Hertzian Contact Radius⚠ unverified
Mechanical / Lubrication · Compute the Hertzian contact radius for two spheres
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 | a | m | Radius of the circular contact area, in metres (m) |
The science & history
Understanding the Parameters
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Normal load $F$ — the contact radius grows as $F^{1/3}$: tripling the load only enlarges the contact circle by ~44 %. Because pressure is load over area ($\propto F/a^2 \propto F^{1/3}$), contact pressure also rises slowly with load — why rolling-element contacts survive surprisingly high loads.
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Effective radius $R$ — the reduced radius combining both bodies' curvatures, $1/R = 1/R_1 + 1/R_2$ (a convex-on-convex contact has a smaller, sharper effective radius; convex-in-concave, like a ball in a race, has a larger $R$ and a bigger, gentler contact). Larger $R$ (conforming surfaces) spreads the load over a bigger area.
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Effective modulus $E$ — the reduced elastic modulus $E^*$ from both materials, $1/E^* = (1-\nu_1^2)/E_1 + (1-\nu_2^2)/E_2$. Stiffer materials (higher $E^*$) deform less, giving a smaller contact circle and higher pressure.
Derivation (Approaching a Proof)
Heinrich Hertz (1882) solved the elastic contact of two curved bodies by finding the pressure distribution that makes the deformed surfaces exactly conform over the contact patch. For two spheres, the elastic deflection and the geometric overlap are matched by assuming a semi-ellipsoidal (Hertzian) pressure distribution
$$p(r) = p_{\max}\sqrt{1 - (r/a)^2}.$$
Integrating this pressure gives the total load $F = \tfrac{2}{3}p_{\max}\pi a^2$, and requiring the elastic deformation (from the theory of elasticity) to match the sphere geometry over radius $a$ yields the contact radius
$$a = \left(\frac{3 F R}{4 E^*}\right)^{1/3} = 0.908\,\left(\frac{F R}{E^*}\right)^{1/3},$$
since $(3/4)^{1/3} \approx 0.908$. The cube root comes from balancing the elastic restoring force (which grows faster than linearly with penetration) against the applied load. With $a$ known, the peak pressure is $p_{\max} = 3F/(2\pi a^2)$ (see Hertzian Contact Pressure).
Dimensional check. $\left(\dfrac{F R}{E}\right)^{1/3} = \left(\dfrac{\text{N}\cdot\text{m}}{\text{Pa}}\right)^{1/3} = (\text{m}^3)^{1/3} = \text{m}$. ✓
History and Development
Heinrich Hertz derived elastic contact theory in 1882 (as a young researcher, alongside his more famous electromagnetism work). It is the foundation of rolling-element bearing, gear, cam, and wheel–rail contact analysis — everywhere curved surfaces touch under load. The contact radius feeds the contact pressure and, via the Elastohydrodynamic Film theory, the lubricant film that separates such contacts.
Related Concepts: Hertzian Contact Pressure, Contact Stress Herzian, Hertzian Max Pressure, Elastohydrodynamic Film, Gear Contact Stress, Contact Stress Basics
Notes: Sphere/point contact ($0.908$ constant). Use reduced radius $1/R = \sum 1/R_i$ and reduced modulus $1/E^* = \sum (1-\nu_i^2)/E_i$. Line (cylinder) contact uses a half-width, not this circular radius.