Towards Simpler Modelling Expressions for the Mechanical Characterization of Soft Materials

Aims: The aim of this paper is to develop a new, simple equation for deep spherical indentations. Background: The Hertzian theory is the most widely applied mathematical tool when testing soft materials because it provides an elementary equation that can be used to fit force-indentation data and determine the mechanical properties of the sample (i.e., its Young's modulus). However, the Hertz equation is only valid for parabolic or spherical indenters at low indentation depths. For large indentation depths, Sneddon's extension of the Hertzian theory offers accurate force-indentation equations, while alternative approaches have also been developed. Despite ongoing mathematical efforts to derive new accurate equations for deep spherical indentations, the Hertz equation is still commonly used in most cases due to its simplicity in data processing. Objective: The main objective of this paper is to simplify the data processing for deep spherical indentations, primarily by providing an accurate equation that can be easily fitted to force-indentation data, similar to the Hertzian equation. Methods: A simple power-law equation is derived by considering the equal work done by the indenter using the actual equation. Results: The mentioned power-law equation was tested on simulated force-indentation data created using both spherical and sphero-conical indenters. Furthermore, it was applied to experimental force-indentation data obtained from agarose gels, demonstrating remarkable accuracy. Conclusion: A new elementary power-law equation for accurately determining Young's modulus in deep spherical indentation has been derived.