Hertz Contact of Spherical Surfaces

For axisymmetric Hertz contact of two spherical surfaces, a normal force $F$ produces a circular contact area of radius $a$:

$$a=\left(\frac{3FR^*}{4E^*}\right)^{1/3}.$$

The maximum contact pressure is:

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

The pressure distribution is:

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

• $F$ — normal force;
• $a$ — contact radius;
• $R^*$ — reduced radius of curvature;
• $E^*$ — reduced elastic modulus;
• $p_0$ — maximum contact pressure;
• $p(r)$ — contact pressure at radial coordinate $r$;
• $r$ — radial coordinate within the circular contact patch.

The formulas apply within the assumptions of classical elastic Hertz theory.