Hertzian contact stress

Remarks

The proof uses the fact that a pressure $p_0 \sqrt{1 - r^2/a^2}$ on a disc of an elastic half-space produces a surface displacement that is a quadratic function of $r$ inside the disc. This matches the quadratic gap $r^2/(2R)$ between two spheres near their touching point, and the total force fixes $a$. Integrating the pressure over the disc gives $F = \tfrac{2}{3} \pi a^2 p_0$.

Example: a steel ball of radius $10$ mm on a flat steel plate, so $R_2 = \infty$ and $R = 10$ mm. With $E = 210$ GPa and $\nu = 0.3$ for both, $E^* \approx 115$ GPa, and a force of $100$ N gives $a \approx 0.19$ mm and $p_0 \approx 1.4$ GPa.

Friction and adhesion are ignored. When the surfaces attract each other, the JKR model replaces this result. The response $F \propto \delta^{3/2}$ is nonlinear even though the material is linear, because the contact region grows with the load.

Going deeper

For a student

Hertz's solution is the model case of contact mechanics, where the contact region is an unknown of the problem. It builds on the Boussinesq problem for a point load on a half-space, superposed over the contact disc. The peak pressure $p_0$ and the subsurface shear stress it creates are used to check bearings against yield and fatigue. The force law $F \propto \delta^{3/2}$ is used for collisions of elastic bodies.