Hertzian Indentation Stress Field Equations

Indentation with hard spherical indenters (so called Hertzian indentation) is a well-known technique to probe the mechanical properties of ceramics and other brittle materials. Using contact mechanics, it is possible to calculate the stress distribution in the sample and use it to express maximum tensile or compressive stresses as a function of load and contact radius. The equations describing the full stress distribution are not usually mentioned in the literature, and surprisingly most of the classical references disclosing them are either mistaken or incomplete. We recovered the full and detailed equations for the stress field distribution and numerically implemented them using the computer language Python. We point out the relationships between the different expressions that have been published, check the consistency with existing results, and discuss the sources of confusion.