An educational perspective on sphere–elastic half-space contact: Linking Hertz and Sneddon models

Abstract

The interaction between a rigid sphere and an elastic half-space is a classical problem in contact mechanics with fundamental applications in materials science and nanotechnology. In most cases, the Hertz equation is used to model the force–indentation data in order to determine the Young's modulus of the sample of interest. However, the validity of the Hertz equation is restricted to small indentation depths compared to the radius of the sphere. On the other hand, Sneddon derived equations that are valid regardless of the indentation depth. In other words, these equations are accurate for both small and large indentation depths. Although it is well known that the Hertz equation remains valid only in the limit of small indentation depths, the explicit mathematical demonstration of its derivation as a special case of Sneddon's solution is often omitted or insufficiently explained in textbooks and the literature. In this paper, we demonstrate—using a Taylor series expansion of Sneddon's equations—that the Hertz relationship is a limiting case of Sneddon's solutions for small indentation depths. This analysis is beneficial for students and early-career researchers, as it explains some of the most fundamental concepts of contact mechanics related to the characterization of soft materials. In particular, it shows why the Hertzian equation, despite being an approximation, is usually preferred over Sneddon's solutions. The analysis presented in this paper is also extended to the case of very deep spherical indentations. In this regime, the sphere–half-space interaction resembles that of a flat punch–half-space interaction, with the indentation depth reduced by a constant factor. Special attention is given to the physical significance of this constant offset factor for educational and pedagogical purposes.