Contact Mechanics · Contact stresses
Subsurface contact stresses: where damage starts below the surface
Under a ball or roller pressed on a flat, the largest shear stress sits below the surface, so cracks can start out of sight.
Where is the stress highest under a ball or roller?
Colour is the stress inside the flat; the ring marks the peak.
Try it: switch to Roller: the ring drops from 0.48a to 0.79b.
What does sliding friction do to the stress below the surface?
Now the ball slides to the right and friction drags the surface.
Try it: press Play: the ring climbs, then jumps to the surface.
Where should a coating interface sit?
The dashed line is where a hard coating meets the metal below.
Try it: set the thickness to 10 µm: the line turns blue.
What to take away
The peak is below the surface
Ball: 0.31 p0 at 0.48a deep. Roller: 0.30 p0 at 0.79b (no friction, ν = 0.3).
Friction pulls it up
Past μ of about 0.25 to 0.3 the peak is at the surface, behind the contact.
Keep interfaces off the peak
An interface at the max shear depth carries the most shear.
More detail: the equations, the numbers and the limits
Contact size and pressure (Hertz)
With 1/E* = (1 − ν12)/E1 + (1 − ν22)/E2: for a ball of radius R on a flat under load W, a = (3WR/4E*)1/3 and p0 = 3W/2πa2. For a long roller with load W′ per unit length, the half-width is b = (4W′R/πE*)1/2 and p0 = 2W′/πb. The pressure is elliptical: p = p0(1 − r2/a2)1/2.
The stress field under the ball
For the normal pressure the page uses the closed form of Huber (1904) as written by Johnson (1985, chapter 3). For sliding, the friction traction is q = μp along the sliding direction. The page adds up Cerruti's point-force solution over the contact with a polar quadrature centred on each point. This is the same field that Hamilton (1983) wrote in closed form. Checks: the Huber field agrees with the same quadrature of Boussinesq's solution to 4 decimals; the surface stress under the contact matches σx = −π(4 + ν)μp0x/8a (Hamilton and Goodman 1966); and the quadrature matches an adaptive double integral (SciPy) to 4 decimals.
The stress field under the roller
McEwen's (1949) closed form, in plane strain (σy = ν(σx + σz)), checked against a numerical sum of Flamant line loads to 4 decimals.
What the page computes (ν = 0.3)
No friction: ball max shear 0.310 p0 at 0.481a; von Mises 0.620 p0 at 0.481a (so first yield at p0 = 1.61 Y, Y the yield stress). Roller: max shear 0.300 p0 at 0.786b; von Mises 0.558 p0 at 0.704b. With friction, the largest value moves to the surface near the trailing edge at μ = 0.29 (ball, max shear), 0.30 (ball, von Mises), 0.25 (roller, max shear) and 0.26 (roller, von Mises). Johnson quotes about 0.3 for the ball and about 0.25 for the roller. For the ball the surface value is largest right at the trailing edge, where the page uses the closed-form surface stresses. The trailing-edge tension of the ball is (1 − 2ν)p0/3 + π(4 + ν)μp0/8.
Limits
Both bodies are elastic, smooth and homogeneous, and the contact is small compared with the parts. The map is the plane through the centre along the sliding direction. The sliding field assumes full sliding (q = μp everywhere); a rolling contact with small traction has stick and slip zones instead (see rolling contact traction). Roughness adds its own stress peaks just under each asperity. A coating with a different stiffness changes the field, and the page does not model that: it only shows where the interface sits in the field of the plain flat.
Elastic constants: bearing steel E = 208 GPa, ν = 0.30 (Harris and Kotzalas); alumina 370 GPa, 0.22; Ti-6Al-4V 114 GPa, 0.34 (Boyer et al.); Al 6061-T6 68.9 GPa, 0.33 (ASM Handbook). The stress map uses the ν of the flat.
On this site: Real contact area · Rolling traction · Fatigue · Scratch testing · Wear mechanisms · Fretting
Questions people ask
Why is the maximum shear stress below the surface in Hertz contact?
At the surface under the centre, all three stresses are compressive and nearly equal, so their difference (the shear) is small. Deeper down the vertical stress fades slowly but the sideways stresses fade fast. Their difference, and so the shear, peaks at about half the contact radius deep.
How deep is the maximum shear stress under a ball or a roller?
For a frictionless ball on a flat, 0.48a deep with a value of 0.31 p0 (ν = 0.3). For a long roller, 0.79b deep with 0.30 p0. The depth grows with the contact size, so a heavier load or a larger ball moves it deeper.
What is the difference between pitting and spalling?
Both mean material breaking out of a surface after many contact cycles (rolling contact fatigue). Usage varies between fields. In the bearing literature, a spall is usually a deeper flake from a crack that started below the surface, and pits or micropits are shallow and start from surface cracks. Gear engineers often say pitting for both, and Tallian's failure atlas calls both spalls but sorts them by where the crack started.
At what friction coefficient does yield start at the surface?
In the elastic model on this page, the largest von Mises stress moves to the surface at μ of about 0.30 for a ball and 0.26 for a roller (ν = 0.3). With the Tresca (max shear) measure the numbers are 0.29 and 0.25. Johnson quotes about 0.3 for the ball and 0.25 for the roller.
Why do brittle materials crack in a ring around a ball contact?
Just outside the contact edge the surface is stretched outward, with a radial tension of (1 − 2ν)p0/3. Glass and ceramics are weak in tension, so a ring crack starts there and grows down into a cone (the Hertzian cone crack). Sliding raises the tension at the trailing edge.
How thick should a hard coating be for a given contact?
Work out the max shear depth for the real load and radius first. An interface at that depth sees the highest shear and can peel. A coating much thinner than the depth leaves the substrate to carry the peak, so the substrate must be hard enough; a much thicker one keeps the peak inside the coating.
References
Show the 9 references
- K. L. Johnson, Contact Mechanics, Cambridge University Press (1985): chapter 3 (the Boussinesq and Cerruti point-force solutions), chapter 4 (Hertz contact and the stresses below it) and chapter 7 (sliding contact).
- M. T. Huber, Zur Theorie der Berührung fester elastischer Körper, Annalen der Physik 319, 153 to 163 (1904). doi:10.1002/andp.19043190611
- E. McEwen, Stresses in elastic cylinders in contact along a generatrix (including the effect of tangential friction), Philosophical Magazine 40, 454 to 459 (1949). doi:10.1080/14786444908521733
- G. M. Hamilton and L. E. Goodman, The stress field created by a circular sliding contact, Journal of Applied Mechanics 33, 371 to 376 (1966). doi:10.1115/1.3625051
- G. M. Hamilton, Explicit equations for the stresses beneath a sliding spherical contact, Proceedings of the Institution of Mechanical Engineers, Part C 197, 53 to 59 (1983). doi:10.1243/PIME_PROC_1983_197_076_02
- T. E. Tallian, Failure Atlas for Hertz Contact Machine Elements, 2nd ed., ASME Press (1999).
- K. Holmberg and A. Matthews, Coatings Tribology: Properties, Mechanisms, Techniques and Applications in Surface Engineering, 2nd ed., Elsevier (2009): stresses in coated contacts.
- T. A. Harris and M. N. Kotzalas, Essential Concepts of Bearing Technology, 5th ed., CRC Press (2007): bearing steel E = 208 GPa, ν = 0.30; subsurface stresses and fatigue.
- Other elastic constants: R. Boyer, G. Welsch and E. W. Collings, Materials Properties Handbook: Titanium Alloys, ASM International (1994) for Ti-6Al-4V; ASM Handbook vol. 2 (1990) for Al 6061-T6; alumina (99.5%) E = 370 GPa, ν = 0.22 from manufacturers' data sheets.
BibTeX
@misc{tripathy2026subsurfacestress,
author = {Tripathy, Manisha},
title = {Subsurface Contact Stress Lab},
year = {2026},
howpublished = {\url{https://untetheredatom.com/tribology/subsurface-contact-stresses}},
note = {Interactive web tool}
}