Hertzian contact

Hertzian contact is the small-contact solution for smooth bodies obeying linear elasticity, with friction and adhesion neglected. For a sphere of radius $R$ pressed into a flat surface by indentation $\delta$, the contact radius $a$ and compressive normal contact force $F$ satisfyThe reduced elastic modulus $E^{\ast }$ includes both bodies. The contact pressure is largest at the centre and falls to zero at the edge:These formulas describe an ideal contact patch, rather than a uniformly loaded disk.^[ref]

The increasing contact area explains the nonlinear load law even though the material itself obeys linear elasticity. In a probe experiment, $F^{2/3}$ should be proportional to $\delta$ in the Hertzian regime. Systematic departures can signal adhesion, plastic deformation, a thin specimen or a contact patch too large for the assumed local geometry.