Hertzian Contact Pressure⚠ unverified

Understanding the Parameters

• Normal load $F$ — the peak pressure grows only as $F^{1/3}$ (because the contact area itself expands with load). Doubling the load raises the peak pressure by just 26 % — the reason concentrated rolling contacts tolerate enormous forces without immediate yielding.

• Effective radius $R$ — enters as $R^{-2/3}$: sharper (smaller-$R$) contacts concentrate load into a tiny area and spike the pressure. Conforming contacts (ball in a matched race, large $R$) spread the load and lower the peak — a key reason bearing races are ground to closely match the ball radius.

• Effective modulus $E^*$ — stiffer materials give a smaller contact and higher pressure ($p\propto E^{*2/3}$). Steel-on-steel contacts run at very high pressure; this is why bearing steels are so hard and clean.

• Relation to average pressure — the peak is 1.5× the average pressure over the contact circle ($p_{\text{avg}} = F/\pi a^2$), a direct consequence of the domed (ellipsoidal) pressure profile.

Derivation (Approaching a Proof)

Hertz's theory assumes a semi-ellipsoidal pressure distribution over the circular contact of radius $a$ (Hertzian Contact Radius):

$$p(r) = p_{max}\sqrt{1 - (r/a)^2}.$$

The total load is the integral of this pressure over the contact area:

$$F = \int_0^a p_{max}\sqrt{1 - (r/a)^2}\,(2\pi r)\,\mathrm{d}r = \tfrac{2}{3}\,p_{max}\,\pi a^2.$$

Solving for the peak pressure:

$$p_{max} = \frac{3F}{2\pi a^2}.$$

This shows the peak is $\tfrac32$ the average $F/(\pi a^2)$. Substituting the Hertzian contact radius $a = 0.908\,(FR/E^*)^{1/3}$ eliminates $a$ and gives the closed form in load, radius, and modulus:

$$p_{max} = 0.578\left(\frac{F E^{*2}}{R^2}\right)^{1/3}.$$

The $F^{1/3}$, $R^{-2/3}$, $E^{*2/3}$ exponents are the fingerprints of elastic point contact — all flowing from the cube-root growth of the contact radius with load. The maximum shear stress occurs below the surface (at ~0.48 a depth), which is where rolling-contact fatigue cracks initiate.

Dimensional check. $\left(\dfrac{F E^2}{R^2}\right)^{1/3} = \left(\dfrac{\text{N}\,\text{Pa}^2}{\text{m}^2}\right)^{1/3}$; with $\text{N} = \text{Pa}\cdot\text{m}^2$ this is $(\text{Pa}^3)^{1/3} = \text{Pa}$. ✓

History and Development

Heinrich Hertz (1882) derived the pressure distribution and peak; Lundberg and Palmgren (1947) connected the sub-surface Hertzian shear stresses to rolling-contact fatigue, founding modern bearing life theory (Bearing Rating Life L10). Hertzian contact pressure is the basis of gear pitting resistance (Gear Contact Stress), cam and wheel–rail stress, and the load side of elastohydrodynamic lubrication.

Related Concepts: Hertzian Contact Radius, Contact Stress Herzian, Hertzian Max Pressure, Gear Contact Stress, Elastohydrodynamic Film, Bearing Rating Life L10

Notes: Sphere/point contact; peak = 1.5 × average. Use reduced $R$ and $E^*$. Max shear is sub-surface (rolling-fatigue origin). Line (cylinder) contact has a different constant ($p_{max}\propto\sqrt{F'E^*/R}$).