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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.

The reduced elastic modulus combines the elastic compliance of two contacting isotropic bodies:Here $E_i$ is each body's Young's modulus and ${\nu }_i$ its Poisson's ratio. Both bodies deform under the same contact load, so their contributions to the indentation add. The factors involving Poisson's ratio account for the lateral constraint inherent in the local contact geometry.^[ref]

For identical materials, $E^{\ast }=E/[2(1-{\nu }^2)]$; against an effectively rigid probe, $E^{\ast }=E/(1-{\nu }^2)$ for the specimen. Thus even with a rigid probe, an indentation fit does not directly return Young's modulus. This matters when comparing Hertzian contact measurements with tensile testing, which imposes a different pattern of stress.