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# Hertzian Contact Stress

Over a century ago Hertz established a theory of contact mechanics that is still used today by engineers working in tribology. To make his theory accessible for engineering applications, the program calculates a fast and accurate solution.
This Demonstration presents the Hertzian contact stress distribution and elastic deformations. It uses a color function to visualize the contact zone between a statically loaded pair of contact models, a sphere and a torus. In bearing engineering, the equalized terms are between balls and an inner or outer ring.

### DETAILS

Snapshot 1: an approximate point contact zone— two surfaces touch at a single point
Snapshot 2: a typical elliptical contact zone between a ball and an inner ring in a ball bearing assembly
Snapshot 3: a typical elliptical contact zone between a ball and an outer ring in a ball bearing assembly
All calculations are in standard international (SI) or metric system units.
is the geometry parameter of contact body one (radius of ball in red).
is the geometry parameter of contact body two (radius of raceway in green).
is the geometry parameter of contact body two (the distance from raceway bottom to shaft axes).
and are moduli of elasticity of the contact bodies one and two.
and are the Poisson's ratios of contact bodies one and two.
is the force between the two contact bodies (ball and raceway).
The top-left plot shows in 3D the geometry of the two contact bodies, especially relevantt for ball bearing engineering analysis.
The top-right plot shows the contact coefficient data based on numerical methods to solve elliptic integrals equations.
ρ is the curvature of the contact body surface, that is the reciprocal of the curvature radius of body, .
is the curvature sum, . The subscripts I and II are for the first and second contact bodies. The subscripts 1 and 2 are for the first and second planes of the contact bodies.
is the curvature difference, .
is the dimensionless semimajor axis of the contact ellipse.
is the dimensionless semiminor axis of the contact ellipse.
is the dimensionless parameter of the contact deformation.
The bottom-left plot shows the contact zone with a color function; this visualizes Hertzian contact zone and stress distribution.
The bottom-right table summarizes the calculation.
is the semimajor axis of the projected elliptical contact area.
is the semiminor axes of the projected elliptical contact area.
is the area of the projected elliptical contact area.
is the ratio of the semimajor to the semiminor axis (/) of the elliptical contact area.
is the maximum Hertztian contact stress.
is the average Hertztian contact stress.
is the deformation of the contact zone.
For more details, see Chapter 7 of F. Wu, Manipulate@Mathematica, Beijing: Tsinghua, 2010.
References:
[1] H. R. Hertz, "Über die Berührung fester elastischer Körper (On Contact Between Elastic Bodies)," Journal für die reine und angewandte Mathematik 92, 1881 pp. 156–171.
[2] T. A. Harris, "Contact Stress and Deformation", Rolling Bearing Analysis, 4th ed., New York: Wiley, 2001 pp. 183–204.

### PERMANENT CITATION

Contributed by: Frederick Wu
With corrections by: Mark Tausch and Jaebum Jung
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