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 whose first contact in this plane is either a point (punctual contact) or a line (lineic 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 the two parts, a contact surface is created through which the forces are transmitted from one part to the other. It is an ellipse for the point contact family and an elongated rectangle for the line contact family. These localized forces generate a specific stress distribution in the contact region which can cause permanent deformations or damage; it is important to be able to predict them.
The application of Hertz’s theory at this contact makes it possible to predict the shape and dimensions of the contact surface, the distribution of stresses on the surface and in the sub-layer in the vicinity of the contact; it is thus possible to determine in each of the solids the most stressed zone and to choose the material or the suitable surface treatments or coatings.
To carry out a study, it is necessary to know the following information:
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the geometries of the two parts in the vicinity of the contact (curvatures);
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their relative positioning;
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the contact force normal to the common tangent plane;
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the properties of elasticity of the two solids (Young’s modulus and Poisson’s ratio) of the materials in contact.
For dimensioning, the elasticity, rupture or fatigue limits may be necessary.
In this article, we will first address the hypotheses of Hertzian contacts and the definition of the geometric and mechanical quantities describing these contacts. One will then present the solution of the general case of the punctual contact and its application to the particular contact sphere/plane, then the solution of the general case of the linear contact cylinder/cylinder with parallel axes. In a second part, the state of surface and in-depth stresses is described using analytical formulas. The principles of dimensioning/choice of materials are deduced from this. Lastly, one will qualitatively approach the effect of a tangential force added to the normal loading.
