Green's function for symmetric loading of an elastic sphere with application to contact problems

TL;DR

This paper derives a compact Green's function for symmetric loading on elastic spheres, enabling precise analysis of contact problems and comparing different load distributions with classical Hertz theory.

Contribution

It introduces a closed-form Green's function capturing singularities and extends solutions to distributed loads, improving contact problem modeling.

Findings

Green's function expression captures singularity in closed form.

Hertz contact theory is valid up to about 10 degrees contact angle.

Displacements are smaller and contact surfaces are non-flat for larger angles.

Abstract

A compact form for the static Green's function for symmetric loading of an elastic sphere is derived. The expression captures the singularity in closed form using standard functions and quickly convergent series. Applications to problems involving contact between elastic spheres are discussed. An exact solution for a point load on a sphere is presented and subsequently generalized for distributed loads. Examples for constant and Hertzian-type distributed loads are provided, where the latter is also compared to the Hertz contact theory for identical spheres. The results show that the form of the loading assumed in Hertz contact theory is valid for contact angles up to about 10 degrees. For larger angles, the actual displacement is smaller and the contact surface is no longer flat.