Frictional Hertzian Contact: From Variational Inequalities to Wear Mechanisms in Bearings – Lukáš Kapera (University of Ostrava)

*****Abstract*****

Frictional contact problems are widespread in engineering, particularly
in the design and lifespan prediction of rolling-element bearings. While
these problems are grounded in well-known physical principles, their
mathematical formulation and numerical solution present significant
challenges.

This lecture bridges the gap between rigorous functional analysis and
practical wear prediction. The first part of the talk focuses on the
mathematical framework of contact mechanics. We will introduce the
Signorini problem and transition through the complexities of Tresca and
Coulomb friction models. Special attention will be given to solvability
issues, the shift from Variational Inequalities (VI) to
Quasi-Variational Inequalities (QVI), and the mathematical consequences
of friction regularization.

The second part demonstrates these concepts on a classic Hertzian
contact model: a rigid cylinder on an elastic half-plane. We will
compare two mathematically similar but physically distinct loading
scenarios. The static case (purely normal force) will be shown to model
fretting wear, whereas the semi-static case (introducing a tangential
force component) drives pitting wear. By analyzing the shift in the
depth of the maximum subsurface shear stress, we will illustrate how
subtle changes in the contact formulation fundamentally alter the
predicted mechanical degradation.

*****Lukáš Kapera (Speaker profile)*****

Was born in Opava, Czech Republic, in 1981. He received the Ing. degree
(M.Sc. equivalent) in applied mathematics from VŠB – Technical
University of Ostrava, Czech Republic, where he is currently a full-time
Ph.D. candidate and working as a researcher. His research interests
include optimization methods, mathematical modeling, and contact
problems in linear elasticity. He is primarily focused on solving
real-world industrial problems, utilizing MATLAB for the implementation
of numerical algorithms.