Contact stresses and Hertz assumptions

Contact stresses

When two curved elastic bodies are pressed together, they touch over a very small area, and high compressive stresses (contact or Hertz stresses) develop at and below the surface. They govern the life of ball and roller bearings, gears, cams, wheels on rails, and rollers. Failure is by pitting, spalling or surface cracking, usually starting below the surface.

Assumptions (Hertz)

1. The materials are homogeneous, isotropic and linearly elastic.
2. The contact area is very small compared with the dimensions of the bodies and their radii of curvature.
3. Surfaces are smooth and frictionless, so only normal pressure acts.
4. Loads are static and act normal to the contact area.
5. Deformations are small. In general the contact area is elliptical; it is a circle for two spheres and a narrow strip for long parallel cylinders.

Point contact: two spheres

With radii $R_1, R_2$, load $F$, and $\dfrac{1}{E^*} = \dfrac{1-\nu_1^2}{E_1} + \dfrac{1-\nu_2^2}{E_2}$, $\dfrac{1}{R} = \dfrac1{R_1} + \dfrac1{R_2}$:

$$a = \left(\frac{3FR}{4E^*}\right)^{1/3},\qquad p_{max} = \frac{3F}{2\pi a^2}$$

The contact area is a circle of radius $a$, with a hemispherical pressure distribution.

Line contact: two parallel cylinders (length $l$)

$$b = \sqrt{\frac{4FR}{\pi l\,E^*}},\qquad p_{max} = \frac{2F}{\pi b\,l}$$

The contact is a strip of width $2b$ with a semi-elliptical pressure distribution.

For a cylinder on a plane, $R = R_1$.

Stresses

The principal stresses are compressive and are greatest on the load axis, at the surface. They fall off rapidly with depth.

• Line contact: $\sigma_z = \sigma_y = -p_{max}$ and, along the cylinder axis, $\sigma_x = -2\nu p_{max}$.
• Spheres: the stress normal to the surface is $-p_{max}$, and the in-plane radial and tangential stresses are $-\dfrac{(1+2\nu)}{2}\,p_{max}$ (about $-0.8\,p_{max}$ for $\nu = 0.3$).

The maximum shear stress is not on the surface but at a depth below it: $\tau_{max} \approx 0.30\,p_{max}$ at $z \approx 0.78b$ for line contact, and $\approx 0.31\,p_{max}$ at $z \approx 0.48a$ for spheres ($\nu = 0.3$). This subsurface shear explains why fatigue cracks start below the surface.