Frictional contact mechanics

This page is mainly concerned with the second scale: getting basic insight in the stresses and deformations in and near the contact patch, without paying too much attention to the detailed mechanisms by which they come about.

[3] They include Leonardo da Vinci, Guillaume Amontons, John Theophilus Desaguliers, Leonhard Euler, and Charles-Augustin de Coulomb.

Later, Nikolai Pavlovich Petrov, Osborne Reynolds and Richard Stribeck supplemented this understanding with theories of lubrication.

Further the fundamental solutions by Boussinesq and Cerruti are of primary importance for the investigation of frictional contact problems in the (linearly) elastic regime.

In 1958, Kenneth L. Johnson presented an approximate approach for the 3D frictional problem with Hertzian geometry, with either lateral or spin creepage.

Among others he found that spin creepage, which is symmetric about the center of the contact patch, leads to a net lateral force in rolling conditions.

In 1967, Joost Jacques Kalker published his milestone PhD thesis on the linear theory for rolling contact.

[6] This theory is exact for the situation of an infinite friction coefficient in which case the slip area vanishes, and is approximative for non-vanishing creepages.

With respect to road-tire interaction, an important contribution concerns the so-called magic tire formula by Hans Pacejka.

Over time, these grew into finite element approaches for contact problems with general material models and geometries, and into half-space based approaches for so-called smooth-edged contact problems for linearly elastic materials.

[5] Further Johnson collected in his book a tremendous amount of information on contact mechanics and related subjects.

[10] Finally the proceedings of a CISM course are of interest, which provide an introduction to more advanced aspects of rolling contact theory.

[11] Central in the analysis of frictional contact problems is the understanding that the stresses at the surface of each body are spatially varying.

Note that wear occurs only where power is dissipated, which requires stress and local relative displacement (slip) between the two surfaces.

The size and shape of the contact patch itself and of its adhesion and slip areas are generally unknown in advance.

For the direction perpendicular to the interface, the normal contact problem, adhesion effects are usually small (at larger spatial scales) and the following conditions are typically employed: Mathematically:

In the tangential directions the following conditions are often used: The precise form of the traction bound is the so-called local friction law.

When viewed from the high side, the tension drops exponentially, until it reaches the lower load at

Note further that the location of the slip area depends on the initial state and the loading process.

A static equilibrium is obtained in which elastic deformations occur as well as frictional shear stresses in the contact interface.

The sphere largely shifts back to its original position, except for frictional losses that arise due to local slip in the contact patch.

The stress distribution in the equilibrium state consists of two parts: In the central, sticking region

is precisely as large such that a static equilibrium is obtained with shear stresses at the traction bound in this so-called slip area.

This demonstrates that the state in the contact interface is not only dependent on the relative positions of the two bodies, but also on their motion history.

So in the final situation tangential stresses remain in the interface, in what looks like an identical configuration as the original one.

Consider a cylinder that is rolling over a plane (half-space) under steady conditions, with a time-independent longitudinal creepage

If the cylinder and plane consist of the same materials then the normal contact problem is unaffected by the shear stress.

) it is: The size of the adhesion area depends on the creepage, the wheel radius and the friction coefficient.

When considering contact problems at the intermediate spatial scales, the small-scale material inhomogeneities and surface roughness are ignored.

The half-space approach is an elegant solution strategy for so-called "smooth-edged" or "concentrated" contact problems.

In railway applications one wants to know the relation between creepage (velocity difference) and the friction force .
Illustration of an elastic rope wrapped around a fixed item such as a bollard. The contact area is divided into stick and slip zones, depending on the loads exerted on both ends and on the loading history.
Rolling contact between a cylinder and a plane. Particles moving through the contact area from right to left, being strained more and more until local sliding sets in.