Beyond Hertz: Accurate Analytical Force–Indentation Equations for AFM Nanoindentation with Spherical Tips

The Hertz equation is the most widely used equation for data processing in AFM nanoindentation experiments on soft samples when using spherical indenters. Although valid only for small indentation depths relative to the tip radius, it is usually preferred because it directly relates applied force to indentation depth. Sneddon derived accurate equations relating force and contact radius to indentation depth for shallow and deep indentations, but they are rarely used in practice. This paper presents analytical approaches to solving Sneddon’s nonlinear system. Using Taylor series expansions and a simple equation linking applied force, average contact radius, and indentation depth, we derive a two-term equation that directly relates force to indentation depth. This expression is accurate for h ≤ 1.5 R, where h is the indentation depth and R is the indenter radius, making it applicable to most practical AFM measurements on soft materials. It should be used instead of the Hertzian model for extracting Young’s modulus, thereby enhancing measurement accuracy without increasing the complexity of data processing. In addition, the results are generalized to produce a series solution that is valid for large indentation depths. The newly derived equations proposed in this paper are tested on both simulated and experimental data from cells, demonstrating excellent accuracy.