Hertzian Contact Pressure, Sphere on a Flat

Heinrich Hertz was twenty-three, working on the optics of stacked glass lenses over the Christmas holiday of 1880, and wondering how the elastic flattening where two lenses touch would disturb the interference fringes. The paper he published in 1882 solved the general problem of two elastic bodies pressed together, and it became — entirely by accident, since Hertz went on to more famous work with electromagnetic waves — the foundation of every rolling bearing, gear tooth and cam follower ever designed.

For a sphere on a flat the result is compact. The contact is a circle of radius $a = \sqrt[3]{3FR/(4E^*)}$, the pressure over it is a hemisphere, and the peak at the centre is $p_0 = 3F/(2\pi a^2)$ — exactly $3/2$ times the mean, since a hemisphere has two thirds the volume of the cylinder that contains it. Combining the two gives the form on this page, $p_0 = \sqrt[3]{6FE^{*2}/(\pi^3 R^2)}$.

The cube root is the first thing to internalise. Pressure goes as $F^{1/3}$, because as the load grows the contact patch grows with it and shares the extra out. Double the load and the peak pressure rises only 26 %. That weak dependence is the reason a ball bearing is possible at all: contact pressures of a gigapascal or two are perfectly routine in hardened steel, and the load would have to change by a factor of eight to move them twofold. Run it the other way and it becomes a warning — the LOAD goes as the cube of the pressure, so a contact designed close to its pressure allowable has very little load margin left.

The second thing is that the failure is not at the surface. Directly under a Hertzian contact the material is in triaxial compression, which is a benign state; hydrostatic pressure alone does not cause yielding. What causes yielding is shear, and the maximum shear stress in a spherical contact is about $0.31 p_0$ located BELOW the surface, roughly half a contact radius down. That is where plastic flow starts, and it is where rolling-contact fatigue cracks nucleate. It is also why a spalled bearing race looks as though a flake was prised out from underneath: it was. Subsurface inclusions and non-metallic stringers sit exactly in the worst place, which is why bearing steel is vacuum-degassed and why cleanliness ratings appear on bearing datasheets at all.

The third thing is that this is an ELASTIC result, with a definite end. As the load rises, that subsurface shear stress reaches the material's yield in shear, and first yield occurs at roughly $p_0 \approx 1.6\sigma_y$. Push further and the plastic zone grows to the surface; by about $p_0 \approx 3\sigma_y$ the contact is fully plastic, and that condition — mean pressure equal to about three times the yield strength — is precisely the definition of indentation hardness. So $H \approx 3\sigma_y$ is not a coincidence but the same calculation seen from the other end, and it is why Archard's equation can use hardness as the pressure at a plastically-deforming asperity. Static brinelling in a bearing is this limit being exceeded, and the dents it leaves make a bearing noisy for the rest of its life.

Two definitional traps. $E^*$ is a pair property, defined by $1/E^* = (1-\nu_1^2)/E_1 + (1-\nu_2^2)/E_2$; for steel on steel it comes to about 115 GPa, and it is dominated by the more compliant body, which is the useful part — putting something soft against something hard drops the contact pressure for the same load. Watch which convention a source uses, because $E' = 2E^*$ appears widely and changes every coefficient by a factor of two. And $R$ is the EFFECTIVE radius: $1/R = 1/R_1 + 1/R_2$, so a flat contributes nothing and a CONCAVE seat contributes a negative term, making $R$ large and the pressure low. That is exactly why a ball bearing's races are ground as grooves slightly larger than the balls rather than left flat — conforming the counterface is the cheapest contact pressure available.

Finally, the assumptions Hertz needed: frictionless surfaces, contact small compared with the bodies, both materials linear-elastic and homogeneous, and profiles smooth on the scale of the contact. The last one is the interesting failure. Real surfaces are rough, so a "Hertzian" contact is in truth a cloud of much smaller asperity contacts inside the nominal circle, each at a far higher local pressure. Hertz gives the right answer for the assembly and understates what any individual asperity is doing — which is the bridge back to friction and wear, where the asperities are the whole story.