Quasi-conformal versus Hertz superposition: a comparison for two-dimensional contact

Resumo

Hertz contact theory results in a relation between the normal contact force and the rigid indentation of the contact surfaces, the force is proportional to the indentation to the power of 3/2, and a semi-ellipsoidal traction distribution. It is a valid theory for non-conformal contact between smooth surfaces (curves in 2D) of elastic bodies. This force model can be derived by considering the Taylor polynomials of degree 2 at the contact point as approximations for the surfaces/curves. In the 2D case, it depends only on the curvature radii of the curves and is undefined when they are equal, the equality of radii violates the non-conformality.
The Hertzian force is usually applied to the contact of solid particles as a model compensating for the small local deformations that occur in the contact region. We consider the 2D contact of a rigid convex particle with a rigid concavity enforcing the contact constraint with a barrier method, an approach that prevents overlapping of the bodies, resulting in a force that acts on the contact point. The geometries and the motion are specially designed to show the coalescence of two contact points into one. It illustrates two possible issues with convex-concave contact, the non-uniqueness of contact points and the conformality. We compare the results with a deformable finite element model for equivalent bodies and motion showing that there is a transition between a Hertzian and a non-Hertzian behavior. We conclude that a criterion for the conformality of curves must include the ever-present gap when a barrier method is used to enforce the constraint and the Hertzian force is not suitable for the coalescence of contact points.