Hertz Contact Pressure Distribution

For a circular Hertz contact area of radius $a$, the pressure varies as:

$$p(r)=p_0\sqrt{1-\frac{r^2}{a^2}},\qquad 0\le r\le a.$$

The maximum pressure at the center is:

$$p_0=\frac{3F}{2\pi a^2}.$$

• $p(r)$ — contact pressure at distance $r$ from the center;
• $p_0$ — maximum contact pressure;
• $a$ — contact radius;
• $r$ — radial coordinate within the contact patch;
• $F$ — normal force.

At the contact boundary $p(a)=0$, while the maximum $p_0$ occurs at $r=0$.