the force-indentation relationship, the size and shape of the contact area, and the pressure distribution. Additionally, the study generalizes the Method of Dimensionality Reduction to non-axisymmetric profiles.