IDB-HTZ-036
Contact mechanics · point & line contact · pitting
Hertz contact stress
The contact stress between curved bodies — point and line contact formulas, the effective modulus and radius, the subsurface shear that drives pitting, and allowable limits.
Abstract
When curved bodies touch — a ball on a race, a gear tooth on its mate, a cam on a follower, a wheel on a rail — they contact over a very small area, so the local (Hertzian) stress is far higher than the nominal load suggests. The peak contact pressure and the shear it produces just below the surface are what limit rolling elements.
Section 1 explains Hertz contact. Section 2 is point contact (spheres/balls). Section 3 is line contact (cylinders/rollers). Section 4 is the effective modulus and radius. Section 5 covers subsurface shear and allowable stress. Section 6 is design notes and a checklist.
1.What Hertzian contact is
Two non-conforming curved surfaces pressed together deform elastically into a small contact patch — a circle (point contact) or a narrow strip (line contact). Across that patch the pressure is semi-elliptical, peaking at p₀ in the centre. Because the area is tiny, p₀ is very large, and the maximum shear stress sits a little below the surface — which is where rolling-contact fatigue (pitting/spalling) initiates.
2.Point contact (spheres / balls)
For two spheres (or a ball on a flat, R₂ → ∞):
- Contact radius
a = (3 F R* / 4 E*)^{1/3} - Peak pressure
p₀ = 3 F / (2 π a²)(= 1.5 × the average pressure)
p₀ rises only with the cube root of load, so doubling the load raises peak pressure ~26% — but it also rises fast as radius shrinks, which is why small balls and sharp crowns are highly stressed.
3.Line contact (cylinders / rollers)
For two parallel cylinders carrying load per unit length w = F / L:
- Contact half-width
b = √(4 w R* / (π E*)) - Peak pressure
p₀ = 2 w / (π b) = √(w E* / (π R*))
Line contact spreads load along the roller, so for the same total load it gives a lower p₀ than point contact — the reason roller bearings carry more than ball bearings of the same size. Watch edge loading: misalignment concentrates w at the roller ends, so rollers are crowned.
4.Effective modulus and radius
Both contact bodies share the deformation, combined as:
1/E* = (1 − ν₁²)/E₁ + (1 − ν₂²)/E₂1/R* = 1/R₁ ± 1/R₂use + for two convex surfaces, − for a convex body in a concave one (a ball in its race), which makes R* large and lowers stress.
So a ball seated in a conforming raceway groove is far less stressed than the same ball on a flat. Steel-on-steel: E* ≈ 113 GPa (E = 207 GPa, ν = 0.3 each).
5.Subsurface shear and allowable stress
- The maximum shear is ≈ 0.30 · p₀, located about 0.48 a (point) or 0.78 b (line) below the surface. Rolling fatigue cracks start there and work to the surface as pits.
- Static limit (brinelling): permanent denting begins around p₀ ≈ 4000–4200 MPa for hardened bearing steelthe basis of the static load rating C₀.
- Cyclic limit (pitting): far lower; case-hardened gear/bearing flanks are typically designed to ~1300–1700 MPa contact stress for long life (per ISO 6336 / bearing L10).
| Situation | Indicative p₀ limit |
|---|---|
| Hardened bearing steel, static (brinelling) | ~4000–4200 MPa |
| Case-hardened gear flank, long life | ~1300–1700 MPa |
| Through-hardened steel, moderate cycles | ~1000–1400 MPa |
Values are indicative — use ISO 6336 (gears) or bearing L10 ratings for real life.
6.Design notes and checklist
- Lower p₀ by increasing the effective radius (crowning, conforming grooves), spreading to line contact, or reducing load.
- Raise the limit with harder, cleaner steel (case-hardening, low inclusions) and a good EHL oil film (a full film dramatically extends pitting life).
- Mind subsurface cleanlinessinclusions at the max-shear depth are crack nuclei; bearing steels are vacuum-degassed for this reason.
- Checklist: find E* and R* → compute a/b and p₀ (point or line) → compare to the static limit (brinelling) and the cyclic limit (pitting) for the material and life → if high, add crowning/conformity, harden, or improve lubrication. Pairs with the Bearing, Gear and Fatigue references.