Hertz Contact Theory. Point or Line Contacts

When two solids with curved boundaries come into contact, they define a common tangent plane called the contact plane. The family of Hertzian contacts corresponds to contacts where the first contact in this plane is either a point (point contact) or a line (line contact). Heinrich Rudolf Hertz proposed the first elements of a solution between 1881 and 1895.

Under the effect of a force normal to the tangent plane common to both parts, a contact surface is created through which forces are transmitted from one part to the other. This is an ellipse for point contacts and an elongated rectangle for line contacts. These localized forces generate a specific distribution of stresses in the contact region, which can lead to permanent deformation or damage, and it is important to be able to predict them.

The application of Hertz theory at this contact enables the shape and dimensions of the contact surface to be predicted, as well as the distribution of surface and sub-surface stresses in the vicinity of the contact. This makes it possible to determine the most stressed zone in each solid, and to select the appropriate material or surface treatments or coatings.

To carry out a study, you need to know the following information:

• the geometries of the two parts in the vicinity of the contact (curvatures) ;

• their relative positioning ;

• the contact force normal to the common tangent plane ;

• the elastic properties of the two solids (Young's modulus and Poisson's ratio) of the materials in contact.

For dimensioning, elasticity, fracture or fatigue limits may be required.

In this article, we'll start by discussing the hypotheses of Hertzian contacts and the definition of the geometric and mechanical quantities describing these contacts. The solution of the general case of point contact and its application to the particular sphere/plane contact will then be presented, followed by the solution of the general case of linear cylinder/cylinder contact with parallel axes. In the second part, the state of surface and depth stresses is described using analytical formulas. Principles of dimensioning/material selection are deduced. Finally, the effect of a tangential force added to the normal loading is discussed qualitatively.