Contact Stress Herzian⚠ unverified
Mechanical / Lubrication · Compute the Hertzian contact stress
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| F | F | N | 1.0 | Normal contact load |
| R | R | m | 1.0 | Effective (equivalent) radius of curvature |
| E | E | Pa | 1.0 | Effective elastic modulus |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | σ | Pa | Maximum Hertzian contact stress, in pascals (Pa) |
The science & history
Understanding the Parameters
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Load $F$, radius $R$, modulus $E^*$ — combine into the universal Hertzian group $(FE^{*2}/R^2)^{1/3}$, with the same exponents and physical meaning as in Hertzian Contact Pressure: stress rises slowly with load ($F^{1/3}$), steeply with sharper curvature ($R^{-2/3}$), and with stiffer materials ($E^{*2/3}$).
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The coefficient (0.418 vs 0.578) — encodes the contact geometry. Two spheres (point contact, circular patch) give 0.578; two cylinders (line contact, rectangular patch) give the classic $p_{\max} = 0.418\sqrt{F'E^*/R}$ with $F'$ the load per unit length. Because bearings and gears involve both (balls = point, rollers/gear teeth ≈ line), the right coefficient and functional form matter for accuracy.
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Why it governs life — the peak contact stress sets the amplitude of the sub-surface alternating shear that drives pitting/spalling (rolling-contact fatigue). Keeping it below the material's surface-fatigue strength is the surface-durability design criterion.
Derivation (Approaching a Proof)
For point (sphere) contact the derivation is exactly that of Hertzian Contact Pressure: the semi-ellipsoidal pressure integrated to the load gives $p_{\max} = 3F/(2\pi a^2)$, and substituting $a = 0.908(FR/E^*)^{1/3}$ yields $\sigma = 0.578\,(FE^{*2}/R^2)^{1/3}$.
For line (cylinder) contact, the contact patch is a long strip of half-width $b$ and length $L$. Hertz theory gives a semi-elliptical pressure across the strip, with
$$b = \sqrt{\frac{4 F' R}{\pi E^*}}, \qquad p_{\max} = \frac{2 F'}{\pi b} = \sqrt{\frac{F' E^*}{\pi R}} \approx 0.418\,\sqrt{\frac{F' E^*}{R}},$$
where $F' = F/L$ is the load per unit length. The 0.418 coefficient comes from this line-contact result ($\sqrt{1/\pi}\approx 0.564$; the standard tabulated form with the material constants folded in gives 0.418). The registry writes the constant 0.418 but with the cube-root point-contact grouping — a mixing of the two cases, so use it as an approximate Hertzian magnitude and select the correct form for the actual geometry. Either way, the peak stress scales with the Hertzian material/geometry group and governs surface fatigue.
Dimensional check. $\left(\dfrac{F E^2}{R^2}\right)^{1/3}$: with $\text{N}=\text{Pa}\cdot\text{m}^2$, $\left(\dfrac{\text{Pa}\cdot\text{m}^2\cdot\text{Pa}^2}{\text{m}^2}\right)^{1/3} = (\text{Pa}^3)^{1/3} = \text{Pa}$. ✓
History and Development
Hertz's 1882 contact theory provides both the sphere (0.578) and cylinder (0.418) coefficients, tabulated in Shigley, Roark, and tribology references. Line-contact stress governs cylindrical/tapered rollers and gear teeth; point-contact governs balls. The distinction is important in bearing and gear rating (AGMA/ISO), where the correct geometry factor sets the surface-fatigue (pitting) life.
Related Concepts: Hertzian Contact Pressure, Hertzian Contact Radius, Hertzian Max Pressure, Gear Contact Stress, Contact Stress Basics, Elastohydrodynamic Film
Notes: Coefficient is geometry-dependent — 0.418 (line/cylinder, uses load/length and $\sqrt{\cdot}$) vs 0.578 (point/sphere, cube-root); confirm the contact type (see note). Governs pitting/rolling-contact fatigue.