HERTZIAN Pressure

Sprachauswahl/Language selection

Dieser Artikel ist auch auf Deutsch verfügbar HERTZ´sche Pressung

A service provided by
Polymer Service GmbH Merseburg
Tel.: +49 3461 30889-50 E-Mail: info@psm-merseburg.de Web: https://www.psm-merseburg.de
Our further education offers: https://www.psm-merseburg.de/weiterbildung
PSM on Wikipedia: https://de.wikipedia.org/wiki/Polymer Service Merseburg

Heterogeneity of the strain distribution

Physical principles

The first fundamental work on the contact problem is attributed to Joseph Boussinesq (french mathematician and physicist; 1842–1929), who calculated the stress field for a point-loaded elastic half-space. The most important result of his solution is that the stresses at infinity decay as 1/r or 1/r² (where r is the distance from the point of contact). In 1881, HEINRICH HERTZ solved the problem of contact between two elastic bodies with curved surfaces in his work ‘On the Contact of Rigid Elastic Bodies’, which remains authoritative to this day. This classic result still forms a foundation of contact mechanics today.

Describing the elastic contact between two bodies in contact requires, on the one hand, a precise understanding of the pressure distribution within the contact region and, on the other hand, the resulting surface displacements at every point both within and outside the contact area. The first assumption of the theory resulting from these considerations is that each of the bodies involved in the contact is regarded as an elastic half-space. This means that the dimensions of the body are large compared to the dimensions of the contact radius. Thus, the linearity conditions for the Hooke’s law region (see: HOOKE’s law) are satisfied and the boundary conditions of elastic contact theory are simplified:

• The contact stresses are concentrated in the contact area; outside the contact zone, the stresses decrease exponentially,
• The contact stresses themselves do not depend on the shape of the bodies outside the contact zone.

When elastic bodies with curved surfaces (cylinders and/or spheres) are pressed against one another or against a plane, in an idealised case they only touch one another or the plane along a line or at a single point. In reality, however, when the two bodies are brought close to one another or to a plane by an external force, an elliptical or rectangular contact surface is formed at the point of contact as a result of flattening. The size and shape of the contact surfaces, as well as the magnitude and distribution of the mechanical stresses (i.e. the surface pressures) beneath the contact surfaces, can be calculated. The highest stress, which occurs at the centre of the contact surface, is also known as the HERTZIAN pressure.

The magnitude of HERTZ's pressure depends on the force with which the two bodies are pressed against one another or against a surface, on their radii of curvature, and on the moduli of elasticity of the bodies involved (Fig. 1).

Fig. 1:

Schematic representation of HERTZ’s pressure on curved bodies

The conditions for calculating surface stress using HERTZ’s equations are:

• homogeneous and isotropic materials with linear-elastic properties,
• a flat contact surface that is small in comparison to the geometric dimensions of the bodies,
• no shear stresses occur in the contact surface due to the absence of friction,
• consideration of the two bodies as elastic half-spaces.

The HERTZ`s pressure p_max at the point of contact between curved surfaces is generally calculated using the following equation:

$p_{max}\,=\,{\frac {1}{\xi \cdot \eta }}\cdot {\frac {1}{\pi }}\cdot {\sqrt[{3}]{\frac {1,5F\cdot E^{2}\cdot \left(\sum k\right)^{2}}{\left(1-\nu ^{2}\right)^{2}}}}$

mit:

F		Kraft FN or FR between the bodies,
E/(1-ν2)		reduced modulus of elasticity
ξ, η		HERTZ coefficients for the contact of curved surfaces and
K		curvature

For the reduced module, the following applies:

${\frac {E}{1-\nu ^{2}}}=\left({\frac {1-\nu _{1}^{2}}{E_{1}}}+{\frac {1-\nu _{2}^{2}}{E_{2}}}\right)^{-1}$

with:

ν1, ν2		Poisson's ratio of bodies 1 and 2, and
E1, E2		the modulus of elasticity of bodies 1 and 2.

Point contact between spheres

In the simple case of contact between two identical spheres or between a sphere and a plane (assuming identical materials), the following equation applies:

$p_{max}\,=\,{\frac {1}{\pi }}\cdot {\sqrt[{3}]{\frac {nF\cdot E^{2}}{R^{2}\left(1-\nu ^{2}\right)^{2}}}}$

where n = 6 (sphere–sphere) or n = 1.5 (sphere–plane)

'where R is the radius of the sphere and ν is Poisson’s ratio.

Line contact (cylinder–cylinder and cylinder–plane)

In the simple case of contact between two identical cylinders or between a cylinder and a plane (assuming identical materials), the following equation applies:

$p_{max}\,=\,{\sqrt {\frac {nF\cdot E}{\pi Rl\cdot \left(1-\nu ^{2}\right)}}}$

where n = 1 (cylinder–cylinder) or n = 0.5 (cylinder–plane)

where R is the radius of the cylinder and l is the length of contact between the cylinders (or between the cylinder and the plane).

When the elastic limit is exceeded – that is, the maximum HERTZIAN pressure that the material can withstand – highly ductile materials undergo lateral displacement (plastic deformation) accompanied by sustained compression. In more brittle materials, material spalling and cracking may also occur, either in addition to or exclusively as a result of this.

See also

• Deformation
• Elastic modulus
• HOOKE´s law

References

• Hertz, H.: Über die Berührung fester elastischer Körper. Journal für die reine und angewandte Mathematik 92 (1881) pp. 56–171; https://home.uni-leipzig.de/pwm/web/download/Hertz1881.pdf
• Hertz, H.: Über die Berührung fester elastischer Körper und über die Härte. Verhandlungen des Vereins zur Beförderung des Gewerbefleißes. 61 (1982) pp. 449–463; https://www.deutsche-digitale-bibliothek.de/item/SM2OBGYYJKT3VKLJV4FC3VCUTVHTWS7B
• Geike, T.: Theoretische Grundlagen eines schnellen Berechnungsverfahrens für den Kontakt rauer Oberflächen. (Dissertation). Technische Universität Berlin, Berlin (2008) p. 15 (last access on March 22, 2026)
• Enders; S.: Untersuchungen der mechanischen Eigenschaften von spröden Schicht- und Kompaktsystemen durch Deformation kleiner Volumina. (Dissertation). Martin-Luther-Universität Halle-Wittenberg, Halle (2000) pp. 4–6
• Popov, V. L.: Kontaktmechanik und Reibung. Ein Lehr und Anwendungsbuch von der Nanotribologie bis zur numerischen Simulation. Springer, Berlin Heidelberg (2009) pp. 61–68 (ISBN 978-3-540-88836-9; e-Book ISBN 978-3-540-88837-6)
• Kunz, J., Jakober, D., Studer, M.: Kontaktmechanik: Längeneinfluss auf die Abspaltung paralleler Zylinder. Contact Mechanics: Influence of the Length of Parallel Cylinders on the Flattening. Konstruktion 6 (2017) pp. 63–67

Weblink

• Wikipedia – The Free Encyclopedia: Heinrich Hertz (https://en.wikipedia.org/wiki/Heinrich_Hertz)